Twists of Shimura Curves

Abstract

Consider a Shimura curve $X_0^D(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livné on ${\mathbf{Q}}_p$ points when $p\mid D$ and for the first time give criteria for ${\mathbf{Q}}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

Slight update

References in corpus (2)

• On the non-existence of exceptional automorphisms on Shimura curves
• On Quadratic Twists of Hyperelliptic Curves

Cited by in corpus (5)

• On Polyquadratic Twists of X_0(N)
• Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves
• On the non-commutative endomorphism rings of abelian surfaces
• On The $j$-Invariants of CM-Elliptic Curves Defined Over ${\mathbb{Z}}_p$
• Shimura curves and explicit descent obstructions via level structure