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

Abstract

Let p ≥ 3 p \geq 3 be a prime. Let E / Q E/\mathbb {Q} and E ′ / Q E’/\mathbb {Q} be elliptic curves with isomorphic p p -torsion modules E [ p ] E[p] and E ′ [ p ] E’[p] . Assume further that either (i) every G Q G_\mathbb {Q} -modules isomorphism ϕ : E [ p ] → E ′ [ p ] \phi : E[p] \to E’[p] admits a multiple λ ⋅ ϕ \lambda \cdot \phi with λ ∈ F p × \lambda \in \mathbb {F}_p^\times preserving the Weil pairing; or (ii) no G Q G_\mathbb {Q} -isomorphism ϕ : E [ p ] → E ′ [ p ] \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 ℓ ≠ p \ell \neq p . Firstly, we determine the primes ℓ \ell for which the local curves E / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ 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 / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ 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 p . We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form y 2 = x p − ℓ y^2 = x^p - \ell and y 2 = x p − 2 ℓ y^2 = x^p - 2\ell where ℓ \ell is a prime; we also give incremental results on the Fermat equation x 2 + y 3 = z p 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 p -torsion of two non-isogenous elliptic curves defined over Q \mathbb {Q} .