Local diophantine properties of Shimura curves and the $\bar\mathbb{F}_p$-gonality

Abstract

In this paper, we study the local points of small degrees on Shimura curves $X_0^D(N)$ over a totally real field $F$. We then study the $\bar\mathbb{F}_p$-gonality for these Shimura curves in the case $F=\mathbb{Q}$.