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Hasse principle violations for Atkin-Lehner twists of Shimura curves

Authors: Pete L. Clark and James Stankewicz
Journal: Proc. Amer. Math. Soc. 146 (2018), 2839-2851
MSC (2010): Primary 11G18, 11G30
Published electronically: February 21, 2018
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Abstract: Let $ D > 546$ be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields $ l/\mathbb{Q}$ such that the twist of the Shimura curve $ X^D$ by the main Atkin-Lehner involution $ w_D$ and $ l/\mathbb{Q}$ violates the Hasse Principle over $ \mathbb{Q}$. More precisely, the number of squarefree $ d$ with $ \vert d\vert \leq X$ such that the quadratic twist of $ (X^D,w_D)$ by $ \mathbb{Q}(\sqrt {d})$ violates the Hasse Principle is $ \gg $ $ X/\log ^{\alpha _D} X$ and $ \ll X/\log ^{\beta _D} X$ for explicitly given $ 0 < \beta _D < \alpha _D < 1$ such that $ \alpha _D - \beta _D \ensuremath {\rightarrow } 0$ as $ D \ensuremath {\rightarrow } \infty $.

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Pete L. Clark
Affiliation: Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, Georgia 30602-7415

James Stankewicz
Affiliation: University of Bristol Department of Mathematics/Heilbronn Institute for Mathematical Research, Howard House, Queens Avenue, Bristol, BS8 1SN, United Kingdom
Address at time of publication: IDA Center for Computing Sciences, 17100 Science Drive, Bowie, Maryland 20715

Received by editor(s): December 5, 2016
Received by editor(s) in revised form: January 20, 2017, September 14, 2017, October 4, 2017, and October 16, 2017
Published electronically: February 21, 2018
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2018 American Mathematical Society

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