Quaternionic loci in Siegel’s modular threefold

Abstract

Let 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 O in an indefinite quaternion algebra of discriminant D over Q such that the Rosati involution coincides with a positive involution of the form α → µ −1 αµ on O for some µ ∈ O with µ 2 + D = 0. In this paper, we first give a formula for the number of irreducible components in 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.

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

“…In this section we shall introduce and review some properties of Shimura curves on Siegel's modular threefold. The materials are taken from [10,12,13,14].…”

“…In [10], in order to determine the exact number of Shimura curves in Q D,1 , Lin and Yang showed that there is a one-to-one correspondecne between Shimura curves in Q D,1 and GL(2, Z)-equivalence classes of positive definite binary quadratic forms of discriminant −16D such that all integers a represented by the quadratc form satisfy a ≡ 0, 1 mod 4 and…”

“…In this section, we shall review the definition of quadratic forms associated to Shimura curves in Q D,N . These quadratic forms appeared in the works of Hashimoto [5] and Runge [15] and were studied in more details in [10]. Let (A, ρ) be a principally polarized abelian surface over C. The polarization ρ induces an involution f → f † , called the Rosati involution, on the endomorphism ring of A.…”

“…In [12,13], Rotger gave upper and lower bounds for the number r D,1 and in some cases an exact formula for r D,1 . To give an exact formula for r D,1 , Lin and Yang [10] showed that each irreducible component in Q D,1 can be associated with a positive definite quadratic form Q of discriminant −16D such that every positive integer a represented by Q satisfies (1) a ≡ 0, 1 mod 4, and…”

“…Hence the problem of determining r D,1 reduces to that of counting certain GL(2, Z)-equivalence classes of quadratic forms. Note that the work of Rotger [12,13,14] and Lin and Yang [10] can be easily extended to the case N > 1. Moreover, the condition −D,a Q ≃ B is equivalent to that χ pj (Q) = −1 for p j |D and χ pj (Q) = 1 for p j |N .…”