Quaternionic loci in Siegel's modular threefold

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.

40 pages, plus 70+ pages of tables

References in corpus (2)

• Computing Igusa class polynomials
• Equations of hyperelliptic Shimura curves

Cited by in corpus (1)

• Class number relations arising from intersections of Shimura curves and Humbert surfaces