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.
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
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
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.
- Kollár, János (1995), Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, ISBN 978-0-691-04381-4, MR 1341589.
- Solomentsev, E.D. (2001) , "Theta-series", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.