The aim of this course was to explain the language of schemes and to explain the first few pages of Serre's great survey
article Zeta and
L-functions. Along the way we gave Bombieri's very short proof of the Rimeann-Hypothesis for
curves.
Research
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.