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Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer to it. The price for explicitness is: dealing with highly singular curves/manifolds. The context is geometric Langlands over the complex numbers over highly singular (but very simple) curve $X$. The hope is: to relate combinatorics + differential operators to geometric Langlands stuff.

Concepts briefly:

  1. Differential operators $\partial_z^n + \sum_{i=n-1...0} f_i(z)\partial_z^i$ by the standard chain of analogies are thought to "define representations of $Gal(X)$"/"local systems on X". Analytical properties of $f_i(z)$ related to what curve $X$ we consider. The term used - "G-opers".
  2. The arithmetic paradigm Every motivic L-function should coincide with an automorphic L-function is rephrased in geometric setup as follows: consider a manifold/motive fibered over $X: M \to X$. For each such manifold there should exist a local system on $X$ i.e. the representation of the $Gal(X)$ i.e. the linear differential operator $\partial_z^n + \sum_{i=n-1...0} f_i(z)\partial_z^i$ over $X$. Will Sawin's answer gives positive light on existence of such correspondence in general (at least as far as I understand).

So the main correspondence:

$M \to X$ corresponds to differential operators on $X$

One may hope to get examples of the correspondence explicitly in the following setup. Now we want to specify $M$ and $X$ to some very singular manifolds, because we know that for such $X$ GL-opers have simple form: $f_i(z)$ are certain rational functions with only poles at 0.

Setup for explicitness:

  1. Consider $X$ to be cusp like curve: $Spec(C[z]~without~z,z^2,z^3,...z^n)$. Note the standard cusp curve $u^2=v^3$ is of that type: $C[z],without~z$ - just take $u=z^3, v=z^2$. Remark: such generalized cusp curves are standard to consider to deal with ramified class field theory of (Serre "Algebraic Groups and Class Fields" (after Lang)). (Class field theory = GL(1)-Langlands correspondence).
  2. Consider $M$ to be $Spec$ of some subalgebra in polynomials in many variables $C[x_i]$ such that there is a map $C[z]~without~some~z$ to that subalgebra. For example take $C[x_1,x_2],~without~x_1~and~x_2$.

So in short: there is $M \to X$ and both $M$ and $X$ are presented by explicit subalgebras of polynoms.

Point: GL-opers on such $X$ are those where $f_i(z)$ are rational functions with only poles at 0, the order of poles is controlled by $n$ in rather simple way. (See refs below).

Question 1: Can we say something explicit about the differential operators which corresponds to $M \to X$ ? Ideally: present a procedure and explicit outcomes to compute differntial operator starting from any given $M$ of that type.

Note: these operators have irregular singularities, so monodromy is not enough to capture all the information about them, we may expect some restrictions on Stokes matrices also.

Adding arithmetic flavour. Assume $M$ is defined over $Q$ or any other field of algebraic numbers.

Question 2: What restriction on differential operator it would give ?

Field with one element. The curves $X$ = $Spec(C[z]~without~z,z^2,z^3,...z^n)$ are defined over any ring - and in particular over $F_{un}$ - since we do not need to operations $1+1$ which are forbidden in $F_{un}$. Similar - consider subalgebras in polynomials which are defined just by dropping out some monomials i.e. $C[x_i],~without~some~set~of~monomials~x_1^{i_1}x_2^{i_2}...x_n^{i_n}$.

Since symmetric group $S_m$ is thought to be $GL_m(F_{un})$ we might expect something like that:

Question 3: Does the monodromy/Stokes matrices are permutation matrices for differential operators arising from such $M$ ?

PS

Considerations of such singular curves, their $Bun(GL)$ , Hitchin systems etc can be found in our paper: https://arxiv.org/abs/hep-th/0309059. The corresponding GL-opers are given by Talalaev's formula $det(\partial_z - L(z))$ (appeared in https://arxiv.org/abs/hep-th/0404153), see more details and Langlands context here: https://arxiv.org/abs/0711.2236 section 8.2: "The geometric Langlands correspondence over C".

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  • $\begingroup$ It seems : for M->X we should just take direct image of "constant sheaf/D-module/motive" and we will get the desired D-module on "X" - "G-oper". However, that direct image operation might not land us exactly where we want, and it might non-trivial to find equivalent representative in the form of $\sum f_i(z) \partial_z^i$ with control of singularities described in the post. But if we believe in Langlands - it should be like that - since these are the GL-opers and there are no others (up to equvalences). Any way explicit description might not be trivial in such singular case. $\endgroup$
    – Alexander Chervov
    Commented Mar 23 at 14:58
  • $\begingroup$ The other concern we might always get a trivial GL-oper. That means motives of such $Spec(C[x_i] ~without~some~sets~of~monoms)$ might be trivial. But - look on one variable case $C[z]~without~z$, despite its normalization is just $C[z]$ in many respects it behaves like the full elliptic curve $u^2=v^3+av^2+bv+c$, it has similar non-trivial differential and its cohomologies/Jacobian is $C^*$ - quite non-trivial. More generally - things which are deformation stable - should be quite non-trivial for such singular manifolds - because when you deform them - you get very non-trivial ones. $\endgroup$
    – Alexander Chervov
    Commented Mar 23 at 15:06
  • $\begingroup$ Here is separate question to clarify - is there any motive-like theory which can give non-trivial representative to singular manifolds discussed below: mathoverflow.net/q/467581/10446 $\endgroup$
    – Alexander Chervov
    Commented Mar 23 at 17:44
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    $\begingroup$ It seems to me that motivic local systems ( in the global unramified setting) will rarely be opers, so taking opers as the stand in for Galois reps or Langlands parameters seems potentially misleading. In general if the Hodge filtration agrees when the oper filtration that’s an extremely restrictive and unusual type of motive.. $\endgroup$
    – David Ben-Zvi
    Commented Mar 23 at 23:58
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    $\begingroup$ Also I disagree with the interpretation of “motivic L functions are automorphic “ as having to do with relating your M->X to a local system (which is maybe sometimes an oper) - this latter relation is something automatic and not involving the automorphic side of Langlands at all $\endgroup$
    – David Ben-Zvi
    Commented Mar 24 at 0:00

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