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

For $n \times n$ random integer matrices $M_1,\ldots,M_k$, the cokernels of the partial products $\mathrm{cok}(M_1 \cdots M_i), 1 \leq i \leq k$ naturally define a random flag of abelian $p$-groups. We prove that as $n \to \infty$, this…

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…

We study the asymptotic estimation of prime ideals that satisfy certain congruence and argument conditions in imaginary quadratic fields. We also discuss the phenomenon of Chebyshev's bias in the distribution of prime ideals among different…

Let $A$ be an abelian variety over a number field $F$, and suppose that $\mathbb Z[\zeta_n]$ embeds in $\mathrm{End}_{\bar F} A$, for some root of unity $\zeta_n$ of order $n = 3^m$. Assuming that the Galois action on the finite group…

The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of…

We continue the investigation of the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields begun in the previous paper. Building off the work on higher Selmer groups in that part,…

Lenstra's concept of Euclidean ideals generalizes the Euclidean algorithm; a domain with a Euclidean ideal has cyclic class group, while a domain with a Euclidean algorithm has trivial class group. This paper generalizes Harper's variation…

For a number field $K$, the Euler-Kronecker constant $\gamma_K$ associated to $K$ is an arithmetic invariant the size and nature of which is linked to some of the deepest questions in number theory. This theme was given impetus by Ihara who…

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We…

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all…

In this paper we propose a heuristic technique for distributing points on the surface of a unit n-dimensional Euclidean sphere, generated as the orbit of a finite cyclic subgroup of orthogonal matrices, the so called cyclic group codes.…

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…

Let $K/F$ be an unramified quadratic extension of non-Archimedian local fields with residue character not equals to 2. We prove the linear Arithmetic Fundamental Lemma for GL$_4$ with the unit element in the spherical Hecke Algebra. In this…

Let $C \geq 1$. Inspired by the LLL-algorithm, we define strongly $C$-reduced divisors of a number field $F$ which are generalized from the concept of reduced Arakelov divisors. Moreover, we prove that strongly $C$-reduced Arakelov divisors…

Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are…

We introduce a new technique for the study of the distribution of modular symbols, which we apply to congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$…

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erd\H{o}s-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear…

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$.…

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…

This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of…