On order of vanishing of characteristic elements

TL;DR

This paper investigates the order of vanishing of characteristic elements associated with the Selmer group of an elliptic curve over a p-adic Lie extension, linking it to Selmer ranks and their twists across subextensions.

Contribution

It establishes a relationship between the order of vanishing of characteristic elements at Artin representations and the Selmer coranks in intermediate subextensions.

Findings

Order of vanishing relates to Selmer coranks.

Results apply to elliptic curves with good ordinary or multiplicative reduction.

Provides insights into the structure of Selmer groups over p-adic Lie extensions.

Abstract

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field with either good ordinary reduction or multiplicative reduction at each prime of $F$ above $p$. We shall study the characteristic element of the Selmer group of $E$ over a $p$-adic Lie extension. In particular, we relate the order of vanishing of these characteristic element evaluated at Artin representations to the Selmer coranks and their twists in the intermediate subextensions of the $p$-adic Lie extension.