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Twists of Shimura Curves

Published online by Cambridge University Press:  20 November 2018

James Stankewicz*
Affiliation:
Mathematics and Computer Science, Wesleyan University, Middletown, CT, USA. e-mail: stankewicz@gmail.com
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Abstract

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Consider a Shimura curve $X_{0}^{D}\left( N \right)$ 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 ${{\mathbf{Q}}_{p}}$ points when $p|D$ and for the first time give criteria for ${{\mathbf{Q}}_{p}}$ points when $p|N$ . We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

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