On superspecial abelian surfaces and type numbers of totally definite quaternion algebras

Which Quaternion Algebras Act on a Modular Abelian Variety?

Let A/Q be a modular abelian variety. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A over Q̄. We accomplish this by analyzing the representation of Gal (Q̄/Q) on the points of N-torsion of A at primes N which ramify in B and by applying descent techniques to certain covers of Shimura varieties. Our result also appli...

Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

In this talk we introduce and study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space of countably infinite tuples of real numbers endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. We will see that in all these various settings the integral re...

Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture

On Finiteness Conjectures for Endomorphism Algebras of Abelian Surfaces

It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL2type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of S...

Ramification of Quaternion Algebras over Stable Elliptic Surfaces

The aim of this work is to study the ramification of quaternion algebras over the function field of a stable elliptic surface, in particular over the field of complex numbers. Over number fields there are criteria for the ramification of quaternion algebras such as the tame symbol formula. We study how this formula can be interpreted in a geometrical way, and how the ramification relates to the...