Independence between coefficients of two modular forms

Abstract

Let $k$ be an even integer and $S_k$ be the space of cusp forms of weight $k$ on $\SL_2(\ZZ)$. Let $S = \oplus_{k\in 2\ZZ} S_k$. For $f,g\in S$, we let $R(f, g) = \{ (a_f(p), a_g(p)) \in \mathbb{P}^1(\CC)\ |\ \text{$p$ is a prime} \}$ be the set of ratios of the Fourier coefficients of $f$ and $g$, where $a_f(n)$ (resp. $a_g(n)$) is the $n$th 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 $\SL_2(\ZZ)$. We apply it to studying the number of representations of a positive integer by a quadratic form.