Quotients of theta series as rational functions of j1,4 Original Research Article

Let 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 image, the complex upper half plane) is a modular form of weight n/2 for the congruence group image. 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 image (the extended plane of image) 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.