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Register a new userAttaining potentially good reduction in arithmetic dynamics
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Authors
Robert L. Benedetto
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Let K be a non-archimedean field, and let f in K(z) be a rational function of degree d>1. If f has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that f is conjugate over L to a map of good reduction. In particular, if d=2 or d is greater than the residue characteristic of K, the bound is d+1. If K is discretely valued, we give examples to show that our bound is sharp.
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Let K be a non-archimedean field, and let f in K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of f and their preimages, that determines whether or not the dynamical system f on P^1 has potentially good reduction.
We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.
The arboreal Galois group of a polynomial $f$ over a field $K$ encodes the action of Galois on the iterated preimages of a root point $x_0in K$, analogous to the action of Galois on the $ell$-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 - 1$ when the field $K$ and root point $x_0$ satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of $f$. For $K=mathbb{Q}$, our condition holds for infinitely many choices of $x_0$.
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.
Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined, and an application to a question of Ihara is discussed.
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