Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves

TL;DR

This paper demonstrates the existence of infinitely many imaginary quadratic fields that cause violations of the Hasse Principle in certain twisted Shimura curves, revealing new insights into rational points and obstructions.

Contribution

It establishes the infinite occurrence of Hasse Principle violations for Atkin-Lehner twists of Shimura curves over specific quadratic fields.

Findings

Infinitely many imaginary quadratic fields cause Hasse Principle violations.

Violations occur for twists of Shimura curves with discriminant greater than 546.

Results extend understanding of rational points on Shimura curves.

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 $\mathbb Q$.