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.

Comment: 18 pages

• 11G99 - Arithmetic algebraic geometry (Diophantine geometry)
• 14G99 - Arithmetic problems in algebraic geometry; Diophantine geometry

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