Modular varieties of D-elliptic sheaves and the Weil-Deligne bound

TL;DR

This paper investigates the growth of rational points on higher-dimensional modular varieties of D-elliptic sheaves over finite fields, relating it to Betti numbers, and constructs a new sequence of asymptotically optimal curves.

Contribution

It generalizes known results from modular curves to higher dimensions and introduces a new sequence of curves that are asymptotically optimal.

Findings

Growth of rational points matches Betti number growth asymptotically

Established a higher-dimensional analogue of classical results

Produced a new sequence of asymptotically optimal curves

Abstract

We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we also produce a new asymptotically optimal sequence of curves.