Abstract
The family of abelian varieties over C which have the endomorphism √–d of type (n, 1) is parameterized on the complex ball B~n~. Let G be a modular group for such a family equipped with a certain level structure. The quotient is called the Picard modular variety of level G. There are many studies on Picard modular varieties (for example, see [2]). But concrete models are known only for the cases d = 1, 3 and 163 (see [3], [6], [7] and [9]). In this paper, we study the Picard modular surface for d = 2.
In Section 1, we give an explicit modular embedding Φ : B~n~ → H~n+1~ for d ≡ 1, 2 mod 4. This is inspired by the modular embedding given in [7].
In Section 2, we study several congruence subgroups of the Picard modular group G(ℤ) for ℚ(√–2 ) and relations among them. Furthermore we show some special properties of the theta constants restricted on Φ(B~n~) with d = 2.
In Section 3, we state the main theorem. Namely,
(1) We give a birational morphism from \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \overline {{\bf B}_2/ G_2(\mathbb Z)} $\end{document} onto a hypersurface in the weighted projective space P(1, 2, 3, 4).
(2) We describe the image as an explicit hypersurface of degree 6 defined over ℤ[√2 ].