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The Airy differential equation $$y''(x)\ = \ xy(x)$$ is one of the simplest irregular differential equations (so not determined by its monodromy data, there is more structure, the Stokes data). Associated to this, Katz constructs an Airy sheaf $\text{Ai}$ on the affine line $\mathbf{A}^1_{\mathbf{F}_q}$; $\text{Ai}$ is the Fourier transform of some nonconstant lisse sheaf. However, I don't know anything about Katz's theory, and this question is an attempt to learn more about it, in an example.

Question: What is the evidence that this is the correct sheaf analogue of the Airy differential equation? e.g.

  1. (How) does this $\ell$-adic sheaf see the Stokes data of the differential equation?
  2. You can write the Airy DE as a limit of DE's which have regular singular points. Does the Katz construction work in families i.e. apply it to the family of DE's $P_ty(x)=0$ where setting $t=0$ gives back Airy?
  3. Other properties I might not have thought of.
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    $\begingroup$ For 2, what limiting family are you thinking of? $\endgroup$
    – Will Sawin
    Commented Nov 4, 2021 at 15:32
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    $\begingroup$ @WillSawin Using the coordinate $u=1/x$ about the irregular point $x=\infty$, we have $$y''(x)\ +\ xy(x)\ =\ (u^4y''(u)+2u^3y'(u))\ +\ (y(u)/u)$$ which is irregular at $u=0$ because when you divide through by $u^4$, the $y$ term has a pole of order $>2$. However, we have $$(1+\varepsilon x^n) y''(x)\ +\ xy(x)\ =\ (u^n+\varepsilon)(u^4y''(u)+2u^3y'(u))/u^n\ +\ (y(u)/u)$$ so if we normalise we now get $$y''(u)\ +\ \frac{2u^{n-1}}{u^n+\varepsilon}y'(u)\ +\ \frac{u^{n-5}}{u^n+\varepsilon}y(u)\ =\ 0.$$ $\endgroup$
    – Pulcinella
    Commented Nov 4, 2021 at 18:10
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    $\begingroup$ The coefficients all have simple poles for $\varepsilon\ne 0$, so $$(1+\varepsilon x^n)y''(x)\ =\ xy(x)$$ is a family of regular ODEs degenerating to the Airy equation, whenever $n>4$. $\endgroup$
    – Pulcinella
    Commented Nov 4, 2021 at 18:12

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(1.) The part of the Stokes data we can see is the restriction of the differential equation to the field of formal Laurent series at each point. There is a classification of differential equations over this field, and a classification of Galois representations of a field of formal Laurent series over a finite field, and the two classifications are pretty similar.

When we compare these classifications, the Airy sheaf and Airy differential equation match up at each point. The most important aspect of this is the slopes - the Airy differential equation has slopes $3/2$ at $\infty$, and so does the Airy sheaf, for the suitable definition of slope in each context. Furthermore, both objects have no singularities away from infinity.

(2.) Doesn't this work, by a similar argument, for every differential equation with regular singularities?

Deformations of sheaves are a bit more restricted then deformations of differential equations. In fact, we have two different notions of deformations of sheaves. In one, the deformation space has coordinates in the coefficient field of the sheaf ($\ell$-adic, in this case), while in the other, the deformation space has coordinates in the base field of the variety the sheaf lives on (a field of characteristic $p$, in this case). Only by combining the two can we obtain as much power as deformations of a differential equation, where both relevant fields are $\mathbb C$ and there's a single complex variety parameterizing both types of deformations.

(3.) Here are some additional properties:

a. The Airy differential equation is preserved by the change of variables $x \mapsto \omega x$ where $\omega^3=1$. The Airy sheaf is preserved under the same change of variables.

b. The Fourier transform of the Airy differential equation (i.e. the differential equation satisfied by the Fourier transform of the Airy function) is a differential equation of rank $1$, with no singularities except $\infty$, with slope $3$ at $\infty$. The Fourier-Deligne transform of the Ary sheaf is a sheaf of rank $1$, with no singularities except $\infty$, with slope $3$ at $\infty$.

In fact, properties a and b characterize the Airy differential equation up to scaling $x$ (and changes of variables in $y$), and they also characterize the Airy sheaf up to scaling $x$.

c. The image of the action of the differential Galois group on solutions of the Airy differential equation is $SL_2$, and the monodromy group of the Airy sheaf (i.e. the Zariski closure of the image of the Galois group on sections of the Airy sheaf) is also $SL_2$.

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  • $\begingroup$ Thank you for this. I just have a couple of questions: a. do you have a reference for the definitions of slope of an ODE and of a sheaf, and b. do you have any reference for this notion of deformation of a sheaf which uses both the ring of coeffients and the base? $\endgroup$
    – Pulcinella
    Commented Nov 12, 2021 at 21:46

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