Reductions of Hecke correspondences on Anderson modular objects

TL;DR

This paper explores properties of a conjectural moduli space of Anderson t-motives, providing formulas for Hecke correspondences and their reductions, with geometric interpretations and conjectures on Hodge numbers, drawing analogies to Shimura varieties.

Contribution

It formulates properties and formulas for Hecke correspondences on a conjectural space of Anderson t-motives, including reductions and geometric interpretations, extending analogies with Shimura varieties.

Findings

Formulas for $ p$-Hecke correspondences and their reductions.

Geometric interpretation of these correspondences.

Conjectural formulas for Hodge numbers of cohomology submotives.

Abstract

We formulate some properties of a conjectural object $X_{fun}(r,n)$ parametrizing Anderson t-motives of dimension $n$ and rank $r$. Namely, we give formulas for $\goth p$-Hecke correspondences of $X_{fun}(r,n)$ and its reductions at $\goth p$ (where $\goth p$ is a prime of $\Bbb F_q[\theta]$). Also, we describe their geometric interpretation. These results are analogs of the corresponding results of reductions of Shimura varieties. Finally, we give conjectural formulas for Hodge numbers (over the fields generated by Hecke correspondences) of middle cohomology submotives of $X_{fun}(r,n)$.