Symplectic criteria for elliptic curves, revisited

2025/09/24 by Kraus, Alain, Freitas, Nuno, Sánchez-Rodríguez, Ignasi
#11G05 (Primary) 11G07 #11Y99 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.19938

Abstract

Let ℓ and p ≥ 3 be different primes. Let E/ℚ_ℓ and E'/ℚ_ℓ be elliptic curves with isomorphic p-torsion. Assume that E has potentially multiplicative reduction. We classify when all G_{ℚ_ℓ}-isomorphisms ϕ: E[p] → 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 ≥ 5 congruences between rational elliptic curves with conductor ≤ 500 000 that satisfy the hypothesis of either of our criteria at some prime~ℓ.

Citations

• Global methods for the symplectic type of congruences between elliptic\n curves
• On the symplectic type of isomorphims of the p-torsion of elliptic\n curves
• Invariant theory for the elliptic normal quintic, I. Twists of X(5)

Related

• On the symplectic type of isomorphims of the p-torsion of elliptic\n curves
• Global methods for the symplectic type of congruences between elliptic\n curves
• Elliptic mod \ℓ Galois representations which are not minimally\n elliptic
• Divisibility of orders of reductions of elliptic curves
• Local points on twists of X(p) with applications