Let p be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that Gal(Qp/Qp) is the étale fundamental group of a certain object Z which is defined over an algebraically closed field. This object is the quotient of the “punctured perfectoid open disk” by an action of the group Qp . The proof of this fact combines two themes: the tilting equivalence for ...

Tuber rot disease is a major constraint to white yam (Dioscorea rotundata) production, accounting for 50-60% of annual yield losses in Nigeria. The main method of control using synthetic fungicides is being discouraged due to human and environmental health hazards. The potential of Oryza sativa husk (OSH) and Quercus phillyraeoides (QP) extracts for the in vitro and in vivo control of six virul...

Given a prime p, the Fermat quotient qp(u) of u with gcd(u, p)= 1 is defined by the conditions qp(u)≡ u p−1 − 1 p mod p, 0≤ qp(u)≤ p − 1. We derive a new bound on multiplicative character sums with Fermat quotients qp(`) at prime arguments `. 2010 Mathematics subject classification: primary 11A07; secondary 11L40, 11N25.

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary ...

Fix an odd prime p, an algebraic closure Qp, and an isomorphism C ∼ → Qp. Fix an integer N ≥ 1 prime to p, and let T be the polynomial algebra over Z generated by the operators T!, ! ! Np, Up and 〈d〉 , d ∈ (Z/NZ). Set W = Spf(Zp[[Zp ]]), and let W = W rig be the rigid analytic space of characters of Zp together with its universal character χW : Z × p → O(W ) ; we embed Z in W (Qp) by mapping k ...

Let Qp be an algebraic closure of Qp and C its p-adic completion. Let K be a finite extension of Qp contained in Qp and set GK = Gal(Qp/K). A Qp-representation (resp. a C-representation) of GK is a finite dimensional Qp-vector space (resp. C-vector space) equipped with a linear (resp. semi-linear) continuous action of GK . A banach representation of GK is a topological Qp-vector space, whose to...

A direct application of Zorn’s lemma gives that every Lipschitz map f : X ⊂ Qp → Qp has an extension to a Lipschitz map f̃ : Qp → Qp . This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps S ⊂ Rn → R`. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the p-adic ...

Soit p un nombre premier impair, K = Qp et soient Kn = Qp(μpn) le corps des racines p-ièmes de l’unité dans une clôture algébriqueQp de Qp, K∞ la réunion des Kn et G∞ = Gal(K∞/K). Soit une courbe elliptique E définie sur Qp. Si E(Kn) est le groupe des points de E définis sur Kn, la limite projective des E(Kn) pour les applications de trace a d’abord été étudiée lorsque E a bonne réduction ordin...

For a linear group G acting on an absolutely irreducible variety X over Q, we describe the orbits of X(Qp) under G(Qp) and of X(Fp((t))) under G(Fp((t))) for p big enough. This allows us to show that the degree of a wide class of orbital integrals over Qp or Fp((t)) is ≤ 0 for p big enough, and similarly for all finite field extensions of Qp and Fp((t)).