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Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves

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arxiv 1612.01440 v1 pith:AOLBYUQY submitted 2016-12-05 math.NT

Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves

classification math.NT
keywords mathbbatkin-lehnerhasseprincipleshimuraalgebracurvecurves
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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 $\mathbb Q$.

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