Quotients of theta series as rational functions of j(l,4)

Abstract

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 -(A)(z):=Sigma(x epsilon zn)e(pi izA[X]) (z epsilon h, the complex upper half plane) is a modular form of weight n/2 for the congruence group Gamma(1)(4) = {y epsilon SL2(Z)\y =((1)(0)(1)*) mod4}. If n greater than or equal to 24 and A[X], B[X] are two quadratic forms in Q(n, 1), then the quotient -(A)(z)/-(B)(z) is a modular function for Gamma(1)(4). Since we can identify the field of modular functions for Gamma(1)(4) with the function field K(X-1(4)) over the modular curve X-1(4)=Gamma(1)(4)\h* (the extended plane of h) with genus 0, in this paper, we express it as a rational function j(1,4) which is a field generator over C of K(X-1(4)) and defined by j(1,4)(z):=-(2)(2z)(4)/-(3)(2z)(4). Here, -(2) and -(3) denote the classical Jacobi theta functions. (C) 1999 Elsevier Science B.V. All rights reserved.