Related papers: A heuristic for ray class groups of quadratic numb…

In number theory, great efforts have been undertaken to study the Cohen-Lenstra probability measure on the set of all finite abelian $p$-groups. On the other hand, group theorists have studied a probability measure on the set of all…

We determine the large-genus limiting distribution of the 4-rank of the Picard group of hyperelliptic curves over a fixed finite field $\mathbb F_q$ of odd characteristic. This is a function field analogue of a result of Fouvry and…

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…

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…

Let A be an n by n random matrix with iid entries taken from the p-adic integers or Z/NZ. Then under mild non-degeneracy conditions the cokernel of A has a universal probability distribution. In particular, the p-part of an iid random…

We describe a probability distribution on isomorphism classes of principally quasi-polarized p-divisible groups over a finite field k of characteristic p which can reasonably be thought of as "uniform distribution," and we compute the…

This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result…

Ellenberg, Venkatesh, and Westerland have established a weak form of the function field analogue of the Cohen--Lenstra heuristic, on the distribution of imaginary number fields with $\ell$-parts of their class groups isomorphic to a fixed…

For a Galois extension $K/\mathbb{F}_q(t)$ of Galois group $\Gamma$ with $\gcd(q,|\Gamma|)=1$, we define an invariant $\omega_K$, and show that it determines the Weil pairing of the curve corresponding to $K$ and it descends to the…

We prove a generalization of the Davenport-Heilbronn theorem to quotients of ideal class groups of quadratic fields by the primes lying above a fixed set of rational primes $S$. Additionally, we obtain average sizes for the relaxed Selmer…

We develop a criterion for a normal basis, and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$.…

Cohen, Lenstra, and Martinet have given conjectures for the distribution of class groups of extensions of number fields, but Achter and Malle have given theoretical and numerical evidence that these conjectures are wrong regarding the Sylow…

We define a group with local data over a number field $K$ as a group $G$ together with homomorphisms from decomposition groups ${\rm Gal}(\overline{K}_p/K_p)\to G$. Such groups resemble Galois groups, just without global information.…

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur…

We observe that Anzanello's work on the proportion of derangements in affine classical groups over finite fields is related to symplectic and orthogonal Cohen-Lenstra type distributions on integer partitions. This leads to a proof of three…

We consider the probability theory, and in particular the moment problem and universality theorems, for random groups of the sort of that arise or are conjectured to arise in number theory, and in related situations in topology and…

Let us consider the following matrix $B_n$. The columns of $B_n$ are indexed with $[n]=\{1,2,\dots,n\}$ and the rows are indexed with $[n]^3$. The row corresponding to $(x_1,x_2,x_3)\in [n]^3$ is given by $\sum_{i=1}^3 e_{x_i}$, where…

We compute all the moments of a normalization of the function which counts unramified $H_{8}$-extensions of quadratic fields, where $H_{8}$ is the quaternion group of order 8, and show that the values of this function determine a constant…

We investigate the distribution of $\alpha p$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponent…

We present the first algorithm for computing class groups and unit groups of arbitrary number fields that provably runs in probabilistic subexponential time, assuming the Extended Riemann Hypothesis (ERH). Previous subexponential algorithms…