at Rennes University
Mini-course: Ideal classes and abelian varieties over finite fields.
- The course will take place on 12-13-14-15 November 2018.
- Each lecture will be 2h long.
- This mini-course will be divided in two parts. In the first we will discuss how to compute all ideal classes of a non-maximal order in étale Q-algebras (i.e. products of number fields). In the second part we will introduce abelian varieties over finite fields and we will describe them by mean of a much more concrete category, which, under certain assumptions, can be described in terms of fractional ideals.
- 1) Introduction - Étale Q-algebras, orders, fractional ideals (invertible and non).
- 2) Ideal classes, Picard Groups and Ideal Class Monoids.
- 3) Abelian varieties over finite fields - basic definitions, Honda-Tate theory.
- 4) Categorical descriptions - Theorems of Deligne and Centeleghe-Stix and computations using ideal classes.
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- - In the 4th lecture and in the magma examples there is also material regarding the connection between conjugacy classes of integral matrices and ideal class monoid, which I did not have time to cover in class.