Finiteness of function field-valued points on exceptional Shimura varieties
Benjamin Bakker, Ananth N. Shankar, Jacob Tsimerman
TL;DR
The paper proves a finiteness phenomenon for function-field abelian varieties and extends it to general Shimura varieties: for curves $C$ over a finite field, there are only finitely many principally polarized abelian schemes of relative dimension $g$ over $C$ up to $p$-power isogeny, with a finite-type analogue when passing to $ar{k}$. The central method reduces to $p$-adic data via $F$-crystals and $p$-divisible groups, establishing finiteness of occurring $F$-isocrystals with $G$-structure through Litt’s results and companions, and then moving along $p$-Hecke orbits using the ordinary locus to replace isogenies by Hecke correspondences. The general Shimura-variety statement asserts that, for sufficiently large $p$, the space of generically ordinary morphisms $C o S_k$ (resp. $C o S_{ar{k}}$) is finite (resp. finite type) up to $p$-Hecke orbits, with finiteness enhanced to a finite type result under maximal algebraic monodromy. Overall, the work provides a robust characteristic-$p$ framework for controlling families of maps into Shimura varieties by reducing to bounded Griffiths degrees via $p$-adic and crystalline structures, yielding consequences for moduli problems in arithmetic geometry.
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.