teorema de ferman

8/13/2019 Teorema de Ferman

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Fermats Last Theorem

Henri Darmon([email protected])

Department of MathematicsMcGill University

Montreal, QCCanada H3A 2K6

Fred Diamond([email protected])

D.P.M.M.S.Cambridge University

Cambridge, CB2 1SBUnited Kingdom

Richard Taylor([email protected])

Mathematics InstituteOxford University

24-29 St. GilesOxford, OX1 3LB

United Kingdom

September 9, 2007

The authors would like to give special thanks to N. Boston, K. Buzzard, andB. Conrad for providing so much valuable feedback on earlier versions of thispaper. They are also grateful to A. Agboola, M. Bertolini, B. Edixhoven, J.Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, J. Manohar-mayum, K. Ribet, D. Rohrlich, M. Rosen, R. Schoof, J.-P. Serre, C. Skinner,

D. Thakur, J. Tilouine, J. Tunnell, A. Van der Poorten, and L. Washingtonfor their helpful comments.

Darmon thanks the members of CICMA and of the Quebec-Vermont Num-ber Theory Seminar for many stimulating conversations on the topics of thispaper, particularly in the Spring of 1995. For the same reason Diamond isgrateful to the participants in an informal seminar at Columbia Universityin 1993-94, and Taylor thanks those attending the Oxford Number TheorySeminar in the Fall of 1995.

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Parts of this paper were written while the authors held positions at otherinstitutions: Darmon at Princeton University, Diamond at the Institute forAdvanced Study, and Taylor at Cambridge University. During some of the pe-riod, Diamond enjoyed the hospitality of Princeton University, and Taylor thatof Harvard University and MIT. The writing of this paper was also supportedby research grants from NSERC (Darmon), NSF # DMS 9304580 (Diamond)and by an advanced fellowship from EPSRC (Taylor).

This article owes everything to the ideas of Wiles, and the arguments pre-sented here are fundamentally his [W3], though they include both the work[TW] and several simplifications to the original arguments, most notably thatof Faltings. In the hope of increasing clarity, we have not always statedtheorems in the greatest known generality, concentrating instead on what isneeded for the proof of the Shimura-Taniyama conjecture for semi-stable ellip-tic curves. This article can serve as an introduction to the fundamental papers[W3] and [TW], which the reader is encouraged to consult for a different, andoften more in-depth, perspective on the topics considered. Another usefulmore advanced reference is the article [Di2] which strengthens the methods of[W3] and [TW] to prove that every elliptic curve that is semistable at 3 and 5is modular.

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IntroductionFermats Last Theorem

Fermats Last Theorem states that the equation

xn + yn =zn, xyz = 0has no integer solutions when n is greater than or equal to 3. Around 1630,Pierre de Fermat claimed that he had found a truly wonderful proof of thistheorem, but that the margin of his copy of Diophantus Arithmeticawas too

small to contain it:Cubum autem in duos cubos, aut quadrato quadratum in duosquadrato quadratos, et generaliter nullam in infinitum ultra qua-dratum potestatem in duos ejusdem nominis fas est dividere; cujusrei demonstrationem mirabile sane detexi. Hanc marginis exiguitasnon caperet.

Among the many challenges that Fermat left for posterity, this was to provethe most vexing. A tantalizingly simple problem about whole numbers, itstood unsolved for more than 350 years, until in 1994 Andrew Wiles finally

laid it to rest.Prehistory: The only case of Fermats Last Theorem for which Fermat actu-ally wrote down a proof is for the case n= 4. To do this, Fermat introducedthe idea of infinite descentwhich is still one the main tools in the study ofDiophantine equations, and was to play a central role in the proof of FermatsLast Theorem 350 years later. To prove his Last Theorem for exponent 4, Fer-mat showed something slightly stronger, namely that the equation x4 +y4 =z2

has no solutions in relatively prime integers with xyz= 0. Solutions to suchan equation correspond to rational points on the elliptic curve v2 = u3 4u.Since every integern3 is divisible either by an odd prime or by 4, the resultof Fermat allowed one to reduce the study of Fermats equation to the casewhere n= is an odd prime.

In 1753, Leonhard Euler wrote down a proof of Fermats Last Theorem forthe exponent = 3, by performing what in modern language we would calla 3-descent on the curve x3 +y3 = 1 which is also an elliptic curve. Eulersargument (which seems to have contained a gap) is explained in [Edw], ch. 2,and [Dic1], p. 545.

It took mathematicians almost 100 years after Eulers achievement to han-dle the case = 5; this was settled, more or less simultaneously, by Gustav

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Peter Lejeune Dirichlet [Dir] and Adrien Marie Legendre [Leg] in 1825. Theirelementary arguments are quite involved. (Cf. [Edw], sec. 3.3.)In 1839, Fermats equation for exponent 7 also yielded to elementary meth-

ods, through the heroic efforts of Gabriel Lame. Lames proof was even moreintricate than the proof for exponent 5, and suggested that to go further, newtheoretical insights would be needed.

The work of Sophie Germain: Around 1820, in a letter to Gauss, SophieGermain proved that if is a prime and q= 2 +1 is also prime, then Fermatsequation x +y =z with exponent has no solutions (x,y,z) with xyz= 0(mod ). Germains theorem was the first really general proposition on Fer-

mats Last Theorem, unlike the previous results which considered the Fermatequation one exponent at a time.

The case where the solution (x,y,z) tox+ y =z satisfiesxyz= 0 (mod) was called the first caseof Fermats Last Theorem, and the case where divides xyz, the second case. It was realized at that time that the first casewas generally easier to handle: Germains theorem was extended, using similarideas, to cases wherek +1 is prime and k is small, and this led to a proof thatthere were no first case solutions to Fermats equation with prime exponents100, which in 1830 represented a significant advance. The division betweenfirst and second case remained fundamental in much of the later work on the

subject. In 1977, Terjanian [Te] proved that if the equation x

2

+ y

2

=z

2

hasa solution (x,y,z), then 2divides eitherxor y, i.e., the first case of FermatsLast Theorem is true for even exponents. His simple and elegant proof usedonly techniques that were available to Germain and her contemporaries.

The work of Kummer: The work of Ernst Eduard Kummer marked thebeginning of a new era in the study of Fermats Last Theorem. For the firsttime, sophisticated concepts of algebraic number theory and the theory ofL-functions were brought to bear on a question that had until then beenaddressed only with elementary methods. While he fell short of providinga complete solution, Kummer made substantial progress. He showed how

Fermats Last Theorem is intimately tied to deep questions on class numbersof cyclotomic fields which are still an active subject of research. Kummersapproach relied on the factorization

(x+ y)(x+ y) (x+ 1 y) =z

of Fermats equation over the ring Z[] generated by the th roots of unity.One observes that the greatest common divisor of any two factors in the prod-uct on the left divides the element (1), which is an element of norm .

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Since the product of these numbers is a perfect -th power, one is tempted toconclude that (x + y), . . . , (x + 1 y) are each-th powers in the ring Z[] upto units in this ring, and up to powers of (1 ). Such an inference would bevalid if one were to replace Z[] byZ, and is a direct consequence ofuniquefactorization of integers into products of primes. We say that a ring R hasproperty UF if every non-zero element ofR is uniquely a product of primes,up to units. Mathematicians such as Lame made attempts at proving Fer-mats Last Theorem based on the mistaken assumption that the rings Z[]had property U F. Legend even has it that Kummer fell into this trap, al-though this story now has been discredited; see for example [Edw], sec. 4.1. Infact, property U F is far from being satisfied in general: one now knows thatthe ringsZ[] have property UFonly for


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where h+

is the class number of the real subfield Q()+

, and h

is defined ash/h+. Essentially because of the units in Q()

+, the factor h+ is somewhatdifficult to compute, while, because the units in Q()

+ generate the group ofunits in Q() up to finite index, the term h

can be expressed in a simple

closed form. Kummer showed that if dividesh+, then dividesh . Hence,

dividesh if and only if dividesh. This allowed one to avoid the difficulties

inherent in the calculation ofh+. Kummer then gave an elegant formula forh

by considering the Bernoulli numbersBn, which are rational numbers definedby the formula

x

ex

1

= Bnn!

xn.

He produced an explicit formula for the class number h, and concluded thatif does not divide the numerator of B2i, for 1 i (3)/2, then isregular, and conversely.

The conceptual explanation for Kummers formula for h lies in the workof Dirichlet on the analytic class number formula, where it is shown that hcan be expressed as a product of special values of certain (abelian) L-series

L(s, ) =n=1

(n)ns

associated to odd Dirichlet characters. Such special values in turn can beexpressed in terms of certain generalized Bernoulli numbers B1,, which arerelated to the Bernoulli numbersBivia congruences mod. (For more details,see [Wa].)

These considerations led Kummer to initiate a deep study relating congru-ence properties of special values ofL-functions and of class numbers, whichwas to emerge as a central concern of modern algebraic number theory, andwas to reappear in a surprisingly different guise at the heart of Wilesstrategy for proving the Shimura-Taniyama conjecture.

Later developments: Kummers work had multiple ramifications, and led

to a very active line of enquiry pursued by many people. His formulae re-lating Bernoulli numbers to class numbers of cyclotomic fields were refinedby Kenneth Ribet [R1], Barry Mazur and Andrew Wiles [MW], using newmethods from the theory of modular curves which also play a central role inWiles more recent work. (Later Francisco Thaine [Th] reproved some of theresults of Mazur and Wiles using techniques inspired directly from a readingof Kummer.) In a development more directly related to Fermats Last Theo-rem, Wieferich proved that if2 does not divide 21 1, then the first caseof Fermats Last Theorem is true for exponent . (Cf. [Ri], lecture VIII.)

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There were many other refinements of similar criteria for Fermats Lasttheorem to be true. Computer calculations based on these criteria led to averification that Fermats Last theorem is true for all odd prime exponents lessthan four million [BCEM], and that the first case is true for all 8.858 1020[Su].

The condition thatis a regular prime seems to hold heuristically for about61% of the primes. (See the discussion on p. 63, and also p. 108, of [Wa], forexample.) In spite of the convincing numerical evidence, it is still not knownif there are infinitely many regular primes. Ironically, it is not too difficult toshow that there are infinitely many irregular primes. (Cf. [Wa].)

Thus the methods introduced by Kummer, after leading to very strongresults in the direction of Fermats Last theorem, seemed to become mired indifficulties, and ultimately fell short of solving Fermats conundrum1.

Faltings proof of the Mordell conjecture: In 1985, Gerd Faltings [Fa]proved the very general statement (which had previously been conjecturedby Mordell) that any equation in two variables corresponding to a curve ofgenus strictly greater than one had (at most) finitely many rational solutions.In the context of Fermats Last Theorem, this led to the proof that for eachexponent n3, the Fermat equation xn +yn =zn has at most finitely manyinteger solutions (up to the obvious rescaling). Andrew Granville [Gra] and

Roger Heath-Brown [HB] remarked that Faltings result implies Fermats LastTheorem for a set of exponents of density one.However, Fermats Last Theorem was still not known to be true for an

infinite set of prime exponents. In fact, the theorem of Faltings seemed ill-equipped for dealing with the finer questions raised by Fermat in his margin,namely of finding a complete list of rational points onal lof the Fermat curvesxn +yn = 1 simultaneously, and showing that there are no solutions on thesecurves when n3 except the obvious ones.Mazurs work on Diophantine properties of modular curves: Althoughit was not realized at the time, the chain of ideas that was to lead to a proof

of Fermats Last theorem had already been set in motion by Barry Mazurin the mid seventies. The modular curves X0() and X1() introduced insection 1.2 and 1.5 give rise to another naturally occurring infinite familyof Diophantine equations. These equations have certain systematic rationalsolutions corresponding to the cusps that are defined over Q, and are analogous

1However, W. McCallum has recently introduced a technique, based on the methodof Chabauty and Coleman, which suggests new directions for approaching Fermats LastTheorem via the cyclotomic theory. An application of McCallums method to showing thesecondcase of Fermats Last Theorem for regular primes is explained in [Mc].

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to the so-called trivial solutions of Fermats equation. Replacing Fermatcurves by modular curves, one could ask for a complete list of all the rationalpoints on the curves X0() and X1(). This problem is perhaps even morecompelling than Fermats Last Theorem: rational points on modular curvescorrespond to objects with natural geometric and arithmetic interest, namely,elliptic curves with cyclic subgroups or points of order . In [Maz1] and [Maz2],B. Mazur gave essentially a complete answer to the analogue of Fermats LastTheorem for modular curves. More precisely, he showed that if = 2, 3, 5and 7, (i.e., X1() has genus>0) then the curve X1() has no rational pointsother than the trivial ones, namely cusps. He proved analogous results forthe curvesX0() in [Maz2], which implied, in particular, that an elliptic curveover Q with square-free conductor has no rational cyclic subgroup of order overQ if is a prime which is strictly greater than 7. This result appeared afull ten years before Faltings proof of the Mordell conjecture.

Freys strategy: In 1986, Gerhard Frey had the insight that these construc-tions might provide a precise link between Fermats Last Theorem and deepquestions in the theory of elliptic curves, most notably the Shimura Taniyamaconjecture. Given a solution a +b = c to the Fermat equation of primedegree , we may assume without loss of generality that a 1 (mod 4) andthat b 0 (mod 32). Frey considered (following Hellegouarch, [He], p. 262;cf. also Kubert-Lang [KL], ch. 8,2) the elliptic curve

E :y2 =x(x a)(x+ b).

This curve issemistable, i.e., it has square-free conductor. Let E[] denote thegroup of points of order on Edefined over some (fixed) algebraic closure QofQ, and let L denote the smallest number field over which these points aredefined. This extension appears as a natural generalization of the cyclotomicfields Q() studied by Kummer. What singles out the field L for specialattention is that it hasvery little ramification: using Tates analytic descriptionofEat the primes dividingabc, it could be shown thatLwas ramified only at 2

and, and that the ramification ofL at these two primes was rather restricted.(See theorem 2.15 of section 2.2 for a precise statement.) Moreover, the resultsof Mazur on the curve X0() could be used to show that L is large, in thefollowing precise sense. The spaceE[] is a vector space of dimension 2 over thefinite fieldF with elements, and the absolute Galois group GQ = Gal (Q/Q)acts F-linearly onE[]. Choosing anF-basis forE[], the action is describedby a representation

E, : Gal (L/Q)GL2(F).

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Mazurs results in [Maz1] and [Maz2] imply that E, is irreducible if > 7(using the fact that E is semi-stable). In fact, combined with earlier resultsof Serre [Se6], Mazurs results imply that for >7, the representation E, issurjective, so that Gal (L/Q) is actually isomorphic to GL2(F) in this case.

Serres conjectures: In [Se7], Jean-Pierre Serre made a careful study of mod Galois representations : GQ GL2(F) (and, more generally, of repre-sentations into GL2(k), where k is any finite field). He was able to make veryprecise conjectures (see section 3.2) relating these representations to modularforms mod . In the context of the representations E, that occur in Freysconstruction, Serres conjecture predicted that they arose from modular forms

(mod) of weight two and level two. Such modular forms, which correspond todifferentials on the modular curveX0(2), do not exist becauseX0(2) has genus0. Thus Serres conjecture implied Fermats Last Theorem. The link betweenfields with Galois groups contained inGL2(F) and modular forms mod stillappears to be very deep, and Serres conjecture remains a tantalizing openproblem.

Ribets work: lowering the level: The conjecture of Shimura and Taniya-ma (cf. section 1.8) provides a direct link between elliptic curves and modularforms. It predicts that the representation E, obtained from the -divisionpoints of the Frey curve arises from a modular form of weight 2, albeit a form

whose level is quite large. (It is the product of all the primes dividing abc,where a +b =c is the putative solution to Fermats equation.) Ribet [R5]proved that, if this were the case, then E, would also be associated with amodular form mod of weight 2 and level 2, in the way predicted by Serresconjecture. This deep result allowed him to reduce Fermats Last Theorem tothe Shimura-Taniyama conjecture.

Wiles work: proof of the Shimura-Taniyama conjecture: In [W3]Wiles proves the Shimura-Taniyama conjecture for semi-stable elliptic curves,providing the final missing step and proving Fermats Last Theorem. Aftermore than 350 years, the saga of Fermats Last theorem has come to a spec-

tacular end.The relation between Wiles work and Fermats Last Theorem has been

very well documented (see, for example, [R8], and the references containedtherein). Hence this article will focus primarily on the breakthrough of Wiles[W3] and Taylor-Wiles [TW] which leads to the proof of the Shimura-Taniyamaconjecture for semi-stable elliptic curves.

From elliptic curves to -adic representations: Wiles opening gambitfor proving the Shimura-Taniyama conjecture is to view it as part of the more

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general problem of relating two-dimensional Galois representations and mod-ular forms. The Shimura-Taniyama conjecture states that ifE is an ellipticcurve overQ, thenEis modular. One of several equivalent definitions of mod-ularity is that for some integer Nthere is an eigenform f=

anq

n of weighttwo on 0(N) such that

#E(Fp) =p+ 1 apfor all but finitely primes p. (By an eigenform, here we mean a cusp formwhich is a normalized eigenform for the Hecke operators; see section 1 fordefinitions.)

This conjecture acquires a more Galois theoretic flavour when one considersthe two dimensional -adic representation

E, :GQGL2(Z)obtained from the action of GQ on the -adic Tate module of E: TE =lim

E[ln](Q). An -adic representation ofGQ is said to arise from an eigen-

form f=

anqn with integer coefficientsan if

tr ((Frob p)) =ap,

for all but finitely many primes p at which is unramified. Here Frobp is aFrobenius element atp (see section 2), and its image under is a well-definedconjugacy class.

A direct computation shows that #E(Fp) = p+ 1tr (E,(Frobp)) forall primes p at which E, is unramified, so that E is modular (in the sensedefined above) if and only if for some , E, arises from an eigenform. Infact the Shimura-Taniyama conjecture can be generalized to a conjecture thatevery -adic representation, satisfying suitable local conditions, arises from amodular form. Such a conjecture was proposed by Fontaine and Mazur [FM].

Galois groups and modular forms

Viewed in this way, the Shimura-Taniyama conjecture becomes part of a muchlarger picture: the emerging, partly conjectural and partly proven correspon-dence between certain modular forms and two dimensional representationsof GQ. This correspondence, which encompasses the Serre conjectures, theFontaine-Mazur conjecture, and the Langlands program for GL2, represents afirst step toward a higher dimensional, non-abelian generalization of class fieldtheory.

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Two-dimensional representations ofGQ: In the first part of this century,class field theory gave a complete description ofGabQ , the maximal (continu-ous) abelian quotient ofGQ. In fact the Kronecker-Weber theorem asserts thatGabQ

= p Zp , and one obtains a complete description of all one-dimensionalrepresentations ofGQ. In the second half of this century much attention hasfocused on attempts to understand the whole group GQ, or more precisely todescribe all its representations. Although there has been a fair degree of suc-cess in using modular forms to construct representations ofGQ, less is knownabout how exhaustive these constructions are. The major results in the lat-ter direction along these lines are the work of Langlands [Ll2] and the recentwork of Wiles ([W3] completed by [TW]). Both concern two-dimensional rep-resentations ofGQ and give significant evidence that these representations areparametrised (in a very precise sense) by certain modular forms. The purposeof this article is to describe both the proven and conjectural parts of this the-ory, give a fairly detailed exposition of Wiles recent contribution and explainthe application to Fermats Last theorem. To make this description somewhatmore precise let us distinguish three types of representation.

Artin representations and the Langlands-Tunnell theorem: Contin-uous representations : GQ GL2(C) are called (two-dimensional) Artinrepresentations. Such representations necessarily have finite image, and are

therefore semi-simple. We restrict our attention to those which are irreducible.They are conjectured to be in bijection (in a precise way) with certain new-forms (a special class of eigenforms). Those which are odd (i.e. the deter-minant of complex conjugation is1), should correspond to weight 1 holo-morphic newforms. Those which are even should correspond to certain non-holomorphic (Maass) newforms. Two partial but deep results are known.

(a) (Deligne-Serre) Iff is a holomorphic weight one newform then the cor-responding Artin representation can be constructed ([DS]).

(b) (Langlands-Tunnell) If is a two dimensional Artin representation with

soluble image then the corresponding modular form exists ([Ll2] and[Tu]).

The proof of the latter result is analytic in nature, invoking the trace formulaand the theory ofL-functions.

-adic representations and the Fontaine-Mazur conjecture: By an -adic representation we shall mean any continuous representation : GQGL2(K) which is unramified outside a finite set of primes and where K is afinite extension ofQ (generalizing slightly the notion of-adic representation

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for some0, the0-adic representationE,0 is modular. In particular it sufficesto verify that either E,3 or E,5 is modular. Hence the Shimura-Taniyamaconjecture can be reduced to (part of) the Fontaine-Mazur conjecture for = 3and 5. We have seen that for these primes part of Serres conjecture is known,so it turns out it suffices to prove results of the form Serres conjecture for implies the Fontaine-Mazur conjecture for . This is the direction of Wileswork, although nothing quite this general has been proven yet.

Deformation theory: Thus the problem Wiles faces is to show that if isan odd -adic representation which has irreducible modular reduction andwhich is sufficiently well behaved when restricted to the decomposition group

at , then is modular. In fact he only proves a weakened version of such aresult, but one which is sufficient to conclude that all semistable elliptic curvesare modular.

Wiles approaches the problem by putting it in a more general setting. Onthe one hand he considers lifts of to representations over complete noetherianlocal Z-algebras R. For each finite set of primes , one can consider lifts oftype ; these are lifts which are well-behaved on a decomposition group at ,and whose ramification at primes not in is rather restricted. In particular,such a lift is unramified outside Swhere S is the set of ramified primesof . A method of Mazur (see [Maz3]) can then be used to show that if is

absolutely irreducible, then there is a representation

univ :GQGL2(R)

which is universal in the following sense. If: GQGL2(R) is a lift of oftype , then there is a unique local homomorphism RR such that isequivalent to the pushforward ofuniv . Thus the equivalence classes of type lifts toGL2(R) can beidentifiedwith Hom(R, R). The local ringRis calledthe universal deformation ringfor representations of type .

On the other hand Wiles constructs a candidate for a universal modularlifting of type

mod :GQGL2(T).The ring T is constructed from the algebra of Hecke operators acting ona certain space of modular forms. The universal property of R gives amap R T. The problem thus becomes: to show that this map is anisomorphism2. In fact, it can be shown to be a surjection without great dif-

2Maps of this kind were already considered in [Maz3] and [BM], and it is conjectured in[MT] that these maps are isomorphisms in certain cases, though not in exactly the situationsconsidered by Wiles.

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ficulty, and the real challenge is to prove injectivity, i.e., to show, in essence,that R is not larger than T.By an ingenious piece of commutative algebra, Wiles found a numerical

criterion for this map to be an isomorphism, and for the ring T to be alocal complete intersection. This numerical criterion seems to be very closeto a special case of the Bloch-Kato conjecture [BK]. Wiles further showed(by combining arguments from Galois cohomology and from the theory ofcongruences between modular forms) that this numerical criterion was satisfiedif the minimal versionTof this Hecke algebra (obtained by taking =, i.e.,allowing the least possible amount of ramification in the deformations) was acomplete intersection. Finally in [TW] it was proved thatT is a completeintersection.

Outline of the paper

Chapter 1 recalls some basic notions from the classical theory of elliptic curvesand modular forms, such as modular forms and modular curves over C and Q,Hecke operators and q-expansions, and Eichler-Shimura theory. The Shimura-Taniyama conjecture is stated precisely in section 1.8.

Chapter 2 introduces the basic theory of representations ofGQ. We describe

Mazurs deformation theory and begin our study of the universal deformationrings using techniques from Galois cohomology and from the theory of finiteflat group schemes. We also recall some basic properties of elliptic curves,both to explain Freys argument precisely and illustrate the uses of -adicrepresentations.

Chapter 3 explains how to associate Galois representations to modularforms. We then describe what was known and conjectured about associatingmodular forms to Galois representations before Wiles work. After introducingthe universal modular lifts of certain mod representations, we give the proofof Wiles main theorems, taking for granted certain results of a more technicalnature that are proved in the last two chapters.

Chapter 4 explains how to prove the necessary results concerning the struc-ture of Hecke algebras: the generalization by Taylor and Wiles of a result ofde Shalit, and the generalization by Wiles of a result of Ribet.

Chapter 5 establishes the fundamental results from commutative algebradiscovered by Wiles, following modifications of the approach of Wiles andTaylor-Wiles proposed by Faltings and Lenstra.

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Contents1 Elliptic curves and modular forms 16

1.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2 Modular curves and modular forms overC . . . . . . . . . . . 221.3 Hecke operators and Hecke theory . . . . . . . . . . . . . . . . 281.4 The L-function associated to a cusp form . . . . . . . . . . . . 331.5 Modular curves and modular forms overQ . . . . . . . . . . . 341.6 The Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . 391.7 The Shimura construction . . . . . . . . . . . . . . . . . . . . 441.8 The Shimura-Taniyama conjecture . . . . . . . . . . . . . . . 47

2 Galois theory 502.1 Galois representations . . . . . . . . . . . . . . . . . . . . . . 502.2 Representations associated to elliptic curves . . . . . . . . . . 552.3 Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . 592.4 Representations ofGQ . . . . . . . . . . . . . . . . . . . . . . 622.5 The theory of Fontaine and Laffaille . . . . . . . . . . . . . . . 692.6 Deformations of representations . . . . . . . . . . . . . . . . . 732.7 Deformations of Galois representations . . . . . . . . . . . . . 762.8 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Modular forms and Galois representations 843.1 From modular forms to Galois representations . . . . . . . . . 843.2 From Galois representations to modular forms . . . . . . . . . 883.3 Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 933.4 Isomorphism criteria . . . . . . . . . . . . . . . . . . . . . . . 983.5 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 1003.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4 Hecke algebras 1064.1 Full Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2 Reduced Hecke algebras . . . . . . . . . . . . . . . . . . . . . 1114.3 Proof of theorem 3.31 . . . . . . . . . . . . . . . . . . . . . . . 1194.4 Proof of theorem 3.36 . . . . . . . . . . . . . . . . . . . . . . . 1244.5 Homological results . . . . . . . . . . . . . . . . . . . . . . . . 133

5 Commutative algebra 1375.1 Wiles numerical criterion . . . . . . . . . . . . . . . . . . . . 1385.2 Basic properties of A and A . . . . . . . . . . . . . . . . . . 140

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5.3 Complete intersections and the Gorenstein condition . . . . . 1435.4 The Congruence ideal for complete intersections . . . . . . . . 1485.5 Isomorphism theorems . . . . . . . . . . . . . . . . . . . . . . 1495.6 A resolution lemma . . . . . . . . . . . . . . . . . . . . . . . . 1525.7 A criterion for complete intersections . . . . . . . . . . . . . . 1535.8 Proof of Wiles numerical criterion . . . . . . . . . . . . . . . 1535.9 A reduction to characteristic . . . . . . . . . . . . . . . . . . 1545.10 J-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

1 Elliptic curves and modular forms

1.1 Elliptic curves

We begin with a brief review of elliptic curves. A general reference for theresults discussed in this section is [Si1] and [Si2].

An elliptic curveEover a field Fis a proper smooth curve over Fof genusone with a distinguishedF-rational point. IfE /Fis an elliptic curve and ifis a non-zero holomorphic differential on E/F then E can be realised in theprojective plane by an equation (called a Weierstrass equation) of the form

(W) Y2Z+ a1XY Z+ a3Y Z2 =X3 + a2X

2Z+ a4XZ2 + a6Z

3

such that the distinguished point is (0 : 1 : 0) (sometimes denotedbecauseit corresponds to the point at infinity in the affine model obtained by settingZ= 1) and = dx

2y+a1x+a3. We also define the following quantities associated

to (W):b2 =a

21+ 4a2 b4 = 2a4+ a1a3 b6 = a

23+ 4a6

b8 =a21a6+ 4a2a6 a1a3a4+ a2a23 a24

= 9b2b4b6 b22b8 8b34 27b26j = (b22 24b4)3/.

One can check that the equation (W) defines an elliptic curve if and only if

is nonzero. One can also check that such equations define elliptic curveswhich are isomorphic over F if and only if they give the same quantityj . Thusj only depends on E so we will denote it jE. The quantity depends onlyon the pair (E, ) so we shall denote it (E, ). If u belongs to F thenu12(E,u) = (E, ).

An elliptic curve E/Fhas a natural structure of a commutative algebraicgroup with the distinguished F-rational point as the identity element.

An algebraic map between two elliptic curves which sends the distinguishedpoint of one to the distinguished point of the other is automatically a morphism

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of algebraic groups. A map between elliptic curves which has finite kernel (andhence, is generically surjective) is called an isogeny.

Elliptic curves over C: If F = C, then the curve E is isomorphic as acomplex analytic manifold to the complex torus C/, where is a latticein C, i.e., a discrete Z-submodule ofC of rank 2. The group law on E(C)corresponds to the usual addition in C/. In terms of , an affine equationfor E in A2(C) is given by

y2 = 4x3 + g2x+ g3,

where

g2=60 {0}

1z4

, g3=140 {0}

1z6

.

In terms of this equation, the map from C/ to E(C) sends z to (x, y) =((z), (z)), where(z) is theWeierstrass-functionassociated to the lattice. (Cf. [Si1], ch. VI.) The inverse map is given by integrating the holomorphicdifferential, i.e., sending P E(C) to the image of

inC/, where is

any path on E(C) fromto P, and is the lattice of periods , where

ranges over the integral homology H1(E(C),Z). Replacing byu changes to u, so that is determined byEonly up to homotheties. We scale sothat one of its Z-generators is 1, and another,, has strictly positive imaginarypart. This gives the analytic isomorphism:

E(C)C/1, .The complex number in the complex upper half planeH is well defined,modulo the natural action ofS L2(Z) onHby Mobius transformations. (Thusthe set of isomorphism classes of elliptic curves over C can be identifiedwiththe quotientH/SL2(Z).)

The map z e2iz identifies C/1, with C/qZ, where q= e2i is themultiplicative Tate period. The analytic isomorphism

E(C)C/qZhas the virtue of generalizing to the p-adic setting in certain cases, as we willsee shortly.

Note that|q| < 1. The invariant j can be expressed in terms of q by aconvergent power series with integer coefficients:

j=q1 + 744 + 196884q+ . (1.1.1)The following basic facts are a direct consequence of the analytic theory:

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Proposition 1.1 The subgroup E[n](C) of points of ordern onE(C) is iso-morphic (non-canonically) to Z/nZZ/nZ. More generally, ifF is any fieldof characteristic zero, the subgroup E[n](F) is contained inZ/nZ Z/nZ.Proof: The analytic theory shows thatE(C) is isomorphic as an abstract groupto a product of two circle groups, and the first statement follows. The secondstatement follows from the Lefschetz principle (cf. [Si1], ch. VI,6).

Proposition 1.2 The endomorphism ringEnd C(E) of an elliptic curve overCis isomorphic either to Z or to an order in a quadratic imaginary field. Thesame is true if one replacesC by any field of characteristic0.

Proof: An endomorphism ofE(C)C/ induces multiplication by complexnumber on the tangent space. Hence EndC(E) is isomorphic to the ringof C satisfying . Such a ring is isomorphic either to Z or to aquadratic imaginary order. The corresponding statement for fields of charac-teristic 0 follows as in the proof of proposition 1.1.

If End C(E) Qis a quadratic imaginary field, we say that Ehas complexmultiplication.

Remark 1.3 It follows from the arithmetic theory of complex multiplication(cf. [Si2], ch. 1) that any elliptic curve Ewith complex multiplication is defined

over an abelian extension of the quadratic imaginary field K= End C(E) Q.IfE is defined over Q, then Khas class number one. There are only finitelymany elliptic curves over Qwith complex multiplication, up to twists (i.e.,C-isomorphism).

Elliptic curves over Qp: Now suppose thatE is an elliptic curve definedover the p-adic field Qp. There is an equation

(Wmin) Y2Z+ a1XY Z+ a3Y Z2 =X3 + a2X

2Z+ a4XZ2 + a6Z

3

for Ewith the property aiZp for all i and|| is minimal amongst all suchequations for E. Although (Wmin) is not unique, the associated discriminantdepends only on E and is denoted minE . Moreover the reduction of (W

min)modulo the uniformizer p defines a projective curve E, which is independentof the particular minimal equation chosen. If (W) is any equation for Ewithcoefficients in Zp and with discriminant , then

minE divides .

IfEis a smooth curve we say that Ehas good reductionat p. IfEhas aunique singular point which is a node we say thatEhas multiplicative reductionatp. Otherwise Ehas a unique singular point which is a cusp and we say that

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E has additive reductionat p. IfE has good or multiplicative reduction wesay that it has semi-stablereduction at p, or simply that E issemi-stable.If (W) defines a smooth curve mod p then E has good reduction at p

and (W) is a minimal equation. If 0 modp but b22= 24b4mod p, thenmodulo p the equation (W) defines a curve with a node. In this caseE hasmultiplicative reduction at pand (W) is a minimal equation.

Curves with good reduction: In that case p does not divide minE , and thereduction Eis an elliptic curve over Fp.

Ifq is any power ofp, and Fq is the field with q elements, we define theintegerNq to be the number of solutions to the equation (W

min) in the projec-

tive plane P2

(Fq). Thus Nq is the order of the finite group E(Fq). We definethe integer aq by the formula

aq =q+ 1 Nq.

The integers aq are completely determined by ap: more precisely, we have

(1 apps +p12s)1 = 1 + apps + ap2p2s + ap3p3s + .(1.1.2)

We call the expression on the left the (local) L-function associated to E

over Qp, and denote it by L(E/Qp, s). Concerning the size ofap we have thefollowing fundamental result of Hasse, whose proof can be found in [Si1], ch.V,1:Theorem 1.4|ap| 2p.

A further division among curves of good reduction plays a significant rolein our later discussion. We say that Ehas (good) ordinaryreduction ifp doesnot divide ap, and that it has supersingularreduction ifpdivides ap.

When Ehas good reduction at p, we define its local conductor at pto bemp(E) = 0.

Curves of multiplicative reduction: Elliptic curves over Qp which have multi-plicative reduction at p can be understood by using the p-adic analytic de-scription discovered by Tate. More precisely, we can formally invert the powerseries (1.1.1) expressing j in terms of q, to obtain to a power series for q inj1, having integercoefficients:

q= j1 + 744j2 + 750420j3 + 872769632j4 + .(1.1.3)

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IfEhas multiplicative reduction, then j Qp is non-integral, and hence thepower series (1.1.3) converges, yielding a well-defined value ofq in pZp. Thisis called Tates p-adic period associated to E over Qp. Note that we havevp(q) =vp(j) =vp(minE ).

We say that E has split (resp. non-split) multiplicative reduction at p ifthe two tangent lines to the node on E(Fp) have slopes defined over Fp (resp.Fp2).

Proposition 1.5 (Tate) There is ap-adic analytic isomorphism

: Qp/qZ E(Qp),

which has the property that

((x)) = (x()), GQp,

where: GQp 1 is the trivial character, ifEhas split multiplicative reduction; the unique unramified quadratic character of GQp, if E has non-split

multiplicative reduction.

The proof of this proposition is explained in [Si2], ch. V, for example.We define the L-function L(E/Qp, s) to be

L(E/Qp, s) =

(1 ps)1 ifEhas split reduction,(1 +ps)1 ifEhas non-split reduction.

(1.1.4)

In both cases the conductor mp(E) is defined to be 1.

Curves of additive reduction: IfEhas additive reduction atp, we simply define

L(E/Qp, s) = 1. (1.1.5)

The conductor mp(E) is defined to be 2, ifp > 3. When p= 2 or 3, it isdetermined by a somewhat more complicated recipe, given in [Ta].

Elliptic curves over Q: Let E be an elliptic curve defined over Q. Inparticular Emay be viewed as a curve over Qp for every p, and we define its(global) conductor by

NE=p

pmp(E).

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The curve E is said to be semi-stable if it is semi-stable over all p-adic fieldsQp. Note that E is semi-stable if and only if its conductor NEis square-free.Using the fact thatQ has class number 1, one can show that Ehas aglobal

minimal Weierstrass model (Wmin) which gives the equation of a minimalWeierstrass model over each Qp. The associated discriminant, denoted

minE ,

depends only on E. The associated differential, denoted NeronE , is called theNeron differential. It is well-defined up to sign.

The following, known as the Mordell-Weil theorem, is the fundamentalresult about the structure of the group of rational points E(Q). (Cf. Forexample [Si1].)

Theorem 1.6 The group E(Q) is a finitely generated abelian group. Hence

E(Q)T Zr,

whereT is the (finite) torsion subgroup ofE(Q), andr0 is the rank ofEoverQ.

Concerning the possible structure ofT, there is the following deep result ofMazur, a variant of which also plays a crucial role in the proof of FermatsLast Theorem:

Theorem 1.7 If E/Q is an elliptic curve, then its torsion subgroup is iso-morphic to one of the following possibilities:

Z/nZ, 1n10, n= 12, Z/2nZ Z/2Z, 1n4.

The proof is given in [Maz1] (see also [Maz2]). Thanks to this result, thestructure of the torsion subgroup T is well understood. (Recently, the tech-niques of Mazur have been extended by Kamienny [Kam] and Merel [Mer] toprove uniform boundedness results on the torsion of elliptic curves over generalnumber fields.)

Much more mysterious is the behaviour of the rank r . It is not known ifr

can be arbitrarily large, although results of Mestre [Mes] and Nagao [Na] showthat it is greater or equal to 13 for infinitely many elliptic curves over Q. Itturns out that many of the deep results on E(Q) and on r are based on therelation with L-functions.

We define the global L-function of the complex variable sby:

L(E/Q, s) =p

L(E/Qp, s). (1.1.6)

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Exercise 1.8 Using theorem 1.4, show that the infinite product defining theL-function L(E/Q, s) converges absolutely on the right half plane Real(s) >3/2.

Conjecture 1.9 (Birch-Swinnerton-Dyer) TheL-functionL(E/Q, s)hasan analytic continuation to the entire complex plane,and in particular is ana-lytic ats= 1. Furthermore:

ords=1L(E/Q, s) =r.

There is also a more precise form of this conjecture, which expresses the leading

coefficient ofL(E/Q, s) at s = 1 in terms of certain arithmetic invariants ofE/Q. For more details, see [Si1], conj. 16.5.

As we will explain in more detail in section 1.8, the analytic continuationof L(E/Q, s) now follows from the work of Wiles and Taylor-Wiles and astrengthening by Diamond [Di2], for a very large class of elliptic curves overQ, which includes all the semi-stable ones.

Abelian varieties: Elliptic curves admit higher-dimensional analogues, calledabelian varieties, which also play a role in our discussion. Analytically, the setof complex points on an abelian variety is isomorphic to a quotient Cg/,where is a lattice in Cg of rank 2g, satisfying the so-called Riemann period

relations. A good introduction to the basic theory of abelian varieties can befound in [CS] and [We1].

1.2 Modular curves and modular forms over C

Modular curves: The groupSL2(Z) of two by two integer matrices of deter-minant one acts by fractional linear (Mobius) transformations on the complexupper half plane

H={zC|Im (z)> 0},equipped with its standard complex analytic structure. The principal con-

gruence group (N) of level N is the subgroup of matrices in SL2(Z) whichreduce to the identity matrix modulo the positive integer N. A subgroup ofSL2(Z) is called acongruence groupif it contains (N) for some N. The levelof is the smallest N for which this is true. The most important examples ofcongruence groups are:

The group 0(N) consisting of all matrices that reduce moduloNto anupper triangular matrix.

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The group 1(N) consisting of all matrices that reduce modulo N to amatrix of the form

1 0 1

.

The principal congruence group (N) of levelNconsisting of all matricesthat reduce modulo Nto the identity.

Notice the natural inclusions of normal subgroups (N) 1(N) 0(N).The quotient 0(N)/1(N) is canonically isomorphic to (Z/NZ)

via

a bc d

d modN.

For any subgroupHof (Z/NZ), we denote by H(N) the group of matricesin 0(N) whose image in 0(N)/1(N) belongs to H.

If is a congruence subgroup ofSL2(Z), define Y to be the quotient ofthe upper half planeH by the action of . One equipsY with the analyticstructure coming from the projection map :H Y. (More precisely, ify = (), and G is the stabilizer of in , then the local ringOY,y isidentified with the local ring of germs of holomorphic functions atwhich areinvariant under the action ofG.) This makes Y into a connected complexanalytic manifold of dimension one, i.e., a Riemann surface. If is 0(N) (resp.

1(N), or (N)), we will also denoteYbyY0(N) (resp.Y1(N), orY(N)). Onecompactifies Y by adjoining a finite set ofcuspswhich correspond to orbitsofP1(Q) =Q {} under . Call X the corresponding compact Riemannsurface. (For more details, notably on the definition of the analytic structureon X at the cusps, see for example [Kn], p. 311, or [Shi2], ch. 1.) It followsfrom the definition of this analytic structure that the fieldK of meromorphicfunctions on X is equal to the set of meromorphic functions onHsatisfying

(Transformation property): f() =f(), for all ;

(Behaviour at the cusps): For all

S L2(Z), the function f() has a

Puiseux series expansion

m anq

n/h in fractional powers ofq= e2i.

Riemanns existence theorem (cf. for example [For], ch. 2) asserts that theanalytic structure on X comes from an algebraic one, i.e., the field K is afinitely generated extension ofC of transcendence degree 1. Thus we can, andwill, viewX as a complex algebraic curve over C. If is 0(N) (resp. 1(N),or (N)), we will also denote X byX0(N) (resp.X1(N), or X(N)).

Examples and exercises:

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1. For N= 1, the curve X0(N) =X1(N) =X(N) is a curve of genus 0, andits field of functions is the ring C(j), wherej is the classical modular function,

j() =q1 + 744 + 196884q+ , q= e2i.(Cf., for example, [Se4], ch. 7.)

2. For H P1(Q) = H, define G to be the stabilizer of in P SL2(Z),and let e = #(G/(G )). Show that e depends only on the -orbitof in H, and that e = 1 for all but finitely many in H/. Using theRiemann-Hurwitz formula (cf. [Ki], sec. 4.3) show that the genus of X isgiven by

g() = 1 [P SL2(Z) : ] +12

H/

(e 1).

Use this to compute the genus of X0(p), X1(p), and X(p) for p prime. Fordetails, see [Shi2], sec. 1.6 or [Ogg].

3. For = (2), show that X is isomorphic to P1, and that Y is isomorphic

to P1 {0, 1, }. Show that /1 is the free group on the two generatorsg1=

1 20 1

and g2=

1 02 1

.

4. Define a homomorphism (2)

Z/nZZ/nZ, by sending g1 to (1, 0)

and g2 to (0, 1), and let denote its kernel. Show that is not in general acongruence subgroup and that the curve Y:=H/ is birationally isomorphicto the Fermat curve of degree n with affine equation xn + yn = 1.

Moduli interpretations: The points in Y =H/ can be interpreted aselliptic curves over Cwith some extra levelN structure. More precisely,

I f = 0(N), then the -orbit of H corresponds to the complextorus E = C/1, with the distinguished cyclic subgroup of order Ngenerated by 1

N. Thus, points onY0(N) parametrize isomorphism classes

of pairs (E, C) where E is an elliptic curve over C and C is a cyclic

subgroup ofEof order N.

If = 1(N), then the -orbit of corresponds to the complex torusE=C/1, with the distinguished point of orderNgiven by 1

N. Hence,

points on Y1(N) parametrize isomorphism classes of pairs (E, P) wherenowP is a point on Eof exact order N.

Remark 1.10 One checks that the above rules set up a bijection betweenpoints on Y and elliptic curves with the appropriate structures, and that the

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projectionY1(N)Y0(N) sending 1(N) to 0(N)becomes the forget-ful map sending (E, P) to (E, P).

Remark 1.11 (This remark will be used in section 1.3 when discussing Heckeoperators.) Define an n-isogeny of -structures to be an n-isogeny of theunderlying elliptic curves which sends one -structure to the other. Ifp is aprime not dividing N, then there are exactly p+ 1 distinct p-isogenies from(C/, 1, 1

N), whose images are the pairs:

C/+ i

p , 1 ,

1

N (i= 0, . . . , p 1), C/p, 1, p

N .Ifp divides N, then there are only p distinct p-isogenies from (C/, 1, 1

N),

since (C/p, 1, pN

) is not a 1(N)-structure (the pointp/Nnot being of exactorderNon the complex torus C/p, 1).

Modular forms: Letk be an even positive integer. A modular formof weightk on is a holomorphic function f onHsatisfying:

(Transformation property): f() = (c+ d)kf(), for all =

a bc d

.

(Behaviour at the cusps): For all P SL2(Z), the function (c +d)kf() has a Puiseux series expansion

0 anq

n/h in fractional powersofq = e2i. We call

anq

n/h the Fourier expansion off at the cusp1(i).

A modular form which satisfies the stronger property that the constantcoefficient of its Fourier expansion at each cusp vanishes is called acusp form.We denote by Mk() the complex vector space of modular forms of weight kon , and by Sk() the space of cusp forms on . (For examples, see [DI],sec. 2.2 and the references therein, especially, [Shi2], ch. 2.)

This article is mainly concerned with modular forms of weight 2, and hencewe will focus our attention on these from now on. A pleasant feature of the casek = 2 is that the cusp forms in S2() admit a direct geometric interpretationas holomorphic differentials on the curve X.

Lemma 1.12 The map f()f := 2if()d is an isomorphism betweenthe spaceS2()and the space

1(X)of holomorphic differentials on the curveX.

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Sketch of proof: One checks that the transformation property satisfied byf()under causes the expressionf()dto be -invariant, and that the conditionof vanishing at the cusps translates into holomorphicity off()d. (Note, forexample, that 2id =dq/q, so that f is holomorphic at i precisely whenf(q) vanishes at q= 0.)

As a corollary, we find:

Corollary 1.13 The spaceS2() is finite-dimensional, and its dimension isequal to the genusg ofX.

Proof: This follows directly from the Riemann-Roch theorem, cf. [Ki], sec. 6.3.

To narrow still further the focus of our interest, we will be mostly concerned

with the cases = 0(N) and 1(N). A slightly more general framework issometimes convenient, so we suppose from now on that satisfies

1(N)0(N).

Such a group is necessarily of the form H(N) for some subgroup H of(Z/NZ). Because the transformation + 1 belongs to the formsin S2() are periodic functions onH of period 1, and hence their Fourierexpansions at i

are of the form

f() =n>0

anqn, q= e2i, anC.

The Petersson inner product: The spaces S2() are also equipped with anatural Hermitian inner product given by

f, g= i82

X

fg =H/

f()g()dxdy,

where =x+ iy. This is called the Petersson inner product.

The diamond operators: Suppose now that = 1(N) and let d be an ele-ment of (Z/NZ). The mapdwhich sends an elliptic curve with -structure(E, P) to the pair (E,dP) gives an automorphism ofY which extends toX.

It is called the diamond operator. For inH and =

a bc d

in 0(N), we

haved() = ().

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Henced acts on S2(), identified with the holomorphic differentials on X,by the ruledf() = (c+ d)2f

a+ b

c+ d

.

In geometric terms, the diamond operators are the Galois automorphisms ofthe natural (branched) covering X1(N) X0(N) whose Galois group isisomorphic to 0(N)/1(N) = (Z/NZ)/1. Given an even Dirichletcharacter : (Z/NZ) C, say that f is a modular form of level N andcharacter if it belongs to the -eigenspace in S2(1(N)) under this action.LetS2(N, ) denote the space of all such forms. Thus a function f inS2(N, )

is a cusp form on 1(N) which satisfies the stronger transformation property:

f

a+ b

c+ d

=(d)(c+ d)2f(), for all

a bc d

0(N).

Note that if1 is the trivial character, then S2(N, 1) is canonically identifiedwith S2(0(N)), which we will also denote by S2(N). Finally note the directsum decomposition:

S2(1(N)) =S2(N, ),where the sum ranges over all the even Dirichlet characters modulo N.

Exercise 1.14 Show that if f() belongs to S2(N), then f(a) belongs toS2(mN), for each integer adividing m.

Jacobians of modular curves: Let Vbe the dual space

V =S2() := Hom(S2(),C).

It is a complex vector space of dimension g = genus(X). The integral ho-mology = H1(X,Z) maps naturally to Vby sending a homology cycle cto the functional c defined by c(f) = c f. The image of is a lattice inV, i.e., a Z-module of rank 2g which is discrete (cf. [Mu1], cor. 3.8). Fix abase point 0 H, and define the Abel-Jacobi map AJ :X(C)V / byAJ(P)(f) =

P0

f. Note that this is well-defined, i.e., it does not depend onthe choice of path on X from0 to P, up to elements in .

We extend the map AJ by linearity to the group Div (X) of divisors onX, and observe that the restriction of AJ to the group Div

0(X) of degree0 divisors does not depend on the choice of base-point 0. Moreover we havethe Abel-Jacobi theorem:

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Theorem 1.15 The map

AJ: Div0(X)V /

has a kernel consisting precisely of the group P(X) of principal divisors onX. HenceAJ induces an isomorphism fromPic

0(X) := Div0(X)/P(X)

to V /.

For the proof, see [Mu1], ch. 3. The quotient V / is a complex torus, and isequal to the group of complex points of an abelian variety. We denote thisabelian variety by J, the Jacobian variety ofX over C. If = 0(N) or

1(N), we will also write J0(N) or J1(N) respectively for the Jacobian J.

1.3 Hecke operators and Hecke theory

We maintain our running assumption that satisfies

1(N)0(N).

Ifp is a prime not dividing the level N, we define the Hecke operator Tp onS2() by the formula

Tp(f) =1p

p1i=0

f

+ ip

+ppf(p).

We give a more conceptual description ofTp in terms of remark 1.11 in thecase = 1(N). We have

Tp(f)=

i (f),

where i() = +ip

, and () =pp represent the p+ 1 curves with -structure that are images of (C/

, 1

, 1N

) by a p-isogeny, and the i are the

pull-back maps on differential forms onH. (An isogeny of elliptic curves with-structure is simply an isogeny between the underlying curves which sendsone -structure to the other.) Such a description makes it evident that Tp(f)belongs to S2(), iffdoes. In terms of the Fourier expansion off=

anq

n,the formula for the operator Tp on S2(N, ) is given by:

Tp(f) =p|n

anqn/p +p(p)

anqpn.

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Ifp dividesN, then we define the Hecke operator Upanalogously, by summingagain over all the cyclic p-isogenies of -structures. Since there are only p ofthem, the formula becomes simpler:

Up(f) =1

p

p1i=0

f

+ i

p

=p|n

anqn/p.

The reader is invited to check that the Hecke operators of the form Tp orUq commute with each other, and also that they commute with the diamondoperators introduced in the previous section.

We extend the definition of the Hecke operators to operators Tpn, with

n >1, by the inductive formulae

Tpn+1 =TpTpn ppTpn1, if (p, N) = 1,and Tpn =U

np otherwise. We then define the operator Tn, where n=

peii is

written as a product of powers of distinct primes pi, by

Tn =i

Tpeii .

This definition makes the Hecke operators multiplicative, i.e., TmTn =Tmn if(m, n) = 1. (A more conceptual definition of the Hecke operator Tn is that

Tn(f) is obtained by summing the pullback of f over the maps describingall the cyclic n-isogenies of -structures.) The relations among the differentHecke operators can be stated succinctly by saying that they obey the following(formal) identity:

p|N(1 Tpps + pp12s)1

p|N

(1 Upps)1 =n

Tnns.

(1.3.1)

The reader can consult [DI], sec. 3 and the references therein (especially [Shi2],ch. 3 or [Kn]) for more details and different points of view on Hecke opera-tors. Let T be the subring of EndC(S2()) generated over C by all the Hecke

operatorsTp for p|N,Uq for q|N, andd acting on S2().Definition 1.16 A modular form f is an eigenform if it is a simultaneouseigenvector for allthe Hecke operators in T, i.e., if there exists a C-algebrahomomorphism : TC such that T f=(T)f, for allT T.A direct calculation shows that the coefficients an of an eigenform f can berecovered from the homomorphism by the formula:

an(f) =a1(f)(Tn).

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It follows that the first Fourier coefficienta1 of a non-zero eigenform is alwaysnon-zero, and that the non-trivial eigenspaces for T are all one-dimensional:

Proposition 1.17 Given a non-zero algebra homomorphism : T C,there is exactly one eigenform f up to scaling, which satisfies T f = (T)f,for allT T.Sketch of proof: The proof of the existence offis an exercise in commutativealgebra (localize S2() at the kernel of), and the uniqueness is clear fromthe formula above.

We call an eigenform satisfying a1= 1 a normalizedeigenform.

Atkin-Lehner theory: It is natural to ask whether S2() can be decomposedinto a basis consisting of distinct normalized eigenforms. Unfortunately, thisis not always possible, as the following exercise illustrates.

Exercise 1.18 Suppose that p3 divides N exactly. Let T be the algebra ofHecke operators (generated by the operators Tq with q|N/p3, and Uq withq|N/p3) acting on S2(N/p3). Let f =

n=1 anq

n be a T-eigenform of levelN/p3 inS2(N/p

3). Show that the space Sfspanned by the formsf(),f(p),f(p2), and f(p3) is contained in S2(N), and is stable for the action of theHecke operatorsTq,q|N, and Uq,q|N. Show that Sfhas no basis of simulta-neous eigenforms for the Hecke algebra T of level N, so that the action ofTon Sfis not semi-simple.

Let T0 denote the subalgebra ofT generated only by the good Hecke oper-atorsTq with q|N, andd.Proposition 1.19 Ifqdoes not divideN, the adjoint of the Hecke operatorTq with respect to the Petersson scalar product is the operatorq1Tq, and theadjoint ofq isq1. In particular, the Hecke operators commute with theiradjoints.

Proof: See [Kn], th. 9.18 and 8.22, or [Ogg]. Proposition 1.19 implies, by the spectral theorem for commuting operators

that commute with their adjoints:

Proposition 1.20 The algebraT0 is semi-simple (i.e, it is isomorphic to aproductC C of a certain number of copies ofC), and there is a basisofS2() consisting of simultaneous eigenvectors for the operatorsTq.

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Thus, T0

has the merit of being semi-simple, while T is not in general. Thecost of replacing T by T0, however, is that one loses multiplicity one, i.e.,the eigenspaces for T0 need not be one-dimensional. For example, the spaceSfdefined in the previous exercise is contained in a single eigenspace for T

0.The theory of Atkin-Lehner [AL] gives essentially a complete understanding

of the structure of the algebraT, and the structure of the space of eigenforms.To motivate the main result, observe that the problem in the exercise aboveseems to be caused by forms of level Nthat are coming from forms of lowerlevelN/p3 by a straightforward operation, and are therefore not genuinelyof level N. They are the analogues, in the context of modular forms, of non-primitive Dirichlet characters.

Definition 1.21 We define theoldsubspace ofS2() to be the space spannedby those functions which are of the form g(az), where g is in S2(1(M)) forsome M < N and aM divides N. We define the new subspace of S2()to be the orthogonal complement of the old subspace with respect to thePetersson scalar product. A normalized eigenform in the new subspace iscalled a newform of levelN.

The following result is the main consequence of the theory of Atkin-Lehner. Itgives a complete answer to the question of what is the structure of the algebraTacting on S2().

Theorem 1.22 Iffis in the new subspace ofS2() and is an eigenvector forall the operators inT0, then it is also an eigenform forT, and hence is uniqueup to scaling. More generally, iff is a newform of levelNf|N, then the spaceSf defined by

Sf={gS2() such thatT g=f(T)g, for allT T0}is stable under the action of all the Hecke operators inT. It is spanned by thelinearly independent forms f(az) where a ranges over the divisors ofN/Nf.Furthermore, we have

S2() =fSf,where the sum is taken over all newformsfof some levelNf dividingN.

See [AL] for the proof in the case = 0(N), and [La2], ch. VIII for thegeneral case. (See also [DI], sec. 6 for an overview.)

Exercise 1.23 Consider the case where = 0(22). Show thatX0(22) is ofgenus 2, and hence that S2(22) has dimension 2. Show that S2(22) is equalto Sf, where f= (()(11))

2 is a newform of level 11, so that in particularthere are no newforms of level 22 on . Show that T0 is isomorphic to C inthis case, and that Tis isomorphic to the semisimple algebra C C.

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Action on homology and Jacobians: Note that the Hecke operators acton V =S2() by duality. One checks (cf. [Kn], props. 11.23, 11.24) that the

sublattice ofVis stable under the action of all the Hecke operators Tn, andof the diamond operatorsd. Therefore the operators Tn andd give riseto endomorphisms of the torus V /, and hence the Jacobian variety J, ina natural way. The involution gives rise to an involution on X(C)(which is complex conjugation on the model ofX over R deduced from theQ- model defined in section 1.5). Since complex conjugation is continuous italso acts on the integral homology =H1(X(C),Z). Let

+ and be thesublattices of on which complex conjugation acts by +1 and1. These aresublattices of of rank g which are stable under the Hecke operators, sincecomplex conjugation commutes with the Hecke action.

A more algebraic description of the action of Tp on J is given via thenotion of an algebraic correspondence. A correspondence on a curve X is adivisorCon X Xwhich is taken modulo divisors of the form{P} XandX {Q}. Let 1 and 2 denote the projections ofX X onto each factor.Then the correspondence C induces a map on divisors ofX, by setting

C(D) =2(11 (D) C).

(For the definition of the intersection D1 D2 of two divisors, see [We1].) Themap Cpreserves divisors of degree 0, and sends principal divisors on X to

principal divisors. It gives a well defined algebraic endomorphism of the Jaco-bian variety Jac (X). Given a correspondenceC, its transposeC is defined tobe the divisor ofX Xobtained by interchanging the two factors ofX X.One can define a natural notion of composition for correspondences, and theset of correspondences forms a ring. The general theory of correspondencesand the proofs of the above facts are given in [We1], particularly the secondchapter.

The Hecke correspondence Tn is defined as the closure in X X of thelocus of points (A, B) inY Y such that there is a degree n cyclic isogeny ofelliptic curves with -structure fromAto B . For example, ifpis a prime not

dividingN, thenTpis an algebraic curve inX1(N)X1(N) which is birationalto X1(N)0(p). The induced map on divisors in this case satisfies

Tp((E, P)) =

(E/C,P mod C)

where the sum runs over the subgroups CofEhaving orderp. Note also thatif (A, B) belongs to Tp, then the isogeny dual to A B gives a p-isogenyfrom B topA, so that

Tp =p1Tp.

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1.4 The L-function associated to a cusp formFor this section, let f in S2(1(N)) be a cusp form with Fourier expansionat i given byn anqn. One has the following estimate for the size of theFourier coefficients an:

Theorem 1.24 The coefficientsanC satisfy the inequality

|an| c(f)0(n)

n,

wherec(f) is a constant depending only onf, and0(n) denotes the numberof positive divisors ofn.

Sketch of proof: This follows from proposition 1.51 of section 1.7 which relatesthe p-th Fourier coefficients of eigenforms, for p a prime not dividing the levelof , to the number of points on certain abelian varieties over the finite fieldFp. The estimates of Hasse and Weil for the number of points on abelianvarieties over finite fields (stated in theorem 1.4 of section 1.1 for the specialcase of elliptic curves; see [We1],IV for the general case) thus translate intoasymptotic bounds for the Fourier coefficients of these eigenforms. We notethat the cruder estimate|an|= O(n), which is enough for the purposes of thissection, can be derived by a more elementary, purely analytic argument; cf.

[Ogg], ch. IV, prop. 16.

The L-function associated to f is defined by the formula:

L(f, s) =

anns.

As in exercise 1.8, one can show that the infinite sum definingL(f, s) convergesabsolutely in the right half-plane Re (s)> 3

2. A much better insight is gained

into the functionL(f, s) by noting that it is essentially the Mellin transform ofthe modular formf. More precisely, if we set (f, s) =Ns/2(2)s(s)L(f, s),then we have

(f, s) =Ns/2

0

f(iy)ysdy/y (1.4.1)

Exercise 1.25 Check the formula above.

This integral representation for L(f, s) gives the analytic continuation ofL(f, s) to the entire complex plane. The modular invariance off translatesinto a functional equation for L(f, s): more precisely, let wN be the Atkin-Lehner involution defined by wN() =1/N . The reader may check that

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wN induces an involution ofX, and hence of 1

(X) =S2(). One finds thatL(f, s) satisfies the functional equation:

(f, s) =(wN(f), 2 s).

For a proof of this, see [Ogg], ch. V, lemma 1. Eigenforms for T in S2(N, )have a great importance in the theory because their associated L-functionshave an Euler product expansion, in addition to an analytic continuation andfunctional equation:

Theorem 1.26 Iff = anqn is a normalized eigenform inS2(N, ) for all

the Hecke operators, then the L-function L(f, s) =

anns

has the Eulerproduct expansionp|N

(1 apps + (p)p12s)1p|N

(1 apps)1.

Proof: This follows directly from equation (1.3.1) of section 1.3. Iffis a newform of levelN, then it is also an eigenform for wN, so that the

functional equation may be viewed as relatingL(f, s) andL(f, 2 s). We canalso state the following more precise version of theorem 1.24 (see lemma 3.2of [Hi2] for example for parts (b), (c) and (d)).

Theorem 1.27 Suppose thatf is a newform of levelNf and let N denotethe conductor of its character.

(a) Ifp does not divideNf then|ap| 2p.(b) If p||Nf and p does not divide N then a2p = 0(p) where 0 is the

primitive character associated to .

(c) Ifp dividesNf andp does not divideNf/N then|ap|=p.

(d) Ifp

2

dividesNf andp dividesNf/N thenap= 0.

1.5 Modular curves and modular forms over Q

Modular curves: For between 0(N) and 1(N), the modular curve Xhas a model over Q. We describe such a model in the case of = 0(N); theconstruction for general follows from similar considerations.

The key remark here is that, as was noted in section 1.2, the complexpoints on the curve Y0(N) have a natural interpretation as moduli of elliptic

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curves together with a cyclic subgroup of order N, given by sending the point H/0(N) to the pair (C/1, , 1/N).Consider the universal elliptic curve

Ej :y2 + xy=x3 36

j 1728 x 1

j 1728 .It is an elliptic curve over the function field Q(j), withj -invariantj . Letd bethe order ofP1(Z/NZ), and letC1, . . . , C ddenote the set of all cyclic subgroupsofEj of order N, defined over Q(j), an algebraic closure ofQ(j). Fix one of

these subgroups, C. The Galois group Gal (Q(j)/Q(j)) permutes the Ci in anatural way. LetFNbe the smallest extension ofQ(j) (viewed as embedded in

Q(j)) with the property that (C) =C, for all Gal (Q(j)/FN). It can beseen (cf. [Shi2], thm. 6.6) that the Galois action on theCi is transitive so thatFN/Q(j) is of degree d, and that it is a regularextension, i.e., FN Q = Q.Geometrically,FNcan be viewed as the function field of a curve X/Q over Q,with the inclusion ofQ(j) corresponding to a map fromX/Q to the projectivej-line over Q. The pair (Ej, C) is an elliptic curve over Q(j) with a subgroupof order Ndefined over FN. Using (Ej, C), each complex point ofXgives anelliptic curve over C with a subgroup of order N, provided the point doesnot lie over j = 0, 1728 or. The resulting map to X0(N) extends to anisomorphism fromX toX0(N). The curveXthus constructed, together with

this identification, is the desired model ofX0(N) over Q.More concretely, the functions j = j() and jN = j(N) are related by apolynomial equation N(j, jN) = 0 with coefficients in Q, of bidegree d. ThefieldFN is the function field of the affine curve N(X, Y) = 0, and the mapping (j(), j(N )) gives a birational equivalence betweenH/0(N) and thecomplex curve defined by the equation N. In practice it is not feasible towrite down the polynomial N, except for certain very small values ofN. Tostudy the models over QofX0(N), more indirect methods are needed, whichrely crucially on the moduli interpretation ofX0(N). Similar remarks hold forX1(N).

Models over Z: The work of Igusa [Ig], Deligne-Rapoport [DR], Drinfeld [Dr],and Katz-Mazur [KM] uses the moduli-theoretic interpretation to describe acanonical proper model for X over Spec Z. These models allow us to talkabout the reduction ofX over finite fields Fp, for p prime. The curve hasgood reduction at primes p not dividing N, with the non-cuspidal points ofX/Fp corresponding to elliptic curves over

Fp with -structure. The singularfibers at primes p dividing N can also be described precisely; an importantspecial case (see [DR]) is that of 0(N) withp exactly dividingN. For furtherdiscussion and references, see [DI], sec. 8, 9.

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From now on, when we writeX,X0(N), orX1(N), we will mean the curveover Q which are the models described above for the complex curves definedin section 1.2.

Remark 1.28 When considering q-expansions, it is more convenient to usea different set of models over Z for these complex curves. We define X1 (N)in the case of 1(N) as a model over Zwhich parametrizes pairs (E, i) wherei is an embedding ofN in the (generalized) elliptic curve E. (So X

1 (N) is

the model denotedX(N) in [DI], sec. 9.3, assuming N >4.) For between0(N) and 1(N), we defineX

as the corresponding quotient ofX

1 (N). This

model has good reduction at primes p not dividing N, but unlike the models

mentioned above, its fibers at primes p dividingNare smooth and irreducible,but not proper. In the case of = 0(N), the curve X

,Q can be identified

withX0(N). However, this is not the case in general: the cuspis a rationalpoint ofX,Q but not necessarily ofX.

Jacobians: Weils theory [We1] of the Jacobian shows that the Jacobians Jdefined in section 1.2 as complex tori also admit models over Q. When wespeak ofJ, J0(N) and J1(N) from now on, we will refer to these as abelianvarieties defined over Q. Thus, the points in J(K), for any Q-algebra K, areidentified with the divisor classes on X of degree 0, defined over K.

We let J/Z, denote the Neron model of the Jacobian J over Spec(Z).Using this model we define J/A for arbitrary rings A. In particular we canconsider J/Fp, the reduction of the Jacobian in characteristicp, which is closelyrelated to the reduction of the integral model of the curveXmentioned above.In particular, ifp does not divide the level of , then J/Fp can be identifiedwith the Jacobian ofX/Fp. For a treatment of the case = 0(N) with pexactly dividing N, see the appendix of [Maz1]; for more general discussionand references, see [DI], sec. 10, especially sec. 10.3.

Hecke operators: The Hecke operators have a natural moduli interpretation,

which was already touched upon in section 1.3. In particular, one finds thatthe operator Tn arises from a correspondence which is defined over Q, andgives rise to an endomorphism of the Jacobian J which is defined over Q.This in turn gives rise to an endomorphism of the Neron modelJ/Z, and wecan then consider the endomorphism Tn on the reduction of the Jacobian incharacteristic p. Recall that ifp is a prime not dividing N, we may identifythis reduction with the Jacobian ofX/Fp. (Cf. [MW], ch. 2, sec. 1, prop. 2.)Furthermore, one can show that the moduli-theoretic interpretation of theHecke operator remains valid in characteristic p; i.e., the endomorphismTn of

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J/Fp is induced by a map on divisors satisfying, for all ordinary elliptic curvesAwith -structure:Tn(A) =

i

i(A),

where the sum is taken over all cyclic isogenies of degree n. (See [DI], sec. 10.2for further discussion and references.)

This description allows one to analyse, for example, the Hecke operatorTp over Fp, when (p, N) = 1. Let us work with = 1(N), to illustratethe idea. For a varietyX over Fp, let Xbe the Frobenius morphism on Xdefined by raising coordinates to the pth power. Thus if (E, P) corresponds

to a point ofX1(N)/Fp, then Eis an isogeny of degree p from (E, P) to thepair (E, P) =X1(N)(E, P). The graph ofX1(N) in (X1(N) X1(N))/Fp isa correspondence of degree p, which we call F. LetF be the transpose of thiscorrespondence. The endomorphism F ofJ induced by F is the Frobeniusendomorphism J, and the endomorphism F

is the dual endomorphism (inthe sense of duality of abelian varieties). Now consider the divisor

F((E, P)) = (E1, P1) + + (Ep, Pp),

where the (Ei, Pi) are elliptic curves with -structure in characteristicp. SinceEi is an isogeny of degree p from (Ei, Pi) to (E, P), we also have the dual

isogeny from (E, P) t o (Ei, pPi). If E is ordinary at p, then the points(E, P), (E1, pP1), . . . , (Ep, pPp) are a complete list of the distinct curveswith -structure which are p-isogenous to (E, P). Hence one has the equalityof divisors on X1(N)/Fp:

Tp((E, P)) = (E, P) + (E1, pP1) + + (Ep, pPp) = (F+ pF)((E, P)).

Since the ordinary points are dense on X1(N)/Fp, we deduce that Tp = (F+pF) as endomorphisms of J1(N)/Fp. This equation, known as Eichler-Shimura congruence relation, plays a central role in the theory. (For moredetails, see [DI], sec. 10.2, 10.3.)

Theorem 1.29 Ifp|Nthen the endomorphismTp ofJ/Fp satisfies

Tp= F+ pF.

Remark 1.30 This was proved by Eichler [Ei] to hold for all but finitely manyp in the case of 0(N), and by Shimura ([Shi1], see also [Shi2], ch. 7) in thecase of 1(N). The fact that it holds for all p not dividing N follows fromwork of Igusa [Ig].

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Modular forms: In the same way that modular curves have models overQ and over Z, the Fourier coefficients of modular forms also have naturalrationality and integrality properties. We start by sketching the proof of:

Theorem 1.31 The spaceS2() has a basis consisting of modular forms withinteger Fourier coefficients.

Proof: The Hecke operators act on the integral homology + in a way thatis compatible with the action on S2() and respects the natural (Poincare)duality between these two spaces. Hence, if{n}nN is a system of eigenvaluesfor the Tn, then the n are algebraic integers in some finite extension KofQ,

and the system{n}nN is a system of eigenvalues for the Tn for any Galoisautomorphism ofQ/Q. Hence, we have shown:

Proposition 1.32 If f S2(M, ) is a newform of some level M dividingN, then its Fourier coefficients lie in a finite extension K ofQ. Moreover,if Gal(Q/Q) is any Galois automorphism, then the Fourier series fobtained by applying to the Fourier coefficients is a newform inS2(M,

).

The explicit description of S2() given in section 1.3 implies that S2() isspanned by forms having Fourier coefficients which are algebraic integers insome finite (Galois) extensionKofQ, and that the space of forms with Fourier

coefficients in K is stable under the natural action of Gal (K/Q) on Fourierexpansions. An application of Hilberts theorem 90 shows thatS2() has abasis consisting of forms with rational Fourier expansions, and the integralityof the Fourier coefficients of eigenforms yields the integrality statement oftheorem 1.31.

We define S2(,Z) to be the space of modular forms with integral Fouriercoefficients in S2(). Theorem 1.31 states thatS2(,Z) C= S2(). Givenany ring A, we define

S2(, A) =S2(,Z) A,and define S2(N, A) and S2(N, , A) (where now is a character with values

in A) in the obvious way. IfA is contained in C, the q-expansion principlebelow allows us to identify S2(, A) with the set of modular forms in S2()with Fourier coefficients in A.

Theq-expansion principle: Because the modular curve X0(N) has a modelover Q, the space of modular formsS2(N) =

1(X0(N)) has a natural rationalstructure, given by considering the differential forms on X0(N) defined overQ. The fundamentalq-expansion principle (see [DR], ch. 7, or [Kat], sec. 1.6)says that these algebraic structures are the same as those obtained analytically

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by considering q-expansions at. More generally, using the model X

, weobtain theq-expansion principle over Z for cusp forms on (cf. [DI], sec. 12.3).

Theorem 1.33 The map S2() 1(X) defined by f f induces anisomorphism fromS2(,Z)to

1(X ). Furthermore, ifA is flat overZ orN isinvertible inA, then the induced map S2(, A)1(X,A)is an isomorphism.Furthermore, ifAis a subring ofC, then this isomorphism identifiesS2(, A)with set of modular forms inS2() having coefficients inA.

1.6 The Hecke algebra

It follows directly from the formulas for the Hecke operators acting on q-expansions that the Tn leave S2(0(N),Z) stable, as well as the subspaceof S2(N, ) with coefficients in Z[]. Using the q-expansion principle (the-orem 1.33), one can also show ([DI], sec. 12.4) that the diamond operatorspreserve the spaces of cusp forms on with integral Fourier expansions, andhence that the space S2(,Z) is preserved by all the Hecke operators.

We define TZ to be the ring generated over Z by the Hecke operators Tnand d acting on the spaceS2(,Z). More generally, ifAis any ring, we defineTA to be the A-algebra TZ A. This Hecke ring acts on the spaceS2(, A)in a natural way. Before studying the structure of the Hecke rings TA as we

vary the ringsA, we note the following general result (Cf. [Shi2], ch. 3.):Lemma 1.34 The spaceS2(, A)

= HomA(S2(, A), A) is a freeTA-moduleof rank one.

Sketch of proof: One checks that the pairing TZ S2(,Z)Z defined by(T, f) a1(T f) sets up a perfect, TZ-equivariant duality between TZ andS2(,Z). The result for arbitrary Afollows.

Hecke rings over C: If A = C, then the structure of the ring T = TCis completely described by theorem 1.22. More precisely, ifTC,f denotes theimage of the Hecke algebra acting on the space Sfdefined in section 1.3, then

TC =fTC,f,where the direct sum ranges over all distinct newforms f of some level Nfdividing N. Furthermore, the algebra TC,f can be described explicitly. Inparticular, iffis a newform of levelN thenTC,fis isomorphic toC, but ifNfis not equal toNthen the ringTC,fneed not be a semi-simple algebra overC.

Lemma 1.34 in the case A= C says that V =S2() is a free TC-module

of rank one, but we also have:

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Lemma 1.35 The moduleS2() is a freeTC-module of rank one.Proof: Let g1, . . . , gt be a complete system of newforms of levels N1, . . . , Ntdividing N. One can check that the form

g= g1(N/N1) + + gt(N/Nt)

generates S2() as a TC-module. The mapTCS2() defined by TT ggives an isomorphism from TC to S2(), as TC-modules.

Remark 1.36 Lemmas 1.34 and 1.35 imply that TCis aGorensteinC-algebra

(of finite rank), i.e., HomC(TC,C) is isomorphic to TC as a TC-module.

Hecke rings over Q: Let [f] be the Galois orbit (under the action ofGQ =Gal(Q/Q)) of a normalized newform f of some level Nf dividing N, and letKfbe the field extension ofQ generated by the Fourier coefficients off. Thespaceg[f]Sg is a vector space of dimension [Kf : Q]0(N/Nf), which isspanned by modular forms with rational Fourier coefficients. Let S[f] be theQ-subspace of forms ing[f]Sg with rational Fourier coefficients. The spaceS[f] is stable under the action ofTQ, and letting TQ,[f] be the image ofTQacting on S[f], we obtain the direct sum decomposition

TQ =[f]TQ,[f],

where the sum is taken over the distinct GQ-orbits of normalized newformsf of some level Nf dividing N. IfNf is equal to N, then the algebra TQ,[f]is isomorphic to the field Kf. IfNf is a proper divisor ofN, then, as in thecomplex case, the algebraTQ,[f] is a (not necessarily semi-simple) algebra overQ of rank 0(N/Nf)[Kf :Q]. The nature of the fields Kf, and in general thestructure ofTQ, is very poorly understood at this stage; for example, one doesnot know how to characterize the number fields that occur as a Kf for somef(but they are all known to be totally real or CM fields).

The ring TQ acts naturally on the rational homology H1(X,Q) = Q,and we have

Lemma 1.37 The module Q is free of rank two overTQ.Sketch of proof: The modules + CV and CVare free of rankone over TC, by lemma 1.34. This implies that

+ Qand Qare bothfree of rank one over TQ.

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Hecke rings overZ: The ring TZis a certain (not necessarily maximal) orderin TQ. One still has an injection

TZ [f]TZ,[f],where now TZ,[f] denotes the ring generated over Z by the Hecke operatorsacting on S[f]. Of course the structure ofTZ is even more mysterious thanthat ofTQ! The ring TZ acts naturally on , but it is not the case in generalthat is free of rank two over TZ, i.e., that the integral analogue of lemma 1.37is true. (See remark 1.42 below.)

Hecke rings overQ: The study of the algebrasTZ andTQ arises naturally

because of the Hecke action on the Tate moduleT(J),T(J) := lim

(J)[

n],

where the inverse limit is taken with respect to the multiplication by maps.The action ofTZ on T(J) is compatible with that ofGQ, and it is this pair ofactions on the Tate module which is used to associate two-dimensional Galoisrepresentations to modular forms.

It will sometimes be more convenient to consider the ring TQ and its actionon

V=

T(J)

ZQ.

We first record a useful duality property enjoyed by the Tate modules. TheWeil pairings on the groups J[

n] for n1 induce a perfect pairing , :T(J) T(J)Z.

Since each Hecke operator T is adjoint to wT w where w = wN is the Atkin-Lehner involution, we have the following lemma.

Lemma 1.38 The map x x where x(y) =x,wy defines an isomor-phism ofTZ-modules,

T(J)=T(J)

= HomZ(T(J),Z),and hence an isomorphism ofTQ-modules

V=V = HomQ(V,Q).The following lemma allows us to regard TQ as a coefficient ring for a

two-dimensional Galois representation

Gal(Q/Q)Aut TQ (V)=GL2(TQ).

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Lemma 1.39 The moduleVis free of rank2 overTQ.Proof: Lemma 1.37 implies directly that the moduleV= Qis free of ranktwo over TQ.

Corollary 1.40 The ringTQ is a GorensteinQ-algebra; i.e.,

TQ = HomQ(TQ,Q)

is free of rank one overTQ.

Proof: Choosing a basis forVover TQ, we obtain an isomorphism

TQTQ =TQ TQ.Decomposing TQ as

iRi where each factor Ri is a finite-dimensional local

Q-algebra, we obtain an isomorphism

Ri Ri=Ri Rifor each i. At least one of the four maps Ri Ri deduced from thisisomorphism must be surjective, and by counting dimensions, we see that itmust be injective as well. It follows that TQ is isomorphic to T

Q

. Recall that for primespnot dividingN, the JacobianJhas good reduction

mod p, and the Eichler-Shimura relation, theorem 1.29, states that on J/Fp,we have

Tp= F+ pF.For primes p not dividing N, we may identifyT(J) with the -adic Tatemodule of the reduction (see [ST]) and consider the Frobenius endomorphismFon the free rank two TQ-module V. As a consequence of the Eichler-Shimurarelation, we find:

Theorem 1.41 Forp not dividingN , the characteristic polynomial ofF ontheTQ-moduleV is

X2

TpX+

pp.

Proof: (We are grateful to Brian Conrad for showing us this argument.) SinceF F =p, it follows from the Eichler-Shimura relation that

F2 TpF+ pp= 0.To conclude that this is in fact the characteristic polynomial, it suffices tocompute the trace ofF. To do so, we use the TQ isomorphism

V V

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defined by the

teorema de ferman

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