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. Abelian ladic representations and elliptic curves mcgill university. 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. Review of abelian ladic representations and elliptic curves by jp. Abelian varieties, ladic representations, and lindependence m. Galois representations attached to elliptic curves algant. The lseries of the complex representation is congruent the lseries of that elliptic curve modulo some prime ideal lying above 5.
Review of abelian l g n c university of california, berkeley. 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 notes. We would like to show you a description here but the site wont allow us. If is a rational prime distinct from the characteristic of k, the. Galois representations attached to elliptic curves 6 1.
Exterior square l functions, herve jacquet and joseph shalika. The initial chapters are devoted to the abelian case complex multiplication, where one finds a nice correspondence between. Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point o. Elliptic curves are believed to provide good security. Jeanpierre serre abelian ladic representations and elliptic curves, j. Silvermanlower bound for the canonical height on elliptic curves. The current interest in qcurves, it is fair to say, began with. Review of abelian l adic representations and elliptic curves kenneth a. A lower bound for the canonical height on elliptic curves. Abelian ladic representation and elliptic curves semantic scholar. 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. This circumstance presents a welcome excuse for writing about the subject, and for placing serres book. Galois representations attached to eigenforms with nebentypus. Mcgill university lecture notes written with the collaboration of.
Galois action on division points of abelian varieties with real multiplications. 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. The iwasawa main conjectures for gl 2 springerlink. Support problem for the intermediate jacobians of ladic representations article in journal of number theory 1001. Support problem for the intermediate jacobians of ladic. Nonabelian congruences between lvalues of elliptic. A qcurve over k is an elliptic curve over k which is isogenous to all its galois conjugates.
Pink let a be an abelian variety of dimension g over a global. Abelian ladic representations and elliptic curves 1st. Chapter iv l adic representations attached to elliptic curves 1 preliminaries 1. Math department, berkeley ca 94720 addisonwesley has just reissued serres 1968 treatise on ladic representations in their advanced book classics series. N u m b e r2 2 2i ng r u n d l e h r e nd e r mathematischen wissenschaften. On the other hand, in 60, mazur discussed various questions and conjectures involving cm points, padic heights, and twovariable padic lfunctions attached to elliptic curves.
Galois representations of elliptic curves and abelian. Modularity of some elliptic curves over totally real fields. Abelian ladic representations and elliptic curves, any edition, for example, research notes in mathematics 7, a k peters, 1998. Abelian l adic representations and elliptic curves. 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. A brief introduction to galois representations attached to. 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. Automorphic forms and the cohomology of vector bundles on shimura varieties, michael harris. Abelian ladic representations and elliptic curves crc. On the vanishing of cohomologies of padic galois representations associated with elliptic curves jerome t.
Recently i read serres wonderful book abelian ladic representations and elliptic curves. Professor siman wong let f be a number field and let a be an abelian algebraic group. Elkies, n elliptic curves with 3adic galois representation surjective mod 3 but not mod 9 preprint, 2006 2. Modularity of some elliptic curves over totally real fields the harvard community has made this.
We end with a short discussion of ladic representations of elliptic curves and understand. Before discussing ladic representations of gq further, let us take a moment to look at ladic representations of gq 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. Review of abelian ladic representations and elliptic curves. Elliptic curves with surjective adelic galois representations. By the kronecker weber theorem, every abelian extension of q lies inside some cy. On some local to global phenomena for abelian varieties core. Superisolated elliptic curves and abelian surfaces in.
Elliptic curves are projective and hence compact in many senses, and there are many nonisomorphic elliptic curves. The motivation for considering these varieties comes. Abelian ladic representations and elliptic curves revised reprint of the 1968 original. Other readers will always be interested in your opinion of the books youve read. Many objects that arise in number theory are naturally galois representations. In the classical case of adic galois representations associated to elliptic curves, there are theorems concerning when these representations are surjective. Buy abelian ladic representations and elliptic curves research notes in mathematics on. 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. We then apply elliptic curves to two cryptographic problemsfactoring integers and constructing publickey cryptosystems. The action of galois on the tate modules of an elliptic curve gives rise to a family of 2dimensional adic representations. The initial chapters are devoted to the abelian case complex multiplication, where one. Review of abelian ladic representations and elliptic curves kenneth a. Also, for these adic representations, conjugacy invariants such as the trace and determinant of the images of frobenius elements can be computed. All rational elliptic curves arise from modular forms. We exhibit a modular curve of level 6 that completes a set of modular curves which parametrise nonserre curves. Serre, abelian ladic representations and elliptic curves. Abelian ladic representations and elliptic curves jean pierre serre on. 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. Doi link for abelian l adic representations and elliptic curves. 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. Constructing elliptic curves from galois representations.
F over which the jacobian of our modular curves still have relevant abelian surface factors with mordellweil rank 0. Surjectivity of mod 2 n representations of elliptic curves. Abelian varieties, ladic representations, and lindependence people. Constructing elliptic curves from galois representations volume 154 issue 10 andrew snowden, jacob tsimerman. In fact, the isomorphy class of e is, over ks, determined by an element je. An old lecture notes on elliptic curves and modular forms describes how to compute the equation of a given elliptic curve. Jones n almost all elliptic curves are serre curves. A lower bound for the canonical height on elliptic curves over abelian extensions.
Elliptic curves we introduce elliptic curves and describe how to put a group structure on the set of points on an elliptic curve. Abelian varieties over function fields and independence of. Math department, berkeley ca 94720 addisonwesley has just reissued serres 1968 treatise on l adic representations in their advanced book classics series. Abelian l adic representations and elliptic curves book. Moving away from elliptic curves, we now consider a wider class of examples of p. 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. Abelian varieties, ladic representations, and lindependence.
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