Infinitely many primes of basic reduction for some abelian fourfolds

TL;DR

This paper proves that certain genus 4 curves with specific automorphisms have Jacobians with basic reduction at infinitely many primes, extending known results from elliptic and genus 2 curves to more complex cases.

Contribution

It extends the infinite primes of basic reduction result to genus 4 curves with automorphisms of order 5, using complex uniformization and analysis of Shimura varieties.

Findings

Identified geodesics in the upper half plane related to the Shimura variety

Determined points with complex multiplication by quadratic extensions of cyclotomic fields

Proved the infinitude of primes where Jacobians have basic reduction

Abstract

If $E$ is an elliptic curve, defined over $\mathbb{Q}$ or a number field having at least one real embedding, then Elkies proved that $E$ has supersingular reduction at infinitely many primes $p$. Baba and Granath extended this result to certain curves $C$ of genus $2$ with field of moduli $\mathbb{Q}$, under a condition on the endomorphism ring of the Jacobian. In this paper, we extend these results to certain curves of genus $4$ having an automorphism of order $5$, proving that the Jacobians of these curves have basic reduction (as defined by Kottwitz) for infinitely many primes $p$. To do this, we study the complex uniformization of the Deligne--Mostow Shimura variety $\mathrm{Sh}$ associated with the one dimensional family of these curves. By analyzing the real points on $\mathrm{Sh}$, we compute three geodesics in the upper half plane that are edges of a fundamental triangle for the action of the unitary similitude group. Using representations of quadratic forms, we determine the points on $\mathrm{Sh}$ which represent curves whose Jacobians have complex multiplication by certain quadratic extensions of the cyclotomic field $\mathbb{Q}(\zeta_5)$. We conclude by studying the equidistribution of these points and the reduction of these CM cycles on the Shimura variety.