Annamaria Iezzi

Laboratoire Jean Kuntzmann, Université Grenoble Alpes

Intro talk: Elliptic curves and endomorphism rings

We will begin this symposium by introducing the simplest examples of abelian varieties: elliptic curves. My focus will be on the endomorphism ring, an important algebraic object associated with an elliptic curve (and an abelian variety more generally). Through various examples, we will explore the structure of this ring in different cases. It will quickly become clear that computing the endomorphism ring is not always an easy task and, in some cases, the difficulty of this problem is actually crucial for the security of protocols in isogeny-based cryptography.

Research talk: Computing the endomorphism ring of low-dimensional abelian varieties over finite fields

For an elliptic curve E defined over a finite field, the endomorphism ring is either an order in an imaginary quadratic field (if E is ordinary) or a maximal order in a quaternion algebra (if E is supersingular). In this talk we will explore the different approaches to computing the endomorphism ring in both cases. Time permitting we will also briefly discuss the corresponding problem for abelian surfaces over finite fields.

Slides

Elisa Lorenzo García

Université de Neuchâtel, Université de Rennes

Intro talk: Parametrising the moduli spaces M2 of genus 2 curves and A2 of ppas

We will start by introducing genus 2 curves and their invariants, called Igusa invariants. They produce a parametrisation of the moduli space M2. This moduli space can be embedded into the moduli space A2 of principally polarised abelian surfaces. We will discuss how to extend Igusa invariants to also parametrise A2. We will also look closer at the border of these spaces (i.e. at their compactifications). We will end up by studying, over the complex numbers, the locus in A2 of abelian surfaces with real multiplication as initiated by Humbert.

Research talk: An arithmetic intersection for squares of elliptic curves with complex multiplication

(joint work with Christophe Ritzenthaler and Fernando Rodríguez Villegas)
Let C be a genus 2 curve with Jacobian isomorphic to the square of an elliptic curve with complex multiplication by a maximal order in an imaginary quadratic field of discriminant −d < 0. We show that if the stable model of C has bad reduction over a prime p then p ≤ d/4. We give an algorithm to compute the set of such p using the so-called refined Humbert invariant introduced by Kani. Using results from Kudla-Rapoport and the formula of Gross-Keating, we compute for each of these primes p its exponent in the discriminant of the stable model of C.

Stefano Marseglia

Laboratoire J.A. Dieudonné, Université Côte d'Azur

Intro talk: An introduction to abelian varieties over finite fields

Abelian varieties are connected projective varieties whose set of points forms a group. They can be considered an higher dimensional analogues of elliptic curves. We will give an overview of the main tools to study (unpolarized) abelian varieties of any dimension, with focus on finite fields, starting from the basic definitions and ending at the classification up-to-isogeny.

Research talk: Classifying abelian varieties over finite fields with commutative endomorphism algebra

Consider an isogeny class C of abelian varieties defined over F_q with commutative F_q-endomorphism algebra. Building on Tate's Theorem, we provide an equivalence of categories between C and certain modules over the ring Z[π,q/π], where π is the Frobenius endomorphism of the isogeny class. These modules are well-suited for concrete computations. For example, we can enumerate them up to isomorphism and we can easily reconstruct the endomorphism rings occurring in the isogeny class. We will discuss some concrete examples of isogeny classes that are not ordinary, not almost ordinary, nor over the prime field F_p, in which the endomorphism rings exhibit some exotic behaviours.
Reference: "Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms"