Infinitely many primes of basic reduction for some abelian fourfolds

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.