On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure
type: Item
instance of: scholarly article
title: On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure
zbMATH Open document ID: 0856.11023
DOI: 10.1007/BF02622113
published in: Mathematische Zeitschrift
publication date: 1996-07-14T00:00:00Z
full work available at URL: https://eudml.org/doc/174847
review text: In [Invent. Math. 3, 75-135 (1967; Zbl 0219.14024)] \textit{D. Mumford} gave a satisfactory answer to the possibility to embed the moduli space \(\Gamma_g (n, 2n) \setminus \mathbb{H}_g\) of principally polarized abelian varieties with level \(n\)-structure in some projective space through theta-nullwerte in case \(n\geq 4\). In a recent paper the author settled the case \(n=2\) showing the map to be generically injective. Now in the present paper it is first shown by using the Gauß map as before that the map \(\Gamma_g (3, 6)\setminus \mathbb{H}_g\to \mathbb{P}^{3^g-1}\) given by the theta-nullwerte \(\tau\mapsto \vartheta {m' \brack 0} (3\tau, 0)\), \(m'\in ({1\over 3} \mathbb{Z}/ \mathbb{Z})^g\) is an embedding, too. Moreover, evaluating the gradients at the origin of the analogous theta-functions \(\vartheta {m' \brack 0} (n\tau, nz)\), \(m'\in (n^{-1} \mathbb{Z}/ \mathbb{Z})^g\), which are modular forms with respect to the representation \(\text{det}^{1/2} \otimes \text{St}\), St being the tautological one, and obvious multiplier system \(v_n\), yields for \(n\geq 3\) a mapping \(\Phi_n: \Gamma_g (n, 2n) \setminus \mathbb{H}_g\to G(g, n^g)\) into the Grassmannian of \(g\)-planes, the differential of which is always injective in case \(n\geq 5\) and with well-understood ramification locus in case \(n=4\), \(g\geq 2\).
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 861948
zbMATH Keywords: level theta structure
zbMATH Keywords: Siegel modular groups
zbMATH Keywords: Siegel modular forms
zbMATH Keywords: moduli space of principally polarized abelian varieties
zbMATH Keywords: theta-functions
zbMATH Keywords: differential
author: Riccardo Salvati Manni
MaRDI profile type: Publication
cites work: Q3882739
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cites work: On the projective varieties associated with some subrings of the ring of Thetanullwerte
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description: scientific article; zbMATH DE number 861948