On The Weil Conjectures

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Rezaee, Fatemeh | 2013

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 47302 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Jafari, Amir
Abstract:
” The story of the "Weil conjectures" is a marvelous example of mathematical imagination, and one of the most striking instances exhibiting the fundamental unity of mathematics.”In 1949,Andre Weil stated some conjectures on the zeta function of Algebraic varieties over finite fields .These conjectures were analogue of the properties of Riemann zeta function ,in particular Riemann hypothesis.In fact ,Weil built a bridge between Diophantine structure on varieties over finite fields (Counting of rational points on varieties) and cohomological structure of them over the field of complex numbers(topology of variety).In this thesis, first we state Weil’s motivations for these conjectures and state exact statement of the conjectures. Then, we prove these conjectures for the case of curves and explain Grothendieck and Etale cohomology that uses in the proof of the weil conjectures.Then,we explain the main ideas of Weil conjectures proof via l-adic cohomology .Finally, we try to explain Deligne’s proof of the Riemann hypothesis for varieties over finite fields
Keywords:
Zeta Potential ; Riemann-Rokh Theorem ; Riemann Hypothesis ; Etal Cohomology ; Algebraic Varieties on Finite Fields

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