We address several questions arising from the birch and swinnertondyer bsd conjecture in this general context. By the riemannroch theorem, an elliptic curve can be described ex. In particular, the proof of the conjecture of birch and swinnertondyer for elliptic curves over q of analytic rank at most one grosszagier, kolyvagin depends essentially on their properties. Q, in other words for rational points on the curve. Heegner point these keywords were added by machine and not by the authors. The birch and swinnertondyer conjecture fore,asstatedbytate50,conj.
The birchswinnertondyer conjecture and heegner points. Kolyvagin later used heegner points to construct euler systems, and used this to prove much of the birchswinnertondyer conjecture for rank 1 elliptic curves. Heegner points, starkheegner points, and values of lseries. Historically, the cm points on modular curves were rst used by heegner. Elliptic curves over finite fields are used in some cryptographic applications as well as for integer factorization. Heegner points and the arithmetic of elliptic curves over. The faltingsmoriwaki modular height and isogenies of elliptic curves ikoma, hideaki, journal of mathematics of kyoto university, 2008. In this case, it is the euler system of heegner points. Pdf heegner points, stark heegner points, and values of l. The birch and swinnertondyer form ula for elliptic curves of analytic rank one. Stark heegner points on elliptic curves defined over imaginary quadratic fields trifkovic, mak, duke mathematical journal, 2006.
The birch and swinnertondyer conjecture clay mathematics institute. Lest i forget, i should stress that when i say elliptic curve i will always mean elliptic curve. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. Combined with the general grosszagier formula yzz and the ppart of the birch and swinnertondyer formula for. The birch and swinnertondyer formula for elliptic curves. Buy heegner modules and elliptic curves lecture notes in mathematics on free shipping on qualified orders. Read download rational points on elliptic curves pdf pdf. The number of independent basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve. Periods and points attached to quadratic algebras, by massimo bertolini, henri darmon and peter green. We show this conjectureholdswhenever ehasarational3isogeny.
We study heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by shimura curves attached to a rather general type of quaternionic orders. We establish a congruence formula between padic logarithms of heegner points for two elliptic curves with the same mod. Heegner points and elliptic curves of large rank over. Heegner points on both modular curves and elliptic curves over global fields of any characteristic form the topic of this research monograph. Fast factorization of integers there is an algorithm for factoring integers that uses elliptic curves and is in many respects better than previous algorithms. A theorem of mordell in 1922 asserts that the abelian group epqq is finitely gen erated. It places a special emphasis on the construction of rational points on elliptic curves, the birch and swinnertondyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. Faster computation of heegner points on elliptic curves. The birchswinnertondyer conjecture and heegner points mit math. On the 2part of the birch and swinnertondyer conjecture for. This method was crucial to the proof of grosszagier, which proves the rank one case of the birch. An elementary and computational approach to heegner points. More precisely under the birch and swinnertondyer conjecture conjecture birch and swinnertondyer l0e. But a1 is a line, hence p0 consists of a single point called the point at in.
Rational points on elliptic curves alexandru gica1 april 8, 2006 1notes, lateximplementation and additional comments by mihai fulger. Elliptic curves with complex multiplication and the. Typically, the general idea in these applications is that a known algorithm which makes use of certain finite groups is rewritten to use the groups of rational points of elliptic curves. Its rational points form a strass form y group, which we denote by eq.
From this product, one naturally obtain a product of the values of riemann zeta function. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a. Beilinsonkato and beilinsonflach elements, colemanrubin. The heegner point zk appearing in this formula and also appearing in. Ranks of elliptic curves the goal of this seminar is to give an account of the main ideas entering in the proof of the following recent result by bhargava, skinner and wei zhang see bsz and the references therein, stating that theorem 0. Elliptic curves with complex multiplication and the conjecture of birch and swinnertondyer karl rubin. The birch and swinnertondyer formula for elliptic curves of. However, heegner points are futile in analytic rank greater than one since they are torsion. The birch and swinnertondyer bsd conjecture is one of the millennium problems. We develop this loosely formulated program in the simple but nontrivial setting where ais an elliptic. An elliptic curve ekis the projective closure of a plane a ne curve y2 fx where f2kx is a monic cubic polynomial with distinct roots in k. These points are the key to the most decisive progress. Heegner modules and elliptic curves lecture notes in.
Chow heegner points on cm elliptic curves 3 the relations if any between the resulting systems of chow heegner points and leading terms of lseries attached to a. Cambridge journalof mathematics volume5,number3,369434,2017 the birch and swinnertondyer formula for elliptic curves of analytic rank one dimitar jetchev. Heegner points and elliptic curves of large rank over function. Pdf heegner points, stark heegner points, and values of. In mathematics, a heegner point is a point on a modular elliptic curve that is the image of a quadratic imaginary point of the upper halfplane. Pdf elliptic curves of rank two and generalised kato classes. Rational points on elliptic curves institute of mathematics. Some remarks on heegner point computations by mark. Birch and swinnertondyerconjecture shafarevichtategroups let e be an elliptic curve over q and let k be a quadratic imaginary. Every line through two of these points also passes through a third inflection point. Congruences between heegner points and quadratic twists of elliptic curves danielkrizandchaoli abstract. Elliptic curves have been used to shed light on some important problems that, at. Heegner points and the arithmetic of elliptic curves. Elliptic curves, second edition dale husemoller springer springer new york berlin heidelberg hong kong london milan paris tokyo.
The birch and swinnertondyer formula for elliptic curves of analytic. We also establish the explicit grosszagier formulae for these heegner points and give variants of the birch and swinnertondyer conjecture of the related elliptic curves. In an attempt to extend rubins theorem to elliptic curves without complex. The main theme of the book is the theory of complex multiplication, heegner points, and some conjectural variants. Here pis a prime number, x pefqrp8sis the pprimary part of tateshafarevich. Elliptic curves overqare equipped with a systematic collection of heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. We study the arithmeticof e over ring class extensions of k,withparticular focus on the case when e has analytic rank at least 2 over q.
This comparison of elliptic units and heegner points has intriguing consequences for the construction of rational points on cm elliptic curves via the special values of the katz padic lfunction of an imaginaryquadratic. We discuss old and new theoretical evidence toward our conjecture and in particular we fully prove it, under mild conditions. We formulate a multivariable padic birch and swinnertondyer conjecture for pordinary elliptic curves a over number fields k. Dick gross and i were invited to talk about heegner points from a historical point of view. They were defined by bryan birch and named after kurt heegner, who used similar ideas to prove gausss conjecture on imaginary quadratic fields of class number one. Heegner point computations via numerical padic integration. The main theme of the book is the theory of complex multiplications, heegner points, and some conjectural variants.
It generalizes the onevariable conjecture of mazur, tate, and teitelbaum, who studied the case k q and the phenomenon of exceptional zeros. To set the stage for heegner points, one may compare the state of the theory of elliptic curves over the rationals, eq for short, in the 1960s and in the 1970s. A positive proportion of elliptic curves over q satis es the birch and swinnertondyer conjecture. Using heegner points on elliptic curves, we construct points of in. E is an elliptic curve defined over q, then the algebraic rank of e equals the. Rubin 1991 showed that for elliptic curves defined over an imaginary quadratic field k with complex multiplication by k, if the lseries of the elliptic curve was not zero at s 1, then the ppart of the tateshafarevich group had the order predicted by the birch and swinnertondyer conjecture, for all primes p 7. Based on the numerical evidence that he gathered, he observed that the heights of these points. Heegner points and elliptic curves of large rank over function fields, by henri darmon, 317322 pdf file. The heegner module of an elliptic curve is an original concept introduced in this text. It places special emphasis on the construction of rational points on elliptic curves, the birch and swinnertondyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. Birch let e he an elliptic curve defined over the rationals q. Heegner points on elliptic curves with a rational torsion point 5 by the work of gross and zagier gz for a squarefree d and the work of zhang zh for a general d, we know that if p. Department of mathematics, ohio state university, 231 w. Given an elliptic curve eover q, a celebrated conjecture of goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 resp.
Rational points on modular elliptic curves henri darmon. We give an overview of the theory of heegner points for elliptic curves. A survey 3 there is also an analogous short exact sequence which gives a cohomological description of rational points on an elliptic curve eover a number eld f. We study heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by shimura curves attached to a rather general type of quaternionic. Moreover, in gz86, grosszagier show that heegner points satisfy a formula that links their height to the value in 1 of the. Heegner points, stark heegner points, and values of lseries henri darmon abstract.
So kolyvagins theorem can be applied to every elliptic curve over q. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of kolyvagingrosszagier and the weak parity theorem of tim and vladimir. Over the complex numbers, every elliptic curve has nine inflection points. Elliptic curves and analogies between number fields and function fields, by douglas ulmer, 285315 pdf file editorial comment. Elliptic curves points on elliptic curves elliptic curves can have points with coordinates in any. Brown proved the birchswinnertondyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic. Heegner points at eisenstein primes and twists of elliptic curves danielkrizandchaoli abstract. The birch and swinnertondyer conjecture admits an extension to elliptic curves twisted by artin representations which arises very naturally in the context of the modu larity questions framed above. This process is experimental and the keywords may be updated as the learning algorithm improves. Heegner points and rankin lseries msri publications volume 49, 2004 heegner points and representation theory benedict h. This essay starts by first explaining, for elliptic curves defined overq, the statement of the conjecture of birch and swinnertondyer.
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