scholarly journals Irreducible components of the eigencurve of finite degree are finite over the weight space

2020 ◽  
Vol 2020 (763) ◽  
pp. 251-269
Author(s):  
Shin Hattori ◽  
James Newton
Keyword(s):  
Positive Integer ◽  
Irreducible Component ◽  
Weight Space ◽  
Finite Degree ◽  
Irreducible Components ◽  
Special Cases

AbstractLet p be a rational prime and N a positive integer which is prime to p. Let {\mathcal{W}} be the p-adic weight space for {{\mathrm{GL}}_{2,\mathbb{Q}}}. Let {\mathcal{C}_{N}} be the p-adic Coleman–Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of {\mathcal{C}_{N}} which is of finite degree over {\mathcal{W}} is in fact finite over {\mathcal{W}}.Combined with an argument of Chenevier and a conjecture of Coleman–Mazur–Buzzard–Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

Some Properties of Generalized Euler Numbers

1981 ◽  
Vol 33 (3) ◽  
pp. 606-617 ◽  
Author(s):  
D. J. Leeming ◽  
R. A. Macleod
Keyword(s):  
Positive Integer ◽  
Roots Of Unity ◽  
Euler Numbers ◽  
Special Cases ◽  
Image Position

We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by(1.1)where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is(1.2)Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).

scholarly journals Conjectures and results about parabolic induction of representations of $${\text {GL}}_n(F)$$

2020 ◽  
Vol 222 (3) ◽  
pp. 695-747
Author(s):  
Erez Lapid ◽  
Alberto Mínguez
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Irreducible Representations ◽  
General Linear ◽  
Parabolic Induction ◽  
Geometric Conditions ◽  
Irreducible Components ◽  
Special Cases

Abstract In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schröer condition that occurs in the conjectural characterization of $$\square $$ □ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

scholarly journals PARABOLIC VECTOR BUNDLES AND EQUIVARIANT VECTOR BUNDLES

2002 ◽  
Vol 13 (09) ◽  
pp. 907-957 ◽  
Author(s):  
IGNASI MUNDET I RIERA
Keyword(s):  
Slope Stability ◽  
Irreducible Component ◽  
Complex Manifold ◽  
Vector Bundles ◽  
Full Subcategory ◽  
Natural Numbers ◽  
Normal Crossing ◽  
Irreducible Components

Given a complex manifold X, a normal crossing divisor D ⊂ X whose irreducible components D1, …, Ds are smooth, and a choice of natural numbers [Formula: see text], we construct a manifold [Formula: see text] with an action of a torus Γ and we prove that some full subcategory of the category of Γ-equivariant vector bundles on [Formula: see text] is equivalent to the category of parabolic vector bundles on (X, D) in which the lengths of the filtrations over each irreducible component of D are given by [Formula: see text]. When X is Kaehler, we study the Kaehler cone of [Formula: see text] and the relation between the corresponding notions of slope-stability.

scholarly journals Right weak Engel conditions on linear groups

2019 ◽  
Vol 114 (1) ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Positive Integer ◽  
Linear Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Linear Groups ◽  
Locally Finite ◽  
Class A ◽  
Special Cases ◽  
Prüfer Rank

AbstractIf X is a group-class, a group G is right X-Engel if for all g in G there exists an X-subgroup E of G such that for all x in G there is a positive integer m(x) with [g, nx] ∈ E for all n ≥ m(x). Let G be a linear group. Special cases of our main theorem are the following. If X is the class of all Chernikov groups, or all finite groups, or all locally finite groups, then G is right X-Engel if and only if G has a normal X-subgroup modulo which G is hypercentral. The same conclusion holds if G has positive characteristic and X is one of the following classes; all polycyclic-by-finite groups, all groups of finite Prüfer rank, all minimax groups, all groups with finite Hirsch number, all soluble-by-finite groups with finite abelian total rank. In general the characteristic zero case is more complex.

scholarly journals An Efficient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups

2017 ◽  
Vol 18 (2) ◽  
pp. 0215 ◽  
Author(s):  
Demerson Nunes Gonçalves ◽  
Tharso D Fernandes ◽  
C M M Cosme
Keyword(s):  
Positive Integer ◽  
Quantum Computation ◽  
Prime Number ◽  
Abelian Groups ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Prime Factorization ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem

The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups $\Z_{N}\rtimes\Z_{q^s}$ where $N$ is an integer with a special prime factorization, $q$ prime number and $s$ any positive integer.

Decomposition and Classification of Generic Quantum Markov Semigroups: The Gaussian Gauge Invariant Case

2012 ◽  
Vol 19 (02) ◽  
pp. 1250010 ◽  
Author(s):  
Franco Fagnola ◽  
Skander Hachicha
Keyword(s):  
Irreducible Component ◽  
Discrete System ◽  
Jump Process ◽  
Invariant State ◽  
Markov Semigroups ◽  
Markov Jump ◽  
Gauge Invariant ◽  
Irreducible Components ◽  
Invariant States

We study the structure of generic quantum Markov semigroups, arising from the stochastic limit of a discrete system with generic Hamiltonian interacting with a Gaussian gauge invariant reservoir. We show that they can be essentially written as the sum of their irreducible components determined by closed classes of states of the associated classical Markov jump process. Each irreducible component turns out to be recurrent, transient or have an invariant state if and only if its classical (diagonal) restriction is recurrent, transient or has an invariant state, respectively. We classify invariant states and study convergence towards invariant states as time goes to infinity.

scholarly journals PARALLEL WEIGHT 2 POINTS ON HILBERT MODULAR EIGENVARIETIES AND THE PARITY CONJECTURE

2019 ◽  
Vol 7 ◽  
Author(s):  
CHRISTIAN JOHANSSON ◽  
JAMES NEWTON
Keyword(s):  
Modular Forms ◽  
Weight Space ◽  
Small Radius ◽  
Real Field ◽  
Central Character ◽  
Hilbert Modular Forms ◽  
Irreducible Components ◽  
Hilbert Modular ◽  
Parity Conjecture ◽  
Totally Real

Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension $1+[F:\mathbb{Q}]$ ) cuspidal Hilbert modular eigenvariety has the property that many (all, if $[F:\mathbb{Q}]$ is even) irreducible components contain a classical point with noncritical slopes and parallel weight $2$ (with some character at $p$ whose conductor can be explicitly bounded), or any other algebraic weight.

scholarly journals Some Results on the Sandor-Smarandache Function

2021 ◽  
Vol 13 (1) ◽  
pp. 73-84
Author(s):  
S. M. S. Islam ◽  
A. A. K. Majumdar
Keyword(s):  
Positive Integer ◽  
Simple Form ◽  
Open Problem ◽  
Type Function ◽  
Special Cases

Sandor introduced a new Smarandache-type function, denoted by SS(n), and is called the Sandor-Smarandache function. When n is an odd (positive) integer, then SS(n) has a very simple form, as has been derived by Sandor himself. However, when n is even, then the form of SS(n) is not simple, and remains an open problem. This paper finds SS(n) for some special cases of n. Particular attention is given to values of the general forms SS(2mp), SS(6mp), SS(60mp) and SS(420mp), where m is any (positive) integer and p is an odd prime. Some particular cases have been treated in detail. In Section 4, some remarks are observed.

scholarly journals On the deduction of the class field theory from the general reciprocity of power residues

2000 ◽  
Vol 160 ◽  
pp. 135-142
Author(s):  
Tomio Kubota ◽  
Satomi Oka
Keyword(s):  
Field Theory ◽  
Number Field ◽  
Algebraic Number ◽  
Quadratic Extension ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Finite Degree ◽  
Reciprocity Law ◽  
Special Cases ◽  
Cyclotomic Extension

AbstractWe denote by (A) Artin’s reciprocity law for a general abelian extension of a finite degree over an algebraic number field of a finite degree, and denote two special cases of (A) as follows: by (AC) the assertion (A) where K/F is a cyclotomic extension; by (AK) the assertion (A) where K/F is a Kummer extension. We will show that (A) is derived from (AC) and (AK) only by routine, elementarily algebraic arguments provided that n = (K : F) is odd. If n is even, then some more advanced tools like Proposition 2 are necessary. This proposition is a consequence of Hasse’s norm theorem for a quadratic extension of an algebraic number field, but weaker than the latter.

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