Modular Elliptic Curves and Fermat's Last Theorem

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM {=

W.n

=n

465

jey6)}

{ (1

e yZ)

:

and VXn

tE

One checks eadily hat pa is continuous omomorphismnd thatthe deformation p,>] unchanged we add coboundary a. We need to check that [p,>] s Selmer deformation. Let 7Y Gal(Qp/Qunr)and Gal(Qunr/Qp). Consider he exact sequenceof (9[g]modules -)

(VAn/Wn)o

(V-

where is a submoduleof VAn/Vln)7H. Since the actionof Dp on VXn/V~n via characterwhich nontrivialmodA (it equals X2X1 modA and X1 X2), 0. Then we have an exact diagramof we see that Xg and H1(g, X) 0-modules

'(91

(V1

"LI)1)

'I

H'(9,

(V~n/WA~)t

H1 (Qp, VAn/Wn)

1(Qpunr,n/Xn By hypothesis he image of is zero in H1(Qq'Tnrhnf/jW>). Hence it is in the image of H1(g, (VXn/Wo~)7H).hus we can assume that it is represented H1(QpV~n/W>?) by cocycle,whichmaps to Vn/W~n; i.e., f(Dp) VA WAn f(Ip) 0. The differenceetween and the image of is VAn. By subtractinghe coboundary i} for ome coboundary la i-f as cocycles {f F-* oU u} from globallywe get a new such that mapping to VXn/W Thus a(Dp) V1,n,oa(Ip) W\n nd it is now easy to check hat [p,] is Selmerdeformation po. Since [p,] is a Selmerdeformation hereis a unique map of local (9we must check the algebras is,>: Rz -* On [E] inducing t. (If

ANDREW WILES

466

other onditions lso.) Since =_pf,Amod we see thatrestrictingp, to PD givesa homomorphism 0-modules, mu,:PD -- FO/A

suchthat

''

(P2)

0. Thus we have defined map p:

y(p: ~e (Qs/Q, VAn)

,,

Hom e(PD/PD,(9/Afl)

It is straightforward checkthat this s map of 0-modules. To checkthe injectivity supposethat .p,(pD) == 0. Then factors hrough /p-D (9 and being 0-algebrahomomorphismhisdetermines ,>. Thus [pfA] [paC]. IfA-'pA pf,A, hen mode is seen to be centralby Schur's emma and so maybe takento be I. simple alculationnow shows hat is coboundary. To see that is surjective hoose Homo(pP)/p2, /An). Then pT: Gal(QE/Q) -- GL2(Rv/(p2, ker )) is inducedby representative of the universal eformationchosento equal pf,Awhenreducedmod PD) and we define map aog: Gal(Qr/Q) -+ VAn by ()

PT g) pf, (g)1

pD/(p2, ker I) pv/(p , ker I)

pD/(p2,ker I) pD/(p2,ker I)

VAn

wherepfA(g) is viewed GL2(RD/(pD,ker )) via the structuralmap RD (RD beingan (9-algebra nd the structuralmap being ocal because of theexistence f section). The right-handnclusion omesfrom PD/ PD,kerT)

)/A

((D/AnE 1)

E.

Then ca is readily een to be a continuous ocyclewhose cohomology lass T. Moreover, he constructionsre lies in HSe(QE/Ql VA\n). Finally p(aT) compatiblewith hange n, i.e., forVA\n V,+i andA: /D 0/An+l. We nowrelatethe ocal cohomology roupswe have defined the theory of Fontaine nd in particular o the groupsof Bloch-Kato [BK]. We will distinguish heseby writing for hecohomology roupsofBloch-Kato. None ofthe resultsdescribed the rest of this sectionare used in the restof the paper. Theyserveonlyto relatetheSelmergroupswe havedefined and later compute) o themore tandardversions.Using he attice ssociatedto pf,Awe (94 withGalois actionvia Adpf,A. Let obtainalso lattice Oz QP be the associatedvector pace and identify with V/T. Let pr: V -* be

the naturalprojection nd define ohomologymodulesby Hk(Qp, V)

Hk(Qp, V)

ker: H1(QpV)

=

Bcrys), ~~~~~~~~~QP

-+

pr %H4(Qp, V))

H1(Qp, V),

(jn) (HF(Qp, V)) HF(Qp, V~n) H1(Qp, Vsn), is the natural map and the two groups the definition where n: VXn ofHF(Qp, V) are defined singcontinuous ochains. Similardefinitionspply to V* HomQP V, Qp(1)) and indeed to any finite-dimensionalontinuous p-adic representationpace. The reader s cautionedthat the definition HF(Qp, VAn) is dependenton the lattice (or equivalently V). Under certainconditionsBloch and Kato show, using the theoryof Fontaine and Lafaille,that this is independent f the lattice (see [BK, Lemmas 4.4 and 4.5]). In anycase we will consider whatfollows fixed atticeassociated to p PfA, Ad p, etc. Henceforth e wil onlyuse thenotationH1(QP, -) when the underlying ector pace is crystalline. (i) If po is flatbutnotordinarynd pf,A associated to p-divisible roup hen orall PROPOSITION

Hf QpV>,n) HF (QpVAn)).

|I

nd pf,A s associated o p-divisible

(ii) If pf,A ordinary,etpfA group, hen orall n,

HF (QpV,\n)

HSe(QpVn).

Proof. Beginningwith (i), we defineHf(QpV) {E H1(QpV) H1 (Qp, V). Then s(a/An) Hf Qp,V) for all n} where H1(Qp, V) we see that in case (i), Hf Qp,V) is divisible.So it is enoughto show that Hk(QP, V)

Hfl Qp IV).

We have to comparetwo construction ssociatedto a nonzero lement of H1 Qp, V). The firsts to associate an extension (1.9)

--

--

-6

-O

of K-vectorspaces withcommuting ontinuousGalois action. If we fixan with8(e) &(of) with the action on is defined oe cocycle representing The secondconstructioneginswith he mageofthesubspace (a) in H1 Qp, V). By the analogue of Proposition .2 in the local case, there is an 0-module isomorphism H1 (QP, V)

Homo PR/p2,K/()

468

ANDREW WILES

where R is the universaldeformationingof po viewed as representation of Gal(Qp/Qp) on 0-algebras and PR is the ideal of R corresponding pD (i.e., its inverse mage in R). Since $& associated to (a) is a quotient PR/(PR, a) of PR/pi which s a free0-module of rank one. We thenobtain a homomorphism Pa: Gal(Qp/Qp) -) GL2 (R/(iR, a)) inducedfrom he universal eformationwe pick representationn the universalclass). This is associated to an 0-module ofrank which ensoredwith givesa K-vector pace E' (K)4 which an extension (1.10)

-E

-?

-?

-?

K2 has the Galois representationf,A viewed ocally). where In the first onstruction HF(QP, V) if nd only heextension 1.9) is crystalline,s the extension iven (1.9) is sumofcopiesofthemoreusual extensionwhereQp replaces in (1.9). On theotherhand (a) Hf (Qp,, if and only he secondconstructionan be madethrough fl, equivalently and only E' is therepresentationssociatedto p-divisible roup. (A priori, the representationssociated to p, only has the property hat on all finite quotients comes from finite latgroupscheme. However theorem Raynaud [Rayl] says that then p, comesfrom p-divisible roup. For more details on Rfl, he universal latdeformationingof the local representation po, see [Ram].) Now the extensionE' comes from p-divisible roup f and only t is crystalline;f.[Fo, ?6]. So we haveto showthat (1.9) is crystalline ifand only 1.10) is crystalline. One obtains 1.10) from1.9) as follows.We view as HomK(U, U) and let ker: HomK(UU

) OU

U}

where he map is the naturalone f(w). (All tensorproducts this proofwill be as K-vector paces.) Then as K[Dp]-modules

E'

(E

U)/X.

To checkthis,one calculatesexplicitlywiththe definitionf the action on (given bove on e) and on E' (given the proof Proposition .1). It follows from tandardproperties crystalline epresentationshatif is crystalline, so is and also E'. Conversely, can recover fromE' as follows. ConsiderE' (E U)/(X U). Then there is natural map inducedby the direct um decomposition up: (det) -- E' (det) ED ym2U. Here det denotes 1-dimensional ector pace over with Galois action via detpf,A,. ow we claim that is injective (det). For

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM

if

then p(f) =

forwhichW1AW2

w2

(Wi

in det

where W1,W2are

W2Owl)

K. So if p(f)

then

in

=0

f(Wl)08)w2-f(w2)08)W1

469

basis for

U.

whence 0. So o is injective f(w2) But this is false unless f(wi) det and if itselfwerenot injective hen E wouldsplitcontradicting on a 7& So 'p injective nd we have exhibited (det) as subrepresentation which is crystalline. We deduce that E is crystallineif E' is. This

of E'

completes heproof i). VAn) jn1 To prove ii) we checkfirst hat HSe(QP, H7e(QP, wasalreadyused 1.7)). Wenexthaveto show hat (Qp, V) where he latter defined HSe(QP ,V)

V)) (this H~e(QP, V)

ker: H1 (Qp IV) -) H1 (Qunr,V/VO)

with V0 the subspace of on which acts via E. But this follows rom he computations Corollary .8.4 of [BK]. Finallywe observe hat pr (HSeQP V))

HSeQP V)

although he inclusionmaybe strict, nd pr (Hk(QP, V))

Hk(QP,

V)

by definition. his completes heproof.

C1

These groupshave the property hat for (1.11)

H1(Qp, VVr

r,

s)) = HF(Qp, VAr)

HF7

whereir,s: Vs is the natural njection. The same holdsforV* and V* in place V,\r of V\r and V1\A hereV)*r defined V*r

and similarly for V*.

Hom(V\r q pr)

Both results are immediate from the definition (and

indeedwerepartof the motivation or he definition). We also givea finiteevelversion a result Bloch-Katowhich s easily be Galois stable deducedfrom he vector pace version.As before et lattice so that

04. Define

HI (Q under the natural inclusion i:

T) --+ V,

(Q7

V))

and likewise for the dual lattice T*

Homzp(V, Qp/Zp)(1)) in V*. (Here V* Hom(V, Qp(l)); throughout his paper we use M* to denote a dual of M with a Cartiertwist.) Also write

470

ANDREW WILES

-- T/A' forthe natural projectionmap, and forthe mapping prn: induceson cohomology.

PROPOSITION 1.4.

If pf,A, associatedto p-divisible roup the ordinarycase is allowed)then (i) prr (H1 (Qp, T))

HF1Qp, T/An) and similarly orT*, T*/An.

(ii) HF(Qp, VAn) the orthogonal omplement f HF(Qp, VAhn)nder Tate local duality etween Qp, VAn) nd H1 Qp, VAin)ndsimilarly orWAn and WA*n eplacing Vxn and VA*n.

More generally heseresultsholdforany crystalline epresentation ' in place of and A' uniformizer K' whereK' is anyfinite xtension Qp withK' EndGal(V/Q)V. Proof. We first observe that

HF(Qp, T/An). Now

HF(Qp, T))

from he constructionwe may identify /Anwith VAn. A resultof BlochKato ([BK, Prop. 3.8]) says that HF (Qp, V) and HF(Qp, V*) are orthogonal complements nderTate local duality.It follows ormallyhat HF(QP, V*n) and prn HFk(Qp,T)) are orthogonal omplements,o to prove he proposition it is enoughto showthat (1.12) Now if (1.13)

HF(Qp,

HFp(Qp, VAn)

VA'n)

dimK Hk(Qp, V) and s

H1(Qp VAn).

dimK HF(Qp, V*) then

dimK HO(Qp, V)

dimKHO(Qp, V*) + dimK V.

Fromthedefinition, (1.14)

HF (Qpa VAn)

Q/)Any) #ker{H1 (Qp, VAn) H1 (Qp, V)}.

The secondfactor equal to {V (Qp)/An (Qp)}. Whenwewrite (Qp)div for hemaximaldivisible ubgroup f (Qp) this the same as V(Q )/V(Q )div)/An

V(QP)/V(Qp)diV)An

V(Qp)An/# (V(Qp)div )An.

Combining hiswith 1.14) gives (1.15)

HF(Qp, VAn)

O/Afn)y H0(Q

VAn) /

Q/An)dimKHO(Qp,

V)

This, togetherwith n analogousformula or HF(Qp, Vi*n) nd (1.13), gives (QP, VAn) H4(Q

Van)

O/A )4.#H0(Qp,

VAin) H0(Qp, VAn).

MODULAR

ELLIPTIC

AND FERMAT'S LAST THEOREM

CURVES

471

H2 (Qp, VAn) he assertion 1.12) nowfollows rom As #Ho (Qp, V*) the formula or he Euler characteristic Vain. The proof orWarn, indeedmoregenerally or ny crystalline epresentation, the same. We also give a characterization f the orthogonal complementsof Hie(Qp, WAn) nd H"e(QpS I/n), underTate's local duality. We write hese duals as e* Qp Wan) and e* (Qp, Vn) respectively. et :p H1 (Qp W~n)

H'(Qp, W~n/(W~n)

be the natural map where (W~n)i is the orthogonal complement of W%-T in

WAn, nd let Xni be defined the imageunderthe compositemap im:

/(ZX)pn

(-

H1(Q

O/An

tpn (9/An)

(Qp7 WA~n/(WA~n) where in the middle term tLpn O/An to be identifiedwith (W~n)1/(W~n)0. e the image OfZX/(ZX Pn Similarlyfwe replace W*n by Vn we let (O/An)2 in H1 (Qp, V*'n/(W~n)0),and we replace ,owby the analogous map (pv. PROPOSITION

1.5. HA *(QpWX*n)

Hse*(QpV>*n)

Proof. This can be checked

-1 (Xni) f01 (yni)

dualizing he sequence

Hstr(Qp, WArn) HSe(QP7 W~n) ker: {H1(Qp, W\n/(W\n)

H(Qunr, WAn/(WAn)},

where Hltr(Qp, WAn) = ker: H1(Qp, The W\n) -- H1(Qp, WAn/(W~n)0) H1 Qp, Win/(W*n)1). By the first erm s orthogonal to ker: H1 (Qp, Wn)

naturality the cup productpairingwithrespect quotients nd subgroups the claim then reduces to the well knownfact that underthe cup product pairing

H1(Qp, ipn)

H1(Qp, Z/pn) _+Z/pn

theorthogonal omplement theunramified omomorphismss the imageof H1 (Qp, Ippn). The proof for VAn essentially the the units ZpX/(Z )pn same. 0

472

ANDREW WILES

2. Some computations of cohomology groups We now make some comparisons f ordersof cohomology roupsusing the theorems Poitou and Tate. We retainthe notation nd conventions Section though t will be convenient state the first wo propositions moregeneral ontext.Suppose that

I7 H1(QqX)

JJLq

qes is subgroup,where is a finitemoduleforGal(Qr/Q) of p-power rder. We define * to be the orthogonal omplement f under he perfect airing (local Tate duality) H1(QqX)

Hl(QqX*)

qEF2

qEF2

where X*

Hom(X, pp.).

Let

Ax: H1(Q/QX)

--

QP/ZP

JJ1(QqX)

qEF,

be the localizationmap and similarlyAx* forX*. Then we set HL(Q/QX)

A-1(L),

HL*(Q/QX*)

=AX1*(L*).

The followingesultwas suggestedby resultof Greenberg cf. [Grel]) and is simpleconsequence f the theorems Poitou and Tate. Recall that is alwaysassumedodd and that E. PROPOSITION

1.6.

#HL(Q/QX) where

hq

{ho

#HL (QE/QX*)

fJ qEF

hoo

#HO(Qq,X*)/[Hl(Qq,X):Lq]

#H0(R, X*) #HO(QX)/#HO(QX*).

Proof. Adapting heexactsequenceofPoitouand Tate (cf. Mi2,Th. 4.20]) we geta seventerm xact sequence HLj(Q/QX)

qEF

H2(Qq X)

H1(Q/QX)

Hj H1(QqX)/Lq qEF,

H2(Q/QX)

HL*(QE/QX*))

L+ HO(Q/Q, X*)A

0,

MODULAR ELLIPTIC

CURVES

473


whereMA Hom(M, Qp/Zp). ow using ocal duality nd global Eulercharacteristicscf. [Mi2,Cor. 2.3 and Th. 5.1]) we easily obtain the formul the proposition.We repeatthat in the above propositionX can be arbitrary [] p-power rder. We wishto applythe proposition o investigate Di. Let D I, , (9, M) be a standarddeformationheory s in Section and define corresponding groupLn LDn by setting H1(Qq, Vn)

Ln~q Then HID(QE/Q,

forq p

HDq(Qq, van)

for

H.1(Qp

for

1Vn)

and

and

qf

q

p.

Vain) HLn (Q /Q,Vain) nd we also define HD*(Q/Q

V*,Vn) HL.(QE/Q,

V*)

We willadopt the conventionmplicitn the above that ifwe considerA' thenHE,(QE//Q, VAn) laces no local restrictionn the cohomology lassesat Thus in HD* (Q//Q, \n)we willrequire byduality) hat primes A'-S. thecohomology lass be locallytrivial t -E. We need now some estimatesfor he local cohomology roups. Firstwe consider n arbitrary initeGal(QE/Q)-module X: PROPOSITION 1.7.

If

moduleofp-power rder,

E, and X is an arbitraryiniteGal(QE/Q)-

#HL, QEuq/QX)/#HL(Q/QX)

#H0(QqX*)

whereL' = Le for E E and L- H1 (Qq ,X). Proof. Consider heshort xact sequenceof nflation-restriction: HHL(QE/QX)

HLj,(QEuq/QX)

-- Hom(Ga1(QEUq/QE),

H1(Qunr, X)Ga1(Q nr/Qq)

X)Gal(QF/Q)

H1(Qunr)Ga1(Q

nr/Qq)

The proposition ollowswhenwe note that #H0(Qq,

X*)

#H1 (Qunr, X)Gal(Qunr/

Now we return o the studyof Vainnd Wan. PROPOSITION 1.8.

If

(q 7&p) and

VAinhen hq

1.

ANDREW WILES

474

Proof. This is (A) thenwe have Lnq

is of type

straightforwardalculation. For example

ker{H1 (Qq, Vain)

H1 (Qq, W\n/W2?n) H1(Qnr,

O/A>n)}.

Usingthe long exact sequenceof cohomology ssociatedto rn W,) 0/Wn Wa one obtains formulafor the order of Lnq in terms of #Ht(Qq, WAn), #Hi QqI WAn WAC etc. Using ocal Euler characteristicsheseare easily reduced to ones involving O(Qq, WAgn)tc. and the resultfollows asily. The calculationof hp is more delicate. We contentourselveswith an inequality somecases. PROPOSITION

(i) If X

1.9.

VAnthen

# O/A)3n Ho QpVA*n)/ Ho QV)*n) in theunrestrictedase.

hp

(ii) If

VAXnhen

# (0/A)n hpho in theordinary ase.

H0(Qp, (Vord)*)/#Ho(Q,W;n)

(iii) If

VAnor WArnhen hp hoo

(iv) If

in the strict case. VAnor WAn then hp hco in theflatcase. V,\n henhpho, #Ho (Q, Vv*\n) Ln VAn or WArn hen hphoo

in theSelmercase. (v) If

(vi) If

HF(Qp,X) and pf,A risesfrom

#IHI(QpI (W n)*)/#Ho(Q Wan)

ordinary -divisible roup.

n. We have Proof. Case (i) is trivial.Consider hencase (iii) with longexact sequenceof cohomology ssociated to the exact sequence: (1.16)

->

Wn

Vxn

->

V,\n/WOn

O-.

In particular hisgivesthemap u in the diagram 1(Qp, VAn) \6 Z=Hl(Qpnr/Qp,

where

(VAn/W~n)H) ) H1(Qp, VXn/WXn)~Hl(Qpnr, VXn/WXn)g

Gal(Qpnr/Qp),7-

Gal(Qp/Qunr) and 6 is defined

make the

triangle commute. Then writing hi(M) for #HI(Qp, M) we have that #Z


ho(Vn/W0) and #im6

(#imu)/(#Z). longexact sequenceassociatedto (1.16) gives #imu

(1.17) Hence

h1(VW? )h2(V

475

simple calculationusing the n)

h2(WA~h2(V,\n/IW5~)

[H1 Qp,VAn):n,p] #im6 #(O/A)3nho(V*,\n)/ho(WXn*). VAnnd the case WAn similar. The inequality (iii) follows or Case (ii) is similar. case (iv) we just need mu which s givenby (1.17) withWAn eplacingV\n. case (v) we havealreadyobserved Section that in the flatcase. Moreover Raynaud's results mply hat #Ho(Qp, V*n) #Hf (Qp, Vn) can be computed o be #(Q/A)2nfrom Hf Qv, VAn) Hf Qp V)An Homo(PR/PR K/O)An where is theuniversal ocal flatdeformationing po for -algebras. Using the relation _RRXl (9 whereRfl s the correspondinging for W(k)W(k)

algebras, nd themaintheorem [Ram] Theorem4.2) which omputesRfl, we can deducetheresult. We now prove vi). Fromthe definitions HF4(Qp V) ,\n

(#O/Al)r #H0(Qp, WAn)

if Pf,AIDp does not split

if PfAID~ Splits (#O/An')r where dimKHF(Qp, V). This we can computeusing the calculations in the 2 in the non-split ase and r [BK, Cor. 3.8.4]. We find hat splitcase and (vi) follows asily. 3. Some results on subgroups of GL2(k) We now give two group-theoreticesultswhichwill not be used until Chapter3. Although hesecould be phrased purelygroup-theoreticerms it willbe moreconvenient continue o work the setting Section 1, i.e., withpo as in (1.1) so that mpo s subgroup GL2(k) and detpo is assumed odd. LEMMA 1.10. If impo has orderdivisible

then:

and rivial with m,p) (i) It contains elementYooforder on any abelianquotient f mpo. (ii) It contains n element o(G) with ny prescribedmage in the Sylow 2-subgroup f impo)/(impo)' and with heratio of the eigenvaluesnot equal to w(of). Here (impo)' denotes he derived ubgroup f impo).)

ANDREW WILES

476

The same resultshold fthe mageoftheprojective epresentation associatedto po is isomorphic A4,S4 or A5. impo and let Z denote the centerof G. Then we Proof. (i) Let have a surjectionG' (G/Z)' wherethe denotesthe derivedgroup. By Dickson's classification the subgroups GL2(k) containing n element f order , (G/Z) is isomorphic PGL2(k') orPSL2(k') for omefinite ieldk' of characteristic orpossibly A5 when 3, cf. Di, ?260]. In each case wecan find, nd then ift o G', an element orderm with m, p) and 3, and PSL2(F3) A4 or PGL2(F3) exceptpossibly n the case S4. However thesecases (G/Z)' has orderdivisible so the 2-Sylow ubgroup ofG' has ordergreater han2. Since t has at mostone element exact order (theeigenvalueswouldbothbe -1 since t is inthe kernel thedeterminant and hencethe elementwouldbe -I) it mustalso have an element order4. The argument theA4, S4 and A5 cases is similar. (ii) Since po is assumedabsolutelyrreducible, impo has no fixed ine. We claim that the same then holds forthe derivedgroup G'. For otherwise sinceG' G we could obtain secondfixed ineby taking gv) where v) is the original ixed ineand is a suitable lement G. Thus G' wouldbe contained in the groupofdiagonalmatricesfor suitable basis and either t would be central n whichcase would be abelian or its normalizer GL2(k), and hencealso G, wouldhave orderprime p. Since neither thesepossibilities is allowed,G' has no fixed ine. By Dickson'sclassification fthe subgroups GL2(k) containing n elementof order the image of impo in PGL2(k) is isomorphic PGL2(k') or PSL2(k') for ome finite ieldk' ofcharacteristic or possibly A5 when 3. The onlyone of thesewith quotientgroup oforder is PSL2(F3) when 3. It follows hat [G: G'] except thisone case whichwe treat separately.So assumingnowthat [G: G'] we see that G' contains nontrivialunipotent lementu. Since G' has no fixed inetheremustbe another noncommuting nipotent lement in G'. Pick a basis forPOIG' consisting of theirfixedvectors. Then let be an element f Gal(Qr/Q) forwhich he imageof po(r) in G/G' is prescribed nd let Po(T) (a d). Then

(a

)(1sax)(

has det(6) detpo(r) and trace sa (ra/3 c) br/3 a d. Since p > we can choose this trace to avoid any two givenvalues (by varying unless forall r. But ra/3 cannot be zero forall as otherwise 0. So we can find 6 forwhichthe ratio of the eigenvalues s not

w(T), det(6) being, of course, fixed.


477

Now suppose that impo does not have orderdivisibleby but that the associatedprojective epresentation has image isomorphic S4 or A5, so necessarily 7& . Pick an element suchthat the imageof po(r) in G/G' is any prescribed class. Since this fixes both det po(r) and w(r) we have to show

thatwe can avoid at mosttwoparticular aluesofthe tracefor r. To achieve thiswe can adapt our first hoiceof by multiplyingy anyelement G'. So picka G' as in (i) whichwe can assume thesetwocases has order3. Pick a basis for o,by extendingcalars necessary, thata -+ (a a- ). Then one checks asilythat Po(r) (c d) we cannothavethetracesofall ofT, ar and a2 lying set of the form Tt} unless 0. Howeverwe can ensure that po(r) does not satisfy hisby firstmultiplying by suitableelement G' sinceG' notcontained thediagonalmatrices it is not abelian). In the A4 case, and in the PSL2(F3) 3, we use A4 case when differentrgument. both cases we find hatthe 2-Sylow ubgroup G/G' is generated y an element inthecentre G. Either power z is suitable candidateforpo(a) or else we mustmultiply he powerof by an element G', the ratio of whose eigenvalues s not equal to 1. Such an element xists because in G' the onlypossibleelementswithout hispropertyre {TI} (such elementsnecessarily ave determinant and orderprimeto p) and we know that #G' as was noted the proof part (i). Remark.By well-knownesult the finite ubgroups PGL2 (Fp) this lemmacovers ll po whose magesare absolutely rreducible nd forwhich -5 is not dihedral. Let K1 be the splitting ieldof po. Then we can view WAand W* as Gal(K1((p)/Q)-modules. We need to analyze theircohomology.Recall that we are assumingthat po is absolutely rreducible.Let -5be the associated projective epresentation PGL2(k). The followingropositions based on thecomputations [CPS]. PROPOSITION

1.1 1.

Supposethatpo is absolutelyrreducible. hen H1 K1 ((p)/Q1WA*) 0.

Proof. If the imageof po has orderprime the lemma trivial. The subgroups GL2 k) containing n element order which re notcontained in a Borelsubgrouphave been classified Dickson[Di, ?260]or [Hu, I.8.27] Their images inside PGL2(k') wherek' is the quadratic extensionof k are conjugateto PGL2(F) or PSL2(F) forsome subfield of k', or they are isomorphico one oftheexceptionalgroupsA4,S4, A5.

Assume hen hat hecohomologyroupH1(Ki((p)/Q, WA) 0. Then

by considering he inflation-restriction equence withrespectto the normal

478

ANDREW WILES

subgroupGal(Ki ((p)/KI) we see that K1. Next,sincetherepresentation is (absolutely) irreducible, he centerZ of Gal(Ki/Q) is contained n the diagonalmatrices nd so acts trivially WA. So by consideringhe nflationrestrictionequencewithrespect o Z we see that Z acts trivially n (p (and on We). So Gal(Q((p)/Q) is quotientof Gal(Ki/Q)/Z. This rulesout all we onlyhaveto consider hecase where he cases when , and when image of the projective epresentations isomorphic s a group to PGL2(F) for ome finite ield characteristic (Note that S4 PGL2(F3).) Extending calars commuteswithformationf duals and H1, so we may k. If and #F then assume without oss of generality H1(PSL2(F), WA) by resultsof [CPS]. Then if p5 s the projective representationssociated to po suppose that g-1 im PGL2(F) and let PSL2(F)g-1. Then WA WAover and H1 0. WA)(F (g- Hg, gA1(WA F)) (1.18) We deducealso thatH1(impo, WA) 0. Finallywe consider hecase where F3. am grateful Taylorfor he followingrgument. irstwe consider he actionofPSL2(F3) on WA xplicitly 0}. byconsideringheconjugation ctionon matrices A M2(F3): trace One sees thatno suchmatrix fixedby all the elements f order2, whence H1(PSL2(F3), WA)

H1(Z/3,

(W,)C2xc2)

whereC2 C2 denotes henormal ubgroup order in PSL2 F3) A4. Next we verify hat there unique copy of A4 in PGL2 (F3) up to conjugation. Forsuppose thatA, GL2(F3) are suchthatA2 B2 I withthe mages ofA, representingistinct ontrivialommutinglements fPGL2(F3). We can choose (O _0)by suitablechoiceof basis, i.e., by suitable conjugation. Then B is diagonalor antidiagonal s it commuteswith up to scalar, and as B, A are distinct PGL2(F3) we have (? -') for ome a. By conjugating y diagonalmatrix whichdoes not change A) we can 1. The groupgeneratedby {A, B} in PGL2(F3) is its own assume that centralizer it has indexat most in its normalizer SinceN/ A, B) S3 there s a uniquesubgroup f in which A, B) has index3 whence he mage oftheembedding A4 in PGL2(F3) is indeedunique up to conjugation).So arguing in (1.18) by extendingcalars we see thatH1 impo, WA*) when F3 also.

The followingemmawas pointedout to me by Taylor. It permitsmost dihedral ases to be coveredbythemethods Chapter and [TW]. LEMMA 1.12.

Supposethatpo is absolutelyrreduciblend that

(a) po is dihedral the case where he mage

Z/2

Z/2 is allowed),


(b) PoK is absolutelyrreducible here

479

(A(-1)(P1)/2P).

Thenfor any positive nteger and any irreducible alois stable subspace there xists n element Gal(Q/Q) such that ofWA (i) f3o(a)

1,

(ii)

fixesQ((pn),

(iii)

has an eigenvalue on X.

Proof. If fio dihedral henpo IndH for omeH of ndex 2 in G, where Gal(Ki/Q). (As before,K1 is the splitting ieldof po.) Here H can be takenas the full nversemageofanyofthenormal ubgroups f ndex 80 IndG(X/X') where is the defininghedihedralgroup. Then WA quadraticcharacter -* C/H and X' is the conjugateof by any element H. Note that 54X' since has nontrivialmage PGL2(k). and conditions i) and (ii) hold, observe such that 8(cr) To find that M((pn) is abelian where is the quadratic fieldassociated to 8. So conditions i) and (ii) can be satisfied fio s non-abelian. fio s abelian (i.e., the magehas theform /2 Z/2), thenwe use hypothesisb). If ndG(X/X') is reducible verk thenWA k is a sum ofthreedistinct uadraticcharacters, none of which s the quadraticcharacter ssociatedto L, and we can repeat the argument y changing he choiceof forthe othertwo characters. If H. This is absolutely rreduciblehenpickany a IndG x/x') -H satisfiesi) and can be madeto satisfyii) if b) holds. Finally, ince we see that has tracezero and o2 1 in its action on X. Thus it has an eigenvalue qual to 1.

Chapter In this chapterwe studythe Hecke rings. In the first ectionwe recall some of the well-known roperties f these ringsand especiallythe Gorenstein propertywhose proof s rather echnical, epending a characteristic version f the q-expansion rinciple. the second sectionwe computethe relations etween heHeckerings s the evel s augmented.The purpose s t find hechange the r7-invariant the level ncreases. In thethird ectionwe statetheconjecture elating he deformationings of Chapter and the Hecke rings. Finallywe end with the critical tep of showing hat if the conjecture true at a minimal evel then it is true at all levels. By the resultsof the appendixthe conjecture equivalent o the

ANDREW WILES

480

equalityof the 71-invariantor he Hecke rings nd the p/p2-invariantor he deformation ings. In Chapter 2, Section 2, we compute the change n the 7a-invariantnd in Chapter 1, Section 1, we estimated he change the p/p2_ invariant.

1. The Gorenstein property X1 N)/Q be the modularcurve For any positive nteger let Xj (N) over correspondingo the grouprl(N) and let J1(N) be its Jacobian.Let T1 N) be thering fendomorphisms J1 N) which s generated verZ bythe standardHecke operators Ti Tl* for N, Uq Uq*for N, (a) (a)* for a, N) 1}. For precisedefinitionsf these see [MW1, Ch. 2, ?5]. In particular one identifieshecotangent pace ofJi N) (C) withthe space of cusp forms weight on rF N) then he action nducedbyT1 N) is theusual one on cusp forms. We let

(a)

(a, N)

1}.

The group (Z/NZ)* acts naturallyon X1 N) via and forany subXH(N)/Q be the quotientX1(N)/H. group (Z/NZ)* we let XH(N) Xo(N) corresponding the group (Z/NZ)* we have XH(N) Thus for ro N). In Section2 it willsometimes e conveniento assume that decomHIHq in (Z/NZ)* -_f Z/qrZ)* wherethe product poses as a product is over the distinctprimepowersdividingN. We let JH(N) denotethe Jacobian of XH(N) and note that the above Hecke operators ct naturally JH(N) also. The ringgenerated y these Hecke operators s denotedTH(N) and sometimes,f and N are clear from he context,we abbreviatethis to T. Let be a prime 3. Let m be a maximal deal of TH(N) with m. Then associated to m there s a continuous dd semisimpleGalois representation pm, (2.1)

pm: Gal(Q/Q)

-- GL2(T/m)

unramifiedutsideNp which atisfies tracepm(Frob

Tq,

detpm(Frob

(q)q

in Gal(Q/Q). for ach prime Np. Here Frobq denotes Frobenius The representation (resp. N) we unique up to isomorphism. resp. Up m). This implies cf.,for xample, say thatm s ordinary Tp theorem of [Wil]) that for ur fixeddecomposition roupDp at p, P

Dp

D(O

0 X2

2)

fora suitablechoiceof basis, with X2 unramifiednd X2(Frobp) Tp mod (resp. equal to Up). In particularPm ordinary the sense of Chapter


481

providedXi 54X2- We willsay that Dp-distinguished m s ordinary nd Xi X2. (In practiceXi is usuallyramifiedo this mposesno extracondition.) We cautionthe reader hat Pm ordinaryn the senseof Chapter thenwe can onlyconcludethat N. Dp-distinguished Let Tmdenote hecompletion at o thatTm s a direct actor fthe complete emi-local ingTp Zp. Let be thepointsof the associated m-divisibleroup JH(N) (Q)m

JH(N) (Q)poo (0 Tm. Tp

It is knownthat HomzP(D, Qp/Zp) is a rank Tm-module, .e., that Qp (Tm Qp)2. Briefly is enough to show that H1(XH(N), C) is zp zp free frank over and thisreducesto showing hat S2 (rH(N), C), the space ofcusp forms fweight on rH(N), is free f rank over 0 C. One showsthenthat if fi, ... f4} s complete et ofnormalized ewforms in S2(rH(N), C) of levels mj,...,m, then if we set di N/mi, the form C-module. fi(diz) is basis vector S2(rH(N), C) as a If is ordinary hen [Wil], itself straightforwardeneralization of Proposition and (11) of [MW2], showsthat (forour fixeddecomposition roupDp) there s a filtration byPontrjagin uals ofrank Tm-modulesin the senseexplainedabove) (2.2)

Do

__DE _0

whereDo is stable underDp and the nduced ctionon DE is unramified ith Frobp Upon it if and Frobp equal to the unit rootofx2 -Tpx p(p) in Tm if N. We can describeDo and DE as follows. Pick of Let e:Dp ZIp which nduces a generator Gal (Qp((Npoo)/Qp((Np)).

be the cyclotomic character. Then DO

ker oa E(a))diV,

the kernel being

taken nside and 'div' meaning he maximaldivisible ubgroup. Although in [Wil] this filtration givenonly for factorAf of Ji N) it is easy to deduce the resultforJH N) itself. We note that this filtrations defined without eferenceo characteristic and also that m s Dp-distinguished, (resp. DE) can be described s the maximalsubmoduleon which 1(of) is topologicallynilpotentforall of Gal(Qp/Qp) (resp. quotienton which f- X2(07) is topologically ilpotent or ll of Gal(Qp/Qp)), where%i(of) any ifting Xiy(o) o Tm. The Weil pairing , on JH(N)(Q)pM satisfies he relatio tx,y) (x, t*y) forany Hecke operatort. It is moreconvenient o use an adapted pairingdefined follows. Let w(, for primitiveNth root of 1, be the involution fXi (N)/Q(() defined [MW1,p. 235]. This induces n involution of XH(N)/Q(() also. Then we can define new pairing , ] by setting for

WILES

482

fixed hoiceof (2.3)

[x,Iy] (x,wy).

Then [tx, y] [x, y] for ll Hecke operators In particularwe obtain an inducedpairing n Dpa. The following heorem s the crucial resultof this section. It was first provedby Mazur in the case ofprime evel Ma2]. It has sincebeengeneralized in [Til], [Ril] [M Ri], [Gro]and [El], but the fundamentalrgument emains that of Ma2]. For summaryee [E1, ?9]. However ome ofthe cases we need are not covered theseaccountsand we will present hese here. THEOREM 2.1.

(i) If

and Pm s irreduciblehen

JH(N) (Q) [m] --(T/M)2.

(ii) If

and Pm s irreduciblend m is Dp-distinguishedhen JH Np) (Q) [m]

(In case (ii) m is a maximal ideal of

(T/)2 TH(Np).)

COROLLARY 1. In case (i), JH(N)(Q)m. T2m M.

In case (ii), JH(Np)(Q)m

Tm

T2

and Tam (JH(N)(Q)>

T. and Tam JH(Np) (Q))

T.

(where

TH(NP)m)

COROLLARY 2. In either

cases (i) or (ii) Tm is

Gorenstein ing.

In each case the firstsomorphisms Corollary follow rom he theorem togetherwith the rank resultalluded to previously.Corollary and the second somorphisms corollary then follow applyingduality 2.4). (In the proof nd in all applicationswe will onlyuse the notionof Gorenstein Zp-algebra s defined the appendix. For finite lat ocal Zp-algebras he notions of Gorenstein ingand GorensteinZp-algebra re the same.) Here Tam JH(N) (Q)) Tap (JH(N) (Q)) Tm is the m-adic Tate module of JH(N).

We shouldalso pointout that althoughCorollary gives representation from hem-adicTate module PTm: Gal(Q/Q) -* GL2 Tm)

thiscan be constructed a muchmore lementary ay. (See [Ca3] for nother argument.)For,the representationxistswithTm replacing whenwe use the fact hat Hom(Qp/Zp,D) was free rank2. A standard rgument


483

usingthe Eichler-Shimuraelations mpliesthat this representation ' with valuesin GL2(Tm Q) has the property hat tracep'(Frob )

Te,

detp'(Frob )

f(f)

forall Np. We can normalize his representation y picking complex conjugation and choosing basis suchthatp'(c) (' _), and thenbypicking a r forwhichp'(r) (a br) with b,c, 0(m) and by resealing he basis so that b, 1. (Note that the explicitdescription fthe tracesshowsthat if Pm then mod m = brmCrm where is also normalized o that pm(c) (o _)b1c1 such that b~c, 0 0(m) comes from P.m(T)= (lrm dbrn,). The existenceof the irreducibility pm.) With thisnormalizationne checks hat p' actually takes values in the (closed) subring Tm generated ver Zp by the traces. One can evenconstruct he representationirectly rom he representations Theorem0.1 usingthis ringwhich s reduced. This is the metho of Carayol whichrequires lso the characterization p by the traces and determinants (Theorem of [Ca3]). One can also often nterprethe Uq operators terms of for usingthe 7rq 7r(gq) theorem f Langlands cf. [Cal]) and the Up operator case (ii) usingTheorem2.1.4 of [Wil]. Proof of theorem).The importantechnique or roving uchmultiplicityone results s due to Mazur and is based on the q-expansion rinciple characteristic Sincethekernel JH N) (Q) -* Ji (N) (Q) is an abelian groupon whichGal(Q/Q) acts through n abelian extension Q, the ntersection ith kerm is trivialwhen pm irreducible.So it is enoughto verify he theorem forJ1(N) in part (i) (resp. Ji(Np) in part (ii)). The methodforpart (i) was developedby Mazur in [Ma2, Ch. II, Prop. 14.2]. It was extended the case of Fo(N) in [Ril, Th. 5.2] which ummarizesMazur's argument.The case of F1(N) is similar cf. [El, Th. 9.2]). A\. Let us first Now consider ase (ii). Let A(p) {(a): l(N)} for ome A(p). This assumethatA(p) is nontrivialmodm, .e.,that6-1 case is essentially overed [Til] (and also in [Gro]). We briefly eview he argument oruse later. Let Qp((p), (p being primitiveth rootofunity, and let be thering ntegers thecompletion the maximalunramified extension K. Usingthe factthat A(p) is nontrivialmod togetherwith Proposition , p. 269 of [MW1]we find hat (Np)m/tO (Fp)

(Pic?0t

Pico E1)m (Fp)

wherethe notation takenfrom MW1] loc. cit. Here E't and El, are the twosmooth rreducibleomponents fthe special fibre fthecanonicalmodel of X1(Np)l0 described [MW1, Ch. 2]. (The smoothness n this case was proved [DR].) Also Jl Np)6t1 denotesthe canonicaletale quotientofthe r-divisiblegroupover0. This makessensebecause J1(Np)m does extendto

ANDREW WILES

484

p-divisible roupover (again bya theorem Deligne and Rapoport [DR] and because A(p) is nontrivialmodm). It is ordinary s follows rom2.2) when we use the main theorem f Tate ([Ta]) since Do and 'DE clearly orrespond to ordinary -divisible roups. Now the q-expansion rinciple mplies hat dimp m'] where {Ho (EP, Q1

Ho EtI Q1)}

and m' defined y embedding /m Fp and settingm' ker: Fp -Fp under the map at mod m. Also T acts on PicoEA Pico Et, the abelian variety artofthe closed fibre f the Neronmodel of J1(Np) IO, and hence also on its cotangent pace X. (For a proof hat X[m'] is at mostonedimensional,which readily dapted to this case, see Lemma 2.2 below. For similarversions n slightly impler ontexts ee [Wi3,?6] or [Gro,?12].) Then the Cartiermap induces injection cf. Prop. 6.5 of [Wi3]) 6: {Pic0 im Pico E6t}[] (p)

Ip

C

X.

The composite can be checked o be Hecke invariantcf. Prop. 6.5 of [Wi3]. In checking he compatibilityorUp use the formulas Theorem5.3 of [Wi3]but notethecorrection [MW1,p. 188].) It follows hat

T/ra

J,(Np)m/0Fp) [m] as

T-module. This shows that if is the Pontrjagin dual of Tm ince H/m T/m. Thus Ji(Np)m10(Fp) thenH JiNp)m1(Fp)

Hom(Tm/p, Z/pZ)

[p]

Now our assumption hat m Dp-distinguishednablesus to identify

'Do= J,Np)m (Qp)

XDE

j, (Np)6t

For the groupson the right re unramifiednd those on the left re dual to groupswhere nertia cts via a character f finite rder dualitywithrespect to Hom( , Qp/Zp(l))). So 'Do[p]

Tm/p, DE[p]

Hom (T./p, Z/pZ)

as Tm-modules,heformerollowingrom he atterwhenweuse dualityunder the pairing ,]. In particular Dp-distinguished, (2.4)

'D[p]

Tm/p

Hom T./p, Z/pZ).

We now use an argument Tilouine [Til]. We pick complexconjugation on Pm o we may decompose D[p] into T-. This has distinct igenvalues eigenspaces for -r:

D[p]

D[p]+

D[p]-.

MODULAR ELLIPTIC

CURVES

AND FERMAT'S

LAST THEOREM

485

SinceTm/p nd Hom Tm/p,Z/pZ) are bothindecomposableHecke-modules, by the Krull-Schmidtheorem his decomposition as factorswhichare isomorphic o those (2.4) up to order.So in the decomposition D[m]

D[m]+ GDD[m]

one ofthe eigenspaces s isomorphic T/m and the other o (Tm/p)[m]. ut sincePm irreducible is easy to see byconsidering [m] Hom(D [m], etPm) that has the same number eigenvalues qual to +1 as equal to -1 in D[m],

whence #(Tm/p) [m]

#(T/m). This shows that D[m]+

D[m]-

T/m as

required. Now we consider he case where \(p) is trivialmod m. This case was treated but only forthe group Fo(Np) and Pm new' at p-the crucial restriction eingthe last one) in [M Ri]. Let Xl (N, p)/Q be the modularcurve corresponding I1 (N) Fo(p) and let Ji(N,p) be its Jacobian. Then since the composite ofnatural maps J1 N, p) -) Ji Np) -+ Ji N, p) is multiplication and since A(p) is trivialmod we see that by an integer rime J, N. p)m(Q) -_J,Np)

Q)

It willbe enough hento use J1 N, p), and thecorrespondinging and ideal m. The curveX1(N,p) has a canonical model X1(N,p)z whichover Fp consistsof two smooth curves SEt and E2 intersectingransversallyt the supersingular oints (again this is a theorem f Deligne and Rapoport; cf. [DR, Ch. 6, Th. 6.9], [KM] or [MW1]formoredetails). We willuse themodels described [MW1, Ch. II] and in particular he cusp ox will lie on E?. Let Q denotethe sheafofregulardifferentials X1(N,P)/FP (cf. [DR, Ch. ?2], [M Ri, ?7]). OverFp, N, p) j; ordinary oublepoint ingularities, the differentials ay be identified iththe meromorphicifferentialsn the normalization N, p)j SEt EA whichhave at mostsimplepoles at the supersingularoints the ntersectionoints f hetwocomponents)nd satisfy if x1 and x2 are the twopointsabove such supersingular resxj resX2 point.We needthe followingemma: LEMMA 2.2.

dimT/.HO(X1(N,p)/FpA)

[m]

1.

Proof. Firstwe remark hat the actionof the Hecke operatorUp here most convenientlyefined singan extension rom haracteristicero. This is explainedbelow. We will first howthat dimT/. HO Xi (N, P)/Fp [m] 1, this being the essential step. If we embed T/m Fp and then set m' ker: T Fp Fp (the map givenby atmodim)then it is enough o showthat dimppHO(X1 N, p)/li, Q) [m'] 1. Firstwe will suppose

486

ANDREW WILES

that there s no nonzeroholomorphic ifferentialn Ho Xi (N, p)/F Q)[i'], i.e., no differentialormwhichpulls back to holomorphic ifferentials E' and EJ. Then if wi and W2 are two differentials Ho(Xi (N, p)/, ,) [n'], the q-expansion rinciple hows hat uW1 AW2has zero q-expansion oo for on some pair (,u,A) $& 0, 0) in F2 and thus is zero on EyL* As ,uwl AW2 EY it is holomorphic Eet. By our hypothesist would thenbe zero which

showsthat wi and W2 are linearly ependent. This use oftheq-expansion rinciple characteristic is crucialand due to Mazur [Ma2]. The point s simply hatall the coefficients the q-expansion are determined elementary ormulae rom he coefficient q provided hat is an eigenformor ll the Heckeoperators.The formulae or he actionof theseoperators characteristic follow rom he formulae characteristic zero. To see this formallyespeciallyforthe Up operator)one checksfirst that Ho(Xi (N, p)/zp,Q), whereQ denotes he sheafofregular ifferentials Xi(N,p)lzp,

behaves well under the base changes Zp -* Fp and Zp

Qp;

cf. [Ma2, ?11.3]or [Wi3, Prop. 6.1]. The action of the Hecke operatorson J1 N, p) induces actionon theconnected omponent ftheNeronmodelof J1 N, P)/QP, so also onitstangent pace and cotangent pace. By Grothendieck dualitythe cotangent pace is isomorphic H0(Xj(N,p)1ZPQ); see (2.5) below. (For summary f the dualitystatements sed in this context, ee [Ma2, ?II.3]. For explicitdualityoverfields ee [AK, Ch. VIII].) This then defines actionof the Heckeoperators thisgroup.To check hat overQp thisgives he standard ctionone usesthecommutativity thediagram fter Proposition .2 in [Mil]. Now assumethat there s a nonzeroholomorphic ifferential H?(Xi(N.p)1V'

[in'].

We claimthatthespace ofholomorphic ifferentialshenhas dimension and that any such differential :$ 0 is actuallynonzeroon EY. The dimension claimfollows rom hesecondassertion y using he q-expansion rinciple.To prove hat :$ on EA we use the formula Up*(x,y)

(Fx, y')

for x,y) (Pico YE Pico E)(Fp), where F denotes the Frobeniusendovariant morphism. The value of y' will not be needed. This formula on the second part of Theorem 5.3 of [Wi3] where the corresponding esult is provedforXi (Np). (A correction o the firstpart of Theorem5.3 was noted in [MW1, p. 188].) One checksthen that the action of Up on Xo Ho EI, Q71) Ho YEt, 1) viewed as subspace of HO Xi (N,p)/ is the same as the action on Xo viewedas the cotangent pace of Pico Ell on EA then Upw on E't. But Up Pic? SEt. Fromthiswe see that if

MODULAR ELLIPTIC

CURVES


487

acts as nonzero calarwhichgives contradiction :$ 0. We can thusassumethatthespace ofm'-torsionolomorphicifferentialsas dimension and is generated y w. So ifW2 is now any differential HO(X1 (N, p)/P, Q) [m'] thenW2 Awhas zeroq-expansion oX for ome choice fA. Then W2 Aw on EA whenceW2 Aw holomorphicnd so W2 Aw. We have now shown in general that dim(HO (Xi(N,p)1F

, Q) [m'])

1.

The singularities X1(N,p)lZP at the supersingular ointsare formally isomorphicver unr to Zunr [X, Y]] /(XY -k) withk 1, or 3 (cf. [DR, Ch. 6, Th. 6.9]). If we consider minimalregularresolutionM1(Np)1ZP then H0(M1(NP)/Fp, 7) HO(X1(N.P)/FP, 7) (see the argument [Ma2, Prop. 3.4]), and similar somorphism oldsforHO Ml (N, p)/ZP,Q) As M1(N,P)/ZP is regular, theorem Raynaud [Ray2] says that the connectedcomponent f the Neron model of J1(N,P)/QP is Ji(N,P)5/z PicO(M1(N,p)1zP). Takingtangent paces at the origin,we obtain (2.5)

Tan(Ji N, p)?Z

H1 Ml N. p)1ZP, OM1 N,p))

Reducingbothsides modp and applyingGrothendieckualitywe get an isomorphism (2.6)

Tan(J1(Np)5/F)

Fp). (To justify he reduction n detail see the arguments [Ma2, ?II. 3]). Since Tan(J1 N, p)?Z is faithful Zp-module t follows hat Hom(H0(Xl(NP)/FP,

H? (Xi (N. A)/FPiQ) [ml

is nonzero.This completes heproof thelemma.

OI

To completethe proofof the theoremwe choose an abelian subvariety of J1 N, p) with multiplicative eduction t p. Specifically et be the connectedpart ofthe kernel J1 N, p) -* J1 N) J1 N) underthe natural map described Section (see (2.10)). Then we have an exact sequence *A -* J1(N,P) -*B O* and J1(N,p) has semistablereduction ver Qp and B has good reduction. By Proposition1.3 of [Ma3] the correspondingequence of connectedgroup schemes -' A[p]ZP] J1 N. p) p]/Z B[IZP -o is also exact,and by Corollary .1 ofthe same proposition he corresponding sequenceoftangent paces ofNeronmodels s exact. Usingthis we maycheck that the naturalmap (2.7)

Tan(J1(N,p)[p]t

Tm Tan(J(Np) Ta(l N P) TP

TP

Tm

ANDREW WILES

488

is an isomorphism, here denotesthe maximalmultiplicative-typeubgroup scheme cf. [Ma3, ?1]). For it is enoughto check uch a relation and B separately nd on it is truebecause the r-divisiblegroup s ordinary. his follows rom 2.2) by the theorem f Tate [Ta] as before. Now (2.6) togetherwiththe lemmashowsthat Tan(Ji(Np))/zP (0 Tm Tm. Tp

We claim that (2.7) togetherwiththis mplies hat as Tm-modules := Ji(N,p) [p]t(Q,)m (Tm/p). To see this is sufficiento exhibit

isomorphism Fp-vector paces

(2.8)

G(Qp)

Tan(G/f

Fp

for ny multiplicative-typeroup cheme finite nd flat) GIzp which s killed by p and moreover o give such an isomorphismhat respectsthe action of endomorphisms Glzp. To obtainsuch an isomorphismbserve hatwe have isomorphisms (2.9)

Hom-, (lp, G) X9

Hompp lp, G) (9 Fp Hom (Tan(tip/. Tan(G/V

whereHomQ denoteshomomorphisms the groupschemesviewedover Qp and similarly orHomF The second somorphisman be checked y reducing to the case lip. Now picking primitive th root of unitywe can identify he left-hand erm (2.9) with G(Qp) Fp. Picking isomorphism Fp

Tan(,ip/l;) with Fp we can identify he last term (2.9) with Tan(G/y ). Thus afterthese choicesare made we have an isomorphism (2.8) which respects he actionofendomorphisms G. On the otherhand the action of Gal(Qp/Qp) on is ramified every 'DQ[p]. (Note that our assumption hat A(p) is trivial subquotient, mod implies hat the actionon V0 p] s ramified n every ubquotient nd on DyE[p] s unramified every ubquotient.) By again examinin and separatelywe see that in fact ZDo[p].For we note that A[p]/A[p]t unramified ecause it is dual to A[p]twhereA is thedual abelian variety.We can now proceedas we did in the case whereA(p) was nontrivialmodm.


489

2. Congruences between Hecke rings Suppose that is primenotdividingN. Let IF(N, q) P1(N) 1o(q) and let Xi (N, q) X1 N, q)/Q be the correspondingurve. The twonatural maps X1 (N, q) -- Xi (N) induced by the maps -and z -- qz on the upper halfplane permit s to define map Ji N) J1 N) -- Ji N, q). Using theorem hara,Ribet shows hatthis map is injective cf. [Ri2,Cor. 4.2]). Thus we can define by Ji N)

(2.10)

J, N) J1

(N, q)

Dualizing,we define by -+B

B-

Ji(N, q)

Ji(N)

Ji(N) -+0.

Let T1 N, q) be the ring of endomorphisms Ji N, q) generatedby the standardHecke operators T1* for I; Nq, Ul* for | Nq, (a) (a)* for (a, Nq) 1}. One can check hat Uq preserves eitherby an explicit alculationor by noting hat is the maximalabelian subvariety J1N, q) with multiplicativeeduction q. We set J2 Ji N) Ji N). More generally,ne can considerJH N) and JH N, q) in place of J1N) and J1(N,q) (whereJH(N,q) corresponds X1(N, q)/H) andwewriteTH(N) and TH(N, q) for he associated Heckerings. In thiscase the corresponding map may have a kernel.However ince the kernel JH N) -+ J1N) does not meet ker forany maximal deal whose associated Pm irreducible, the above sequencesremain xact ifwe restrict M(q)-divisibleroups,M(q) being the maximal deal associated to m of the ring q)(N,q) generatedby the standard Hecke operatorsbut omittingUq. With this minormodification the proofsof the resultsbelow for #& follow rom he cases of full level. We will use the same notation the general ase. Thus 0 is the map and z -- qz on the two factors, J2 JH(N)2 --+ JH(N, q) inducedby -and ker (B willnot be an abelian variety general.) The followingemma is straightforwardeneralization f lemma of Ribet ([Ri2]). Let nq be an integer atisfying q(N) and nq 1(q), and write q) (nq) TH(Nq). LEMMA 2.3 (Ribet). ducible m.

4'(B)

PO(J2).(q)

sc(J2) [Uq

(q)]M(q)

for irre-

Proof. The left-hand ide is (im n ker3), so we compute p1 (im of ker )= ker( p). An explicit alculation howsthat Woc~o=[ q+

-qi1

onJ2

ANDREW WILES

490

where on J2

Tq (q)'.

The matrix ctionhere s on the eft.We also find hat

[~~~~~ Tq

(2.11)

whence -(q)

(U2-(q))o

(

Now suppose that is maximal ideal of TH(N), p and Pm irreducible.We will nowgive slightlytronger esult han that given n the lemma the special case we will also strengthenut p. (The case we will do thisseparately.)Assume then that and Tp m. Let ap be in TH(N)m. We first efine maximal the unit root of x2 Tpx + p(p) ideal TH (N, p) with hesame associatedrepresentation m. To do this consider hering S1

TH(N)[Ul]/(U2

TpUl

p(p))

End(JH(N)2)

whereU1 is the endomorphism JH(N)2 givenby the matrix

Tp -(Pj It is thuscompatiblewiththe action of Up on JH(N, p) whencomparedusing (i, UiNow -p ) is maximal deal of S1 where is any element of TH(N) representinghe class aip TH(N)m/m TH(N)/m. Moreover Si,mi TH(N)m and we let mp be the inverse mageofml in TH(N,p) under the naturalmap TH(N, p) -* S1. One checks hatmp Dp-distinguished. or anystandardHeckeoperator exceptUp i.e., TI, Uqifor :$ or (a)) the imageof t is t. The imageof Up U1. We need to check hat the inducedmap A.

a: TH(NP)m

-+

Silmi

TH(N)m

is surjective.The onlyproblem toshow hatTp s inthe mage. In thepresent context ne can provethis usingthe surjectivity in (2.12) and usingthe factthat the Tate-modulesn the rangeand domainof are free frank by Corollary to Theorem .1. The result henfollows romNakayama's emma s one deduceseasily hatTH(N)m is cyclicTH(N, p)m-module. This argument was suggestedby Diamond. second argument singrepresentationsan be found t theend ofProposition .15. We willnowgivea third nd moredirect proofdue to Ribet (cf. [Ri4, Prop. 2]) but found ndependentlynd shown o us by Diamond.

491


For thefollowingemmawe etTM, for integerM, denote hesubring f End (S2(rF (N))) generated ythe HeckeoperatorsTnforpositive ntegers relatively rime M. Here S2 I1 (N)) denotesthe vector pace ofweight cusp forms n IF (N). Write forT'. It will be enoughto showthat Tp is a redundant perator T', i.e., that TP T. The resultforTH(N)m then follows. LEMMA (Ribet).

If

Suppose that (M,N)

is even thenTM has finite ndex n

1. If

is odd then TM

equal to powerof2.

T.

As the rings re finitelyenerated ree -modules, suffices prove hat TM F1 -are both even. The claim F1 is surjective nless 1 and follows rom 1. TM OF1 -- TM/POF1 is surjective 2. T'

F1 -+

F1 is surjective f

M and

IN.

2N.

Proofof1. Let denote he Tate moduleTal(Ji(N)). Then TM/PO (R Ql) Endz, and choose d so that Zj acts faithfully A. Let R' ldR'

iR.

Zl Consider the Gal(Q/Q)-module

Jl(N)[1d]

uN~d. By

Cebotarevdensity,here s a prime notdividingMNl so thatFrob on B. Usingthe fact that Tr Frobr (r)r(Frobr)'- on for q, we see that Tp

and therefore idR'

Frob and

Tq on J1 N) [Id]. It followsthat Tp Tq is in ld Endz,

iR.

Proofof2. Let be theset of uspforms S2 (r1 (N)) whoseq-expansions at oo havecoefficientsn Z. Recall thatS2(rF N)) SoC and thatS is stable underthe action of (cf. [Shl, Ch. 3] and [Hi4,?4]). The pairing S-+ definedby a, (Tf) is easily checked o induce an isomorphism T-modules Homz (T, Z). The surjectivity T'/lT' map

-+

T/lT is equivalent o the injectivity the dual

Hom(T, Fj)

-*

Hom(T1, Fj).

Now use the isomorphism /iS

Hom(T, Fj) and note that if is in the kernel of -- Hom(T1, Fj), then an(f) al(Tnf) is divisible by for all prime 1. But then hemod form efined y is inthekernel ftheoperator and is thereforerivial is odd. (See Corollary of the main theorem of Ka].) Therefore is in IS. Remark. The argument oes not provethat TMd

Td if d, N) $&

492

ANDREW WILES

We now return o the assumptions hat Pm irreducible, and Tp m. Next we define principal deal (Lxp) of TH(N)m as follows.Since TH(N, p)mp nd TH(N)m are both Gorenstein ings by Corollary of Theorem2.1) we can define n adjointa^ to a: TH(NPp)m

Siml

TH(N)m

in the mannerdescribed the appendixand we set ?p (a a^)(1). Then (zAp) is independent f the choiceof (Hecke-module) airings n TH(N, p),, and TH(N)m. It is equal to the ideal generated y anycompositemap TH(N)m

TH(N)m TH(NP)nvp provided hat/3s an injectivemapofTH (N, p)mpmoduleswithZp torsion-free cokernel. The modulestructure TH(N)m is defined ia a.) PROPOSITION2.4. Assume that is Dp-distinguishednd thatPm irreduciblef evel with N. Then (Ap)

(T2

(p)(1 +p)2)

(a2

(p)).

Proof. Consider he maps on p-adic Tate-modules nducedby o and A':

Tap JH (N) 2)

Tap JH (N, p))

Tap JH (N)

These maps commutewiththe standardHeckeoperatorswiththe exception of Tp or Up (which re not even defined n all the terms). We define S2

Tp U2

TH(N)[U2]/(U22

End (JH(N)2)

whereU2 is the endomorphism JH(N)2 definedby (? j(P)) (O U2

Upo. Again m2

(m, U2

It satisfies

p) is a maximal ideal of S2 and we have,

on restrictingo the ml, mp nd m2-adic ate-modules: Tam2 (JHN2)

(2.12)

V2 Tam JH N))

Tamp (JH(N,p))

-Tam,

(JH(N)2)

vl Tam JH N))

The vertical somorphismsre defined v2: apx) and vi: (apx, px). (Here ap TH(N)m can be viewedas an elementof TH(N)p HITH(N)n wherethe product is taken over the maximal ideals containing p. So vi and v2 can be viewed as maps to Tap (JH(N)2) whose images are respectively am, JH(N)2) and Tam2 JH(N)2).) Now is surjective nd p is injectivewithtorsion-freeokernel the result of Ribet mentionedbefore. Also Tam JH(N)) TH(N)2 and


493

Ta,,p JH(N, p)) TH(N, p)2 by Corollary1 to Theorem2.1. So as I, are maps of TH(N p),p-moduleswe can use this diagramto computeLp as remarkedust prior o the statement f the proposition. The compatibility ofthe Up actionsrequires hat,on identifyinghe completions 1iml and S2,M2 withTH(N)m, we get U1 U2 which s indeed the case.) We find hat VT11

(p)) (Z).

We nowapplyto J1 N, q2) (but the sameanalysis hatwe have ust appliedto J1(N,p). HereXi(A, B) is thecurve orresponding F1(A)nro(B) and J1A, B) its Jacobian. First we need the analogue of hara's result. It is conveniento work n a slightlymoregeneral etting.Let us denotethe maps Xi(Nqr-lqr) X(Nqr'-1) induced by -- z and z -- qz by 7lrl, and 7r2,r X1 Nqr) induced respectively. Similarly we denote the maps Xi (Nqr, qr+l) by -+ z and z -? qz by 7r3,r nd 7r4,rrespectively. Also let ir: Xi (Nqr) X1 Nqr-l, qr) denote the natural map induced by -- z. In the followingemma m s a maximal deal ofTl(Nqr-1) or T1(Nqr) we use M(q) to denote the maximal deal o Ti (Nqr, qr+l) compatiblewith m, the ring T(q) Nqr, qr+l) T, (Nqr, qr+l) being the subring obtained by omitting q from helistofgenerators. LEMMA2.5. If varieties

$&

-O Ji(Nqr-l)

where 6i

((irr

J(Nqr) o 7r)*,

thenthesequenceof abelian

is primeand J1(Nq)

J1(Nq

qr+l)

(7r*r,7r*,r) induces a corre-

(7r2,r 7r)*) and

sponding equenceofp-divisible roupswhichbecomes xact when ocalized anyM(q)for which m s irreducible. Proof.Let Fl(Nqr) denotethe group O(qr-1), B1 and let

P1(N):

l(qr),

O(q)}. Let Bi and B1 be givenby

Fl(Nqr) jFi(Nqr) Fi(Nqr-l)/ri(Nqr)

r(q),

B1

f1(Nqr)/F

r(q). Thus

(Nq r)

SL2 (Z/q) if

is of order powerof ifr > 1. The exact sequencesof nflation-restriction ive: A1 H1 Fi (Nqr) Q /Z)-*H1 (r1(Nqr) n F(q), Qp/Zp)B1

and

togetherwith similarsomorphism ithA1replacingA1 nd B1 replacing 1. We also obtain H1 F1 Nqr-l), -H1 (F1 Nqr) F(q), QP/ZP)Aq. /Z

494

ANDREW WILES

The vanishing f H2(SL2(Z/q), Qp/Zp) can be checkedby restrictingo the Sylow -subgroup hich s cyclic.Notethat A1,nimAl H1 Fl (N qr)nr(q), Qp/Zp)~'q sinceB1 and B1 together enerateAq. Now consider hesequence (2.13) 0 /Z) H1(Fl(Nqr-l), Hl (Fl Nqr Qp/Zp)EDHl (rl (Nqr Qp/Zp)

>>

F(q), Qp/Zp).

Hl (F1 Nqr)

We claim it is exact. To checkthis, suppose that A1(x) -A1(y). Then Hl(Fi(Nqr) Al(X) F(q),QP/ZP)Aq. So Al(x) is the restriction f an x' Hl (IF(Nqr-l), Qp/Zp) whence resi(x') kerAl 0. It follows also that y =-res1 (x').

Now onjugation thematrix(O?) inducessomorphisms -_ (Nqr),

rl(Nqr)

(Nqr)

nfF(q)

Fi(Nqr, qr+l).

So our sequence 2.13) yields he exact sequenceofthe emma, xceptthat we have to changefrom roup cohomology o the cohomology f the associated completecurves. If the groupsare torsion-freehen the differenceetween for _ modNqr+l thesecohomologiess Eisenstein moreprecisely is nilpotent) will vanish when we localize at the preimageof M(q) in the abstractHeckeringgenerated polynomial ingby all the standardHecke then the group r1(M) has torsion. For operators xcludingTq. If 1,2,3 we can restrict F(3), F(4), F(3), respectively, here the cohomology s Eisenstein s the correspondingurveshave genus zero and the groups re torsion-free.hus one onlyneedsto check he actionof the Hecke operators n thekernels f herestriction aps these hree xceptional ases. This can be doneexplicitlynd again they re Eisenstein.This completes he proof fthe lemma. X1 N) induced by -* z and z -* qz by 7r1nd 7r2 espectively.imilarly e denote hemaps X1 N, q2) -* X1 N, q) Let us denote the maps X1 (N, q)

induced by

-*

z and z

-*

qz by 7r3 nd 7r4 espectively.

1) and Ihara's result 2.10) we deduce that

From the lemma (with there s a sequence (2.14) where

-*

J1(N)

-*

J1(N)

(7rl 7r3)* (7r2 13)*

J1(N)

Ji(Nq2)

(7r2 14)* and that the induced map of p-

divisible roupsbecomes njective fter ocalization M(q) which orrespond to irreducible m's.By dualitywe obtain sequence J1(Nq2)

J1(N)3

-*

which s 'surjective' n Tate modules the same sense. More generallywe can prove nalogousresultsforJH(N) and JH(N, q2) although heremay be

MODULAR ELLIPTIC

CURVES

kernelof order divisible by


in JH N)

-*

495

Ji N). However this kernel will

notmeettheM(q)-divisibleroupfor nymaximal deal m(q)whoseassociated Pm irreduciblend hence, in the earlier ases, willnotaffect heresults f after assing p-divisible roupswe localize at suchan m(q).We use thesame notation thegeneral ase when $& so is themap JH(N)3 -* JH(N, q2). We supposenow that s a maximal deal of TH(N) (as alwayswithp m) associatedto an irreducible epresentationnd that is prime, Np. We now define maximal deal mqof TH(N, q2) with the same associated representations m. To do thisconsider he ring Si TH(N)[Ui]/Ul(U?2-Tq End (JH(N)3) U1 q(q)) where he actionofU1 on JH(N)3 is givenbythematrix Tq -(q)

0 Oj

Then U1 satisfies hecompatibility Uq

Ul

One checks his usingthe actions on cotangent paces. For we may identify thecotangent paceswith pacesofcusp formsnd with his dentificationny Heckeoperator induces heusual actionon cusp forms.There s a maximal

ideal ml (U1, m) in Si and Slmi TH(N)m. We let mqdenote the reciprocal image of ml in TH(N, q2) under the natural map TH(N, q2) -* S1.

Next we define principal deal (A') of TH(N)m using the fact that TH(N, q2)m, nd TH(N)m are both Gorenstein ings cf.Corollary to The-

orem 2.1). Thus we se (A')

') where

(a'

a': TH(N, q2 mq

Slm

TH(N)m

is the naturalmap and a" is theadjointwithrespect selectedHecke-module pairings TH(N, q2)mq nd TH(N)m. Note that a' is surjective. To show that the Tq operator inthe mageone can use theexistence ftheassociated 2-dimensional epresentationcf. ?1) in whichTq trace(Frobq) and apply the Cebotarevdensity heorem. PROPOSITION 2.6.

Suppose thatm is a maximal deal ofTH(N) associated to an irreducible m. uppose also that Np. Then (A') = (q

1) (T2

(q)(1 + q)2).

Proof. We provethis n the same manner s we provedProposition .4. Consider he maps on p-adic Tate-modules nducedby and (: (2.15)

Tap (JH N))

Tap (JH (N,

q2))

Tap (JH N)3)

496

ANDREW WILES

These maps commutewith the standardHeckeoperatorswiththe exception ofTq and Uq (which re noteven defined n all the terms).We define S2

TH(N)[U2]/U2(U22-Tq

U2

q(q))

End (JH(N)3)

whereU2 is theendomorphism JH(N)3 givenbythe matrix -(q) Tq Then Uq because

U2as one can verify ycheckingheequality o0) U2 U1 o ) is an isogeny. The formula or is given below. Again m2 (i, U2) is maximal deal of S2 and TH(N)m. On restricting (2.15) to theM2,Mqand m1-adic ate moduleswe get

(2.16)

Tam2(JH(N)3)

>Tamq(JHN, q2)

Tam, JH(N)3)

TU1

U2 Tam(JH(N))

Tam(JH(N)).

The vertical isomorphisms are induced by u2: ((q)z, -Tqz, qz) and ui: z (0, 0, z). Now calculation shows that on JH(N)3 q(q 0o

1)

Tq*

Tq*2~~~~~~qq1 (q)-'l q)

Tq q(q+ 1)

Tq2-(q)(1 Tq

q)

q(q +

Tq*

where Tq*= (q)-1Tq.

We compute henthat (U1

2)=(q-l)(q

1) Tq-(q)(1

q))

Now usingthe surjectivityf and that has torsion-freeokernel (2.16) (by Lemma 2.5) and that Tam JH(N))

and Tam (JH(N, q2)) are each freeof

rank over herespective eckerings Corollary ofTheorem2.1), we deduce the result s in Proposition .4. [1 There is one furtherand completely lementary) eneralization f this result. We let ir: XH(Nq,q2) -* XH(N,q2) be the map given by -* z. Then lr*: JH(N, q2)

-*

JH(Nq, q2) has kernel

cyclic group and as before

this will vanishwhen we localize at M(q) if is associated to an irreducible representation.As before he superscript denotesthe omission Uq from the list of generators TH (Nq, q2) and M(q)denotesthe maximal deal of T(q) (Nq, q2) compatible with m.)


497

We thushave sequence not necessarily xact) -*

-)JH(Nq

JH(N)3

lr* which nduces correspondingequenceofp-divisible roups where whichbecomes xact when ocalizedat an m(q) orrespondingo an irreducible Pm.Here Z is th quotient belian varietyJH(Nq, q2)/ im As before here is a natural urjective omomorphism a: TH(Nq,

Si)me

)mq

TH(N)m

wheremq s the inverse mage of ml in TH(Nq, q2). (We note that one can replace TH(Nq, q2) by TH(Nq2) in the definition f a and Proposition2.7 belowwouldstillhold unchanged.) Since both rings re again Gorensteinwe can define adjointa' and a principal deal (Aq)

a')

(a

Suppose thatm is maximal deal of TH(N) associatedto an irreducibleepresentation.upposethat Np. Then PROPOSITION

2.7.

(Aq)

(q

1)2 (T2

(q)(1

q)2))

The proof s a trivialgeneralizationf that ofProposition .6. Remark .8. We have ncluded heoperatorUqin the definition Tmq TH(Nq, q2)mq s intheapplication theq-expansion rinciple is important in Tmq.To see this we recall to have all the Heckeoperators.HoweverUq that the absolutevalues ofthe eigenvalues (q, of Uq on newforms evel are known cf. [Li]). They satisfy (q, f 2 (q) in Of (the Nq with if is on Fl (N, q), ringof ntegers enerated the Fourier oefficients and c(q, f)I q1/2 if is on rF(Nq) but not on r1(N, q). Also when is a newform level dividing the roots of x2 c(q, f have xf q) absolutevalue q1/2where (q, f) is theeigenvalue Tq and Xf q) of q). Since for on r1(Nqq2), Uqf is a form n Fr(Nq) we see that Uq(Uq2

(q))

JJUq

fESl

c(qf))

fES2

(u

c(q, f)Uq

q(q))

in TH(Nq, q2) 0 where is the set of newforms Ii (Nq) which re not on IF (N, q) and S2 is the set of newformsf eveldividingN. In particular Uq is in mq mustbe zeroin Tmq slightly ifferentituation rises fm s maximal dealof TH(N, q) (q p) which s notassociatedto anymaximal deal of evel (in the senseof having he same associatedPm). thiscase wemayuse themap (r4, 7r) to give (2.17)

JH(N, q)

JH(N, q)

JH(Nfq)

JH(N,q)

JH(N,q).

ANDREW WILES

498

Then

is givenby the matrix

Uq] LUq qj

on JH(N, q)2, where Uq* Uq(q)1- and TT2 (q) on the m-divisiblegroup. The

second of theseformulaes standardas mentioned bove; cf.for xample[Li, Th. 3], sincePm s not associatedto any maximal deal of evelN. For the first consider ny newform of evel divisibleby and observe hatthe Petersson innerproduct (UUq l)f(rz), f(mz)) is zero forany r, (Nq/ levelf) by [Li, Th. 3]. This showsthat Uq*Uqf(rz), prioria linearcombination f on the space of forms rH(N,q) f(miz), is equal to f(rz). So Uq*Uq whichare new at q, i.e. the space spanned by forms f(sz)} where runs evelf. In particularUq* reserves he m-divisible throughnewforms ith group nd satisfies hesame relation n it, again because Pm s not associated to anymaximal deal of evelN. Remark2.9. Assume that Pm s of type (A) at in the terminology Chapter 1, ?1 (whichensuresthat Pmdoes not occur at level N). In this case Tm TH(N, q)m s already generated the standardHeckeoperators withthe omission Uq. To see this,consider he GL2 (Tm) representation Gal(Q/Q) associatedto them-adicTate moduleofJH(N,q) (cf.thediscussion following orollary2 of Theorem2.1). Then this representation already defined ver he ZP-subalgebra t' ofTmgenerated ythetracesofFrobenius elements, .e. by the Te for Nqp. In particular q) Tt'. Furthermore, Tt' is local and complete, nd as TT2 (q), it is enoughto solve X2 (q) in the residuefieldofTt'. But we can even do thisin ko (the minimalfield of definition Pm) by letting be the eigenvalue Frobq on the unique unramifiedank-one ree uotient ko and invoking he lrq 7r(aq) theorem ofLanglands cf. [Cal]). (It is to ensurethat the unramifieduotient s free ofrankone thatwe assumePm o be oftype A).) We assumenowthat Pm s oftype A) at q. DefineSi thistimebysetting TH(N,

Si whereU1 is givenbythe matrix q)[Ui]/Ui(Ui(2.18)

Uq)

End

(JH(N, q)2)

U,

on JH(N, q)2. The map (3 is notnecessarily urjective nd to remedy hiswe q) N, q) whereT(q) N, q) is the subring TH (N, q) introduceM(q) generated ythestandardHeckeoperators utomitting q. We also writeM(q)


499

forthe correspondingmaximal deal of )(Nq, q2). Then on M(q)-divisible groups, 3 and (3 7r* re surjective nd we get naturalrestrictionmap of localizationsTH(Nq, q2)(m(q)) Sl(m(q)). (Note that the image of Uq under thismap s U1 and notUq.) The ideal ml (m,U1) is maximal S1 and so also in Si,(m(q))and we let mqdenotethe inverse mageofml under hisrestriction map. The inversemageofmq TH(Nq, q2) is also maximal deal whichwe again write mq. Since the completions TH(Nq, q2)mq nd Si,m, TH(N, q)m are both Gorenstein ings by Corollary2 of Theorem2.1) we can define principal ideal (Aq) of TH(N, q)m by (a

(Aq) where a: TH(Nq, q2)mq_*Sl,ml

TH(N, q)m is the restriction map induced by the restriction ap on M(q)-localizationsescribed bove. Suppose that is a maximal deal of TH(N, q) associatedto an irreducible of type A). Then PROPOSITION 2.10.

(Aq) = (q

1)2 (q

1).

Proof.The method s a straightforward daptationof hatused for ropositions2.4 and 2.6. We let S2 TH(N, q)[U2]/U2(U2 Uq) be the ringof endomorphisms JH(N,q)2 whereU2 is givenby the matrix [Uq

This satisfies the compatibility 43U2

and observe hat S2,m2 TH(N, q)m. Then we have maps 1ro0

Tam2JH(N, q)2)

/30-r

qg]

Uq 43. We define m2

TanmqJH(Nq, q2))

(m,U2) in S2

Tam,(JH Nt q)

vi

V2

Tam JH(N, q))

Tam(JH(N, q)).

The maps vi and v2 are given by v2: -* (-qz, aqz) and vi: -* (z, 0) where Uq aq in TH(N, q)m. One checks then that v1 o0(c3o7r*)o(7r*oc3)ov2 is equal to -(q 1) (q2 1) or -2(q 1)(q2 1).

The surjectivity that

o7r* n the completions equivalent o the statemen

JHNj q) 2[P]ml is surjective.We can replacethiscondition a similar ne withM(q)substitutedformq nd forml, i.e., the surjectivity JH Nq, 2) p]mq

JH(Nq, 2)[P]m(q)

JH(N, q)2 [P]m(q)

ANDREW WILES

500

By our hypothesis hat Pm oftype A) at it is even sufficient showthat the cokernel JH(Nq, q2)[p] Fp- JH(N,q)2[p] has no subquotient a Galois-modulewhich irreducible,wo-dimensionalnd ramified q. This statement,r rather tsdual,follows rom emma2.5. The injectivity 7r*63 on the completions nd the factthat it has torsion-freeokernel lso follows from emma 2.5 and our hypothesis hat Pm e oftype A) at q. The case thatcorresponds type B) is similar.We assume the analysisoftype B) (and also oftype C) below) that decomposes HI as described thebeginning Section1. We assumethat m s a maximal deal of TH(Nqr) where contains he Sylowp-subgroup of Z/qrZ)* and that

Pm

(2.19)

(Xq

for suitablechoiceofbasis withXq $& and condXq qr. Here we assumealso that Pm irreducible.We use thesequence JH(Nq')

JH(Nqr

(7r/)

06

I*

r~~l ~2O07r'

JH,(Nqrq r+l)

J J(

Np and

xJHN

defined nalogously (2.17) where was as defined Lemma2.5 and where H' is defined s follows.Usingthe notation rH as at the beginning Section1 set Hl H1 for $& and Hq Sp Hq. Then defineH'= rl Hl and let ir' XHi(Nqr, qr+l) -+ XH(Nqr, qr+l) be the natural map z z. Using Lemma2.5 we check hat 62 is injective n theM(q)-divisibleroup.Againwe set S1 TH(Nqr) [Ul]/Ul (U1 Uq) End(JH(Nqr)2) where U1 is given by

the matrix in (2.18). We define ml (m, U1) and let mq be the inverse image of ml in T.H(Nqr, qr+l). The natural map (in which Uq -+ Ui) a: TH,(Nqr, qr+ )m

iSiml

TH(Nqr)m

is surjective ythe following emark. Remark .11. Whenwe assumethatPm s oftype B) then he Uq operator is redundant Tm TH(Nqr)m. To see this,first ssume thatTm reduced and consider he GL2(Tm) representation Gal(Q/Q) associated to the madic Tate module. Pick a aq Iq, the inertiagroup Dq in Gal(Q/Q), such thatXq aq) $& Then because theeigenvalues Uq are distinctmod we can diagonalize herepresentationithrespect aq. IfFrob is a Frobenius Dq, then the GL2 Tm) representationhe imageof Frob normalizes and we can recoverUq as theentry thematrix iving he value of Frob on the unit eigenvector or q. This is by the lrq iq) theorem fLanglandsas before (cf. [Cal]) applied to each of the representationsbtained frommaps Tm (9f,A. Since the representation defined verthe Zp-algebraTtr generated the traces, hesame reasoning ppliedto TM showsthat Uq Tm.


501

If Tm is not reducedthe above argument hows only that there s an operatorvq Ttr such that (Uq vq) is nilpotent. Now TH(Nqr) can be viewedas a ringofendomorphisms S2(FH(Nqr)), the space of cusp forms of weight on FH(Nqr). There s a restriction ap TH(Nqr) -? TH(Nqr)new whereTH(Nqr)new s the imageof TH(Nqr) in the ring f endomorphisms the old part being defined s the sum of two S2(FH(Nqr))/S2(FH(Nqr))old, and z -? qz. One sees that on copies of S2(Fr(Nqr-l)) mapped via z -? m-completionsTm

(TH(Nqr)new)m since the conductor of Pm divisible by

qr. It follows hat Uq Tm satisfies n equation of the form (Uq) 0 where P(x) is a polynomialwithcoefficients W(km) and with distinct oots. By extending calars to (9 (the integers f a local field ontainingW(km))we can (9. assumethat the roots ie in -i Tm W(km)

Since (Uq vq) is nilpotent follows hat P(vq)r 0 for ome r. Then 0. Now consider hemap -? HT(p) since vq Ttrwhich s reduced, (vq) where he product s takenoverthe localizations at the minimalprimes ofT. The map is injective incethe associatedprimes the kernel re all maximal,whencethe kernel s of finite ardinality nd hence zero. Now in

because the roots ai and Vq aj for roots aj, aj of P(x) are distinct. Since Uq Vq for ach it follows hat ai aj for ach whence Uq Vq in each T(p). Hence Uq Vq in also and this finallyshows each T(p), Uq

that Uq Ttr in general.

We can thereforeefine principal deal (a using, previously,hattherings H' (Nqr, qr+l)M and TH(Nqr)m areGorenstein. We compute Aq) in a similarmanner o the type A) case, but using on the space of forms FH(Nqr) which re new at thistimethat Uq* q, i.e., the space spanned by forms f(sz)} wheref runs throughnewforms withqr level To see this et be any newformf eveldivisibleby qr and for observethat the Petersson nnerproduct (UUq q)f (rz), f mz)) any (Nqr/ evelf by [Li, Th. 3(ii)]. This showsthat (Uq*Uq q) f rz), priori linearcombination {f(miz)}, is zero. We obtain the following result. (Lq)

2.12. Suppose that is a maximal deal of TH(Nqr) associatedto an irreducible oftype B) at q, i.e., satisfying2.19) including thehypothesishat contains p. (Again q Np.) Then PROPOSITION

(LAq)

((q

1)2 ).

Finallywe have the case wherePm s oftype C) at q. We assume then that s a maximal deal ofTH(Nqr) where contains heSylowp-subgroup

502

ANDREW WILES

Sp of Z/q'Z)* and that (2.20)

H1(Qqi WA)

whereWA s defined s in (1.6) but withPm eplacing O, .e., WA ado Pm. This time we let mq be the inverse mage of in THI (Nqr) under the natural restrictionmap THI (Nqr) TH(Nqr) with H' defined s in the case oftypeB. We set (Aq)

(a(o&)

wherea: TH' (Nq')m TH(Nq') is the inducedmap on the completions, which before re Gorenstein ings. The proof f the following roposition is analogous (but simpler) o the proof f Proposition .10. (Notice that the proposition oes notrequire hecondition hatPm atisfy 2.20) but this s the case in whichwe will use it.) 2.13. Supposethatm s maximal deal ofTH(Nqr) associatedtoan irreducible with containing heSylow -subgroupf Z/qrZ)*. Then PROPOSITION

(Aq)

(q- 1).

Finally, thissectionwe state Proposition .4 in the case q ? p as this willbe used in Chapter3. Let be prime, Np and let Si denotethering (2.21)

TH(N)[Ul] /{U2

TqUi

(q)q}

End(JH(N)2)

where ): JH(N, q) -- JH(N)2 is the map defined after (2.10) and Ui is the

matrix

Thus, 'Uq

Tq

q)

U1. Also (q) is defined s (nq) wherenq

q(N), nq

l(q).

Let ml be a maximal ideal of Si containing the image of m, where m is a

maximal deal of TH(N) withassociated rreducible m.We will also assume that pm(Frobq)has distinct igenvalues. (We will only need this case and it simplifieshe exposition.) Let mqdenotethe correspondingmaximal deals of TH(N, q) and TH(Nq) underthe naturalrestrictionmaps TH(Nq) TH(N, q) -- SI. The corresponding aps on completions re (2.22)

TH(Nq)m,

TH(N, q)m, Siml

TH(N)m

W(km)

W(k )

wherek+ is theextension fkm enerated the eigenvalues {pm(Frob )}. Thus k+ is either qual to km r its quadraticextension.The maps A, are surjective, he latterbecause Tq is a trace the 2-dimensionalepresentation


503

to GL2(TH(N)m) given afterTheorem2.1 and hence is 'redundant'by the Cebotarevdensity heorem.The completionsre Gorenstein y Corollary to Theorem2.1 and so we define nvariantdeals ofSi,m1 (2.23)

(A) ,(). (aU

&),

(A')

Let aq be the mageofU1 n TH(N)m

(aU f6) (C

09 W(k+) under he ast isomorphism

W(km)

in (2.22). The proof f Proposition .4 yields Suppose thatPm s irreducible here is maximal ideal ofTH(N) and thatpm(iFrob has distinct igenvalues.Then PROPOSITION

2.4'.

(A) (A/)

(agq-(a2 )X (c(2 (q))(q

1).

Remark. Note thatifwe supposealso that q ideal and is an isomorphism (2.22).

1(p) then i\) is the unit

3. The main conjectures As we suggested Chapter 1, in order studythe deformation-theory of po in detail we need to assume that it is modular. That thisshouldalways be so fordetpo odd was conjectured Serre. Serrealso made a conjecture (the 'e'-conjecture)makingprecisewhereone could find lifting po once one assumedit to be modular cf. [Se]). This has now been provedby the combinedefforts f a numberof authors ncludingRibet, Mazur, Carayol, Edixhovenand others. The most difficulttep was to show that if po was unramifiedt prime thenone could find liftingn which did not divide the level. This was proved in slightlyess generality) yRibet. For a precise statement nd completereferences e refer o Diamond's paper [Dia] which removed he last restrictionseferredo in Ribet's survey rticle Ri3]. The followings minor daptationoftheepsilon onjecture our situationwhich can be found [Dia, Th. 6.4]. (We wish to use weight only.) Let N(po) be theprime p partof the conductor po as defined or xample [Se]. THEOREM 2.14.

Suppose thatpo is modular nd satisfies 1.1) (so in particular irreducible) nd is of typeVD (.,2,O,M) with. Se, str or fl. Suppose thatat least one of thefollowing onditions olds i) p 3 or (ii) po is not induced rom character Q(V'/=). Then there xistsa newform of weight and a prime of Of such thatpf,> is of typeV' (,EO',M) for some O', and such that (pf,Amod A) po over Fp. Moreoverwe can assume that has characterXf oforderprimeto and has levelN(po)p6(PO)

ANDREW WILES

504

whereb(Po) if POIDp is associated to finite lat groupscheme over Zp otherwise.Furthermoren theSelmercase and detpo aw, nd 6(po) we can assume that p(f) X2(FRobp) modA in the notationof 1.2) where ap(f) is the eigenvalue f Up. For the rest ofthis chapterwe will assume that po is modularand that if 3 then po is not inducedfrom character fQ(j/=S). Here and in the restof the paper we use the term induced' to signifyhat the representation is inducedafter n extension f scalars to the algebraic losure. For each D {., E, (, M} we will now define Hecke ring TD except where is unrestricted. uppose first hat we are in the flat, elmeror strict cases. Recall that when referringo the flat case we assume that po is not W. Suppose that ordinary nd that detpolI {qj} and that N(po) qi with si 0. If UA, k2 is the representationpace of po we set nq by dimk(U>,)Iq where q is the inertiagroup q. DefineMO and (2.24)

MO= N(po)

where r(po)

11

qj*

nqi=l qj FMU{p}

nq =2

qi, Mzp(PO)

1 if po is ordinary and r(po)

otherwise. Let

be the

subgroup Z/MZ)* generated theSylowp-subgroup Z/qiZ)* for ach oftype A). Let T/(M) as well as byall of Z/qiZ)* for ach qi qi denote heringgenerated the standardHeckeoperators T1 for Mp, (a) for a, Mp) 1}. Let m'denote he maximal deal ofT'(M) associatedto the and given the theorem nd let kmi e the residuefieldT'(M)/m'. Note thatm'does notdependon theparticular hoiceofpair f, A) in theorem .14. Then km_ kowhereko s thesmallest ossiblefield fdefinitionor because kmi generated ythe traces. Henceforth e will dentify withkm'.There is one exceptional ase wherepo is ordinary nd POIDp isomorphic o a sum oftwodistinct nramifiedharacters X1 and X2 in the notation Chapter 1, ?1). If po is notexceptionalwe define (2.25(a))

YD

(M)m'

W(ko)

(9.

If po is exceptionalwe let T4(M) denotethe ringgenerated the operators {T1 for Mp, (a) for a, Mp) 1, Up}. We choose m" to be a maximal ideal ofTV(M) lying bove m'forwhich here an embedding mno (over ko kmi)satisfying X2(Frobp). (Note that X2 is specified y D.) Then in theexceptional ase km"e eitherkoor itsquadraticextension nd we define (2.25(b)) The o

soW(k)

Tv=

T(M)mi

The omission fthe HeckeoperatorsUq for

..

(9.

MO ensures hat TD is reduced.

505


We need to relateTD to Heckeringwithno missing perators order to applythe results f Section1. 2.15. In thenonexceptionalase there a maximal deal T' (M) with m' and ko km,)nd such thatthenatural

PROPOSITION

for TH(M)

map T' (M)m,

TH(M)m is an isomorphism, thus giving TD2

TH(M)m

(9O

W(ko)

In the exceptional case the same statements hold withm" replacing m', TV (M) replacing '(M) and kintreplacing o.

Proof. Forsimplicity describe henonexceptionalase indicating here appropriate he slightmodifications eededin the exceptional ase. To construct wetaketheeigenform obtainedfrom henewform ofTheorem .14 byremovinghe Eulerfactors t all primes {M p}. If po is ordinary and

has level prime to

we also remove the Euler factor (1

.p p-s)

where

3p s the non-unit igenvalue (Of,,. (By 'removing uler factors'we mean take the eigenform hose L-series that of with these Euler factors emoved.) Then fo an eigenform weight on FH(M) (this ensuredbythe choiceof f) with Of,, coefficients.We have a corresponding omomorphism lrf0: TH(M) -* O9f,And we let m 7r-i(A). Since theHeckeoperatorswe have used to generateT'(M) are prime the levelthere s an inclusionwithfinitendex T' (M)c+fl

O9g

where runsoverrepresentatives the Galois conjugacy lasses ofnewforms associatedto FH(M) and wherewe notethatbymultiplicityne O., can also be described s the ring ntegers enerated the eigenvalues the operators in T'(M) actingon g. If we considerTH(M) in place of T'(M) we get similarmap but we have to replacethering by the ring Sg

where p, qj,... Xqr2i

qr

09g[Xql ..., Xqr, XP]/{Yi, zp}tj1

are the distinct rimesdividingMp. Here (Xqj

qi g)) (Xqi

XqriiXqi -aqi(g))

qi 9())

ifqj

level(g)

ifqi

where the Euler factor of at qj (i.e., of its associated L-series) is (1-Oqi (g)qTS) (1- !q (g)q S) in the first ase and (1- aqj (g)qi-S) in the second case, and qijll M/ level(g)). (We allow aqi(g) to be zero here.) Similarly is

ANDREW WILES

506

defined p2-

Zp

ap(g)XXppXg(P) if

M,

level(g)

ifp{M Xp-ap(g) ifplevel(g), p- ap(g) wherethe Euler factorof at is (1 ap(g)p-8 Xg(p)pl-2.) in the first twocases and (1 ap(g)p-8) in the third ase. We thenhave a commutative diagram T',(M)

Og

C0

(2.27) Sg

TH(M)

09g[Xql ,***Xqr, Xp]/{Yi, Zp}1

where the lower map is given on {Uqi, Up or Tp} by Uq% Xqj, Up or M or M). To verify he existenceof such Tp Xp (according homomorphismne considers he actionof TH(M) on the space offorms weight invariant nderrH(M) and uses that I>r=1 j(mjz) is freegenerator as TH(M) 0 C-module where gj} runsoverthe set of newformsnd mj M/level(gj). Now we tensor ll the rings (2.27) with Zp. Then completing he top rowof 2.27) withrespect o m' and the bottomrowwithrespect we get a commutative iagram

U))m

T' (M)mi

{

(2.28)

Sg)

TH(M)m

H(Sg)m.

Here IL runs through he primes bove in each 0g forwhichm' TH'(M) X)9. Now (Sg)m givenby

(2.29) (Sg Zp)m

((Og

\ilp

Zp) [Xq I... *Xqr

Jm

Xp]/{YiX Zp}i

IL under )m

whereA9,j denotesthe productof the factors f the complete emi-local ing Qgqs[Xq1,. ***Xqr,Xp]/{Yi, Zp};r1 in whichXqj is topologically ilpotent or


507

M and in whichXp is a unit fwe are n theordinaryase (i.e.,when and Up s a unit t m n theordinary M). This becauseUqj case. Now if m' then (Ag-,,)m claim hatYi is given to a unitby M. Similarly given to Xqi bifor omebi 0g,,,with 0 if unitbyXp ap(g) where p(g) is theunit oot fx2 ap(g)x PXg(P) in level and and byXp ap(g) if level or M. This 0g,4 if will how hat Ag,,)m (g,ji whenm' -+ nd Ag,,i)m otherwise. or M, itamounts andfor , the laim sstraightforward. For to thefollowing.et Ugly enote he 2-dimensionalg,'vector pacewith Galois ction ia pgq and et nqig, u) dim(Ug,,,)'qi. e wish o check hat Yj unit.Xqj in Ag,,i)mndfrom hedefinition Yi in (2.26)thisreduces to checkinghatri nqig, u)bythe rq ir(oq) theoremcf. Cal]). Weuse here hat xq2 9), /qj g) andaqi g) arep-adic nitswhen hey renonzeroince they reeigenvalues Frob i. Nowbydefinitionhepower qi dividing is the ame s thatdividing (po)qji (cf. 2.21)). By an observation Livne (cf. Liv], Ca2,?1]), qj

(2.30)

ordqilevel

ordqiN(po)qini q,)) fqi

As bydefinitionir I(M/ evel wededuce hatri nqig, as required. We havenow hown hat achAg, 09g,4 whenm'-* iL)and itfollows from2.28) and 2.29)thatwehavehomomorphisms T1 (M)mt TH(M)m 9gj where he nclusionsreoffinitendex.Moreover e have eenthatUqi in TH(M)m for M. We have M. We nowconsiderhe primes are redundantn the sensethat to show hattheoperators q for they ie in T' (M)m', .e., in the ZP-subalgebra of TH(M)m generated by the

{Ti: Mp, (a): (Z/MZ)*}. For as explained Remark2.9 and for

of type (A), Uq T' (M)m' of type (B), Uq T/(M)m' as

of ype C) butnotof ype A), Uq explainedn Remark .11. Forq whence lso by the rq ir(aq) theoremcf. Cal]). For n thiscase nq

then for ach pair (g,1L)withm' IL. If po is strict r Selmer nq(g, ) epresentation(describedfter Upcanberecoveredromhe wo-dimensional

the orollaries Theorem .1) as the igenvalue Frobp n the free, rank one)unramifieduotientcf.Theorem .1.4 of Wil]). As thisrepresentation is definedver he Zp-subalgebraenerated thetraces,t followshatUp is contained this ubring. theexceptionalase Up s in TV4(M)mtt definition. Finallywe have show hatTp alsoredundant thesense xplained abovewhen M. proof f hishasalready eengiven Section (Ribet's

ANDREW WILES

508

lemma). Here we give an alternative rgument singthe Galois representations. We know hat Tp m and it willbe enoughto showthat Tp (m2, ). Writing m or he residuefieldTH(M)m/m we reduceto the followingitua-

tion. If Tp

(mi2,

then there is

TH(M)m/(m

quotient

P)+km[E]

TH(M)m/a

wherekm[E]s the ringof dual numbers so E2 0) withthe property hat Tp As with $& and such that the mageof T' (M)m, lies in km.Let G/Q denotethefour-dimensionalm-vectorpace associated to the representation Pe: Gal(Q/Q)

GL2 (km[6])

inducedfrom herepresentation Theorem2.1. It has theform G/Q ?GO/Q

ED GO/Q

correspondingpace associated po by hypothesis hat GO is the traces ie in km.The semisimplicity G/Q here s obtainedfrom he main theoremof [BLR]. Now G/QP extendsto a finite latgroupscheme GIZP. Explicitly is quotient thegroup chemeJH(M)m[p]/Zp. inceextensions to Zp are unique cf. [Rayl]) we know

G/ZP -G/Z

Go/Z

Now by the Eichler-Shimuraelationwe knowthat in JH(M)/F Tp

(p)F

Since Tp t follows hat + (p)FT on Go/F and hencethesame holds on G/FP. But Tp is an endomorphism G/ZP which zero on the special on G/ZP. follows hat Tp fibre, by [Rayl, Cor. 3.3.6], Tp in km[E] whichcontradicts ur earlierhypothesis.So Tp (m2,p) as required. This completes heproof f the proposition. Oi FRom he proof theproposition is also clearthat the unique maximal ideal ofTH(M) extendingm' and satisfyinghe conditions hat Uq for {M p} and Up po is ordinary. or therestofthis chapter we will alwaysmakethischoiceof (givenpo). Next we defineTD in the case when (ord, ,Q,M). If n is any ordinarymaximal deal (i.e. Up n) of TH(Np) with prime thenHida has constructed 2-dimensional oetherian ocal Heckering Too which is

TH(Np')n:=

lim e TH(Npr)nr

ZpJ[T-algebra satisfyingTO/T

TH(Np)n.

inversemageof under he naturalrestriction ap. Also

Here nr is the

lim(1 Np) -1

MODULAR

and e

limU!.

(2.25(a)), whereD'

ELLIPTIC

CURVES

For an irreducible (Se , (IM), YD

of type

we have definedTpD in

by

TH(MOP)m

W(km)

(9,

the isomorphismomingfrom roposition .15. We willdefine (2.31)

YD

509


eTH(MOPc?)m

by

(9)

W(km)

In particularwe see that (2.32)

YDIT',

i.e., whereD' is the same as but with Selmer'replacingord'. Moreover is height ne prime deal of TD containing (1 T)P' (1 NP)Pn(k-2)) for ny integers 2, then TD/q is associatedto an eigenform 0, naturalway generalizinghe case 2). For moredetails about these rings s well as about A-adicmodularforms ee for xample [Wil] or [Hil]. let Tn For each n TH(Mopl),n. Then by the argument iven after he statement fTheorem .1 wecan construct Galois representation unramifiedutsideMp withvalues GL2(Tn) satisfyingracep (Frob 1) T1, 1. These representationsan be patched detp, (Frobl) 1(1) for 1, Mp) together givea continuou epresentation (2.33)

limp,: Gal(Q/Q)

GL2(TD)

where is the set of primesdividingMp. To see this we need to checkthe commutativity themaps Rs->

Tn

Tn-1

where hehorizontalmaps are inducedby Pnand Pn-1 and the verticalmap is thenaturalone. Nowthecommutativitys valid on elements RE, which re tracesordeterminantsn the universal epresentation,incetrace Frob1) -4 T1 underboth horizontalmaps and similarly ordeterminants.Here RE is the universal eformationingdescribed Chapter withrespectto po viewed withresiduefield km. sufficeshento showthat RE is generated topologically)bytracesand thisreduces o checking hat there re no nonconstant deformations po to E]with races ying n (cf. Mal, ?1.8]). Forthen Rtr denotes he closedW(k)-subalgebra RE generated y thetraceswe see that Rtr --(RE/m2) is surjective, being the maximal deal of RE, fromwhich we easilyconcludethat Rtr RE. To see that the conditionholds, assume

510

ANDREW WILES

that a basis is chosen that po(c) (1 _0)for chosencomplex onjugation and poG(o) with and c, 54 for ome a. (This is possible because po is irreducible.)Then any deformationp]to k[e] an be represented by representation such that p(c) and p(or)have the same properties. follows asilythat if the tracesof lie in then takes values in whence it is equal to po. (Alternativelyne sees thatthe universal epresentationan be defined verRtrby diagonalizing omplex onjugation before.Sincethe two maps Rt-+ Tn_- inducedby the triangle re the same, so the associated representationsre equivalent, nd the universalproperty henimplies the commutativity the triangle.) The representations2.33) werefirst xhibited y Hida and were he original inspiration orMazur's deformationheory. For each D {., 3,0, M} where is not unrestrictedhere s then canonicalsurjectivemap Wp RD

YTD

which nduces herepresentationsescribed fter he corollaries o Theorem .1 W(ko) (or W(km,,) the and in (2.33). It is enoughto check his when exceptional ase). Then one ust has to check hat for verypair (g,pa)which appears in (2.28) theresulting epresentations oftypeD. For thenwe claim that the mageofthe canonicalmap RpD TO II 0g9,, TD wherehere denotes he normalization.In thecase where s ord thisneeds to be checked (9 foreach n.) For this we just need to see that RD is instead forT, W(ko)

generated y traces. (In the exceptional ase we have to show also that Up is in the image. This holds because it can be identified,singTheorem2.1.4 of RD whereu is the eigenvalue Frobp on the [Wil], withthe imageof uniquerankone unramifieduotient RD with igenvalue X2 Frobp) which is specified the definition D.) But we saw above that thiswas true for RD is surjective ecause themap RE. The same thenholdsforRD as RE on reducedcotangent paces is surjective cf. (1.5)). To checkthe condition we haveimposedthefollowing on thepairs (g, u) observefirst hat for conditions n the level and character f suchg's byour choiceof and H: oftype A): qI levelg, detpg,|I

1,

oftype B): condXqlI evelg,detpg,| of type (C): det pg,tI

Xq,

is the Teichmiiller ifting f det po I.

In the first wo cases the desiredform PgqDi then follows rom he 'K(0q) theorem Langlands (cf. [Cal]). The thirdcase is already of 1rq


511

type C). For one can use Theorem2.1.4 of [Wil] in the ordinary ase, theflat ase beingwell-known. The followingonjecture eneralizes fundamentalonjecture f Mazur and Tilouine for (ord,A, W(ko), 0); cf. [MT]. CONJECTURE 2.16.

FpD

isomorphism.

Equivalently hisconjecture ays that the representationescribed fter thecorollaries Theorem .1 (or (2.33) in theordinary ase) is theuniversal one for suitable hoice H, and m. We remind he reader hatthroughout this section we are assumingthat if 3 then po is not induced from character Q(+-/=). and po is repRemark.The case of most nterest o us is when resentationwithvalues in GL2(F3). In this case it is a theorem f Tunnell, extending esults Langlands,that po is alwaysmodular. For GL2(F3) is double coverofS4 and can be embedded GL2(Z[VA/=])whence n GL2(C); cf. [Se] and [Tu]. The conjecturewillbe provedwith mildrestriction po at the end ofChapter3. Remark.Our originalrestrictiono the types (A), (B), (C) forpo was motivated the wishthat the deformationype a) be of minimal onductor among ts twists, b) retainproperty a) underunramifiedase changes. Without hiskindof tabilityt can happenthatafter base change to an extension nramified A, po has smaller conductor'for ome character A. The typical xampleofthis s wherePo = IndQ (x) withq -I(p) and is a ramified haracter ver K, the unramified uadratic extension Qq. What makesthisdifficultorus is thatthere re thennontrivial amifiedocal deformationsIndQPXf for a ramified haracter orderp ofK) whichwe cannotdetectby changeof evel. For the purposesofChapter3 it is convenient digressnow n order introduce slightvariantof the deformationingswe have been considering so far. Suppose that *, (9, M) is a standarddeformation roblem (associated to po) with Se, str or fland suppose that H, MO, and m are defined s in (2.24) and Proposition .15. We choose a finite et ofprimes {q1, ... qr} with qi Mp. Furthermorewe assume that each qi 1(p) and that the eigenvalues ci, /i} of po(Frobqi) are distinctforeach qi Q. This lastcondition nsures hatpo does notoccur theresidual epresentation

of the A-adic representation associated to any newformon FH(M, q..qr)

where ny qi divides he evelofthe form.This can be seen directly y ooking at (Frobqi) in such a representation by usingProposition .4' at the end of Section2. It willbe convenient o assume that theresiduefield 9 contains si, /3iforeach qi.

ANDREW WILES

512

Pick ai for ach i. We let DQ be the deformation roblem ssociatedto representations of Gal(QsuQ/Q) which re oftypeD and which n addition satisfy hepropertyhat at each qj

,(

(2.34)

Xlqi

with X2,qiunramifiednd X2,qiFrobqj) -i mod m for suitable choiceof basis. One checks in Chapter that associatedto DQ there s a universal deformationingRQ. (These new conditions re reallyvariants n type B).) We willonlyneed corresponding eckering n very pecialcase and it is convenientn thiscase to define t usin ll theHeckeoperators.Let us now set N(po)p6(Po)where6(po) is as defined Theorem2.14. Let modenote a maximal ideal of TH(N) given by Theorem2.14 with the property hat pmo PO over Fp relative to a suitable embedding of kM0

k over ko. (In the

exceptional ase we also imposethesamecondition n mo bout the reduction of Up as in thedefinition TD in theexceptional ase before 2.25)(b).) Thus pmo pf,A,mod A over the residuefieldof OfA, for ome choice of and A with of levelN. By dropping ne of the Euler factors t each qj as in the proof f Proposition .15, we obtaina form nd hence a maximal deal mQof TH(Nql ... qr) with the property that pmQ PO over Fp relative to suitable overkM0.The fieldkmQ the extension f ko (or kmin embeddingkmQ theexceptional ase) generated the caj,fi. We set (2.35)

TQ

TH(Nql

...

qr).Q

0

W(k-Q)

(9.

It is easy to see directly or by the arguments Proposition2.15) that TQ is reduced nd that there s an inclusionwithfinitendex

(2.36)

TQ >TQ

9g,1 where heproduct takenoverrepresentatives theGalois conjugacy lasses of eigenforms of level Nqj ... q. with mQ -4 pi. Now define DQ using the choices ai for which Uq, -+ ai under the chosen embedding kQ -- k. Then

each of the 2-dimensionalepresentationsssociated to each factor gt is of using eitherthe Wq 7r(q) type DQ. We can check this foreach theorem cf. [Cal]) as in the case of type (B) or using the Eichler-Shimura relation does notdividethe evelofthe newformssociatedto g. So we get homomorphism 9-algebrasRQ -+ TQ and hence also an 0-algebra map (2.37)

(oQ: RQ- TQ

as RQ is generatedby traces. This is not an isomorphism generalas we in place of M. However t is surjectiveby the arguments have used Proposition .15. Indeed,for N(po)p, we checkthat Uq is in the imageof

MODULAR

ELLIPTIC

CURVES

AND FERMAT'S

LAST THEOREM

513

(pQ using the arguments the second half of the proofof Proposition2.15. For we use the fact that Uq is the image of the value of X2,q Frobq) in the universal epresentation;f. 2.34). For M, but not ofthe previous types,Tq is a trace PTQ and we can applythe Cebotarevdensity heorem to show that it is in the imageof ,oQ. Finally, ifthere is a section a: TQ

ker7r nd let pp denotethe 2-dimensional epresentation GL2 0) obtainedfrom TQ mod pQ. Let V Adpp K/c where is the field f fractions 0. We pick basis -+

0, then set pQ

forPPsatisfying2.34) and then et

{(

(2.38)

CAdpOK/O P()

{(

)} ):a~b~cdE0}?K/0 )

' '

and let V(qi) V/V(qi). Then as in Proposition .2 we have an isomorphism (2.39) wherePRQ 2.40) HDQ (2140)

Hom0(pRQ/p1Q2K/O)

HEQ (QEuQ/QIV)

ker(ir FQ) and thesecondterm defined V) ker: 1/(Q/ ~~~~~~~~~~~~~~q

uQ/Q1 V)

i=1

(Qunr, (qi)

We returnnow to our discussion f Conjecture2.16. We will call a deformationheoryD minimal f {p} and is Selmer, trictor flat. This notionwillbe critical n Chapter3. (A slightlytronger otionofminimality s described n Chapter3 where he Selmer ondition replaced,when possible,by thecondition hat the representationsrisefrom inite latgroup schemes-see the remark fter he proof fTheorem3.1.) Unfortunatelyven up to twist,not everypo has an associatedminimalD even whenpo is flator as explained the remarks fterConjecture .16. However his ordinary could be achieved one replacedQ by suitable finite xtensiondepending on po. Supposenowthat is (normalized)newform, s prime Of above and PfA deformation po oftype where (., Z, (Of,A,M) with Se, stror fl. (Strictly peakingwe may be changing as we wishto choose its field fdefinitiono be k Of,A/A.) uppose furtherhat level(f) where M is defined y (2.24). Now let us set OfA for he restofthissection.There a homomorphism (2.41)

rD,f: T-D 40

514

ANDREW WILES

whosekernel s the prime deal PT,f associatedto f and A. Similarl here s homomorphism Rv C0 whose kernel s the prime deal PR,f ssociatedto f and and whichfactors through rf. Pick perfect airings 0-modules,the secondone TD-bilinear, (2.42)

-0,

(, ): Tv

Tv

O.

In each case we use thetermperfect airing o signifyhatthepairsof nduced --) Homo are isomorphisms. In maps 0) and Tv -* Homo (Tv, addition hesecondone s required o be Tv-linear. The existence hesecond pairing equivalent o the Gorenstein roperty, orollary ofTheorem2.1, as we explain below. Explicitly h is a generator the freeTV-module Homo(TD, 0) we set (tl, t2) h(tit2). prioriTH(M)m (occurringn thedescription Tv inProposition .15) is is Zp-algebra it follows GorensteinW(km)-algebra. The notionofGorenstein -algebra is explained in the appendix.) Indeedthe map Homzp TH(M)m, Zp) trace is easily seen to be an isomorphism, the reduction givenby modp is injective nd theranks reequal. Thus Tv is Gorenstein -algebra. -- Tv be the adjointof withrespectto these pairings. Now let *: Then define principal deal (71) Tv by Homw(k.) (TH(M)m, W(km))

(10

OqDj)

(* (1)).

This is well-definedndependentlyf the pairings nd moreover ne sees that TvD/71 torsion-freesee the appendix). From ts description71) s invariant under xtensions to 0' in an obviousway. Since Tv is reduced r(r1) 0. One can also verifyhat (2.43) up to a unit n 0. if we obtain Di by relaxingcertainof the We will say that Di hypotheses D, i.e., if19= (., E, 0, M) and Di (, 0iOi, M1) we allow E, any O1, M1 (but ofthe same type) and if is Se or str that El in it can be Se, str,ord or unrestricted Di, if is fl Di it can be fl or unrestricted 1i. We use the termrestricted signify hat is Se, str, flor ord. The followingheorem educesconjecture .16 to 'class number' criterion. For an interpretationf the right-hand ide of the inequality the theorem s the orderof cohomology roup, ee Proposition1.2. For an interpretationf the left-handide in terms f the value ofan innerproduct, see Proposition .4.


515

THEOREM 2.17.

type

(E,A,

Assume, as above,thatpf,A s a deformation po of Se, str or fl. Supposethat O9f,), M) with #0/l(mDf)

#PR,f/IRf

Then (i) pD1,:RD1 TM1 is an isomorphismor all (restricted)

1.

(ii) TD1 is a completentersectionover 01 if. is Se, str or fl) for all reD. stricted Proof.Let us write Then we alwayshave

forTD, PT forPTf, PR forPRf and 71for D,f.

#//p #1'T/1T (2.44) (Here and in what followswe sometimeswrite for r(r) ifthe contextmakes thisreasonable.) This is proved s follows.T/rq cts faithfully PT. Hence the Fitting deal of PT as T/rq-modules zero. The same is then true of /p2 as an X)7 (T/71)/pT-module. o the Fitting deal ofPT/p2 as an 0-moduleis contained (r1) nd theconclusion theneasy. So togetherwith the hypothesis the theoremwe get inequalities and henceequalities) #PR/PR >- #PT/PT

c/(1 By Proposition oftheappendixT is complete ntersectionver 9. Part (ii) ofthe theorem henfollows orD. Part (i) follows or from roposition of the appendix. We nowprove nductivelyhat we can deducethe same inequality #0e/7r(7)

(2.45) #PRi,f/PRI1,f 01/1"1,f forD, 9D and R1 RD1. The above argumentwill thenprovethe theorem forV1. We explainthisfirstn the case D, 1Dqwhere Dq iffersrom only in replacing by U q}. Let us writeTq for Dq, PR,q forPRJfwith RDq and rq for 7Pqfj

We recall that Uq

in Tq.

We choose somorphisms Homo Tq, (9) Homo T, (9), (2.46) Tq Gorenstein -algebra. If comingfrom he fact that each of the rings aqo aq: Tq -- T is thenaturalmap we mayconsider he element where he adjoint s withrespect the above isomorphisms. hen it is clear that (2.47)

(q(q))

(JAq)

as principal deals of T. In particular r(rq) Ir(?Aq) in (. Now t follows rom roposition .7 thattheprincipaldeal (Aq) is givenby (2.48)

(Aq)

((q

1)2(T2

(q)(1

q)2))

516

ANDREW WILES

In the statement of Proposition 2.7 we used Zp-pairings HomZ, (T, Zp),

Tq

Homzp (Tq, Zp)

to define (LAq) (aqo &q). However, using the description of the pairings as W(km)-algebras derived from these Zp-pairings in the paragraph following (2.42) we see that the ideal (Aq) is unchanged when we use W(km)-algebra pairings, and hence also when we extend scalars to ( as in (2.42). On the other hand #PR~q/PR7q

#PR/R

O/(q-1)2

(q)(1

(T,

q2)}

byPropositions .2 and 1.7. Combining hiswith 2.47) and (2.48) gives 2.45).

If we use a similar argument to pass from to Dq where this time Dq signifiesthat D is unchanged except for dropping q from M. In each of types (A), (B), and (C) one checks fromPropositions 1.2 and 1.8 that #PR~ /p2

#PR/p2 #H?(Qqi

V*).

This is in agreementwith Propositions2.10, 2.12 and 2.13 which give the

correspondingchange in r1by the method described above. To change from an (9-algebra to an (91-algebra is straightforward the complete intersectionpropertycan be checked using [Kul, Cor. 2.8 on p. 209]),

and to changefrom

to ord we use (1.4) and (2.32). The change from tr

to ord reduces to this since by Proposition 1.1 strict deformationsand Selmer deformations re the same. Note that forthe ord case ifR is local Noetherian ring and R is not unit and not zero divisor, then R is a complete Th. This the intersectionif and only if R/f is (cf. -1 proof of the theorem. with Remark 2.18. If we suppose in the Selmer case that has level we can also consider the ring TH(MO)mo (with Mo as in (2.24) and mo defined n the same way as forTH(M)). This time set To

TH(MO)no

(9,

W(km.0)

TH(M)m

(9.

W(krn)

Define r10, j,Po and with respect to these rings,and let (Ap) oo &p where To and the adjoint is taken with respect to (-pairings on and To. OP: We then have by Proposition 2.4 (2.49)

(rjp)

(n Ap)

(T-(p)(l 22

?p)2))

=

(a2

(p))

as principal ideals of T, where ap is the unit root of X2 -Tpx + p(p)

0.

Remark. For some earlier work on how deformation rings change with E see [Bo].


517

Chapter In this chapterwe prove the main resultsabout Conjecture2.16. We beginby showing hattheboundfor heSelmergroup o which was reduced in Theorem 2.17 can be checked one knowsthat the minimalHecke ring is complete ntersection.Combining his with the main resultof [TW] we complete he proof Conjecture2.16 under hypothesis hat ensures hat a minimalHeckering xists.

Estimates for the Selmer group Let po: Gal(Q/Q) -- GL2(k) be an odd irreducible epresentation hich we willassume s modular.Let be deformationheory type -, A, 9,M) such that po is type D, where is Selmer, trictor flat. We remind he reader that k is assumedto be the residuefieldof (. Then as explained Theorem2.14, we can pick modular ifting f,Aof po of type (altering if necessary nd replacing by ringcontaining f,A,) provided hat po 3. For the rest of this is not induced from character f (V'Z3) if chapter,we will make the assumption hat po is not of thisexceptional ype. Theorem2.14 also specifies certainminimumeveland character or and in particular nsures hat we can pick f to have levelprimeto when PO DP is associated to a finite latgroup chemeoverZp and detpoIP W. In Chapter 2, Section3, we defined ringTo associated to D. Here we make slightmodification f this ring. In the case where is Selmerand we set group cheme nd detpo POIDP (3.1)

TDo

T'H (MO)' W/0

with Mo as in (2.24), defined ollowing2.24) (it is actually subgroup of (Z/Mo Z)*) and m' the maximal deal ofT' (Mo) associated to po. The same proof in Proposition .15 ensures hat there s a maximal deal moof TH(MO) withmo T1 (Mo) m' and such that th naturalmap (3.2)

TDo

T'(Mo)ma

HW(ko)

?(9 )X

TH(MO)

W(ko)

(9

is an isomorphism. he maximal deal mowhichwe choose characterized y theproperties hat pnlo po and Uq mofor {p}. (The value of is determined the otheroperators; ee the proof Tp or of Uq for Proposition .15.) We now defineTDO in generalby the following:

518

ANDREW WILES

TDO is givenby (3.1)

if. is Se and POIDP associated to a finite latgroup cheme verZp and detpolip

W;

(3.3) TDO

TD

if is stror fl, POIDP not associated to finite latgroup chemeoverZp, or detPOIIP W.

We choose pair (f,A) of minimum evel and character rem2.14 and thisgives homomorphismf 9-algebras irf:

T0o -> (9

given by Theo-

of"Ad

We set PT,f ker and similarly let PRf denote he nversemageofPTJf in RD. We define principal deal (rTif) ofTDo by taking n adjoint#'f rf withrespect pairings in (2.42) and write n1T,f

Notethat

l)).

is finite nd 7rf IT, 0 because Tpo is reduced. We also write Trf or rfirTf) ifthecontextmakesthisusage reasonable. We let Vf Ad pp 09K/c) wherepp s the extension scalarsof pf,A (. T f/

THEOREM

PO

3.1. Assumethat is minimal, .e., a,

absolutelyrreducible henrestrictedoQ (i) #HD1(QE/QVf)

#(PT~f/PTf)

(

U {p}, and that ).

Then

#OT

where #(9/U?2 (p)) o0 whenPo is Selmerand POID, is associatedto finite lat group chemeoverZp and detPOIIJ w, and cp otherwise; (ii) ifTD0 is completentersectionver then i) is an equality, To and TD is completentersection. In general, or any (not necessarilyminimal) of Selmer,strictorflat type, nd any pf,A f type #HD,(Qz/Q,Vf) oo ifpo is as above. Remarks.The finiteness as provedby Flach in [Fl] under omerestrictions on f, and by a different ethod. particular, did not consider the strictcase. The bound we obtain in (i) is in fact the actual orderof H,(QE/Q, Vf/)s follows rom he main resultof [TW] whichproves he hypothesisofpart (ii). Then applyingTheorem2.17 we obtain the order f this groupformoregeneralD's associatedto Po under hecondition hat minimal existsassociatedto po. This is stated Theorem3.3.


519

The case where he projective epresentationssociatedto po is dihedral does notalwayshavethepropertyhat twist f t has an associatedminimal D. In thecase where he associatedquadraticfield imaginarywe willgive differentrgument Chapter4. he hat D is minimal,ndicating ssume Proof.We only t theend theslight hangesneededfor hefinal ssertion fthetheorem. Let be finiteetofprimes isjoint rom satisfying 1(p) and po Frobq) Q. For the minimaldeformation having distinct igenvaluesforeach (., A, 9,M), letVQ be thedeformationroblem escribed efore problem (2.34); i.e., it is the refinement (., Q, (9,M) obtainedby imposing he additionalrestriction2.34) at each q Q. (We willassumefor he proof hat (9 is chosen (9/A k contains heeigenvalues fpo(Frob ) for ach q E Q.) We set RD TDO and recallthe definition TQ and RQ fromChapter 2, ?3 (cf. (2.35)). We write forVf and recallthedefinition V(q)following2.38). Also remember that mQ s a maximal deal of TH(Nql ... q,) as in (2.35) forwhichpmQ po k as in over Fp (recallthat this uses the same choiceofembeddingkmQ the definitionf TQ). We use mQ also to denotethe maximal deal of TQ if the contextmakesthisreasonable. and Consider iagram -0

H~D(QE/QV)

HD'Q(QEUQ/QV)

-Q

JJ(Qunrv(q))Ga (Q

nr/Qq)

qEQ

It

T PT/P )

(PTQ/PTQ)

QS. KQ -O

whereKQ is bydefinitionhecokernel thehorizontal equence nd denotes The keyresult s: Homo( , K/O) forK the field f fractions LEMMA 3.2.

satisfying

The map LQ is injective or any finite et of primes

q_ 1(p),T

(q) (1 +q)2modm forall eQ.

Proof.Note thatthe hypotheses f the emmaensure hatpo(Frob ) has distinct igenvalues or ach q Q. First, onsider he ideal aQ ofRQ defined

520

ANDREW WILES

by (3.4)

aQ={ai-1,bicidi-1:

with i Iqqi

b(i)

Then the universal roperty RQ showsthat RQ/aQ to identifyPR/P2)* (PR/P2)

{f

(PRQ/ipQ)*

f(aQ)

R. This permits 0}.

Ifwe prove hesamerelation or he Heckerings, .e.,with and TQ replacing R and RQ thenwe will have the injectivity LQ. We will writeaQ forthe image of aQ in TQ under the map CpQ of (2.37).

It willbe enough check hatfor ny q Q', Q' subsetofQ, TQ//-aq TQ,_{q} where aq is defined in (3.4) but with Q replaced by q. Let N' (po) p'(PO) * rlqiEQ'-{q}qi where6(po) is as defined Theorem2.14. Then take an element Iq Gal(Qq/Qq) whichrestricts o a generator of Gal(Q((Nfq)/Q((N/)). Then det(a) (tq) TQ/ in the representation GL2(TQ/) defined fterTheorem2.1. (Thus tq 1(N') and tq is primitive root modq.) It is easily checked hat (3.5)

JH(N' q)mQ (Q) JH(N'q)mQ,

(Q) [(tq) 1]

Here is stilla subgroup Z/MoZ)*. (We use herethat po is notreducible for he njectivitynd also that po is not nducedfrom character Q(Vz/3) for hesurjectivity hen 3. The latter s to avoidtheramificationoints the covering H(N'q) -4 XH(N', q) oforder which an giveriseto invariant divisors XH(N'q) which re not the imagesof divisors XH(N', q).) Now by Corollary to Theorem2.1 the Pontrjagin uals of the modules in (3.5) are free f ranktwo. It follows hat (3.6)

(TH(N'q)mQ,)2/((tq)

1) -_ TH(N' ,q)mQ,)2.

The hypotheses the emma mply he condition hat po Frobq) has distinct eigenvalues.So applyingProposition .4' (at the end of ?2) and the remark following (or usingthe factremarked Chapter 2, ?3 that this condition implies hat po does not occuras theresidualrepresentationssociatedto any formwhichhas thespecial representation q) we see thatafter ensoringver W(kmQ/) ith theright-handide of 3.6) can be replacedby TQ,{q} thus giving TQ, aq -T Q'-fq} '

since (tq) -1 dq. Repeated inductively his gives the desired relation F1 TQ/aQ T, and completes he proof the lemma.

MODULAR ELLIPTIC

CURVES


521

Supposenowthat is a finiteet ofprimes hosen s in the emma. Recall that from hetheory congruencesProp. 2.4' at the end of?2) 71TQf/1Tf

17 -1),

qEQ

the factors a2- (q)) being unitsby our hypotheses Q. (We onlyneed that the right-handide divides he eftwhich s somewhat asier.) Also,from the theory Fitting deals (see theproof 2.44)) #(PT/PT)

#(O/1TTf) #(0O7/TQf).

#(PTQ/PTQ)

We deducethat rJ/ (q -1)

#KQ where

Sincetherangeof LQ has ordergivenby

#(PT/P2)/#(O/Tf).

as LQ is injective. we compute hat theindexofthe imageof LQ is from emma3.2, consider he kernel AM Keepingourassumption applied to the diagramat the beginning f the proofof the theorem. Then with M chosen arge enoughso that AM annihilates T/P4 (which s finite because is reduced)we get:

J7J

)~ D,(QE/QV[,XM]) HIQ(QSuQ/QSV[AM])

qEQ

4Q (PTQ/PTQ/p*[AM]

(PT /p2)

V(q)[AM])Ga1(Qu"/Qq)

1(Qunr,

LQ

(PT-/p

KQ[AM]

See (1.7) for he ustificationhat AM can be taken nsidethe parentheses the first wo terms. Let XQ 'bQ((PTQ/pTQ)* [AM]). Thenwe can estimate the orderof 5Q(XQ)usingthe factthat the imageof LQ has index at mostt. We get (3.7)

#6Q(XQ)

Now we choose (3.8)

EQ

(II

qEQ

#O/(AM

1))

(t)

(1/#(PTPT))

to be a set ofprimeswiththepropertyhat HV1*QE/Q VAM)

H1 (Qq7 AqM)

qEQ

ANDREW WILES

522

is injective.We also keep the condition hat LQ is injectiveby onlyallowing to containprimes the form iven the lemma. In addition,we require

these 's tosatisfy =_1(pM).

To see that this can be done, suppose that x 5$ . We H'(QE/Q, V;M) [A]

HI(Q/QV*)

II H'(Qq,

qEQ

kereQ and Ax

but

V;M) [A]

fJH'(QqV;)

qEQ

theright-handsomorphismsoming rom ur particular hoicesofq's and the left-handsomorphismrom ur hypothesis n po. The same diagramwill hold ifwe replaceQ by Qo Q qo} and we nowneedto showthatwe can choose qO so that 6Qo(X) 5#

The restrictionmap

H' (Q~/Q,V

Hom(Gal(Q/Ko((p)),V)Ga1(KO(7P)/Q)

has kernelH'(Ko((p)/Q, k(1)) byProposition .11 wherehereKo is the split3 and x tingfield po. Now ifx E H'(Ko((p)/Q, k(1)) and x 5$ then factors hrough n abelian extensionL of Q((3) ofexponent which s nonabelian overQ. In this exceptional ase, must ramify some prime of Q((3), and if lies overtherationalprime 54 thenthe compositemap

HI

((3) /Q 1))

H'(Qunr,k(l))

H'(Qunr, (q/Am)(1))

is nonzero n x. But thenx is not oftypeD* whichgives contradiction. his only eavesthepossibilityhat Q((3, 3V1) but again thismeans that is notoftype D* as locallyat theprime bove 3, is notgenerated y the cube root of unitoverQ3((3). This argument oldswhether r not is minimal. So x, whichwe view in kereQ, gives nontrivialGalois-equivariant omomorphism., Hom(Gal(Q/Ko((p)), V*) whichfactors hrough abelian extensionM. of Ko((p) of exponentp. Specificallywe choose M, to be the minimal uch extension. Assume first hat the projectiverepresentationj3 associatedto po is not dihedral that Sym2 is absolutely rreducible. ick Gal(Mx (pM)/Q) satisfying (3.9)

(i) (ii) (iii)

po(a) has order

with m,p)

1,

fixesQ(detpo) (4pM), fx(orm)7+ Ok

To show that this is possible,observefirst hat the first wo conditions an be achievedby Lemma 1.10(i) and the subsequentremark. Let al be an el-


523

ementsatisfyingi) and (ii) and let a1 denote its image in Gal(Ko((p)/Q). Then (&1) acts on Gal(Mx/Ko((p)) and under this action G decomG1 EDGi where acts trivially G1 and withoutfixedpoints poses as on G'. If is any irreducibleGalois stable k-subspace f (G) ?Fp k then 0 since Sym2 is assumed absolutely rreducible.So also ker(al 1) 0. ker(a 1)If(G) 5$ and thus we can find G1 such that f,(r) Viewing as an element f we thentake At

Gal(Mx((pM)/Ko((p))

Gal(Ko((pM)/Ko((p))

(This decomposition olds because Mx is minimal nd because Sym2 o and 4p are distinct rom he trivial epresentation.) ow rT commuteswith and eitherfx (ri al)m) or fx(af') . Since po(Tiai) po(ai) this gives (3.9) with at least one of

r1al or

al. We may then choose q0 so that

a and wewillthenhaveeQ0 x) 540. Note that conditionsi) and (ii) Frob =_ (p) imply po a) has distinct igenvalues, husgiving both the hypotheses Lemma 3.2. If on the otherhand po is dihedral henwe pick a's satisfying 3oa)

(i)

1,

fixes Q(4PM),

(ii)

(iii) fx am) 54 withm the order po a) (and since is dihedral).The firstwoconditionscan be achievedusingLemma 1.12 and, in addition,we can assumethat a takes the eigenvalue on any given rreducibleGalois stable subspace of Wx k. Arguing above, we find G1 such that fx(r) 0 and we proceedas before.Again, conditions i) and (ii) imply hypotheses Lemma 3.2. So by successively djoining 's we can assumethat is chosen so that eQ is injective. We have thus shown that we can choose

{q1,...

qI}

to be a finite

set ofprimes j i-=(pM) satisfyinghe hypotheses Lemma3.2 as wellas the injectivity FQ (3.8). By Proposition .6,the njectivity FQ mplies hat (3.10)

#HD(QEuQ/Q, VfAM])

ho.

IJ

qEEUQ

q.

Here we are using heconventionxplained fter roposition .6 to defin HD1. Now, as was chosen to be minimal,hq for EI -{p} by Proposition 1.8. Also, hq #(O/AM)2 for Q. If is str or flthen hoohp by Proposition .9 (iv) and (v). If Se, hochp cp by Proposition1.9 (iii). (To computethis we can assume that Ip acts on WAvia w, as otherwisewe

ANDREW WILES

524

get hwhp 1. Then withthishypothesis,Won)* s easilyverified be unramifiedwithFrobp acting as U2(p)-l by the description Pf,A in [Wil, Th. 2.1.4].) On the otherhand,we have constructed lasses which re ramified at primes n Q in (3.7). These are of typeDQ. We also have classesin Hom(Gal(QsuQ/Q), 0/AM)

H1(QruQ/Q, 0/AM)

H1(Q uQ/Q, VAm)

coming from he cyclotomic extension Q(Cql ... (qr). These are of type D and

disjointfrom he classes obtained from 3.7). Combining hese with (3.10)

gives

#Hv(QE/Ql Vf Am])

t

P/p2

*

as required.This provespart (i) ofTheorem3.1. Now ifwe assumethat is a complete ntersection have that by Proposition of the appendix. In the strictor flatcases (and indeed in all cases wherecp 1) this implies hat RD -_ by Proposition of the appendixtogetherwithProposition .2. In the Selmer ase we get

(3.11)

PT #(wr/PT)

O/1,TJ)cp #(01/7T,,f) PT,/T') wherethe centralequality by Remark2.18 and the right-handnequality is from he theory Fitting deals. Now applyingpart (i) we see that the inequality n (3.11) is an equality. By Proposition of the appendix,TD is also a complete ntersection. The final ssertion the theorem proved exactlythe same wayon noting hat we onlyused the minimalityo ensurethat the hq were 1. In and easily computed. The only general, hey re bounded ndependent pointto note that ifpf,As ofmultiplicativeypeat thenpfAIDq does not split.) Remark.The ringTDO defined (3.1) and used in this chapter hould be the deformationing ssociatedto the following eformationroblemDo. One alters onlyby replacing heSelmer ondition the condition hat the deformationse flat n the sense of Chapter 1, i.e., that each deformation of po to GL2(A) has the property hat forany quotientA/a of finite rder, PIDPmoda is the Galois representationssociated to the Qp-points f a finite flatgroup chemeover Zp. (Of course,po is ordinary ere contrast our usual assumption orflatdeformations.) FromTheorem3.1 we deduce our main results bout representations using the main resultof [TW], whichprovesthe hypothesis Theorem3.1 (ii), and thenapplyingTheorem2.17. Moreprecisely,he main result [TW] as explained showsthat is complete ntersectionnd hence that above. The hypothesis f Theorem 2.17 is then given by Theorem 3.1(i), togetherwith the equality (and the centralequalityof (3.11) in the

MODULAR ELLIPTIC

CURVES


525

Selmer case) and Proposition 1.2. Strictlyspeaking, Theorem of [TW] refers to a slightly maller class of D's than those covered by Theorem 3.1 but up to twist every such is covered. It is straightforward o see that it is enough to check Theorem 3.3 forpo up to a suitable twist. THEOREM

3.3.

when estricted

Assume thatpo is modularand absolutely rreducible p)

Assume lso that is oftypeA), (B)

or (C) at each in S. Then themap WE): RE TE of Conjecture .16 is an isomorphismorall associatedto po, i.e., where (.,E,Y,M) with Se, str,fl or ord. In particular Se, str or fl and is any newform for which f,A deformation po of type then #H1(QE/Q, Vf)

#(01/iD,f) 00

where 1D,fs the nvariant efined Chapter priorto (2.43). The condition at in ensures that there is a minimal associated to po. The computation of the Selmer group followsfromTheorem 2.17 and Proposition 1.2. Theorem 0.2 of the introduction follows fromTheorem 3.3, after t is checked that a twistof po as in Theorem 0.2 satisfies the hypotheses of Theorem 3.3. Chapter 4 In this chapter we give a differentand slightlymore general) derivation of the bound for the Selmer group in the CM case. In the firstsection we estimate the Selmer group using the main theorem of [Ru 4] which is based on Kolyvagin's method. In the second section we use a calculation of Hida to relate the rj-invariant special values of an L-function. Some of these computations are valid in the non-CM case also. They are needed if one wishes to give the order of the Selmer group in terms of the special value of an L-function. 1. The ordinary CM case this section we estimate the order of the Selmer group in the ordinary CM case. In Section we use the proof of the main conjecture by Rubin to bound the Selmer group in terms of an L-function. The methods are standard (cf. [de Sh]) and some special cases have been described elsewhere (cf. [Guo]). In Section we use calculation of Hida to relate this to the rj-invariant. We assume that

(4.1)

IndQ i,:

Gal(Q/Q)

-*

GL2(0)

526

ANDREW WILES

is the p-adicrepresentationssociatedto a character i: Gal(L/L) Ox of an imaginary uadraticfield We assumethat is unramifiedn and that factors hrough n extension f whoseGalois grouphas the form ZpET where is a finite roupoforderprime p. The ring is assumedto be the ringof ntegers f a local fieldwithmaximal deal and we also assume that p is a Selmerdeformation po mod A which supposed rreducible ith detpolII, w. In particular follows hatp splits L, po say,and that precisely ne of a, a* is ramified t (K* beingthe character for ny representinghe nontrivial oset Gal(Q/Q)/ Gal(Q/L)). We can supposewithout oss ofgeneralityhat is ramified p. We consider herepresentation oduleV (K/O)4 (where is the field of fractions 0) and the representations via Ad p. In this case splitsas ED KIO) (0) EDKIO where is the quadraticcharacter f Gal(Q/Q) associated to L. We let denote a finite et of primes ncluding ll those which ramify p (and in particular ). Our aim is to computeHs e(Q/Q, V). The decomposition gives correspondingecomposition H1(Qr/Q, V) and we can use it to defineHs e(Q/Q, Y). Since W0 Y (see Chapter1 for hedefinition Wo)

we can define HSle(QF/Q, Y) by HSe(Q/Q,

Y)

ker{H1 (Q/Q,

H1 (Qunr Y/Wo)}.

Y)

Let Y* be the arithmetic ual of Y, i.e., Hom(Y,ppoo) Qp/Zp. Write for Ke/K* and let L(v) be the splitting ieldof v. Then we claim that Gal(L(v)/L) Zp T' withT' finite roupof orderprimeto p. For this it is enoughto show that KK*/e actors hrough groupof orderprime 1. to since 82X-1' Suppose that has order moprwith mop) Then xmoextends o a character f which s thenunramified sincethe same is trueofX. Also it factors hrough abelianextension f withGalois group somorphic Z2 since factors hrough uchan extensionwithGalois T1 with oforderprime p (thecomposite fthe group somorphic splitting ields f and a*). It follows hat xm0s also unramifiedutsidep, whence t is trivial.This proves he claim. Over there s an isomorphism Galois modules *- (KIO) (v) ED KIO) (v-162). In analogyto the above we defineHSe(QF/Ql Y*) by HSe(QF/Q, Y*)

ker{H1 Qr/Q, Y*)

-?

H1 Qunr,Wo)*)}.

Analogousdefinitionspply ifY* is replacedby Yv*nL.lso we say informally if t vanishes n H1 Qunr, WO)*) (resp. that cohomology lass is Selmer


527

Hl(Qunr, (WOn)*)). Let Moo be the maximalabelian p-extension f L(v) un-

ramified utsidep. The following roposition eneralizes CS, Prop. 5.9]. PROPOSITION Hunr(QF/Q,

4.1. Y*)

There s an isomorphism Hom Gal(Moo/L(v)), (K/O)(V))Ga1(L(v)/L)

whereHlnr denotes hesubgroup fclasses which re Selmerat p and unramified verywherelse. equenceas Proof. The sequence s obtainedfrom he nflation-restriction follows. irstwe can replaceH1(Qr/Q, Y*) by {H1 (QF/L, (K/()(v))

H1 (Q/L,

(K/9)(v-162))}

where A Gal(L/Q). The unramifiedondition hen translates nto the requirementhatthe cohomology lass should ie in p* (Qr/L, (K/O)(v162))} Since interchangeshe two groups nsidethe parenthesest is enoughto compute hefirst fthem, .e.,

{Hunr E-p(Qr2/L, (K/c)(v))

Hunr

jnr n _p (Qr/L,

(4.2)

K/O(v)).

The inflation-restriction equenceappliedto thisgivesan exact sequence Hlnr in U-p (L(v)/L, (K/O)(v))

(4.3)

Hunr in Up (Qr/L, (K/O)(v)) -?

Hom (Gal(Mco/L(v))I

(K/Q)(V))Gal(L(v)/L)

The first erm s zero as one easilychecksusing the divisibility f (K/O)(v). Nextnotethat H2 (L(v)/L, (K/Q)(v)) is trivial. 1(A) this s straightforwardcf.Lemma 2.2 of [Rul]). If =1(A) thenGal (L(v)/L) Zp and so it is trivial thiscase also. It follows hat anyclass in thefinal erm f 4.3) lifts o class c in H1 (QE/L, (K/Q)(v)). Let Lo be thesplitting ield fY*. Then MooLo/Lo unramifiedutside and Lo/L has degreeprimeto p. It LI follows hat c is unramifiedutsidep. Now writeHltr(QF/Q, 1n*) where n* thesubgroup Hunr(QF/Q,Yn*) ivenby Hstr(QE/Q YEn)

Hlnr(QF/Q, Yn)

Y\' and similarly orY,) for op

in H1(Qp, Yn*/(Yn)O)}

where Yn*)0 the firsttep n thefiltrationnderDp, thusequal to (Yn/yn?)* or equivalently to (Y*)%n where (Y*)O is the divisible submodule of Y* on which he actionof Ip is via 9. (If p $& one can characterize Yn)0 as the

ANDREW WILES

528

maximal ubmodule n which acts via E2.) A similar efinitionpplieswith Y, replacing ?n*. follows rom examination f the action of on Y),that (4.4)

Hstr(QE/Ql Yn)

Hunr(QE/Q, Yn).

In the cas of Y* we will use the inequality

H1tr(QF/Q,Y*)

(4.5)

Hunr(QF/Qi Y*).

arge the map We also need the factthat for sufficiently (4.6)

Hstr(Q/Qi Yn*) Hstr(Q/Qi Y*) is infective.One can checkthis by replacing hesegroupsby the subgroups of H1(L, (K/O)(V)An) and H1(L, (K/O)(v)) whichare unramified utside p and trivial t p*, a manner imilar o the beginning f the proof f Proposition4.1. The above map is then njectivewhenever he connecting omomorphism Ho (Lp*, (K/O) (v)) -+ H1 (Lp*, K/Q) (V)An) argen. is infective, hichholdsfor ufficiently Now, by Proposition .6, (Q. Yn) Ho Yn) (Yn)*)#H #Hr(Q/Q, HQ #HO (Q, Yn*) () #Hsltr(QE/Q,Y*) and a simplecalculation howsthat Also,HO Q, Yn)

# HQ. En)

i{

if v-=I1mod

#(C)/l-I/(q))

otherwise

where runsthrough set of primes OL prime o cond(v) ofdensity ne. This can be checked inceY* IndQ v) K/c. So, setting

(4.8)

t

we get (4.9) HSe(Q/Q, Y)

infq#(O/(1-v(q)))

-~~~

()

ifvmodA=1

vmodA #,41

Hom Gal (Moo/L(v)), (KI 0)(v

where HO(Qq, Y*) for :&p, ip lim HO(Qp, (Yn?)*).This follows from roposition .1, (4.4)-(4.7) and the elementarystimate

(4.10)

)) #(HSle(Qr/Ql)/Hulnr(QE/Ql

1I fqi

qEE-{p}

which follows fromthe fact that #H1 (Qunr, y)Ga1(Qunr/Qq)

MODULAR ELLIPTIC

CURVES


529

Our objective s tocomputeH'e(Qr/Q, V) and the mainproblems to estimate e(Q-/Q, Y). By (4.5) this n turn educes o theproblem estimating Hom(Gal(MOO/L(v)), (K/O)(V))Ga1(L(v)/L)).

This order can be computed

using he mainconjecture' stablished Rubinusing deas ofKolyvagin. cf. [Ru2] and especially Ru4]. In the former eference ubin assumes that the class number is prime p.) We could nowderive he resultdirectly rom thisby referring [de Sh, Ch. 3], but we willrecallsomeofthe stepshere. Let Wf enotethe number f roots ofunity of L such that mod (J integral deal of OL). We choose an primeto such that Wf 1. Then there grossencharacter of satisfying((a)) for mod (cf. [de Sh, II.1.4]). According Weil, after ixing embedding Qp we can associate p-adic characterWp (cf. [de Sh, II.1.1 (5)]). We choose an embedding orrespondingo prime bove and thenwe find forsome of finite rder and conductorprimeto p. Indeed Wp nd are both unramified p* and satisfyWpIi, KII where is the cyclotomic character nd I. is an inertia roup p. Without ltering we can evenchoose so thattheorder is prime p. This is byour hypothesishat factored through extension fthe form with of orderprime p. To see thispickan abeliansplitting ield Wp nd whoseGalois grouphas the form G' with pro-p-groupnd G' of orderprimeto p. Then we see that WPIGas conductor ividingfp'. Also the onlyprimeswhichramify Zpextension ie above so our hypothesis n ensures hat KIG as conductor dividingfp'. The same is thentrue of the p-partof whichtherefore as conductor djust so that has orderprime f. to as claimed. We will not however hoose so that is as this would requirefp? to be divisibleby condx. Howeverwe willmake the assumption, by altering if necessary, ut stillkeeping primeto p, that both and Wp have conductor dividing fp

Thus we replace fp ? by l.c.m.{f, cond

The grossencharacter (or moreprecisely NF/L) is associated to (unique) elliptic urve defined ver L(f), therayclass field conductor f,with complexmultiplication y CL and isomorphic ver C to CIOL (cf. [de Sh, II. Lemma 1.4]). We mayeven fix Weierstrassmodelof overOF whichhas good reduction t all primes bove p. Foreach primeX3of above p we have a formal roup ET, and this is relativeLubin-Tategroupwith respectto FT3over Lp (cf. [de Sh, Ch. II, ?1.10]). We let A ASE be the logarithm fthisformal roup. Let UOO the productofthe ocal unitsat the primes of L(fpoo); .e.,

Uo=fJ q3lp

uCo

where

Uoo,q=limUn,T,

530

ANDREW WILES

each Un't beingthe principal ocal units L(fpn)qp. (Note that the primes of L(f) above are totallyramified L(fpoo) o we stillcall them q3}.) We wishto define ertainhomomorphisms on Up. These werefirstntroduced in [CW] in the case where he local fieldFsp s Qp. Assumefor he moment hat Fq3 Qp. thiscase Eq is isomorphic the Lubin-Tategroup ssociatedto 7rx xP where7r ~p(p). Then lettingwn chosen o that [ir]an) wi~n-1t was shown be nontrivial ootsof [7rn]x) in [CW] that to each element = limun UOq there orresponded unique suchthat fu(wn) Un forn 1. The definition power eriesfu(T) e Zj[T of 6ke (k 1) in this case was then k,9(U)=

A'(T)

dT

logfu T)

T=O

(9watisfying It is easy to see that k,q3gives homomorphism: ooUoo,-Ox is the character iving where9: Gal (FIF) 09(0)k k,(6) skq3(e) the actionon E[p']. The construction thepower eries [CW] does not extendto the case or to the case where t is defined ver where he formal rouphas height an extension Qp. morenatural pproachwas developedbyColeman [Co] whichworks general. See also [Iwi].) The correspondingeneralizations 6k weregiven nsomewhat reater enerality [Ru3] nd then n full enerality by de Shalit [de Sh]. We now summarize hese results, hus returningo the general ase whereFq is notassumedto be Qp. To an element limun UOOwe can associate a power eriesfuv,(T) O,3[[T]] whereOC3 thering ntegers Few;ee [de Sh, Ch. II ?4.5]. (More precisely uqp(T) s the q3-componentfthepower eriesdescribed here.)For X3 we will choose the primeabove corresponding our chosenembedding c* Qp. This power eries atisfiesUndo (fu,)(wn) for ll 0, n= 0(d) whered [Fqe: .] and {wn} is chosenas before s an inverse ystem irn division oints Em.Wedefine homomorphism k: Uoo (4.11)

Sk(U)

6k, p(U) =d

-- ()

by

logfu'(T) ET

~~~~T=O

Then

(4.12)

5k UT)

9(Q)kSk(U)

for

Gal(F/F)

where again denotesthe actionon E[p']. Now pp on Gal(F/F). We actuallywant a homomorphism UO,witha transformationroperty orreon Gal(F/F). Let on all of Gal(L/L). Observethat sponding


531

be a set of cosetrepresentativesorGal(L/L)/ Gal (LIF) and define (4.13)

dI2(U)

dES

-1(of)62(uU) Oq3[v].

Each term s independent f the choic of coset representativey (4.8) and it is easily checked hat (2(U')

V(Of)4D2(U).

It takes ntegral aluesin (9q [v]. Let UO v) denotethe productofthe groups of local principalunits at the primes bove ofthe fieldL(v) (by whichwe mean projective imitsof local principalunits as before). Then 12 factors throughUO v) and thusdefines continuous omomorphism (D2:UOO,,(VJ) +CP Let COO e the groupofprojectiveimits elliptic nits L(v) as defined in [Ru4]. Then we have crucialtheorem Rubin (cf. [Ru4], Ru2]), proved using deas ofKolyvagin: There is an equalityof characteristicdeals as THEOREM 4.2. Zp[[Gal(L(v)/L)]] -modules: charA Gal (Moo/L(v))) charA(UO v)/CJ,). Let vo mod A. For any Zp[Gal(L(vo)/L)]-module we writeX(vO) (9 on which he actionofGal(L(vo)/L) is via for hemaximalquotient zp theTeichmiillerift vo. Since Gal(L(v)/L) decomposes nto directproduct of pro-pgroupand a groupoforderprime p, Gal L(v)IL(vo)) Gal L(v)IL) Gal(L(vo)IL), we can also consider ny Zp[[Gal(L(v)/L)]]-module lso as Zp[Gal(L(vo)/L)]module. In particularX(vO) s a moduleover Zp[Gal(L(vo)/L)](v0) 0. Also A(vO) [[T]]. Now according o resultsof Iwasawa ([1w2, ?12], [Ru2, Theorem5.1]), 0-linearly to UOO(v)(vO) freeA(vO)-modulef rank one. We extend Uoo(v) Ozp (9 and it then factors hroughUoo(v)(v?). Suppose that u is a generator UO v) vO) nd ,3an element C( ?). Then y- 1)u fi or ome f T) 0[[T]] and -y topologicalgenerator Gal (L(v)/L(vo)). Computing (2 on both u and ,3 gives (4.14) v(-y) 1) (2(0)/P2(U)Next we let e(a) be the projective imit of ellipticunits in limLpn for a some ideal prime to 6fpdescribed [de Sh, Ch. II, ?4.9]. Then by the power n lim xn We proposition Chapter I, ?2.7 of [de Sh] this s

532

ANDREW WILES

let 31i !3(a)1/12 e the projection e(a)1/12 o UOO nd take ,3 Normol3 where he norm s from fpo to L(v). A generalization f the calculatio [CW] whichmaybe found [de Sh, Ch. II, ?4.10]showsthat (4.15)

(D23)

(root of unity)Q-2 (Na

v(a)) Lf 2,

OT9[v]

where is a basisfor he 9L-module periods ourchosenWeierstrassmodel ofE/F. (Recall thatthiswas chosen havegood reduction t primes bove p. The periodsare thoseofthe standardNerondifferential.) lso hereshould be interpreteds the grossencharacterhose associatedp-adic character, ia the chosen mbedding )-k QP, is v, and vPs the complex onjugateofv. The only restrictions e have placed on are that (i) is prime to p; (ii) wf 1; and (iii) cond pv . Now let fop' be the conductor f v withto prime o p. We show now that we can choose such that Lf(2, P)/Lf02, vP) p-adic unit unless vo 1 in whichcase we can choose it to be t as defined in (4.4). We can clearlychoose to be a unit if vo #4 as -P(q)v(q) Normq2 or ny ideal q prime o fop.Note that if vo 1 thenalso 3. Also if vo

1 then we see that

inf# {O/{JLoq(21P)/Lfo(2P)}} since PF-2 =-

We can compute D2(u) by choosing special local unitand showing hat (D2(U) s p-adicunit,but t s sufficientor s to know hat t is integral.Then sinceGal (Moo/L(v)) has no finite -submoduleby result f Greenberg; ee and that [Gre2, nd of ?4]) we deduce from heorem #Hom(Gal (Moo/1L(v)),I K/c)) (z,))Ga1(L(v)1L) ?l

#O/Q-2Lfo2, I)

(#O/Q-2Lfo (2,v))

ifvo7&

t if vo

Combining hiswith 4.9) gives:

HSe(QE/Ql Y)

O/Q-2Lfo(2, ))

1.

fJ qEE

HO (Qp, (YO)*). where HI(Qq, Y*) (for p), tp Since V (K/O)(f EDK/(9 we need also a formula or ker{H1(QE/Q, (K/O)(0)

K/c) -- Hl(Qunr, K/( )(0) ED /()}

This is easilycomputed o be (4.16)

#(O/hL)-

qEE-{p}

4q

ELLIPTIC CURVES AND

where #H0(Qq, ((K/O)(0) Combining hesegives: PROPOSITION

533

LAST

K/O)*) and hL is theclass number OL.

4.3.

#HSe(QE/QV) ? #(9/Q2L10(2,v)) #(O(/hL) where

OH(Qq,V*) (for #4), tp

qEE

#HO(Qp,(YO)*)-

2. Calculation of rq We needto calculate xplicitlyhe nvariantsD,f introduced Chapter2, ?3in special case. Let po be an irreducibleepresentation in (1.1). Suppose that f is newform weight and levelN, prime Of above and pf,A deformation po. Let be the kernel thehomomorphism N) -* Of /A arisingfrom We write forTl(N)m (9, where 9 = Qf,A and km W(km,)

N. We assume here that is the the residuefieldof m. Assume that residuefield f and thatit is chosento containkm.Then by Corollary of Theorem2.1, T1 N)m is Gorenstein nd it follows hat T is also Gorenstein O-algebra see thediscussion ollowing2.42)). So we can use perfect airings (the secondone T-bilinear) OxO

-,

(,):

TxT

-O

(9 is the natural map, we set to define n invariant of T. If 7r: (r7) (*ir(1))where is the adjointof 7rwithrespectto the pairings. It is well-defined an ideal of T, depending nlyon 7r. Furthermore,s we noted in Chapter2, ?3, 7r(r) (?7, 7)up to a unit and as noted n the appendix

77

Ann

T[[p] where

ker r. We now give an explicit formula for 77

developedbyHida (cf. [Hi2]for survey his earlier esults)by interpreting (, ) in termsof the cup productpairingon the cohomology X1 N), and then termsof the Petersson nnerproductof with tself.The following account whichdoes notrequire heCM hypothesis) adapted from Hi2]and we refer hereformoredetails. Let (4.17)

):H1 (Xi(N), IOf) H1 (Xi(N), I)

Of

be the cup productpairingwith Of as coefficients.We sometimes rop the C fromX1(N)IC or Ji(N)IC if the contextmakesit clear that we are referring the complexmanifolds.) In particular tx,y) (x,t*y) for all x, and for ach standardHecke correspondence We use the action of t on H1 Xi (N), Of) givenby t*x and simplywrite for *x. This is thesame

534

ANDREW WILES

as the action induced by t* T1(N) on H1(Jl(N),Of) H'(Xi(N), Of Let pf be the minimal rime f T1 N) 0 Of associatedto f (i.e., the kernel H-* /ctt(f) where T1(N) Of -- Of givenby t1 ct(f)f), and let Lf Eanqn let fP

H1 X1 N), Of) [pf].

Then fP again newformnd we define LfP by replacing by fP in the definition Lf. (Note herethat Of OfP as theserings re the integers fieldswhich re either otallyreal or CM by a resultof Shimura.Actuallythis s not essential s we could replace Of by any ringofintegers ontainingt.) Then the pairing , inducesanotherby restriction If

Eanqn.

(,): Lf LfP

(4.18)

Of.

ReplacingOf (and theOf-modules)bythe ocalization fOf at p (ifnecessary) we can assume that Lf and Lfp are freeof rank 2 and directsummands s Of-modules f the respective ohomology roups. Let 61, be basis of Lf. Then also 61,62 is basis of Lfp Lf. Here complexconjugation cts on H1(Xi(N), Of) via its actionon Of. We can thenverify hat (6, 6) :=det(bi, Ej) is an element f Of (or its localization t p) whose mage Of,> is givenby defined r(q2) (unit). To see this, onsider modified airing (x, y)

(4.19)

(x,way)

wherewC s defined s in (2.4). Then (tx,y) operators Furthermore det(bi, j)

det(6i, 6j)

(x, ty) for ll x,

and Hecke

cdet(6i,6j)

for ome p-adic unit (in Of). This is because wC(LfP) Lf and wC Lf) Lf (One can check this, forexample, using the explicit bases described below.) Moreover, yTheorem2.1, (X1(N),

Z)

ft1(N)

H1(Xi(N), Of) ?Tl(N)?of

mlf~m

T1N2

T2.

Thus (4.18) can be viewed after ensoringwith Of,, and modifying as in (4.19)) as perfect airing T-modules nd so thisserves o compute r(iq2) as explainedearlier the square comingfrom he factthat we have rank module). To give moreuseful xpression or 6, 6) weobserve hat and fP can be C) via viewed s elements H1 (X1 N), C) HN), z)dz, fP fPdz. Then {If,fP} form basis forLf Oof C. SimilarlyIf, fP} form basis

MODULAR ELLIPTIC

CURVES


(ffP), W2 forLfp ?Of C. Definethe vectorswi M2(C). Then writing W1 C6 and W2 C6 with

(Wa) :=det((fi,f))

535

(f, fP) and write If,f2 fP we set

(6,6) det(CC).

Now (w, c) is given xplicitly terms fthe non-normalized) eterssonnner

product,):

(W, ')

-4(f, f)2

where f, f) fs/r1(N)ffdxdy. To computedet(C) we consider ntegrals verclasses H1 (Xi(N), Of). By Poincare dualitythere exist classes C1,C2 Hl(Xi(N), Of) such that det 60) is unit in Of. Hence det generates he same Of-moduleas is generatedby {det (fc f)} for ll suchchoicesof classes (Cl, C2) and with

{fi, f2} {f, fI}. Letting be generatorf heOf-moduledet (fc fi)} we have the following ormula f Hida: PROPOSITION

4.4.

7r(iR2) (f, f)2/Uf

(unit in Of,,A).

Now we restrict o the case where po IndQ sio forsome imaginary quadraticfield which unramifiedt and some kX-valued haracter of Gal(L/L). We assume that Po is irreducible,.e., that so 5$ ",, where for any representing he nontrivial coset of Ko(aT-lb6) KO'a(b) Gal(L/Q)/ Gal(L/L). In additionwe wish to assume that Po ordinary nd w. In particular splits n L. These conditionsmply hat, detPo is

prime of

above p, soi(a)

a-1 mod on U. afterpossible replacement of svo

by voa Here the U. are the unitsofL. and sincesvo character, he restriction ofsvo o an inertiagroup induces homomorphism U.. We assume now that p is fixed nd so chosento satisfy hiscongruence.Our choice of so will mply hatthegrossencharacterntroduced elowhas conductor rime

top.

We choose (primitive) rossencharacter on togetherwithan embedding c-* Qp corresponding the prime above such thatthe nduced p-adiccharacter pp as the properties: (i) ppmodp

so (p

maximal deal o Qp).

(ii) fpp actors hrough abelian extension somorphic Zp T with of finite rderprime p. (iii) p((a))

for _1 (f) for ome ntegraldeal prime p.

To obtain it is necessary irst o define pp. Let Mo denotethe maximal abelian extension which s unramifiedutsidep. Let 0: Gal(Mx/L) -+ QpX be any characterwhichfactors hrough Zp-extension nd inducesthe

536 homomorphism

l(p)}. Then set ',p

ANDREW WILES

a-1 on U,

Gal(M,,/L) whereUp,1 {u Up: 'coO,and pick a grossencharacte such that ((p)p (np. --*

Note that our choice of here s not necessarilyntended o be the same as the choiceofgrossencharacter Section1. Now let f, be the conductor f and let be the ray class field f conThen over there s an elliptic urve,unique up to isomorphism, ductor with omplexmultiplication y OL and period atticefree, rankone overOL and with ssociatedgrossencharacterp NF/L. The curveE/F is th extension ofscalarsof uniqueelliptic urveE/F+ whereF+ is the real subfield f of index2. (See [Shl, (5.4.3)].) OverF+ thiselliptic urvehas onlythep-power isogenies ftheform pm for Z. To see thisobserve hat is unramified at and po is ordinary that the only sogenies degreep over are ones thatcorrespond o divisionbyker and ker ' wherepp' (p) in L. Over F+ thesetwosubgroups re interchangedycomplex onjugation,whichgives the assertion.We let E/O9F+ (p) denote Weierstrassmodel overOF+,(p),the localizationofOF+ at p, withgood reduction the primes bove p. Let WE be a Neron differential E/OF+<( Let be a basis for he OL-moduleof

periods of WE. Then

forsome p-adic unit in FX.

According o theorem fHecke, is associatedto cusp form Al such way that the L-seriesL(s, cp)and L(s, fg) are equal (cf. [Sh4,Lemma 3]). Moreover ince was assumed primitive, newform.Thus the fgs integer cond IAL/Q NormL/Qcondcp) primeto p and there s homomorphism 'Of: Ti(N)-4?Rf

Of

0cp

satisfying Tl) (p(c) (P(C) if inert cc L, (I N) and o)f (Ti) in (1 N). Also of (l(l)) (p((l))?ol) where is the quadraticcharacter associated to L. Using the embedding in Qp chosen above we get primeA of Of above p, maximal deal m of T1 N) and a homomorphism such that the associated representation f,A educesto Tl(N), f,pomodA. Let po kerf f:T1 (N) -+ Of and let Af Ji N)/poJ1 N) be the abelianvariety ssociatedto f byShimura.OverF+ there s an isogeny Af F+ (E/F+ )d where [Of: Z] (see [Sh4,Th. 1]). To see thisonechecks hatthep-adicGaloisrepresentationsssociatedto the Tate moduleson each side are equivalent to (Ind ,op)Ozp Kf,pwhereKf,p Of Qp and where pr:Gal(F/F) is the p-adic character ssociated to and restricted o F. (One compares traceFrob?) in the two representationsor Np and split completely F+; cf.thediscussion fterTheorem2.1 for he representation Af.)


537

Now picka nonconstanmap N) IF+ -+EIF+ whichfactors hroughAf/F+. Let be the compositeof F+ and the normal closureof Kf viewed n C. Let be a Nerondifferential E/OF+(p). Extending calarsto we can write 7r:

ir*WE

wherewfu

00

n=l

an(fo)qfln

aEHom(Kf ,C)

a,

aawfa

for ach a. By suitably hoosing we can assume

that aid =$ Then there xist Ai OM and ti T1(N) suchthat forsome

Aiti-7r*WE ClWf

Cl

M.

We consider he map (4.20)

7r:Hi (Xi (N)/C, Z)

Hi (E/C, Z)

O9M,(p)

O9M,(p)

givenby 7r' Ai(7r i). Even if7r' s not surjectivewe claim that th mage OM* This is of 7r' lwayshas the formHi(EIC, Z) aOM,(p) for ome because tensoredwithZp 7r' an be viewedas Gal(Q/F+)-equivariant map ofp-adic Tate-modules, nd theonlyp-power sogenies n E/F+ havetheform ?pm for ome Z. It follows hatwe can factor r' (1 0 a) for ome other urjective a: Hi(Xi(N)/C, Z)

Om -+

(E/C, Z) 0) OM,

nowallowing to be in OM,(p) Now define on QE by a-lAiti r* where-*QE/C QJ1(N)/C is the map induced by 7r nd ti has the usual action on QJ1(N)C

Then a*(WE)

(4.21)

CWf orsome

Ja*(wE)=

M and

a(y)

forany class Hi(Xi(N)/C, OM). We note that (on homology s in (4.20)) also comes from map of abelian varieties J1(N)/F+ 'z OM E/F+ ?Z OM althoughwe have not used this to define **. We claim nowthat OM,(p). We can compute *(WE) by considering a**(WE 01)

,Ztlr* 0 a-'Ai on QE0F+

OM to QJ1(N)/F+ Q0 OF+ OM 9F+,(p). Then there re isomorphisms 8f ii(N)1,1-0

OM

and then mapping the image in

QJ1 N)/M

Q3

Hom(OM, J1 (N)101)

Now let us writeO1 for

)/c~lecv2 Hom~c)M, 11(N) -82 N)1~0

)

ANDREW WILES

538

where6 is the differentf M/Q. The first somorphisman be described follows. et e(-y):J1(N) -+ J1(N) Opm or OM be themap 'y. Then ti(w)(-y) e(-y)*w. imilar dentificationsccurforE in place ofJi(N). OM it is enough observe hatby So to check hat a* WE 1) Q1 its construction comesfrom homomorphism (N)/01 Om -k E101 OM. It follows hat we can compare he periodsoff and ofWE. For fP we use the factthat fly Pdz fCY dz wherec is the OM-linear map on homology oming rom omplex onjugation n the curve.We deduce: PROPOSITION 4.5.

Uf

Q2.(1/(p-adic nteger)).

We now give an expression or f,, f,) in termsof the L-function f p. This was first bservedby Shimura Sh2] although he preciseformwe want was givenby Hida. PROPOSITION 4.6.

(1--

(ft2'fib16

27 X) LN(1,4)

where is the character fig nd its restriction L; is the quadratic haracter ssociatedto L; denotes hattheEuler factors orprimesdividing have been LN( removed; such that qq' with cond and q,q' S. is the set ofprimes primesofL, notnecessarily istinct. Proof. One beginswith a formula fPetersson hat for weight on F1(N) says

(f,

(47)-2

whereD(s, f, fP)

(2) n=1

(1)

SL2(Z): Ii(N)

Ian 2n-s

n=1

eigenformf

(+1)] Res8=2D(s, f,fP)

anqn (cf. [Hi3, 5.13)]). One checks

that,removing he Euler factors primesdividingN, DN(S,

f,P)

LN(S,

p2k)LN(S

1, 0k)(QN(s

1)/(QN(2s

2)

at of the form by usingLemma 1 of [Sh3]. For each Euler factor (1- aq q-8) wegetalso an Euler factor D(s, f, fP) oftheform1- qaq q8). When fl this can only happen for split prime q whereq' divides the conductor p but does not,or for ramified rime whichdoes notdivide the conductor Ap. this case we get a term ql-8) since I9 (q)12 q. Putting ogetherhepropositions this ectionwenowhave a formula or 7r(?7)s defined t thebeginning fthis ection.Actually is more onvenient


539

to give a formula or 7r(?JM), n invariant efined the same way but with (9 replacingTl(N)m (9 where pMo with MO Ti(M)ml and M/N is ofthe form

lb

IJ q2.

qESW

qI M0

Here ml is defined therequirementshat Pm1 po, Uq f (q p) over ko takingUp -ap and there s an embedding whichwe fix) km1 whereap is the unit eigenvalue Frobp in pfLA. So if is the eigenform obtainedfromf by 'removing he Euler factors' t (M/N) (q p) and removing he non-unit uler factor t we have AM *(1) where7w: T, (9 corresponds f' and theadjoint s takenwithrespect T1 M)ml (9 W(km1 )

to perfect airings f T1 and (9 withthemselves s 0-modules,the first ne assumedTi-bilinear. Property ii) offp ensures hat is as in (2.24) with (Se, A, 0, q) where is the set ofprimesdividingM. (Note that S. is precisely he set of in the notation Chapter2, ?3.) As in Chapter2, primes forwhichnq ?3 there s a canonicalmap (4.22)

RV

TV

Ti(M)ml

W(km1 )

which s surjectiveby the arguments the proofof Proposition2.15. Here we are considering slightlymoregeneral ituation han that in Chapter 2, ?3 as we are allowingpo to be inducedfrom character Q(i/=3). In this special case we define to be T, (M)m, 0( 0. The existence the map W(km1,)

in (4.22) is proved in Chapter2, ?3. For thesurjectivity,otethat for ach (with p) Uq is zero in TV as Uq m1 for ach such so that we can apply Remark 2.8. To see that Up is in the image of Rv we use that it is the eigenvalue Frobp on the unique unramifieduotientwhich s free f rankone in therepresentation described fter he corollaries o Theorem2.1 (cf. Theorem2.1.4 of [Wil]). To verify his one checksthat TV is reduced or alternatively ne can apply the methodof Remark2.11. We deduce that T, the W(kmi)-subalgebra f T1(M)mi generated y thetraces, nd it follows henthatit is in theimageof RV. We also need t givea definition (ord,A, 0), and po is inducedfrom character Q(V/=3). TV where For this we use (2.31). Now we take M=Np

fJ

540

The argumentsn the proof f Theorem2.17 showthat ir(,M) is divisibleby ir(r/)(a2 (p))

171

qESFp

1)

where is the uniteigenvalue Frobp in Pf,A\.he factor t is givenby remark .18 and at q it comesfrom heargument Proposition .12 but with H' 1. Combining hiswithPropositions .4, 4.5, and 4,6,we have that

X)

(4.23) lr(r7M)s divisible Q2LN 2,

()

-(p))

171q-

qIN

1).

We deduce: #(O/r(?JM)) #Hs

THEOREM 4.7.

Proof. As explained Chapter2, ?3 it is sufficient prove he nequality #((9/lr(?JM)) #HSe(Qr/Q7V) as theoppositeone is immediate.For this suffices compare 4.23) withProposition .3. Since LN(2, FI) = LN(2, v) = LN(2, 92X)

(note that the right-handerm s realby Proposition .6) it suffices pair up the Euler factors forq I N in (4.23) and in the expression or he upper boundof Hse(Q/Q, V). We now deduce the main theorem n the CM case using the methodof Theorem2.17. Suppose thatpo as in (1.1) is an irreducible epresentationof odd determinantuch thatpo Ind? so for a character o of an imaginary uadratic xtension of which unramified p. Assume also that: THEOREM 4.8.

(i) detpo

(ii) po

'p

=W;

ordinary.

Thenfor every

(-,E,0,0)

such thatpo is of typeVD ith Rv

and TD is

Se or ord,

Tvz

completentersection.

COROLLARY. For any po as in the theoremupposethat p: Gal(Q/Q)

-)

GL2(0)

is a continuous epresentation ithvalues in the ring of integers local field,unramifiedutside finite et ofprimes, atisfying po whenviewed as representations GL2((Fp). Supposefurther hat:

MODULARLLIPTIC URVESAND FERMAT'S ASTTHEOREM (i) (ii) detp

541

is ordinary; Xek-

with offinite rder,

2.

Thenp is associatedto a modular ormof weight

Chapter In this chapterwe provethe main results bout elliptic urves nd especially showhow to remove he hypothesis hat the representationssociated to the 3-division oints houldbe irreducible. Application to elliptic curves The key resultused is the following heorem f Langlandsand Tunnell, extending arlierresultsof Hecke in the case wherethe projective mage is dihedral. THEOREM 5.1 (Langlands-Tunnell). Suppose that p:Gal(Q/Q) GL2(C) is continuous rreducible epresentation hose mage is finite nd solvable. Supposefurther hatdet is odd. Then thereexists a weight ne newform such thatL(s,f) L(s,p) up tofinitelymanyEulerfactors. Langlands actually proved [La] a much moregeneralresultwithout restrictionn the determinantr the number ield which n our case is Q). However n the crucialcase where heimage PGL2(C) is S4, the resultwas onlyobtainedwithan additionalhypothesis.This was subsequently emoved by Tunnell n [Tu]. Suppose thenthat po: Gal(Q/Q)

--

GL2(F3)

is an irreducible epresentationf odd determinant. We now show, using the theorem, hat this representations modular the sense that over F3, po pg,j,mod gL or ome pair (g, -) with somenewform weight (cf. [Se, ?5.3]). Thereexistsa representation

i: GL2(F3) -* GL2

[VZ])

GL2(C).

By composing with n automorphism GL2(F3) ifnecessarywe can assume that inducesthe identity reductionmod (1 vi2). So if we consider

ANDREW WILES

542

0PO: Gal(Q/Q) -+ GL2 C) we obtain an irreducible epresentation hich s

easilyseen to be odd and whose mage s solvable. Applying he theoremwe find newform ofweight ne associated o thisrepresentation.ts eigenvalues lie in [VA/]. Now picka modularform ofweight ne suchthat -1(3). For example,we can take El, whereE1, is the Eisenstein erieswith Mellintransformivenby (s) ((s, X) for thequadraticcharacter ssociated f mod3 and usingthe Deligne-Serreemma ([DS, to Q(VA/=). Then fE Lemma6.11])wecan find n eigenform ofweight with hesameeigenvalues as modulo a primepu' bove (1 X/=2). There is newform of weight whichhas the same eigenvalues s g' for lmost ll T1's, nd we replace 9', eu') by (g, u) for ome prime I- bove (1 j/2). Then the pair (g, u) satisfies ur requirementsor suitablechoiceof u compatiblewithps'). We can apply this to an elliptic urve defined verQ by considering E[3]. We now howhow n studying lliptic urves urrestriction irreducible representations thedeformationheory an be circumvented. THEOREM 5.2.

All semistable lliptic urvesoverQ are modular.

Proof. Suppose that is a semistableellipticcurve over Q. Assume first hat the representation E,3 on E[3] is irreducible. Then if Po PE,3 restricted Gal(Q/Q (vZ/=)) werenotabsolutelyrreducible,he mageofthe restriction ouldbe abelianoforderprime o 3. As the semistablehypothesis impliesthat all the inertiagroupsoutside 3 in the splitting ieldof Po have orderdividing thismeansthat the splitting ield fPo is unramifiedutside 3. However, (V/Z3) has no nontrivialbelianextensions nramifiedutside and oforderprime o 3. So Po itselfwouldfactor hrough n abelian extension of and this is a contradiction Po is assumed odd and irreducible.So Po restricted o Gal(Q/Q(VE/=)) is absolutely rreducible nd PE,3 is then modularbyTheorem .2 (proved t the end ofChapter3). By Serre's sogeny theorem, is also modular in the senseofbeing factor f the Jacobianof modular urve). So assumenowthat PE,3 is reducible.Then we claim thatthe representationP-E,5 the 5-division oints s irreducible. his is because Xo(15) (Q) has onlyfourrationalpointsbesidesthe cusps and thesecorrespond nonsemistable urveswhich anycase are modular;cf. [BiKu, pp. 79-80]-. f we knew hat -E,5 was modularwe couldnowprove he theorem the same way we did knowing hat PE,3 was modular nce we observe hat PE,5 restricted Gal(Q/Q(x/5)) is absolutely rreducible.This irreducibilityollows similar argument o the one for -E,3 since the only nontrivial belian extension

(Vs) unramifiedutside andoforder rime

is Q((5) which abelian

overQ. Alternatively, is enoughto checkthat thereare no elliptic urves forwhich E,5 is an inducedrepresentationverQ(v/5) and E is semistable


543

at 5. This can be checked n the supersingular ase usingthe description f PE,5 ID5 in particulart s inducedfrom character theunramifieduadratic extension f Q5 whose restrictiono inertia the fundamental haracter level 2) and in the ordinary ase it is straightforward. Considerthe twistedformX(p)lQ of X(5)/Q defined s follows. Let X(5)/Q be the (geometricallyisconnected) urvewhosenon-cuspidal oints classify lliptic urveswithfull evel 5 structure nd let the twisted urvebe defined thecohomology lass (evenhomomorphism)

H1(Gal(L/Q), Aut givenby PE,5: Gal(L/Q) GL2(Z/5Z) AutX(5)/L whereL denotesthe splitting ield PE,5. Then E defines rationalpointon X(p)/Q and hence also of an irreducible omponent f it whichwe denote C. This curveC is smooth X(p)lQ X(5)/Q is smooth. It ha genuszero since the same is trueofthe irreducibleomponents X(5)-Q. A rationalpointon (necessarily on-cuspidal) orrespondso an elliptic curveE' over with isomorphism '[5] E[5] as Galois modules cf. [DR, VI, Prop. 3.2]). We claim that we can choose such a point with the two propertieshat (i) the Galois representationE',3 is irreducible nd (ii) E' (or quadratictwist)has semistable eduction 5. The curveE' (or quadratic twist)willthen atisfyll theproperties eededto applyTheorem .2. (Forthe primes we ust use thefact hatE' is semistable t == E-,5(Iq) 5.) So E' willbe modular nd hence so too wil PE',5 To picka rationalpointon satisfyingi) and (ii) we use the Hilbert rreducibilityheorem. or,to ensure onditioni) holds,weonlyhaveto eliminate thepossibilityhatthe mageofPE',3 is reducible.But thiscorresponds E' beingthe image ofa rationalpointon an irreducibleovering ofdegree 4. Let Q(t) be the function ield f C. We havethereforen irreducible olynomial f(x, t) Q (t) x] of degree and we need to ensurethat formany values to in Q, f(x, to) has no rationalsolution. Hilbert'stheorem nsures that there xists t1 such that f(x, ti) is irreducible.Then we pick prime 5 suchthat f(x, t1) has no rootmod P1. (This is easilyachievedusing he P1 Cebotarevdensity heorem; f. [CF, ex. 6.2, p. 362].) So finallywe pick any to E which pl-adicallyclose to t1 and also 5-adically lose to the original value of t givingE. This last condition nsures hat E' (corresponding to) or quadratictwisthas semistable eduction 5. To see this,observe hat sinceJE $A 1728,we can find family (j): y2 X3- 92(j)x 93(j) with rational unctions 2(ij, 93(j) which refinite and with he -invariant E(jo) equal to whenever hegi(jo) are finite.Then E is givenby quadratic twist f E(jE) and so after changeoffunctions ftheform 2(i) u2g2(j), E and that the U3g3(j) with u Qx we can assume that E(jE) 93(j) close enough5-adically equationE(jE) is minimal t 5. Then for

544

ANDREW WILES

stillminimal nd semistable 5, since criterion or his, theequation theequationE(j') E(j') is stillminimal or ord5(c4(E(j'))) for integralmodel, model, s thateither rd5(A(E(j'))) 0. So up to quadratictwist E' is also semistable. quadratictwistE' This kindofargument of argument an be appliedmore applied moregenerally. generally. THEOREM 5.3.

Suppose thatE thatE is an elliptic urvedefined urvedefined ver Q with thefollowing thefollowing roperties: (i) has goodor good or multiplicativeeduction multiplicativeeduction t 3, 5, (ii) For 3,5 and foranyprime anyprime -1 modp modp either E,p Dq is reducible over orPEIpIIq is irreducible ver P. Then is modular. one can carryover Proof. The main pointto point to be checked that one carryover condicurveE'. For thiswe use that that for nyodd tion ii) to the new curveE'. ny odd prime PEXpDq

is absolutely rreducible nd -EXPIq is absolutely educible

and 3

#PE)p(Iq)

acquiresgood acquiresgood reduction veran abelian 2-power xtension f extension Qq. Qqnrbut not overan abelian extension thenthat q _-1(3) and thatE' that E' does not satisfy ondition ii) at Suppose thenthat that also q (for 3). Then we claim that #PE',3(Iq). For otherwisePEj,3(Iq) has its normalizer normalizer GL2(F3) contained Borel,whence Borel, whence -El,3(Dq) would urhypothesis. be reduciblewhich reduciblewhich ontradicts ur above equivalence hypothesis. usingthe usingtheabove we deduce,by passingvia -E',5 PE,5, that E also does not satisfy ypothesis 3. (ii) at We also need to ensure hat PE',3 is absolutely rreducible ver Q(VW3). This we can do byobserving hatthepropertyhatthe hatthe mageof -E',3 ies n the of a rationalpoint Sylow2-subgroup GL2(F3) implies hat E' is the image ofa ofnontrivial egree.We on certain rreducibleovering rreducible overing thenargue egree.We can thenargue in the same way we did in the previous heorem o eliminate he possibility that -E',3 was reducible, histimeusingtwoseparatecoverings o ensure hat educiblenorcontained n Sylow2-subgroup. the imageof -E',3 is neither educiblenor Finally one also has to show that if both PE,5 is reducible nd PE,3 is induced from character (v/Z3)then is modular. (The case where reduciblehas already been considered.) Taylor has pointedout both were reduciblehas pointed out oth theseconditions conditions re classified the non-cuspidal that curves atisfying oththese rationalpoints rationalpointson on modularcurve modularcurve somorphic Xo(45)/Wg,and this is an ellipticcurve sogenous Xo(15) with rank zero over Q. The non-cuspidal rationalpoints rationalpointscorrespond urvesofconductor conductor 38. El correspond o modular lliptic urvesof

MODULAR ELLIPTIC

CURVES

AND FERMAT'S

LAST THEOREM

545

Appendix Gorenstein rings and local complete intersections ring iscretevaluationring Suppose that is a complete iscretevaluation and thatp: thatp: -+ is a suriectiveocal0-algebrahomomorphism 0-algebra homomorphismetween omSupposefurther hatPT is a prime deal of plete ocal Noetherian -algebras. Supposefurther PROPOSITION 1.

such that T/PT (i)

O(X1,

and let Ps

...

,Xr1/(fi,

..

W-1(PT). Assume that

,fr-u)

where is the ize ofa minimal et of

0-generators fPT/PT, PT/P2 and that hese re finitely isomorphisms/p2 (ii) induces isomorphisms/p2 has ranku. ranku. whosefreepart parthas generated -moduleswhosefree Then is an isomorphism. First we consider he case where Proof. Firstwe

generators x1,.

0. We may assume the assume that thatthe

their residues in T/PT .,Xr lie in PT by subtracting theirresidues

(ii) we mayalso may also write

(9JX1,.

Xrll(gl,*

0. By

v9,)

generatedby the allowingrepetitions necessary)and Ps generatedby (by allowingrepetitions Writing fi images of {x1,.. .,Xr}. Let p (X1,... ,Xr) in 0Jx1,. .xr] the Fitting deal as an 0-module of Eaijxj mod p2 with aij 0, we see that the PT/PT is givenby givenby (PT/Pp) det(aij)

with

nonzeroby the hypothesis hat and that this is nonzeroby gi =Ebijxj modp2, then Fo(ps/p2)

{det(bij):

I, #I=r,

0. Similarly, each {1, ...,

ome choice By (ii) again we see that det(aij) det(bij) as ideals of for omechoice of I. After enumbering may assume that Io {1, . r} Then each gi for some rij 0x1,.. .,Xr]J and we (i 1,... ,r) can be written gi Erijfi forsome have mod p. det(bij) det(rij) det(aij) modp. Hence det(rij) is unit,whence matrix.Thus the fi's can unit,whence rij) is an invertiblematrix.Thus T. be expressed terms f the gi's and so that they We can extend histo the case $& by pickingX1,.. Xr-u so thatthey generatepT/p2)tors. Then we can write ach fi Er-uaIj modp2 and likewise or he gi's. The arguments beforebut applied to the argument s now ust as beforebut Fitting deals of pT/p2)tors.

546

ANDREW WILES

For the next propositionwe propositionwe continue assume that is a complete aluationring.Let Let be local 0-algebra which s a module s finite discrete aluationring. finite and free ver 9. In addition,we addition,we assume the existence f an isomorphismf T-modules --0 Homr T, 0). We call a local 0-algebra which s finite nd free nd satisfies hisextracondition extra condition Gorenstein -algebra (cf. ?5 of [Til]). Now suppose that p is prime deal of such that T/p X. Let o3: -- Tlp be the naturalmap naturalmap and define principal deal of by (AT)

C3(1))

where3: where 3: --) is the adjoint of /3 ithrespectto perfect -pairingson and T, and wherethe pairingof pairingof with itself s T-bilinear. (By perfect -( of finite ankwe ankwe mean pairing free -module M offinite pairing such that boththe both the nducedmaps nducedmaps -) Homo(M, 0) areisomorphisms. are isomorphisms.When we are thusrequiring thusrequiringhat hat thisbe thisbe an isomorphism T-modules lso.) The ideal (AT) is independent the pairings.Also pairings.Also T/TT is torsion-free an 0-module,as 0-module, as can be seen by applyingHom applyingHom 0) to the sequence -+

-T

--+

-+0,

to obtain homomorphism/T/T Hom(p,0). Hom(p,0). This also showsthat OT) Annp. If we let l(M) denotethe length an 0-module M, then P/P

1(0/t)

(wherewe (wherewe write 7T- or/3(qT)) because is faithful /IqT-module. For brief ccount of the the relevant roperties Fitting deals see the appendixto appendix to [MW1].) Indeed, writing R(M) for he Fitting deal of as an R-module, we have FT(P) (r/T) FT/p,(P/P2) (AT) FT/1T (P) and we then use the fact that that the length f an 0-module is equal to the aluationring. particularwhenp/p2 length 09/FO(M) as is a discrete aluationring.

is torsion-module torsion -module hen 7T 0.

ora Gorenstein -algebra to be a complete nterWe need criterion ora section. We will say that a local 0-algebra which s finite nd freeover freeover is a complete ntersection ver if there is an 0-algebra isomorphism ome r. Such ring necessarily GorenOj[xi,. .,Xrj/(fi *... fr) for omer. stein0-algebra stein0-algebra and {fi, .. fir} necessarily regular equence. That (i) =X (ii) in the following roposition due to Tate (see A.3, conclusion in the appendix [M Ro].) PROPOSITION 2.

Assume that is a completediscrete valuationrmng completediscretevaluation rmng nd freeover over and and that is a local Gorenstein -algebra which s finite ndfree

MODULAR ELLIPTIC

CURVES


547

and PT/PT is a torsion thatPT is a primeideal of T such thatT/PT 0-module. Then thefollowing wo conditions re equivalent: (i)

is a completentersectionver0.

(ii) l(PT/PT)

0-modules.

1(0//7T-)

Proof.To prove hat (ii) =X (i), picka complete ntersection over (so assumedfinite nd flat ver0) suchthata: S-ET and suchthat ps/p2 PT/P2 r-1 (PT). The existenceof such an S seems to be well known wherePs (cf. [Ti2, ?6]) but here s an argument uggestedby N. Katz and H. Lenstra (independently). of 0[xi,. ..,xr]/(fl,.. .,f) with PT the image in it follows that also (Xi, ... Xr). Since T is local and finite and freeover (90X1 X Xr]j/(fi, X Eaijf; fs)). We can pick 91,..., gr such that gi Write

withaij

(9 and suchthat (fiI.

..-

f8,p2)

(9gi..

igrp2).

We thenmodify i, .. gr bytheadditionofelements &I} of f, .. ., f8)2 nd set (gj gi a1,,..., gr gr ar). SinceT is finite ver0, there xists n such that foreach i, xjv can be written n as a polynomial hi Xi .. r) of total degree ess than N. We can assume also that is chosengreater han

Then set (x' the total degree of gi foreach i. Set ai hi(xi, ...,x))2. (9 .. Xr]/(g1 g). Then S is finite ver byconstructionnd also (.g

where A) is the maximal deal of0. It follows sincedim(S/A) dim(S) that {9g, g9} is regular equenceand hencethatdepth(S) dim(S) 1. In particular he maximal0-torsionsubmoduleof is zero sinceit is also finiteength -submoduleofS. Now 0/(i-s) l(ps/p2) by (i) X, (ii) and 0/(ft), since l(01/(7s)) l(PT/p2) by hypothesis. ick isomorphisms I(0/(ftr))

Homo T, ),

Homo S, 0)

as T-modulesand S-modules,respectively.The existenceof the latter for complete ntersectionsver is wellknown; f.conclusion of TheoremA.3 of [M Ro]. Then we have a sequenceof maps, in which and ,3 denote the adjointswithrespect these somorphisms:

oA

-0.

One checks hat & is map of S-modules T being givenan S-action via a) and in particular hat o is multiplication an element of T. Now (,3 0,L) (ft7) in and (,3 a) (NT) in C, we a) (vs) in (9. As (as) is an isomorphism. follows havethat t is a unitmodPT and hencethat

ANDREW WILES

548

ker im is that z- T, as otherwise S-modules,which ontradicts being ocal.

nontrivial ecomposition [1

Remark.Lenstra has made a important mprovemento this proposition by showing hat replacing-ZT y /3(ann ) givesa criterion alid for ll local (9-algebraswhich re finite nd free ver 9, thu without he Gorenstein hypothesis.

PRINCETON UNIVERSITY, PRINCETON, NJ REFERENCES

[AK] [BiKu] [Bo] [BH] [BK] [BLR] [CF] [Cal] [Ca2] [Ca3] [CPS] [CS]

[CW]

[CO] [DR] [DS] [Dia] [Di]

A. ALTMAN and S. KLEIMAN, An Introduction o Grothendieckuality Theory, ol. 146, SpringerLecture Notes in Mathematics,1970. B. BIRCH and W. KUYK (eds.), Modular Functions of One Variable IV, vol. 476, SpringerLecture Notes in Mathematics,1975. Boston, N., Families of Galois representations-Increasing he ramification, uke Math. J. 66, 357-367. W. BRUNS and J. HERZOG, Cohen-Macaulay Rings, Cambridge University ress, 1993. S. BLOCH and K. KATO, L-Functions and Tamagawa Numbers of Motives, The Grothendieck estschrift, ol. (P. Cartieret al. eds.), Birkhduser, 990. N. BOSTON, H. LENSTRA, and K. RIBET, Quotients of group ringsarisingfrom wodimensionalrepresentations, R. Acad. Sci. Paris t312, Ser. 1 (1991), 323-328. J. W. S. CASSELS and A. FR6LICH (eds.), AlgebraicNumberTheory,Academic Press, 1967. H. CARAYOL, Sur les representations -adiques associees aux formesmodulairesde Hilbert,Ann. Sci. Ec. Norm.Sup. IV, Ser. 19 (1986), 409-468. Sur les representations aloisiennesmodulo attachees aux formesmodulaires,Duke Math. J. 59 (1989), 785-801. Formesmodulaireset representations aloisiennesa valeursdans un anneau local complet, p-Adic Monodromy nd the Birch-Swinnerton-Dyeronjecture eds. B. Mazur and G. Stevens), Contemp. Math., vol. 165, 1994. E. CLINE, B. PARSHALL, and L. SCOTT, Cohomologyof finitegroups of Lie type I, Publ. Math. IHE 45 (1975), 169-191. J. COATES and C. G. SCHMIDT, Iwasawa theory or hesymmetricquare of an elliptic curve,J. reine und angew. Math. 375/376 (1987), 104-156. J. COATES and A. WILES, On p-adic L-functions nd ellipticunits,Ser. A26, J.Aust. Math. Soc. (1978), 1-25. R. COLEMAN, Division values in local fields, nvent.Math. 53 (1979), 91-116. P. DELIGNE and M. RAPOPORT, Schemas de modulesde courbes lliptiques, Springer Lecture Notes in Mathematics,Vol. 349, 1973. P. DELIGNE and J-P. SERRE, Formes modulairesde poids 1, Ann. Sci. Ec. Norm. Sup. IV, Ser. (1974), 507-530. F. DIAMOND, The refined onjectureof Serre,to appear in Proc. 1993 Hong Kong Conf. on Modular Formsand Elliptic Curves. L. E. DICKSON, Linear Groupswith n ExpositionoftheGalois Field Theory, eubner, Leipzig, 1901.


549

V. DRINFELD, Two-dimensional -adic representations the fundamental roupof curve over a finite ieldand automorphicforms GL(2), Am. J. Math. 105 (1983), 85-114. B. EDIXHOVEN, The weight Serre's conjectureon modular forms,nvent. Math. [El] 109 (1992), 563-594. L'action de l'algebrede Hecke sur les groupes de composantesdes jacobi[E2] ennes des courbes modulairesest "Eisenstein", n Courbes Modulaireset Courbes de Shimura,Asterisque196-197 (1991), 159-170. M. FLACH, A finitenessheorem or he symmetricquare of n elliptic urve, nvent. [Fl] Math. 109 (1992), 307-327. J.-M. ONTAINE, Sur certains ypesde representations-adiquesdu groupede Galois [Fo] d'un corps local; construction 'un anneau de Barsotti-Tate,Ann. of Math. 115 (1982), 529-577. G. FREY, Links betweenstable ellipticcurvesand certaindiophantine quations, [Fr] Annales Universitatis araviensis1 (1986), 1-40. R. GREENBERG, Iwasawa theory or -adic representations, dv. St. Pure Math. 17 [Grel] (1989), 97-137. On the structure fcertainGalois groups, nvent.Math. 47 (1978), 85-99. [Gre2] B. H. GROSS, A tamenesscriterion orGalois representationsssociated to modular [Gro] formsmodp, Duke Math. J. 61 (1990), 445-517. L. Guo, General Selmer groups and criticalvalues of Hecke L-functions,Math. [Guo] Ann. 297 (1993), 221-233. Y. HELLEGOUARCH, Pointsd'ordre ph sur escourbeselliptiques,Acta Arith.XXVI [He] (1975), 253-263. H. HIDA, Iwasawa modules attached to congruences f cusp forms,Ann. Sci. Ecole [Hil] Norm.Sup. (4) 19 (1986), 231-273. Theoryofp-adic Hecke algebras and Galois representations,ugaku Ex[Hi2] positions2-3 (1989), 75-102. Congruencesof cusp forms nd special values of their zeta functions, [Hi3] Invent.Math. 63 (1981), 225-261. On p-adic Hecke algebras forGL2 overtotallyreal fields,Ann. ofMath. [Hi4] 128 (1988), 295-384. B. HUPPERT, Endliche Gruppen , Springer-Verlag,967. [Hu] Y. IHARA, On modular curvesover finite ields, n Proc. Intern.Coll. on discrete [Ih] subgroupsof Lie groupsand applicationto moduli,Bombay,1973,pp. 161-202. K. IWASAWA,Local Class Field Theory,OxfordUniversity ress, Oxford,1986. [Iw1] On Zi-extensionsof algebraic numberfields,Ann. of Math. 98 (1973), [Iw2] 246-326. N. KATZ, resulton modular forms characteristic in Modular Functions of [Ka] One VariableV, Springer N. M. 601 (1976), 53-61. E. KUNZ, Introductiono Commulative lgebra nd AlgebraicGeometry,irkhaiiser, [Kul] 1985. Almostcomplete ntersectionsre not Gorenstein,J. Alg. 28 (1974), 111[Ku2] 115. N. KATZ and B. MAZUR, ArithmeticModuli ofElliptic Curves,Ann. ofMath. Studies [KM] 108, PrincetonUniversity ress, 1985. R. LANGLANDS, Base ChangeforGL (2), Ann. of Math. Studies,PrincetonUniver[La] sityPress 96, 1980. W. Li, Newforms nd functional quations,Math. Ann. 212 (1975), 285-315. [Li] R. LIVNE, On the conductors mod Galois representationsomingfrommodular [Liv] forms, of No. Th. 31 (1989), 133-141. B. MAZUR, DeformingGalois representations, Galois Groups over Q, vol. 16, [Mal] MSRI Publications,Springer,New York,1989. [Dr]

550 [Ma2]

ANDREW WILES

Modular curves and the Eisensten ideal, Publ. Math. IHES 47 (1977),

133-186. Rational isogeniesofprimedegree, nvent.Math. 44 (1978), 129-162. [Ma3] [M Ri] B. MAZUR and K. RIBET, Two-dimensionalrepresentationsn the arithmetic modular curves, CourbesModulaireset Courbes de Shimura,Asterisque196-197 (1991), 215-255. [M Ro] B. MAZUR and L. ROBERTS, Local Euler characteristics,nvent. Math. 9 (1970), 201-234. [MT] B. MAZUR and J. TILOUINE, Representations aloisiennes,differentiellese Kihler et conjecturesprincipales, ubl. Math. IHES 71 (1990), 65-103. [MW1] B. MAZUR and A. WILES, Class fields f abelian extensions Q, Invent.Math. 76 (1984), 179-330. [MW2] , On p-adic analytic families f Galois representations, omp. Math. 59 (1986), 231-264. J. S. MILNE, Jacobian varieties, ArithmeticeometryCornell and Silverman, [Mil] eds.), Springer-Verlag,986. [Mi2] Arithmetic uality Theorems,Academic Press, 1986. R. RAMAKRISHNA, On a variation f Mazur's deformation unctor, omp. Math. 87 [Ram] (1993), 269-286. M. RAYNAUD, Schemas en groupes de type (p,p, .. ,p), Bull. Soc. Math. France [Rayl] 102 (1974), 241-280. [Ray2] Specialisationdu foncteur Picard, Publ. Math. IHES 38 (1970), 27-76. K. A. RIBET, On modularrepresentations Gal(Q/Q) arising rommodularforms, [Ril] Invent.Math. 100 (1990), 431-476. [Ri2] ' Congruencerelationsbetweenmodularforms, roc. Int. Cong. of Math. 17 (1983), 503-514. [Ri3] Report on mod representations Gal(Q/Q), Proc. of Symp. in Pure Math. 55 (1994), 639-676. [Ri4] Multiplicities p-finitemodp Galois representations Jo(Np), Boletin da Sociedade Brasileirade Matematica, Nova Serie 21 (1991), 177-188. K. RUBIN, Tate-Shafarevich roupsand L-functions ellipticcurveswithcomplex [Rul] multiplication,nvent.Math. 89 (1987), 527-560. [Ru2] The 'main conjectures'of wasawa theoryfor maginary uadratic fields, Invent.Math. 103 (1991), 25-68. [Ru3] Elliptic curveswith complexmultiplication nd the conjectureof Birch and Swinnerton-Dyer,nvent.Math. 64 (1981), 455-470. More main conjectures'for maginary uadratic fields, RM Proceedings [Ru4] and LectureNotes, 4, 1994. M. SCHLESSINGER, Functorson Artinrings,Trans. A.M.S. 130 (1968), 208-222. [Sch] R. SCHOOF, The structureof the minus class groups of abelian number fields, [Scho] in Seminairede Theorie des Nombres,Paris (1988-1989), Progress Math. 91, Birkhauser 1990), 185-204. J-P. SERRE, Sur les representationsmodulairesde degre 2 de Gal(Q/Q), Duke [Se] Math. J. 54 (1987), 179-230. [de Sh] E. DE SHALIT, Iwasawa Theory of Elliptic Curves with Complex Multiplication, Persp. in Math., Vol. 3, Academic Press, 1987. G. SHIMURA, Introduction the ArithmeticTheory of AutomorphicFunctions, [Shl] Iwanami Shotenand PrincetonUniversity ress, 1971. On the holomorphy certainDirichlet eries,Proc. London Math. Soc [Sh2] (3) 31 (1975), 79-98. The special values ofthe zeta function ssociated withcusp forms, omm. [Sh3] Pure and Appl. Math. 29 (1976), 783-804.

Modular Elliptic Curves and Fermat's Last Theorem

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