On Galois representations arising from geometry
par
Pierre Grisvard
Institut Henri Poincaré
A classical result of Belyi asserts that a smooth projective curve over the complex numbers is defined over a number field if and only if it admits a finite morphism to \P^1 which is branched at most above 0,1 and the point at infinity. If G_\Q denotes the absolute Galois group of \Q, this notably implies that the natural morphism from G_\Q to the outer automorphisms of the étale fundamental group of \P^1_{\bar{\Q}} minus 0,1 and \infty is injective. On the other hand, an important problem in arithmetic geometry is to classify all representations of Galois groups of number fields that arise from the étale cohomology of algebraic varieties. A famous conjecture of Fontaine and Mazur predicts that Galois representations which satisfy reasonable geometric conditions actually arise from the cohomology of varieties. Building on the ideas presented in the previous talk, I will try to explain a recent result of Petrov, which essentially shows that all Galois representations coming from geometry can be realised as linear representations on the étale fundamental group of \P^1_{\bar{\Q}} minus 0,1 and \infty. I will also explain how this result (which may be interpreted as a motivic generalisation of Belyi's theorem), coupled with another result of Petrov on p-adic local systems, refines the conjecture of Fontaine and Mazur.