{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T00:16:23Z","timestamp":1771460183757,"version":"3.50.1"},"edition-number":"1","reference-count":0,"publisher":"Princeton University Press","isbn-type":[{"value":"9780691182148","type":"print"},{"value":"9780691184432","type":"electronic"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,2,19]]},"abstract":"<p>A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. 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 G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (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 G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting \u2113-adic sheaves. Using this theory, the authors articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck\u2013Lefschetz trace formula, the book shows that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.<\/p>","DOI":"10.23943\/princeton\/9780691182148.001.0001","type":"edited-book","created":{"date-parts":[[2019,9,19]],"date-time":"2019-09-19T09:15:53Z","timestamp":1568884553000},"source":"Crossref","is-referenced-by-count":4,"title":["Weil's Conjecture for Function Fields"],"prefix":"10.23943","author":[{"given":"Dennis","family":"Gaitsgory","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jacob","family":"Lurie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"10341","published-online":{"date-parts":[[2019,9,19]]},"container-title":[],"original-title":["Weil's Conjecture for Function Fields"],"language":"en","deposited":{"date-parts":[[2022,7,28]],"date-time":"2022-07-28T14:21:20Z","timestamp":1659018080000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/princeton-scholarship-online\/book\/43676"}},"subtitle":["Volume I (AMS-199)"],"short-title":[],"issued":{"date-parts":[[2019,2,19]]},"ISBN":["9780691182148","9780691184432"],"references-count":0,"URL":"https:\/\/doi.org\/10.23943\/princeton\/9780691182148.001.0001","relation":{"is-identical-to":[{"id-type":"doi","id":"10.1515\/9780691184432","asserted-by":"subject"},{"id-type":"doi","id":"10.2307\/j.ctv4v32qc","asserted-by":"subject"}]},"subject":[],"published":{"date-parts":[[2019,2,19]]}}}