number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, a Poincaré series is a

mathematical series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mat ...

generalizing the classical theta series that is associated to any

discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...

of symmetries of a

complex domain In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

, possibly of

several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex number, complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several ...

. In particular, they generalize classical

Eisenstein series Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be general ...

. They are named after

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

. If Γ is a finite group acting on a domain ''D'' and ''H''(''z'') is any

meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. ...

on ''D'', then one obtains an automorphic function by averaging over Γ: :

\sum_ H(\gamma(z)).

However, if Γ is a

, 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 :

\theta_k(z) = \sum_ (J_\gamma(z))^k H(\gamma(z))

where ''J''_γ is the

Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...

of the group element γ,Or a more general

factor of automorphy In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. Factor ...

as discussed in . 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 2''k'' of a

Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations ...

Γ is defined by the series :

\theta_k(z) = \sum_ (cz+d)^H\left(\frac\right)

the summation extending over congruence classes of fractional linear transformations :

\gamma=\begina&b\\c&d\end

belonging to Γ. Choosing ''H'' to be a

character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

of the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...

of order ''n'', one obtains the so-called Poincaré series of order ''n'': :

\theta_(z) = \sum_ (cz+d)^\exp\left(2\pi i n\frac\right)

The latter Poincaré series converges absolutely and uniformly on compact sets (in the upper halfplane), and is a

modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...

of weight 2''k'' for Γ. Note that, when Γ is the full

modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fraction ...

and ''n'' = 0, one obtains the Eisenstein series of weight 2''k''. In general, the Poincaré series is, for ''n'' ≥ 1, a

cusp form In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. Introduction A cusp form is distinguished in the case of modular forms for the modular g ...

Notes

References

*. *. {{DEFAULTSORT:Poincare series (modular form) Automorphic forms Modular forms Mathematical series