$\mathscr{D}$-elliptic sheaves and the Hasse principle

TL;DR

This paper investigates the arithmetic of $ abla$-elliptic sheaves and their moduli spaces, providing criteria for the absence of rational points and demonstrating violations of the Hasse principle over certain quadratic extensions.

Contribution

It introduces explicit criteria for the non-existence of rational points on Drinfeld--Stuhler varieties and constructs infinite families of quadratic extensions where the Hasse principle fails.

Findings

Criteria for non-existence of rational points on $X^D$

Explicit infinite families of quadratic extensions violating the Hasse principle

Arithmetic properties of $ abla$-elliptic sheaves

Abstract

Let $p$ be a rational prime, $q>1$ a power of $p$ and $F=\mathbb{F}_q(t)$. For an integer $d\geq 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\infty$ and has invariant $\mathrm{inv}_x(D)=1/d$ at any place $x$ of $F$ at which $D$ ramifies. Let $X^D$ be the Drinfeld--Stuhler variety, the coarse moduli scheme of the algebraic stack over $F$ classifying $\mathscr{D}$-elliptic sheaves. In this paper, we establish various arithmetic properties of $\mathscr{D}$-elliptic sheaves to give an explicit criterion for the non-existence of rational points of $X^D$ over a finite extension of $F$ of degree $d$. As an application, for $d=2$, we present explicit infinite families of quadratic extensions of $F$ over which the curve $X^D$ violates the Hasse principle.