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Density of Arithmetic Representations of Function Fields

Hélène Esnault, Moritz Kerz

TL;DR

This work proposes and analyzes a density conjecture for arithmetic points in the deformation space of étale representations over positive characteristic fields. It develops the Zariski topology on semi-simple representation spaces via pseudorepresentations, proves aWeak Conjecture for curves with irreducible residual representations, and establishes a Strong Conjecture for the tame rank-2 case on X = P^1 ackslash {0,1,∞}, showing arithmetic points correspond to quasi-unipotent monodromy at infinity. The authors provide reductions to curves and explicit geometric arguments, including a novel approach to de Jong’s conjecture in the targeted case that sidesteps the Langlands program. These results link deformation theory, arithmetic points, and Lefschetz-type phenomena in positive characteristic, with implications for étale cohomology and the Hard Lefschetz theorem.

Abstract

We propose a conjecture on the density of arithmetic points in the deformation space of representations of the étale fundamental group in positive characteristic. This? conjecture has applications to étale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus \{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication in Epiga.

Density of Arithmetic Representations of Function Fields

TL;DR

This work proposes and analyzes a density conjecture for arithmetic points in the deformation space of étale representations over positive characteristic fields. It develops the Zariski topology on semi-simple representation spaces via pseudorepresentations, proves aWeak Conjecture for curves with irreducible residual representations, and establishes a Strong Conjecture for the tame rank-2 case on X = P^1 ackslash {0,1,∞}, showing arithmetic points correspond to quasi-unipotent monodromy at infinity. The authors provide reductions to curves and explicit geometric arguments, including a novel approach to de Jong’s conjecture in the targeted case that sidesteps the Langlands program. These results link deformation theory, arithmetic points, and Lefschetz-type phenomena in positive characteristic, with implications for étale cohomology and the Hard Lefschetz theorem.

Abstract

We propose a conjecture on the density of arithmetic points in the deformation space of representations of the étale fundamental group in positive characteristic. This? conjecture has applications to étale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve . v2: very small typos corrected.v3: final. Publication in Epiga.

Paper Structure

This paper contains 11 sections, 17 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. The Zariski topology on the set of semi-simple representations
  3. The density conjectures
  4. The deformation space of pseudorepresentations
  5. Characteristic polynomials
  6. Compatibility with restriction and induction
  7. Proof of Theorem B
  8. Proof of Theorem A
  9. Some Applications
  10. The Hard Lefschetz theorem in positive characteristic.
  11. Our proof of Theorem B on $X_0={\mathbb P}^1\setminus \{0,1,\infty\}$ for $\bar{\rho}$ tame and $r=2$ implies de Jong's conjecture deJ01 in this case

Key Result

Proposition 2.1

There exists an integer $\tilde{m} >0$ and a family $\underline{\tilde{g}}\in G^{\tilde{m}}$ such that for any finite family $\underline{g}\in G^{ m}$ which contains $\underline{\tilde{g}}$ we have:

Theorems & Definitions (51)

  • Proposition 2.1
  • Remark 2.2
  • Remark 3.1
  • Conjecture 3.2: Weak Conjecture
  • Conjecture 3.3: Strong Conjecture
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • Theorem 3.7: Theorem A
  • Theorem 3.8: Theorem B
  • ...and 41 more