Quaternionic loci in Siegel's modular threefold

Abstract

Let QD be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order O in an indefinite quaternion algebra of discriminant D over Q such that the Rosati involution coincides with a positive involution of the form alpha & x21a6;mu-1 alpha over bar mu on O for some mu is an element of O with mu 2+D=0. In this paper, we first give a formula for the number of irreducible components in QD, 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.