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34
Large Torsion Subgroups Of Split Jacobians Of Curves Of Genus Two Or Three
 FORUM MATH
, 1998
"... We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus2 curves over Q whose Jac ..."
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Cited by 33 (7 self)
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We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus3 curve ) = 0, whose Jacobian has 864 rational torsion points.
Elliptic curves with large rank over function fields
, 2001
"... We produce explicit elliptic curves over Fp(t) whose MordellWeil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and SwinnertonDyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. I ..."
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Cited by 28 (1 self)
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We produce explicit elliptic curves over Fp(t) whose MordellWeil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and SwinnertonDyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic curves over number fields.
Elliptic Curve Discrete Logarithms and the Index Calculus
"... . The discrete logarithm problem forms the basis of numerous cryptographic systems. The most effective attack on the discrete logarithm problem in the multiplicative group of a finite field is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller ..."
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Cited by 23 (4 self)
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. The discrete logarithm problem forms the basis of numerous cryptographic systems. The most effective attack on the discrete logarithm problem in the multiplicative group of a finite field is via the index calculus, but no such method is known for elliptic curve discrete logarithms. Indeed, Miller [23] has given a brief heuristic argument as to why no such method can exist. IN this note we give a detailed analysis of the index calculus for elliptic curve discrete logarithms, amplifying and extending miller's remarks. Our conclusions fully support his contention that the natural generalization of the index calculus to the elliptic curve discrete logarithm problem yields an algorithm with is less efficient than a bruteforce search algorithm. 0. Introduction The discrete logarithm problem for the multiplicative group F q of a finite field can be solved in subexponential time using the Index Calculus method, which appears to have been first discovered by Kraitchik [14, 15] in the 192...
The Xedni Calculus And The Elliptic Curve Discrete Logarithm Problem
 Designs, Codes and Cryptography
, 1999
"... . Let E=Fp be an elliptic curve defined over a finite field, and let S ..."
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Cited by 19 (1 self)
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. Let E=Fp be an elliptic curve defined over a finite field, and let S
The analytic rank of J0(q) and zeros of automorphic Lfunctions
"... We study zeros of the Lfunctions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the Lfunctions at the central critical point s = 1 2 is absolutely bounded. On the Birch and SwinnertonDyer conjecture, this provides ..."
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Cited by 16 (11 self)
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We study zeros of the Lfunctions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the Lfunctions at the central critical point s = 1 2 is absolutely bounded. On the Birch and SwinnertonDyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the same order of magnitude as what is expected to be true. 1
The Average Rank Of An Algebraic Family Of Elliptic Curves
 J. REINE ANGEW. MATH
, 1997
"... Let E=Q(T ) be a oneparameter family of elliptic curves. Assuming various standard conjectures, we give an upper bound for the average rank of the fibers E t (Q) with t 2 Z, improving earlier estimates of FouvryPomykala and Michel. We also show how certain assumptions about the distribution of zer ..."
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Cited by 16 (2 self)
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Let E=Q(T ) be a oneparameter family of elliptic curves. Assuming various standard conjectures, we give an upper bound for the average rank of the fibers E t (Q) with t 2 Z, improving earlier estimates of FouvryPomykala and Michel. We also show how certain assumptions about the distribution of zeros of Lseries might help explain the experimentally observed fact that the average rank of the fibers appears to be strictly larger than the naive expected value of rank E(Q(T )) + 1/2.
Investigations of zeros near the central point of elliptic curve Lfunctions
"... We explore the effect of zeros at the central point on nearby zeros of elliptic curve Lfunctions, especially for oneparameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman’s Specialization Theorem, for t sufficiently large the Lfunction of each curve Et in t ..."
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Cited by 16 (4 self)
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We explore the effect of zeros at the central point on nearby zeros of elliptic curve Lfunctions, especially for oneparameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman’s Specialization Theorem, for t sufficiently large the Lfunction of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 oneparameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the normalized differences are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from oneparameter families of rank 0 or 2 over Q. 1
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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Cited by 14 (3 self)
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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...
Elliptic curves and analogies between number fields and function fields
 HEEGNER POINTS AND RANKIN LSERIES, EDITED BY HENRI DARMON AND SHOUWU
, 2004
"... Wellknown analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works ..."
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Cited by 13 (0 self)
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Wellknown analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works on Heegner points and Gross–Zagier formulas in the function field context; these works lead to a complete proof of the conjecture of Birch and SwinnertonDyer for elliptic curves of analytic rank at most 1 over function fields of characteristic> 3. In the second part of the paper, we review the fact that the rank conjecture for elliptic curves over function fields is now known to be true, and that the curves which prove this have asymptotically maximal rank for their conductors. The fact that these curves meet rank bounds suggests interesting problems on elliptic curves over number fields, cyclotomic fields, and function fields over number fields. These