6. Hurwitz Spaces
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Epsilon factors
Problem 6.1.
[Mark Shusterman] Fix a finite group $G$ and an irreducible, nontrivial complex representation $\rho$. study \begin{align*} \sum_k \epsilon(1/2, \rho_k), \end{align*} where the sum is over $G$-extensions $k$ over $\mathbb F_q(t)$ and $\rho_k: \operatorname{Gal}(\overline{\mathbb F}_q(t)/\mathbb F_q) \to G \xrightarrow{\rho} \operatorname{GL}_r(\mathbb C)$.
in the case $G = \mathbb Z/\ell \mathbb Z$, equidistribution of $\epsilon(1/2, \rho_k)$ is known by work of Florea and others.
What is the topological interpretation of this? Relevant to this is to understand the cohomology of certain $1$-dimensional representations of $\pi_1(\operatorname{Hur}^G_n)$.
For cubic characters, there was a recent annals paper by Dunn and Radziwell establishing a secondary term. So potentially one could try to investigate secondary stability in this case. -
Representation stability
Problem 6.2.
[Aaron Landesman] Consider the ordered Hurwitz space where one marks branch points. is there representation stability? There is some work in progress by Ellenberg-Shusterman and Himes-Miller-Wilson-Hoang in the non-splitting case.
Can you do the general case? Can you find the stable value? -
Rack Hurwitz spaces
Problem 6.3.
[Aaron Landesman] Is there a common generalization of Malle’s conjecture and the Davis-Schlank conjecture that gives asymptotics for $c$ extensions for a rack $c$? The relevant homological stability result is per component is known due to a result of Landesman-Levy over $\overline{\mathbb F}_q$. Namely, there is a hurwitz space $\operatorname{Hur}^c_n$ parameterizing rack covers “branched over a degree $n$ divisor,” see Definition 2.2.2 of the paper by Landesman-Levy on ”Homological stability for Hurwitz spaces and applications.“ part of this question is to explain what the relevant variety over $\mathbb F_q$. The challenge is to descend this to the same $\mathbb F_q$ for all $n$. Mark Shusterman has some first thoughts on how one might define this. One of the interesting questions would be to understand the relevant number of components. What are arithmetic consequences of objects not coming from Hurwitz spaces. Does the Hurwitz space for racks have a moduli interpretation, which one can use to define it over a general base, perhaps with the order of the structure group inverted? -
Cohomology coming from quantum shuffle algebras
Problem 6.4.
[Craig Westerland] Note that racks give braided vector spaces, which give quantum shuffle algebras whose cohomology agrees with that of the Hurwitz space associated to $c$, $\operatorname{Hur}^c_n$. If we consider braided vector spaces whose quantum shuffle algebra has well-understood cohomology (for example, involving nichols algebras) do we get arithmetic applications?
One prerequisite to this question would be whether we can descend the relevant local systems to $\mathbb F_q$.
Basically, someone has computed some relevant results in cohomology, and this question is asking whether there are relevant arithmetic statistical questions. There are many applications of Nichols algebras of diagonal type by people including Androskiewitsch, Angiono, Witherspoon, etc.
will sawin points out the following: certain nichols algebras may have cohomology which are relatively simple to understand over function fields, and this may give a hint for what interesting special cases of relevant arithmetic questions over $\mathbb q$ could be; they could give a hint on where it might be possible to make progress.
Cite this as: AimPL: Moments in families of L-functions over function fields, available at http://aimpl.org/momentsoverff.