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

Abstract

Let $p\ge 3$ be a prime. Let $E/\mathbb{Q}$ and $E^{\prime }/\mathbb{Q}$ be elliptic curves with isomorphic $p$-torsion modules $E[p]$ and $E^{\prime }[p]$. Assume further that either (i) every $G_{\mathbb{Q}}$-modules isomorphism $\varphi :E[p]\to E^{\prime }[p]$ admits a multiple $\lambda \cdot \varphi$ with $\lambda \in {\mathbb{F}}_p^{\times }$ preserving the Weil pairing; or (ii) no $G_{\mathbb{Q}}$-isomorphism $\varphi :E[p]\to E^{\prime }[p]$ 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 $\ell \ne p$. Firstly, we determine the primes $\ell$ for which the local curves $E/{\mathbb{Q}}_{\ell }$ and $E^{\prime }/{\mathbb{Q}}_{\ell }$ 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 $E^{\prime }/{\mathbb{Q}}_{\ell }$, 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 $\mathbb{Q}$.

theorem numbers differ from published version; updated references and a few other small changes; 104 pages

References in corpus (2)

• Integral Tate modules and splitting of primes in torsion fields of elliptic curves
• A remark on Tate's algorithm and Kodaira types

Cited by in corpus (1)

• On the equation $x^2+d{y}^6=z^p$ for square-free $1\le d\le 20$