Mathematics > Number Theory
[Submitted on 2 Jul 2018]
Title:Quaternionic loci in Siegel's modular threefold
View PDFAbstract: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.
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