Some remarks on the moduli space of principally polarized abelian varieties with level (2,4)-structure. (English) Zbl 0785.14026

Sasaki, Ryuji

Compos. Math. 85, No. 1, 87-97 (1993).

Let \(A_ g(n,2n)\) be the quotient space of the Siegel upper half-space of degree \(g\), \(H_ g\) by the congruence subgroup \(\Gamma(n,2n)\). This is the moduli space of principally polarized abelian varieties with level \((n,2n)\) structure. If \(n \geq 2\) there is a holomorphic map \(\varphi_ n\) from \(A_ g(n,2n)\) to \(\mathbb{P}^ N\), \(N=n-1\), defined by the evaluation at 0 of the theta functions \(\theta {a \brack 0}(n \tau, nz)\), \(a \in (n^{-1} \mathbb{Z}/ \mathbb{Z})^ g\).
At the moment in which this paper was accepted for the publication it was known that \(\varphi_ n\) is an immersion for all \(n \geq 4\) [see D. Mumford, C.I.M.E., \(3^ 0\) Ciclo Varenna, 1969 Quest. Algebraic Varieties, 29-100 (1970; Zbl 0198.258)] and \(\varphi_ 3\) is injective [see Ch. Birkenhake and H. Lange, J. Reine Angew. Math. 407, 167-177 (1990; Zbl 0727.14025)]. – In this paper the author proves the injectivity of the map \(\varphi_ 2\) on the set of hyperelliptic points for all \(g\); this implies the generic injectivity of the map. Moreover he also shows that \(\varphi_ 2\) is injective for \(g=3\). The reviewer has to mention that he proved the injectivity of \(\varphi_ 2\) for all \(g\), only after that this paper was accepted for publication [see the reviewer, “Moduli space of ppav with level 2 (theta) structure and reducible points”, Am. J. Math. 116 (1994)].

Reviewer: R.Salvati-Manni (Roma)

MSC:

14K25	Theta functions and abelian varieties
11F46	Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Keywords:

moduli space of principally polarized abelian varieties; theta functions

Citations:

Zbl 0198.258; Zbl 0727.14025

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