Superspecial primes for QM abelian surfaces over real number fields

TL;DR

The paper extends Elkies' infinitude results for supersingular primes to genus $2$ Jacobians with QM by the maximal order of discriminant $6$ over number fields with a real embedding, relaxing bad-prime hypotheses at $2$ and $3$. It achieves this by combining the Shimura-curve framework with arithmetic intersections of Heegner divisors on integral models of the QM locus, translating valuations of $P_D(j_0)$ into superspecial reduction data. Using carefully chosen discriminants $D$ and primes $p$, the authors ensure the Legendre symbol $(D/p)=-1$ and positive $p$-adic valuations of $P_D(j_0)$, guaranteeing superspecial reduction at infinitely many primes. The work broadens the scope from $b Q$ to real-embedded number fields, provides explicit local-condition analyses at $2$ and $3$, and outlines potential further generalizations via alternative discriminants and cycles. These results advance understanding of supersingular phenomena for higher-dimensional abelian varieties with QM and their arithmetic intersections on Shimura curves.

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.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 2.2: MR2704678*3.3.1
• Remark 2.3
• Lemma 2.4: MR2704678*3.4.1, 3.4.2
• Lemma 3.1
• proof
• Definition 3.2
• Example 3.3