Calcul effectif de la cohomologie des faisceaux constructibles sur le site étale d'une courbe

Abstract: This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the \'etale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or nodal curve, which in turn allows us to give an explicit description of the functor $\mathrm{R}\Gamma(X,-)\colon \mathrm{D}^b_c(X,\mathbb{Z}/n\mathbb{Z})\to \mathrm{D}^b_c(\mathbb{Z}/n\mathbb{Z})$. This description is functorial in the scheme $X$ and the given complex of constructible sheaves. In particular, if $X$ and the sheaf $\mathcal{F}$ are obtained by base change from a subfield, we describe the Galois action on the complex $\mathrm{R}\Gamma(X,\mathcal{F})$. We give precise bounds on the number of operations performed by the algorithm computing $\mathrm{R}\Gamma(X,\mathcal{F})$. We also give an explicit description of cup-products in the cohomology of locally constant constructible sheaves over smooth projective curves. Finally, we show how to use these algorithms in order to compute the cohomology groups of a constant sheaf on a smooth surface fibered over the projective line.