Related papers: Periods of rational maps modulo primes

K3 surfaces with non-symplectic involution are classified by open sets of seventy-five arithmetic quotients of type IV. We prove that those moduli spaces are rational except two classical cases.

We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of…

In this paper, we consider rational functions $f$ with some minor restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a…

Let $\alpha $ be a $p \times q$ interval matrix with $p \geq q$ and with the endpoints of all its entries in the set of the rational numbers. We prove that, if $\alpha $ contains a rank-$r$ real matrix with $r \in \{2, q-2,q-1,q\}$, then it…

For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular attention to the case when phi commutes with a…

Let f : X --> X be a dominant rational map of a projective variety defined over a number field. An important geometric-dynamical invariant of f is its (first) dynamical degree d_f= lim SpecRadius((f^n)^*)^{1/n}. For algebraic points P of X…

Let $p$ be a prime number and $k$ be a positive integer not divisible by $p$. We describe the Heller translates of the periodic Lie module $\mathrm{Lie}(pk)$ in characteristic $p$ and show that it has period $2p-2$ when $p$ is odd and $1$…

For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always…

Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the…

In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over $\mathbb{C}$. In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic…

In this paper, we study the orbit intersection problem for the linear space and the algebraic group in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic and let $\Phi_1, \Phi_{2}: K^d…

Let p>2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fractional field of the Witt ring of k. Let G and H be finite flat commutative group schemes killed by p over O_K and…

We study the set of periods of degree 1 continuous maps from sigma into itself, where sigma denotes the space shaped like the letter sigma (i.e., a segment attached to a circle by one of its endpoints). Since the maps under consideration…

It is proved that, for a prime number $p$, showing that an $n$-element matroid is not representable over $GF(p)$ requires only $O(n^2)$ rank evaluations.

Let S be a closed oriented surface of genus $g\geq 0$ with $n\geq 0$ punctures and $3g-3+n\geq 5$. Let $Q$ be a connected component of a stratum in the moduli space Q(S) of area one meromorphic quadratic differentials on S with n simple…

Let F be a real quadratic field, and let R be an order in F. Suppose given a polarized abelian surface (A,\lambda) defined over a number field k with a symmetric action of R defined over k. This paper considers varying A within the…

We give a complete characterization of degree two rational maps with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space. We detail an algorithm by which…

Let $\mathbb{A}[n]$ be the group of $n$-torsion points of a commutative algebraic group $\mathbb{A}$ defined over a number field $F$. For a prime ideal $\mathfrak{p}$, we let $N_{\mathfrak{p}}(\mathbb{A}[n])$ be the number of…

This paper is an extension of an earlier paper that dealt with global dynamics in autonomous triangular maps. In the current paper, we extend the results on global dynamics of autonomous triangular maps to periodic non-autonomous triangular…

In this note we study rational curves on degree $p^r+1$ Fermat hypersurface in $\PP^{p^r+1}_k$, where $k$ is an algebraically closed field of characteristic $p$. The key point is that the presence of Frobenius morphism makes the behavior of…