Shin Hattori: D-elliptic sheaves and the Hasse principle

Abstract:

Let \(p\) be a rational prime, \(q > 1\) a \(p\)-power integer, and \(F = \mathbb{F}_q(t)\). Let \(d \ge 2\) be an integer and \(D\) a central division algebra over \(F\) of dimension \(d^2\) which splits at the infinity and such that for any place \(x\) of \(F\) at which \(D\) ramifies, the invariant of \(D\) at \(x\) is \(1/d\).

A \(D\)-elliptic sheaf is a system of locally free sheaves equipped with an action of a sheafified version of \(D\). They are parametrized by the Drinfeld–Stuhler variety. When \(d=2\), it is also called the Drinfeld–Stuhler curve, and can be considered as a function field analogue of a quaternionic Shimura curve over \(\mathbb{Q}\).

For the latter curves, in the 1980s Jordan proved a criterion for the non-existence of quadratic points on them, and gave an example of a quaternionic Shimura curve \(X\) and a quadratic field \(K\) for which \(X\) has no \(K\)-rational points but has \(K_v\)-rational points for any place \(v\) of \(K\). This property of having local points without having global points is often called a violation of the Hasse principle.

In this talk, I will explain how to generalize Jordan's criterion to Drinfeld–Stuhler varieties to obtain similar examples of quadratic extensions \(K/F\) over which the Drinfeld–Stuhler curve violates the Hasse principle. This is a joint work with Keisuke Arai, Satoshi Kondo and Mihran Papikian.