Recall how Deligne associates a filtered sheaf to an irregular connection: along each direction coming into the singularity, you can filter the space of local analytic solutions by growth rate.  (Maybe it would be better to say something about filtering the differential Galois group, in terms of what comes next).  This filtration of a sheaf on a circle can be traded for a constructible sheaf on an annulus — the “analytic halo”.  We like this sheaf better because its singular support is a knot, and we study knots.

OK, now there’s some mysterious analogy between irregular singularities in characteristic zero and wild ramification of $\ell$-adic sheaves in characteristic p.  I basically don’t understand this analogy at all, (but just started reading Nick Katz’s book on Exponential Sums and Differential Equations and may hopefully gain some enlightenment here).  I want to make it less mysterious.  In particular I want to see Stokes rays, an analytic halo, and maybe even a knot at a wild ramification point.  I want this knot to be ”the same knot”.

The first step is having some sort of rigid analytic halo.  I have only the vaguest sense of how this should go, but philosophically the idea is that if I think of $Spec \mathbb{F}_p((t))$ as a circle with some infinitesimal thickness, then I can make a whole family of punctured disks by taking (maybe Artin-Schreier) covers.  As you go deeper in this tunnel, you kill more and more of the wild ramification of the $\ell$-adic sheaf.  This sounds exactly like the concentric circles in the usual analytic halo — as you go in, you forbid more and more of the functions.

Maybe there is a rigid analytic space of some flavor which interpolates between these circles.  A rigid analytic halo?

Maybe it makes sense at any point of this space to specialize the original Galois representation.  So you define a constructible sheaf on this space which is everywhere something like “the tame part”.  The part of this representation which is tame gets bigger and bigger as you go in, so what you have is a constructible sheaf.  Maybe it makes sense to take its singular support.  Maybe if all the yoga is right, the tunnel of perfection is “topologically an annulus”, and the singular support of this sheaf is really a “knot in its conormal bundle”…

(@ A. C., this is my answer to your request to connect number theory and legendrian knots 🙂 )

 

A further fantasy connected to some discussions with Emily Norton re. how Cherednik algebras and finite groups of Lie types have some representation theoretic similarities (see this paper of Rouquier, and the introduction of Emily’s paper), which could e.g. be proven by a discussion like the above but maybe can be attacked separately:

Conjecture: The moduli of $\ell$-adic representations of $Gal(\mathbb{F}_p((t)))$ with a given knot type is the (horocycle transform of) the Deligne-Lusztig variety for the corresponding braid, just like the moduli of ODE with a given knot type is the (horocycle transform of) the open Bott-Samelson variety of this braid.

(probably this conjecture is actually a well known fact if true, except that one has to untranslate the knot into whatever data they usually use to classify wild ramification)