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
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if , and
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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
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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
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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
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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
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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 |
Bers space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bers_space&oldid=46029