Project : We fix a very concrete goal: check if there exists a curve of genus 5 over the finite field F_47 with 113 rational points. If you are wondering why this problem, you can check this video:
On the way, we will develop tools to address this question which are useful in many other contexts and which open doors to many exciting new problems, not related with finite fields. We will work on two approaches:
1) A direct one through good models of genus 5 curves over finite fields. The generic description is known from a recent article but it would be nice to see how to complete it. We will also look at special strata with many automorphisms and see if we can deal with some cases of our initial questions thanks to this (as indeed all of them would have non-trivial automorphism groups);
2) Schottky locus description: explicit equations describe the locus containing genus 5 curves inside the moduli space of principally polarized abelian varieties of dimension 5. They are given in terms of the so-called theta constants over the complex but the theory holds over any field. Other descriptions of the locus exist as well and, as first step, it would be nice to have a comparison and an implementation of them. Once this is understood, we can test if some well-chosen abelian varieties have a chance to be the Jacobians of our genus 5 curves.
The project relies on a good understanding of the geometry of curves for 1) and on knowledge of theta functions for the beginning of 2) and of the work developed in https://arxiv.org/abs/2004.08315 for the most advanced parts.
Although we will recall and study some parts of the necessary theory during the two weeks, it is hopeless to arrive unprepared to work on this project. If you want to participate, you will need to study the recorded courses
which nicely cover all the necessary material.
In order to help you, a chat platform is open so that you can ask your questions to the speakers. Send an email to christophe.ritzenthaler@univ-rennes1.fr to register on this platform.
