Néron-Tate heights on the Jacobians of high-genus hyperelliptic curves

We use Arakelov intersection theory to study heights on the Jacobians of high-genus hyperelliptic curves. The main results in this thesis are: 1) new algorithms for computing Neron-Tate heights of points on hyperelliptic Jacobians of arbitrary dimension, together with worked examples in genera up to 9 (pre-existing methods are restricted to genus at most 2 or 3). 2) a new definition of a naive height of a point on a hyperelliptic Jacobian of arbitrary dimension, which does not make use of a projective embedding of the Jacobian or a quotient thereof. 3) an explicit bound on the difference between the Neron-Tate height and this new naive height. 4) a new algorithm to compute sets of points of Neron-Tate height up to a given bound on a hyperelliptic Jacobian of arbitrary dimension, again without making use of a projective embedding of the Jacobian or a quotient thereof

Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians

In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.Comment: 24 page

Canonical heights on the jacobians of curves of genus 2 and the infinite descent

We give an algorithm to compute the canonical height on a Jacobian of a curve of genus 2. The computations involve only working with the Kummer surface and so lengthy computations with divisors in the Jacobian are avoided. We use this height algorithm to give an algorithm to perform the infinite descent stage of computing the Mordell-Weil group. This last stage is performed by a lattice enlarging procedure