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On graded Lie algebras associated to once-punctured elliptic curves with complex multiplication

Shun Ishii

Abstract

We study a graded Lie algebra arising from the Galois action on the pro-$p$ fundamental group of a once-punctured elliptic curve with complex multiplication. Among other things, we provide a minimal generating set of the rationalized Lie algebra under suitable assumptions. The proof is based on a slight variant of the theory of weighted completion of profinite groups developed by Hain and Matsumoto.

On graded Lie algebras associated to once-punctured elliptic curves with complex multiplication

Abstract

We study a graded Lie algebra arising from the Galois action on the pro- fundamental group of a once-punctured elliptic curve with complex multiplication. Among other things, we provide a minimal generating set of the rationalized Lie algebra under suitable assumptions. The proof is based on a slight variant of the theory of weighted completion of profinite groups developed by Hain and Matsumoto.
Paper Structure (14 sections, 18 theorems, 64 equations, 1 table)

This paper contains 14 sections, 18 theorems, 64 equations, 1 table.

Table of Contents

  1. Introduction and main result
  2. Lie algebra arising from Galois action on fundamental group
  3. Structure of Lie algebra and its application
  4. Case of projective line minus three points
  5. Speculation
  6. Notational conventions
  7. Weighted completion and structure of its Lie algebra
  8. Negatively weighted extension
  9. Weighted completion of profinite group
  10. Comparison of Galois image and weighted completion
  11. Galois action on prounipotent fundamental group
  12. Structure of certain outer automorphism group scheme
  13. Comparison of Lie algebras
  14. Summary of main result for inert primes

Key Result

Lemma 1.1

Let $E'$ be another elliptic curve over a finite extension of $K$ having complex multiplication by $\mathcal{O}_{K}$, and $\mathfrak{g}_{X'}$ the graded Lie algebra defined using the pro-$p$ outer Galois representation of the once-punctured elliptic curve $X'$ associated with $E'$. Then $\mathfrak{g

Theorems & Definitions (44)

  • Lemma 1.1
  • proof
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Remark 1.7
  • Conjecture 1.8: Is23+
  • ...and 34 more