Displaying similar documents to “On (C,1) summability for Vilenkin-like systems”

Bounds on the radius of the p-adic Mandelbrot set

Jacqueline Anderson (2013)

Acta Arithmetica

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Let f ( z ) = z d + a d - 1 z d - 1 + . . . + a 1 z ∈ ℂ 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...

Heights and totally p-adic numbers

Lukas Pottmeyer (2015)

Acta Arithmetica

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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...

On the heights of totally p -adic numbers

Paul Fili (2014)

Journal de Théorie des Nombres de Bordeaux

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Bombieri and Zannier established lower and upper bounds for the limit infimum of the Weil height in fields of totally p -adic numbers and generalizations thereof. In this paper, we use potential theoretic techniques to generalize the upper bounds from their paper and, under the assumption of integrality, to improve slightly upon their bounds.

On the local behaviour of ordinary Λ -adic representations

Eknath Ghate, Vinayak Vatsal (2004)

Annales de l'Institut Fourier

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Let f be a primitive cusp form of weight at least 2, and let ρ f be the p -adic Galois representation attached to f . If f is p -ordinary, then it is known that the restriction of ρ f to a decomposition group at p is “upper triangular”. If in addition f has CM, then this representation is even “diagonal”. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members...