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Started arithmetic topology.
I added a reference to Morishita’s book which makes the cute point that Gauss dealt with both sides of the analogy, i.e., his integral in what might be called abelian gauge field theory for the linking number of two embedded circles, and his work in what’s now abelian class field theory on quadratic reciprocity. Hence the hope for further connections between non-abelian versions of each, non-abelian gauge field theory and Langlands.
Added some detail about the MKR dictionary.
Thanks!
So I suppose one point is that we are to think of a global field as being like a bundle of surfaces over a circle…
From that point of view though, why are primes like knots? Maybe like this: instead of thinking of a prime as being a point, now it is an $S^1$-family of points, hence a section of the bundle of surfaces over the circle, hence a map from the circle to the total space 3-manifold.
Hm, does that make sense?
People most often point to $Spec(\mathbb{F}_p)$ being one-dimensional. Minhyong Kim does that, but has a more subtle account of why this is embedded in a 3-dimensional space in terms of the normal bundle being 2-dimensional.
Yes, but how does that give that primes are knots? I see that it gives that primes are some knots, namely the suitable sections of the bundle over the “circle” $Spec(\mathbb{F}_p)$. But somehow all knots are supposed to correspond to primes?
I must still not quite have the right picture here…
I really cannot believe that “people most often point to $Spec(\mathbb{F}_p)$ being one-dimensional”. At the very least it is a misleading statement: the orthodox view is that it is zero-dimensional, like the spectrum of any other field.
@Urs
I don’t see where you’re getting the idea that all global fields are supposed to be like 3-manifolds fibred over a circle. Morishita [p. 6] says quite clearly that “an algebraic curve $C$ over a finite field $\mathbb{F}_q$ […] is seen homotopically as a surface bundle over a “circle” $Spec(\mathbb{F}_q)$ […] Since a number ring has no constant field, there is no analogy between the structures $\pi_1 (Spec (\mathscr{O}_k))$ and $\pi_1 (C)$”.
(Besides, the Hopf fibration has $S^3$ fibred over $S^2$ with fibre $S^1$, not the other way around!)
Re 7, that was obviously meant in the context of people giving a rationale for a 3-dimensional $Spec(Z)$. E.g., Kim saying
$Spec(F_p)$ has dimension one because it’s a $K(\hat{Z}, 1)$.
Try reading about the Principle of Charity.
So, indeed, Morishita writes it is
as a surface bundle over a “circle”
and this is recently repeated here in a somewhat stronger form:
the geometric analog of a number field or function field in finite characteristic should not be a Riemann surface, but roughly a surface bundle over the circle. This explains the “categorification” (need for a function-sheaf dictionary, which is the weak part of the analogy) that takes place in passing from classical to geometric Langlands
(Curiously, this is in a comment apparently meant to refute my statement that there is “educated guesswork” involved in formulating geometric Langlands – while to me it reads rather like further strengthening this point. )
What I would enjoy is to understand the analogy – or false analogy, as it may turn out – well enough to have it guide one to a formal “inter-geometric” axiomatics that would allow one to turn the handwaving analogies into systematic theorems.
Yes – I can see that curves over $\mathbb{F}_p$ are supposed to be like surface bundles over a circle. This is more or less clear once you accept that $Spec(\mathbb{F}_p)$ is supposed to be like a circle. But my point is that Morishita makes no such claim about number rings! (Recall that there are two kinds of global fields: number fields and function fields.)
As far as I understand it, this “primes are knots” idea can be very quickly motivated like this:
The statement about number fields is the very first line of the MKR dictionary, which I was replying to in #4.
I suppose one issue is that there are two different analogies here. That MKR dictionary and the idea of having something like a 3-manifold fibered over a circle.
I don’t think the two analogies are so separate. A given 3-manifold might be fibred over $S^1$ or it might not.
But here is a simpler objection to thinking about primes as sections of a bundle over $S^1$. Putting things back into the language of schemes, you are asking whether every (closed) point of a variety $X$ over $\mathbb{F}_p$ is necessarily a section of the unique morphism $X \to Spec(\mathbb{F}_p)$, and the answer is no in general: this only recovers the rational points of $X$. In general, a (closed) point of $X$ is a morphism $Spec(\mathbb{F}_{p^n}) \to X$; so back in this point of view where $Spec(\mathbb{F}_q)$ is a circle, what we have is not quite a section of a circle bundle but rather a diagram
$\begin{array}{ccc} S^1 & \to & M \\ & \searrow & \downarrow \\ & & S^1 \end{array}$where the diagonal arrow $S^1 \to S^1$ is the $n$-fold covering map.
I added quite a few things. I don’t know if I got that pairing right in ’Explanations for the analogy’. I corrected the coefficient given in this post. Does it look right?
There seem to be plenty of related ideas out there, including a paper discussing ’arithmetic topological quantum field theory’ which reminds the authors, Gukov, Zagier and others, of arithmetic topology
@Zhen but most 3-manifolds (the hyperbolic ones, which I gather are “most” of them), after Agol’s theorem, are at least virtually fibred over the circle.
re #13: that sounds sensible!
re #14: this is about the new section Explanations for the analogy, right?
re #15: what’s a good link (if not nLab page…)
(in my dictionary, netiquette includes providing pertinent links :-)
Re #16 (re #14), yes. And I redid it according to Knots and Primes: An Introduction to Arithmetic Topology, Masanori Morishita.
He talks there (p. 43) of “modified étale cohomology groups… which take the infinite primes into account”. I haven’t gone into that.
Apparently the pairing is part of Artin-Verdier duality. We have Verdier duality. Any relation?
Thanks for the link. That does sound relevant, doesn’t it. Have given it a minimum entry virtually fibered conjecture and cross-linked a bit.
There are ideas to connect the volumes of hyperbolic manifolds with the discriminant of a number field, as here. Ought to chase up that cusp/infinite place connection too.
So the volumes of hyperbolic manifolds are supposed to be related to the imaginary part of analytically continued Chern-Simons theory (see the links there). This in turn should be given by the “arithmetic Chern-Simons theory” which is the goal of all that talk about arithmetic cohesion etc. elsewhere.
(So, while I have nothing concrete to add for the moment, this just to highlight the bigger story here.)
That’s what’s being described in this MO question, volume being combined with Chern-Simon invariant.
That paper by Gukov, Zagier et al I mentioned elsewhere is all about such such things.
Urs, isn’t it analytically continued Chern-Simons theory we want rather than holomorphic Chern-Simons theory, which works with an odd number of complex dimensions? But they’re related somehow presumably.
For what it’s worth, I added something to volume conjecture. Another immense morass awaits me if I go any further.
Yes, right, sorry, I meant analytically continued CS.
So had to create colored Jones polynomial. Just a stub.
I added links to the paper of Agol et al finalising the proof of the virtual fibering conjecture, and Agol’s ICM talk this year.
added pointer to this, from yesterday:
I added
More broadly, the scope of arithmetic topology is now taken to include the intersection of arithmetic geometry, algebraic topology and low-dimensional topology (see GGW20)
since, as seen in the contents of the article in #29, the meaning of arithmetic topology is broadening.
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