Related papers: The shapes of pure cubic fields

We address the question of existence of absolutely simple abelian varieties of dimension 2 with everywhere good reduction over quadratic fields. The emphasis will be given to the construction of pairs $(K,C)$, where $K$ is a quadratic…

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete…

This paper is a part of ongoing research on order positive fields started some years ago. We prove that the real closure of an order positive field even in non-Archimedean case is also order positive.

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and…

We study real trigonal curves and elliptic surfaces of type $\I$ (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's \emph{dessins d'enfants}. We give a…

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 study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.

We determine conditions that guarantee that a hyperelliptic or plane curve over a field of characteristic not equal to 2 can be defined over its field of moduli. We also give new examples of curves not definable over their fields of moduli.

We use light-cone gauge formalism to study interacting massive and massless continuous-spin fields and finite component arbitrary spin fields propagating in the flat space. Cubic interaction vertices for such fields are considered. We…

We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic…

We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements…

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.

A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tamely ramified extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt…

We extend to characteristic two recent results about isotropy of quadratic forms over function fields. In particular, we provide a characterization of function fields not only of quadratic forms but also more generally of polynomials in…

For every field $k$ of characteristic zero, we determine the groups that act as automorphisms on a smooth cubic surface over $k$. We also determine the groups that act on $k$-rational, stably $k$-rational, or $k$-unirational smooth cubic…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…

We describe a normal form for a smooth intersection of two quadrics in even-dimensional projective space over an arbitrary field of characteristic 2. We use this to obtain a description of the automorphism group of such a variety. As an…

Let $K$ be a number field, which is tame and non totally real. In this article we give a numerical criterion, depending only on the ramification behavior of ramified primes in $K$, to decide whether or not the integral trace of $K$ is…

We study the structure of two-sided vector spaces over a perfect field $K$. In particular, we give a complete characterization of isomorphism classes of simple two-sided vector spaces which are left finite-dimensional. Using this…