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started function field analogy – table, but didn”t get very far before being interrupted now
was aiming for the table in section 2.6 of
I think $k(t)$ stands not for Laurent polynomials, but for the field of rational functions – right?
And presumably $X$ should be $\mathbb{P}^1_k$.
How about another column for the full Weil-ian Rosetta stone? His ’Riemannian’ language.
@Todd, Ah, thanks, sorry. I have changed it to ring of rational functions.
@Zhen Lin: Regarding the $X$: the previous line was supposed to introduce it, I have added it now:
the function field of an algebraic curve $X$ over $k$ is supposed to be to $X$ as a number field is to its spectrum plus archimedean places.
But as I said, I am just trying to implement the table in section 2.6 of
@David:
I don’t really know much of this in detail yet, but I am longing to learn it. I think I understand the rough idea, but I would need to spend some time to see how to write out the entries in the three-columned Rosetta stone.
Anyway, I have started a third column at function field analogy – table.
I would enjoy a lot if we could jointly expand that table further. But I should say that today and for the remainder of the week, I really must be concentrating on something else.
I have added also mention of F1 by expanding the top left entry to
number fields (function fields of curves over F1)
Is that acceptable?
By the way: why just curves?
Shouldn’t there be such a dictionary which in the right column has complex anallytic spaces of dimension $d$, in the middle has $d$+dimensional algebraic spaces over $\mathbb{F}_p$ and on the left has… whatever it is that goes there, whatever it is, I suppose it will be called $d$-dimensional spaces over $\mathbb{F}_1$.
And I suppose there should be on the left “Riemann zeta function” and on the right “zeta function of a Riemann surface” and in the middle.. something. Is that right? I have tentatively added it.
By the way: why just curves?
Might it have something to do with the completions at places being locally compact fields? This is the arena where Pontryagin duality, harmonic analysis, adeles and ideles, and their manifold connections with the analysis of zeta functions come together and the function field analogy seems to be the most successful, but in the nonarchimedean case the only locally compact fields are local completions of function fields of curves over a finite field.
Thanks, Todd. Probably what I am trying to wonder here is this: how far have people had succes with systemartically realizing number fields as “function fields over $\mathbb{F}_1$”? And how much have they progressed with making sense of “absolute algebraic geometry” in this fashion for higher dimensional spaces?
I gather that there are about a dozen proposals for what geometry over $\mathbb{F}_1$ actually is. Which of these has been most successful with making the function field analogy more than an analogy?
Is that acceptable?
I don’t have an answer myself, but it would be very nice if the nLab could make this analogy more explicit still. My idea of $\mathbb{F}_1$ is still pretty rudimentary, even relative to anyone else’s idea being rudimentary.
Briefly from my phone: there is an old MO question by David C about the analogy and Weil’s Rosetta stone, and one reply he got is along the line that it’s all the F1 story. Maybe one could further chase poiters from there.
Then there’s all that business to consider about $Spec(\mathbb{Z})$ being three-dimensional, and quadratic reciprocity relating linking numbers of loops. We discussed something about that at the Cafe, here and here.
Did I see, Urs, you’re speaking at Symmetries and correspondences in number theory, geometry, algebra and quantum computing: intra-disciplinary trends? Not in the list there, but you mention it at your web.
Yes, I was invited to that event. I still am, but a few minutes back I get a message that they are doing some re-organization and, if I understand correctly, then some people including me will stiil speak on the July 8-9 – but the title may now be different.
Regarding quadratic reciprocity:
so we already have as a fact (from the construction and nature of the string orientation of tmf and the story of equivariant elliptic cohomology) that at a deep level the modular functor of Chern-Simons theory is built by passing from Riemann surfaces/algebraic curves over the complex numbers to arithmetic geometry over $\mathbb{Z}$. That business about quadratic reciprocity (e.g. this MO discussion) seems to say that in fact Chern-Simons theoretic structure may be recognized already inside the terminal point of artithmetic geometry.
Would be great to understand this in some actual detail. But at least these two stories seem to be nicely confluent.
By the way: why just curves?
possibly some sort of homotopical reason as well, as Riemann surfaces are $K(\pi,1)$’s. I have a wishful idea that complex surfaces that are 2-types should be a good source of examples to play with, but I have no hard data.
@Urs
Why curves? The simple algebraic reason is that the isomorphism class (!) of a smooth projective algebraic curve (over an algebraically closed field, at least) is completely determined by its function field. This fails for $dim \gt 1$: $\mathbb{P}^1 \times \mathbb{P}^1$ has the same function field as $\mathbb{P}^2$, for example.
Let me sketch how this is done. Let $k$ be an algebraically closed field and let $K$ be a finitely generated field over $k$, of transcendence degree $1$ (over $k$). Then $K$ is a finite extension of $k (t)$. A point of $K$ is a $k$-subalgebra of $K$ that is a discrete valuation ring. For cofinite subsets $U$ of the set of points $X$, we define $\mathcal{O}_X (U)$ to be the intersection of the subalgebras corresponding to $U$. It can be shown that $\mathcal{O}_X$ is a sheaf with respect to the cofinite topology, and that $(X, \mathcal{O}_X)$ is isomorphic to a smooth projective algebraic curve over $k$ (minus the generic point, if working in the language of schemes).
Conversely, if we start with a (not necessarily smooth projective) algebraic curve $(X, \mathcal{O}_X)$ over $k$ and take its function field $K$, it is not hard to show that each point of $X$ defines a discrete valuation $k$-subalgebra of $K$, and when $X$ is smooth projective, every discrete valuation $k$-subalgebra of $K$ arises in this way.
If one tries to apply this story to a number field $K$, one discovers that there are not as many discrete valuations on $K$ as one would like, and it turns out that the missing points correspond to the archimedean places.
Re Urs #15, I hope you get the chance to talk with Minhyong Kim.
Zhen Lin, thanks, that makes sense.
But what I mean is: given the dictionary relating curves over the complex numbers, over finite fields and over, presumably, F1, then it seems clear that there must be analogous dictionaries doing the same kind of base change translation for higher dimensional spaces. Any hints of that?
I have no idea. There are some simple-minded things one can say: for instance, for $dim = 2$, we could look at $\mathbb{Z} [x]$ as being the arithmetic analogue of $k [t, x]$, and one could regard these geometrically as being line bundles over curves. But $\operatorname{Spec} \mathbb{Z}$ is missing its archimedean places, which is (apparently) where Arakelov theory comes in.
Is this why we might need a Galois Theory in Two Variables?
Re #19: I sure am looking forward to speaking with him, yes. BTW, they have updated the conference webpage. It seems the title was shortened to “Symmetries and Correspondences” And the typo in “instra-disciplinary” is apparently there to stay.
I have expanded a bit more at function field analogy – table, trying to bring more of the Langlands story in.
This needs more attention, clearly.
Here is an observation ragarding the question (which keeps haunting me, but which you may want to please ignore if it is getting boring) how to correctly think of p-adic geometry in the context of cohesion.
So the function field analogy says that
is analogous to
But now observe the following: while synthetic differential complex analytic geometry is cohesive, its slice over a Riemann surface $\Sigma$ is never: the non-trivial homotopy type of $\Sigma$ makes the would-be shape modality of the slice fail to preserve the terminal object, and the many points in $\Sigma$ prevent the slice from being local.
As these problems already suggest, one sufficient condition (and probably close to necessary, I’d guess) for an object to make slicing over it preserve cohesion is that it is an atomic object (this is shown in this proposition).
But the formal neighbourhoods $\widehat{\Sigma_x} \hookrightarrow \Sigma$ are just that: the atomic subobjects of $\Sigma$.
So the analog of what in arithmetic geometry is base change from the absolute context to each p-adic context, in the complex-analytic context is base change to all the cohesive contexts obtainable from a Riemann surface.
Now if we run that observation backwards through the function field analogy, then it suggests that it is plausible that while absolute analytic geometry over $Spec_{an}(\mathbb{Z}^{an})$ won’t be cohesive, the analytic geometries over each $\mathbb{Z}_p$ ought to be cohesive…
Is it relevant that $Spec(\mathbb{Z}_p)$ in $Spec(\mathbb{Z})$ has been likened to a knot within a 3-manifold? See resources for this and variants, such as Deninger’s, at MO.
The analogy was discussed at the Cafe here, a message by James being particularly interesting. Minhyong Kim replies to that here.
Also James’s follow-up:
First consider a manifestly geometric example. A point has dimension 0, a punctured disk has dimension 1, and a Riemann surface has dimension two. Going up in dimension comes first from local winding and then globalization.
Similarly, a finite field has dimension one. A local field, such as $Q_p$ or Laurent series $F_q((x))$ has one higher dimension, the extra dimension coming from local winding, or ”ramification” as those in the business call it. A ring of integers in a global field, for example $Z$ or $F_q[x]$, has one more dimension coming from globalization.
So if you allow yourself to think of $Z$ as being the ring of functions on something like a Riemann surface, then it’s all the usual stuff except that now your points are 1-dimensional, so everything else gets bumped up by one.
Thanks for the pointers.
It seems to me that the function field analogy table supports the idea that SpecZ is 2-dimensional. On the far right of the function analogy table it is impossible to have an odd real dimension, since there we are in complex analytic geometry.
At least for standard concepts of dimension in the complex case. The argument for dimSpecZ=3 derives in part from the fact that SpecFp has inner structure, which makes it seem more than a point.
How does the notion of ’atomic object’ work in the arithmetic case then? If atoms are contractible, how does that fit with the 1-dimensionality of $\mathbb{Z}_p$?
We have the $\infty$-topos over the étale site over $Spec(\mathbb{F}_p)$. In there $Spec(\mathbb{F}_p)$ is atomic.
The topos is not $\infty$-contractible, the $\Pi$-left adjoint does not exist on the nose, but its pro-left adjoint does exist. This sends objects to their étale homotopy type which for $Spec(\mathbb{F}_p)$ is apparently a non-trivial 1-type.
So I suppose another way to reply to the actual question which I suppose you raise is: if one were to succeed to exhibit cohesion on any kind of analytic geoemtry over $\mathbb{Z}_p$ or the like, then part of what this means is that the terminal point is contractible and does not exhibit any “spurious dimensions”.
That said, we should talk about a general issue: is the function field analogy indeed more a statement in global analytic geometry than in algebraic geometry? Because at a crucial point when considering the spectrum of the number field, we need to consider it with its archimedean places included. But this means to consider it more in analytic geometry than in algebraic geometry, no?
I gave function field analogy itself an Idea-section.
This certainly needs to be expanded. But now my battery is dying any second.
The étale (1-)topos of a field $k$ is equivalent to the topos of $Gal (k)$-sets, where $Gal (k)$ is regarded as a topological group. So if $k$ is not algebraically closed, then its étale topos has a non-trivial homotopy type (namely, $B Gal (k)$). But it seems strange to say that $B Gal (k)$ is 1-dimensional…
Ah, right, thanks.
But it seems strange to say that $B Gal(k)$ is 1-dimensional…
Probably the idea is that if anything has non-trivial fundamental group, then it’s “at least not 0-dimensional”. Minhyong Kim’s argument in that letter (pdf) is all based on this idea that $Spec(\mathbb{F}_p)$ “is 1-dimensional”, except towards the end, where he talks about cohomological dimension.
I’d think bringing cohomological dimension and homotopy dimension and covering dimension and Heyting dimension and whatever (maybe KO-dimension :-) into the discussion is a good idea. They should all agree for complex curves and will in general disagree in more exotic geometries.
added one more row to the bottom of the function field analogy – table, pointing to the number-theoretic and the function-field version of the Weil conjecture on Tamagawa numbers.
I suppose there could be a complex-analytic version of this, too, discussing some kind of “complex analytic stacky Tamagawa measure” of $Bun_G$. Is there?
alse added more links into the row on zeta functions. I don’t understand this well yet, and what I put there remains tentative. Have to quit now.
I am beginning to think that for the analogy to work well we are to think of $Spec(Z) \cup place_\infty$ as analogous to the Riemann sphere.
Because this way on the far right of the analogy we’d have the branched covers of the Riemann sphere, which give all compact Riemann surfaces.
Moreover, the arithmetic genus of $\mathbb{Q}$ is zero.
And the place at infinity of $\mathbb{Q}$ is probably the “north pole” as for the Riemann sphere.
I am beginning to think that for the analogy to work well we are to think of $Spec(Z) \cup place_\infty$ as analogous to the Riemann sphere.
I think this idea – that by taking all valuations $Spec(\mathbb{Z}) \cup place_\infty$ we get something like a projective completion, thus extending the function field analogy – must be somewhat prevalent. There is a nice comment by Matthew Emerton which I’ll link to here, since every now and then I want to come back to it myself.
Thanks, Todd!
I may well be unaware of the prevalent knowledge on the analogy. If you have any (further) reading suggestions, I’d much appreciate it.
(I started looking a bit more at $F_1$-literature, hoping to find more of a systematic picture there. But if it is there, then at least I haven’t found it yet.)
This brings back memories of a previous discussion where I tried to explain how $\operatorname{Spec} \mathbb{Z}$ is like a curve without much success.
Wait, if I remember well the disagreement there was that in the context of smooth and supergeometry I claimed it wrong to say that there is a hidden spurious dimension to everything because allegedly the integers in the coefficients are functions on a hidden curve.
And I still think that’s wrong. The function field analogy does not say for instance that complex curves have a further hidden dimension, just that the dimensions which they already have also exist in a more, let’s say, skeletal form.
But be that as it may, your expertise is valued.
I certainly don’t believe that $\operatorname{Spec} \mathbb{C}$ is 1-dimensional (or that $\operatorname{Spec} \mathbb{C} [x]$ is 2-dimensional etc.), and I never said so. (At any rate the morphisms go the wrong way; $\operatorname{Spec} \mathbb{C}$ is over $\operatorname{Spec} \mathbb{Z}$ and not the other way around.) But very specifically I do assert that $\operatorname{Spec} \mathbb{Z}$ is 1-dimensional (and that $\operatorname{Spec} \mathbb{Z} [x]$ is 2-dimensional etc.). There are famous pictures in Mumford’s Red Book (Ch. II, §1, Examples C and H) illustrating this.
Sure, that’s what this thread here is about.
We may have been talking past each other, and in any case it may not be worthwhile rehashing it, but what I was and am objecting to (which maybe you never claimed, we might have been talking past each other) is that it is a good idea, as has maybe been claimed by some bigshot somewhere, to look at analytic functions on a supermanifold and say:
“look, there are three types of dimensions there: the ordinary dimensions, the superdimensions, and a hidden dimension on which $\mathbb{Z}$ is the functions”.
This is not correct, and it contradicts the function field analogy.
I started looking a bit more at $F_1$-literature…
Did you see Manin’s latest?
Local zeta factors and geometries under Spec Z
The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez–Villegas transform ([Ro–V]) a polynomial satisfying the functional equation of zeta type and having nontrivial zeros only on the middle line of its critical strip. The second part discusses Chebyshev lambda–structure of the polynomial ring as Borger’s descent data to $\mathbf{F}_1$ and suggests its role in possible relation of $\Gamma_{\mathbf{R}}$–factor to “real geometry over $\mathbf{F}_1$” (cf. also [CoCons2]).
Thanks, David, I’ll look at it now.
Meanwhile, I should say that I have expanded the function field analogy – table a bit more.
I have added the bit about the projective line and the implication that $\mathbb{F}_1$-curves over $Spec(\mathbb{Z})$ are hence analogous to branched covers of the Riemann sphere.
I am thinking of starting a qestion on Mathoverflow asking for feedback on the table. My impression is that this is a classical case of a lot of latent knowledge present in the community, which hasn’t been fully materialized yet.
Worth adding that branch points of a curve correspond to ramified primes in a field extension?
Ah, I see. Could you briefly point me to somehere where this is made explicit?
Oh, never mind, I see it explained already in Wikipedia’s entry on ramification.
Need to add a pointer to the textbook by Michael Rosen, “Number theory in function fields”…
Lots of people seem to like Poonan’s table mentioned here. It’s over three pages (pp. 32-34) of the notes Lectures on rational points on curves.
Yes, thanks, that’s the table that I started with when producing the one on the nLab now.
I (or somebody) should add more of the items that Poonan has but which are not yet on the nLab, At least maybe class field group/Picard group, … though that’s sort of implicit in the entry on Langlands which we have, but Poonan does not.
as mentioned in #44, I sent this to MO http://mathoverflow.net/q/176798/381 for further discussion
I put in a row under class field theory dealing with the Hilbert reciprocity law (doesn’t exist yet), dealing with number fields, Artin reciprocity law (subsumes Hilbert, but definitely covers the function field of a curve over a finite field), and Weil reciprocity law. The last has a nice interpretation in terms of 2-gerbes on $\mathbf{B}\mathbb{C}^\time$, due to Brylinski and McLaughlin (section 3 of Geometry of degree 4 characteristic classes II). Zoran had already started Weil reciprocity law, and I intend to put a bit more in, but have to do something else just now.
Thanks!
I have added some content to Artin reciprocity law. Also started geometric class field theory, but nothing much there yet.
Have restructured the function field analogy – table by splitting the “section” on Galois theory and class field theory into two, and renaming the following section that used to be “automorphy and bundles” into “non-abelian class field theory”.
I have added
to function field analogy – table two rows with pointers to Fermat quotient and to Lambda-ring;
to function field analogy a new section Formalizations with a few paragraphs briefly indicating how Borger’s absolute geometry formalizes the idea that $Spec(\mathbb{Z})$ is like a ring of polynomials (see also a comment in another thread) and how it formalizes the analogy between geometry over number fields and over function fields.
Note to myself for later: need to understand and then add the story involving the analogy between certain products over all places yielding 1, and products of monodromy/contour integration around all suitable points being trivial.
(If anyone recognized from these keywords which story I mean, I’d be grateful for pointers to decent accounts. There is an MO-discussion which is ever so briefly alluding to this: “If Spec(Z) is like a Riemann surface, what is the analog of integration along a contour?”
This quote seems apposite:
One begins with a number field and the arithmetic problem of comparing the anisotropic part of the Arthur–Selberg trace formula for different groups. This leads to the combinatorial question of the Fundamental Lemma about integrals over $p$-adic groups. Now not only do the integrals make sense for any local field, but it turns out that they are independent of the specific local field, and in particular its characteristic. Thus one can work in the geometric setting of Laurent series with integrals over loop groups (or synonymously, affine Kac–Moody groups). Finally, one returns to a global setting and performs analogous integrals along the Hitchin fibration for groups over the function field of a projective curve. In fact, one can interpret the ultimate results as precise geometric analogues of the stabilization of the original trace formula.
(page 4 of http://www.ams.org/journals/bull/2012-49-01/S0273-0979-2011-01342-8/S0273-0979-2011-01342-8.pdf)
The function field analogy turns up on page 10, for instance:
As we will recount below, one of the most intriguing aspects of the (currently known) proof of the Fundamental Lemma is its essential use of function fields and the extensive analogy between them and number fields.
I am starting to add theta function-entries to the function field analogy – table, to go along with the zeta functions that they give rise to under integration transform. So far I have added an entry for Jacobi theta function corresponding to Riemann zeta function in the case over $\mathbb{Q}$; and Hecke theta function, corresponding to Dedekind zeta function in the case of general number fields.
So how to make sense of David Ben-Zvi’s comment:
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?
Oh, the continuation
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 — if you study the corresponding QFT on such three-manifolds, you get structures much closer to those of the classical Langlands correspondence.
couldn’t be to do with $Spec(\mathbb{Z})$ being three-dimensional, could it? Or as a 3-manifold with a flow on it, as some would have it?
Cor! About that Scholze he mentions, there’s a citation:
Already in his master’s thesis at the University of Bonn, Scholze gave a new proof of the Local Langlands Conjecture for general linear groups.
re#58, yeah, I suppose that’s the idea. (We are not doing educated guesses here, are we ;-))
re#59 yup, Scholze is being hailed all around. Somehow his writing-up though seems to be lagging behing, or else I am looking in the wrong places. For instance where would that discussion of moduli spaces of shtukas being Rapoport-Zink spaces be laid out in text?
That raises at least a couple of pages we need: Rapoport-Zink space, shtuka.
Hmm, sometimes I dream that diving into these advanced concepts will miraculously help understanding. Anyway I put up the stubbiest of stubs for both in #61.
Well, this is deeply in frog territory ;-)
Check out again Hartl 06 and search the text for the (many) occurences of “shtuka” and (some) of Rapoport-Zink space. That may still not make you slap your forehead, but at least it’s getting more in the direction of (speaking of which elsewhere) “conceptual mathematics”.
Ah, so that reference I was complaining above for not being able to identify must be Scholze-Weinstein 12
the sphere spectrum is the higher homotopy version of the commutative ring of integers. Is there a higher homotopy version of the finite fields, and the polynomial ring on the finite fields generated by a singleton?
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