Related papers: Automorphisms of the Generalized Fermat curves

Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to…

Let $p\ge 7$ be a prime, and $m\ge 5$ an integer. A natural generalization of Bring's curve valid over any field $\mathbb{K}$ of zero characteristic or positive characteristic $p$, is the algebraic variety $V$ of…

Let $\mathbb{F}_q$ be the finite field of order $q=p^h$ with $p>2$ prime and $h>1$, and let $\mathbb{F}_{\bar{q}}$ be a subfield of $\mathbb{F}_q$. From any two $\bar{q}$-linearized polynomials $L_1,L_2 \in \overline{\mathbb{F}}_q[T]$ of…

Let $S$ be a $p$-subgroup of the $\K$-automorphism group $\aut(\cX)$ of an algebraic curve $\cX$ of genus $\gg\ge 2$ and $p$-rank $\gamma$ defined over an algebraically closed field $\mathbb{K}$ of characteristic $p\geq 3$.In this paper we…

A closed Riemann surface $\mathcal X$, of genus $g \geq 2$, is called a generalized superelliptic curve of level $n \geq 2$ if it admits an order $n$ conformal automorphism $\tau$ so that $\mathcal X/\langle \tau \rangle$ has genus zero and…

Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this…

For a field $\mathbb{K}$ of characteristic $p\ge5$ containing $\mathbb{F}_{p}^{\operatorname{alg}}$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $\mathbb{K}\left(s,t\right)$ of two variables $s$…

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this…

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N \geq k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result…

Let $n$, $k$ and $t$ be integers with $1\leq t< k \leq n$. The \emph{generalized Kneser graph} $K(n,k,t)$ is a graph whose vertices are the $k$-subsets of a fixed $n$-set, where two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|<t$. The…

In this paper, we determine the reduced automorphism groups of hyperelliptic curves of a small genus in characteristic $2$, when they are of $2$-rank $0$. Such a curve is an Artin-Schreier curve defined in the form $y^2-y=f(x)$ for a…

Let $S$ be a $p$-subgroup of the $\mathbb K$-automorphism group $Aut(\mathcal X)$ of an algebraic curve $\mathcal X$ of genus $g\ge 2$ and $p$-rank $\gamma$ defined over an algebraically closed field $\mathbb{K}$ of characteristic $p\geq…

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced to the determination of the sets of rational…

The study of algebraic curves $\cX$ with numerous automorphisms in relation to their genus $g(\cX)$ is a well-established area in Algebraic Geometry. In 1995, Irokawa and Sasaki \cite{Sasaki} gave a complete classification of curves over…

In this paper we describe and characterize the fundamental group of the complement to generic fiber-type curves, i.e. unions of (the closure of) finitely many generic fibers of a component-free pencil $F=[f:g]:\mathbb C\mathbb…

We study canonical models of $\left(\mathbb{Z}/k\mathbb{Z}\right)^n$- covers of the projective line, tamely ramified at exactly $n+1$ points each of index $k$, when $k,n\geq 2$ and the characteristic of the ground field $K$ is either zero…

Let $p\geq 3$ be a prime number. A Fermat curve over $\mathbb{Q}$ of exponent $p$ is defined by an equation of the shape $ax^p+by^p+cz^p=0$, where $a,b,c$ are non-zero rational numbers. We prove in this article that there exist infinitely…

Let $(t_n)_{n\ge0}$ be the well konwn $\pm1$ Thue-Morse sequence $$+1,-1,-1,+1,-1,+1,+1,-1,\cdots.$$ Since the 1982-1983 work of Coquet and Dekking, it is known that $\sum_{k<n}t_ke^\frac{2k\pi i}{3}$ is strongly related to the famous Koch…