Related papers: Galois points for double-Frobenius nonclassical cu…

We prove that the Galois equidistribution of torsion points of the algebraic torus $\mathbb{G}_{m}^d$ extends to the singular test functions of the form $\log{|P|}$, where $P$ is a Laurent polynomial having algebraic coefficients that…

We study birational transformations belonging to Galois points. Let $P$ be a Galois point for a plane curve $C$ and $G_P$ be a Galois group at $P$. Then an element of $G_P$ induces a birational transformation of $C$. In general, it is…

Using class field theory one associates to each curve C over a finite field, and each subgroup G of its divisor class group, unramified abelian covers of C whose genus is determined by the index of G. By listing class groups of curves of…

We use a recent characterization of minimal value set polynomials and $q$- Frobenius nonclassical curves to construct curves that generalize the Hermitian curve. The genus $g$ and the number $N$ of $\mathbb{F}_q$-rational points of the…

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the…

The number of rational points of a plane non-singular algebraic curve X defined over a finite field is computed, provided that the generic point of X is not an inflexion and that X is Frobenius non-classical with respect to conics.

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear $q$-difference equation with rational function coefficients. This Galois…

Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies…

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q…

For any smooth Hurwitz curve $\mathcal{H}_n: \, XY^n+YZ^n+X^nZ=0$ over the finite field $\mathbb{F}_{p}$, an explict description of its Weierstrass points for the morphism of lines is presented. As a consequence, the full automorphism group…

We classify the number of $k$-rational lines and conic fibrations on del Pezzo surfaces over a field $k$ in terms of relatively minimal surfaces and establish rational curve analogues of the inverse Galois problem for del Pezzo surfaces. We…

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized…

We show how the Galois-Picard_Vessiot theory of differential equations and difference equations, and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the…

We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are…

For a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer-Manin obstructions. Given a Galois extension of the ground field one can consider similar…

We study the trace form $q_L$ of $G$-Galois algebras $L/K$ when $G$ is a finite group and $K$ is a field of characteristic different from $2$. We introduce in this paper the category of $2$-reduced groups and, when $G$ is such a group, we…

The paper is concerned with the following version of Hilbert's irreducibility theorem: if $\pi: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and…

The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail…

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple…

Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We…