  We propose a conjecture on the density of arithmetic points in the
deformation space of representations of the \'etale fundamental group in
positive characteristic. This? conjecture has applications to \'etale
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.

Algebraic Geometry;Number Theory