Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations

In this paper we prove identities involving the classical Jacobi theta functions of the form ∑c(i1, i2, i3, i4)θ1(z|τ)1 θ2(z|τ)2 θ3(z|τ)3 θ4(z|τ)4 = 0 with c(i1, i2, i3, i4) ∈ K[Θ], where K is a computable field and Θ := { θ 1 (0|τ) : k ∈ N } ∪ { θ j (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.