{"product_id":"weils-conjecture-for-function-fields-dennis-gaitsgory-9780691182148","title":"Weil's Conjecture for Function Fields: Volume I (Ams-199)","description":"\u003cp\u003eA central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field \u003ci\u003eK\u003c\/i\u003e in terms of the behavior of various completions of \u003ci\u003eK\u003c\/i\u003e. This book looks at a specific example of a local-to-global principle: Weil's conjecture on the Tamagawa number of a semisimple algebraic group \u003ci\u003eG\u003c\/i\u003e over \u003ci\u003eK\u003c\/i\u003e. In the case where \u003ci\u003eK\u003c\/i\u003e is the function field of an algebraic curve \u003ci\u003eX\u003c\/i\u003e, this conjecture counts the number of \u003ci\u003eG\u003c\/i\u003e-bundles on \u003ci\u003eX\u003c\/i\u003e (global information) in terms of the reduction of \u003ci\u003eG\u003c\/i\u003e at the points of \u003ci\u003eX\u003c\/i\u003e (local information). The goal of this book is to give a conceptual proof of Weil's conjecture, based on the geometry of the moduli stack of \u003ci\u003eG\u003c\/i\u003e-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ℓ-adic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of \u003ci\u003eG\u003c\/i\u003e-bundles (a global object) as a tensor product of local factors. \u003c\/p\u003e\u003cp\u003e\u003c\/p\u003eUsing a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e Dennis Gaitsgory, Jacob Lurie\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 0691182140\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9780691182148\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Princeton University Press\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 02\/19\/2019\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 320\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Paperback\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 1.15lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.20h x 6.10w x 1.10d","brand":"Dennis Gaitsgory","offers":[{"title":"Paperback","offer_id":43981106381055,"sku":"9780691182148","price":84.0,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_b9a23c6d-f6d3-481a-b450-da3233ae5aeb.jpg?v=1683253690","url":"https:\/\/www.whiterainbookhouse.com\/products\/weils-conjecture-for-function-fields-dennis-gaitsgory-9780691182148","provider":"WR Book House","version":"1.0","type":"link"}