Poincaré series (modular form)
Encyclopedia
In number theory
, a Poincaré series is a mathematical series
generalizing the classical theta series that is associated to any discrete group
of symmetries of a complex domain, possibly of several complex variables
. In particular, they generalize classical Eisenstein series
. They are named after Henri Poincaré
.
If Γ is a finite group acting on a domain D and H(z) is any meromorphic function
on D, then one obtains an automorphic function by averaging over Γ:
However, if Γ is a discrete group
, then additional factors must be introduced in order to assure convergence of such a series. To this end, a Poincaré series is a series of the form
where Jγ is the Jacobian determinant of the group element γ, and the asterisk denotes that the summation takes place only over coset representatives yielding distinct terms in the series.
The classical Poincaré series of weight 2k of a Fuchsian group
Γ is defined by the series
the summation extending over congruence classes of fractional linear transformations
belonging to Γ. Choosing H to be a character of the cyclic group of order n, one obtains the so-called Poincaré series of order n:
The latter Poincaré series converges absolutely and uniformly on compacta (in the upper halfplane), and is a modular form of weight 2k for Γ. Note that, when Γ is the full modular group and n = 0, one obtains the Eisenstein series of weight 2k. In general, the Poincaré series is, for n ≥ 1, a cusp form
.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a Poincaré series is a mathematical series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
generalizing the classical theta series that is associated to any discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
of symmetries of a complex domain, possibly of several complex variables
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
. In particular, they generalize classical Eisenstein series
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...
. They are named after Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
.
If Γ is a finite group acting on a domain D and H(z) is any meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
on D, then one obtains an automorphic function by averaging over Γ:
However, if Γ is a discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
, then additional factors must be introduced in order to assure convergence of such a series. To this end, a Poincaré series is a series of the form
where Jγ is the Jacobian determinant of the group element γ, and the asterisk denotes that the summation takes place only over coset representatives yielding distinct terms in the series.
The classical Poincaré series of weight 2k of a Fuchsian group
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
Γ is defined by the series
the summation extending over congruence classes of fractional linear transformations
belonging to Γ. Choosing H to be a character of the cyclic group of order n, one obtains the so-called Poincaré series of order n:
The latter Poincaré series converges absolutely and uniformly on compacta (in the upper halfplane), and is a modular form of weight 2k for Γ. Note that, when Γ is the full modular group and n = 0, one obtains the Eisenstein series of weight 2k. In general, the Poincaré series is, for n ≥ 1, a cusp form
Cusp form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
.