Quotients of theta series as rational functions of j1,4

AbstractLet Q(n, 1) be the set of even unimodular positive definite integral quadratic forms in n-variables. Then n is divisible by 8. For A[x] in Q(n, 1), the theta series 0A(z): = ∑XϵZneπizA¦X¦ (z ϵ h, the complex upper half plane) is a modular form of weight n2 for the congruence group Γ1(4) = {γ ϵ SL2(Z)¦γ  1∗01mod 4}. If n ≥ 24 and A[X], B[X] are two quadratic forms in Q(n, 1), then the quotient θA(z)θB(z) is a modular function for Γ1(4). Since we can identify the field of modular functions for Γ1(4) with the function field K(X1(4)) over the modular curve x1(4) = Γ1(4)h∗ (the extended plane of h) with genus 0, in this paper, we express it as a rational function j1,4 which is a field generator over C of K(X1(4)) and defined by j1.4(z): = θ2(2z)4(θ3(2z)4. Here, θ2 and θ3 denote the classical Jacobi theta functions