{"product_id":"weils-conjecture-for-function-fields-volume-i-9780691182148","title":"Weil's Conjecture for Function Fields: Volume I","description":"\u003cp\u003e • Author(s): Dennis Gaitsgory\u003cbr\u003e • Publisher: Princeton University Press\u003cbr\u003e • Publisher Imprint: Princeton University Press\u003cbr\u003e • BISAC: Geometry - Algebraic\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003e\u003cspan\u003eA central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eK\u003c\/span\u003e\u003cspan\u003e in terms of the behavior of various completions of \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eK\u003c\/span\u003e\u003cspan\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 \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eG\u003c\/span\u003e\u003cspan\u003e over \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eK\u003c\/span\u003e\u003cspan\u003e. In the case where \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eK\u003c\/span\u003e\u003cspan\u003e is the function field of an algebraic curve \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eX\u003c\/span\u003e\u003cspan\u003e, this conjecture counts the number of \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eG\u003c\/span\u003e\u003cspan\u003e-bundles on \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eX\u003c\/span\u003e\u003cspan\u003e (global information) in terms of the reduction of \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eG\u003c\/span\u003e\u003cspan\u003e at the points of \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eX\u003c\/span\u003e\u003cspan\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 \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eG\u003c\/span\u003e\u003cspan\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 \u003c\/span\u003e\u003cspan class=\"a-text-italic\"\u003eG\u003c\/span\u003e\u003cspan\u003e-bundles (a global object) as a tensor product of local factors.\u003cbr\u003e\u003cbr\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.\u003c\/span\u003e\u003c\/p\u003e","brand":"Princeton University Press","offers":[{"title":"Paperback","offer_id":45268911882391,"sku":"9780691182148","price":5698.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9780691182148.webp?v=1769236780","url":"https:\/\/atlanticbooks.com\/products\/weils-conjecture-for-function-fields-volume-i-9780691182148","provider":"Atlantic Books","version":"1.0","type":"link"}