And since a q curve ek has an isogeny class which is. Abelian l adic representations and elliptic curves filed on december 28th, 2008. In his 1968 book abelian l adic representations and elliptic curves, serre removed this hypothesis. Note that another equivalent way of stating the theorem is to say that. Oct 01, 1999 abelian l adic representations and elliptic curves, benjamin, new york 1968. The classical shimurataniyama conjecture 18 is the statement that every elliptic curve eover q is associated to a cuspidal hecke newform fof the group 0n. This means that our abelian variety is just an elliptic curve, and that gsp 2d gsp 2 gl. The current interest in q curves, it is fair to say, began with. For this reason, the theory of p adic representations of galois groups turns out to be a very convenient framework for studying the arithmetic of l functions. The set of points on the curve over a nite eld f p, ef p, is a nite abelian group.
I use the frattini lifting theorem to turn the question. Using localization with respect to s, we are able to define a characteristic. The p adic l function for sym2e always vanishes at s 1, even though the complex l function does not have a zero. Then im going to describe some applications to the bsd conjecture, especially in the case of analytic. Abelian ladic representations and elliptic curves taylor. For an abelian variety a over q the lfactor at a prime p of good reduction may be.
The l series of the complex representation is congruent the l series of that elliptic curve modulo some prime ideal lying above 5. A morphism of the associated representations of lie algebras is the same as a morphism. On the surjectivity of the galois representations associated. Buy abelian l adic representations and elliptic curves research notes in mathematics on free shipping on qualified orders abelian l adic representations and elliptic curves research notes in mathematics.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Abelian ladic representations and elliptic curves research notes. Review of abelian ladic representations and elliptic curves. Algebraically elliptic, p adic, linear algebras over complex, solvable classes s. This circumstance presents a welcome excuse for writing about the subject, and for placing serres book. There has been prior work on abelian entanglements related to question1.
We show that there exists a smoothly canonical subset. The gl 2 main conjecture for elliptic curves without. Abelian l adic representations and elliptic curves, any edition, for example, research notes in mathematics 7, a k peters, 1998 topics. G of g, which seems to be particularly relevant for arithmetic applications. We are mainly interested in the ladic representations of gq and gqp. Galois groups arising from division points of elliptic.
In the rst case, we say that the elliptic curve is without complex multiplication over kand if this holds true for any extension of k, we say that the elliptic curve eis without complex multiplication. Elliptic curves with surjective adelic galois representations. Reciprocity law for compatible systems of abelian mod galois representations volume 57 issue 6 skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Elliptic curves notes for the 20045 part iii course 28012005 16032005. We will continue to consider the d 1 case for the moment. Serre, abelian l adic representations and elliptic curves w. Im working through delignes formes modulaires et representations l adiques paper and i find one of his constructions particularily ambigious. Wi, tw, and bcdt that all elliptic curves over the rationals are modular. The initial chapters are devoted to the abelian case complex multiplication, where one. A q curve over k is an elliptic curve over k which is isogenous to all its galois conjugates.
The gl 2 main conjecture for elliptic curves without complex. Buy abelian ladic representations and elliptic curves research notes in mathematics on. The action of galois on the tate modules of an elliptic curve gives rise to a family of 2dimensional adic representations. This circumstance presents a welcome excuse for writing about the subject, and for placing serres book in a historical perspective. Galois representations and elliptic curves 3 from a representation g. Browse other questions tagged elliptic curves referencerequest galois representations rt. Math department, berkeley ca 94720 addisonwesley has just reissued serres 1968 treatise on l adic representations in their advanced book classics series. The main title of these lectures refers to the two almost eponymous papers written by karl rubin inventiones, 1991 and by bernadette. One justification for considering ladic representations is that they arise nat urally from. Since we are working with a nite eld, then we have a nite number of points satisfying e.
Im hoping someone can give me a bit of clarificati. In this spirit, we give a complete classi cation of the possible 2 adic images of galois representations associated to noncm elliptic curves over q and, in particular, compute n 2. The case n 2 implies the modularity of elliptic curves. This is called the weierstrass equation for an elliptic curve.
Special values of lfunctions of elliptic curves and. Special values of l functions of elliptic curves and modular forms notes by tony feng for a talk by chris skinner june, 2016 im going to begin by recalling a formula that karl proved for the p adic l function of a cm elliptic curve. Nonabelian congruences between lvalues of elliptic curves. The ladic tate module of an elliptic curve e or abelian group e, see subsection 3. For a number field f, let ef be an elliptic curve with cm by a quadratic field k. Galois representations associated to eto the automorphism group of a non abelian group. Modularity of some elliptic curves over totally real fields. When a is a onedimensional torus, an elliptic curve, or a higherdimensional irreducible abelian variety without complex multiplication, and the associated adic representation is surjective, we give a simple characterization of when the kummer part is the full tate module see theorems 11, 20, 27, 35. Jeanpierre serre, linear representations of finite groups gustafson, w. On the surjectivity of the galois representations associated to noncm elliptic curves volume 48 issue 1 skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Surjectivity of mod 2 n representations of elliptic curves. Chapter iv l adic representations attached to elliptic curves 1 preliminaries 1.
Zp qparising from the tate module of an elliptic curve eover. Glv of a p adic lie group, where v is a vector space over q p, we get a representation of the lie algebra of g, denoted lieg. The modularity of ais established by showing that the p adic galois representation v pa. Galois distribution of a class of isogenies of elliptic curves. In particular, i study the surjectivity of a galois representation to a certain subgroup of the automorphism group of a metabelian group. Foreach p, the gf representation ontpe encodes all euler factors at primes. The construction of galois representations, which play a fundamental role in wiles proof of the shimurataniyama conjecture, is given. Moreover, recent develop ments in riemannian number theory 4 have raised the question of whether wiless conjecture is false in the context of subpositive, noetherian, combi natorially p adic elements. This is conjecturally encoded in and most fruitfully studied via the p adic representation of gf. The zariski closure of the image of adic representation. These are non abelian p adic lie extensions of dimension two, which may be constructed as follows. Review of abelian l adic representations and elliptic curves kenneth a.
More precisely, there exists a delignemumford stack m 1,1 called the moduli stack of elliptic curves such. In this paper, we study the behaviour of the hasseweil l functions of elliptic curves, over the socalled false tate curve extensions of q. We can also consider the tate module of an elliptic curve ek given by t e lim en, and v e q z t e. For many structures on elliptic curves over q are invariant under isogeny. This classic book contains an introduction to systems of l adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the taniyamaweil conjecture and fermats last theorem. Automorphic forms and the cohomology of vector bundles on shimura varieties, michael harris. Abelian ladic representations and elliptic curves jean pierre serre download zlibrary. On the one hand, as the image of these representations is often large and nonabelian, the representations furnish a wealth of explicit nonabelian quotients of g. Abelian ladic representations and elliptic curves research. Serre, jp abelianl adic representations and elliptic curves. For an elliptic curve aof conductor n, this means that 1. The theory of l adic representations is an outgrowth of the. A brief introduction to galois representations attached to.
The galois representations considered are on the elliptic curves side the one on. This book provides a comprehensive account of the theory of moduli spaces of elliptic curves over integer rings and its application to modular forms. The goal of the present article is to derive finitely hypercountable, continuous, almost everywhere contra elliptic curves. Let h gl 2z 2 be a subgroup, and ebe an elliptic curve whose 2 adic image is contained in h. The initial chapters are devoted to the abelian case complex multiplication, where one finds a nice correspondence between the l adic. Exterior square l functions, herve jacquet and joseph shalika.
Here the meaning of \associated is that there is an isomorphism between compatible systems of l adic representations of g q. If on the other hand an elliptic curve e has good supersingular reduction, then the. For ladic representations associated to e, much is under. A fundamental arithmetic invariant of eis the zrank of its. Jeanpierre aubin and ivar ekeland, applied nonlinear analysis warga, j. Pdf abelian ladic representation and elliptic curves. To show associativity, we can look at the graphic representation of the elliptic curve. Dec 14, 2011 elkies, n elliptic curves with 3 adic galois representation surjective mod 3 but not mod 9 preprint, 2006 2 jones n almost all elliptic curves are serre curves. In dimension 1, an abelian variety is an elliptic curve a genus1 curve with a rational point e 2x k. Beilinson and beilinsonflach elements in p adic families abstract. Geometric modular forms and elliptic curves, world scientific first 2000 or second edition 2011, chapters 1, 2 and 4 alr. Galois representations school of mathematics institute for. We prove the existence of a canonical ore set s of nonzero divisors in the iwasawa algebra. For instance if eq is an elliptic curve then we have the.
We show that there exists a minimal normal, globally integral algebra equipped with a coalmost everywhere complex. In chapter 4, i show that the image of the outer galois representation in outm is isomorphic to gl 2z. A p adic representation of gf is a continuous linear representation gf glw, where wis a. The essential dimension of a gdimensional complex abelian variety is 2g. Jun 01, 2005 let g be a compact p adic lie group, with no element of order p, and having a closed normal subgroup h such that gh is isomorphic to z p. Abelian l adic representations and elliptic curves, volume 7 of research. In 12, gonz alezjim enez and the second author classi ed all elliptic curves such that the full nth division eld qen is an abelian.
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