THE WEIL CONJECTURES I By Pierre Deligne (translated by Evgeny Goncharov)
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[email protected] I attempted to write the full translation of this article to make the remarkable proof of Pierre Deligne available to a greater number of people. Overviews of the proofs can be found elsewhere. I especially recommend the notes of James Milne on Etale Cohomology that also contain a justifica- tion for the theory underlying this article and proofs of the results used by Deligne. The footnotes are mostly claims that some details appear in Milne, clarifications of some of the terminology or my personal struggles. I have also made a thorough overview of the proof together with more detailed explanations - https: // arxiv. org/ abs/ 1807. 10812 . Enjoy! Abstract In this article I prove the Weil conjecture about the eigenvalues of Frobenius endomor- phisms. The precise formulation is given in (1.6). I tried to make the demonstration as geometric and elementary as possible and included reminders: only the results of paragraphs 3, 6, 7 and 8 are original. In the article following this one I will give various refinements of the intermediate results and the applications, including the hard Lefschetz theorem (on the iterated cup products by the cohomology class of a hyperplane section). The text faithfully follows from the six lectures given at Cambridge in July 1973. I thank N.Katz for allowing me to use his notes. Contents 1 The theory of Grothendieck: a cohomological interpretation of L-functions 2 2 The theory of Grothendieck: Poincare duality 7 arXiv:1807.10810v2 [math.AG] 26 Jan 2019 3 The main lemma (La majoration fondamentale) 10 4 Lefschetz theory: local theory 13 5 Lefschetz theory: global theory 15 6 The rationality theorem 20 7 Completion of the proof of (1.7) 23 8 First applications 26 ∗Cambridge University, Center for Mathematical Sciences, Wilberforce Road, Cambridge, UK yNational Research University Higher School of Economics (NRU HSE), Usacheva 6, Moscow, Russia.