CM points on Shimura curves via QM-equivariant isogeny volcanoes

Frederick Saia

Vol. 332 (2024), No. 2, 321–384

DOI: 10.2140/pjm.2024.332.321

Abstract

We study complex multiplication points on the Shimura curves $X_0^D{(N)}_{∕\mathbb{Q}}$ and $X_1^D{(N)}_{∕\mathbb{Q}}$, parametrizing abelian surfaces with quaternionic multiplication and extra level structure. A description of the locus of points with CM by a specified order is obtained for general level, via an isogeny-volcano approach in analogy to work of Clark and Saia in the $D=1$ case of modular curves. This allows for a count of all points with CM by a specified order on such a curve, and a determination of all primitive residue fields and primitive degrees of such points on $X_0^D{(N)}_{∕\mathbb{Q}}$. We leverage computations of least degrees towards the existence of sporadic CM points on $X_0^D{(N)}_{∕\mathbb{Q}}$.

Keywords

Shimura curve, quaternionic multiplication, abelian surface, complex multiplication, isogeny

Mathematical Subject Classification

Primary: 11G10, 11G15, 11G18

Milestones

Received: 24 July 2023

Revised: 6 September 2024

Accepted: 2 November 2024

Published: 6 December 2024