Twists of Shimura Curves

Abstract

AbstractConsider a Shimura curve XD0(N) over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on Qp points when p |D and for the first time give criteria for Qp points when p |N. We also give congruence conditions for roots modulo p of Hilbert class polynomials.