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I saw that even Wikipedia doesn’t know these links, so I noted them down at Weil conjecture on Tamagawa numbers:
The announced proof by Jacob Lurie and Dennis Gaitsgory via nonabelian Poincaré duality of the Weil conjecture on Tamagawa numbers was announced in
and details are at
Perhaps it should be mentioned that the number field case was already proved by Langlands, Lai, Kottwitz, and Gaitsgory-Lurie proved the function field case.
True, I have added a a super-brief comment to that extent. Somebody should expand on it and provide some actual content here.
How big a deal is this conjecture? I’d not heard of it until I saw Lurie’s course. Someone mentioned this proof as potential Fields Medal fuel, and I’m in no position to even estimate the validity of that statement.
From a Rosetta stone point of view (p. 4), what is the equivalent in Weil’s third column of Riemannian surfaces?
I added some statements to Weil conjecture on Tamagawa numbers.
Thanks! That’s excellent.
I made some more hyperlinks come out.
@adeel do you know what the consensus is regarding the work of Lurie-Gaitsgory on this?
@David, do you mean whether it’s been accepted as a complete proof, or whether it’s “potential Fields medal fuel”?
I mean as a complete proof.
I’m not really qualified to answer, but I can at least say that I’ve never heard anyone expressing any doubts about the proof.
I wonder what their work has to do with
The aim of this article is threefold: to announce results, produced in the setting of A1-homotopy theory, on the motivic cohomology (and hence its many realizations) of quotients in the sense of Mumford’s geometric invariant theory or GIT [ADK], generalizing what was known for singular cohomology of GIT quotients; to explain how methods used over the last three decades to study the singular cohomology of the moduli spaces M(n,d) can thence be adapted to study the motivic cohomology of moduli spaces of bundles on a smooth projec- tive curve C over an algebraically closed field k; and at the same time to extend our understanding of the link between Yang-Mills theory and Tamagawa numbers remarked on by Atiyah and Bott, by re-examining some of their essentially homotopy theoretic considerations in a more algebraic modern light.
In Weil conjecture on Tamagawa numbers, there is reference to the genus of a quadratic form and a link to genus, however at that entry no mention seems to be made of quadratic forms. I was hoping to gain a little understanding of the Weil conjecture on Tamagawa numbers but hit a brickwall at this point. Am I being thick and this meaning of genus is essentially on that linked page but is hidden by terminological links that escape me?
In Weil conjecture on Tamagawa numbers, there is reference to the genus of a quadratic form and a link to genus, however at that entry no mention seems to be made of quadratic forms. I was hoping to gain a little understanding of the Weil conjecture on Tamagawa numbers but hit a brickwall at this point. Am I being thick and this meaning of genus is essentially on that linked page but is hidden by terminological links that escape me?
Lurie defines (in lecture 1) two quadratic forms over $\mathbf{Z}$ to be of the same genus if they are isomorphic after extension of scalars to $\mathbf{Z}/n\mathbf{Z}$ for all $n\ge 1$.
Ah, but my point was not that of looking for the definition (it is on Wikipedia in fact), but that the entry at genus does not say what the genus of a quadratic form is, and does not explain, for instance, if it is linked to the other well-known uses of the word ’genus’. Often by going in the direction of the nPOV, there is some clarification of concepts and their relationship. I was wondering if here something similar can be done. I was therefore asking if someone more knowledgable than me in the matter could check genus to see if the entry needs amending, or if a new entry on genus of a quadratic form is required. (Sometimes, in a hurry, one of us puts in a link to explain some word but the link does not lead to an entry that explains the word! I know, I’ve done that myself several times.)
I have another thought. I think the definition is nearer to the other meaning of genus, i.e. of a surface. This then gave me a vague idea that there might be a connection with Grothendieck-Teichmuller theory / Dessins d’Enfants. I have not checked anywhere if such is known or not.
I see, that’s a valid point. I don’t know anything about quadratic forms myself, or the notions of genus covered at that page, so I can’t say anything.
… that makes two of us. ;-)
I would like to add to the Idea section of Weil conjecture on Tamagawa numbers something that says more about the appearance of $G(F) \backslash \prod'_{x \in X} G(\mathcal{F_Bun_G$. Before adding anything there, I’ll experiment here, starting with a 0th order approximation. This is not to claim any insight but on the contrary to fish for comments that would eventually help materialize a good Idea-section.
So here might be an attempt:
The Weil conjecture on Tamagawa numebrs, concerns the “sizes” (cardinalities) of the arithmetic geometry of certain algebraic groups $G$ and their coset spaces, and in particular of certain double quotient spaces – or rather quotient stacks – which are naturally identified with the moduli stack of G-principal bundles over certain algebraic curves.
Originally this was formulated for $G$ the automorphism group of some quadratic form, but the conjecture makes sense and its proof holds for more general $G$.
A key fact here, which also governs the relation between the number theoretic Langlands correspondence and the geometric Langlands correspondence, is that for $\Sigma$ some complex curve, then the double quotient stack
$G(F) \backslash \prod^{\prime}_{x \in X} G(F_x) / \prod_{x \in X} G(\mathcal{O}_x)$has a special geometric meaning. Here
$F\coloneqq \mathbb{C}(t_x)$ is the ring of rational functions on $X$,
$F_x \coloneqq \mathbb{C}((t_x))$ is the ring of formal Laurent series around $x\in X$
$\mathcal{O}_x \coloneqq \mathbb{C}[ [ t_x ] ]$ is the ring of formal power series around $x \in X$
$\prod'$ denotes the “restricted product” meaning that only finitely many terms in the product are non-trivial.
The secret meaning of this double quotient is revealed by noticing that every $G$-principal bundle on $X$ trivializes on the complement of a finite number of points. This means that we get a “sufficiently good” cover of $X$ by choosing as one patch such a complenent, and as the other patches small disks, and in fact just formal disks, around the given points.
This way then a Cech cocycle for a $G$-principal bundle on $X$ is just a $G$-valed function on the pointed disks obtained from the given disks by removing the center. The collection of such functions is just the term that appears in the middle of the above double quotient. Moreover, a gauge transformation between two such cocycles is then given by a $G(\mathcal{F_G$-valued function on the disjoint union of all the patches. The collection of these is hence given by the elements in $G(F)$, which are the functions on the “big patch”, and functions on all the small disks, which form $\prod_{x \in X} G(\mathcal{O}_x)$. Quotienting out gauge transformations from the Cech cocycles hence yields the above double quotient.
In conclusion, this stacky double quotient is a Cech-presentation of the moduli stack of G-principal bundles on $X$
Now since the Tamagawa numbers give a measure of the size of the coset spaces appearing here, they give a measure of the size of the stack appearing on the right.
This state of affairs becomes more pronounced as we pass (base change) from complex analytic geometry to arithmetic geometry, because in arithmetic geometry over a finite field, the moduli stack will take values in “finite groupoids” or at least in tame groupoids, which means that the “measures” here reduce to counting measures and in fact just to groupoid cardinality.
Passing the above story to arithmetic geoemtry means to invoke the function field analogy.
(Am being urgently interrupted. To be continued…)
Added a small part of the above as a remark here.
Fixed one typo, properly I hope.
Also, $X$ and $\Sigma$ are being mixed. Is one preferable?
And, is that right in $G(K_X)\backslash G(\mathbb{A}_X)//G(\mathcal{O})$ to have single and double slashes?
Thanks for cleaning up after me, I left a mess.
Right, so I changed all the $\Sigma$s to $X$s (I like $\Sigma$s for algebraic curves that play the role of $\Sigma$s in $\Sigma$-models, such as here, but here to fit with the rest of the entry it should be $X$).
Regarding the double slash: there must be a stacky quotient somewhere, for the last statement about the groupoid cardinality of $Bun_G$ at least. But let me check to make the notation consistent…
Don’t we generally dislike single slashes as decategorifications?
Yes, and there is double slashing implicit in what Lurie writes, after all he talks about the actual stack $Bung_G$, computing the groupoid cardinality of its value on $Spec(\mathbb{F}_q)$.
I suppose both sides should really be double slashes, though it might be that the right action is faithful (?). I’d have to check. Unfortunately, right now I both urgently need to do something else and am actually ill-situated to do much work either way. I’ll leave further editing here for later.
I added the following reference:
Thanks! I was going to add this when Aaron mentioned it in the MO chat, but I didn’t get around to.
Wow, they are really good! And give an inkling as to the scope of the theorem, more so than trying to read the lecture notes of the course.
I added the following reference, which just appeared on Lurie’s web page:
Thanks, that’s excellent.
(snip)
Finally :-) Spreading the proof over multiple pdf wasn’t great for readability…
Edit: but only “half” the proof (350-odd pages…)
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