Exceptional splitting of reductions of abelian surfaces

Abstract

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces having real multiplication. Similar to Charles' theorem on exceptional isogeny of reductions of a given pair of elliptic curves and Elkies' theorem on supersingular reductions of a given elliptic curve, our theorem shows that a density-zero set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

18 pages. We generalize the main result in v1 (only for abelian surfaces over the field of rationals) to the case of abelian surfaces with real multiplication over any number fields. This improvement is due to the new treatment of finite contribution in arithmetic intersection number in section 3

References in corpus (3)

• A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures
• Borcherds products with prescribed divisor
• A New Northcott Property for Faltings Height

Cited by in corpus (5)

• Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
• Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
• A New Northcott Property for Faltings Height
• Picard rank jumps for K3 surfaces with bad reduction
• Just-likely intersections on Hilbert modular surfaces