Weil's Conjecture for Function Fields
- Volume I (AMS-199)
indgår i Annals of Mathematics Studies serien
- Indbinding:
- Paperback
- Sideantal:
- 320
- Udgivet:
- 19. februar 2019
- Størrelse:
- 235x156x26 mm.
- Vægt:
- 550 g.
- 8-11 hverdage.
- 20. november 2024
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Ingen binding og kan opsiges når som helst.
Beskrivelse af Weil's Conjecture for Function Fields
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: WeilΓÇÖs 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 WeilΓÇÖs 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 Γäô-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 G-bundles (a global object) as a tensor product of local factors.
Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies WeilΓÇÖs conjecture. The proof of the product formula will appear in a sequel volume.
Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies WeilΓÇÖs conjecture. The proof of the product formula will appear in a sequel volume.
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