AbstractsMathematics

In this thesis, I mainly study the forms of a smooth projective variety over a finite field k and the attached Hasse-Weil zeta functions. I also study the forms of a scheme. The study begins with understanding the relationship between etale cohomology and the Hasse-Weil zeta function of a smooth projective variety over k. In order to classify forms of a quasi-projective variety V over a perfect field K, I study nonabelian cohomology and Galois descent to give a proof of the bijection between the equivalence classes of K'/K-forms of V and H1 (Gal(K'/K), AutK' (V)), where K'/K is some Galois extension. I also present explicitly forms of elliptic curves and their corresponding Hasse-Weil zeta functions. The second part of my thesis is focused on forms of a scheme, especially in the affine case. This is a generalization of forms of a variety. I define an etale form of a scheme and generalize Milne's definition of the first Cech cohomology of a non-abelian sheaf to any (not necessarily abelian) presheaf. I prove there exists an injective map in the affine case from the set of equivalence classes of affine etale forms into the first Cech cohomology of a contravariant functor. I prove that the definition of an etale form of a scheme is compatible with the definition of a form of a variety over a perfect field. I also prove that the first Galois cohomology can be canonically identified with the first Cech cohomology when the base is Spec k for some perfect, field k.