location:  Publications → journals → CJM
Abstract view

# Twists of Shimura Curves

Published:2013-09-04
Printed: Aug 2014
• James Stankewicz,
Mathematics and Computer Science, Wesleyan University, Middletown, CT, USA
 Format: LaTeX MathJax PDF

## 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-Livné 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.
 Keywords: Shimura curves, complex multiplication, modular curves, elliptic curves
 MSC Classifications: 11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G35 - Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 11G15 - Complex multiplication and moduli of abelian varieties [See also 14K22] 11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]