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 ...

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 ...

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...

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)).

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)).

The ideals of the ring Zp are {0} and pZp, n ≥ 0. From this it follows that Zp is a discrete valuation ring, a principal ideal domain with exactly one maximal ideal, namely pZp; Zp is the valuation ring of Qp with the valuation vp. For n ≥ 1, Zp/pZp is isomorphic as a ring with Z/pZ. |x|p = p−vp(x), dp(x, y) = |x− y|p. With the topology induced by the metric dp, Qp is a locally compact abelian ...