Plectic p-adic invariants

TL;DR

This paper introduces new p-adic invariants for modular elliptic curves over certain number fields, inspired by plectic conjectures, and provides numerical evidence for their role in understanding the Mordell--Weil group.

Contribution

It defines novel p-adic invariants for elliptic curves with multiplicative reduction, inspired by plectic conjectures, and supports their significance with numerical experiments.

Findings

Numerical experiments suggest these invariants relate to the Mordell--Weil group.

New p-adic invariants are constructed for specific elliptic curves.

Inspiration from Nekovar and Scholl's plectic conjectures guides the work.

Abstract

For modular elliptic curves over number fields of narrow class number one, and with multiplicative reduction at a collection of p-adic primes, we define new p-adic invariants. Inspired by Nekovar and Scholl's plectic conjectures, we believe these invariants control the Mordell--Weil group of higher rank elliptic curves and we support our expectations with numerical experiments.