Theta functions of arbitrary order and their derivatives

Abstract

In this paper we establish the relationships between theta functions of arbitrary order and their derivatives. We generalize our previous work [4] and prove that for any n > 1 the map sending an abelian variety to the set of Gauss images of its points of order 2n is an embedding into an appropriate Grassmannian (note that for n = 1 we only got generic injectivity in [4]). We further discuss the generalizations of Jacobi's derivative formula for any dimension and any order

Geometry of theta divisors --- a survey

“…3 , and the gradients of the theta function at higher torsion points, were studied by Salvati Manni and the first author in [GSM04], [GSM06], where they showed that the values of all such gradients determine a ppav generically uniquely. Furthermore, in [GSM09] Salvati Manni and the first author (motivated by their earlier works [GSM08] and [GSM07] on double point singularities of the theta divisor at 2-torsion points) studied the geometry of these loci further, and made the following The motivation for these conjectures comes from the cases g ≤ 5 discussed above, and also from some degeneration considerations that we will discuss in Section 6.…”

Vector-Valued Modular Forms and the Gauss Map

“…In this paper, we present an easier and more natural method of constructing such differentials forms, providing also a natural bridge between methods of [Fre78] and [SM87]. Our tools will be the gradients of theta functions and expressions in terms of them considered by the third and fifth author in [GSM04,GSM06]. Our result is the following.…”

A generalization of Jacobi's derivative formula to dimension two, II

“…It was shown in [GM2] that for any g, the nullwerte of a hessian determinant of a ratio of certain theta functions of even characteristic was a modular form of degree g and the authors found its value in terms of thetanullwerte. They generalized this to higher order theta functions in [GM3]. In [dJ] de Jong defined a function on the theta divisor of any abelian variety in terms of first and second derivatives of theta functions, that in the case of A τ reduces to X[δ](z, τ ) restricted to Θ, which is essentially the numerator of the y-coordinate on C embedded via φ in A τ .…”