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3. Function field analogues

    1. Counting in the Artin-Schreier case

          In a $\href{https://arxiv.org/abs/2410.23964}{recent \ paper \ of \ Gundlach}$, the author counts the numer of $\mathbb{F}_q(t)$-extensions with abelian Galois group $G$, $q = p^k$, and $p \vert |G|$.

      Add remark

      Problem 3.1.

      [Kiran Kedlaya] There are two natural follow up questions.

      1. The generating series associated to this count has a nice formula, is there a geometric interpretation of this formula?

      2. Are there other groups for which the generating series is similarly nice? As a suggestion, one can try the $\mathbb{F}_p$-points of an algebraic group where one has non-abelian Artin-Schreier.

        • Langlands and field counting

              Recently, a $\href{https://people.mpim-bonn.mpg.de/gaitsgde/GLC/}{series \ of \ five \ papers}$ were released, which claim to proof the geometric Langlands conjecture.

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          Problem 3.2.

          [Brandon Alberts] Malle’s conjecture is known for abelian extensions because of input from class field theory. The Langlands problem is supposed to a non-abelian analogue. What can it tell us about field counting?
            1. Remark. A wide ranging discussion followed this question. It was mentioned that it isn’t clear that using automorphic data makes it easier to count number fields ($\href{https://mathscinet.ams.org/mathscinet/article?mr=2507622}{[MR2507622]}$ was mentioned in a paper as going in the opposite direction). There were questions of which fields are "seen" by conjectures and whether local-global compatibility tells us anything about heuristics.

                • Function field analogues of Smith


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                  Problem 3.3.

                  [Jordan Ellenberg] Are there function field analogue of Smith’s work on Selmer groups? What geometry would underlie such work? Are the relevant moduli spaces nice or do they have nice homology?

                    • 4-ranks of quadratic extensions in the function field case


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                      Problem 3.4.

                      [Melanie Wood] Can we compute moments for the 4-rank of quadratic extensions in the function field case? What simplifies from the number field case? What geometry underlies the computation?

                        • Components of Hurwitz space


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                          Problem 3.5.

                          [Melanie Wood] What does analyzing components of Hurwitz spaces tell us about Alberts’s twisted Malle conjecture?

                            • Small monodromy and genus theory

                                  In the function field setting, you can rephrase Cohen-Lenstra in terms of connected components of certain moduli spaces. When $\ell$ is odd these spaces have two connected components because of large (full symplectic group) monodromy, but when $\ell = 2$ the number of connected components goes to infinity because the monodromy is small (symmetric group $S_{2g + 2}$).

                              Add remark

                              Problem 3.6.

                              [David Zureick-Brown] Are there other “small monodromy” explanations for phenomenon like genus theory (i.e. large lower bounds on class groups coming from ramification)?

                                  Cite this as: AimPL: Nilpotent counting problems in arithmetic statistics, available at http://aimpl.org/nilpotentarithstat.