$\mathcal{D}$-elliptic sheaves and odd Jacobians

TL;DR

This paper investigates rational divisors on modular curves associated with $\\mathcal{D}$-elliptic sheaves and demonstrates that their Jacobians' Tate-Shafarevich groups often have non-square orders, revealing new arithmetic properties.

Contribution

It introduces a criterion for the existence of rational divisors on these curves and shows that Tate-Shafarevich groups frequently have non-square orders, advancing understanding of their arithmetic structure.

Findings

Existence of rational divisors on certain modular curves.

Tate-Shafarevich groups of Jacobians often have non-square orders.

Infinitely many cases with non-square Tate-Shafarevich group orders.

Abstract

We examine the existence of rational divisors on modular curves of $\mathcal{D}$-elliptic sheaves and on Atkin-Lehner quotients of these curves over local fields. Using a criterion of Poonen and Stoll, we show that in infinitely many cases the Tate-Shafarevich groups of the Jacobians of these Atkin-Lehner quotients have non-square orders.