scholarly journals On the non-vanishing of Poincaré series

1989 ◽  
Vol 32 (1) ◽  
pp. 131-137 ◽  
Author(s):  
C. J. Mozzochi
Keyword(s):  
Fuchsian Group ◽  
Poincaré Series ◽  
Matrix Group ◽  
Poincare Series ◽  
Congruence Group ◽  
Image Position

R. A. Rankin [2] and J. Lehner [1] considered the non-vanishing of Poincaré series for the classical modular matrix group and for an arbitrary fuchsian group, respectively.In this paper we consider the non-vanishing of Poincaré series for the congruence group

scholarly journals On the non-vanishing of Poincaré series

1980 ◽  
Vol 23 (2) ◽  
pp. 225-228 ◽  
Author(s):  
J. Lehner
Keyword(s):  
Complete System ◽  
Poincaré Series ◽  
Matrix Group ◽  
Poincare Series ◽  
Image Position

Let M = SL(2, Z) be the classical modular matrix group. One form of the Poincaré series on M ishere z ∈ H={z = x + iy: y >0}, q ≧ 2 and m ≧ 1 are integers, and the summation is over a complete system of matrices (ab: cd) in M with different lower row. The problem of the identical vanishing of the Poincaré series for different values of m and q goes back to Poincaré.

The Poincaré Series Of Stretched Cohen-Macaulay Rings

1980 ◽  
Vol 32 (5) ◽  
pp. 1261-1265 ◽  
Author(s):  
Judith D. Sally
Keyword(s):  
Local Ring ◽  
Poincaré Series ◽  
Local Rings ◽  
Embedding Dimension ◽  
Loop Spaces ◽  
Gorenstein Rings ◽  
Poincare Series ◽  
Image Position

There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré serieswhere k = R/m, has been settled. (For an example of a local ring with non-rational Poincaré series see the recent paper by D. Anick, “Construction of loop spaces and local rings whose Poincaré—Betti series are nonrational”, C. R. Acad. Sc. Paris 290 (1980), 729-732.) In this note, we compute the Poincaré series of a certain family of local Cohen-Macaulay rings and obtain, as a corollary, the rationality of the Poincaré series of d-dimensional local Gorenstein rings (R, m) of embedding dimension at least e + d – 3, where e is the multiplicity of R. It follows that local Gorenstein rings of multiplicity at most five have rational Poincaré series.

scholarly journals INTEGRATION AND CELL DECOMPOSITION IN P-MINIMAL STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 1124-1141 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
EVA LEENKNEGT
Keyword(s):  
Cell Decomposition ◽  
Poincaré Series ◽  
Weak Version ◽  
Poincare Series ◽  
Skolem Functions ◽  
Direct Corollary ◽  
Image Position ◽  
And Function ◽  
Constructible Functions

AbstractWe show that the class of ${\cal L}$-constructible functions is closed under integration for any P-minimal expansion of a p-adic field $\left( {K,{\cal L}} \right)$. This generalizes results previously known for semi-algebraic and subanalytic structures. As part of the proof, we obtain a weak version of cell decomposition and function preparation for P-minimal structures, a result which is independent of the existence of Skolem functions. A direct corollary is that Denef’s results on the rationality of Poincaré series hold in any P-minimal expansion of a p-adic field $\left( {K,{\cal L}} \right)$.

The Poincaré series of $\mathbb C\setminus\mathbb Z$

1999 ◽  
Vol 19 (1) ◽  
pp. 1-20 ◽  
Author(s):  
JON AARONSON ◽  
MANFRED DENKER
Keyword(s):  
Random Walk ◽  
Geodesic Flow ◽  
Fuchsian Group ◽  
Poincaré Series ◽  
Poincare Series ◽  
Asymptotic Type

We show that the Poincaré series of the Fuchsian group of deck transformations of ${\mathbb C}\setminus{\mathbb Z}$ diverges logarithmically. This is because ${\mathbb C}\setminus{\mathbb Z}$ is a ${\mathbb Z}$-cover of the three horned sphere, whence its geodesic flow has a good section which behaves like a random walk on ${\mathbb R}$ with Cauchy distributed jump distribution and has logarithmic asymptotic type.

scholarly journals Kleinian groups with unbounded limit sets

1972 ◽  
Vol 13 (1) ◽  
pp. 24-28
Author(s):  
A. F. Beardon
Keyword(s):  
Limit Point ◽  
Automorphic Forms ◽  
Poincaré Series ◽  
Kleinian Groups ◽  
Limit Sets ◽  
Poincare Series ◽  
Crucial Step ◽  
Ordinary Point ◽  
Image Position ◽  
Automorphic Functions

The easiest way to construct automorphic functions is by means of the Poincaré series. If G is a Kleinian group with ∞ an ordinary point of G and if k ≧ 4, thenwhere Vz=(az+b)/(cz+d) and ad-bc=1. The convergence of this series is the crucial step in showing that the Poincaré series converges and is an automorphic form on G If ∞ is a limit point of ∞ the series in (1) may diverge and one can derive automorphic forms on ∞ from the Poincaré series of some conjugate group. These constructions are described in greater detail in /3, pp. 155–165].

scholarly journals The vanishing of Poincaré series

1980 ◽  
Vol 23 (2) ◽  
pp. 151-161 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Modular Form ◽  
Linear Combination ◽  
Cusp Form ◽  
Modular Group ◽  
Poincaré Series ◽  
Complex Field ◽  
Poincare Series ◽  
Holomorphic Modular Form ◽  
Image Position

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series.Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} andThe summation is over all matriceswith different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.

Notions De Base Pour L'Arthmetique De Fq((1/t))

1988 ◽  
Vol 40 (04) ◽  
pp. 817-832 ◽  
Author(s):  
Yves Hellegouarch
Keyword(s):  
Poincaré Series ◽  
Poincare Series ◽  
Image Position

Le but de ce travail est de définir un certain nombre d'objets de F q ((1/t) et de fonctions qui sont analogues à des objets classiques de R(ou C) et à des fonctions classiques R → R (ou C → C) : par exemple ζ(l), ζ(2), etc., fractions continues, domaine fondamental du “demi-plan” de Poincaré, séries d'Eisenstein, fonctions multiplement périodiques. Un tel travail avait été entrepris autrefois par E. Artin [2], mais dans un autre esprit, et la fonction zéta que j'ai définie dans [7] n'est pas celle d'Artin, mais celle de Carlitz [3]. L'intersection de la thèse d'Artin et de ce travail n'est donc pas vide, mais elle n'est pas non plus très étendue bien que je me sois limité aux définitions et aux propriétés les plus élémentaires. En revanche, par son esprit, ce travail se rapproche davantage du point de vue de Carlitz.

scholarly journals Poincaré series, 3d gravity and averages of rational CFT

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Viraj Meruliya ◽  
Sunil Mukhi ◽  
Palash Singh
Keyword(s):  
Poincaré Series ◽  
Simons Theory ◽  
Partition Functions ◽  
Moody Algebra ◽  
Modular Invariant ◽  
Poincare Series ◽  
Single Genus ◽  
3D Gravity ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

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