Albert-Ludwigs-University Freiburg

Department of Mathematics

Section of Pure Mathematics

Arithmetic Geometry Group

Stephen Meagher

Department of Mathematics
Albert-Ludwigs-University Freiburg
Eckerstrasse 1, Room 418
79104 Freiburg im Breisgau, Germany

Telephone: +49 (0) 761 203-5598

Here is my cv.


Summer Term 2009


My research area is arithmetic geometry, which is a mixture of number theory and algebraic geometry. During my thesis I worked on problems related to rational points on curves over finite fields. I approached this problem via the Jacobian variety of an algebraic curve. This leads to issues to do with when an Abelian variety is a Jacobian and trying to detect this from the null values of the associated theta functions of the Abelian variety. Thus quickly one is led into arithmetic problems to do with moduli spaces of curves and abelian varieties of low genus and problems to do with forms or twists of algebraic varieties.

In Chapter 1 of my thesis I specifically dealt with a problem of Serre on determining, from algebraic or analytic theta nulls, when a three dimensional Abelian variety is a Jacobian over its ground field. Christophe Ritzenthaler, Gilles Lachaud and Alexey Zykin also worked on this problem but from a different perspective.

In Chapter 2 of my thesis I studied twists of genus three curves over finite fields and how one can computer their numbers of points in terms of a 1-cocycle. An expanded version of this work has been completed recently with Jaap Top.

More recently I have been working with Robert Carls on the relations satisfied by the theta nulls of ordinary abelian varieties in a given isogeny class. Interestingly we can actually write down equations which characterise these theta nulls. We hope to apply this the computational problem of comptuting an Abelian varietiey with a specified Zeta function and also to theoretical problems to with how an isogeny class lies inside the moduli space.


PhD Thesis

Last Modified June 2010