Resolving the incompleteness of Weil-Petersson metric on Teichmüller spaces by taking metric and geodesic completion results in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson metric. We construct a geodesic completion of the Teichmüller space through the formalism of Coxeter complex with the Teichmüller space as its non...

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use...

— We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though stil...

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...

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer 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. Asymptoti...

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the SteinWatkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.

The main result of this note is that there are at most seven rational points (including the one at infinity) on the curve CA with the affine equation y = x+A (where A is a tenth power free integer) when the Mordell-Weil rank of the Jacobian of CA is one. This bound is attained for A = 18 .

In this paper we examine diierences between the two standard methods for computing the 2-Selmer group of an elliptic curve. In particular we focus on practical diierences in the timings of the two methods. In addition we discuss how to proceed if one fails to determine the rank of the curve from the 2-Selmer group. Finally we mention brieey ongoing research i n to generalizing such methods to t...