A higher-order generalization of Jacobi’s derivative formula and its algebraic geometric analogue

Abstract

We generalize Jacobi’s derivative formula for odd m by writing an m × m determinant of higher order derivatives at 0 of theta functions in 1 variable with characteristic vectors with entries in 1 2mZ as an explicit constant times a power of Dedekind’s η-function. We do so by deriving it from an algebraic geometric version that holds in characteristic not dividing 6m. Introduction In the vast pantheon of theta function identities, a central position is held by Jacobi’s derivative formula. Recall that for τ ∈ h = {x+ iy | y > 0}, and a, b ∈ R, we define the theta function in one variable z ∈ C with characteristic vector [ a b ] by (1) θ [ a b ] (z, τ) = ∑ n∈Z eπi(n+a) τ+2πi(n+a)(z+b). A characteristic vector [ a b ] with a, b ∈ 1 2Z is called a theta characteristic, which is called odd or even depending on whether θ [ a b ] (z, τ) is an odd or even function of z. Modulo 1 there is a unique odd theta characteristic δ := [ 1/2 1/2 ] , and three even ones, 1 := [ 0 0 ] , 2 := [ 1/2 0 ] , 3 := [ 0 1/2 ] . Manuscrit reçu le 6 février 2020, révisé le 2 février 2021, accepté le 18 mai 2021. 2010 Mathematics Subject Classification. 14K25, 14H42. Mots-clefs. Theta functions, elliptic curves.