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A criterion for potentially good reduction in nonarchimedean dynamics

Robert L. Benedetto (2014)

Acta Arithmetica

Let K be a nonarchimedean field, and let ϕ ∈ 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 ϕ and their preimages, that determines whether or not the dynamical system ϕ: ℙ¹ → ℙ¹ has potentially good reduction.

Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics

Yûsuke Okuyama (2013)

Acta Arithmetica

We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on a variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and...

Bounds on the radius of the p-adic Mandelbrot set

Jacqueline Anderson (2013)

Acta Arithmetica

Let f ( z ) = z d + a d - 1 z d - 1 + . . . + a 1 z p [ z ] be a degree d polynomial. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p ≥ d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for 1/2d ≤ p < d. We also explore a one-parameter family of cubic polynomials over ℚ₂ to illustrate that the p-adic Mandelbrot set can be quite complicated when p < d, in contrast with the simple and...

Effective equidistribution of S-integral points on symmetric varieties

Yves Benoist, Hee Oh (2012)

Annales de l’institut Fourier

Let K be a global field of characteristic not 2. Let Z = H G be a symmetric variety defined over K and S a finite set of places of K . We obtain counting and equidistribution results for the S-integral points of Z . Our results are effective when K is a number field.

Galois extensions of height-one commuting dynamical systems

Ghassan Sarkis, Joel Specter (2013)

Journal de Théorie des Nombres de Bordeaux

We consider a dynamical system consisting of a pair of commuting power series under composition, one noninvertible and another nontorsion invertible, of height one with coefficients in the p -adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results from the theory of the field of norms, we also show that the dynamical system must include a torsion series. From an earlier...

Heights and totally p-adic numbers

Lukas Pottmeyer (2015)

Acta Arithmetica

We study the behavior of canonical height functions h ̂ f , associated to rational maps f, on totally p-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of h ̂ f on the maximal totally p-adic field if the map f has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset X in the compositum of all number fields of degree at most d such that f(X) = X for some non-linear polynomial f. This answers a...

Lifting the field of norms

Laurent Berger (2014)

Journal de l’École polytechnique — Mathématiques

Let K be a finite extension of Q p . The field of norms of a p -adic Lie extension K / K is a local field of characteristic p which comes equipped with an action of Gal ( K / K ) . When can we lift this action to characteristic 0 , along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of ( ϕ , Γ ) -modules, and give a condition for the existence of certain types of lifts.

On a dynamical Brauer–Manin obstruction

Liang-Chung Hsia, Joseph Silverman (2009)

Journal de Théorie des Nombres de Bordeaux

Let ϕ : X X be a morphism of a variety defined over a number field  K , let  V X be a K -subvariety, and let  𝒪 ϕ ( P ) = { ϕ n ( P ) : n 0 } be the orbit of a point  P X ( K ) . We describe a local-global principle for the intersection  V 𝒪 ϕ ( P ) . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of  V ( K ) are Brauer–Manin unobstructed for power maps on  2 in two cases: (1)  V is a translate of a torus. (2)  V is a line and  P has a preperiodic coordinate. A key tool in the proofs is the classical...

The minimal resultant locus

Robert Rumely (2015)

Acta Arithmetica

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map γ o r d ( R e s ( φ γ ) ) factors through a function o r d R e s φ ( · ) on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in P ¹ K , or on a segment, and the minimal resultant locus is contained in the tree in P ¹ K spanned by the fixed points and poles...

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