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Weil conjecture on Tamagawa numbers

Contents

Idea

Let qq be a positive-definite quadratic form over the ring of integers Z\mathbf{Z}. The mass of qq is a weighted count of the number of quadratic forms in the genus of qq, up to isomorphism (weighted by multiplicity). The Smith-Minkowski-Siegel mass formula gives a (complicated but computable) formula for the mass of qq.

Over number fields, ideas of Tamagawa and Weil allow a reformulation of this formula as the statement that the Tamagawa number of a certain algebraic group associated to qq is equal to 1. Weil’s conjecture is then the statement, now a theorem of Robert Langlands, K. F. Lai and Robert Kottwitz, that the Tamagawa number of any semisimple simply-connected algebraic group is equal to 1.

There is analogue of the conjecture for function fields, and it has been proved by Dennis Gaitsgory and Jacob Lurie.

The statement

Number field case

Let qq be a positive-definite quadratic form over the ring of integers Z\mathbf{Z}.

Definition

The mass of qq is the sum

(1) q1|O q(Z)| \sum_{q'} \frac{1}{|O_{q'}(\mathbf{Z})|}

taken over the positive-definite quadratic forms qq' in the genus of gg.

Let A\mathbf{A} be the ring of adeles, a locally compact commutative ring containing Q\mathbf{Q} as a discrete subring.

Definition

Let O q(A)O_q(\mathbf{A}) denote the automorphism group of q Aq_\mathbf{A}, the base change of qq to A\mathbf{A}. Let SO q(A)O q(A)SO_q(\mathbf{A}) \subset O_q(\mathbf{A}) denote the subgroup of automorphisms with determinant 1.

SO q(A)SO_q(\mathbf{A}) is a locally compact topological group containing SO q(Q)SO_q(\mathbf{Q}) as a discrete subgroup and SO q(Z^×R)SO_q(\hat{\mathbf{Z}} \times \mathbf{R}) as a compact open subgroup.

Theorem

(Tamagawa-Weil reformulation of Siegel mass formula). Let μ Tam\mu_{\mathrm{Tam}} denote the Tamagawa measure. Then

(2)μ Tam(SO q(Q)\SO q(A))=2. \mu_{\mathrm{Tam}}(SO_q(\mathbf{Q}) \backslash SO_q(\mathbf{A})) = 2.

Equivalently,

(3)μ Tam(Spin q(Q)\Spin q(A))=1, \mu_{\mathrm{Tam}}(Spin_q(\mathbf{Q})\backslash Spin_q(\mathbf{A})) = 1,

where Spin q\Spin_q is the 2-fold universal cover of SO qSO_q.

Theorem

(Langlands-Lai-Kottwitz, “Weil conjecture”). Let GG be a semisimple simply-connected algebraic group over Q\mathbf{Q}. Then

(4)μ Tam(G(Q)\G(A))=1. \mu_{\mathrm{Tam}}(G(\mathbf{Q})\backslash G(\mathbf{A})) = 1.

Function field case

Let XX be a smooth projective curve over the finite field F qF_q, for some prime qq. Let K XK_X denote the function field of XX. For xXx \in X, write O xO_x for the completion of the local ring at xx and K xK_x for its fraction field.

Definition

The ring of adeles of K XK_X is defined as

(5)A X= x resK x xK x, \mathbf{A}_X = \prod^{res}_x K_x \subset \prod_x K_x,

i.e. the subgroup consisting of elements {g x} xX\{g_x\}_{x \in X} such that gG(O x)g \in G(O_x) for all but finitely many xx.

A X\mathbf{A}_X is a locally compact commutative ring with discrete subring K XA XK_X \subset \mathbf{A}_X.

Let G 0G_0 be a semisimple simply-connected linear algebraic group over K XK_X. Then G 0(K X)G 0(A)G_0(K_X) \subset G_0(\mathbf{A}) is a discrete subgroup of the locally compact group G 0(A)G_0(\mathbf{A}). One defines a Tamagawa measure on G(A)G(\mathbf{A}) in a similar way as usual, i.e. by choosing a differential form and multiplying the forms on G 0(K x)G_0(K_x) (xXx \in X). Then the function field version of Weil's conjecture is

Theorem

(Gaitsgory-Lurie, “Weil conjecture for function fields”). Let XX be a smooth projective curve over the finite field F qF_q, for some prime qq. Then

(6)μ Tam(G 0(K X)\G 0(A X))=1. \mu_{\mathrm{Tam}}(G_0(K_X)\backslash G_0(\mathbf{A}_X)) = 1.

This was proved by Dennis Gaitsgory and Jacob Lurie. They reformulated the conjecture as a statement about the cohomology of the moduli stack of G-principal bundles Bun G(X)Bun_G(X) on XX, in view of the function field analogy:

Remark

Under the function field analogy, a global field K XK_X such as a function field or a number field is interpreted as the field of global sections of the rational functions on an arithmetic curve XX over a finite field 𝔽 q\mathbb{F}_q or “over 𝔽 1\mathbb{F}_1” (the would-be field with one element), respectively. Moreover, under this analogy

  • the ring of adeles 𝔸 X\mathbb{A}_X is the ring of functions on all punctured formal disks in XX subject to the condition that all but at most finitely many of them extend to the un-punctured disk;

  • accordingly G(𝔸 X)G(\mathbb{A}_X) is the group of GG-valued such functions;

  • the quotient K X\𝔸 XK_{X}\backslash \mathbb{A}_{X} is hence the quotient of such functions on punctured formal disks around finitely many points by the functions on Σ\Sigma with these finitely many points removed; and similarly G(K X)\G(𝔸 X)G(K_X)\backslash G(\mathbb{A}_X) is the quotient of group-valued such function;

  • the ring 𝒪\mathcal{O} is the ring of functions on all formal disks in Σ\Sigma;

  • hence the further double quotient stack

    Bun G(X)=G(K X)\G(𝔸 X)//G(𝒪) Bun_G(X) = G(K_X)\backslash G(\mathbb{A}_X)//G(\mathcal{O})

    is the groupoid of Cech cocycles with Cech coboundaries between them for GG-principal bundles relative to covers of Σ\Sigma with patches being the complement of finitely many points and the formal disks around these points.

For more on this see at moduli space of bundles – over curves.

First they proved a Grothendieck-Lefschetz trace formula for Bun G(X)Bun_G(X), generalizing work of Kai Behrend:

Theorem

(Gaitsgory-Lurie, “Grothendieck-Lefschetz trace formula for Bun G(X)Bun_G(X)”). Let XX be a smooth projective curve over the finite field F qF_q, for some prime qq. Then

(7)|Bun G(X)(F q)|q dim(Bun G(X))= i0(1) iTr(Frob 1H i(Bun¯ G(X);Q ) \frac{|Bun_G(X)(\mathbf{F}_q)|}{q^{\dim(Bun_G(X))}} = \sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell)

where Bun¯ G(X)\overline{Bun}_G(X) denotes the base change of Bun G(X)Bun_G(X) to the algebraic closure of F q\mathbf{F}_q, where ||{\vert -\vert} denotes groupoid cardinality, and where Frob:Bun¯ G(X)Bun¯ G(X)Frob : \overline{Bun}_G(X) \to \overline{Bun}_G(X) denotes the Frobenius map.

Then they proved the following result, via nonabelian Poincaré duality which provides a local-global principle.

Theorem

(Gaitsgory-Lurie). Let XX be a smooth projective curve over the finite field F qF_q, for some prime qq. Then

(8) i0(1) iTr(Frob 1H i(Bun¯ G(X);Q )= xXq ddeg(x)|G(κ(x))| \sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell) = \prod_{x \in X} \frac{q^{d \cdot \deg(x)}}{|G(\kappa(x))|}

References

Work of Gaitsgory-Lurie

A proof of the function field case is discussed in

The proof was announced in

and is outlined in the lecture notes

  • Jacob Lurie, Tamagawa Numbers via Nonabelian Poincare Duality (282y), lecture notes, 2014 (web)

See also the shorter lecture notes

Previous work

The idea of the relationship between Tamagawa numbers and moduli spaces of vector bundles goes back to Günter Harder, who primarily considered the case G=SL nG = SL_n.

  • Günter Harder, Eine Bemerkung zu einer Arbeit von P. E. Newstead., Journal für die reine und angewandte Mathematik, 242 (1970): 16-25, eudml.

  • Günter Harder, M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Mathematische Annalen, 212 (1975), Issue 3, pp 215-248.

The moduli stack of principal bundles was studied in more generality in

  • Kai Behrend, Ajneet Dhillon, On the Motive of the Stack of Bundles, 2005, arXiv.

  • Kai Behrend, Ajneet Dhillon, Connected components of moduli stacks of torsors via Tamagawa numbers., Canad. J. Math, 2009, arXiv.

See also

  • Aravind Asok, Brent Doran, Frances Kirwan. Yang–Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics. Bulletin of the London Mathematical Society (2008), arXiv.

Revised on December 31, 2014 17:11:44 by Adeel Khan (127.0.0.1)