相关论文: On maximal curves in characteristic two

We prove that if q is a power of an odd prime then there is no genus-2 curve over F_q whose Jacobian has characteristic polynomial of Frobenius equal to x^4 + (2-2q)x^2 + q^2. Our proof uses the Brauer relations in a biquadratic extension…

We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic…

Let $X$ be a connected, smooth, and projective curve of genus $g$ over an algebraically closed field of characteristic $p >0$. This paper investigates a characteristic-$p$ analogue of a well-known fact concerning flat vector bundles in…

Let $G$ be an embedded graph and $A$ an edge subset of $G$. The partial dual of $G$ with respect to $A$, denoted by $G^A$, can be viewed as the geometric dual $G^*$ of $G$ over $A$. If $A=E(G)$, then $G^A=G^*$. Denote by $\gamma(G^A)$ the…

We prove that if $X$ is a complex projective K3 surface and $g>0$, then there exist infinitely many families of curves of geometric genus $g$ on $X$ with maximal, i.e., $g$-dimensional, variation in moduli. In particular every K3 surface…

Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general…

A real morphism $f$ from a real algebraic curve $X$ to $\mathbb{P}^1$ is called separating if $f^{-1}(\mathbb{R} \mathbb{P}^1) = \mathbb{R} X$. A separating morphism defines a covering $\mathbb{R} X \to \mathbb{R} \mathbb{P}^1$. Let $X_1,…

For a plane curve, a point on the projective plane is said to be Galois if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. We present upper bounds for the number of Galois…

Let $\mathcal{X}$ be an irreducible algebraic curve defined over a finite field $\mathbb{F}_q$ of characteristic $p>2$. Assume that the $\mathbb{F}_q$-automorphism group of $\mathcal{X}$ admits as an automorphism group the direct product of…

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves.

The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known…

We propose a detailed study of a canonical bound which relates the numbers of rational points of a curve over a finite field with that over its quadratic extension. Alternative proofs which make a connection with the variance enable to…

Let N_q(g) the maximal number of points on a genus g curve over F_q. We prove that N_3(5)=13.

A new genus $g=g(X,\ce)$ is defined for the pairs $(X,\ce)$ that consist of $n$-dimensional compact complex manifolds $X$ and ample vector bundles $\ce$ of rank $r$ less than $n$ on $X$. In case $r=n-1$, $g$ is equal to curve genus. Above…

Let $C$ be a genus two hyperelliptic curve over a totally real field $F$. We show that the mod 2 Galois representation $\bar{\rho}_{C,2}\colon\mathrm{Gal}(\bar{F}/F)\to \mathrm{GSp}_4(\mathbb{F}_2)$ attached to $C$ is automorphic when the…

In this paper the number of $\mathbb{F}_q$-isomorphism classes of Legendre elliptic curves over the finite fields $\mathbb{F}_q$ is enumerated.

Let $\mathcal{X}$ be a (projective, non-singular, geometrically irreducible) curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. If the $p$-rank $\gamma(\mathcal{X})$ equals…

The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite field $\mathbb{F}_q$ arises naturally from the classical Dickson invariant of the projective linear group $PGL(3,\mathbb{F}_q)$. The DGZ curve is an (absolutely…

In this note we discuss techniques for determining the automorphism group of a genus $g$ hyperelliptic curve $\X_g$ defined over an algebraically closed field $k$ of characteristic zero. The first technique uses the classical $GL_2…

We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also…