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arXiv · 1208.3594

Twists of Shimura Curves

James Stankewicz
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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.

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James Stankewicz. 2012-08-17. Twists of Shimura Curves. https://doi.org/10.4153/cjm-2013-023-8

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