Let p be a rational prime, q>1 a p-power integer and F=F_q(t). Let d>=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 it can be considered as a function field analogue of a quaternionic Shimura curve over Q.

For the latter curves, in 1980s Jordan proved a criterion for the non-existence of quadratic points on them and gave an example of a quaternionic Shimura curve 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.

Organizer: Fu-Tsun Wei (NTHU)