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On the Kodaira dimension of moduli spaces of abelian varieties with non-principal polarizations. (English) Zbl 0848.14021

Barth, Wolf (ed.) et al., Abelian varieties. Proceedings of the international conference held in Egloffstein, Germany, October 3-8, 1993. Berlin: Walter de Gruyter. 293-302 (1995).
Let \({\mathcal A}_{g,T}\) be the moduli space of complex abelian varieties of dimension \(g\) with a polarization of type \(T = (t_1, \dots, t_g)\), where \(t_1, \dots, t_g\) are positive integers such that \(t_1 |\cdots |t_g\). The purpose of this paper is to prove that the quasi-projective variety \({\mathcal A}_{g,T}\) is of general type for \(g \geq 16\), and for \(g \geq 8\) if each \(t_i\) is odd and a sum of two squares (it has been proved by Freitag, Tai and Mumford that \({\mathcal A}_{g, (1, \dots, 1)}\) is of general type for \(g \geq 7)\). The variety \({\mathcal A}_{g,T}\) can be constructed as the quotient of the Siegel upper half-space \(S_g\) by a certain subgroup \(\Gamma_T\) of \(Sp (2g,\mathbb{Q})\). One needs to construct enough \(k\)-canonical forms on a smooth projective model of \({\mathcal A}_{g,T}\) and for that, by now standard techniques, it is enough to construct one non-zero \(\Gamma_T\)-modular form on \(S_g\) of weight \(k(g + 1)\), that vanishes to order \(> k\) at infinity.
This is done by considering an embedding \(h\) of \(S_g\) into the quaternionic half-space \(S_{g,H}\), equivariant for the respective actions of \(\Gamma_T \) and of a certain subgroup \(\Gamma_{q,H}\) of \(GL(2g, {\mathfrak O})\) (where \({\mathfrak O}\) is the ring of Hurwitz quaternions), independent of \(T\), such that for each \(\Gamma_{g,H}\)-modular form \(F\) on \(S_{g,H}\), the composition \(F \circ h\) is a \(\Gamma_T\)-modular form of the same weight, that vanishes to the same order at infinity. Using analogs of theta constants, one constructs a non-zero \(\Gamma_{g,H}\)-modular form \(F\) of weight \(2^{2g + 5} n\) (for some integer \(n)\) that vanishes to order at least \(2^{2g + 1} n\) at infinity. For \(g \geq 16\), one has \(2^{2g + 1} n (g + 1) > 2^{2g + 5} n\), hence \((F \circ h)^{g + 1}\) is the required modular form.
When each \(t_i\) is odd and a sum of two squares, the technique is similar: one considers an equivariant embedding \(h\) of \(S_g\) into the Hermitian half-space \(S_{g,C}\), and one constructs a non-zero \(\Gamma_{g,C}\)-modular form \(F\) of weight \(2^{g + 2} (2^g + 1) (g + 1)\) that vanishes to order at least \(2^{2g - 1} (g + 1)\) at infinity. For \(g \geq 8\), one has \(2^{2g - 1} (g + 1) > 2^{g + 2} (2^g + 1)\), hence \(F \circ h\) is the required modular form.
For the entire collection see [Zbl 0817.00023].

MSC:

14K10 Algebraic moduli of abelian varieties, classification
14K25 Theta functions and abelian varieties
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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