Two generalizations of Jacobi’s derivative formula

Abstract

Abstract. In this paper we generalize Jacobi's derivative formula, considered as an identity for theta functions with characteristics and their derivatives, to higher genus/dimension. By applying the methods developed in our previous paper [GSM04], several generalizations to Siegel modular forms are obtained. These generalizations are identities satisfied by theta functions with characteristics and their derivatives at zero. Equating all the coefficients of the Fourier expansion of these relations to zero yields non-trivial combinatorial identities.

Jacobians with a vanishing theta-null in genus 4

“…As discussed in [14], F (τ ) is a scalar modular form of weight g + 2 with respect to Γ g (4, 8) (recall that theta constants have weight 1/2). We can now compute the degree of S and thus finish the proof of the theorem.…”

“…Another potential approach to this problem would be to use theorem 6 from [14] to express the B in terms of Jacobian determinants of odd theta functions, with syzygetic characteristics. If one could then prove the appropriate generalization of Jacobi's derivative formula, conjectured for all genera and proven for genus 4 in [19], this expression in terms of Jacobian determinants could then be rewritten as an algebraic expression in terms of theta constants, and then could perhaps be compared to Schottky-Jung [9] equations for theta constants, or could be used to give a conjectural local algebraic solution to the Schottky problem.…”

“…In a recent paper [14], we proved some identities between the simplest types of theta series with harmonic polynomial coefficients, which are generalizations of the classical Jacobi's derivative formula. Proving these involves the evaluation at zero of the derivatives of first and second order of classical theta functions, i.e.…”

Theta functions of arbitrary order and their derivatives

“…Recently in [5] we found di¤erent generalizations of Jacobi's derivative formula to higher genus, involving second order derivatives of theta functions at zero.…”

Periods of second kind differentials of $(n,s)$-curves

“…E.g. subtracting the two expressions (3.19) written for different indices i and j, we get (using the classical Rosenhain derivative formula [Ros851] New interesting generalisations of the Jacobi derivative formula were recently found by Grushevsky and Salvati Manni [GM05], who also presented a detailed list of references to the other known generalizations in their paper. We do not discuss here the relevance of the formulae obtained here to the results [GM05] but plan to consider this question in a separate publication.…”