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On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves

Shun Ishii

Abstract

In this paper, we compare a certain field arising from the pro-$p$ outer Galois representation associated to a once-punctured CM elliptic curve over an imaginary quadratic field $K$ with the maximal pro-$p$ Galois extension of the mod-$p$ ray class field $K(p)$ of $K$ unramified outside $p$. We prove that these two fields coincide for every prime $p$ which satisfies certain assumptions, assuming an analogue of the Deligne-Ihara conjecture. This may be regarded as an analogue of a result of Sharifi on the kernel of the pro-$p$ outer Galois representation associated to the projective line minus three points.

On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves

Abstract

In this paper, we compare a certain field arising from the pro- outer Galois representation associated to a once-punctured CM elliptic curve over an imaginary quadratic field with the maximal pro- Galois extension of the mod- ray class field of unramified outside . We prove that these two fields coincide for every prime which satisfies certain assumptions, assuming an analogue of the Deligne-Ihara conjecture. This may be regarded as an analogue of a result of Sharifi on the kernel of the pro- outer Galois representation associated to the projective line minus three points.
Paper Structure (13 sections, 41 theorems, 143 equations)

This paper contains 13 sections, 41 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structure of graded Lie algebra
  4. Elliptic analogues of Soulé characters
  5. Main result
  6. Two-variable filtrations on profinite groups
  7. Two-variable filtration on free pro-$p$ group of rank two
  8. Two-variable filtration on pro-$p$ mapping class group
  9. Two-variable filtration on Galois group
  10. Proof of main result
  11. Construction of elements
  12. Group-theoretic lemmas and end of proof of main result
  13. Pure locality of $p$-th cyclotomic extension of imaginary quadratic field

Key Result

Theorem 1.1

Let $p$ be an odd regular prime and suppose the Deligne-Ihara conjecture holds for $p$. Then the fixed field of the pro-$p$ outer Galois representation associated to the projective line minus three points coincides with $\Omega^{\mathrm{cyc}}$.

Theorems & Definitions (101)

  • Theorem 1.1: Sharifi Sh
  • Remark 1.1.1
  • Theorem A: =Theorem \ref{['thm:main']}
  • Remark 1.1.2
  • Example 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.2.1
  • ...and 91 more