Related papers: A dynamical Shafarevich theorem for twists of rati…

Let $F : \mathrm{End}_{\mathbb{F_p}}(\mathbb{G}_{a/K}^d)$ be an additive polynomial mapping over a global function field $K/\mathbb{F}_q$, and let $P \in \mathbb{G}_a^d(K)$. Following Silverman, consider $\delta := \lim_{n \in \mathbb{N}}…

Let $K$ be a number field and $S$ a fixed finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we study the cycles for rational maps of $\mathbb{P}_1(K)$ of…

Let P^n denote the n-dimensional projective space defined over the algebraic closure of a finite field F_q, let V contained P^n be a complete intersection defined over F_q of dimension r and singular locus of dimension at most s, and let…

For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher…

Given a prime $p$, a number field $\K$ and a finite set of places $S$ of $\K$, let $\K_S$ be the maximal pro-$p$ extension of $\K$ unramified outside $S$. Using the Golod-Shafarevich criterion one can often show that $\K_S/\K$ is infinite.…

Let K be an algebraic number field. For a degree d rational morphism of projective n-space defined over K let R denote its minimal resultant ideal. For a fixed height function on the moduli space of dynamical systems this paper shows that…

Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer…

Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega_{X/K}$ is ample. Let $R:={\rm Zar}(X)(K)\cap X)$…

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…

Let $X$ be an Enriques surface defined over a number field $K$. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational points of $X$ is Zariski dense.

The purpose of this note is to give a short, selfcontained proof of the following result: A complex surface which is diffeomeorphic to a rational surface is rational.

In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there…

Given an uncountable algebraically closed field $K$, we proved that if partially defined function $f\colon K \times \dots \times K \dashrightarrow K$ defined on a Zariski open subset of the $n$-fold Cartesian product $K \times \dots \times…

Let $K$ be a field and $V$ be a set of rank one valuations of $K$. The corresponding Tate-Shafarevich group of a $K$-torus $T$ is $Sha(T , V) = \ker\left(H^1(K , T) \to \prod_{v \in V} H^1(K_v , T)\right)$. We prove that if $K = k(X)$ is…

In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivity of the shape derivative and we show it…

We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected commutative linear algebraic groups $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon G\longrightarrow G$…

The theory of moduli of morphisms on P^n generalizes the study of rational maps on P^1. This paper proves three results about the space of morphisms on P^n of degree d > 1, and its quotient by the conjugation action of PGL(n+1). First, we…

We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only…

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let $\pi$ be a unitary, cuspidal, automorphic representation of $GL_n(\A_K)$. Let $S$ be a set of finite places of $K$, such that the sum $\sum_{v\in…

Let $F$ be a finite field with characteristic greater than two. Define a \emph{Besicovitch set} in $F^4$ to be a set $P \subseteq F^4$ containing a line in every direction. The \emph{Kakeya conjecture} asserts that $|P| \approx |F|^4$. A…