On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure (Q1910196)

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 \(n=4\), \(g\geq 2\).