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On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises

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On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Breuil, Christophe, Brian Conrad, Fred Diamond, and Richard L. Taylor. 2001. On the modularity of elliptic curves over Q: Wild 3- adic exercises. Journal of the American Mathematical Society 14(4): 843-939. Published Version doi:10.1090/S0894-0347-01-00370-8 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3626807 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume Numb er Xxxx XXXX Pages S XX ON THE MODULARITY OF ELLIPTIC CURVES OVER Q WILD ADIC EXERCISES CHRISTOPHE BREUIL BRIAN CONRAD FRED DIAMOND AND RICHARD TAYLOR Introduction In this pap er building on work of Wiles Wi and of Wiles and one of us RT TW we will prove the following two theorems see x Theorem A If E is an el liptic curve then E is modular Q Theorem B If GalQQ GL F is an irreducible continuous representation with cyclotomic determinant then is modular We will rst remind the reader of the content of these results and then briey outline the metho d of pro of Z If N is a p ositive integer then we let N denote the subgroup of SL consisting of matrices that mo dulo N are of the form The quotient of the upp er half plane by N acting by fractional linear transformations is the complex manifold asso ciated to an ane algebraic curve Y This curve has a natural mo del Y N which N Q C for N is a ne mo duli scheme for elliptic curves with a p oint of exact order N We will let X N denote e curve which contains Y N as a dense Zariski op en subset the smo oth pro jectiv and level N is a holomorphic function f on the upp er half Recall that a cusp form of weight k complex plane H such that for all matrices a b N c d k and all z H we have f az bcz d cz d f z k and jf z j Im z is b ounded on H N of cusp forms of weight k and level N is a nite dimensional complex vector space If The space S k f S N then it has an expansion k X inz f z c f e n n Received by the editors and in revised form January Mathematics Subject Classication Primary G Secondary F Key words and phrases Elliptic curve Galois representation mo dularity The rst author was supp orted by the CNRS The second author was partially supp orted by a grant from the NSF The at Rutgers third author was partially supp orted by a grant from the NSF and an AMS Centennial Fellowship He was working University during much of the research The fourth author was partially supp orted by a grant from the NSF and by the Miller Institute for Basic Science c American Mathematical So ciety and we dene the Lseries of f to b e X s Lf s c f n n n or each prime p jN there is a linear op erator T on S N dened by F k p p X k k f jT z p f z ip p cpz d f apz bcpz d p i for any a b SL Z c d with c mo d N and d p mo d N The op erators T for p jN can b e simultaneously diagonalised on p If f is an eigenform then the the space S N and a simultaneous eigenvector is called an eigenform k corresp onding eigenvalues a f are algebraic integers and we have c f a f c f p p p Let b e a place of the algebraic closure of Q in C ab ove a rational prime and let Q denote the algebraic closure of Q thought of as a Q algebra via If f S N is an eigenform then there is a unique k continuous irreducible representation GalQQ GL Q f is unramied at p and tr Frob a f The existence of is due such that for any prime p jN l f f p p f to Shimura if k Sh to Deligne if k De and to Deligne and Serre if k DS Its irreducibility is due to Rib et if k Ri and Deligne and Serre if k DS Moreover is o dd in the sense that det of complex conjugation is Also is p otentially semistable at in the sense of Fontaine We f can cho ose a conjugate of which is valued in GL O and reducing mo dulo the maximal ideal and f Q semisimplifying yields a continuous representation GalQQ GL F f which up to isomorphism do es not dep end on the choice of conjugate of f GL No w supp ose that G Q is a continuous representation which is unramied outside nitely Q many primes and for which the restriction of to a decomp osition group at is p otentially semistable in we can asso ciate b oth a pair of Ho dgeTate numb ers and a Weil the sense of Fontaine To j Q Q Gal Deligne representation of the Weil group of Q We dene the conductor N of to b e the pro duct over p of the conductor of j and of the conductor of the WeilDeligne representation asso ciated Gal Q Q p p We dene the weight k of to b e plus the absolute dierence of the two Ho dgeTate to j Q Q Gal It is known by work of Cara numb ers of j yol and others that the following two conditions are Q Q Gal equivalent for some eigenform f and some place j f for some eigenform f of level N and weight k and some place j f that When these equivalent conditions are met we call modular It is conjectured by Fontaine and Mazur if G GL Q is a continuous irreducible representation which satises Q is unramied outside nitely many primes j is p otentially semistable with its smaller Ho dgeTate numb er Gal Q Q and in the case where b oth Ho dgeTate numb ers are zero is o dd then is mo dular FM GalQQ GL F Serre Se denes the Next consider a continuous irreducible representation conductor N and weight k of We call modular if for some eigenform f and some place f j We call strongly modular if moreover we may take f to have weight k and level N It is known olo ch and others that for is strongly from work of Mazur Rib et Carayol Gross Coleman and V mo dular if and only if it is mo dular see Di Serre has conjectured that all o dd irreducible are strongly mo dular Se Now consider an elliptic curve E Let denote the representation of GalQQ on the resp E Q E Q Let N E denote the conductor of E It is known that adic Tate mo dule resp the torsion of E the following conditions are equivalent The Lfunction LE s of E equals the Lfunction Lf s for some eigenform f The Lfunction LE s of E equals the Lfunction Lf s for some eigenform f of weight and level N E For some prime the representation is mo dular E For all primes the representation is mo dular E There is a nonconstant holomorphic map X N C E C for some p ositive integer N There is a nonconstant morphism X N E E which is dened over Q The implications and are tautological The implication follows from the characterisation of LE s in terms of The implication follows from a theorem of E Carayol Ca and a theorem of Faltings Fa The implication follows from a construction of Shimura Sh and a theorem of Faltings Fa The implication seems to have b een rst noticed by Mazur Maz When these equivalent conditions are satised we call E modular It has b ecome a standard conjecture that all elliptic curves over Q are mo dular although at the time this conjecture was rst suggested the equivalence of the conditions ab ove may not have b een clear Taniyama made a suggestion along the lines as one of a series of problems collected at the TokyoNikko conference However his formulation did not make clear whether f should b e a mo dular form or some in Septemb er more general automorphic form He also suggested that constructions as in and might help attack this problem at least for some elliptic curves In private conversations with a numb er of mathematicians Shimura suggested that assertions along the lines of and might b e including Weil in the early s true see Sh and the commentary on a in We The rst time such a suggestion app ears in print is Weils comment in We that assertions along the lines of and follow from the main result of that pap er a construction of Shimura and from certain reasonable supp ositions and natural assumptions That assertion is true for CM elliptic curves follows at once from work of Hecke and Deuring Shimura Sh went on to check assertion for these curves approach to Theorem A is an extension of the metho ds of Wiles Wi and of Wiles and one of us Our RT TW We divide the pro of into three cases p is irreducible we show that is mo dular j If E E Gal QQ p p is absolutely irreducible we show that is If j is reducible but j E E E GalQ Q Gal QQ mo dular p p If is reducible and is absolutely reducible then we show that E is j j E E Gal Gal QQ QQ isogenous to an elliptic curve with j or and so from tables of mo dular invariant elliptic curves of low conductor is mo dular is mo dular and then that is In each of cases and there are two steps First we prove that E E mo dular In case this rst step is our Theorem B and in case it is a celebrated theorem of Langlands and Tunnell L T In fact in b oth cases E obtains semistable reduction over a tame extension of Q and the deduction of the mo dularity of from that of was carried out in CDT by an extension of the E E metho ds of Wi and TW In the third case we have to analyse the rational p oints on some mo dular curves help in CDT of small level This we did with Elkies b e as in that theorem Twisting by a quadratic It thus only remained to prove Theorem B Let character we may assume
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