On the non-commutative endomorphism rings of abelian surfaces

TL;DR

This paper investigates the endomorphism rings of abelian surfaces, proving that infinitely many quaternion algebras cannot occur as their endomorphism algebras over Q, by analyzing rational points on Shimura curves.

Contribution

It demonstrates that infinitely many quaternion algebras are excluded as endomorphism algebras of abelian surfaces over Q, using descent obstructions on Shimura curves.

Findings

No rational points on infinitely many Atkin-Lehner quotient Shimura curves for certain quaternion algebras.

Development of new Diophantine and analytic techniques for Shimura curves.

Confirmation of conjectural finiteness related to endomorphism rings of abelian surfaces.

Abstract

A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may think of this as a higher-dimensional version of the Gauss Class Number problem. Before now, no one has ruled out quaternion algebras over $\mathbf{Q}$ not already ruled out by the Albert classification. We rule out infinitely many such quaternion algebras by showing that for infinitely many $D$, the Atkin-Lehner quotient Shimura curve $X^D/w_D$ has no $\mathbf{Q}$-rational points. Our principal method is to use the level structure maps above $X^D$ to create torsors for use in the descent obstruction. Numerous Diophantine and analytic results on Shimura curves are also proved.