ELLIPTIC CURVES AND PARAMODULAR FORMS

Abstract

There is a lifting from a non-CM elliptic curve $E/\mathbb{Q}$ to a cuspidal paramodular newform $f$ of degree $2$ and weight $3$ given by the symmetric cube map. We find a description of the level of $f$ in terms of the coefficients of the Weierstrass equation of $E$. In order to compute the paramodular level, we need a detailed description of the local representations $\pi_p$ of $\GL(2,\mathbb{Q}_p)$ attached to $E/\mathbb{Q}_p$, where $\pi\cong\bigotimes\limits_p\pi_p$ is the cuspidal automorphic representation of $\GL(2,\mathbb{A}_{\mathbb{Q}})$ associated with $E/\mathbb{Q}$. We use the available description of the local representations of $\GL(2,\mathbb{Q}_p)$ attached to $E$ for $p \ge 5$ and determine the local representation of $\GL(2,\mathbb{Q}_3)$ attached to $E$. In fact, we study the representations of $\GL(2, K)$ attached to $E/K$ for any non-archimedean local field $K$ of characteristic $0$ and residue characteristic $3$.