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 [ Grushevsky , S ., Salvati Manni , R ., Gradients of odd theta functions, J. reine angew. Math. 573 (2004), 43-59.] and prove that for any n > 1 the map sending an abelian variety to the set of Gauss images of its points of order 2 n is an embedding into an appropriate Grassmannian (note that for n = 1 we only got generic injectivity in [ Grushevsky , S ., Salvati Manni , R ., Gradients of odd theta functions, J. reine angew. Math. 573 (2004), 43-59.]). We further discuss the generalizations of Jacobi's derivative formula for any dimension and any order.