Elliptic function based algorithms to prove Jacobi theta function relations

Abstract

In this paper we prove identities involving the classical Jacobi theta functions of the form∑c(i1,i2,i3,i4)θ1(z|τ)i1θ2(z|τ)i2θ3(z|τ)i3θ4(z|τ)i4=0 with c(i1,i2,i3,i4)∈K[Θ], where K is a computable field and Θ:={θ1(2k+1)(0|τ):k∈N}∪{θj(2k)(0|τ):k∈N and j=2,3,4}. We give two algorithms that solve this problem. The second algorithm is simpler and works in a restricted input class.