Finiteness of function field-valued points on exceptional Shimura varieties

TL;DR

This paper proves finiteness results for points on certain Shimura varieties over finite fields, showing that the moduli spaces of abelian schemes and morphisms are finite or of finite type up to p-power isogeny or p-Hecke orbits.

Contribution

It establishes finiteness of points on Shimura varieties over finite fields and generalizes previous results to arbitrary Shimura varieties for large primes.

Findings

Finiteness of principally polarized abelian schemes over curves in finite fields.

Finiteness of moduli space of such schemes over algebraic closures.

Finiteness of morphisms to Shimura varieties up to p-Hecke orbits.

Abstract

Let $C/k$ be a smooth curve over a finite field of characteristic $p>0$. We prove that there are finitely many principally polarized abelian schemes of given dimension $g$ over $C$ up to $p$-power isogeny. For curves over $\overline{k}$, we prove that the moduli space of such abelian schemes is finite type up to $p$-power isogeny. Moreover, we generalize this result to arbitrary (not necessarily abelian type) Shimura varieties $S$ and sufficiently large primes $p$ in terms of $S$: The space of generically ordinary morphisms $C\to S_{k}$ (resp. $C\to S_{\overline{k}})$ is finite (resp. finite type) up to $p$-Hecke orbits.