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A complex [[Banach space|Banach space]] of holomorphic automorphic forms introduced by L. Bers (1961). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103801.png" /> be an open set of the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103802.png" /> whose boundary consists of more than two points. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103803.png" /> carries a unique complete conformal metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103804.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103805.png" /> with curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103806.png" />, known as the hyperbolic metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103807.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103808.png" /> be a properly discontinuous group of conformal mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b1103809.png" /> onto itself (cf. also [[Kleinian group|Kleinian group]]; [[Conformal mapping|Conformal mapping]]). Typical examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038010.png" /> are Kleinian groups (cf. also [[Kleinian group|Kleinian group]]), that is, a group of Möbius transformations (cf. also [[Fractional-linear mapping|Fractional-linear mapping]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038011.png" /> acting properly discontinuously on an open set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038012.png" />. By the conformal invariance, the hyperbolic area measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038014.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038015.png" /> is projected to an area measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038016.png" /> on the orbit space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038017.png" />. In other words, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038020.png" /> is the natural projection.
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Fix an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038021.png" />. A [[Holomorphic function|holomorphic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038023.png" /> is called an automorphic form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038028.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038029.png" /> is invariant under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038030.png" /> and hence may be considered as a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038031.png" />. The Bers space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038033.png" />, is the complex Banach space of holomorphic automorphic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038034.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038036.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038037.png" /> such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038039.png" /> belongs to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038040.png" /> with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038041.png" />. The norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038042.png" /> is thus given by
+
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 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038043.png" /></td> </tr></table>
+
A complex [[Banach space|Banach space]] of holomorphic automorphic forms introduced by L. Bers (1961). Let  $  D $
 +
be an open set of the Riemann sphere  $  {\widehat{\mathbf C}  } = \mathbf C \cup \{ \infty \} $
 +
whose boundary consists of more than two points. Then  $  D $
 +
carries a unique complete conformal metric  $  \lambda ( z )  | {dz } | $
 +
on  $  D $
 +
with curvature  $  - 4 $,
 +
known as the hyperbolic metric on  $  D $.
 +
Let  $  G $
 +
be a properly discontinuous group of conformal mappings of  $  D $
 +
onto itself (cf. also [[Kleinian group|Kleinian group]]; [[Conformal mapping|Conformal mapping]]). Typical examples of  $  G $
 +
are Kleinian groups (cf. also [[Kleinian group|Kleinian group]]), that is, a group of Möbius transformations (cf. also [[Fractional-linear mapping|Fractional-linear mapping]]) of  $  {\widehat{\mathbf C}  } $
 +
acting properly discontinuously on an open set of  $  {\widehat{\mathbf C}  } $.
 +
By the conformal invariance, the hyperbolic area measure  $  \lambda ( z )  ^ {2}  dx  dy $(
 +
$  z = x + iy $)
 +
on  $  D $
 +
is projected to an area measure  $  d \mu $
 +
on the orbit space  $  D/G $.
 +
In other words, let  $  d \mu ( w ) = \lambda ( z )  ^ {2}  dx  dy $,
 +
$  w = \pi ( z ) $,
 +
where  $  \pi : D \rightarrow {D/G } $
 +
is the natural projection.
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038044.png" />, and
+
Fix an integer  $  q \geq 2 $.
 +
A [[Holomorphic function|holomorphic function]]  $  \varphi $
 +
on  $  D $
 +
is called an automorphic form of weight  $  - 2q $
 +
for  $  G $
 +
if $  ( \varphi \circ g ) \cdot ( g  ^  \prime  )  ^ {q} = \varphi $
 +
for all  $  g \in G $.
 +
Then  $  \lambda ^ {- q } | \varphi | $
 +
is invariant under the action of  $  G $
 +
and hence may be considered as a function on  $  D/G $.  
 +
The Bers space  $  A _ {q}  ^ {p} ( D,G ) $,
 +
where  $  1 \leq  p \leq  \infty $,
 +
is the complex Banach space of holomorphic automorphic forms  $  \varphi $
 +
of weight  $  - 2q $
 +
on  $  D $
 +
for  $  G $
 +
such that the function  $  \lambda ^ {- q } | \varphi | $
 +
on  $  D/G $
 +
belongs to the space  $  L _ {p} $
 +
with respect to the measure  $  \mu $.  
 +
The norm in  $  A _ {q}  ^ {p} ( D,G ) $
 +
is thus given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038045.png" /></td> </tr></table>
+
$$
 +
\left \| \varphi \right \| = ( {\int\limits \int\limits } _ {D/G }  {\lambda ^ {- pq } \left | \varphi \right |  ^ {p} }  {d \mu } ) ^ {1/p }
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038046.png" />. Automorphic forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038047.png" /> are said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038049.png" />-integrable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038050.png" />, and bounded if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038051.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038052.png" /> is trivial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038053.png" /> is abbreviated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038054.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038055.png" /> is isometrically embedded as a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038056.png" />.
+
if $  1 \leq  p < \infty $,
 +
and
 +
 
 +
$$
 +
\left \| \varphi \right \| = \sup  _ {D/G } \lambda ^ {- q } \left | \varphi \right |
 +
$$
 +
 
 +
if  $  p = \infty $.  
 +
Automorphic forms in $  A _ {q}  ^ {p} ( D,G ) $
 +
are said to be $  p $-
 +
integrable if $  1 \leq  p < \infty $,  
 +
and bounded if $  p = \infty $.  
 +
When $  G $
 +
is trivial, $  A _ {q}  ^ {p} ( D,G ) $
 +
is abbreviated to $  A _ {q}  ^ {p} ( D ) $.  
 +
Note that $  A _ {q}  ^  \infty  ( D,G ) $
 +
is isometrically embedded as a subspace of $  A _ {q}  ^  \infty  ( D ) $.
  
 
==Some properties of Bers spaces.==
 
==Some properties of Bers spaces.==
  
 +
1) Let  $  {1 / p } + {1 / {p  ^  \prime  } } = 1 $.
 +
The Petersson scalar product of  $  \varphi \in A _ {q}  ^ {p} ( D,G ) $
 +
and  $  \psi \in A _ {q} ^ {p  ^  \prime  } ( D,G ) $
 +
is defined by
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038057.png" />. The Petersson scalar product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038059.png" /> is defined by
+
$$
 
+
\left ( \varphi , \psi \right ) = {\int\limits \int\limits } _ {D/G }  {\lambda ^ {- 2q } \varphi {\overline \psi \; } }  {d \mu } .
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038060.png" /></td> </tr></table>
+
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038061.png" />, then the Petersson scalar product establishes an anti-linear isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038062.png" /> onto the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038063.png" />, whose operator norm is between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038065.png" />.
+
If $  1 \leq  p < \infty $,  
 +
then the Petersson scalar product establishes an anti-linear isomorphism of $  A _ {q} ^ {p  ^  \prime  } ( D,G ) $
 +
onto the dual space of $  A _ {q}  ^ {p} ( D,G ) $,  
 +
whose operator norm is between $  ( q - 1 ) ( 2q - 1 ) ^ {- 1 } $
 +
and $  1 $.
  
2) The Poincaré (theta-) series of a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038066.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038067.png" /> is defined by
+
2) The Poincaré (theta-) series of a holomorphic function $  f $
 +
on $  D $
 +
is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038068.png" /></td> </tr></table>
+
$$
 +
\Theta f = \sum _ {g \in G } ( f \circ g ) \cdot ( g  ^  \prime  )  ^ {q}
 +
$$
  
whenever the right-hand side converges absolutely and uniformly on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038069.png" /> (cf. [[Absolutely convergent series|Absolutely convergent series]]; [[Uniform convergence|Uniform convergence]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038070.png" /> is an automorphic form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038071.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038072.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038073.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038074.png" /> gives a continuous linear mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038075.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038076.png" /> of norm at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038077.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038078.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038079.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038080.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038081.png" />.
+
whenever the right-hand side converges absolutely and uniformly on compact subsets of $  D $(
 +
cf. [[Absolutely convergent series|Absolutely convergent series]]; [[Uniform convergence|Uniform convergence]]). Then $  \Theta f $
 +
is an automorphic form of weight $  - 2q $
 +
on $  D $
 +
for $  G $.  
 +
Moreover, $  \Theta $
 +
gives a continuous linear mapping of $  A _ {q}  ^ {1} ( D ) $
 +
onto $  A _ {q}  ^ {1} ( D,G ) $
 +
of norm at most $  1 $.  
 +
For every $  \varphi \in A _ {q}  ^ {p} ( D,G ) $
 +
there exists an $  f \in A _ {q}  ^ {p} ( D ) $
 +
with $  \| f \| \leq  ( 2q - 1 ) ( q - 1 ) ^ {- 1 } \| \varphi \| $
 +
such that $  \varphi = \Theta f $.
  
3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038082.png" /> be the set of branch points of the natural projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038083.png" />. Assume that: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038084.png" /> is obtained from a (connected) closed [[Riemann surface|Riemann surface]] of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038085.png" /> by deleting precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038086.png" /> points; and ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038087.png" /> consists of exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038088.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038089.png" /> (possibly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038090.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038091.png" />). For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038092.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038093.png" /> be the common multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038094.png" /> at points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038095.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038096.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038097.png" /> and
+
3) Let $  B $
 +
be the set of branch points of the natural projection $  \pi $.  
 +
Assume that: i) $  D/G $
 +
is obtained from a (connected) closed [[Riemann surface|Riemann surface]] of genus $  g $
 +
by deleting precisely $  m $
 +
points; and ii) $  \pi ( B ) $
 +
consists of exactly $  n $
 +
points $  p _ {1} \dots p _ {n} $(
 +
possibly, $  m = 0 $
 +
or $  n =0 $).  
 +
For each $  k = 1 \dots n $,  
 +
let $  \nu _ {k} $
 +
be the common multiplicity of $  \pi $
 +
at points of $  \pi ^ {- 1 } ( p _ {k} ) $.  
 +
Then $  A _ {q}  ^ {p} ( D,G ) = A _ {q}  ^  \infty  ( D,G ) $
 +
for $  1 \leq  p \leq  \infty $
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038098.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm dim} } A _ {q}  ^  \infty  ( D,G ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b11038099.png" /></td> </tr></table>
+
$$
 +
=  
 +
( 2q - 1 ) ( g - 1 ) + ( q - 1 ) m + \sum _ {k =1 } ^ { n }  \left [ q \left ( 1 - {
 +
\frac{1}{\nu _ {k} }
 +
} \right ) \right ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380100.png" /> denotes the largest integer that does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380101.png" />.
+
where $  [ x ] $
 +
denotes the largest integer that does not exceed $  x $.
  
4) Consider the particular case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380102.png" /> is the unit disc. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380103.png" /> is a [[Fuchsian group|Fuchsian group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380104.png" />. It had been conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380105.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380106.png" />, until Ch. Pommerenke [[#References|[a6]]] constructed a counterexample. In [[#References|[a5]]] D. Niebur and M. Sheingorn characterized the Fuchsian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380107.png" /> for which the inclusion relation holds. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380108.png" /> is finitely generated, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380109.png" />.
+
4) Consider the particular case where $  D $
 +
is the unit disc. Then $  G $
 +
is a [[Fuchsian group|Fuchsian group]] and $  \lambda ( z ) = ( 1 - | z |  ^ {2} ) ^ {- 1 } $.  
 +
It had been conjectured that $  A _ {q}  ^ {1} ( D,G ) \subset  A _ {q}  ^  \infty  ( D,G ) $
 +
for any $  G $,  
 +
until Ch. Pommerenke [[#References|[a6]]] constructed a counterexample. In [[#References|[a5]]] D. Niebur and M. Sheingorn characterized the Fuchsian groups $  G $
 +
for which the inclusion relation holds. In particular, if $  G $
 +
is finitely generated, then $  A _ {q}  ^ {1} ( D,G ) \subset  A _ {q}  ^  \infty  ( D,G ) $.
  
5) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380110.png" /> be a Fuchsian group acting on the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380111.png" />. It also preserves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380112.png" />, the outside of the unit circle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380113.png" /> is conformal on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380114.png" /> and can be extended to a [[Quasi-conformal mapping|quasi-conformal mapping]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380115.png" /> onto itself such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380116.png" /> is a Möbius transformation for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380117.png" />, then its [[Schwarzian derivative|Schwarzian derivative]]
+
5) Let $  G $
 +
be a Fuchsian group acting on the unit disc $  D $.  
 +
It also preserves $  D  ^ {*} = {\widehat{\mathbf C}  } \setminus  {\overline{D}\; } $,  
 +
the outside of the unit circle. If $  f $
 +
is conformal on $  D  ^ {*} $
 +
and can be extended to a [[Quasi-conformal mapping|quasi-conformal mapping]] of $  {\widehat{\mathbf C}  } $
 +
onto itself such that $  f \circ g \circ f ^ {- 1 } $
 +
is a Möbius transformation for each $  g \in G $,  
 +
then its [[Schwarzian derivative|Schwarzian derivative]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380118.png" /></td> </tr></table>
+
$$
 +
Sf = {
 +
\frac{f ^ {\prime \prime \prime } }{f  ^  \prime  }
 +
} - {
 +
\frac{3}{2}
 +
} \left ( {
 +
\frac{f ^ {\prime \prime } }{f  ^  \prime  }
 +
} \right )  ^ {2}
 +
$$
  
belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380119.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380120.png" />. Moreover, the set of all such Schwarzian derivatives constitutes a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380121.png" /> including the open ball of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380122.png" /> centred at the origin. This domain can be regarded as a realization of the [[Teichmüller space|Teichmüller space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380123.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380124.png" />, and the injection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380125.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110380/b110380126.png" /> induced by the Schwarzian derivative is referred to as the Bers embedding.
+
belongs to $  A _ {2}  ^  \infty  ( D  ^ {*} ,G ) $
 +
with $  \| {Sf } \| \leq  6 $.  
 +
Moreover, the set of all such Schwarzian derivatives constitutes a bounded domain in $  A _ {2}  ^  \infty  ( D  ^ {*} ,G ) $
 +
including the open ball of radius $  2 $
 +
centred at the origin. This domain can be regarded as a realization of the [[Teichmüller space|Teichmüller space]] $  T ( G ) $
 +
of $  G $,  
 +
and the injection of $  T ( G ) $
 +
into $  A _ {2}  ^  \infty  ( D  ^ {*} ,G ) $
 +
induced by the Schwarzian derivative is referred to as the Bers embedding.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kra,  "Automorphic forms and Kleinian groups" , Benjamin  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lehner,  "Discontinuous groups and automorphic functions" , Amer. Math. Soc.  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Lehner,  "Automorphic forms"  W.J. Harvey (ed.) , ''Discrete Groups and Automorphic Functions'' , Acad. Press  (1977)  pp. 73–120</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Nag,  "The complex analytic theory of Teichmüller spaces" , Wiley  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Niebur,  M. Sheingorn,  "Characterization of Fuchsian groups whose integrable forms are bounded"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 239–258</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Kra,  "Automorphic forms and Kleinian groups" , Benjamin  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lehner,  "Discontinuous groups and automorphic functions" , Amer. Math. Soc.  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Lehner,  "Automorphic forms"  W.J. Harvey (ed.) , ''Discrete Groups and Automorphic Functions'' , Acad. Press  (1977)  pp. 73–120</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Nag,  "The complex analytic theory of Teichmüller spaces" , Wiley  (1988)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Niebur,  M. Sheingorn,  "Characterization of Fuchsian groups whose integrable forms are bounded"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 239–258</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR></table>

Latest revision as of 10:58, 29 May 2020


A complex Banach space of holomorphic automorphic forms introduced by L. Bers (1961). Let $ D $ be an open set of the Riemann sphere $ {\widehat{\mathbf C} } = \mathbf C \cup \{ \infty \} $ whose boundary consists of more than two points. Then $ D $ carries a unique complete conformal metric $ \lambda ( z ) | {dz } | $ on $ D $ with curvature $ - 4 $, known as the hyperbolic metric on $ D $. Let $ G $ be a properly discontinuous group of conformal mappings of $ D $ onto itself (cf. also Kleinian group; Conformal mapping). Typical examples of $ G $ are Kleinian groups (cf. also Kleinian group), that is, a group of Möbius transformations (cf. also Fractional-linear mapping) of $ {\widehat{\mathbf C} } $ acting properly discontinuously on an open set of $ {\widehat{\mathbf C} } $. By the conformal invariance, the hyperbolic area measure $ \lambda ( z ) ^ {2} dx dy $( $ z = x + iy $) on $ D $ is projected to an area measure $ d \mu $ on the orbit space $ D/G $. In other words, let $ d \mu ( w ) = \lambda ( z ) ^ {2} dx dy $, $ w = \pi ( z ) $, where $ \pi : D \rightarrow {D/G } $ is the natural projection.

Fix an integer $ q \geq 2 $. A holomorphic function $ \varphi $ on $ D $ is called an automorphic form of weight $ - 2q $ for $ G $ if $ ( \varphi \circ g ) \cdot ( g ^ \prime ) ^ {q} = \varphi $ for all $ g \in G $. Then $ \lambda ^ {- q } | \varphi | $ is invariant under the action of $ G $ and hence may be considered as a function on $ D/G $. The Bers space $ A _ {q} ^ {p} ( D,G ) $, where $ 1 \leq p \leq \infty $, is the complex Banach space of holomorphic automorphic forms $ \varphi $ of weight $ - 2q $ on $ D $ for $ G $ such that the function $ \lambda ^ {- q } | \varphi | $ on $ D/G $ belongs to the space $ L _ {p} $ with respect to the measure $ \mu $. The norm in $ A _ {q} ^ {p} ( D,G ) $ is thus given by

$$ \left \| \varphi \right \| = ( {\int\limits \int\limits } _ {D/G } {\lambda ^ {- pq } \left | \varphi \right | ^ {p} } {d \mu } ) ^ {1/p } $$

if $ 1 \leq p < \infty $, and

$$ \left \| \varphi \right \| = \sup _ {D/G } \lambda ^ {- q } \left | \varphi \right | $$

if $ p = \infty $. Automorphic forms in $ A _ {q} ^ {p} ( D,G ) $ are said to be $ p $- integrable if $ 1 \leq p < \infty $, and bounded if $ p = \infty $. When $ G $ is trivial, $ A _ {q} ^ {p} ( D,G ) $ is abbreviated to $ A _ {q} ^ {p} ( D ) $. Note that $ A _ {q} ^ \infty ( D,G ) $ is isometrically embedded as a subspace of $ A _ {q} ^ \infty ( D ) $.

Some properties of Bers spaces.

1) Let $ {1 / p } + {1 / {p ^ \prime } } = 1 $. The Petersson scalar product of $ \varphi \in A _ {q} ^ {p} ( D,G ) $ and $ \psi \in A _ {q} ^ {p ^ \prime } ( D,G ) $ is defined by

$$ \left ( \varphi , \psi \right ) = {\int\limits \int\limits } _ {D/G } {\lambda ^ {- 2q } \varphi {\overline \psi \; } } {d \mu } . $$

If $ 1 \leq p < \infty $, then the Petersson scalar product establishes an anti-linear isomorphism of $ A _ {q} ^ {p ^ \prime } ( D,G ) $ onto the dual space of $ A _ {q} ^ {p} ( D,G ) $, whose operator norm is between $ ( q - 1 ) ( 2q - 1 ) ^ {- 1 } $ and $ 1 $.

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

$$ \Theta f = \sum _ {g \in G } ( f \circ g ) \cdot ( g ^ \prime ) ^ {q} $$

whenever the right-hand side converges absolutely and uniformly on compact subsets of $ D $( cf. Absolutely convergent series; Uniform convergence). Then $ \Theta f $ is an automorphic form of weight $ - 2q $ on $ D $ for $ G $. Moreover, $ \Theta $ gives a continuous linear mapping of $ A _ {q} ^ {1} ( D ) $ onto $ A _ {q} ^ {1} ( D,G ) $ of norm at most $ 1 $. For every $ \varphi \in A _ {q} ^ {p} ( D,G ) $ there exists an $ f \in A _ {q} ^ {p} ( D ) $ with $ \| f \| \leq ( 2q - 1 ) ( q - 1 ) ^ {- 1 } \| \varphi \| $ such that $ \varphi = \Theta f $.

3) Let $ B $ be the set of branch points of the natural projection $ \pi $. Assume that: i) $ D/G $ is obtained from a (connected) closed Riemann surface of genus $ g $ by deleting precisely $ m $ points; and ii) $ \pi ( B ) $ consists of exactly $ n $ points $ p _ {1} \dots p _ {n} $( possibly, $ m = 0 $ or $ n =0 $). For each $ k = 1 \dots n $, let $ \nu _ {k} $ be the common multiplicity of $ \pi $ at points of $ \pi ^ {- 1 } ( p _ {k} ) $. Then $ A _ {q} ^ {p} ( D,G ) = A _ {q} ^ \infty ( D,G ) $ for $ 1 \leq p \leq \infty $ and

$$ { \mathop{\rm dim} } A _ {q} ^ \infty ( D,G ) = $$

$$ = ( 2q - 1 ) ( g - 1 ) + ( q - 1 ) m + \sum _ {k =1 } ^ { n } \left [ q \left ( 1 - { \frac{1}{\nu _ {k} } } \right ) \right ] , $$

where $ [ x ] $ denotes the largest integer that does not exceed $ x $.

4) Consider the particular case where $ D $ is the unit disc. Then $ G $ is a Fuchsian group and $ \lambda ( z ) = ( 1 - | z | ^ {2} ) ^ {- 1 } $. It had been conjectured that $ A _ {q} ^ {1} ( D,G ) \subset A _ {q} ^ \infty ( D,G ) $ for any $ G $, until Ch. Pommerenke [a6] constructed a counterexample. In [a5] D. Niebur and M. Sheingorn characterized the Fuchsian groups $ G $ for which the inclusion relation holds. In particular, if $ G $ is finitely generated, then $ A _ {q} ^ {1} ( D,G ) \subset A _ {q} ^ \infty ( D,G ) $.

5) Let $ G $ be a Fuchsian group acting on the unit disc $ D $. It also preserves $ D ^ {*} = {\widehat{\mathbf C} } \setminus {\overline{D}\; } $, the outside of the unit circle. If $ f $ is conformal on $ D ^ {*} $ and can be extended to a quasi-conformal mapping of $ {\widehat{\mathbf C} } $ onto itself such that $ f \circ g \circ f ^ {- 1 } $ is a Möbius transformation for each $ g \in G $, then its Schwarzian derivative

$$ Sf = { \frac{f ^ {\prime \prime \prime } }{f ^ \prime } } - { \frac{3}{2} } \left ( { \frac{f ^ {\prime \prime } }{f ^ \prime } } \right ) ^ {2} $$

belongs to $ A _ {2} ^ \infty ( D ^ {*} ,G ) $ with $ \| {Sf } \| \leq 6 $. Moreover, the set of all such Schwarzian derivatives constitutes a bounded domain in $ A _ {2} ^ \infty ( D ^ {*} ,G ) $ including the open ball of radius $ 2 $ centred at the origin. This domain can be regarded as a realization of the Teichmüller space $ T ( G ) $ of $ G $, and the injection of $ T ( G ) $ into $ A _ {2} ^ \infty ( D ^ {*} ,G ) $ 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
How to Cite This Entry:
Bers space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bers_space&oldid=46029
This article was adapted from an original article by M. Masumoto (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article