A SIMPLE PROOF OF QUOTIENTS OF THETA SERIES AS RATIONAL FUNCTIONS OF J

Abstract

For two even unimodular positive definite integral quadratic forms A[X], B[X] in n-variables, J. K. Koo [1, Theorem 1] showed that θA(τ )/θB(τ ) is a rational function of J, satisfying a certain condition. Where θA(τ ) and θB(τ ) are theta series related to A[X] and B[X], respectively, and J is the classical modular invariant. In this paper we give a simple proof of Theorem 1 of [1].

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