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:

${\displaystyle \sum _{\gamma \in \Gamma }H(\gamma (z)).}$

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

${\displaystyle \theta _{k}(z)=\sum _{\gamma \in \Gamma ^{*}}(J_{\gamma }(z))^{k}H(\gamma (z))}$

where J_γ is the Jacobian determinant of the group element γ,[1] 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

${\displaystyle \theta _{k}(z)=\sum _{\gamma \in \Gamma ^{*}}(cz+d)^{-2k}H\left({\frac {az+b}{cz+d}}\right)}$

the summation extending over congruence classes of fractional linear transformations

${\displaystyle \gamma ={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$

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:

${\displaystyle \theta _{k,n}(z)=\sum _{\gamma \in \Gamma ^{*}}(cz+d)^{-2k}\exp \left(2\pi in{\frac {az+b}{cz+d}}\right)}$

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