G-fonctions et cohomologie des hypersurfaces singulières

Our object of study is the arithmetic of the differential modules W(l) (l ∈ ℕ - {0}), associated by Dwork's theory to a homogeneous polynomial f(λ, X) with coefficients in a number field. Our main result is that W(l) is a differential module of type G, c'est-à-dire, a module whose solutions are G-functions. For the proof we distinguish two cases: the regular one and the non regular one. Our method gives us an effective upper bound for the global radius of W(l), which doesn't depend on "l" but only on the polynomial f(λ, X). This upper bound is interesting because it gives an explicit estimate for the coefficients of the solutions of W(l). In the regular case we know there is an isomorphism of differential modules between W(1) and a certain De Rham cohomology group, endowed with the GaussManin connection, c'est-à-dire, our module "comes from geometry". Therefore our main result is a particular case of André's theorem which assert that at least in the regular case, all modules coming from geometry are of type G.