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We develop the basic theory of the Weyl symbolic calculus of pseudodifferential operators over the $p$ -adic numbers. We apply this theory to the study of elliptic operators over the $p$ -adic numbers and determine their asymptotic spectral behavior.

Quasi-semi-stable representations

Xavier Caruso, Tong Liu (2009)

Bulletin de la Société Mathématique de France

Fix $K$ a $p$ -adic field and denote by $G_{K}$ its absolute Galois group. Let $K_{\infty}$ be the extension of $K$ obtained by adding $p^{n}$ -th roots of a fixed uniformizer, and $G_{\infty} \subset G_{K}$ its absolute Galois group. In this article, we define a class of $p$ -adic torsion representations of $G_{\infty}$ , calledquasi-semi-stable. We prove that these representations are “explicitly” described by a certain category of linear algebraic objects. The results of this note should be considered as a first step in the understanding of the structure of quotient...

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex normal number...

Quelques applications du théorème de densité de Chebotarev

Jean-Pierre Serre (1981)

Publications Mathématiques de l'IHÉS

Quelques « formules de masse » raffinées en degré premier

Chandan Singh Dalawat (2012)

Bulletin de la Société Mathématique de France

Pour un corps local à corps résiduel fini de caractéristique $p$ , nous donnons quelques raffinements de la formule de masse de Serre en degré $p$ qui nous permettent de calculer par exemple la contribution des extensions cycliques, ou celles dont la clôture galoisienne a pour groupe d’automorphismes un groupe donné à l’avance, ou possède des propriétés de ramification également données à l’avance.

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