Related papers: Class number and regulator computation in cubic fu…

The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach…

We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and…

We describe a new algorithm for computing the ideal class group, the regulator and a system of fundamental units in number fields under the generalized Riemann hypothesis. We use sieving techniques adapted from the number field sieve…

Arising from the factorizations of Dedekind zeta-functions of cubic fields, we obtain Artin $L$-functions of certain two-dimensional representations. In this paper, we study the value-distribution of such Artin $L$-functions for families of…

We compute a complete set of isomorphism classes of cubic fourfolds over $\mathbb{F}_2$. Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all…

In various contexts, the zeta function of an object splits into a product of $L$-functions. We categorify this product formula for quadratic covers of objects in the following contexts: quadratic extensions of number fields, ramified double…

In this paper, assuming the weak Schanuel Conjecture (WSC), we prove that for any collection of pairwise non-arithmetically equivalent totally real number fields, the residues at $s=1$ of their Dedekind zeta functions form a linearly…

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism…

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 show how to construct infinite families of explicitly determined cubic number fields whose class group has a subgroup isomorphic to $(\mathbb{Z}/2)^8$ using degree $1$ del Pezzo surfaces. We illustrate the method and provide an example…

We compute the $p$-adic regulator of cyclic cubic extensions of $\mathbb Q$ with discriminant up to $10^{16}$ for $3<p<100$, and observe the distribution of the $p$-adic valuation of the regulators. We find that for almost all primes, the…

The Dedekind zeta function of a quadratic number field factors as a product of the Riemann zeta function and the $L$-function of a quadratic Dirichlet character. We categorify this formula using objective linear algebra in the abstract…

In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in…

Let $k$ be a number field. In this paper, we give a formula for the mean value of the square of class numbers times regulators for certain families of quadratic extensions of $k$ characterized by finitely many local conditions. We approach…

Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.

We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time…

The number of non-isomorphic cubic fields L sharing a common discriminant d(L) = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a multiplet M(d) = (L(1) ,..., L(m)). In this paper, the…

We construct some families of quadratic fields whose class numbers are divisible by $3.$ The main tools used are a trinomial introduced by Kishi and a parametrization of Kishi and Miyake of a family of quadratic fields whose class numbers…

The theory of continued fractions of functions $ \sqrt D $ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt D)$ over $k={\bf F}_q(T)$. For five series of real quadratic function…

We improve the error terms in the Davenport-Heilbronn theorems on counting cubic fields to $O(X^{2/3 + \epsilon})$. This improves on separate and independent results of the authors and Shankar and Tsimerman. The present paper uses the…