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.

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 The normalizer of Γ0(N) in PSL(2, ℝ)

1990 ◽  
Vol 32 (3) ◽  
pp. 317-327 ◽  
Author(s):  
M. Akbas ◽  
D. Singerman
Keyword(s):  
Positive Integer ◽  
Modular Group ◽  
Prime Divisors ◽  
Möbius Transformations ◽  
The Matrix ◽  
Image Position ◽  
Mobius Transformations

Let Γ denote the modular group, consisting of the Möbius transformationsAs usual we denote the above transformation by the matrix remembering that V and – V represent the same transformation. If N is a positive integer we let Γ0(N) denote the transformations for which c ≡ 0 mod N. Then Γ0(N) is a subgroup of indexthe product being taken over all prime divisors of N.

scholarly journals Dirichlet and Poincaré series

1985 ◽  
Vol 27 ◽  
pp. 39-56 ◽  
Author(s):  
A. Good
Keyword(s):  
Functional Equation ◽  
Number Theory ◽  
Algebraic Geometry ◽  
Entire Function ◽  
Cusp Form ◽  
Modular Forms ◽  
Modular Group ◽  
Positive Proportion ◽  
Fundamental Discriminant ◽  
Image Position

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet seriesdefines an entire function of s and satisfies the functional equationThe most penetrating statements that have been made on T(n) and LΔ(s)are:Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

scholarly journals On the representation of a number as a sum of squares

1978 ◽  
Vol 19 (2) ◽  
pp. 173-197 ◽  
Author(s):  
Karl-Bernhard Gundlach
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Remainder Term ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Hilbert Modular Group ◽  
Totally Positive ◽  
Hilbert Modular ◽  
Image Position ◽  
Totally Real

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the formwhere Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact thatis a modular form for a certain congruence subgroup of the modular group, andwithwhere Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

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

ARITHMETIC PROPERTIES OF TRACES OF SINGULAR MODULI ON CONGRUENCE SUBGROUPS

2010 ◽  
Vol 06 (08) ◽  
pp. 1755-1768 ◽  
Author(s):  
SOON-YI KANG ◽  
CHANG HEON KIM
Keyword(s):  
Modular Form ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Arithmetic Properties ◽  
Holomorphic Modular Form ◽  
Singular Moduli ◽  
Arbitrary Level

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.

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 Galois action on some ideal section points of the abelian variety associated with a modular form and its application

1983 ◽  
Vol 91 ◽  
pp. 19-36 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Modular Form ◽  
Abelian Variety ◽  
Cusp Form ◽  
Modular Group ◽  
Jacobian Variety ◽  
Modular Curve ◽  
Galois Action ◽  
New Form

For an integer N let X1(N) be the modular curve defined over Q which corresponds to the modular group Γ1(N) To each primitive cusp form f ═ Σ amqm, a1═1, (Γ normalized new form in the sense of [1]) on Γ1(N) of weight 2, there corresponds a factor Jf of the jacobian variety of X1(N) (cf. Shimura [19]).

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].

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