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 &Gamma; 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 &Gamma;:
 * $$\sum_{\gamma\in\Gamma} H(\gamma(z)).$$

However, if &Gamma; 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
 * $$\theta_k(z) = \sum_{\gamma\in\Gamma^*} (J_\gamma(z))^k H(\gamma(z))$$

where J&gamma; is the Jacobian determinant of the group element &gamma;, 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 &Gamma; is defined by the series
 * $$\theta_k(z) = \sum_{\gamma\in\Gamma^*} (cz+d)^{-2k}H\left(\frac{az+b}{cz+d}\right)$$

the summation extending over congruence classes of fractional linear transformations
 * $$\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$

belonging to &Gamma;. Choosing H to be a character of the cyclic group of order n, one obtains the so-called Poincaré series of order n:
 * $$\theta_{k,n}(z) = \sum_{\gamma\in\Gamma^*} (cz+d)^{-2k}\exp\left(2\pi i n\frac{az+b}{cz+d}\right)$$

The latter Poincaré series converges absolutely and uniformly on compact sets (in the upper halfplane), and is a modular form of weight 2k for &Gamma;. Note that, when &Gamma; 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.