Reductions of Hecke correspondences on Anderson modular objects

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 $X_{fun}(r,n)$.