Infinitely many supersingular primes for some Mumford's abelian fourfolds

TL;DR

This paper extends the known infinitude of supersingular primes from elliptic curves to certain abelian fourfolds and Kuga-Satake varieties, using advanced deformation and Shimura variety techniques.

Contribution

It generalizes Elkies' result to higher-dimensional abelian varieties within Mumford's families and related Shimura varieties.

Findings

Proves infinitely many supersingular primes for specific abelian fourfolds.

Utilizes deformation spaces and Shimura variety analysis.

Builds on Madapusi's and Shimura's foundational work.

Abstract

Elkies proved the infinitude of supersingular primes for elliptic curves over real number fields. We generalize Elkies' result to some abelian fourfolds in Mumford's families, and more generally, to certain families of Kuga-Satake abelian varieties. The proof relies on the study of local deformation spaces at closed points of the integral model of a Hodge-type Shimura variety, based on the work of Madapusi, and on the analysis of real points of a Shimura curve, based on the work of Shimura.