Quaternionic loci in Siegel's modular threefold

TL;DR

This paper investigates the structure of quaternionic loci within Siegel's modular threefold, providing formulas for irreducible components and explicit rational parameterizations for genus zero components.

Contribution

It offers a new formula for counting irreducible components of quaternionic loci and explicitly parameterizes genus zero components using Hauptmoduln.

Findings

Derived a formula for the number of irreducible components in quaternionic loci.

Provided explicit rational parameterizations for genus zero components.

Strengthened previous results by Rotger on these loci.

Abstract

Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $\mathcal O$ in an indefinite quaternion algebra of discriminant $D$ over $\mathbb Q$ such that the Rosati involution coincides with a positive involution of the form $\alpha\mapsto\mu^{-1}\overline\alpha\mu$ on $\mathcal O$ for some $\mu\in\mathcal O$ with $\mu^2+D=0$. In this paper, we first give a formula for the number of irreducible components in $\mathcal Q_D$, strengthening an earlier result of Rotger. Then for each irreducible component of genus $0$, we determine its rational parameterization in terms of a Hauptmodul of the associated Shimura curve.