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This book explains a result called the Modularity Theorem:All rational elliptic curves arise from modular forms.Taniyama first suggested in the 1950’s that a statement along these lines mightbe true, and a precise conjecture was formulated by Shimura. A paper of Weil[Wei67] provides strong theoretical evidence for the conjecture. The theoremwas proved for a large class of elliptic curves by Wiles [Wil95] with a keyingredient supplied by joint work with Taylor [TW95], completing the proofof Fermat’s Last Theorem after some 350 years. The Modularity Theoremwas proved completely by Breuil, Conrad, Taylor, and the first author of thisbook [BCDT01]. Different forms of it are stated here in Chapters 2, 6, 7, 8,and 9.To describe the theorem very simply for now, first consider a situationfrom elementary number theory. Take a quadratic equationQ:x2=d, d∈Z,d=0,and for each prime numberpdefine an integerap(Q),ap(Q)=(the number of solutionsxof equationQworking modulop)−1.The valuesap(Q) extend multiplicatively to valuesan(Q) for all positive in-tegersn, meaning thatamn(Q)=am(Q)an(Q) for allmandn.Since by definitionap(Q) is the Legendre symbol (d/p) for allp>2, onestatement of the Quadratic Reciprocity Theorem is thatap(Q) depends onlyon the value ofpmodulo 4|d|. This can be reinterpreted as a statement thatthe sequence of solution-counts{a2(Q),a3(Q),a5(Q),…}arises as a systemof eigenvalues on a finite-dimensional complex vector space associated to theequationQ.LetN=4|d|,letG=(Z/NZ)∗be the multiplicative group of

viiiPrefaceinteger residue classes moduloN,andletVNbe the vector space of complex-valued functions onG,VN={f:G−→C}.For each primepdefine a linear operatorTponVN,Tp:VN−→VN,(Tpf)(n)={f(pn)ifpN,0ifp|N,where the productpn∈Guses the reduction ofpmoduloN.Consideraparticular functionf=fQinVN,f:G−→C,f(n)=an(Q)forn∈G.This is well defined by Quadratic Reciprocity as stated above. It is immediatethatfis an eigenvector for the operatorsTp,(Tpf)(n)={f(pn)=apn(Q)=ap(Q)an(Q)ifpN,0ifp|N=ap(Q)f(n) in all cases.That is,Tpf=ap(Q)ffor allp. This shows that the sequence{ap(Q)}is asystem of eigenvalues as claimed.The Modularity Theorem can be viewed as giving an analogous result.Consider a cubic equationE:y2=4×3−g2x−g3,g2,g3∈Z,g32−27g23=0.Such equations defineelliptic curves, objects central to this book. For eachprime numberpdefine a numberap(E) akin toap(Q) from before,ap(E)=p−(the number of solutions (x,y) of equationEworking modulop).One statement of Modularity is that again the sequence of solution-counts{ap(E)}arises as a system of eigenvalues. Understanding this requires somevocabulary.Amodular formis a function on the complex upper half plane that sat-isfies certain transformation conditions and holomorphy conditions. Letτbea variable in the upper half plane. Then a modular form necessarily has aFourier expansion,f(τ)=∞∑n=0an(f)e2πinτ,an(f)∈Cfor alln.Each nonzero modular form has two associated integerskandNcalled itsweightand itslevel. The modular forms of any given weight and level form

Prefaceixa vector space. Linear operators called theHecke operators, including an op-eratorTpfor each primep, act on these vector spaces. Aneigenformis amodular form that is a simultaneous eigenvector for all the Hecke operators.By analogy to the situation from elementary number theory, the Modular-ity Theorem associates to the equationEan eigenformf=fEin a vectorspaceVNof weight 2 modular forms at a levelNcalled theconductorofE.The eigenvalues offare its Fourier coefficients,Tp(f)=ap(f)ffor all primesp,andaversionofModularityisthatthe Fourier coefficients give the solution-counts,ap(f)=ap(E) for all primesp.(0.1)That is, the solution-counts of equationEare a system of eigenvalues, like thesolution-counts of equationQ, but this time they arise from modular forms.This version of the Modularity Theorem will be stated in Chapter 8.Chapter 1 gives the basic definitions and some first examples of modularforms. It introduces elliptic curves in the context of the complex numbers,where they are defined as tori and then related to equations likeEbut withg2,g3∈C. And it introducesmodular curves, quotients of the upper halfplane that are in some sense more natural domains of modular forms than theupper half plane itself. Complex elliptic curves are compact Riemann surfaces,meaning they are indistinguishable in the small from the complex plane. Chap-ter 2 shows that modular curves can be made into compact Riemann surfacesas well. It ends with the book’s first statement of the Modularity Theorem,relating elliptic curves and modular curves as Riemann surfaces:If the com-plex numberj= 1728g32/(g32−27g23)is rational then the elliptic curve is theholomorphic image of a modular curve. This is notatedX0(N)−→E.Much of what follows over the next six chapters is carried out with an eyeto going from this complex analytic version of Modularity to the arithmeticversion (0.1). Thus this book’s aim is not to prove Modularity but to state itsdifferent versions, showing some of the relations among them and how theyconnect to different areas of mathematics.Modular forms make up finite-dimensional vector spaces. To compute theirdimensions Chapter 3 further studies modular curves as Riemann surfaces.Two complementary types of modular forms areEisenstein seriesandcuspforms. Chapter 4 discusses Eisenstein series and computes their Fourier ex-pansions. In the process it introduces ideas that will be used later in the book,especially the idea of anL-function,L(s)=∞∑n=1anns.

xPrefaceHeresis a complex variable restricted to some right half plane to make theseries converge, and the coefficientsancan arise from different contexts. Forinstance, they can be the Fourier coefficientsan(f) of a modular form. Chap-ter 5 shows that iffis a Hecke eigenform of weight 2 and levelNthen itsL-function has anEuler factorizationL(s,f)=∏p(1−ap(f)p−s+1N(p)p1−2s)−1.The product is taken over primesp,and1N(p)is1whenpN(true for allbut finitely manyp) but is 0 whenp|N.Chapter 6 introduces theJacobianof a modular curve, analogous to acomplex elliptic curve in that both are complex tori and thus have Abeliangroup structure. Another version of the Modularity Theorem says that everycomplex elliptic curve with a rationalj-value is the holomorphic homomorphicimage of a Jacobian,J0(N)−→E.Modularity refines to say that the elliptic curve is in fact the image of aquotient of a Jacobian, theAbelian varietyassociated to a weight 2 eigenform,Af−→E.This version of Modularity associates a cusp formfto the elliptic curveE.Chapter 7 brings algebraic geometry into the picture and moves towardnumber theory by shifting the environment from the complex numbers to therational numbers. Every complex elliptic curve with rationalj-invariant canbe associated to the solution set of an equationEwithg2,g3∈Q. Modularcurves, Jacobians, and Abelian varieties are similarly associated to solutionsets of systems of polynomial equations overQ, algebraic objects in contrastto the earlier complex analytic ones. The formulations of Modularity alreadyin play rephrase algebraically to statements about objects and maps definedby polynomials overQ,X0(N)alg−→E,J0(N)alg−→E, Af,alg−→E.We discuss only the first of these in detail sinceX0(N)algis a curve whileJ0(N)algandAf,algare higher-dimensional objects beyond the scope of thisbook. These algebraic versions of Modularity have applications to numbertheory, for example constructing rational points on elliptic curves using pointscalled Heegner points on modular curves.Chapter 8 develops theEichler–Shimura relation, describing the HeckeoperatorTpin characteristicp. This relation and the versions of Modularityalready stated help to establish two more versions of the Modularity Theorem.One is the arithmetic version thatap(f)=ap(E) for allp,asabove.Fortheother, define theHasse–WeilL-functionof an elliptic curveEin terms of thesolution-countsap(E) and a positive integerNcalled theconductorofE,

PrefacexiL(s,E)=∏p(1−ap(E)p−s+1N(p)p1−2s)−1.Comparing this to the Euler product form ofL(s,f) above gives a version ofModularity equivalent to the arithmetic one:TheL-function of the modularform is theL-function of the elliptic curve,L(s,f)=L(s,E).As a function of the complex variables, bothL-functions are initially definedonly on a right half plane, but Chapter 5 shows thatL(s,f) extends ana-lytically to all ofC. By Modularity the same now holds forL(s,E). This isimportant because we want to understandEas an Abelian group, and theconjecture of Birch and Swinnerton-Dyer is that the analytically continuedL(s,E) contains information about the group’s structure.Chapter 9 introduces-adic Galois representations, certain homomor-phisms of Galois groups into matrix groups. Such representations are asso-ciated to elliptic curves and to modular forms, incorporating the ideas fromChapters 6 through 8 into a framework with rich additional algebraic struc-ture. The corresponding version of the Modularity Theorem is:Every Galoisrepresentation associated to an elliptic curve overQarises from a Galoisrepresentation associated to a modular form,ρf,∼ρE,.This is the version of Modularity that was proved. The book ends by discussingtwo broader conjectures that Galois representations arise from modular forms.Many good books on modular forms already exist, but they can be daunt-ing for a beginner. Although some of the difficulty lies in the material itself,the authors believe that a more expansive narrative with exercises will helpstudents into the subject. We also believe that algebraic aspects of modu-lar forms, necessary to understand their role in number theory, can be madeaccessible to students without previous background in algebraic number the-ory and algebraic geometry. In the last four chapters we have tried to do soby introducing elements of these subjects as necessary but not letting themtake over the text. We gratefully acknowledge our debt to the other books,especially to Shimura [Shi73].The minimal prerequisites are undergraduate semester courses in linear al-gebra, modern algebra, real analysis, complex analysis, and elementary num-ber theory. Topics such as holomorphic and meromorphic functions, congru-ences, Euler’s totient function, the Chinese Remainder Theorem, basics ofgeneral point set topology, and the structure theorem for modules over aprincipal ideal domain are used freely from the beginning, and the SpectralTheorem of linear algebra is cited in Chapter 5. A few facts about represen-tations and tensor products are also cited in Chapter 5, and Galois theory is

xiiPrefaceused extensively in the later chapters. Chapter 3 quotes formulas from Rie-mann surface theory, and later in the book Chapters 6 through 9 cite steadilymore results from Riemann surface theory, algebraic geometry, and algebraicnumber theory. Seeing these presented in context should help the reader ab-sorb the new language necessary en route to the arithmetic and representationtheoretic versions of Modularity.We thank our colleagues Joe Buhler, David Cox, Paul Garrett, Cris Poor,Richard Taylor, and David Yuen, Reed College students Asher Auel, RachelEpstein, Harold Gabel, Michael Lieberman, Peter McMahan, and John Saller,and Brandeis University student Makis Dousmanis for looking at drafts. Com-ments and corrections should be sent to the second author atjerry@reed.edu