Hasse principle violations for atkin-lehner twists of shimura curves

Pete L. Clark, James Stankewicz

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)


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→∞.

Original languageEnglish
Pages (from-to)2839-2851
Number of pages13
JournalProceedings of the American Mathematical Society
Issue number7
Publication statusPublished - 1 Jul 2018


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

Cite this