On the symplectic type of isomorphims of the p-torsion of elliptic curves

Abstract

Let $p \geq 3$ be a prime. Let $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ be elliptic curves with isomorphic $p$-torsion modules $E[p]$ and $E'[p]$. Assume further that either (i) every $G_\mathbb{Q}$-modules isomorphism $\phi : E[p] \to E'[p]$ admits a multiple $\lambda \cdot \phi$ with $\lambda \in \mathbb{F}_p^\times$ preserving the Weil pairing; or (ii) no $G_\mathbb{Q}$-isomorphism $\phi : E[p] \to E'[p]$ preserves the Weil pairing. This paper considers the problem of deciding if we are in case (i) or (ii). Our approach is to consider the problem locally at a prime $\ell \neq p$. Firstly, we determine the primes $\ell$ for which the local curves $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ contain enough information to decide between (i) or (ii). Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$, to decide between (i) and (ii). We show that our results give a complete solution to the problem by local methods away from $p$. We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form $y^2 = x^p - \ell$ and $y^2 = x^p - 2\ell$ where $\ell$ is a prime; we also give incremental results on the Fermat equation $x^2 + y^3 = z^p$. As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the $p$-torsion of two non-isogenous elliptic curves defined over $\mathbb{Q}$.