### Abstract

Let D > 546 be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields l/ℚ such that the twist of the Shimura curve X^{D} by the main Atkin-Lehner involution w_{D}and l/ℚ violates the Hasse Principle over ℚ. More precisely, the number of squarefree d with |d| ≤ X such that the quadratic twist of (X^{D},_{wD}) by ℚ(√d) violates the Hasse Principle is ≫ X/ log^{αD}X and ≪ X/ log^{βD}X for explicitly given 0 < β_{D}< α_{D}< 1 such that α_{D}− β_{D}→ 0 as D→∞.

Original language | English |
---|---|

Pages (from-to) | 2839-2851 |

Number of pages | 13 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Jul 2018 |

## Fingerprint Dive into the research topics of 'Hasse principle violations for atkin-lehner twists of shimura curves'. Together they form a unique fingerprint.

## Cite this

Clark, P. L., & Stankewicz, J. (2018). Hasse principle violations for atkin-lehner twists of shimura curves.

*Proceedings of the American Mathematical Society*,*146*(7), 2839-2851. https://doi.org/10.1090/proc/14001