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In [[#References|[a2]]], M. Eichler conceived the  "Eichler cohomology theory"  (but not the designation) while studying  "generalized Abelian integrals"  (now called  "Eichler integrals" ; see below).
 
In [[#References|[a2]]], M. Eichler conceived the  "Eichler cohomology theory"  (but not the designation) while studying  "generalized Abelian integrals"  (now called  "Eichler integrals" ; see below).
  
The setting for this theory is that of automorphic forms, with multiplier system, on a discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200701.png" /> of fractional-linear transformations (equivalently, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200702.png" />-matrices; cf. also [[Automorphic form|Automorphic form]]; [[Fractional-linear mapping|Fractional-linear mapping]]). One may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200703.png" /> consists of real fractional-linear transformations, that is, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200704.png" /> fixes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200705.png" />, the upper half-plane. A fundamental region, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200706.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200707.png" /> is required to have finite hyperbolic area; this is equivalent to the two conditions (taken jointly):
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The setting for this theory is that of automorphic forms, with multiplier system, on a discrete group $\Gamma$ of fractional-linear transformations (equivalently, of $( 2 \times 2 )$-matrices; cf. also [[Automorphic form|Automorphic form]]; [[Fractional-linear mapping|Fractional-linear mapping]]). One may assume that $\Gamma$ consists of real fractional-linear transformations, that is, that $\Gamma$ fixes $\mathcal{H}$, the upper half-plane. A fundamental region, $\mathcal{R}$, of $\Gamma$ is required to have finite hyperbolic area; this is equivalent to the two conditions (taken jointly):
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200708.png" /> is finitely generated;
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i) $\Gamma$ is finitely generated;
  
ii) each real point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e1200709.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007010.png" /> is a parabolic point (a cusp) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007011.png" />, that is, it is fixed by a cyclic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007013.png" /> with parabolic generator.
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ii) each real point $q$ of $\overline { \mathcal{R} }$ is a parabolic point (a cusp) of $\Gamma$, that is, it is fixed by a cyclic subgroup $\Gamma _ { q }$ of $\Gamma$ with parabolic generator.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007014.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007015.png" /> be a multiplier system in weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007016.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007017.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007018.png" /> is integral, this means simply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007021.png" /> is multiplicative:
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Let $k \in \mathbf{Z}$ and let $\mathbf{v}$ be a multiplier system in weight $k$ with respect to $\Gamma$. Since $k$ is integral, this means simply that $|\mathbf{v} ( M ) | = 1$ for all $M \in \Gamma$, and $\mathbf{v}$ is multiplicative:
  
<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/e/e120/e120070/e12007022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \mathbf{v} ( M _ { 1 } , M _ { 2 } ) = \mathbf{v} ( M _ { 1 } ) \mathbf{v} ( M _ { 2 } ) , M _ { 1 } , M _ { 2 } \in \Gamma. \end{equation}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007024.png" /> a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007025.png" />, define the slash operator
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For $M \in \Gamma$ and $\varphi$ a function on $\mathcal{H}$, define the slash operator
  
<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/e/e120/e120070/e12007026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} ( \varphi | _ { k } ^ { \mathbf{v} } M ) ( z ) = {\bf v} ( M ) ( cz + d ) ^ { - k } \varphi ( M z ). \end{equation}
  
In this notation, the characteristic transformation law satisfied by an automorphic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007028.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007029.png" /> and multiplier system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007030.png" /> can be written
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In this notation, the characteristic transformation law satisfied by an automorphic form $f$ on $\Gamma$ of weight $k$ and multiplier system $\mathbf{v}$ can be written
  
<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/e/e120/e120070/e12007031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} f | _ { k } ^ { \mathbf{v} } M = f , \forall M \in \Gamma. \end{equation}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007032.png" /> denote the vector space of automorphic forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007033.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007034.png" /> and multiplier system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007035.png" />, the collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007036.png" /> satisfying (a3) and that are holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007037.png" /> and meromorphic at each parabolic cusp of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007038.png" /> (in the usual local variable, cf. also [[Analytic function|Analytic function]]; [[Meromorphic function|Meromorphic function]]). One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007039.png" /> is an entire automorphic form if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007040.png" /> is holomorphic at each parabolic cusp. An entire automorphic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007041.png" /> is called a cusp form if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007042.png" /> vanishes at each parabolic cusp. As usual, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007043.png" /> denotes the space of entire automorphic forms and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007044.png" /> denotes the subspace of cusp forms. For groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007045.png" /> of the kind considered here, a suitable version of the [[Riemann–Roch theorem|Riemann–Roch theorem]] shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007046.png" /> has finite dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007047.png" />.
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Let $\{ \Gamma , k , \mathbf{v} \}$ denote the vector space of automorphic forms on $\Gamma$ of weight $k$ and multiplier system $\mathbf{v}$, the collection of $f$ satisfying (a3) and that are holomorphic on $\mathcal{H}$ and meromorphic at each parabolic cusp of $\overline { \mathcal{R} }$ (in the usual local variable, cf. also [[Analytic function|Analytic function]]; [[Meromorphic function|Meromorphic function]]). One says that $f \in \{ \Gamma , k , \mathbf{v} \}$ is an entire automorphic form if $f$ is holomorphic at each parabolic cusp. An entire automorphic form $f$ is called a cusp form if $f$ vanishes at each parabolic cusp. As usual, $C ^ { + } ( \Gamma , k , \mathbf{v} )$ denotes the space of entire automorphic forms and $C ^ { 0 } ( \Gamma , k , \mathbf{v} )$ denotes the subspace of cusp forms. For groups $\Gamma$ of the kind considered here, a suitable version of the [[Riemann–Roch theorem|Riemann–Roch theorem]] shows that $C ^ { + } ( \Gamma , k , \mathbf{v} )$ has finite dimension over $\mathbf{C}$.
  
 
To describe the genesis of Eichler cohomology it is helpful to introduce the Bol identity [[#References|[a1]]]:
 
To describe the genesis of Eichler cohomology it is helpful to introduce the Bol identity [[#References|[a1]]]:
  
<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/e/e120/e120070/e12007048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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\begin{equation} \tag{a4} D ^ { k + 1 } \{ ( c z + d ) ^ { k } F ( M z ) \} = \end{equation}
  
<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/e/e120/e120070/e12007049.png" /></td> </tr></table>
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\begin{equation*} = ( c z + d ) ^ { - k - 2 } F ^ { ( k + 1 ) } ( M z ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007052.png" /> is any fractional-linear transformation of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007053.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007054.png" /> is a function with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007055.png" /> derivatives ((a4) is easily derived from the Cauchy integral formula, cf. [[Cauchy integral theorem|Cauchy integral theorem]], or proved by induction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007056.png" />). As a consequence of (a4), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007057.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007058.png" />.
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where $k \in \mathbf{Z}$, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007051.png"/>, $M = \left( \begin{array} { c c } { * } &amp; { * } \\ { c } &amp; { d } \end{array} \right)$ is any fractional-linear transformation of determinant $1$, and $F$ is a function with $k + 1$ derivatives ((a4) is easily derived from the Cauchy integral formula, cf. [[Cauchy integral theorem|Cauchy integral theorem]], or proved by induction on $k$). As a consequence of (a4), if $F \in \{ \Gamma , - k , \mathbf{v} \}$, then $F ^ { ( k + 1 ) } \in \{ \Gamma , k + 2 , \mathbf{v} \}$.
  
There is a second consequence of (a4), more directly relevant to the case under consideration: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007060.png" /> (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007061.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007062.png" /> satisfies
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There is a second consequence of (a4), more directly relevant to the case under consideration: if $f \in \{ \Gamma , k + 2 , \mathbf{v} \}$ and $F ^ { ( k + 1 ) } = f$ (for example, $F ( z ) = ( 1 / k ! ) \int _ { i } ^ { z } f ( \tau ) ( z - \tau ) ^ { k } d \tau$), then $F$ satisfies
  
<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/e/e120/e120070/e12007063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \tag{a5} F | _ { - k } ^ { \mathbf{v} } M = F + p _ { M } , \forall M \in \Gamma, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007064.png" /> is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007065.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007066.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007067.png" /> is called an Eichler integral of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007068.png" /> and multiplier system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007069.png" />, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007070.png" />, and with period polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007072.png" />. Eichler integrals generalize the classical Abelian integrals (cf. [[Abelian integral|Abelian integral]]), which occur as the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007074.png" />. As an immediate consequence of (a5), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007075.png" /> satisfies the cocycle condition
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where $p_{M}$ is a polynomial in $z$ of degree at most $k$. $F$ is called an Eichler integral of weight $-  { k }$ and multiplier system $\mathbf{v}$, with respect to $\Gamma$, and with period polynomials $p_{M}$, $M \in \Gamma$. Eichler integrals generalize the classical Abelian integrals (cf. [[Abelian integral|Abelian integral]]), which occur as the case $k = 0$, $\mathbf{v} \equiv 1$. As an immediate consequence of (a5), $\{ p _ { M } : M \in \Gamma \}$ satisfies the cocycle condition
  
<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/e/e120/e120070/e12007076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007076.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a6)</td></tr></table>
  
Consider the cocycle condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007077.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007078.png" /> of polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007079.png" />. A collection of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007080.png" /> satisfying (a6) is called a cocycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007081.png" />. A coboundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007082.png" /> is a collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007083.png" /> such that
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Consider the cocycle condition for $\{ p _ { M } : M \in \Gamma \}$ in the space $P ( k )$ of polynomials of degree at most $k$. A collection of polynomials $\{ p _ { M } \in P ( k ) : M \in \Gamma \}$ satisfying (a6) is called a cocycle in $P ( k )$. A coboundary in $P ( k )$ is a collection $\{ p _ { M } \in P ( k ) : M \in \Gamma \}$ such that
  
<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/e/e120/e120070/e12007084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
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\begin{equation} \tag{a7} p _ { M } = p | _ { - k } ^ { \mathbf{v} } M - p , M \in \Gamma , \end{equation}
  
with a fixed polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007085.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007086.png" /> defined by (a7) satisfies (a6). The Eichler cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007087.png" /> is now defined to be the quotient space: cocycles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007088.png" /> modulo coboundaries in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007089.png" />.
+
with a fixed polynomial $p \in P ( k )$. Note that $\{ p _ M\}$ defined by (a7) satisfies (a6). The Eichler cohomology group $H ^ { 1 } = H ^ { 1 } ( \Gamma , k , \mathbf{v} ; P ( k ) )$ is now defined to be the quotient space: cocycles in $P ( k )$ modulo coboundaries in $P ( k )$.
  
To state Eichler's cohomology theorem of [[#References|[a2]]] one must introduce the notion of a  "parabolic cocycle" . Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007090.png" /> be the (necessarily finite) set of inequivalent parabolic cusps in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007091.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007092.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007093.png" /> be the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007094.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007095.png" /> with parabolic generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007096.png" /> (cf. also [[Stabilizer|Stabilizer]]). One says that the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007097.png" /> is parabolic if the following holds: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e12007099.png" />, there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070100.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070101.png" />.
+
To state Eichler's cohomology theorem of [[#References|[a2]]] one must introduce the notion of a  "parabolic cocycle" . Let $q _ { 1 } , \dots , q _ { t }$ be the (necessarily finite) set of inequivalent parabolic cusps in $\overline { \mathcal{R} }$. For $1 \leq h \leq t$, let $\Gamma _ { h }$ be the stabilizer of $q_h$ in $\Gamma$ with parabolic generator $Q _ { h }$ (cf. also [[Stabilizer|Stabilizer]]). One says that the cocycle $\{ p _ { M } \in P ( k ) : M \in \Gamma \}$ is parabolic if the following holds: For each $h$, $1 \leq h \leq t$, there exists a $p _ { h } \in P ( k )$ such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070101.png"/>.
  
Coboundaries are of course parabolic cocycles, so one may form the quotient group: parabolic cocycles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070102.png" /> modulo coboundaries in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070103.png" />. This is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070104.png" />, called the parabolic Eichler cohomology group and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070105.png" />.
+
Coboundaries are of course parabolic cocycles, so one may form the quotient group: parabolic cocycles in $P ( k )$ modulo coboundaries in $P ( k )$. This is a subgroup of $H ^ { 1 } ( \Gamma , k , \mathbf{v} ; P ( k ) )$, called the parabolic Eichler cohomology group and denoted by $\tilde { H } ^ { 1 } = \tilde { H } ^ { 1 } ( \Gamma , k , {\bf v} ; P ( k ) )$.
  
Eichler's theorem [[#References|[a2]]], p. 283, states: The vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070107.png" /> are isomorphic under a canonical mapping.
+
Eichler's theorem [[#References|[a2]]], p. 283, states: The vector spaces $C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) \oplus C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$ and $\widetilde { H } ^ { 1 } ( \Gamma , k , \mathbf v ; P ( k ) )$ are isomorphic under a canonical mapping.
  
The discussion above, leading to (a6), shows how to associate a unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070109.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070110.png" />, by forming a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070111.png" />-fold anti-derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070112.png" />. The key to the proof of Eichler's theorem lies in the construction of a suitable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070113.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070114.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070115.png" />. Eichler accomplishes this by attaching to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070116.png" /> an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070117.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070118.png" /> with poles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070119.png" />, and then passing to the cocycle of period polynomials of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070120.png" />-fold anti-derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070121.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070122.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070123.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070124.png" /> is then defined by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070125.png" />. The proof that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070126.png" /> is one-to-one follows from Eichler's generalization of the Riemann period relation for Abelian integrals to the setting of Eichler integrals.
+
The discussion above, leading to (a6), shows how to associate a unique element $\beta ( f )$ of $\widetilde { H } ^ { 1 }$ to $f \in C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$, by forming a $( k + 1 )$-fold [[anti-derivative]] of $f$. The key to the proof of Eichler's theorem lies in the construction of a suitable mapping $\alpha ( g )$ from $g \in C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$ to $\widetilde { H } ^ { 1 }$. Eichler accomplishes this by attaching to $g$ an element $\hat{g}$ of $\{ \Gamma , k + 2 , \mathbf{v} \}$ with poles in $\overline { \mathcal{R} }$, and then passing to the cocycle of period polynomials of a $( k + 1 )$-fold anti-derivative of $\hat{g}$. The mapping $\mu$ from $C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) \oplus C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$ to $\widetilde { H } ^ { 1 }$ is then defined by means of $\mu ( g , f ) = \alpha ( g ) + \beta ( f )$. The proof that $\mu$ is one-to-one follows from Eichler's generalization of the Riemann period relation for Abelian integrals to the setting of Eichler integrals.
  
The proof can be completed by showing that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070127.png" />. The essence of Eichler's theorem is that every parabolic cocycle can be realized as the system of period polynomials of some unique Eichler integral of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070128.png" /> and multiplier system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070129.png" />, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070130.png" />.
+
The proof can be completed by showing that $\operatorname { dim } \tilde { H } ^ { 1 } = \operatorname { dim } C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) + \operatorname { dim } C ^ { 0 } ( \Gamma , k + 2 ,\mathbf{v} )$. The essence of Eichler's theorem is that every parabolic cocycle can be realized as the system of period polynomials of some unique Eichler integral of weight $-  { k }$ and multiplier system $\mathbf{v}$, with respect to $\Gamma$.
  
R.C. Gunning [[#References|[a3]]] has proved a related result, from which Eichler's theorem follows as a corollary: The vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070132.png" /> are isomorphic under the mapping of Eichler's theorem.
+
R.C. Gunning [[#References|[a3]]] has proved a related result, from which Eichler's theorem follows as a corollary: The vector spaces $C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) \oplus C ^ { + } ( \Gamma , k + 2 , \mathbf{v} )$ and $H ^ { 1 } ( \Gamma , k , \mathbf{v} ; P ( k ) )$ are isomorphic under the mapping of Eichler's theorem.
  
Proving Gunning 's theorem first and then deriving Eichler's theorem from it has the advantage that the calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070133.png" /> is substantially easier than that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070134.png" />; this, because in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070135.png" /> there is no restriction on the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070136.png" /> associated to the parabolic generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070138.png" />.
+
Proving Gunning 's theorem first and then deriving Eichler's theorem from it has the advantage that the calculation of $\operatorname{dim} \, H ^ { 1 }$ is substantially easier than that of $\operatorname{dim} \tilde { H } _ { 1 }$; this, because in $H ^ { 1 }$ there is no restriction on the elements of $P ( k )$ associated to the parabolic generators $Q _ { h }$, $1 \leq h \leq t - 1$.
  
There are various proofs of Gunning 's theorem and its corollary, in addition to those in [[#References|[a2]]], [[#References|[a3]]]. See, for example, [[#References|[a4]]], [[#References|[a11]]], [[#References|[a14]]]. (G. Shimura [[#References|[a14]]] has refined Eichler's theorem by working over the real rather than the complex field.) In [[#References|[a6]]], Chap. 5, [[#References|[a7]]], [[#References|[a8]]], and [[#References|[a13]]], analogous results are proved for the more general situation in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070139.png" /> is a finitely generated [[Kleinian group|Kleinian group]]. I. Kra has made further contributions to this case ([[#References|[a9]]], [[#References|[a10]]]).
+
There are various proofs of Gunning 's theorem and its corollary, in addition to those in [[#References|[a2]]], [[#References|[a3]]]. See, for example, [[#References|[a4]]], [[#References|[a11]]], [[#References|[a14]]]. (G. Shimura [[#References|[a14]]] has refined Eichler's theorem by working over the real rather than the complex field.) In [[#References|[a6]]], Chap. 5, [[#References|[a7]]], [[#References|[a8]]], and [[#References|[a13]]], analogous results are proved for the more general situation in which $\Gamma$ is a finitely generated [[Kleinian group|Kleinian group]]. I. Kra has made further contributions to this case ([[#References|[a9]]], [[#References|[a10]]]).
  
The literature contains several results describing the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070141.png" /> that arise when the space of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070142.png" /> is replaced by a larger space of analytic functions [[#References|[a3]]], Thm. 3, [[#References|[a5]]], Thms. 1; 2, [[#References|[a7]]], Thm. 5. Gunning [[#References|[a3]]], Thms. 4; 5, discusses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070143.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070144.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070145.png" />, as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120070/e120070146.png" />. For an overview see [[#References|[a5]]].
+
The literature contains several results describing the cohomology groups $H ^ { 1 }$ and $\widetilde { H } ^ { 1 }$ that arise when the space of polynomials $P ( k )$ is replaced by a larger space of analytic functions [[#References|[a3]]], Thm. 3, [[#References|[a5]]], Thms. 1; 2, [[#References|[a7]]], Thm. 5. Gunning [[#References|[a3]]], Thms. 4; 5, discusses $H ^ { 0 }$ and $H ^ { p }$, for $p &gt; 1$, as well as $H ^ { 1 }$. For an overview see [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Bol,  "Invarianten linearer Differentialgleichungen"  ''Abh. Math. Sem. Univ. Hamburg'' , '''16''' :  3–4  (1949)  pp. 1–28</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Eichler,  "Eine Verallgemeinerung der Abelschen Integrale"  ''Math. Z.'' , '''67'''  (1957)  pp. 267–298</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.C. Gunning,  "The Eichler cohomology groups and automorphic forms"  ''Trans. Amer. Math. Soc.'' , '''100'''  (1961)  pp. 44–62</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.Y. Hussemi,  M.I. Knopp,  "Eichler cohomology and automorphic forms"  ''Illinois J. Math.'' , '''15'''  (1971)  pp. 565–577</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.I. Knopp,  "Some new results on the Eichler cohomology of automorphic forms"  ''Bull. Amer. Math. Soc.'' , '''80'''  (1974)  pp. 607–632</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Kra,  "Automorphic forms and Kleinian groups" , Benjamin  (1972)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I. Kra,  "On cohomology of Kleinian groups"  ''Ann. of Math.'' , '''89''' :  2  (1969)  pp. 533–556</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  I. Kra,  "On cohomology of Kleinian groups - II"  ''Ann. of Math.'' , '''90''' :  2  (1969)  pp. 576–590</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I. Kra,  "On cohomology of Kleinian groups - III"  ''Acta Math.'' , '''127'''  (1971)  pp. 23–40</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Kra,  "On cohomology of Kleinian groups - IV"  ''J. d'Anal. Math.'' , '''43'''  (1983-84)  pp. 51–87</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J. Lehner,  "Automorphic Integrals with preassigned period polynomials and the Eichler cohomology"  A.O.L. Atkin (ed.)  B.J. Birch (ed.) , ''Computers in Number Theory, Proc. Sci. Research Council Atlas Symp. no. 2'' , Acad. Press  (1971)  pp. 49– 56</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Lehner,  "Cohomology of vector-valued automorphic forms"  ''Math. Ann.'' , '''204'''  (1973)  pp. 155–176</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  J. Lehner,  "The Eichler cohomology of a Kleinian group"  ''Math. Ann.'' , '''192'''  (1971)  pp. 125–143</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  G. Shimura,  "Sur les intégrales attachées aux formes automorphes"  ''J. Math. Soc. Japan'' , '''11'''  (1959)  pp. 291–311</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  G. Bol,  "Invarianten linearer Differentialgleichungen"  ''Abh. Math. Sem. Univ. Hamburg'' , '''16''' :  3–4  (1949)  pp. 1–28</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Eichler,  "Eine Verallgemeinerung der Abelschen Integrale"  ''Math. Z.'' , '''67'''  (1957)  pp. 267–298</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  R.C. Gunning,  "The Eichler cohomology groups and automorphic forms"  ''Trans. Amer. Math. Soc.'' , '''100'''  (1961)  pp. 44–62</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  S.Y. Hussemi,  M.I. Knopp,  "Eichler cohomology and automorphic forms"  ''Illinois J. Math.'' , '''15'''  (1971)  pp. 565–577</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M.I. Knopp,  "Some new results on the Eichler cohomology of automorphic forms"  ''Bull. Amer. Math. Soc.'' , '''80'''  (1974)  pp. 607–632</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  I. Kra,  "Automorphic forms and Kleinian groups" , Benjamin  (1972)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  I. Kra,  "On cohomology of Kleinian groups"  ''Ann. of Math.'' , '''89''' :  2  (1969)  pp. 533–556</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  I. Kra,  "On cohomology of Kleinian groups - II"  ''Ann. of Math.'' , '''90''' :  2  (1969)  pp. 576–590</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  I. Kra,  "On cohomology of Kleinian groups - III"  ''Acta Math.'' , '''127'''  (1971)  pp. 23–40</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  I. Kra,  "On cohomology of Kleinian groups - IV"  ''J. d'Anal. Math.'' , '''43'''  (1983-84)  pp. 51–87</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  J. Lehner,  "Automorphic Integrals with preassigned period polynomials and the Eichler cohomology"  A.O.L. Atkin (ed.)  B.J. Birch (ed.) , ''Computers in Number Theory, Proc. Sci. Research Council Atlas Symp. no. 2'' , Acad. Press  (1971)  pp. 49– 56</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J. Lehner,  "Cohomology of vector-valued automorphic forms"  ''Math. Ann.'' , '''204'''  (1973)  pp. 155–176</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  J. Lehner,  "The Eichler cohomology of a Kleinian group"  ''Math. Ann.'' , '''192'''  (1971)  pp. 125–143</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  G. Shimura,  "Sur les intégrales attachées aux formes automorphes"  ''J. Math. Soc. Japan'' , '''11'''  (1959)  pp. 291–311</td></tr></table>

Latest revision as of 17:20, 2 January 2021

In [a2], M. Eichler conceived the "Eichler cohomology theory" (but not the designation) while studying "generalized Abelian integrals" (now called "Eichler integrals" ; see below).

The setting for this theory is that of automorphic forms, with multiplier system, on a discrete group $\Gamma$ of fractional-linear transformations (equivalently, of $( 2 \times 2 )$-matrices; cf. also Automorphic form; Fractional-linear mapping). One may assume that $\Gamma$ consists of real fractional-linear transformations, that is, that $\Gamma$ fixes $\mathcal{H}$, the upper half-plane. A fundamental region, $\mathcal{R}$, of $\Gamma$ is required to have finite hyperbolic area; this is equivalent to the two conditions (taken jointly):

i) $\Gamma$ is finitely generated;

ii) each real point $q$ of $\overline { \mathcal{R} }$ is a parabolic point (a cusp) of $\Gamma$, that is, it is fixed by a cyclic subgroup $\Gamma _ { q }$ of $\Gamma$ with parabolic generator.

Let $k \in \mathbf{Z}$ and let $\mathbf{v}$ be a multiplier system in weight $k$ with respect to $\Gamma$. Since $k$ is integral, this means simply that $|\mathbf{v} ( M ) | = 1$ for all $M \in \Gamma$, and $\mathbf{v}$ is multiplicative:

\begin{equation} \tag{a1} \mathbf{v} ( M _ { 1 } , M _ { 2 } ) = \mathbf{v} ( M _ { 1 } ) \mathbf{v} ( M _ { 2 } ) , M _ { 1 } , M _ { 2 } \in \Gamma. \end{equation}

For $M \in \Gamma$ and $\varphi$ a function on $\mathcal{H}$, define the slash operator

\begin{equation} \tag{a2} ( \varphi | _ { k } ^ { \mathbf{v} } M ) ( z ) = {\bf v} ( M ) ( cz + d ) ^ { - k } \varphi ( M z ). \end{equation}

In this notation, the characteristic transformation law satisfied by an automorphic form $f$ on $\Gamma$ of weight $k$ and multiplier system $\mathbf{v}$ can be written

\begin{equation} \tag{a3} f | _ { k } ^ { \mathbf{v} } M = f , \forall M \in \Gamma. \end{equation}

Let $\{ \Gamma , k , \mathbf{v} \}$ denote the vector space of automorphic forms on $\Gamma$ of weight $k$ and multiplier system $\mathbf{v}$, the collection of $f$ satisfying (a3) and that are holomorphic on $\mathcal{H}$ and meromorphic at each parabolic cusp of $\overline { \mathcal{R} }$ (in the usual local variable, cf. also Analytic function; Meromorphic function). One says that $f \in \{ \Gamma , k , \mathbf{v} \}$ is an entire automorphic form if $f$ is holomorphic at each parabolic cusp. An entire automorphic form $f$ is called a cusp form if $f$ vanishes at each parabolic cusp. As usual, $C ^ { + } ( \Gamma , k , \mathbf{v} )$ denotes the space of entire automorphic forms and $C ^ { 0 } ( \Gamma , k , \mathbf{v} )$ denotes the subspace of cusp forms. For groups $\Gamma$ of the kind considered here, a suitable version of the Riemann–Roch theorem shows that $C ^ { + } ( \Gamma , k , \mathbf{v} )$ has finite dimension over $\mathbf{C}$.

To describe the genesis of Eichler cohomology it is helpful to introduce the Bol identity [a1]:

\begin{equation} \tag{a4} D ^ { k + 1 } \{ ( c z + d ) ^ { k } F ( M z ) \} = \end{equation}

\begin{equation*} = ( c z + d ) ^ { - k - 2 } F ^ { ( k + 1 ) } ( M z ), \end{equation*}

where $k \in \mathbf{Z}$, , $M = \left( \begin{array} { c c } { * } & { * } \\ { c } & { d } \end{array} \right)$ is any fractional-linear transformation of determinant $1$, and $F$ is a function with $k + 1$ derivatives ((a4) is easily derived from the Cauchy integral formula, cf. Cauchy integral theorem, or proved by induction on $k$). As a consequence of (a4), if $F \in \{ \Gamma , - k , \mathbf{v} \}$, then $F ^ { ( k + 1 ) } \in \{ \Gamma , k + 2 , \mathbf{v} \}$.

There is a second consequence of (a4), more directly relevant to the case under consideration: if $f \in \{ \Gamma , k + 2 , \mathbf{v} \}$ and $F ^ { ( k + 1 ) } = f$ (for example, $F ( z ) = ( 1 / k ! ) \int _ { i } ^ { z } f ( \tau ) ( z - \tau ) ^ { k } d \tau$), then $F$ satisfies

\begin{equation} \tag{a5} F | _ { - k } ^ { \mathbf{v} } M = F + p _ { M } , \forall M \in \Gamma, \end{equation}

where $p_{M}$ is a polynomial in $z$ of degree at most $k$. $F$ is called an Eichler integral of weight $- { k }$ and multiplier system $\mathbf{v}$, with respect to $\Gamma$, and with period polynomials $p_{M}$, $M \in \Gamma$. Eichler integrals generalize the classical Abelian integrals (cf. Abelian integral), which occur as the case $k = 0$, $\mathbf{v} \equiv 1$. As an immediate consequence of (a5), $\{ p _ { M } : M \in \Gamma \}$ satisfies the cocycle condition

(a6)

Consider the cocycle condition for $\{ p _ { M } : M \in \Gamma \}$ in the space $P ( k )$ of polynomials of degree at most $k$. A collection of polynomials $\{ p _ { M } \in P ( k ) : M \in \Gamma \}$ satisfying (a6) is called a cocycle in $P ( k )$. A coboundary in $P ( k )$ is a collection $\{ p _ { M } \in P ( k ) : M \in \Gamma \}$ such that

\begin{equation} \tag{a7} p _ { M } = p | _ { - k } ^ { \mathbf{v} } M - p , M \in \Gamma , \end{equation}

with a fixed polynomial $p \in P ( k )$. Note that $\{ p _ M\}$ defined by (a7) satisfies (a6). The Eichler cohomology group $H ^ { 1 } = H ^ { 1 } ( \Gamma , k , \mathbf{v} ; P ( k ) )$ is now defined to be the quotient space: cocycles in $P ( k )$ modulo coboundaries in $P ( k )$.

To state Eichler's cohomology theorem of [a2] one must introduce the notion of a "parabolic cocycle" . Let $q _ { 1 } , \dots , q _ { t }$ be the (necessarily finite) set of inequivalent parabolic cusps in $\overline { \mathcal{R} }$. For $1 \leq h \leq t$, let $\Gamma _ { h }$ be the stabilizer of $q_h$ in $\Gamma$ with parabolic generator $Q _ { h }$ (cf. also Stabilizer). One says that the cocycle $\{ p _ { M } \in P ( k ) : M \in \Gamma \}$ is parabolic if the following holds: For each $h$, $1 \leq h \leq t$, there exists a $p _ { h } \in P ( k )$ such that .

Coboundaries are of course parabolic cocycles, so one may form the quotient group: parabolic cocycles in $P ( k )$ modulo coboundaries in $P ( k )$. This is a subgroup of $H ^ { 1 } ( \Gamma , k , \mathbf{v} ; P ( k ) )$, called the parabolic Eichler cohomology group and denoted by $\tilde { H } ^ { 1 } = \tilde { H } ^ { 1 } ( \Gamma , k , {\bf v} ; P ( k ) )$.

Eichler's theorem [a2], p. 283, states: The vector spaces $C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) \oplus C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$ and $\widetilde { H } ^ { 1 } ( \Gamma , k , \mathbf v ; P ( k ) )$ are isomorphic under a canonical mapping.

The discussion above, leading to (a6), shows how to associate a unique element $\beta ( f )$ of $\widetilde { H } ^ { 1 }$ to $f \in C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$, by forming a $( k + 1 )$-fold anti-derivative of $f$. The key to the proof of Eichler's theorem lies in the construction of a suitable mapping $\alpha ( g )$ from $g \in C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$ to $\widetilde { H } ^ { 1 }$. Eichler accomplishes this by attaching to $g$ an element $\hat{g}$ of $\{ \Gamma , k + 2 , \mathbf{v} \}$ with poles in $\overline { \mathcal{R} }$, and then passing to the cocycle of period polynomials of a $( k + 1 )$-fold anti-derivative of $\hat{g}$. The mapping $\mu$ from $C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) \oplus C ^ { 0 } ( \Gamma , k + 2 , \mathbf{v} )$ to $\widetilde { H } ^ { 1 }$ is then defined by means of $\mu ( g , f ) = \alpha ( g ) + \beta ( f )$. The proof that $\mu$ is one-to-one follows from Eichler's generalization of the Riemann period relation for Abelian integrals to the setting of Eichler integrals.

The proof can be completed by showing that $\operatorname { dim } \tilde { H } ^ { 1 } = \operatorname { dim } C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) + \operatorname { dim } C ^ { 0 } ( \Gamma , k + 2 ,\mathbf{v} )$. The essence of Eichler's theorem is that every parabolic cocycle can be realized as the system of period polynomials of some unique Eichler integral of weight $- { k }$ and multiplier system $\mathbf{v}$, with respect to $\Gamma$.

R.C. Gunning [a3] has proved a related result, from which Eichler's theorem follows as a corollary: The vector spaces $C ^ { 0 } ( \Gamma , k + 2 , \overline{\mathbf{v}} ) \oplus C ^ { + } ( \Gamma , k + 2 , \mathbf{v} )$ and $H ^ { 1 } ( \Gamma , k , \mathbf{v} ; P ( k ) )$ are isomorphic under the mapping of Eichler's theorem.

Proving Gunning 's theorem first and then deriving Eichler's theorem from it has the advantage that the calculation of $\operatorname{dim} \, H ^ { 1 }$ is substantially easier than that of $\operatorname{dim} \tilde { H } _ { 1 }$; this, because in $H ^ { 1 }$ there is no restriction on the elements of $P ( k )$ associated to the parabolic generators $Q _ { h }$, $1 \leq h \leq t - 1$.

There are various proofs of Gunning 's theorem and its corollary, in addition to those in [a2], [a3]. See, for example, [a4], [a11], [a14]. (G. Shimura [a14] has refined Eichler's theorem by working over the real rather than the complex field.) In [a6], Chap. 5, [a7], [a8], and [a13], analogous results are proved for the more general situation in which $\Gamma$ is a finitely generated Kleinian group. I. Kra has made further contributions to this case ([a9], [a10]).

The literature contains several results describing the cohomology groups $H ^ { 1 }$ and $\widetilde { H } ^ { 1 }$ that arise when the space of polynomials $P ( k )$ is replaced by a larger space of analytic functions [a3], Thm. 3, [a5], Thms. 1; 2, [a7], Thm. 5. Gunning [a3], Thms. 4; 5, discusses $H ^ { 0 }$ and $H ^ { p }$, for $p > 1$, as well as $H ^ { 1 }$. For an overview see [a5].

References

[a1] G. Bol, "Invarianten linearer Differentialgleichungen" Abh. Math. Sem. Univ. Hamburg , 16 : 3–4 (1949) pp. 1–28
[a2] M. Eichler, "Eine Verallgemeinerung der Abelschen Integrale" Math. Z. , 67 (1957) pp. 267–298
[a3] R.C. Gunning, "The Eichler cohomology groups and automorphic forms" Trans. Amer. Math. Soc. , 100 (1961) pp. 44–62
[a4] S.Y. Hussemi, M.I. Knopp, "Eichler cohomology and automorphic forms" Illinois J. Math. , 15 (1971) pp. 565–577
[a5] M.I. Knopp, "Some new results on the Eichler cohomology of automorphic forms" Bull. Amer. Math. Soc. , 80 (1974) pp. 607–632
[a6] I. Kra, "Automorphic forms and Kleinian groups" , Benjamin (1972)
[a7] I. Kra, "On cohomology of Kleinian groups" Ann. of Math. , 89 : 2 (1969) pp. 533–556
[a8] I. Kra, "On cohomology of Kleinian groups - II" Ann. of Math. , 90 : 2 (1969) pp. 576–590
[a9] I. Kra, "On cohomology of Kleinian groups - III" Acta Math. , 127 (1971) pp. 23–40
[a10] I. Kra, "On cohomology of Kleinian groups - IV" J. d'Anal. Math. , 43 (1983-84) pp. 51–87
[a11] J. Lehner, "Automorphic Integrals with preassigned period polynomials and the Eichler cohomology" A.O.L. Atkin (ed.) B.J. Birch (ed.) , Computers in Number Theory, Proc. Sci. Research Council Atlas Symp. no. 2 , Acad. Press (1971) pp. 49– 56
[a12] J. Lehner, "Cohomology of vector-valued automorphic forms" Math. Ann. , 204 (1973) pp. 155–176
[a13] J. Lehner, "The Eichler cohomology of a Kleinian group" Math. Ann. , 192 (1971) pp. 125–143
[a14] G. Shimura, "Sur les intégrales attachées aux formes automorphes" J. Math. Soc. Japan , 11 (1959) pp. 291–311
How to Cite This Entry:
Eichler cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Eichler_cohomology&oldid=12564
This article was adapted from an original article by M.I. Knopp (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article