on deformation rings and hecke rings

Annals of Mathematics

On Deformation Rings and Hecke RingsAuthor(s): Fred DiamondSource: Annals of Mathematics, Second Series, Vol. 144, No. 1 (Jul., 1996), pp. 137-166Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2118586 .

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Annals of Mathematics, 144 (1996), 137-166

On deformation rings and Hecke rings

By FRED DIAMOND*

1. Introduction

Let e be an odd prime and let p be a continuous, irreducible representation

Gal(Q/Q) -- GL2(F)

where F is an algebraic closure of Fe. We suppose throughout that p is modular in the sense that it arises from a modular form (see [16]).

It is a theorem of Wiles [19], completed by work of Taylor and Wiles [18], that certain lifts of p to characteristic zero arise from modular forms. Wiles applies this result in the case that p is the representation arising from the ?-torsion of an elliptic curve over Q and proves the Shimura-Taniyama-Weil conjecture for semistable elliptic curves [19, Theorem 0.4].

In Wiles' central result [19, Theorem 3.3], the representation p is subject to three sets of technical hypotheses which do not intervene in his application to semistable elliptic curves. These are:

(Hi) hypotheses on the restriction of p to decomposition groups at primes p # t;

(H2) hypotheses on the restriction of p to a decomposition group at e;

(H3) the condition that PIGal(Q/K) remain irreducible where K is the quadratic extension of Q such that IDiscK/QI = e.

Under the hypotheses of (H2), Wiles formulates a conjecture [19, Conjec- ture 2.16] identifying certain Hecke rings as universal deformation rings for Galois representations. He goes on to prove the conjecture provided (Hi) and (H3) are also satisfied. The purpose of this article is to extend that result

*This research was supported by a United Kingdom ESPRC grant and was conducted at the Institute for Advanced Study. The author also wishes to thank Richard Taylor and Andrew Wiles for enlightening correspondence and conversations regarding their work, and Brian Conrad, Ken Ribet, Karl Rubin, Alice Silverberg, Richard Taylor, Jerry Tunnell, and the paper's referee for suggesting corrections and improvements to the exposition.



138 FRED DIAMOND

[19, Theorem 3.3] by removing the hypotheses of (Hi), i.e., those on the local behavior of p at primes p #4 e. We prove:

THEOREM 1.1. Wiles' Conjecture 2.16 holds if p satisfies (H3) above.

The hypotheses and definitions needed for the statement of Wiles' conjecture are woven into [19], but see Theorem 5.3 below for a closely related consequence of Theorem 1.1 whose statement is more self-contained. Using Theorem 5.3, we can apply Wiles' method to a larger class of elliptic curves than those considered in [19, Chapter 5]. We conclude:

THEOREM 1.2. If E is an elliptic curve over Q with semistable reduction at 3 and 5, then E is modular.

We still require semistability at 3 and 5 because of the requirements (H2) on the local behavior of p at e which are incorporated in the hypotheses of [19, Conjecture 2.16]. More work is needed even to provide a suitable formulation of Wiles' conjecture under less restrictive hypotheses.

Remark 1.3. Rubin and Silverberg have observed that it follows from Theorem 1.2 that if E is an elliptic curve over Q all of whose points of order two are rational, then E is modular. For a discussion of this, see [7], where the modularity of such curves is in fact established as a corollary of results of Wiles.

Recall that Wiles' proof of [19, Theorem 3.3] proceeds by considering certain "minimal" deformations for p. Wiles then compares the corresponding universal deformation ring to a certain Hecke ring. Briefly, the latter ring is defined by considering the endomorphism ring generated by Hecke operators on the space of weight two cusp forms of the "minimal" level from which p can arise (in the sense of a slight variant of Serre's conjecture [16]), and then completing this endomorphism ring at a certain maximal ideal. Using his result with Taylor [18, Theorem 1] that certain Hecke rings are complete intersections, Wiles proves that the minimal deformation ring and Hecke ring are isomorphic. The isomorphism between "nonminimal" deformation rings and Hecke rings is then established using an inductive argument developed in [19, Chapter 2].

Without the hypotheses of (Hi), one finds that modular forms of the minimal level for p may give rise to lifts of p with additional ramification at primes p _ -1 mod e (see [8]). To adapt Wiles' method, we will define a minimal Hecke ring using endomorphisms of a space of modular forms with specified local behavior at such primes p. Some of the ingredients needed for the analogue of the "minimal" case of [19, Theorem 3.3] can then be deduced from the work of Wiles and Taylor in [19] and [18]. The rest are supplied as in



ON DEFORMATION RINGS AND HECKE RINGS 139

[8] by working with forms on quaternion algebras and appealing to the Jacquet- Langlands correspondence [11]. Otherwise, the approach in the minimal case is essentially the same as that of [19, Chapter 3] and [18], but we incorporate some simplifications based on suggestions of Faltings explained in [18, Appendix]. Wiles' conjecture, Theorem 1.1, is then deduced from its "minimal" analogue using the methods and results of [19, Chapter 2].

2. Deformation rings

We write GQ for Gal(Q/Q) and fix a decomposition group Dp C GQ for each prime p. We write 4p for the inertia subgroup of Dp. We shall sometimes regard Q as embedded in Qp and identify Dp with the image of Gal(Qp/Qp) in GQ. For a Ze-algebra A, we write EA: GQ -+ Z' -, Ax for the cyclotomic character.

Let Z denote the ring of algebraic integers in C. Fix a homomorphism w: Z -- F. Let F be a number field contained in C. Let K (respectively, 0) denote the completion of F (respectively, OF = Z n F) at the kernel of =IOF. Let A be a uniformizer in 0 and identify the residue field k = O/AO with its image in F. For an 0-algebra A, we sometimes use to denote the Teichmfiller lift kX ,Ox -, Ax.

Assume that F is sufficiently large so that the eigenvalues of all elements of p(GQ) are contained in k. We shall later assume F is sufficiently large in other respects, but the requirements will always depend only on p. We write PO for the representation GQ -- GL2(k) determined up to isomorphism by P PO ?k F. Let us also fix an unramified character '0: De -+ kX and assume that det PO D, # 'i2. Let E denote the set of primes p ?& e at which PO is ramified.

We impose the following restrictions on P0:

* PO Ga( /1K) is irreducible where K is the quadratic subfield of Q(ce).

* POID1 is finite or f0-Selmer in the sense of [6, ?6].

* If p e E and POIDP is reducible, then pf' # 0. Remark 2.1. Note that for arbitrary po, the last condition is satisfied by

a twist of po by a character unramified outside E.

Remark 2.2. Recall that we consider POID1 to be finite if it is associated to a finite flat group scheme over Ze and det(po)lie is cyclotomic.

Let P denote the set of primes p =-1 mod e such that Po IDP is irreducible and PG I, is reducible. In [19, Theorem 3.3] and in [18], it is assumed that P is empty; we focus on the case that P is not empty.



140 FRED DIAMOND

Following [19, Chapter 3] and [18], we will consider a finite set of primes Q such that if q is in Q, then q _ 1 mod e, p is unramified at q and p(Frobq) has distinct eigenvalues. For each such q, we fix an ordering (aq, /3q) of the eigenvalues.

For a complete local Noetherian 0-algebra A with residue field k, we consider deformations a: GQ - GL2(A) of Po satisfying:

* det(of)&E1 has finite order prime to e;

* if p Z E U Q U {f}, then a is unramified at p;

* if p E Z and POIDP is reducible, then as an A-module, aIP is a free rank one summand of a;

* if PO D, is finite, then so is a; otherwise a is VO-Selmer (in the sense of

[6]).

We can then define a ring RQ and a universal deformation

rQ: GQ -GL2(RQ).

We omit the precise formulation of the universal property as well as the proof of existence as these are now standard and can be found in [13], [14], and [19]. We recall that RQ is a complete local Noetherian 0-algebra with residue field k and that rQ mod nQ = Po where nQ is the maximal ideal of RQ. Standard arguments also furnish a canonical isomorphism between HpUQ(GQ, ad0 (Po)) and the tangent space

Homk(nQ/((nQ)2, A), k)

of RQ/ARQ. Recall that if S is a finite set of primes different from e, then H (GQ, ad?(po)) is defined as the preimage of

11 H (DV, ado (po)) vSf

under the natural map

H1(GQ, ado(po)) - J7 H1(Dv ado(po)) vVS

The groups H1 (Dv, ado(po)) are defined in [18, ?4], but we recall [6, Proposi- tion 6.1] that if det PolIe = k I e and PO is Selmer but not finite, then all Selmer deformations are "strict" in the sense of [19]. In particular, it follows that H' (De, ado(po)) may always be defined as in case 3 in [18] if POIDe is Selmer but not finite.

Next we consider the ramification of the universal deformation rQ at a prime q E Q. We prove that rQ IDq is the direct sum of two characters Dq RQ. First choose or E Dq such that or mod Iq = Frobq. Then rQ (o) has distinct




eigenvalues a and : in R' with a mod nQ = aq and 3 mod nQ = fq. So with respect to some basis, rQ(Jf) = (0 ,3). Next observe that rQIIq factors through the maximal pro-t quotient of Iq. Let r be a topological generator for this quotient and write rQ (-r) = (a d) with respect to this basis. Applying rQ to the equation CrT1T-l = Tq- we find b E btRQ and c E dRQ (recall q _ 1 mod t) and therefore b = c = 0. We conclude that

rQIDq, XaEDX0

where X(> mod nQ and Xy mod nQ are the unramified characters Dq -+ kX

defined by Frobq & + aq and Frobq H+ /q. Moreover x~ I1q and X, I1q factor through the maximal pro-t quotient Lq of the image of Iq in the abelianization of Dq. We define Oq: Zq -- R' by (XOX-1) IAq* We let LQ = HqeQ Aq and define 6Q: LQ -R as the product of the 6q*

Now let us consider rQ I D for primes p e P. For such primes POI Dp is induced from a character G: H -- kX where H is the unique subgroup of index two in Dp containing IP. Moreover POIP is the sum of the distinct characters 'plip and 'lip where (p is (p composed with conjugation by Frobp. For each

p in P, we fix a choice of (p and let Ap denote the maximal pro-t quotient of the image of IP in the abelianization of H. An argument similar to the one for primes in Q, but beginning with o' E IP such that p(o') $ p(op'), shows that

rQIIp -p p I E 0 1

for a unique character 6p: Ap -+ R'. Write Ap for the product of the Ap and Op: Ap -RQ for the product of the Op.

We define a deformation ring Rk and universal deformation rQ exactly as we defined RQ and rQ, except that now we require oji, I p E (p for primes p e P. We let nQ denote the maximal ideal of R&. The tangent space of Rk/ARk is now given by H'(GQ, ad0 (po))

LEMMA 2.3. There is an integer s > dimk Ho (GQ,ado (po)) such that for each integer m > 1, there is a set of primes Qm satisfying:

* #Qm=s.

* For each q E Qm, q --1 mod tm, PO is unramified at q and po(Frobq) has distinct eigenvalues.

* H'(GQ,ado(po)) = H'm (GQ,ado(po)).

This is proved exactly as in [18, ?4] and [19, Chapter 3]. Let us record the relationships among the various deformation rings we

have defined. Using the maps OQ and Op, we regard RQ as an algebra over



142 FRED DIAMOND

the group ring O[AOQ x Ap] = O[AQ] 00 O[ZAp]. Letting JQ denote the augmentation ideal of O[AQ] we may identify RQ/JQRQ with Ro as O[Ap]- algebras. Similarly letting Jp be the augmentation ideal in O[[Ap] we have RQ/JpRQ = Rk by which we regard Rk as an (9[Ap]-algebra. Thus

k = R0/JpR0 = RQ/JpQRQ = R i/JQRk

where JpQ = Jp + JQ is the augmentation ideal of O[[Ap x AQ]. Let us recall the definition of certain Galois cohomology groups associated

to liftings o: GQ -* GL2(O) of Po* We assume that of is a deformation of the type considered in the definition of R'. Let MO = ado(L) 0G (K/O) and

M = ad(L) (go (K/O) r-M M (K/0)

where L is a model for u. We let Mn (respectively, M7?) denote the kernel of A' in M (respectively Mo.) Let S be a finite set of primes different from ?. Define H (GQ, Mn) as the preimage of

vSS

under the natural map

H1 (GQ, Mn) I|J H1 (Dv, Mn)I vqts

and let H (GQ, M) denote the union of the H (GQ, Mn). Here H (Dv, Mn) is defined as the kernel of H1 (Dv, Mn) -?H1 (Iv, Mn) if v =$ L. If Po is O/)o- Selmer, we define Hse (Dv, Mn) as in Chapter 1 of [19], and if PO is not finite we let H1 (Dv, Mn) = H'e(Dv, Mn). If PO is finite, we define H1 (De, Mn) as in Chapter 1 of [19] using extensions of finite flat group schemes. Recall that Wiles defines H1 (De, Mn) this way only if Po is not also Selmer, but the definition carries over without change as does the property that H1 (De, Mn) is the preimage of H1 (De, Mn+l) in H1(De, Mn). Moreover if we define H1 (De, M) C

H' (De, M) as the union of the images of H1 (De, Mn), then Wiles' argument proving [19, Proposition 1.3(i)] yields that in general H'(De, M) C H (De, M), where H (De, M) is defined by Bloch and Kato in Section 3 of [1]. Equality then follows on noting that H} (De, M) is divisible and that #H1 (De, M1) > #Hf (De, M1). (The former order is computed in the proof of [19, Proposi- tion 1.9], the latter in the proof of [18, Lemma 5].) Combining this observation with [19, Proposition 1.3], we have:

LEMMA 2.4. If Po is finite, then

#Hf(DeaMn) = #(O/Ano)#H0(DeMn).

If Po is also Selmer, then H (DeMn) C H1e(DeMn)




We define Hs(GQ, M7?) and Hh(GQ, MO) by replacing M with MO in the discussion above. Again we find that H (De, M0) = H' (Di, M7?) if Po is finite. In particular

Hf (De, Mn) Hf (De, Mno) G H1 (Di, A-n7/(9)

where H1 (De, A-nO/O) is the group of unramified homomorphisms. Thus in general

(2.1) HS(GQ, Mn) H'(GQ, Mno) (D H (GQ, A-nO/O)

where H (GQ, A-n(O/) is the group of homomorphisms unramified outside S. We have a similar decomposition of H (GQ, M).

3. Hecke rings

We briefly recall a definition for the Hecke ring TQ considered in [18], which for Q $ 0 is a slight variant of the ring denoted TQ in [19, Chapter 3]. Recall that Q is a finite set of primes q satisfying q -1 mod e, p is unramified at q, and p(Frobq) has distinct eigenvalues (q, /3q). Let NQ = NO6 FlqEQ q where No is the conductor of p and 6 is 0 or 1 according to whether Po IDe is finite. Then rQ, defined in [18, ?1], is a certain group intermediate to rF(NQ) and ro(NQ). We let T(rQ) be the subring of End(S2(FQ)) generated by the Hecke operators:

* Tr and (r) for primes r not dividing NQe;

* Uq for primes q dividing NQ/NO.

Since Po is assumed to be modular we conclude from the work of Ribet and others [6, Theorem 6.4] that there is an eigenform f in S2(T'Q) for the action of T(FQ) such that the composite T(rQ) Zw F, where the first homomorphism is defined by mapping a Hecke operator to its eigenvalue, satisfies:

* Tr ~-+ tr(po(Frobr)) if r does not divide NQi;

* r(r) ~-4 det(po(FRobr) if r does not divide NQe;

* Uei -* ibo (Frobe) if 6 = 1;

* Uq alq for primes q E Q.

These conditions characterize a nonzero homomorphism

(3.1) w: T(rQ) - k,

which induces a surjective homomorphism T(rQ) 00 -* k of (9-algebras. We let mQ denote the kernel of the latter homomorphism and we may identify TQ with (T(rQ) 09 O)mQ (see the proof of [18, Lemma 1]).



144 FRED DIAMOND

Recall that we have defined ?&q as a quotient of Iq, but note that it is naturally isomorphic to the maximal pro-e quotient of Gal(Q(Cq)/Q) ' (Z/qZ)x which is the definition used in [18]. Then TQ becomes an 9[ZAQ]-algebra via the isomorphism of AQ with a subgroup of ro(NQ)/rQ. Recall also that XQ denotes the character

GQ -_ 1?&Q C-+ Q[jAQ]X

introduced in [18, ?1] and that X-1/2 is its composite with the inverse of the XQ automorphism of O[AQ] defined by d X-4 d-2 for d E AQ.

We shall write pQ for the representation GQ -- GL2(TQ) denoted p' in [18, ?1]. Thus if r is a prime not in E U Q U e, then the following hold:

* pQ is unramified at r.

* trpQ(Frobr) = XQ(/2F(lobr)Tr.

* det pQ (Frobr) = X1 (Probr)r(r).

In particular, pQ is a deformation of po. From the universality of rQ and the properties of pQ listed in [18], we deduce that there is a homomorphism of 0[ZAQ]-algebras

qQ: RQ ' TQ

such that pQ r- Q ?RQ TQ. As TQ is generated by the image of trpQ (see the proofs of [19, Proposition 2.15] and [18, Lemma 6]), OQ is surjective.

Our main objective is to prove that T0 is a complete intersection and q0 is an isomorphism. When P is empty the first assertion is the main result of [18] and the second one is [19, Theorem 3.3] (or its variant mentioned in the remark preceding the theorem). Recall however that we are not assuming that P is empty. Note that the only difference this makes in Sections 1 and 2 of [18] is that for primes p in P, pQ does not necessarily have the property that pQ(Ip) Po(IP).

We consider TQ as an (9[Ap]-algebra by the composite

([z'p] -* RQ -' TQ.

Define TO = TQ/JpTQ and let

0b : R' -- TO Q :Q QT be the induced surjection of O[[AQ]-algebras. Note that

PQ = PQ ?TQ TQ _ rQ ?RQ TQ

has the property that pO(Ip) Po(Ip). We shall find that TO behaves well in several ways, but nQ in several ways, but note that a priom6 it is not obviously torsion-free as an




0-module. For some purposes it will be more convenient to use a Hecke ring defined in terms of endomorphisms of a suitable space of modular forms.

Let UQ denote the open compact subgroup of H GL2(Zr) C GL2(A??) consisting of those g such that for each r dividing NQ, gr = ( c with c E NQZr and dmodNQZr in the image of rQ/Fr(NQ) (Zr/NQZr)x. Recall that S2(rQ) is isomorphic as a T(rQ)-module to

@ (7oo)UQ

irEH

where H is the set of weight two cuspidal automorphic representations r = 0,irX

From now on we assume that the field F is Galois over Q and contains the values of VIP) and e2ri/(p+l) for all p E P. Define up: Ip -- GL2(F) as the restriction of ,p i9 Up. Let SQ be the subspace of s2(rQ) defined by

iD (7 0)UQ

1rEIP

where HIb is the set of ir in H such that

(1rp) |I p -Up OF C

for each p E P, where u(irp) denotes the representation of WQp given by the local Langlands correspondent of irp. Then SQ is stable under the action of T(rQ) whose image in End(SQ) we denote T(SQ). Define

T' = (T (S) 0 ()mQ,

which is nonzero by the following lemma.

LEMMA 3.1. The homomorphism w: T(rQ) k of (3.1) factors through T(SQ).

This is essentially proved in [8], but not stated there in the generality we need. For the convenience of the reader, we shall give a variant of the argument in Section 4 below. Let PQ = PQ ?TQ Tb where g9: TQ -* is the natural surjective (9[AQ]-algebra homomorphism.

It will be convenient to introduce some more Hecke rings in the case Q = 0. For each character L = Hf Op: Ap -- FX, we let Hbp be the set of 7r in H such that

7 p ) I Ip- (pP |p IP pp ?D (p| I4 ~~ ) OF C

for each p E P. Define Sb, T(Sb) and T, by replacing HRb with Hb above. Recall that we are assuming for the moment that Q = 0, so that if 4 is trivial, these are simply Sk, T(Sb) and Tb. Finally let HIP be union of the HVI over all



146 FRED DIAMOND

characters fL and define Sd, T(Sg) and Tb by replacing b with b above. Note that

Su Sb

and that there is a natural surjection T(S) - T(Sb) for each fL by which we may regard this as a decomposition into T(Sd)-modules. Note that T(SM) eT(Sb ) is injective.

LEMMA 3.2. Let a denote the kernel of the natural surjection T(Sc)0-)

T(S0). Then for each nontrivial '/: Ap -+ FX,

AnnT(Sb)a C a+,

where ap is the kernel of T(SM) -- T(Sb).

Proof. Note that if p is a minimal prime of T(SM) which contains ap = AnnT(S )Sb, then there is an eigenform in Sb for T(SM) such that p is the

kernel of the homomorphism T(S&) -> Z defined by mapping a Hecke operator to its eigenvalue. By strong multiplicity one, such eigenforms for nontrivial fL cannot be Galois conjugate to eigenforms in Sb for T(SM), and therefore the set of minimal primes containing a is disjoint from the set of minimal primes containing ctp. It follows that a is the kernel of

T(Sg) - f T(SM) VDa

and that the annihilator of a, which is the kernel of

T(SM) - 1 T(S)s,

is contained in ap for each nontrivial fb.

Now let us return to the case of arbitrary Q and examine the relationships among the various Hecke rings. First recall from Chapter 2 of [19] that restriction to the space of forms which are "old" at q defines a surjective homomorphism

T(rQ) -' T(rQ{q})[Uq]/(Uq2 - TqUq + q(q)).

Tensoring with (9 and localizing at mQ we obtain a surjective homomorphism TQ -' TQ_{q} of 0-algebras and hence a surjection hQ: TQ -+ T0. Similarly we define a surjection hb : T- - Tb yielding a commutative diagram

RQ? TQ 9 Tb ~~thQ I 4TQ A 4 T

R0 0 T0 ~~~~~~~~~0 T0




Commutativity of the first square follows from the fact that p- PQ ?TQ T0. In particular, hQ: TQ -+ T0 is O[[Ap]-linear. Since Ap is the kernel of pb, it follows that the image of Jp in Tb is trivial and therefore that gb: TQ -Tb factors through TO. We have now constructed a commutative diagram of surjective RQ-algebra homomorphisms

TQ -Q TO 9Q Tb

lbhQ lhQ Q

T0 ?~TV tT0

for each Q. The maps from RQ to T0 (respectively, TO, TO) factor through R0 (respectively, Rb, Rb). Moreover, recall from Corollary 1 of [18, Theorem 2] that hQ induces an isomorphism TQ/JQTQ T0. Hence:

LEMMA 3.3. hO induces an isomorphism TO/JQTO Z To.

We record another immediate consequence of [18, Theorem 2].

LEMMA 3.4. There is an integer d independent of Q such that To can be generated by d elements as an (9[AQ]-module.

The crucial lemma concerning our Hecke rings is the following; it will play the same role as the generalization of de Shalit's result in [18].

LEMMA 3.5. For any ideal I of O[L\Q], the map O[AQ]/I - To/ITO is injective.

We postpone the proof until Section 4, as we will need to recall some of the tools used in [8].

From now on, we omit the subscript Q if Q = 0. We need two more lemmas in that case, the proof of the first of which we also postpone until Section 4.

LEMMA 3.6. Suppose that w: Tb (9 is a homomorphism of 0-algebras. Then

r(gb'(AnnT (ker(g')))) C (iii(p + 1) (9. PEP /

LEMMA 3.7. The kernel of g: T- Tb is the (9-torsion submodule of TO. The natural map T -+ TV is an isomorphism.

Proof. For the first assertion, it suffices to check that TO90QK and Tb(&0K have the same dimension. For this it is convenient to assume that F contains the eigenvalues of all Hecke eigenforms of level N. To each such eigenform f we



148 FRED DIAMOND

can associate a homomorphism Of: T(r) -9 OF -+ ( defined by Tf = Of (T)f and a representation pf: GQ -+ GL2(0) characterized by

trp (F(robr) = Of (Tr) and det pf (Frobr) = rOf ((r))

for all primes r not dividing Nt. Let 1 denote the set of such eigenforms satisfying:

* pf (o k-po;

* if t divides N, then Of(Ut) = 'bo(Frobf) mod AO.

Then T (go K is isomorphic to ef EK, and

PO0 go K: GQ -+ GL2(T (go K) @ GL2(K) f E

is given by ef? (pE f 00 K). Note that this determines Ap -+ T (go K. We conclude that

TO Oo K- (T0o K)/Jp(T oK) K fer"

where (D is the set of f in 1 such that pf (ZAp) is trivial, or equivalently Pf I |I, - (p P) I . On the other hand, T (goK is isomorphic to ef ebb K where (DI is, by the theorem of Deligne, Langlands and Carayol [3, Theoreme A], the set of f E D such that for some y E Gal(F/Q),

(pf (go K)II- (up () F,y K)

for all p E P. Since pf (go k _ po for f E D, we conclude (DI = (D and hence the lemma.

The comparison of T and TV is similar. O

4. More Hecke rings

In this section we shall prove Lemmas 3.1, 3.5, and 3.6. First note that if P = 0, then Lemmas 3.1 and 3.6 are obvious and that Lemma 3.5 is immediate from [18, Theorem 2]. If P #& 0, we shall prove the lemmas by working with modular forms for a quaternion algebra B ramified at exactly the primes in P (and at oc if #P is odd).

As in [18, ?2], we first introduce for technical reasons an auxiliary prime R not dividing 6Nt furnished by [8, Lemma 3]. Recall that this is a prime R with the property that every t-adic lifting of p is unramified at R. Let N' = NQR




and IF' = EQ n 0r1(R). We let T(I'7) be the subring of End(S2(IF')) generated by the Hecke operators:

* Tr and (r) for primes r not dividing NQ ;

* Uq for primes q dividing N I/NO.

Restriction to the R-old subspace of S2(I7') induces a surjective homomorphism

(4.1) T(]F)- T(EQ)[X]/(X2 - TRX + R(R))

where X is the image of UR. A choice of eigenvalue AR for po (FrobR), or equivalently a choice of root in k for

X2 - w(TR)X + Rw((R))

where w is as in (3.1), yields a homomorphism

a'T(rl/ ) k.

Remark 4.1. One can check that the prime R may be chosen so that po(FrobR) has distinct eigenvalues unless ? = 3 and po has projective image isomorphic to C2 x C2 or A4. We shall not, however, assume that this is the case.

Let m' denote the kernel of the resulting 0-algebra homomorphism

T(F )0O--k

and write T' for (T(IF') 0 (),' . By tensoring (4.1) with ( and localizing at Q Q ~~~Q M/ we obtain an isomorphism

(4.2) TQ[X]/(X2-TRX+ R(R))

according to whether or not po(FrobR) has distinct eigenvalues. (The injectivity of the localized map follows for example from the fact that

(UR - (R)) E(d) d

annihilates the kernel of (4.1) where the sum is over the elements of the kernel of (Z/NI Z)x > (Z/NQZ)x. On the other hand, our choice of R ensures that this operator is not in the kernel of rw'. The calculation of the localization of the polynomial ring over T(J7Q) is straightforward.)

Define the open compact subgroup UQ of FL GL2(Zr) by replacing NQ (respectively, FQ) by NQ (respectively, I' ) in the definition of UQ. Similarly, define SQ by replacing UQ with UQ in the definition of and let T(SQ)



150 FRED DIAMOND

denote the image of T(rF) in End(SQ). We shall prove below that Lemma 3.1 holds with w replaced by w' and T(SQ) replaced by T(SQ). Lemma 3.1 then follows since w' factors through

T(rI>) -- T(S1) -- T(SQ)[X]/(X2-TRX + R(R)).

Furthermore, the 0-algebras TQ and TQ = (T(SQ) 0 0),/ are related as in (4.2). We conclude that:

* To prove Lemma 3.1, we may replace a by w' and T(SQ) by T(SQ); moreover we may assume Q = 0.

* To prove Lemma 3.5, we may replace To by T , hence by TQ.

In the case Q = 0, we define Su' (respectively, S,' for each ?b: Ap -? FX) as the subspace of S2(rF) obtained by substituting U' for U in the definition of Sb (respectively St). We let T(S4') (respectively, T(S')) denote the image of T(J7') in the corresponding space of endomorphisms and write TV' (respectively, Tf,) for the localization at m' of the corresponding ring tensored with (. Let

g9' denote the natural map from TV' to Tb'. Recalling that T TV is an isomorphism (Lemma 3.7) and describing the rings TV' and TL' as in (4.2), we obtain a commutative diagram

TV' T T/' - T[X]/(X2-TRX + R(R)) lb Ig' or ~or l/ I Tb/ Tb Tb/ -~ IbX]I(X2 -TRX + R(R))

where the rightmost downward arrow is induced by gK. Note also that in either case we have

(4.3) g9 (AnnT (kerg ) ) Tb' = gb'(AnnVb, (kerg"')).

We now begin the transition to the setting of quaternion algebras by recalling some results of Gerardin [9] which identify the representations corresponding via Jacquet-Langlands to those in llb and Wi. See [8] for statements of Gerardin's results in the form we shall use.

Let B be a quaternion algebra ramified at exactly the primes in P. For each prime r not in P, choose an isomorphism B 0 Zr = Br M2(Zr). For each prime p in P, choose an embedding Qp2 a Bp where Qp2 denotes the unramified quadratic extension of Qp. Let wp be an element of Bp such that W2 = p and such that wpxw; = xa for all x in Qp2 where a is the nontrivial automorphism of Qp2. Let Zp2 denote the ring of integers of Qp2 and let mp > 0 be such that (p(p: Gal(Qp/Qp2) -- kX has conductor (pZp2)2mp+l or (pZ 2)2mp+2. Consider the open compact subgroup of Bx defined by

Ut = Z2 + '2mp+lZ 2. P P2 +P.




We let Kp: U-t GLpbp (C) be the restriction of the irreducible representation of Qp Ut which in [9, Section 5] and [8, Section 2] is denoted Cop, Op: Q C2 -x

being the character corresponding to (p by the Artin map. Thus the image of KP is finite and 6p is O or 1 according to whether (p has conductor an odd or even power of pZp 2. Recall that in the case that 6P = 0, Kp coincides with Op on Z'P2. In the case that 6p = 1, let

p= p2 +W P

and choose a nontrivial character ip/ of Zx of order ?. Then Kp is a summand

of IndU p+ where s+ is a certain character of U+ which coincides with Op,4p on Z x . Enlarging F if necessary, we assume that Kp has a model Mp over OF,

Ut and in the case bp = 1 that Mp is a summand of Ind + M+ where M+ is a model for Kp+

Let H denote the maximal subgroup of Zx of ?-power index, and let

Ut = H + .72mP+l Z 2

Thus Ut is the maximal subgroup of Ut of ?-power index; it is normal in Ut and the Artin map determines an isomorphism Azr, UJt/Ul. Note that Kp is irreducible; moreover, so is its restriction to Ut P.

We define the Up = HJPp Ut-module

M = 0PEpMP

where the tensor product is over OF. Note that the Artin maps together with our choice of embeddings of Qp2 in Bp for each p determine an isomorphism Ap- Up/Up where Up = HP Ut. Recall from [8, Lemma 4] that for a character 4': AP -- Ox, we have Mp = M0OF 4' as a model for the representation of Ut defined by replacing (p with (pbp for each p. We note also that M = HomoF (M, OF) is a model for the representation defined by replacing (p with its inverse. For an OF-algebra A, we write M(A) for M (OF A, and we define Mp(A), M(A) and Mp(A) analogously.

Let Vp = VJp Vp where for each p E P, Vp = kerKp n ut. We let TQ be the open compact subgroup of (B 0 AP)x GL2(A ) corresponding under our fixed isomorphisms to the projection of the open compact subgroup UC c GL2(A?) defined in Section 3. Also let U$Q correspond to the image of U_ where U_ is defined in the same way as U' but without the restriction on dq for q E Q. Thus there is an isomorphism U/Q/U/ LQ determined by our choices of M2(Zq) r Bq. We then define open compact subgroups of (B 0Ao)x by

Ut =At UUtU=U tU Ua ui v VU" Qt p tU1P Qi-, Ut=U1 and VQ-= PUQ.



152 FRED DIAMOND

As usual we omit the subscript Q when Q = 0. Note the inclusions

V c UW c Ut and VQ c Ut c Ut

We let Gt = Ut/VU- t /VQ and GI = U/V C Gt.

We view M as a module for Gt. Next we define spaces of automorphic forms for B which will come into

play. Let S(V) denote (E3,r/ E r (7r /)OV

where II' is the set of infinite-dimensional weight two automorphic representations ir'= Star' of (B ? A)x. Now let

St = HomCGt (M(C), S(V)).

Similarly we define St (respectively, St, St ) by replacing Gt (respectively M, V) with GI (respectively M+, VQ). Note that there is a canonical isomorphism

St m eSt _ ,

The space St is naturally endowed with an action of the abstract Hecke Q

algebra TQ, defined as the polynomial ring over Z in the variables:

* Tr and (r) for primes r not dividing N' ;

* Uq for primes q dividing N I/No.

Similarly, the spaces St, St and St are modules for T = T0. We write T(St T(St), T(St) and T(St) for the images in the corresponding rings of endomorphisms. Note that the action of LQ UtlUt on St naturally factors

through T(S$). We also find that the action of T on SI commutes with that of AP?p - Ut/Ut, so we may regard the decomposition St '- GeSt as one of

T(St)-modules which induces natural surjections T(SI) - T(St) for each 'b.

Recall from [8] that the reason for introducing Kp is to detect those automorphic representations which correspond, in the sense of Jacquet-Langlands, to representations ir such that the local Langlands correspondent o(p1) is of a prescribed type. More precisely, let ir = 0irp be an automorphic representation of GL2 (A) such that r1, is discrete series for all p E P, and let ir' = 0 r be the corresponding automorphic representation of (B 0 A) x. Then ir is in HI if and only if there is a nontrivial homomorphism Kp CrpIut for each p. Thus the Jacquet-Langlands correspondence provides a noncanonical isomorphism

(4.4) 0 Homcut (M(C), (Ir)) ) - (Pr )m Ir' EI' IrErI




of modules for GL2(A??P) (Bo AP,') x with m. > 0 if and only if ir E Il. Taking U$Q invariants we obtain an isomorphism of TQ-modules from which we deduce that the annihilator in TQ of SQ coincides with that of St. This Q yields a canonical isomorphism

T(kS/t) -- T(SQ")

of Z[AQ]-algebras (unless both spaces St and SQ are zero). In the case Q = 0, we may replace M by Mp, to obtain an isomorphism analogous to (4.4) above, so that T(St) T(S,'). Moreover these extend to an isomorphism with t replacing t and b replacing I, so that we obtain a commutative diagram

T(St) T(S4') (4.5) l l

T (St) T (S+')

for each 'b. We write Tt, T1, Tt and Tt for the localization at ? of the corresponding Hecke ring. (We know that Si and therefore S4' and SI are nonzero; we have not yet shown that St and S,' are nonzero.)

Next we construct certain Hecke modules whose properties will enable us to prove suitable translations of Lemmas 3.1, 3.5, and 3.6 in the context of quaternion algebras. These modules are essentially constructed in [8], the main difference being that here we will not use etale cohomology.

Let us first assume #P is odd, so that B is a definite quaternion algebra. For an OF-algebra A we let S(V, A) denote the set of functions

BX\Bx/V -- A.

Recall that the double coset space BX\Bx/V is finite and that S(VA) is naturally endowed with an action of the abstract Hecke algebra T. The action of T commutes with the natural action of Gt defined by right multiplication. We define

Lt (A) = HomAGt (M(A), S(V, A))

which we view as a T via the Hecke action on S(V, A). Our introduction of the auxiliary prime R ensures that Bx nxUtx-1 is trivial for all x in Bx and hence that Gt acts without fixed points on BX \B x/V. It follows that S(V, OF) is a free OFGt-module and that Lt (OF) (goF A - Lt (A) is a free A-module for any A. We similarly define T-modules Lt (A) (respectively, LI(A)) by replacing M with Mp (respectively, Gt with GI), as well as the TQ-module LQ(A) by replacing V with VQ.

Recall that there is a natural surjection S(V, C) -- S(V) whose kernel I(V) is the set of functions BX \B x /V -- C which factor through det. The surjection respects the action of T and Gt. Note that HOmCGt (M(C), I(V)) = 0.



154 FRED DIAMOND

This holds since the CGt-module I(V) decomposes as a sum of one-dimensional spaces on which Gt acts via a character which factors through det. So the assertion is obvious from the irreducibility of M(C) unless each Kp is one- dimensional. But in that case we use that the restriction of r, to 2is given

-~~~~~~~~~~~~J - Wetushv by Up composed with the Artin map and Up $ on U1 f Z_2 We thus have natural isomorphisms of T-modules

Lt (C) St Lt (C) St

L (C) St.

Therefore the T-action on Lt(OF) (respectively, Lt (OF), Lt(OF)) makes it a faithful module for T(St) (respectively, T(St), T(St)). Similarly Lt (C) St as a TQ-module so that Lt (OF) becomes a faithful T(St )-module. We use subscript (#) to denote localization at ?, and let Ao = OF,(e) = OF 0 Z(e). We let Lt the free Z(e)-, faithful Tt-module Lt(Ao) 2 Lt(OF) 0 Z(e) and we similarly define 4t, LP and Lt . The natural maps LP Lt are Tt-linear.

Before defining the Hecke modules in the case of indefinite B, we recall the definitions of the relevant Shimura curves. Much of the notation will follow Hida [10], where the reader can find the definitions of the Hecke actions on cohomology groups in terms of double coset operators (see [10, Section 7] and [17, Chapter 8]).

Recall that we assume P 7 0, so B is a quaternion algebra. We fix an isomorphism B 0 R rvA M2(R) and let UOO be the stabilizer of i in GLj+(R) under its usual action on the upper half complex plane Z. For an open compact subgroup U of Bx, we let X(U) denote the manifold

BX\(B 0 A)X/UU0.

When det(U) = Z, X(U) is connected and can be viewed as a Riemann sur- face via the usual isomorphism with r\Z where r = BX n GL+(R)U. Since det(Ut) = det(U~t) = Z, this is the case for U = Ut, UI, Ut and Ut and we denote the corresponding discrete subgroups of B x by rt, rt, rt and rt Q Q- Moreover our introduction of the auxiliary prime R ensures that these groups are torsion-free and hence can be identified with the fundamental groups of the corresponding surfaces. We also have natural etale coverings

X(VQ) X(V) {Gt jGt

X(Ut) X(Ut)

AQ P ,14 . | . 1




the horizontal covers are Galois with the indicated transformation groups act- ing naturally on the right.

For an OF-algebra A, we define a sheaf

Mt (A) = Bx \Mt(A) x (B 0 A) x /Ut Uoc

on X(Ut) where Mt(A) denotes HomA(M(A), A) with its natural right Ut- action. The cohomology group

Lt(A) = Hl(X(Ut),Mt(A))

is endowed with a natural action of T. and is canonically isomorphic to

Hl(Ft, M(A))

where rt acts via its inclusion in Ut. Similarly we define T-modules Lt (A) and Lt(A) and the TQ-module Lt (A).

It is simplest to present the following lemma in much more generality than will be needed. Recall that AO denotes the localization of OF at e.

LEMMA 4.2. Let r be any one of the groups rt, rF, rt or rt. Let N be any one of M(Ao), Mfp(Ao), M(Ao) or Mg,(Ao) viewed as a r-module via IF - Ut. Then H1(r,N) is torsion-free over Z(e) and HZ(r,N) = 0 for i :$ 1. If A is an Ao-algebra, then the natural map

Hz(r,N) (?Ao A -* Hz(r,N (AO A)

is an isomorphism.

Proof. We first check that Ho (Ut4 IMP) = 0 where denotes reduction mod A. If Kp is one-dimensional this is clear since 7Kp is nontrivial on Zx2 nf ut If Kp is p-dimensional, then we have

Ut Uti Ut ResjMp c ResjIndu + rV InduP ResuJp+L

where U-=H + 1P2mP+3Z 2 is the maximal subgroup of Up+ of t-power index. Hence

H0(UtMP) c H?(Up, ) =

since -gp+ is nontrivial on Zx n up-U Next note that we may replace (p by pbOp for any a in Gal(F/Q) and any

4'p of i-power order. Since the image of r in Ut contains U , we conclude that H0(r, N 0 k') = Ho(I, N O k') = 0, where k' -Ao/ and I is the product of the maximal ideals of Ao. Since r is the fundamental group of a two-dimensional orientable manifold, we conclude that H(, N 0 k') = 0 for all i 5$ 1 (see [2, VIII.10], for example). It follows that Hz(r, N) 0 for all i :$ 1 and that



156 FRED DIAMOND

H1 (F, N) is torsion-free over Z(e) and hence flat over AO. The last assertion of the lemma then follows from the Kunneth formula, for example. C1

The Eichler-Shimura isomorphism H1(X(V), C) S 5(V) G S(V) is compatible with the actions of T and Gt. We deduce from this an isomorphism Lt (C) -_ St (0 St of T-modules. Similarly we have isomorphisms Lt(C) St ? St and Lt (C) St S St of T-modules and an isomorphism Lt (C) St (S0 St of TQ-modules. Thus the natural T action makes the free Z(e)-module Lt (Ao) into a faithful Tt-module which we again denote Lt. Note that for an AO-algebra A, we have a natural isomorphism Lt (A) - Lt0AOA as Tt-modules. We define Lt4 Lt and Lt similarly and note that the analogous assertion holds.

Lemma 3.1 is immediate from the following which is a slight strengthening of a special case of [8, Theorem 9]. It is similar to a result of Carayol [4, Lemme 1] (see also [6, Lemma 2.2]).

LEMMA 4.3. If a: T- F is a homomorphism, then a factors through gt: Tt t Tt.

Proof. Let m denote the kernel of a. Then m contains the annihilator of St for some 4, hence m contains the annihilator of Lt for some 0. Therefore m is in the support of

Lt 0AO k' Lt (k') Lt(k') where k' = OF/ (OF. (The first isomorphism uses Lemma 4.2 in the indefinite case; the second is obvious.) Reversing the steps we conclude that m contains the annihilator of St. C1

Remark 4.4. We now know that Tt is nonzero, hence so are TQ and

Q. We shall use the following lemma to deduce Lemma 3.6. The proof is

inspired by a method of Ribet [15].

LEMMA 4.5. Let at denote the kernel of Tt -+ Tt, let bt = AnnTtat and let C = rHp(p + 1). The finitely generated faithful Tt-module Lt is free over Z(e) and satisfies btLt C CLt.

Proof. We consider the natural Tt-equivariant maps

Lt (A) r Lt (A) Lt (A) 1 Lt (A)

for Ao-algebras A. These are defined by the restriction and trace maps on sheaf cohomology and are compatible with the restriction and transfer maps on group cohomology.




We claim that resA is infective and trA is surjective. In the case of definite B the infectivity is obvious and the surjectivity is immediate from the fact that S(V, A) is a free AGt-module. In the case of indefinite B, the infectivity of resA follows from the Hochshild-Serre spectral sequence and the vanishing of HO(rF, M(A)) established in Lemma 4.2. By duality (see [2, VIII.10]), the surjectivity of trA is equivalent to that of the natural map

H1 (rt M(A)) - Hi (rFt M(A)).

But this follows from the Hochschild-Serre spectral sequence for homology and the vanishing of Ho (rt, M(A)) H2(Ft M(A)).

We omit the subscript A for A = AO. Note that res has torsion-free coker- nel by Lemma 4.2 and the injectivity of resk/, and that tr o res is multiplication by #Z\p. Therefore tr induces a Tt-equivariant isomorphism

Lt/(im(res) (0 ker(tr)) - Lt/#z\pLt. Note also that under the decomposition Lt(F) '_ 0+pLt (F), we have

ker(trF) = C00+1Lp(F). By Lemma 3.2 and (4.5), we have that bt annihilates ker(trF) and we conclude that btLt C im(res). It follows that btLt is contained in #/LpLt = CLt.

To deduce Lemma 3.6, first note that in Lemma 4.5 we can replace Tt (respectively, Tt) with T(Sb') 0 Z(e) (respectively, T(Sb') 0 Z(e)). We can then tensor with 0 and localize at m' to produce a finitely generated faithful Tb'-module L, free over 0, such that b'L C CL where

b/ = 95/(AnnV/(kerg5/ ) .

It follows from (4.3) that L is also a finitely generated faithful Tb-module satisfying bL C CL where

b = gb (AnnT(kerg~'))

in the notation of Lemma 3.6. Let p denote the kernel of ir: T- 0. Then

ir(b)(L 0Tb X 0) C C(L Tb,7 0).

Since T' is reduced, we have L KTb X K L. which is nonzero since L is faithful. It follows that ir(b) C Co.

Finally we shall deduce Lemma 3.5 from the following lemma, whose proof in the indefinite case is based on an alternate proof of [18, Proposition 1] suggested by Faltings.

LEMMA 4.6. The finitely generated faithful Tt -module Lt is free over Z(e) [AQ] a

Proof. In the case of definite B, this is immediate from the observation that S(VQ, OF) is free over Z [Gt x AQ]. In the case of indefinite B we have



158 FRED DIAMOND

already proved that Lt is free over Z(e) in Lemma 4.2. Therefore (for example by [2, VI(8.7)] which holds with Z(e) replacing Z), it suffices to check that

HZ(1GQ L) = 0 for i > 0. But this is immediate from Lemma 4.2 and the Hochschild-Serre spectral sequence. C1

To prove Lemma 3.5, note that we can replace Tt by T(SQ) 0 Z(e) Then LQ = (Lt 09 0)m/ is a finitely generated faithful TQ-module which is free over 0[AQ]. We therefore have a sequence of homomorphisms of [L[AQ]-modules

0[AQ] -t T' -* Endo[A,]LQ.

Note that the composite remains injective when we tensor over O[[AQ] with O[AQ]/I for any ideal I of O[L\Q]. Therefore

O[AQ]/I -4 T" /IT" *~~~~~~~~~~~ *Q

is injective.

5. Isomorphisms

We now prove the main results. The methods are essentially those of Wiles and Taylor [19], [18], but following a suggestion of Faltings, we combine the methods of Chapter 3 of [19] and Section 3 of [18] to give a more direct proof that the minimal deformation ring and Hecke ring are isomorphic complete intersections. Faltings' argument is explained in the appendix of [18], but we shall give a complete proof here since several modifications are needed to adapt the method to our context.

THEOREM 5.1. RA TO T' is a complete intersection.

Proof. Fix an integer s and a set Qm for each integer m satisfying the conditions in Lemma 2.3. Let x1, X2, .. ., x, be elements of n' which generate k as an 0-algebra. For each m > 1, choose preimages xm,1, Xm,2, ... , xms in n' . As R'm /AR'Qm - R'/ARk induces an isomorphism of tangent spaces, the {Xmi} generate R'm as an 0-algebra. Choose also generators 6m,1.i.. , 6m,s for \Qm. Let A denote the power series ring 0 [[X1, X2,. . ., X8]] and let B denote 0[[Y1,Y2, . .,Y]]. Define A -* Rk by Xi -* xmi, define B -( O[/\Qm] by

Yi --mi -1, and regard Rk, TO, Rk and Tm as C = A' 0B-algebras. Now define:

* = TO/AnTO for n > 1;

* Tmn = To m/mngTm for m > n > 1;

* Ri = R'/ (A, nbker(Rk -4 Ti));




* n = im(Rm - Tmn GD R1) for m > n > 1,

where the maps - , T1 and k Tm,n are induced by q q$0 and q5Qm.

For m > n > 1, we now have a sequence Sm,n of surjective C-algebra homomorphisms

OMnT hmwn Rm,n Imn )Tn4T

where qm,n is projection onto the first coordinate, and hm,n is induced by hQ. Moreover the natural projection maps for n > n' > 1 yield a commutative diagram of surjective homomorphisms of C-algebras

Rmn ) Tmn ) Tn

Rm,n' ) Tm,ni ) Tn.

Note that by Lemma 3.4, the order of Tm,n is bounded independently of m, hence so is that of Rm,n since R1 is finite. Denote the bound D(n) and observe that for each n there are only finitely many isomorphism classes of sequences of surjective C-algebra morphisms

U *V -*Tn with #U < D(n). We view two such sequences as isomorphic if there are C-algebra isomorphisms U -- U' and V -* V' such that the diagram

U-* V Tn

1 I 11 U' - V' )n

commutes. Thus for each n there are only finitely many isomorphism classes of sequences Smn for m > n. Note also that Sm,1 is naturally isomorphic to

Rq5 T = T,

where q1 is induced by q. Beginning with any integer m(l) > 1, inductively define a sequence m(n)

with the properties:

* m(n) > n;

* Sm(n),n-l Sm(n-1),n-1;

* Sm(n),n Sm,n for infinitely many m > n.

This gives for each n > 2 a commutative diagram of C-algebra homomorphisms

Rm(n),n Tm(n),n Tn

I I I Rm(n-1),n-1 4 Tm(n-1),n-1 Tn-1.



160 FRED DIAMOND

Taking inverse limits with respect to the vertical maps we obtain a sequence of surjective C-algebra homomorphisms

Roo ? Too o TO.

As A is complete and A -? Rm(n),n is surjective for all n, we see that A - Ro is surjective. By Lemma 3.4, TOO is finitely generated as a B-module. Next note that the kernel of B 0' OIQm(n),n] is contained in mn. Applying Lemma 3.5, we see that the map B/mng - Tm(n)n is injective and hence so is B -* To. Thus Too has Krull dimension s + 1 and

A Roo Too.

By Lemma 3.3, the map hm,n induces an isomorphism

Tm(n),n/JBTm(n),n A Tn

for each n where JB = (Y1, Y2,... , YS)B and it follows that hoo induces an isomorphism

TOO/JBTOO A TO.

We have now proved that TO is a complete intersection in the sense that it is isomorphic to A/(f, ... , f8) for some power series f, . . . , fS in A. As TO is finitely generated as an 0-module, [12, Theorem 17.4] shows that

fi, I .* I f8 A

is a regular A-sequence. Therefore TO is torsion-free over 0 and g: TO T-

is an isomorphism by Lemma 3.7. Finally, consider the commutative diagram

R00 ) *00 1 1 ft1 0 T1,

where the vertical arrows are surjections defined by the isomorphism of Sm(l),, with R1 -T = T1. Since 0o is an isomorphism, mg is in the kernel of

Roo - R1 and T00/mBT,0 A T1 by Lemma 3.3, we conclude that q1 is an isomorphism. Since TO is torsion-free, it follows that q induces an isomorphism k --b TO. E

Now that we have identified the "minimal" universal deformation ring as a Hecke ring, we proceed as in Chapter 2 of [19] to relax the restrictions on the deformations.

THEOREM 5.2. R A T is a complete intersection.




Proof. We may assume that F contains the eigenvalues of all eigenforms in S2(]F) for T(S2(rP)). Hence there is an 0-algebra homomorphism

7rI: TI -* 0(

Let IT denote the kernel and let PT (respectively, OR, PR ) denote the preimage of P4 in T (respectively, R, R1). Let

rb =b7rI(AnnT P4) and = 7r(AnnTPT)

where 7r = orbo g1. Recall from [19, Chapter 2] that T is Gorenstein and 7r - 0. We also know that TI is Gorenstein since it is a complete intersection. Hence Lemma 3.6 implies that

71 = (?7I)(7r(AnnT(kerg'))) C Crb

where C = Hpep(p + 1).

Therefore #(0/7i) ? #(01/C)#(0/r17). By Wiles' isomorphism criterion [19, Appendix], it suffices to check that

#(p2p) < #(1CIC)#(PI/(PI )2.

Consider the (9[GQ]-module MO = ado(a) (o (K/0) where

a = Po 0VTb o 0: GQ -* GL2(0)

As in [19, Proposition 1.1], we have isomorphisms

Homo(PR/ , K/0) H p(GQ MO) Homo(p' /(b' )2, KIO) H1 (GQ, MO)

where H (GQ, MO) was defined following Lemma 2.4. Thus we have only to check that

#H (IpMO)Dp/Ip < #(O/(p + 1)0)

for each p E P and positive integer n. By local Tate duality and triviality of the Euler characteristic, this group has the same order as H0(Dp, (Mno)*). Note that alDp is induced from a character (: H -+ Ox where H is the subgroup of index two in Dp which contains Ip and (/(' mod A is ramified. Thus ado (CT) IDp is given by Ind DP (/') E E where E is the unramified quadratic character of DP. Writing On for (9/AnC, we now have that

H? (Dp, (IndHp en ((/() )*)- H? (H. O~n (E/( ) ) vanishes and H0(DP, (9n(E)*) is isomorphic to (9/(p + 1)0 for large n since EE(Frobp) = -p. E

Next we compare our deformation rings to those considered by Wiles. We first show that R0 can be identified with the deformation ring denoted RDO in

[19] where

Do = (.,l3U{e},fl3-P)



162 FRED DIAMOND

with = Se if e divides N0 and = f otherwise. The only differences between the two deformation problems are that for R0 there is a global condition on the determinant, and that the local conditions at M in the definition of R-D are sometimes stronger than those for R0. Observe first that the determinant condition implies the stronger local conditions, so that there is a canonical surjective homomorphism R-D -- R0 ' T. To prove that it is an isomorphism we may assume that 0 is sufficiently large that there is a homomorphism 7r: T -* T- 0, and it suffices to check that the groups

Hp (GQI MO) C H1 (GQ, Mn) are equal where Hl,0 (GQ, Mn) is defined as in [19]. But one easily verifies from the definitions that

Hi (GQ, Mn) = Hl,0(GQ, Mn) We can then apply (2.1) together with the observation that H'p(GQ, AV-no/0) is trivial since p -1 mod e for p in P.

We have now shown that RDO -* T is an isomorphism. Unless P0 is both finite and Selmer, we let D = Do. If PO is finite and Selmer, there is one more step required to relate our deformation problems to those considered by Wiles; we must replace Do = (f, E U {e}, (, M - P) with D = (Se, E U {e}, C, M - P). We consider the rings TDO and TY in the notation of [19]. Note that TDO = T and recall that there are natural surjections RD --* T-D T. We now apply Wiles' isomorphism criterion again to check that RD D TD is a complete intersection. Again we may assume that there is a map 7r: T -T -* 0 and let a = P0 ?T (. Write

where 5 is unramified and 4l1e = EolIe. Write 0 -* -* L -* L2 -O 0 be the corresponding filtration of a model L for a. The rings TD and T are Gorenstein and [19, Proposition 2.4] shows that 7T, = cenT where

ce = ((4'-41E)(Frobe) - 1).

(Note that ce = u7r(T2 - (f)( + 1)2) 5 0 where u is a unit in 0 .) To apply Wiles' isomorphism criterion it suffices to check that the index of H1 (De, Mn) in Hse(De, Mn) is at most #(O/ceO). This is essentially done in [19, Proposi- tion 1.9], but we extract the argument. Recall that H1e(De, Mn) is defined as the kernel of the natural map

H1 (De, Mn) - H1 (Ie, Mn/Nn)

where Nn = Homo (L2, L1) 0o (A-n/0/). But the kernel of H1 (De, Mn) H1 (De, Mn/N,) has order

#H?(De, Mn)#H1(De, Nn) _ #H?(De, Mn)#Ho(Di, N*)#((9/AnQ) #H?(De, Mn/Nn)#H0(Dj, Nn) #H?(De, Mn/Nn)




and the kernel of H1(De, Mn/Nn) -) H1(Ie, Mn/Nn) has order

#HO(De, Mn/Nn).

Combined with Lemma 2.4, this yields

[Hse(De, Mn): H (De, Mn)] < #H0(De, Nn)

which we observe is at most #(O/ce(). We have now proved that RpD TO is a complete intersection. Note that

among the deformation problems considered by Wiles, D is the inf with respect to the partial ordering introduced before [19, Theorem 2.17]. We can therefore apply Wiles' result to deduce Theorem 1.1 of the Introduction, i.e., that [19, Conjecture 2.16] holds. The requirement that pI$ 5 0 if POID, is reducible is easily removed in the statement by twisting by a character of finite order (see Remark 2.1). We recall our standard notation and hypotheses in the statement of the following consequence of Theorem 1.1.

THEOREM 5.3. Let f be an odd prime, and let 0 be the ring of integers of a finite extension of Qe with uniformizer A and residue field k. Suppose that

a: Gal(Q/Q) -* GL2(0)

is a continuous representation unramified outside a finite set of primes. Sup- pose also that one of the following holds:

* slDe is associated to an indivisible group and det alI, is cyclotomic; or

* iDe ( ($ ~ ,) nwith 0 unramified, X 0 0 mod A and klie = Xekl1 II!

for some X of finite order and some integer k > 2.

If po = a 0o k is modular, and is absolutely irreducible when restricted to

Gal(Q/Q( (-1)(e-1)/2e)), then a is modular.

We now apply Wiles' method [19, Chapter 5] to conclude:

THEOREM 5.4. Let E be an elliptic curve over Q with semistable reduction at 3. Suppose that either E has semistable reduction at 5 or that

PE,3: GQ -* GL2(F3) is absolutely irreducible when restricted to Gal(Q/Q(V7/=))- Then E is modular.

The proof of Theorem 5.4 closely follows that of [19, Theorem 5.2], so we refer the reader there for more details. One first considers the case of an elliptic curve E with semistable reduction at 3 and PE,3 restricted to Gal(q/Q(N/VC)) absolutely irreducible. Then PE,3 is modular by the Langlands-Tunnell Theo- rem and it follows from Theorem 5.3 that E is modular. If E has semistable reduction at 3 and 5 and PE,5 is irreducible, then Wiles' argument allows us to



164 FRED DIAMOND

apply Theorem 5.3 to PE,5 to conclude E is modular. If both PE,3 and PE,5 are reducible, then E is isogenous to a twist of Xo(50), and hence is modular but not semistable at 5. Finally, we must verify that every elliptic curve E with the following three properties is modular:

* PE,5 is reducible;

* PE,3 is irreducible;

* PE,3 restricted to GQ(f?3) is absolutely reducible.

This is established by the lemma below, whose proof is based on an observation of Taylor and is sketched at the end of [19, Chapter 5]. Write E0 for the elliptic curve defined by

Y2+Xy+y =X3+X2+3X-5.

Cremona has verified that this curve, denoted 338E1 in the tables of [5], is modular. A straightforward calculation shows that EO does indeed have the properties listed above.

LEMMA 5.5. If E is an elliptic curve over Q with the three properties listed above, then E is isogenous over Q to a twist of E0, and hence is modular.

Proof. Consider the modular curve Y = r\Z where Z is the complex upper half plane and F is the group of matrices ( a b ) in SL2(Z) satisfying:

* c-Omod5;

* b-=cO mod3, ora-=d--Omod3.

Then Y is a quotient of r(15)\Z, and it has a model over Q which parametrizes equivalence classes of triples (A, C, S) where A is an elliptic curve, C is a subgroup of A of order 5 and S is a (nonordered) set of two distinct subgroups of A of order 3. Suppose now that E is an elliptic curve with the properties listed above. Since H = GQ(fC/=g) is the kernel of det PE,3, the projective image Of PE,31H has order 2 and that of PE,3 is isomorphic to (Z/2Z)2 C PGL2(F3). Identifying this projective image with a group of permutations of the 4 cyclic subgroups of E of order 3, it follows that each GQ-orbit contains exactly 2 elements. Denote these orbits Si and S2, let Co be a subgroup of E of order 5 defined over Q, and write 7r for the isogeny E - E/CO. Then the equivalence classes of triples

(E, CO, SI), (E, CO, S2), (E/Co, ir(E[5]), 7r(Si)) and (E/Co, 7r(E[5]), 7r(S2))

define rational points of Y, and these points are distinct if E does not have complex multiplication. Since E0 does not have complex multiplication, it suffices to prove that Y has at most 4 rational points.




Let Yo(45) denote the modular curve over Q parametrizing pairs (E, C) where C is a cyclic subgroup of E of order 45. Then Yo(45) is a model for ro(45)\Z and the map

(El C) I , (EIC[31, C[15] /C[311 JC[9] /C[311 E[3] /C[31 })

induces an isomorphism Yo(45)/Wg -* Y over Q, where W9 is the involution defined by

(E, C) I , (E/C[9], (C[5] e E[9])/C[9]).

To prove that Yo(45)/Wg has at most 4 rational points, we consider the com- pactified modular curve Xo(45)/Wg. This curve has genus one and is obtained from Yo(45)/Wg by adjoining 4 rational cusps. The curve is isomorphic to its Jacobian, which is isogenous over Q to Jo(15) as each is isogenous to

Jo(45)/(Wg - I)Jo(45).

Since Jo(15) is an elliptic curve of conductor 15, so is Xo(45)/W9; hence it has at most 8 rational points (appealing again to [5]). It follows that Y has at most 4 rational points. O

D.P.M.M.S., CAMBRIDGE UNIVERSITY, CAMBRIDGE, UK

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(Received December 30, 1994)

(Revised October 16, 1995)



on deformation rings and hecke rings

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