Related papers: The average size of $3$-torsion in class groups of…

The Cohen-Lenstra-Martinet heuristics lead one to conjecture that the average size of the $p$-torsion in class groups of $G$-extensions of a number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved this conjecture in the…

We prove a composite case of the Cohen--Lenstra--Gerth heuristics. Specifically, we establish an asymptotic for the average $6$-torsion of the class group of quadratic number fields. We also prove Malle's conjecture for Galois…

Ellenberg--Venkatesh proved in 2007 that $h_3(d) \ll_\epsilon |d|^{1/3 + \epsilon}$, where $h_3(d)$ denotes the size of the $3$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. We improve this bound to $h_3(d) \ll_\epsilon |d|^{\kappa…

Bhargava, Hanke, and Shankar have recently shown that the asymptotic average $2$-torsion subgroup size of the family of class groups of monogenized cubic fields with positive and negative discriminants is $3/2$ and $2$, respectively. In…

We prove upper bounds for the average size of the $\ell$-torsion $\text{Cl}_K[\ell]$ of the class group of $K$, as $K$ runs through certain natural families of number fields and $\ell$ is a positive integer. We refine a key argument, used…

We determine the mean number of 2-torsion elements in class groups of cubic orders, when such orders are enumerated by discriminant. Specifically, we prove that when isomorphism classes of totally real (resp., complex) cubic orders are…

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the…

We study average $2$-torsion in the class group of monogenised fields of odd degree. Bhargava-Hanke-Shankar have recently shown that the average number of non-trivial $2$-torsion elements in the class group of monogenised cubic fields of a…

For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(\mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the…

We compute all the moments of the p-torsion in the first step of a filtration of the class group defined by Gerth for cyclic fields of degree p, unconditionally for p=3 and under GRH in general. We show that it satisfies a distribution…

In this paper we give a conjecture for the average number of unramified $G$-extensions of a quadratic field for any finite group $G$. The Cohen-Lenstra heuristics are the specialization of our conjecture to the case that $G$ is abelian of…

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed…

We study the distribution of $2$-torsion in class groups and narrow class groups of cubic fields and cubic orders subject to prescribed shape conditions. The \emph{shape} of a cubic order in a number field is a natural geometric invariant…

Fix a number field $k$, integers $\ell, n \geq 2$, and a prime $p$. For all $r \geq 1$, we prove strong unconditional upper bounds on the $r$-th moment of $\ell$-torsion in the ideal class groups of degree $p$ extensions of $k$ and of…

For all positive integers $\ell$, we prove non-trivial bounds for the $\ell$-torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular,…

We describe the relations among the $\ell$-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent…

We prove new conditional bounds on the the $m$-torsion of class groups of number fields of any fixed degree, for $m=2$, $3$, $4$, and $5$. Our methods first recast the problem in the language of class groups of Galois modules, which allows…

For every finite $p$-group $G_p$ that is non-cyclic and non-quaternion and every positive integer $\ell\neq p$ that is greater than $2$, we prove the first non-trivial bound on $\ell$-torsion in class group of every $G_p$-extension. More…

We compute the average size of $\mathrm{Cl}_F[2]$ in the family of cubic fields $F = \mathbb{Q}(\sqrt[3]{n})$. Specifically, as $F$ varies over the subfamily of wildly (resp. tamely) ramified fields $\mathbb{Q}(\sqrt[3]{n})$, the average…

The main aim of the present paper is to disprove the Cohen--Lenstra--Martinet heuristics in two different ways and to offer possible corrections. We also recast the heuristics in terms of Arakelov class groups, giving an explanation for the…