this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula,

Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓ-adic Tate

Abstract not found

The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this

. Let (A; ) be a principally polarized abelian variety dened over a global eld k, and let X(A) be its Shafarevich{Tate group. Let X(A) nd denote the quotient of X(A) by its maximal divisible subgroup. Cassels and Tate constructed a non-degenerate pairing X(A) nd X(A) nd ! Q=Z : If A is an elliptic curve, then by a result of Cassels' the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent denitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating non-degenerate pairing on X(A) nd . These criteria are expressed in terms of an element c 2 X(A) nd that is canonically associated to the polarization . In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine eectively whether #X(A) (if nite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperell...

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 genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve ) = 0, whose Jacobian has 864 rational torsion points.

In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af, 1)/ΩA, develop tools for computing with Af, and gather data about f certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of �(Af). We find that there are at least 168 such Af for which the Birch and Swinnerton-Dyer conjecture implies that �(Af) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of �(Af) really divides # �(Af) by constructing nontrivial elements of �(Af) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.

Abstract. It is shown that the obvious method of descending from an element of the 2-Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the Weil-Chatelet group of E. Explicit algorithms for such a method are given. 1.

. Let E=Fp be an elliptic curve defined over a finite field, and let S