Forms of smooth projective varieties and their zeta functions

Institution: | McGill University |
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Department: | Department of Mathematics and Statistics. |

Degree: | MS |

Year: | 2006 |

Keywords: | Mathematics. |

Record ID: | 1787882 |

Full text PDF: | http://digitool.library.mcgill.ca/thesisfile99214.pdf |

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.