Related papers: Upper bounds for the Euclidean minima of abelian n…

The aim of this paper is to give upper bounds for the Euclidean minima of abelian fields of odd prime conductor. In particular, these bounds imply Minkowski's conjecture for totally real number fields of conductor p^r, where p is an odd…

In this paper, we use some of our previous results to improve an upper bound of Bayer-Fluckiger, Borello and Jossen on the Euclidean minima of algebraic number fields. Our bound depends on the degree $n$ of the field, its signature,…

In this paper, we define Euclidean minima for function fields and give some bound for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.

In this note we present algorithms for computing Euclidean minima of cubic number fields; in particular, we were able to find all norm-Euclidean cubic number fields with discriminants -999 < d < 10000.

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…

We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of "close" abelian cubic number fields with class numbers as large as possible. We also give a first step toward an…

We establish an explicit upper bound for the Euclidean minimum of a number field which depends, in a precise manner, only on its discriminant and the number of real and complex embeddings. Such bounds were shown to exist by Davenport and…

We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their…

We establish in this paper a sharp lower bound for the area of a unit vector field $V$ defined on some spherical annuli in the Euclidean sphere $\mathbb{S}^2$.

We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of…

In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite Abelian…

We consider upper and lower bounds on the minimal height of an irrational number lying in a particular real quadratic field.

Following the work of Mestre, we use Weil's explicit formulas to compute explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields. Moreover, we obtain bounds for the conductor of elliptic curves…

This paper is concerned with some Algebraic Geometry codes on Jacobians of genus 2 curves. We derive a lower bound for the minimum distance of these codes from an upper "Weil type" bound for the number of rational points on irreducible…

We consider binary abelian codes of length $p^m q^n$, where $p$ and $q$ are prime rational integers under some restrictive hypotheses. In this case, we determine the idempotents generating minimal codes and either the respective weights or…

Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have been identified,…

We construct small models of number fields and deduce a better bound for the number of number fields of given degree and bounded discriminant.

Upper bounds on the minimum Lee distance of codes that are linear over ${\mathbb Z}_q$, $q=p^t$, $p$ prime are discussed. The bounds are Singleton like, depending on the length, rank, and alphabet size of the code. Codes meeting such bounds…

We give a new lower bound for the essential minimum of subvarieties of abelian varieties with small codimension, under a conjecture about ordinary primes in abelian varieties. This lower bound is already known in the toric case since the…

Algebraic surfaces in the complex projective space with a high number of A-type singularities have been presented in a recent paper. We extend the construction in order to obtain lower bounds for the maximal number of A singularities for…