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Universality of the Galois action on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$

Alexander Petrov

Abstract

We prove that any semi-simple representation of the Galois group of a number field coming from geometry appears as a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective line with $3$ punctures.

Universality of the Galois action on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$

Abstract

We prove that any semi-simple representation of the Galois group of a number field coming from geometry appears as a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective line with punctures.

Paper Structure

This paper contains 13 sections, 31 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Pro-algebraic completion
  3. Belyi's theorem and its immediate consequences
  4. Direct sum and tensor product
  5. Artin motives
  6. Dual representations
  7. First cohomology of local systems
  8. Proof of Theorem \ref{['main theorem intro']}
  9. Variants and questions
  10. Frobenius eigenvalues
  11. Pro-reductive completion
  12. Base points
  13. Tangential base points

Key Result

Proposition 1.1

For a number field $F$, any continuous finite image representation $\rho:G_F\to GL_d({\mathbb Q})$ can be embedded into the space of locally constant functions $\mathop{\mathrm{Func}}\nolimits^{\mathrm{loc.}\mathrm{const.}}(\pi_1^{\mathrm{\acute{e}t}}({\mathbb P}^1_{\overline{F}}\setminus\{0,1,\inft

Theorems & Definitions (62)

  • Proposition 1.1: Proposition \ref{['artin motives']}
  • Theorem 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Example 1.5
  • Example 1.6
  • Proposition 1.7: Proposition \ref{['tensor: tensor product']}
  • Proposition 1.8: Proposition \ref{['H1 main']}
  • Lemma 2.1
  • proof
  • ...and 52 more