CIMPA School on Effective Algebra and the LMFDB
Galois representations and Sato-Tate groups

13-24 January, 2025

Leonardo Colò and David Kohel

Effective Algebra and the LMFDB

The CIMPA school on Effective Algebra and the LMFDB takes place at the Departement of Mathematics, of Makerere University in in Kampala, Uganda, 13-24 January 2025. In addition to general lectures on effective algebra in number theory and arithmetic geometry, specific research projects covering various aspects of the arithmetic of curves, abelian varieties, and applications to cryptography form part of this school

Research Project Galois representations and Sato-Tate groups

The project Galois representations and Sato-Tate groups is a research program for students in the CIMPA school on Effective algebra and the LMFDB, aimed at introducing students to the study of higher dimensional Sato-Tate groups via their explicit irreducible characters. A brief resume of this project with annoted bibliography appears below.

Galois representations and Sato-Tate groups

Worksheets

The research project on Galois representations and Sato-Tate groups focuses on the approach through irreducible characters on compact Lie groups. We proceed from characters on number fields (dimension 0) to elliptic curves (abelian varieties of dimension 1) to genus-2 curves (or their Jacobian surfaces of dimension 2).

Character theory for number fields and annotated version with solutions and Sage code,
Character theory for elliptic curves and annotated version with solutions and Sage code,
Character theory for genus 2 curves and annotated version with solutions and Sage code.

At the conclusion of this school, students should have the tools to understand the role of character theory in arithmetic geometry, approaching the subject from the perspective of arithmetic statistics, and should develop the mathematical and computational tools (using Python/SageMath) to work effectively with this character theory.

Sage worksheets

The following Sage worksheet(s) contain code for working with the explicit characters and their inner products.

Research group presentation

The following beamer template (zip) may be used in preparing the presentation. The final presentation (focusing on the underlying character theory in the context of number fields) is now available.

CIMPA School on Effective Algebra and the LMFDB Galois representations and Sato-Tate groups