Elliptic curves are projective and hence compact in many senses, and there are many nonisomorphic elliptic curves. The current interest in qcurves, it is fair to say, began with. We end with a short discussion of ladic representations of elliptic curves and understand. Also, for these adic representations, conjugacy invariants such as the trace and determinant of the images of frobenius elements can be computed. A lower bound for the canonical height on elliptic curves over abelian extensions. Serre, abelian ladic representations and elliptic curves. Abelian l adic representations and elliptic curves book. Abelian ladic representations and elliptic curves revised reprint of the 1968 original. Review of abelian l adic representations and elliptic curves kenneth a. Before discussing ladic representations of gq further, let us take a moment to look at ladic representations of gq p. Buy abelian ladic representations and elliptic curves research notes in mathematics on. Chapter iv l adic representations attached to elliptic curves 1 preliminaries 1.
An elliptic curve is an abelian variety that is, it has a multiplication defined algebraically, with respect to which it is an abelian group and o serves as the identity element. An elliptic curve over q is a serre curve if its attached galois representation is as large as possible, and it is known that most elliptic curves over q are of this type. Galois representations of elliptic curves and abelian. We should also mention here that for elliptic curves there are some evidences for the existence of such nonabelian padic lfunctions offered in 4, 10 and also some computational evidences. The motivation for considering these varieties comes. Abelian ladic representations and elliptic curves crc. The method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu2,2.
Abelian ladic representations and elliptic curves crc press book this classic book contains an introduction to systems of ladic 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. The lseries of the complex representation is congruent the lseries of that elliptic curve modulo some prime ideal lying above 5. Galois action on division points of abelian varieties with real multiplications. Moving away from elliptic curves, we now consider a wider class of examples of p. Modularity of some elliptic curves over totally real fields. A lower bound for the canonical height on elliptic curves. The initial chapters are devoted to the abelian case complex multiplication, where one. The action of galois on the tate modules of an elliptic curve gives rise to a family of 2dimensional adic representations. Galois representations attached to elliptic curves algant. 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. We prove the one, two, and threevariable iwasawagreenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p. For example, if l is a galois extension of a number field k, the ring of integers o l of l is a galois module over o k for the galois group of lk see hilbertspeiser theorem. In fact, the isomorphy class of e is, over ks, determined by an element je. Elliptic curves are believed to provide good security.
For some large galois extensions lof k containing all ppower roots of. On some local to global phenomena for abelian varieties core. Pink let a be an abelian variety of dimension g over a global. Cmi summer school notes on padic hodge theory preliminary version 5. 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. This circumstance presents a welcome excuse for writing about the subject, and for placing serres book. Other readers will always be interested in your opinion of the books youve read. Abelian varieties, ladic representations, and lindependence people. Galois representations attached to elliptic curves 6 1. Abelian ladic representations and elliptic curves mcgill university.
Newest galoisrepresentations questions mathematics. A qcurve over k is an elliptic curve over k which is isogenous to all its galois conjugates. Exterior square l functions, herve jacquet and joseph shalika. Many objects that arise in number theory are naturally galois representations.
We then apply elliptic curves to two cryptographic problemsfactoring integers and constructing publickey cryptosystems. Jones n almost all elliptic curves are serre curves. Review of abelian l g n c university of california, berkeley. Mcgill university lecture notes written with the collaboration of.
Abelian ladic representations and elliptic curves 1st. Elliptic curves we introduce elliptic curves and describe how to put a group structure on the set of points on an elliptic curve. Gajda and petersen 2011 yes wojciech gajda adam mickiewicz universitypoznaabelian varieties over function. Support problem for the intermediate jacobians of ladic. On the vanishing of cohomologies of padic galois representations associated with elliptic curves jerome t.
An old lecture notes on elliptic curves and modular forms describes how to compute the equation of a given elliptic curve. All rational elliptic curves arise from modular forms. Surjectivity of mod 2 n representations of elliptic curves. Abelian varieties, ladic representations, and lindependence m. A brief introduction to galois representations attached to. The iwasawa main conjectures for gl 2 springerlink. Galois representations attached to eigenforms with nebentypus. Constructing elliptic curves from galois representations volume 154 issue 10 andrew snowden, jacob tsimerman. Constructing elliptic curves from galois representations.
Abelian ladic representations and elliptic curves jean pierre serre on. F over which the jacobian of our modular curves still have relevant abelian surface factors with mordellweil rank 0. Professor siman wong let f be a number field and let a be an abelian algebraic group. If is a rational prime distinct from the characteristic of k, the. Abelian ladic representation and elliptic curves semantic scholar. Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point o. Review of abelian ladic representations and elliptic curves. Math department, berkeley ca 94720 addisonwesley has just reissued serres 1968 treatise on ladic representations in their advanced book classics series. Elliptic curves with surjective adelic galois representations.
Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Galois representations and elliptic curves 3 from a representation g. Abelian varieties over function fields and independence of. Review of abelian ladic representations and elliptic curves kenneth a. Silvermanlower bound for the canonical height on elliptic curves. Elkies, n elliptic curves with 3adic galois representation surjective mod 3 but not mod 9 preprint, 2006 2. Glv of a padic lie group, where v is a vector space over q p, we get a representation of the lie algebra of g, denoted lieg.
Automorphic forms and the cohomology of vector bundles on shimura varieties, michael harris. Abelian varieties, ladic representations, and lindependence. Superisolated elliptic curves and abelian surfaces in. This classic book contains an introduction to systems of ladic representations, a topic of great importance in number theory and algebraic geometry, as reflected. By the kronecker weber theorem, every abelian extension of q lies inside some cy. Math department, berkeley ca 94720 addisonwesley has just reissued serres 1968 treatise on l adic representations in their advanced book classics series. Modularity of some elliptic curves over totally real fields the harvard community has made this. We would like to show you a description here but the site wont allow us. N u m b e r2 2 2i ng r u n d l e h r e nd e r mathematischen wissenschaften.
Nonabelian congruences between lvalues of elliptic. Recently i read serres wonderful book abelian ladic representations and elliptic curves. The initial chapters are devoted to the abelian case complex multiplication, where one finds a nice correspondence between. Doi link for abelian l adic representations and elliptic curves. We exhibit a modular curve of level 6 that completes a set of modular curves which parametrise nonserre curves. The simplest nonabelian example april 8, 20 introduction this talk bears on a result called the modularity theorem. Abelian l adic representations and elliptic curves. Abelian ladic representations and elliptic curves research notes. Jeanpierre serre abelian ladic representations and elliptic curves, j. In the classical case of adic galois representations associated to elliptic curves, there are theorems concerning when these representations are surjective. The theory of ladic representations is an outgrowth of the study of abelian varieties in positive characteristic, which was initiated by hasse and deuring see, e. Review of abelian ladic representations and elliptic curves by jp. Abelian ladic representations and elliptic curves, any edition, for example, research notes in mathematics 7, a k peters, 1998. Support problem for the intermediate jacobians of ladic representations article in journal of number theory 1001.