Seminario Teoría de Números
Symplectic criteria for elliptic curves: the multiplicative reduction case
Ponente: Ignasi Sánchez (Universidad de Barcelona)Fecha: jueves 04 de diciembre de 2025 - 12:30Lugar: Aula 420, Módulo 17, Departamento de Matemáticas, UAM
Resumen:
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 \(E[p] \simeq E'[p]\) 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 Oesterl\'e.
As an application, we determine the symplectic type of all the mod \(p \geq 5\) congruences between all rational elliptic curves available in the LMFDB with conductor \(\leq 500 000\) that satisfy the hypothesis of either of our criteria at some prime~\(\ell\).
