Superspecial primes for QM abelian surfaces over real number fields

2025/11/11 by Fangu Chen, Chen, Fangu
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2511.07814

openalex publication_date 2025/11/11 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28

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 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.

Citations

• Infinitely many supersingular primes for some Mumford's abelian fourfolds
• Infinitely many primes of basic reduction for some abelian fourfolds
• Intersection numbers of Heegner divisors on Shimura curves

Related

• Infinitely many supersingular primes for some Mumford's abelian fourfolds
• Infinitely many primes of basic reduction for some abelian fourfolds
• On superspecial abelian surfaces over finite fields III
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