There is a general circle of ideas according to which true statements about number fields should have analogues in function fields. As best I can tell, the fact that this seems to work is pretty mysterious. The only results I know directly relating the two come from logic, such as Ax-Kochen, and these are limited to first-order statements in restricted languages. But the analogy apparently goes well beyond such statements. Are there any theorems/conjectures/observations that would explain why the analogy is a good one? I am looking for statements that allow one to go directly from the function field case to the number field case or vice versa.
[Math] Explaining the number field-function field analogy
ag.algebraic-geometrynt.number-theory
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(1) Regarding the relationship between geometric Langlands and function field Langlands: typically research in geometric Langlands takes place in the context of rather restricted ramification (everywhere unramified, or perhaps Iwahori level structure at a finite number of points). There are investigations in some circumstances involving wild ramification (which is roughly the same thing as higher than Iwahori level), but I believe that there is not a definitive program in this direction at this stage.
Also, Lafforgue's result was about constructing Galois reps. attached to automorphic forms. Given this, the other direction (from Galois reps. to automorphic forms), follows immediatly, via converse theorems, the theory of local constants, and Grothendieck's theory of $L$-functions in the function field setting.
On the other hand, much work in the geometric Langlands setting is about going from local systems (the geometric incarnation of an everywhere unramified Galois rep.) to automorphic sheaves (the geometric incarnation of an automorphic Hecke eigenform) --- e.g. the work of Gaitsgory, Mirkovic, and Vilonen in the $GL_n$ setting does this. I don't know how much is known in the geometric setting about going backwards, from automorphic sheaves to local systems.
(2) Regarding the status of function field Langlands in general: it is important, and open, other than in the $GL_n$ case of Lafforgue, and various other special cases. (As in the number field setting, there are many special cases known, but these are far from the general problem of functoriality. Langlands writes in the notes on his collected works that "I do not believe that much has yet been done beyond the group $GL(n)$''.) Langlands has initiated a program called ``Beyond endoscopy'' to approach the general question of functoriality. In the number field case, it seems to rely on unknown (and seemingly out of reach) problems of analytic number theory, but in the function field case there is some chance to approach these questions geometrically instead. This is a subject of ongoing research.
Jon,
I should have been more careful in drawing the slide mentioned above (and reproduced below). I didn't want to convey the idea that there is a method to assign to a knot a specified knot (and certainly not one having exactly p crossings...). I only wanted to depict the result that different primes should correspond to different knots.
Here's the little I know about this :
Artin-Verdier duality in etale cohomology suggests that $Spec(\mathbb{Z})$ is a 3-dimensional manifold, as Barry Mazur pointed out in this paper
The theory of discriminants shows that there are no non-trivial global etale extensions of $Spec(\mathbb{Z})$, whence its (algebraic) fundamental group should be trivial. By Poincare-Perelman this then implies that one should view $Spec(\mathbb{Z})$ as the three-sphere $S^3$. Note that there is no ambiguity in this direction. However, as there are other rings of integers in number fields having trivial fundamental group, the correspondence is not perfect.
Okay, but then primes should correspond to certain submanifolds of $S^3$ and as the algebraic fundamental group of $Spec(\mathbb{F}_p)$ is the profinite completion of $\mathbb{Z}$, the first option that comes to mind are circles
Hence, primes might be viewed as circles embedded in $S^3$, that is, as knots! But which knots? Well, as far as I know, nobody has a procedure to assign a knot to a prime number, let alone one having p crossings. What is known, however, is that different primes must correspond to different knots
because the algebraic fundamental groups of $Spec(\mathbb{Z})- \{ p \}$ differ for distinct primes. This was the statement I wanted to illustrate in the first slide.
But, the story goes a lot further. Knots may be linked and one can detect this by calculating the link-number, which is symmetric in the two knots. In number theory, the Legendre symbol, plays a similar role thanks to quadratic reciprocity
and hence we can view the Legendre symbol as indicating whether the knots corresponding to different primes are linked or not. Whereas it is natural in knot theory to investigate whether collections of 3, 4 or 27 knots are intricately linked (or not), few people would consider the problem whether one collection of 27 primes differs from another set of 27 primes worthy of investigation.
There's one noteworthy exception, the Redei symbol which we can now view as giving information about the link-behavior of the knots associated to three different primes. For example, one can hunt for prime-triples whose knots link as the Borromean rings
(note that the knots corresponding to the three primes are not the unknot but more complicated). Here's where the story gets interesting : in number-theory one would like to discover 'higher reciprocity laws' (for collections of n prime numbers) by imitating higher-link invariants in knot-theory. This should be done by trying to correspond filtrations on the fundamental group of the knot-complement to that of the algebraic fundamental group of $Spec(\mathbb{Z})-\{ p \}$ This project is called arithmetic topology
(my thanks to the KnotPlot site for the knot-pictures used in the slides)
(my apologies for cross-posting. MathOverflow was the intended place to reply to Jon's question, but this topic was temporarily removed yesterday.)
Best Answer
I think your statement could usefully be sharpened in a couple of ways.
Firstly, the state-of-the-art is that true statements for function fields are expected to have analogues for number fields. The outstanding example here is naturally the Riemann hypothesis. Where conjectures for number fields are present, the way of working is via the heuristic that the function field analogue may be sought, and then proved. This has been taken a long way for the Langlands philosophy, for example.
The other major point is that historically what came first was analogies between Riemann surface theory and number fields. Hilbert seems to have conjectured the main outlines of class field theory using complex curves and Jacobians as the source of inspiration. Certainly the prior Dedekind-Weber geometric view fed into that. Subsequently we get "global field" as a kind of middle term: curves over finite fields are a little closer to number fields than complex curves. The attitude of Weil's Basic Number Theory is to develop the parallelism to the point of a common vocabulary (which is now widely used). Global fields can be studied successfully using local compactness, would be a succinct summary. Pontryagin duality, for example, can take much of the strain, and this theory is of course of broader application than number theory.
Of course we don't yet know why this works.