Poincaré series (modular form)

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:
:$\sum_ H(\gamma(z)).$
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
:$\theta_k(z) = \sum_ (J_\gamma(z))^k H(\gamma(z))$
where ''J''_γ is the Jacobian determinant of the group element γ,Or a more general factor of automorphy 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 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 of the cyclic group 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 of weight 2''k'' for Γ. Note that, when Γ is the full modular group and ''n'' = 0, one obtains the Eisenstein series of weight 2''k''. In general, the Poincaré series is, for ''n'' ≥ 1, a cusp form.

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