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

About this Title

Nuno Freitas and Alain Kraus

Publication: Memoirs of the American Mathematical Society
Publication Year: 2022; Volume 277, Number 1361
ISBNs: 978-1-4704-5210-0 (print); 978-1-4704-7093-7 (online)
DOI: https://doi.org/10.1090/memo/1361
Published electronically: March 25, 2022
Keywords: Elliptic curves, torsion points, Weil pairing, symplectic isomorphism, local fields

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Table of Contents

Chapters

• 1. Motivation and results
• 2. The existence of local symplectic criteria
• 3. The criterion in the case of good reduction
• 4. Elliptic curves with potentially good reduction
• 5. The morphism $\gamma _E$
• 6. Proof of the criteria
• 7. Applications

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 $\mathbb {Q}$.