Jan Steffen Müller

Computing canonical heights using arithmetic intersection theory

The canonical height ĥ on an abelian variety A defined over a global field k is an object of fundamental importance in the study of the arithmetic of A. For many applications it is required to compute ĥ(P ) for a given point P ∈ A(k). For instance, given generators of a subgroup of the Mordell-Weil group A(k) of finite index, this is necessary for most known approaches to the computation of generators of the Mordell-Weil group A(k). Furthermore, the regulator of A(k), which appears in the statement of the conjecture of Birch and Swinnerton-Dyer, is defined in terms of the canonical height and thus we need the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank.

A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p- adic L-function of a modular abelian variety and showing it satises the appropriate interpolation property. We describe the techniques used to formulate the conjecture and give evidence supporting the conjecture in the case when the modular abelian variety is of dimension 2.

Explicit Kummer surface formulas for arbitrary characteristic

If C is a curve of genus 2 defined over a field k and J is its Jacobian, then we can associate a hypersurface K in P to J , called the Kummer surface of J . Flynn has made this construction explicit in the case that the characteristic of k is not 2 and C is given by a simplified equation. He has also given explicit versions of several maps defined on the Kummer surface and shown how to perform arithmetic on J using these maps. In this paper we generalize these results to the case of arbitrary characteristic.

Computing integral points on hyperelliptic curves using quadratic Chabauty

We give a method for the computation of integral points on a hyperelliptic curve of odd degree over the rationals whose genus equals the Mordell-Weil rank of its Jacobian. Our approach consists of a combination of the p-adic approximation techniques introduced in previous work with the Mordell-Weil sieve.

LOCAL HEIGHTS ON ELLIPTIC CURVES AND INTERSECTION MULTIPLICITIES

In this short note we prove a formula for local heights on elliptic curves over number fields in terms of intersection theory on a regular model over the ring of integers.