Identities between Hecke Eigenforms

Abstract

In this paper, we study solutions to $h=a{f}^2+bfg+g^2$, where $f,g,h$ are Hecke newforms with respect to ${\Gamma }_1(N)$ of weight $k>2$ and $a,b\ne 0$. We show that the number of solutions is finite for all $N$. Assuming Maeda's conjecture, we prove that the Petersson inner product $⟨f^2,g⟩$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $S{L}_2(\mathbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for $S{L}_2(\mathbb{Z})$ of the form $X^2+{\sum }_{i=1}^n{\alpha }_i{Y}_i=0$ all are forced by dimension considerations. We also give a proof using polynomial identities between eigenforms that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$.