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Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
Some remarks on Heegner point computations
, 2004
"... We make some remarks concerning Heegner point computations. One of our goals shall be to give an algorithm to find a nontorsion rational point on a given rank 1 elliptic curve. Much of this is taken from a section in Henri Cohen’s latest ..."
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We make some remarks concerning Heegner point computations. One of our goals shall be to give an algorithm to find a nontorsion rational point on a given rank 1 elliptic curve. Much of this is taken from a section in Henri Cohen’s latest
StarkHeegner points over real quadratic fields
, 2007
"... Motivated by the conjectures of “MazurTateTeitelbaum type” formulated in [BD1] and by the main result of [BD3], we describe a conjectural construction of a global point PK ∈ E(K), where E is a (modular) elliptic curve over Q of prime conductor p, and K is a real quadratic field satisfying suitabl ..."
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Motivated by the conjectures of “MazurTateTeitelbaum type” formulated in [BD1] and by the main result of [BD3], we describe a conjectural construction of a global point PK ∈ E(K), where E is a (modular) elliptic curve over Q of prime conductor p, and K is a real quadratic field satisfying suitable conditions. The point PK is constructed by applying the Tate padic uniformization of E to an explicit expression involving geodesic cycles on the modular curve X0(p). These geodesic cycles are a natural generalization of the modular symbols of Birch and Manin, and interpolate the special values of the HasseWeil Lfunction of E/K twisted by certain abelian characters of K. In the analogy between Heegner points and circular units,
Discretisation for odd quadratic twists
 in Proceedings of the Clay Mathematics Institute Special Week on Ranks of Elliptic Curves and Random Matrix Theory
"... The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point ente ..."
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The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the ShafarevichTate group). We discuss a couple of models and present data on this question. 1.1
Computing rational points on rank 1 elliptic curves via Lseries and canonical heights
 Math. Comp
, 1999
"... Abstract. Let E/Q be an elliptic curve of rank 1. We describe an algorithm which uses the value of L ′ (E,1) and the theory of canonical heghts to efficiently search for points in E(Q) andE(ZS). For rank 1 elliptic curves E/Q of moderately large conductor (say on the order of 10 7 to 10 10)andwitha ..."
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Abstract. Let E/Q be an elliptic curve of rank 1. We describe an algorithm which uses the value of L ′ (E,1) and the theory of canonical heghts to efficiently search for points in E(Q) andE(ZS). For rank 1 elliptic curves E/Q of moderately large conductor (say on the order of 10 7 to 10 10)andwitha generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set E(ZS) contains nontorsion points.
Heegner points, Stark–Heegner points, and values
 of Lseries, inProceedings of the ICM
, 2006
"... Abstract. Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch an ..."
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Abstract. Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and SwinnertonDyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1, arising from the work of GrossZagier and Kolyvagin. In [Da2], it is suggested that Heegner points admit a host of conjectural generalisations, referred to as StarkHeegner points because they occupy relative to their classical counterparts a position somewhat analogous to Stark units relative to elliptic or circular units. A better understanding of StarkHeegner points would lead to progress on two related arithmetic questions: the explicit construction of global points on elliptic curves (a key issue arising in the Birch and SwinnertonDyer conjecture) and the analytic construction of class fields sought for in Kronecker’s Jugendtraum and Hilbert’s twelfth problem. The goal of this article is to survey Heegner points, StarkHeegner points, their arithmetic applications and their relations (both proved, and conjectured) with special values of Lseries attached to modular forms.
THE METHOD OF CHABAUTY AND COLEMAN
"... Abstract. This is an introduction to the method of Chabauty and Coleman, a padic method that attempts to determine the set of rational points on a given curve of genus g ≥ 2. We present the method, give a few examples of its implementation in practice, and discuss its effectiveness. An appendix tre ..."
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Abstract. This is an introduction to the method of Chabauty and Coleman, a padic method that attempts to determine the set of rational points on a given curve of genus g ≥ 2. We present the method, give a few examples of its implementation in practice, and discuss its effectiveness. An appendix treats the case in which the curve has bad reduction. 1. Rational points on curves of genus ≥ 2 We will work over the field Q of rational numbers, although everything we say admits an appropriate generalization to a number field. Let Q be an algebraic closure of Q. For each finite prime p, let Qp be the field of padic numbers (see [Kob84] for the definition). Curves will be assumed to be smooth, projective, and geometrically integral. Let X be a curve over Q of genus g ≥ 2. We suppose that X is presented as the zero set in some P n of an explicit finite set of homogeneous polynomials. We may give instead an equation for a singular (but still geometrically integral) curve in A 2; in this case, it is understood that X is the smooth projective curve birational to this singular curve. Rational points on X can be specified by giving their coordinates. (A little more data may be required if a singular model for X is used.) Let X(Q) be the set of rational points on X. Faltings ’ theorem [Fal83] states that X(Q) is finite. Thus we have the following welldefined problem: Given X of genus ≥ 2 presented as above, compute X(Q). Faltings ’ proof is ineffective in the sense that it does not provide an algorithm for solving this problem, even in principle. In fact, it is not known whether any algorithm is guaranteed to solve the problem. Even the case g = 2 seems hard. Nevertheless there are a few techniques that can be applied: see [Poo02] for a survey. On individual curves these seem to solve the problem often, perhaps even always when used together, though it seems very difficult to prove that they always work. One of the methods used is the method of Chabauty and Coleman.
On the Computation of the Cassels Pairing for certain Kolyvagin classes in the ShafarevichTate group
"... Kolyvagin has shown how to study the ShafarevichTate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit nontrivial elements of the ShafarevichTate group is by proving that a locally trivial Kolyvagin class i ..."
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Kolyvagin has shown how to study the ShafarevichTate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit nontrivial elements of the ShafarevichTate group is by proving that a locally trivial Kolyvagin class is globally nontrivial, which is difficult in practice. We provide a method for testing whether an explicit element of the ShafarevichTate group represented by a Kolyvagin class is globally nontrivial by determining whether the Cassels pairing between the class and another locally trivial Kolyvagin class is nonzero. Our algorithm explicitly computes Heegner points over ring class fields to produce the Kolyvagin classes and uses the efficiently computable cryptographic Tate pairing.
Computing the Cassels Pairing on Kolyvagin Classes in the ShafarevichTate Group
"... Abstract. Kolyvagin has shown how to study the ShafarevichTate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit nontrivial elements of the ShafarevichTate group is by proving that a locally trivial Kolyvagi ..."
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Abstract. Kolyvagin has shown how to study the ShafarevichTate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit nontrivial elements of the ShafarevichTate group is by proving that a locally trivial Kolyvagin class is globally nontrivial, which is difficult in practice. We provide a method for testing whether an explicit element of the ShafarevichTate group represented by a Kolyvagin class is globally nontrivial by determining whether the Cassels pairing between the class and another locally trivial Kolyvagin class is nonzero. Our algorithm explicitly computes Heegner points over ring class fields to produce the Kolyvagin classes and uses the efficiently computable cryptographic Tate pairing. 1