Stroeker and Tzanakis gave convincing numerical and heuristic evidence for the fact that in their Ellog method a certain parameter λ plays a decisive role in the size of the final bound for the integral points on elliptic curves. Furthermore, they provided an algorithm to determine the MordellWeil basis of the curve which corresponds to the optimal choice of λ. In this paper we show that workin...

Introduction. Let C be a (smooth, projective, absolutely irreducible) curve of genus g > 2 over a number field K. Faltings [Fa1, Fa2] proved that the set C(K) of K-rational points of C is finite, as conjectured by Mordell. The proof can even yield an effective upper bound on the size #C(K) of this set (though not, in general, a provably complete list of points); but this bound depends on the ar...

We give an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use Kedlaya’s algorithm to compute Coleman inte...

Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{F}_\infty)$ consisting of rational elliptic curve $E$ and $p$-adic Lie extension $\mathcal{F}_\infty$ $\mathbb{Q}$, is the $p$-primary Selmer group $Sel_{p^\infty}(E/\mathcal{F}_\infty)$ over $\mathcal{F}_\infty$. In this paper, we study arithmetic statistics for algebraic structure group. The results provide insights into asymptotics ...

We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.

CONTENTS We analyze the 128-dimensional Mordell-Weil lattice of a cer1 . Introduction j -a jn elliptic curve over the rational function field k(t)f where k is 2. Statement of Results a finite field of 2 elements. By proving that the elliptic curve 3. Proof of Rank, Discriminant and Tate-Safarevic Group has trivial Tate-Safarevic group and nonzero rational points of 4. Proof of Minimal Norm, Den...

The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve deened over a eld K by 2-descent. The results lead to some simpliications to the method rst presented in (Birch and Swinnerton-Dyer, 1963), and can be applied to give a more eecient algorithm for determining Mordell-Weil groups over Q, as well as being...

The purpose of these notes is to describe the notion of an Euler system, a collection of compatible cohomology classes arising from a tower of fields that can be used to bound the size of Selmer groups. There are applications to the study of the ideal class group, Iwasawa’s main conjecture, Mordell-Weil group of an elliptic curve, X (the Safarevich-Tate group), Birch-Swinnerton-Dyer conjecture,...

We prove that the Mordell-Tornheim zeta value of depth r can be expressed as a rational linear combination of products of the Mordell-Tornheim zeta values of lower depth than r when r and its weight are of different parity.