Galois points for a plane curve in arbitrary characteristic

In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a plane curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given plane curve. In this paper, we survey recent results related to this problem.      

Rational points and Galois points for a plane curve over a finite field

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 Hermitian, Klein quartic or Ballico–Hefez curve. The author proposes a problem: Does the converse hold true? If the curve of genus zero or one has a rational point, we have an affirmative answer.     

On the Number of Galois Points for a Plane Curve in Positive Characteristic

We study Galois points for a plane smooth curve C ? P ² of degree d ≥ 4 in characteristic p > 2. We generalize Yoshihara's result on the number of inner (resp., outer) Galois points to positive characteristic under the assumption that d ? 1 (resp., d ? 0) modulo p. As an application, we also find the number of Galois points in the case that d = p.     

On the number of Galois points for a plane curve in positive characteristic,III

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(p ^e  + 1) of degree p ^e  + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \mathbbF₂{\mathbb{F}_{2}} -rational points in \mathbbP²{\mathbb{P}^{2}}.     

A birational embedding with two Galois points for quotient curves

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti–Korchmáros curve or of the curves constructed by Skabelund. New examples of plane curves with two Galois points are described, as plane models of such quotient curves.     

On the plane section of an integral curve in positive characteristic

If is an integral curve and an algebraically closed field of characteristic 0, it is known that the points of the general plane section of are in uniform position. From this it follows easily that the general minimal curve containing is irreducible. If char, the points of may not be in uniform position. However, we prove that the general minimal curve containing is still irreducible.

     

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of q elements, which are Frobenius nonclassical for different powers of q. This family is an important class of plane curves with many remarkable properties. It contains the Dickson–Guralnick–Zieve curve, which has been recently studied by Giulietti, Korchmáros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.     

Galois points for a plane curve in characteristic two

Let C$C$ be an irreducible plane curve. A point P$P$ in the projective plane is said to be Galois with respect to C$C$ if the function field extension induced by the projection from P$P$ is Galois. We denote by δ^′(C)${\delta }^{\prime }\left(C\right)$ the number of Galois points contained in P²?C${\mathbb{P}}^2?C$. In this article we will present two results with respect to determination of δ^′(C)${\delta }^{\prime }\left(C\right)$ in characteristic two. First we determine δ^′(C)${\delta }^{\prime }\left(C\right)$ for smooth plane curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ^′(C)${\delta }^{\prime }\left(C\right)$. Second we determine δ^′(C)${\delta }^{\prime }\left(C\right)$ for a generalization of the Klein quartic, which is related to an example of Artin–Schreier curves whose automorphism group exceeds the Hurwitz bound. This curve has many Galois points.     

On the number of rational points on a strictly convex curve

Let γ be a bounded convex curve on the plane. Then #(γ ∩ (?/n)²) = o(n ^2/3). This strengthens the classical result due to Jarník [J] (the upper bound cn ^2/3) and disproves the conjecture on the existence of a so-called universal Jarník curve.     

一类椭圆曲线有正整数点的判别条件

设p是适合p≡1（mod81的奇素数．本文主要利用初等方法证明了椭圆曲线y2=px（x2＋1）在P≡9（rood16）时没有正整数点（x,y）;并且对于p≡1（mod16）的情况,给出了该椭圆曲线有整数点的两个判别条件．     

Mapping class group of a plane curve germ

We prove that every topological conjugacy between two germs of singular holomorphic curves in the complex plane is homotopic to another conjugacy which extends homeomorphically to the exceptional divisors of their minimal desingularisations. As an application we give an explicit presentation of a finite index subgroup of the mapping class group of the germ of such a singularity.     

On the number of rational points of a plane algebraic curve

The number of F_q -rational points of a plane non-singular algebraic curve defined over a finite field F_q is computed, provided that the generic point of is not an inflexion and that is Frobenius non-classical with respect to conics. Received: 18 March 2003     

An invariant for quadratic forms valued in Galois Rings of characteristic 4

We introduce an invariant for nonsingular quadratic forms that take values in a Galois Ring of characteristic 4. This notion extends the invariant in ${\mathbb{Z}}_8$ for ${\mathbb{Z}}_4$-valued quadratic forms defined by Brown [E.H. Brown, Generalizations of the Kervaire invariant, Ann. of Math. (2) 95 (2) (1972) 368–383] and studied by Wood [J.A. Wood, Witt's extension theorem for mod four valued quadratic forms, Trans. Amer. Math. Soc. 336 (1) (1993) 445–461]. It is defined in the associated Galois Ring of characteristic 8. Nonsingular quadratic forms are characterized by their invariant and the type of the associated bilinear form (alternating or not).     

On the number of certain Galois extensions of local fields

In this paper, we will calculate the number of Galois extensions of local fields with Galois group or .

     

一种保持周长的平面曲线流

The purpose of this paper is to investigate a new type of evolution problem for closedconvex plane curves which will preserves the perimeter of the curve but expands the enclosedarea and the final limiting curve is a circle in the Hausdorff metric in the plane.