Bers space

A complex Banach space of holomorphic automorphic forms introduced by L. Bers (1961). Let be an open set of the Riemann sphere whose boundary consists of more than two points. Then carries a unique complete conformal metric on with curvature , known as the hyperbolic metric on . Let be a properly discontinuous group of conformal mappings of onto itself (cf. also Kleinian group; Conformal mapping). Typical examples of are Kleinian groups (cf. also Kleinian group), that is, a group of Möbius transformations (cf. also Fractional-linear mapping) of acting properly discontinuously on an open set of . By the conformal invariance, the hyperbolic area measure () on is projected to an area measure on the orbit space . In other words, let , , where is the natural projection.

Fix an integer . A holomorphic function on is called an automorphic form of weight for if for all . Then is invariant under the action of and hence may be considered as a function on . The Bers space , where , is the complex Banach space of holomorphic automorphic forms of weight on for such that the function on belongs to the space with respect to the measure . The norm in is thus given by

if , and

if . Automorphic forms in are said to be -integrable if , and bounded if . When is trivial, is abbreviated to . Note that is isometrically embedded as a subspace of .

Some properties of Bers spaces.

1) Let . The Petersson scalar product of and is defined by

If , then the Petersson scalar product establishes an anti-linear isomorphism of onto the dual space of , whose operator norm is between and .

2) The Poincaré (theta-) series of a holomorphic function on is defined by

whenever the right-hand side converges absolutely and uniformly on compact subsets of (cf. Absolutely convergent series; Uniform convergence). Then is an automorphic form of weight on for . Moreover, gives a continuous linear mapping of onto of norm at most . For every there exists an with such that .

3) Let be the set of branch points of the natural projection . Assume that: i) is obtained from a (connected) closed Riemann surface of genus by deleting precisely points; and ii) consists of exactly points (possibly, or ). For each , let be the common multiplicity of at points of . Then for and

where denotes the largest integer that does not exceed .

4) Consider the particular case where is the unit disc. Then is a Fuchsian group and . It had been conjectured that for any , until Ch. Pommerenke [a6] constructed a counterexample. In [a5] D. Niebur and M. Sheingorn characterized the Fuchsian groups for which the inclusion relation holds. In particular, if is finitely generated, then .

5) Let be a Fuchsian group acting on the unit disc . It also preserves , the outside of the unit circle. If is conformal on and can be extended to a quasi-conformal mapping of onto itself such that is a Möbius transformation for each , then its Schwarzian derivative

belongs to with . Moreover, the set of all such Schwarzian derivatives constitutes a bounded domain in including the open ball of radius centred at the origin. This domain can be regarded as a realization of the Teichmüller space of , and the injection of into induced by the Schwarzian derivative is referred to as the Bers embedding.

References

[a1]	I. Kra, "Automorphic forms and Kleinian groups" , Benjamin (1972)
[a2]	J. Lehner, "Discontinuous groups and automorphic functions" , Amer. Math. Soc. (1964)
[a3]	J. Lehner, "Automorphic forms" W.J. Harvey (ed.) , Discrete Groups and Automorphic Functions , Acad. Press (1977) pp. 73–120
[a4]	S. Nag, "The complex analytic theory of Teichmüller spaces" , Wiley (1988)
[a5]	D. Niebur, M. Sheingorn, "Characterization of Fuchsian groups whose integrable forms are bounded" Ann. of Math. , 106 (1977) pp. 239–258
[a6]	Ch. Pommerenke, "On inclusion relations for spaces of automorphic forms" W.E. Kirwan (ed.) L. Zalcman (ed.) , Advances in Complex Function Theory , Lecture Notes in Mathematics , 505 , Springer (1976) pp. 92–100