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1. Class group statistics

    1. Indecomposable modules in biquadratic class groups


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      Problem 1.1.

      [Jiuya Wang] Which indecomposable $\mathbb{F}_2[C_2 \times C_2]$-modules can arise as direct summands of $\mathrm{Cl}(K)[2]$ for a biquadratic extension $K/\mathbb{Q}$?
        1. Remark. [Carlo Pagano] In $\href{https://mathscinet.ams.org/mathscinet/article?mr=4381932}{[MR4381932]}$, the structure of $\mathrm{Cl}^+(K)^\vee[2]$ is worked out for multi-quadratic fields $K/\mathbb{Q}$. These results maybe be of some use.

            • 3-torsion in cubic fields


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              Problem 1.2.

              [Jiuya Wang] Can we understand the $3$-torsion part of the class group of cubic $S_3$-extensions over $\mathbb{Q}$? Can we understand it in the function field setting?
                1. Remark. [Melanie Wood] It would be interesting to look at the $3$-torsion in case of $S_3$-cubics whose discriminant is squarefree. This guarantees there is no $C_3$ inertia and corresponds to there being no genus part.
                    • Remark. [Peter Koymans] In contrast, it may be useful to consider pure cubic extensions (those obtained by adjoining a cubed root), which may be more approachable because they have a large genus part.

                        • Bounding moments of torsion in class groups

                              In $\href{https://mathscinet.ams.org/mathscinet/article?mr=4801852}{[MR4801852]}$, the authors (building on $\href{https://mathscinet.ams.org/mathscinet/article?mr=3692746}{[MR3692746]}$ and $\href{https://mathscinet.ams.org/mathscinet/article?mr=4238263}{[MR4238263]}$) derive bounds for moments of $\ell$-torsion in class groups without using GRH given a sufficient understanding of primes. $\href{https://arxiv.org/abs/2405.08383}{Lemke \ Oliver - Smith}$ gives input about these primes for primitive extensions and hence derive bounds for moments of $\ell$-torsion in class groups of primitive extensions.

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                          Problem 1.3.

                          [Robert Lemke Oliver] Can we use these ideas to give bounds on moments of $\ell$-torsion in the imprimitive case?

                            • Relating class field towers and class groups


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                              Problem 1.4.

                              [Alexander Smith] For a number field $K/\mathbb{Q}$, how many, as a function of $\mathrm{Cl}(K)[3]$ and a fixed $m \geq 1$, unramified $3$-extensions of $K$ are there with degree $3^m$?

                                • A subgroup of the class group coming from knot theory


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                                  Problem 1.5.

                                  [Alison Miller] Let $K/\mathbb{Q}$ be a quadratic extension with discriminant $D = 1 + 4m$ and consider the subgroup $G$ of $2\mathrm{Cl}(\mathbb{Q}(\sqrt{D}))$ generated by the squares of primes lying above divisors of $m$. What can we say about the statistics of this group? About its $2^{\infty}$-part?
                                    1. Remark. [Alison Miller] An element of $G$ gives me a quadratic form corresponding to a surface that bounds a knot of discriminant related to $D$. Non-trivial elements of $G$ give rise to knots which I cannot tell apart, but for which I can tell the surface apart.

                                        • Error terms in the average 4-part of class group


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                                          Problem 1.6.

                                          [Alexander Smith] We know that $$\sum_{K \text{ imag. quad}, \ |\mathrm{disc}(K)| < X} \# 2\mathrm{Cl}(K)[4] = cX + \mathcal{E}(X)$$ where $\mathcal{E}(X) = o(X)$. Is there no power saving in this error term? Can we show, for example, that $\mathcal{E}(X) > \frac{X}{\log(X)}$ for infinitely many $X$?
                                            1. Remark. [Jürgen Klüners] In $\href{https://mathscinet.ams.org/mathscinet/article?mr=2726105}{[MR2726105]}$, a similar result is shown for a thin subfamily and for this subfamily I expected $\mathcal{E}(X) > \frac{X}{\sqrt{\log(X)}}$ infinitely often. Perhaps (if there is no cancellation) this is the right size for the error term.

                                                • Data for arithmetic statistics


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                                                  Problem 1.7.

                                                  [David Zureick-Brown] What sort of data would be useful to arithmetic statisticians?
                                                      For instance, I have a 1TB file containing the class groups of imaginary quadratic fields. How can that be of use?
                                                    1. Remark. [Arul Shankar] Robert Hough’s $\href{https://stacks.stanford.edu/file/druid:bb259tw6295/thesis-augmented.pdf}{thesis}$ suggests that secondary terms in the average $p$-torsion of class groups of imaginary quadratic fields should be negative (this is known for $p = 3$). The data could be used to investigate this?

                                                        • Structures and statistics of class groups

                                                              In a forthcoming paper (and $\href{https://vimeo.com/showcase/11449719/video/1029831140}{in \ one \ of \ the \ workshop \ talks}$), Yuan Liu describes conjectural distributions for the $p$-part of class groups of number fields with abelian Galois group $\Gamma$. The paper uses the module theory of $\mathbb{Z}_p[\Gamma]$ to produce finer invariants which are the subjects of Liu’s conjectured distributions.

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                                                          Problem 1.8.

                                                          [Yuan Liu] There are three follow up questions:

                                                          1. Can the module classification be extended to the non-abelian case?

                                                          2. What can one say about what these finer invariants "miss" about the class group? (In Liu’s notation, this is asking about the average size of the kernel of $\rho_K$, see Theorem 4 in the linked talk).

                                                          3. Can Smith’s work be applied to prove these conjectured distributions?

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