*Dennis Gaitsgory and Jacob Lurie*

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691182148
- eISBN:
- 9780691184432
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691182148.003.0001
- Subject:
- Mathematics, Mathematical Finance

This introductory chapter sets out the book's purpose, which is to study Weil's conjecture over function fields: that is, fields K which arise as rational functions on an algebraic curve X over a ...
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This introductory chapter sets out the book's purpose, which is to study Weil's conjecture over function fields: that is, fields K which arise as rational functions on an algebraic curve X over a finite field Fq. It reformulates Weil's conjecture as a mass formula, which counts the number of principal G-bundles over the algebraic curve X. An essential feature of the function field setting is that the objects that we want to count (in this case, principal G-bundles) admit a “geometric” parametrization: they can be identified with Fq-valued points of an algebraic stack BunG(X). This observation is used to reformulate Weil's conjecture yet again: it essentially reduces to a statement about the ℓ-adic cohomolog of BunG(X), reflecting the heuristic idea that it should admit a “continuous Künneth decomposition”.Less

This introductory chapter sets out the book's purpose, which is to study Weil's conjecture over function fields: that is, fields K which arise as rational functions on an algebraic curve X over a finite field **F**_{q}. It reformulates Weil's conjecture as a mass formula, which counts the number of principal G-bundles over the algebraic curve X. An essential feature of the function field setting is that the objects that we want to count (in this case, principal G-bundles) admit a “geometric” parametrization: they can be identified with **F**q-valued points of an algebraic stack Bun_{G}(X). This observation is used to reformulate Weil's conjecture yet again: it essentially reduces to a statement about the ℓ-adic cohomolog of Bun_{G}(X), reflecting the heuristic idea that it should admit a “continuous Künneth decomposition”.

*Dennis Gaitsgory and Jacob Lurie*

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691182148
- eISBN:
- 9780691184432
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691182148.001.0001
- Subject:
- Mathematics, Mathematical Finance

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 ...
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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 ℓ-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–Lefschetz 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.Less

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 ℓ-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–Lefschetz 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.

*Dennis Gaitsgory and Jacob Lurie*

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780691182148
- eISBN:
- 9780691184432
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691182148.003.0002
- Subject:
- Mathematics, Mathematical Finance

The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category Dℓ(X). This chapter attempts to ...
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The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category Dℓ(X). This chapter attempts to remedy the situation by introducing a mathematical object Shvℓ (X), which refines the triangulated category Dℓ (X). This object is not itself a category but instead is an example of an ∞-category, which is referred to as the ∞-category of ℓ-adic sheaves on X. The triangulated category Dℓ (X) can be identified with the homotopy category of Shvℓ (X); in particular, the objects of Dℓ (X) and Shvℓ (X) are the same. However, there is a large difference between commutative algebra objects of Dℓ (X) and commutative algebra objects of the ∞-category Shvℓ (X). We can achieve (b') by viewing the complex B as a commutative algebra of the latter sort.Less

The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category D_{ℓ}(X). This chapter attempts to remedy the situation by introducing a mathematical object Shv_{ℓ} (X), which refines the triangulated category D_{ℓ} (X). This object is not itself a category but instead is an example of an ∞-category, which is referred to as the ∞-category of ℓ-adic sheaves on X. The triangulated category D_{ℓ} (X) can be identified with the homotopy category of Shv_{ℓ} (X); in particular, the objects of D_{ℓ} (X) and Shv_{ℓ} (X) are the same. However, there is a large difference between commutative algebra objects of D_{ℓ} (X) and commutative algebra objects of the ∞-category Shv_{ℓ} (X). We can achieve (b') by viewing the complex B as a commutative algebra of the latter sort.