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Abelian surfaces with anti-holomorphic multiplication

Mark Goresky·Yung sheng Tai
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Abstract

For appropriate $N\ge 3$ and $d<0,$ the moduli space of principally polarized abelian surfaces with level $N$ structure and anti-holomorphic multiplication by $\mathcal O_d$ (the ring of integers in $\mathbb Q(\sqrt{d})$) is shown to consist of the real points of a quasi-projective algebraic variety defined over $\mathbb Q$, and to coincide with finitely many copies of the quotient of hyperbolic 3-space by the principal congruence subgroup of level $N$ in $\mathbf{SL}(2, \mathcal O_d).$

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Mark Goresky, Yung sheng Tai. 2001-08-14. Abelian surfaces with anti-holomorphic multiplication. https://arxiv.org/abs/math/0108099

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