# Independence between coefficients of two modular forms

## TL;DR

This paper proves that two modular forms with finitely many ratios of Fourier coefficients are scalar multiples of each other, extending the result to weakly holomorphic forms and applying it to quadratic form representations.

## Contribution

It establishes a rigidity result linking the finiteness of coefficient ratios to scalar multiples for modular forms, including weakly holomorphic forms.

## Key findings

- Finite ratio set implies forms are scalar multiples
- Extension to weakly holomorphic modular forms
- Application to quadratic form representation counts

## 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.