Independence between coefficients of two modular forms

Abstract

Let k be an even integer and Sk be the space of cusp forms of weight k on SL2(Z). Let S=⊕k∈2ZSk. For f,g∈S, we let R(f,g) be the set of ratios of the Fourier coefficients of f and g defined by R(f,g):={x∈P1(C)|x=[af(p):ag(p)]for some primep}, where af(n) (resp. ag(n)) denotes the nth Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R(f,g) is finite, then f=cg for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL2(Z). We apply it to study the number of representations of a positive integer by a quadratic form.