Related papers: Ray class fields of global function fields with ma…
We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…
We develop a criterion for a normal basis, and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$.…
We give a complete characterization of all Galois subfields of the generalized Giulietti--Korchm\'aros function fields $\mathcal C_n / \fqn$ for $n\ge 5$. Calculating the genera of the corresponding fixed fields, we find new additions to…
The notions of spectral measures and spectral classes, which are well known for graphs, are generalized and investigated for oriented hypergraphs.
We study the distribution of algebraic points on curves in abelian varieties over finite fields.
This paper develops explicit class field theory for orders: of rank 1 in any global function field -- Hayes theory -- and of rank 2 in real quadratic function fields -- Real Multiplication. The essential ingredient in the development of the…
We develop class field theory of curves over $p$-adic fields which extends the unramified theory of S. Saito. The class groups which approximate abelian \'etale fundamental groups of such curves are introduced in the terms of algebraic…
Over a global field (number field or function field of a curve over a finite field), theorems for the Galois cohomology of algebraic groups have long been known. For $F$ the function field of a curve over the formal series field…
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the…
In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for…
It is shown that there exists a duality among fields. If a field is dual to another field, the solution of the field can be obtained from the dual field by the duality transformation. We give a general result on the dual fields. Different…
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
This is an extended version of an invited lecture I gave at the Journees Arithmetiques in St. Etienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective)…
We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.
We introduce a generalization of symmetric functions and apply the resulting theory to compute the class in the Grothendieck ring of varieties of the space of geometrically irreducible hypersurfaces of a fixed degree in projective space.
In this paper we introduce geometric tools to study the families of rational vector fields of a given degree over $\mathbb C\mathbb P^1$. To a generic vector field of such a parametric family we associate several geometric objects: a…
Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to…
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear…
We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the…
The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.