Heights and totally p-adic numbers

Abstract

Abstract. 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 question of W.

Two remarks on Narkiewicz's property (P)

“…are finite for all linear polynomials L ∈ Q[x] (see [5]); (III) F/K Galois, such that infinitely many local degrees of F are finite (see [19]).…”

“…A nice example of fields from part (III) are the fields which are generated over Q by elements of bounded degree. See [19] for more information.…”

Les Huit Premiers Travaux de Pierre Liardet

“…On déduit des travaux de Bombieri et Zannier [19] que non seulement le corps engendré par tous les nombres √ p, p décrivant l'ensemble des nombres premiers, a la propriété (P ), mais qu'il en est de même pour le corps engendré par toutes les racines n-ièmes des entiers rationnels, premiers ou non. La note [4] de K. K. Kubota et P. Liardet est citée non seulement dans l'article de Dvornicich et Zannier [22], mais aussi dans ceux de L. Pottmeyer [33] et M. Widmer [39]. [34,35] a démontré que si f ∈ P(S, S) où S est, comme ci-dessus, l'ensemble des nombres de Pisot -Vijayaraghavan, alors il existe m > 0 tel que f (X, X m ) = 0 dans Ω[X].…”

Three counterexamples concerning the Northcott property of fields

“…Pottmeyer [Pot15,Question 4.7] asks whether the Northcott property is implied by other properties like the Narkiewicz property (R) (cf. [CW13, Definition 6.6]).…”

“…Namely, Pottmeyer[Pot15, Theorem 4.3] shows that every Galois extension of Q that has finite local degree at infinitely many prime numbers (in particular, any K as inProposition 1.3) satisfies the so-called universal strong Bogomolov property (USB), which in turn implies (R) and other related properties [Pot15, Lemma 4.2]. The construction of the example in Proposition 1.3 builds on a result of Bombieri and Zannier [BZ01].…”