Twists of Shimura Curves

TL;DR

This paper investigates the local points of twisted Shimura curves over rational numbers, providing new criteria for their existence over local fields and offering a novel proof of a known theorem, along with new results on Hilbert class polynomials.

Contribution

It introduces new criteria for the existence of local points on twisted Shimura curves and extends known results to cases where primes divide the level N.

Findings

Criteria for local points on twisted Shimura curves when p divides D or N

A new proof of Jordan-Livné's theorem for p dividing D

Congruence conditions for roots of Hilbert class polynomials modulo p

Abstract

Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livn\'e on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.