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].

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 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.

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 $\left( {K,{\cal L}} \right)$.

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 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.

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