# Identities between Hecke Eigenforms

## TL;DR

This paper investigates identities between Hecke eigenforms, proving finiteness of solutions to certain quadratic relations, and under Maeda's conjecture, establishes non-vanishing of inner products and constraints on eigenform identities, with implications for algebraicity of the j-function.

## Contribution

It proves finiteness of solutions to quadratic identities among Hecke eigenforms and, assuming Maeda's conjecture, shows non-vanishing inner products and dimension-based restrictions on eigenform identities.

## Key findings

- Finiteness of solutions to $h=af^2+bfg+g^2$ for Hecke newforms.
- Non-vanishing of Petersson inner product $raket{f^2,g}$ under Maeda's conjecture.
- Eigenform identities are constrained by dimension considerations.

## Abstract

In this paper, we study solutions to $h=af^2+bfg+g^2$, where $f,g,h$ are Hecke newforms with respect to $\Gamma_1(N)$ of weight $k>2$ and $a,b\neq 0$. We show that the number of solutions is finite for all $N$. 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 $SL_2(\mathbb{Z})$ 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. We also give a proof using polynomial identities between eigenforms that the $j$-function is algebraic on zeros of Eisenstein series of weight $12k$.