Seminario Teoría de Números

Endomorphism algebras of Siegel-modular abelian varieties

Ponente:  Enric Florit Zacarias (Universitat de Barcelona)
Fecha:  jueves 16 de octubre de 2025 - 11:30
Lugar:  Aula 420, Módulo 17

Resumen:

The construction of Eichler and Shimura associates to a classical modular eigenform an abelian variety of GL(2)-type over Q. Such a variety A/Q is defined by the property that its endomorphism algebra is a number field E of degree [E:Q] = dim A. The correspondence has been made into a bijection by means of modularity theorems, and in particular by work of Ribet, Khare and Wintenberger.

If we move beyond the GL(2) case, Brumer and Kramer have conjectured that an abelian surface A/Q corresponds to a Siegel paramodular eigenform with rational coefficients. This has been recently proven for a large class of abelian surfaces by Boxer, Calegari, Gee and Pilloni. In the converse direction, however, there cannot exist an Eichler-Shimura construction for the Siegel automorphic representations that should correspond to abelian varieties. In particular, we cannot tell the endomorphism algebra of a Siegel-modular abelian variety from the Hecke eigenvalues alone. Moreover, a generalisation of the Brumer-Kramer conjecture to higher dimensional abelian varieties seems to be restricted to certain settings.

In this talk, I will introduce the general framework of abelian varieties of GL(n)-type. Under certain hypotheses, these have a theory of building blocks, inner twists and equivariant pairings. I will use the theory to describe the possible endomorphism algebras that arise from Siegel-modular abelian varieties. This is joint work with Francesc Fité and Xavier Guitart.