{"product_id":"weils-conjecture-for-function-fields","title":"Weil's Conjecture for Function Fields","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.\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\/p\u003e","brand":"Princeton University Press","offers":[{"title":"Default Title","offer_id":54218296033624,"sku":"9780691182148","price":90.99,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0278\/1295\/4195\/files\/9780691182148_6e67b38e-7fd3-4c55-a277-a0e3fd7d6ecf.jpg?v=1771307130","url":"https:\/\/agendabookshop.com\/products\/weils-conjecture-for-function-fields","provider":"Agenda Bookshop","version":"1.0","type":"link"}