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Search: All articles in the CJM digital archive with keyword $p$-adic

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1. CJM Online first

Gras, Georges
Les $\theta$-régulateurs locaux d'un nombre algébrique -- Conjectures $p$-adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that $\operatorname{Reg}_\infty (\eta) \ne 0$. We define the local $\theta$-regulators $\Delta_p^\theta(\eta) \in \mathbb{F}_p$ for the $\mathbb{Q}_p\,$-irreducible characters $\theta$ of $G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \, V_\theta$ is associated with $\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta \geq 1$. Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$ modulo $p$ in the factorization $\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of $\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]} }$ (normalized $p$-adic regulator). From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal L}^\theta \simeq \delta \, V_\theta\big ) \leq p^{- f \delta^2}$ ($f \geq 1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$ is a $p$-adic unit or that $p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$ with $f=\delta=1$); this obstruction may be lifted assuming the existence of a binomial probability law confirmed through numerical studies (groups $C_3$, $C_5$, $D_6$). This conjecture would imply that, for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational. We recall some deep cohomological results that may strengthen such conjectures.

Keywords:$p$-adic regulators, Leopoldt-Jaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$-ramification, probabilistic number theory
Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40

2. CJM Online first

Dubickas, Arturas; Sha, Min; Shparlinski, Igor
Explicit form of Cassels' $p$-adic embedding theorem for number fields
In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

Keywords:number field, $p$-adic embedding, height, polynomial, cyclotomic field
Categories:11R04, 11S85, 11G50, 11R09, 11R18

3. CJM 2009 (vol 62 pp. 34)

Campbell, Peter S.; Nevins, Monica
Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field
We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.

Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groups
Categories:20G25, 20G05

4. CJM 2008 (vol 60 pp. 1067)

Kariyama, Kazutoshi
On Types for Unramified $p$-Adic Unitary Groups
Let $F$ be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field $F_0$, and let $G$ be a symplectic group over $F$ or an unramified unitary group over $F_0$. Following the methods of Bushnell--Kutzko for $\GL(N,F)$, we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in $G$. In particular, we obtain an irreducible supercuspidal representation of $G$ like $\GL(N,F)$.

Keywords:$p$-adic unitary group, type, supercuspidal representation, Hecke algebra
Categories:22E50, 22D99

5. CJM 2006 (vol 58 pp. 897)

Courtès, François
Distributions invariantes sur les groupes réductifs quasi-déployés
Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\`erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis.

Keywords:reductive $p$-adic groups, orbital integrals, invariant distributions
Categories:22E35, 20G40

6. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field
We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types
Categories:20G25, 22E35, 20H25

7. CJM 2004 (vol 56 pp. 897)

Borwein, Jonathan M.; Borwein, David; Galway, William F.
Finding and Excluding $b$-ary Machin-Type Individual Digit Formulae
Constants with formulae of the form treated by D.~Bailey, P.~Borwein, and S.~Plouffe (\emph{BBP formulae} to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be \emph{normal} numbers, {\em i.e.,} that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call \emph{Machin-type BBP formulae}, for which it is relatively easy to determine whether or not a given constant $\kappa$ has a Machin-type BBP formula. In particular, given $b \in \mathbb{N}$, $b>2$, $b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $\kappa$ is a formula of the form $\kappa = \sum_{m} a_m \arctan(-b^{-m})$, $a_m \in \mathbb{Q}$, while when $b=2$, we also allow terms of the form $a_m \arctan(1/(1-2^m))$. Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b \neq 2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.

Keywords:BBP formulae, Machin-type formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$-adic analysis
Categories:11Y99, 11A51, 11Y50, 11K36, 33B10

8. CJM 2003 (vol 55 pp. 711)

Broughan, Kevin A.
Adic Topologies for the Rational Integers
A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.

Keywords:$p$-adic, metrizable, quasi-valuation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbers
Categories:11B05, 11B25, 11B50, 13J10, 13B35

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