A Jacobi meromorphic form

Let L be a complex lattice. Our object of study is a function D-L(z; phi), periodic with period lattice L in the second variable, and analytic in the first variable with normalization condition [GRAPHICS] zD(L)(z; phi) = 1; up to an exponential factor, this function is related to the form F-r(u, v) = theta'(0)theta(u + v)/(theta(u)theta(v)) (see [7], 3), analytic in tau is an element of H (=upper half plane) and in u, v is an element of C, with (u, v) proportional to (z, phi) and theta the Jacobi's triple product. Our main result is that D-L also satisfies a simple additive distribution relation. Indeed if Lambda is a lattice such that L subset of Lambda and [Lambda : L] = l, we have: [GRAPHICS] where t runs over a representative system of Lambda/L. When phi is a torsion point of C/L, we recover known results.