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Hasse principle violations for atkin-lehner twists of shimura curves

Research output: Contribution to journalArticle

  • Pete L. Clark
  • James Stankewicz
Original languageEnglish
Pages (from-to)2839-2851
Number of pages13
JournalProceedings of the American Mathematical Society
Volume146
Issue number7
DOIs
DatePublished - 1 Jul 2018

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 XD by the main Atkin-Lehner involution wDand l/ℚ violates the Hasse Principle over ℚ. More precisely, the number of squarefree d with |d| ≤ X such that the quadratic twist of (XD,wD) by ℚ(√d) violates the Hasse Principle is ≫ X/ logαDX and ≪ X/ logβDX for explicitly given 0 < βD< αD< 1 such that αD− βD→ 0 as D→∞.

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