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

Shun Ishii, Yuki Goto

Abstract

In this paper, we study certain Kummer characters, which we call the elliptic Soulé 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 Soulé characters having values in Tate twists can be written in terms of the Soulé characters and generalized Bernoulli numbers. We apply this result to give a criterion for surjectivity of the elliptic Soulé characters and an analogue of the Coleman-Ihara formula.

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

Abstract

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

Paper Structure

This paper contains 8 sections, 18 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental theta function
  4. Elliptic units and cyclotomic units
  5. Étale regulator map and Soulé cocycles
  6. Properties of elliptic Soulé characters
  7. Proof of the factorization formula
  8. Proofs of corollaries

Key Result

Lemma 1.2

For each $n \geq 1$, the following assertions hold.

Theorems & Definitions (52)

  • Remark 1.1
  • Lemma 1.2: cf. Section \ref{['2.1']} for the proof
  • Definition 1.3
  • Remark 1.4: Relation to Nakamura's work Na95
  • Definition 1.5: Soulé characters
  • Remark 1.6
  • Theorem 1.7: cf. Corollary \ref{['cor:main2']}
  • Remark 1.8
  • Corollary 1.9: cf. Corollary \ref{['cor:main3']}
  • Corollary 1.10: cf. Corollary \ref{['cor:Coleman-Ihara']}
  • ...and 42 more