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Symplectic criteria for elliptic curves, revisited

Alain Kraus, Nuno Freitas, Ignasi Sánchez-Rodríguez

TL;DR

This work completes the local-to-global program for determining the symplectic type of mod $p$ Galois representations attached to elliptic curves by classifying when a single prime yields a uniform symplectic criterion across reduction types. It introduces two new criteria: one for mixed reduction and another for the case both curves have (potentially) multiplicative reduction, addressing gaps left by Kraus–Oesterlé and extending applicability to unramified $p$-torsion. The authors also provide an alternative good-reduction criterion with a practical Magma implementation and prove that their criteria, together with twisting invariance, give a complete list of applicable local criteria, enabling efficient determination of symplectic type in many $E,E',p$ congruence problems. As an application, they compute and classify symplectic types for a wide range of mod $p\ge5$ congruences among rational elliptic curves with conductor up to $5\times 10^5$, producing extensive data and explicit examples, including both irreducible and reducible cases. The results deliver new tools for analyzing congruences of elliptic curves, with potential impact on topics such as level-lowering and Diophantine applications reliant on symplectic behavior of $p$-torsion.

Abstract

Let $\ell$ and $p \geq 3$ be different primes. Let $E/\mathbb{Q}_\ell$ and $E'/\mathbb{Q}_\ell$ be elliptic curves with isomorphic $p$-torsion. Assume that $E$ has potentially multiplicative reduction. We classify when all $G_{\mathbb{Q}_\ell}$-isomorphisms $φ: E[p] \to E'[p]$ have the same symplectic type and prove two new criteria to determine the type in that case. In particular, when both curves have multiplicative reduction, our results cover the case of unramified $p$-torsion which is not covered by the original criterion due to Kraus and Oesterlé. We also give a variant of a symplectic criterion for the case when both $E$ and~$E'$ have good reduction and provide an algorithm to apply it. As an application, we determine the symplectic type of all the mod $p \geq 5$ congruences between rational elliptic curves with conductor $\leq 500 000$ that satisfy the hypothesis of either of our criteria at some prime~$\ell$.

Symplectic criteria for elliptic curves, revisited

TL;DR

This work completes the local-to-global program for determining the symplectic type of mod Galois representations attached to elliptic curves by classifying when a single prime yields a uniform symplectic criterion across reduction types. It introduces two new criteria: one for mixed reduction and another for the case both curves have (potentially) multiplicative reduction, addressing gaps left by Kraus–Oesterlé and extending applicability to unramified -torsion. The authors also provide an alternative good-reduction criterion with a practical Magma implementation and prove that their criteria, together with twisting invariance, give a complete list of applicable local criteria, enabling efficient determination of symplectic type in many congruence problems. As an application, they compute and classify symplectic types for a wide range of mod congruences among rational elliptic curves with conductor up to , producing extensive data and explicit examples, including both irreducible and reducible cases. The results deliver new tools for analyzing congruences of elliptic curves, with potential impact on topics such as level-lowering and Diophantine applications reliant on symplectic behavior of -torsion.

Abstract

Let and be different primes. Let and be elliptic curves with isomorphic -torsion. Assume that has potentially multiplicative reduction. We classify when all -isomorphisms have the same symplectic type and prove two new criteria to determine the type in that case. In particular, when both curves have multiplicative reduction, our results cover the case of unramified -torsion which is not covered by the original criterion due to Kraus and Oesterlé. We also give a variant of a symplectic criterion for the case when both and~ have good reduction and provide an algorithm to apply it. As an application, we determine the symplectic type of all the mod congruences between rational elliptic curves with conductor that satisfy the hypothesis of either of our criteria at some prime~.

Paper Structure

This paper contains 14 sections, 10 theorems, 37 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The existence of symplectic criteria
  4. Auxiliary results
  5. The case of potentially multiplicative reduction
  6. The mixed reduction case
  7. Proof of Theorem \ref{['thm:multiplicative']}
  8. Proof of Theorem \ref{['thm:mixedReduciton']}
  9. The case of good reduction
  10. Proof of Theorem \ref{['thm:summary']}
  11. Applications
  12. Finding congruences mod $p$
  13. Examples
  14. Reducible examples for $p=5$

Key Result

Theorem 1.1

Let $\ell$ and $p\geq 3$ be primes satisfying $\ell \equiv 1 \pmod{p}$. Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}_\ell$ with split multiplicative reduction. Let ${\tilde{j}_i} := j_{E_i} \cdot \ell^{-v(j_{E_i})}$. Suppose that Then, $E_1[p]$ and $E_2[p]$ are $G_{\mathbb{Q}_\ell}$-isomorphic and not simultaneously symplectically and anti-symplectically isomorphic. Furthermore, they a

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • ...and 17 more