Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quaternionic loci in Siegel's modular threefold   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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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg