Superspecial primes for QM abelian surfaces over real number fields

TL;DR

This paper extends the understanding of superspecial primes for QM abelian surfaces over real number fields, generalizing previous results and employing intersection theory on Shimura curves.

Contribution

It broadens the class of fields and local conditions under which superspecial primes for QM abelian surfaces can be studied, using intersection theory techniques.

Findings

Generalization to any number field with a real embedding.

Weakened local conditions at 2 and 3.

Application of intersection theory of Heegner divisors.

Abstract

Baba and Granath generalize Elkies' theorem on infinitude of supersingular primes for elliptic curves to abelian surfaces with quaternionic multiplication of discriminant $6$, whose field of moduli is $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$. We extend the field of moduli to any number field with a real embedding, and weaken the local conditions at $2$ and $3$. The proof relies on the intersection theory of Heegner divisors on Shimura curves.