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2. Number field statistics

    1. Lowers bounds for solvable groups


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      Problem 2.05.

      [Robert Lemke Oliver] In what level of generality (with respect to both ordering invariant and groups covered) can we give lower bounds for the count of $G$-extensions, $G$ solvable? Can we get lower bounds of the form with the right power of $X$ while giving up, at most, some powers of $\log(X)$? For example, for the discriminant ordering can we get a lower bound of the form $\frac{X^{a(G)}}{(\log(X))^A}$?
          The idea here is to separate out from the concentrated case. If $G$ is not concentrated in some $N \lhd G$ then in some sense, if a positive proportion of $G/N$ extensions lie inside $G$-extensions this should give rise to the “right number" of $G$-extensions.

        • Minimal embedding extensions


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          Problem 2.1.

          [Arul Shankar] Let $K$ be a number field, $G = \mathrm{Gal}(K/\mathbb{Q})$, and $\tilde{G}$ a finite solvable group that is an extension of $G$ by $N$. Can we bound the minimal discriminant (as a function of $\mathrm{Disc}(K)$) of an extension $L/K$ that solves the embedding problem $1 \rightarrow N \rightarrow \tilde{G} \rightarrow G \rightarrow 1$? Can we bound other inertial invariants of $L$?
            1. Remark. [Jürgen Klüners] We restrict to the case of solvable groups to avoid issues with the inverse Galois problem.

                • Gundlach’s conjecture

                      Fabian Gundlach has proposed a $\href{https://arxiv.org/abs/2211.16698}{variant \ of \ Malle's \ conjecture}$ in which multiple intertial ordering invariants are allowed to vary independently.

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                  Problem 2.15.

                  [Peter Koymans] In what generality can we prove Gundlach’s conjecture?

                    • Non-inertial height functions


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                      Problem 2.2.

                      [Robert Lemke Oliver] Can we count number fields ordered by non-inertial height functions (i.e., those which are not constructed from Gundlach’s invariants)? For example, can we count number fields ordered by the smallest height of a generating point on $\mathbb{P}^1$?
                        1. Remark. [Melanie Wood] There is some work (see, for example, $\href{https://mathscinet.ams.org/mathscinet/article?mr=3782066}{[MR3782066]}$ or $\href{https://arxiv.org/pdf/2304.01050}{Shankar-Siad-Swaminathan}$) that orders by the height of defining coefficients, though these consider class group statistics.

                            • Fixed invariant counting


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                              Problem 2.25.

                              [Alison Miller] Let $a_1, \dots, a_n$ be natural numbers. What can we say about the number of fields with $\mathrm{inv}_1 = a_1, \dots, \mathrm{inv}_m = a_m$? Here $\mathrm{inv}_j$ are Gundlach’s inertial invariants.

                                • Upper bounds from $3 \cdot 2^k$-torsion estimates


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                                  Problem 2.3.

                                  [Carlo Pagano] One should be able to use $\href{https://arxiv.org/abs/2409.02080}{Koymans-Pagano-Sofos}$ to extract an upper bound of the correct order of magnitude for the number of $D_{12}$-extensions. What else can we do with these techniques?

                                    • Local-to-global principles for hypercentral embedding problems


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                                      Problem 2.35.

                                      [Brandon Alberts] Is there a local-to-global principle for hypercentral embedding problems, like there is for central embedding problems? In principle, a hypercentral extension is a series of central extensions.
                                        1. Remark. [Peter Koymans] Away from roots of unity this can be found in $\href{https://mathscinet-ams-org.myaccess.library.utoronto.ca/mathscinet/article?mr=560411}{[MR560411]}$ (Main Theorem on page 148, journal ordering). In the presence of roots of unity, there should be a counterexample.

                                            • Small nilpotent groups in Malle’s conjecture


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                                              Problem 2.4.

                                              [Brandon Alberts] Currently, there are three nilpotent groups in degree eight for which Malle’s conjecture is not known and that no one is working on. They are 8T8, 8T28, and 8T29. As well, $D_8 \subseteq S_{16}$ may be the smallest nilpotent group for which Malle’s conjecture is not known. Prove Malle’s conjecture for these groups.

                                                • Generalizing the Shankar-Tsimerman heuristic to other groups

                                                      In $\href{https://mathscinet.ams.org/mathscinet/article?mr=4585297}{[MR4585297]}$, the authors give a new heuristic explanation for the Bhargava constant $c_n$ appearing in the count of degree $n$ $S_n$-fields.

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                                                  Problem 2.45.

                                                  How can we adapt this heuristic for counting problems with other Galois groups? What would it say about the leading constant?

                                                    • The logarithm power in Malle’s conjecture

                                                          In a forthcoming paper (and in a $\href{https://vimeo.com/showcase/11449719/video/1030091376}{talk \ at \ the \ workshop}$), Jiuya Wang gave counterexamples to Türkelli proposed constant $b_T$ for the logarithm power in Malle’s conjecture. She then proposed a corrected constant for the logarithm power.

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                                                      Problem 2.5.

                                                      [Jiuya Wang] What can we say about Wang’s proposed correction? Is it true in the function field case?

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