Quaternionic loci in Siegel’s modular threefold
Mathematische Zeitschrift(2019)
摘要
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 ^-1αμ 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.
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关键词
Primary 11G15, Secondary 11F03, 11F46, 11G10
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