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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.

Raabe’s formula for $p$ -adic gamma and zeta functions

Henri Cohen, Eduardo Friedman (2008)

Annales de l’institut Fourier

The classical Raabe formula computes a definite integral of the logarithm of Euler’s $Γ$ -function. We compute $p$ -adic integrals of the $p$ -adic $log Γ$ -functions, both Diamond’s and Morita’s, and show that each of these functions is uniquely characterized by its difference equation and $p$ -adic Raabe formula. We also prove a Raabe-type formula for $p$ -adic Hurwitz zeta functions.

Ramification and moduli spaces of finite flat models

Naoki Imai (2011)

Annales de l’institut Fourier

We determine the type of the zeta functions and the range of the dimensions of the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This gives a generalization of Raynaud’s theorem on the uniqueness of finite flat models in low ramifications.

Ramification groups in Artin-Schreier-Witt extensions

Lara Thomas (2005)

Journal de Théorie des Nombres de Bordeaux

Let $K$ be a local field of characteristic $p > 0$ . The aim of this paper is to describe the ramification groups for the pro- $p$ abelian extensions over $K$ with regards to the Artin-Schreier-Witt theory. We shall carry out this investigation entirely in the usual framework of local class field theory. This leads to a certain non-degenerate pairing that we shall define in detail, generalizing in this way the Schmid formula to Witt vectors of length $n$ . Along the way, we recover a result of Brylinski but with...

Ramification in p-Adic Lie Extensions.

Shankar Sen (1972)

Inventiones mathematicae

Ramification in quartic cyclic number fields $K$ generated by $x^{4} + p x^{2} + p$

Julio Pérez-Hernández, Mario Pineda-Ruelas (2021)

Mathematica Bohemica

If $K$ is the splitting field of the polynomial $f (x) = x^{4} + p x^{2} + p$ and $p$ is a rational prime of the form $4 + n^{2}$ , we give appropriate generators of $K$ to obtain the explicit factorization of the ideal $q 𝒪_{K}$ , where $q$ is a positive rational prime. For this, we calculate the index of these generators and integral basis of certain prime ideals.

Ramification in the Coates-Wiles tower.

Rajiv Gupta (1985)

Inventiones mathematicae

Ramification of local fields with imperfect residue fields. II.

Abbes, Ahmed, Saito, Takeshi (2003)

Documenta Mathematica

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