Identities between Hecke Eigenforms

Abstract

In this dissertation, we study solutions to certain low degree polynomials in terms of Hecke eigenforms. We show that the number of solutions to the equation $h=af^2+bfg+g^2$ is finite for all $N$, where $f,g,h$ are Hecke newforms with respect to $\Gamma_1(N)$ of weight $k>2$ and $a,b\neq 0$. Using polynomial identities between Hecke eigenforms, we give another proof that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$. Assuming Maeda's conjecture, we prove that the Petersson inner product $\langle f^2,g\rangle$ is nonzero, where $f$ and $g$ are any nonzero cusp eigenforms for $SL_2(\mathhbb{Z})$ of weight $k$ and $2k$, respectively. As a corollary, we obtain that, assuming Maeda's conjecture, identities between cusp eigenforms for $SL_2(\mathbb{Z})$ of the form $X^2+\sum_{i=1}^n \alpha_iY_i=0$ all are forced by dimension considerations, i.e., a square of an eigenform for the full modular group is unbiased. We show by an example that this property does not hold in general for a congruence subgroup. Finally we attach our Sage code in the appendix.