Superspecial primes for QM abelian surfaces over real number fields

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.