# Contents

## Idea

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

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

###### Definition

The mass of $q$ is the sum

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

taken over the positive-definite quadratic forms $q'$ in the genus of $g$.

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

###### Definition

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

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

###### Theorem

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

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

Equivalently,

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

where $\Spin_q$ is the 2-fold universal cover of $SO_q$.

###### Theorem

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

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

### Function field case

Let $X$ be a smooth projective curve over the finite field $F_q$, for some prime $q$. Let $K_X$ denote the function field of $X$. For $x \in X$, write $O_x$ for the completion of the local ring at $x$ and $K_x$ for its fraction field.

###### Definition

The ring of adeles of $K_X$ is defined as

(5)$\mathbf{A}_X = \prod^{res}_x K_x \subset \prod_x K_x,$

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

$\mathbf{A}_X$ is a locally compact commutative ring with discrete subring $K_X \subset \mathbf{A}_X$.

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

###### Theorem

(Gaitsgory-Lurie, “Weil conjecture for function fields”). Let $X$ be a smooth projective curve over the finite field $F_q$, for some prime $q$. Then

(6)$\mu_{\mathrm{Tam}}(G_0(K_X)\backslash G_0(\mathbf{A}_X)) = 1.$

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 $Bun_G(X)$ on $X$, in view of the function field analogy:

###### Remark

Under the function field analogy, a global field $K_X$ 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 $X$ over a finite field $\mathbb{F}_q$ or “over $\mathbb{F}_1$” (the would-be field with one element), respectively. Moreover, under this analogy

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

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

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

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

• hence the further double quotient stack

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

is the groupoid of Cech cocycles with Cech coboundaries between them for $G$-principal bundles relative to covers of $\Sigma$ 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 $Bun_G(X)$, generalizing work of Kai Behrend:

###### Theorem

(Gaitsgory-Lurie, “Grothendieck-Lefschetz trace formula for $Bun_G(X)$”). Let $X$ be a smooth projective curve over the finite field $F_q$, for some prime $q$. Then

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

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

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

###### Theorem

(Gaitsgory-Lurie). Let $X$ be a smooth projective curve over the finite field $F_q$, for some prime $q$. Then

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

## 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)

### 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 $G = SL_n$.

• 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.