Related papers: Averages and higher moments for the $\ell$-torsion…

We prove for each integer $\ell\geq 1$ an unconditional upper bound for the size of the $\ell$-torsion subgroup $Cl_K[\ell]$ of the class group of $K$, which holds for all but a zero density set of number fields $K$ of degree $d\in\{4,5\}$…

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,…

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 any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on…

For a prime $\ell$, let $h_\ell(K)$ denote the $\ell$-part of the class number of the number field $K$. We investigate upper bounds for $h_\ell(K)$ when $K$ is quadratic or cubic, particularly in the case in which the discriminant of $K$ is…

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…

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 prove an upper bound for $\ell$-torsion in class groups of almost all fields in certain families of $D_4$-quartic fields. Our key tools are a new Chebotarev density theorem for these families of $D_4$-quartic fields and a lower bound for…

Elementary abelian groups are finite groups in the form of $A=(\mathbb{Z}/p\mathbb{Z})^r$ for a prime number $p$. For every integer $\ell>1$ and $r>1$, we prove a non-trivial upper bound on the $\ell$-torsion in class groups of every…

We prove a generalization of a result of Bhargava regarding the average size $\mathrm{Cl}(K)[2]$ as $K$ varies among cubic fields. For a fixed set of rational primes $S$, we obtain a formula for the average size of $\mathrm{Cl}(K)/\langle S…

In a 2008 paper Ellenberg suggested a strategy to improve the known upper bounds for the $\ell$-torsion part of class groups of number fields of fixed degree $d$. Motivated by this he proposed a question about the number of primitive…

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…

Let $K$ be a multiquadratic extension of $\mathbb{Q}$ and let $\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors \cite{KP} gave a bound for $|\text{Cl}^{+}(K)[2]|$ only in terms of the degree of $K$ and the number of…

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 determine the average size of the 3-torsion in class groups of $G$-extensions of a number field when $G$ is any transitive $2$-group containing a transposition, for example $D_4$. It follows from the Cohen--Lenstra--Martinet heuristics…

There are several recent works where authors have shown that number fields $K$ with `sufficiently many' units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ is absolutely abelian. In this…

For any odd prime $\ell$, let $h_\ell(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for…

We prove that in the function field setting, $\ell$-torsion in the class groups of quadratic fields can be arbitrarily large. In fact, we explicitly produce a family whose $\ell$-rank growth matches the growth in the setting of genus…

There is a known analogy between growth questions for class groups and for Selmer groups. If $p$ is a prime, then the $p$-torsion of the ideal class group grows unboundedly in $\mathbb{Z}/p\mathbb{Z}$-extensions of a fixed number field $K$,…

For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over…