transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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.
Let $q$ be a positive-definite quadratic form over the ring of integers $\mathbf{Z}$.
The mass of $q$ is the sum
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.
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.
(Tamagawa-Weil reformulation of Siegel mass formula). Let $\mu_{\mathrm{Tam}}$ denote the Tamagawa measure. Then
Equivalently,
where $\Spin_q$ is the 2-fold universal cover of $SO_q$.
(Langlands-Lai-Kottwitz, “Weil conjecture”). Let $G$ be a semisimple simply-connected algebraic group over $\mathbf{Q}$. Then
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.
The ring of adeles of $K_X$ is defined as
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
(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
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:
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
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:
(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
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.
(Gaitsgory-Lurie). Let $X$ be a smooth projective curve over the finite field $F_q$, for some prime $q$. Then
A proof of the function field case is discussed in
The proof was announced in
and is outlined in the lecture notes
See also the shorter lecture notes
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.
See also