On factorizations of certain Kummer characters associated to once-punctured elliptic curves with complex multiplication

TL;DR

This paper investigates elliptic Soulé characters from Galois actions on punctured elliptic curves with complex multiplication, expressing them via classical characters and Bernoulli numbers, and explores their surjectivity and related formulas.

Contribution

It introduces a factorization of elliptic Soulé characters in terms of known characters and Bernoulli numbers, providing new insights into their structure and properties.

Findings

Elliptic Soulé characters can be expressed using classical Soulé characters and Bernoulli numbers.

A criterion for the surjectivity of elliptic Soulé characters is established.

An analogue of the Coleman-Ihara formula for these characters is derived.

Abstract

In this paper, we study certain Kummer characters, which we call the elliptic Soul\'e characters, arising from Galois actions on the pro-$p$ fundamental groups of once-punctured elliptic curves with complex multiplication. In particular, we prove that elliptic Soul\'e characters having values in Tate twists can be written in terms of the Soul\'e characters and generalized Bernoulli numbers. We apply this result to give a criterion for surjectivity of the elliptic Soul\'e characters and an analogue of the Coleman-Ihara formula.