Related papers: A new parametrization for ideal classes in rings d…

Ideal class pairings map the rational points of rank $r\geq 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D) \geq \frac{1}{2}(c(E)-\varepsilon)(\log…

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…

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…

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…

This article is the first in a series devoted to computing the class groups of real quadratic fields. We present a new relation between the class number and the index of unit groups. This relation generalizes Hilbert class field theory for…

This article discusses a way for uniquely setting up the valuations for the minimal generators of the maximal ideal of a one dimensional complete reduced and irreducible local algebra over an algebraically closed field, when treated as a…

Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s)^{\ell} \oplus k(-2s+1)$, where $s \geq3$ and…

We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average…

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…

Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I.

We consider the groups of regular circulant matrices over finite fields and integer residue class rings. In both cases we present a formula for the order of these groups. We also make a first step towards finding the algebraic structure of…

Using the methods of quantisation ideals, we construct a family of quantisations corresponding to Case alpha in Sergeev's classification of solutions to the tetrahedron equation. This solution describes transformations between special…

Let $R$ be a local principal ideal ring of length two, for example, the ring $R=\Z/p^2\Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of…

We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard…

We introduce a new topological invariant of a rigidly-compactly generated tensor-triangulated category and two new notions of support. The first is based on smashing subcategories: it is unknown whether the frame of smashing subcategories…

A ringoid is a set with two binary operations that are linked by the distributive laws. We study special classes of ringoids that are congruence-simple or ideal-simple. In particular, we examine generalised parasemifields and…

We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for…

Classes of pairs of ternary quadratic forms parametrize quartic rings by a result of Bhargava. We give an algorithm for finding a pair of ternary quadratic forms that parametrize a given order of a quartic field. We examine a new technique,…

This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of…