Namespaces
Variants
Actions

Difference between revisions of "Jacobi inversion problem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
The problem of inverting the Abelian integrals (cf. [[Abelian integral|Abelian integral]]) of the first kind of an arbitrary algebraic function field (cf. [[Algebraic function|Algebraic function]]). In other words, the problem of inverting the Abelian integrals of the first kind on the compact [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540701.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540702.png" /> that corresponds to a given algebraic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540703.png" />.
+
<!--
 +
j0540701.png
 +
$#A+1 = 70 n = 0
 +
$#C+1 = 70 : ~/encyclopedia/old_files/data/J054/J.0504070 Jacobi inversion problem
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540704.png" /> be a basis of the Abelian differentials of the first kind on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540705.png" /> (cf. [[Abelian differential|Abelian differential]]). The inversion of a single Abelian integral, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540706.png" />, that is, the representation of all possible rational functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540707.png" />, in particular, the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540708.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j0540709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407010.png" />, is only meaningful when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407011.png" />. In this case one speaks of the [[Inversion of an elliptic integral|inversion of an elliptic integral]], and this leads to doubly-periodic elliptic functions (cf. [[Elliptic function|Elliptic function]]). For example, the inversion of an integral of the first kind in Legendre normal form
+
{{TEX|auto}}
 +
{{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/j/j054/j054070/j05407012.png" /></td> </tr></table>
+
The problem of inverting the Abelian integrals (cf. [[Abelian integral|Abelian integral]]) of the first kind of an arbitrary algebraic function field (cf. [[Algebraic function|Algebraic function]]). In other words, the problem of inverting the Abelian integrals of the first kind on the compact [[Riemann surface|Riemann surface]]  $  F $
 +
of genus  $  p \geq  1 $
 +
that corresponds to a given algebraic equation  $  F ( z, w) = 0 $.
  
leads to the Jacobi elliptic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407013.png" /> (cf. [[Jacobi elliptic functions|Jacobi elliptic functions]]).
+
Let  $  \phi _ {1} \dots \phi _ {p} $
 +
be a basis of the Abelian differentials of the first kind on  $  F $(
 +
cf. [[Abelian differential|Abelian differential]]). The inversion of a single Abelian integral, for example,  $  \int _ {c _ {1}  } ^ {w _ {1} } \phi _ {1} \equiv u _ {1} ( w _ {1} ) = z _ {1} $,
 +
that is, the representation of all possible rational functions of  $  w _ {1} $,
 +
in particular, the representation of  $  w _ {1} $
 +
as a function of  $  z _ {1} $,
 +
$  w _ {1} = w _ {1} ( z _ {1} ) $,
 +
is only meaningful when  $  p = 1 $.  
 +
In this case one speaks of the [[Inversion of an elliptic integral|inversion of an elliptic integral]], and this leads to doubly-periodic elliptic functions (cf. [[Elliptic function|Elliptic function]]). For example, the inversion of an integral of the first kind in Legendre normal form
  
As C.G.J. Jacobi already observed (1832), the inversion problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407014.png" /> must be considered for the set of all Abelian integrals of the first kind, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407015.png" />, since one has to obtain functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407016.png" /> periods. In the general case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407017.png" />, the rational statement of the Jacobi inversion problem is as follows: Suppose one is given a system of equalities
+
$$
 +
u _ {1} ( w _ {1} ) = \
 +
\int\limits _ { 0 } ^ { {w _ 1} }
  
<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/j/j054/j054070/j05407018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{dt}{\sqrt {( 1 - t  ^ {2} ) ( 1 - k  ^ {2} t  ^ {2} ) } }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407019.png" /> are fixed points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407021.png" /> are variable points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407023.png" /> are arbitrary but given complex numbers. It is required to find how and under what conditions the system (1) can be inverted, that is, to obtain a representation of all possible symmetric rational functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407025.png" />, as functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407026.png" />.
+
= z _ {1,}  $$
  
Since the Abelian integrals in (1), as functions of the upper limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407027.png" />, depend on the form of the path on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407028.png" /> that joins <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407030.png" />, they are multiple-valued: When the path is varied they can increase by an integer linear combination of periods. Hence it follows that (1) is essentially a system of congruences modulo the periods of the differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407031.png" />. The values of the functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407032.png" /> obtained by solving the Jacobi inversion problem must remain unchanged when the argument is increased by any integer combination of the periods of the differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407033.png" />. Consequently, they are special Abelian functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407034.png" /> independent periods (cf. [[Abelian function|Abelian function]]).
+
leads to the Jacobi elliptic function  $  w _ {1} = \mathop{\rm sn}  z _ {1} $(
 +
cf. [[Jacobi elliptic functions|Jacobi elliptic functions]]).
  
For the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407035.png" />, that is, for an [[Elliptic integral|elliptic integral]], the elliptic functions solving the inversion problem can be constructed by using the comparatively-simple Jacobi theta-functions in a single complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407036.png" />, and meromorphic elliptic functions can be constructed as quotients of entire theta-functions. It is also possible to solve the general Jacobi inversion problem by using the theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407037.png" /> of the first order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407038.png" /> complex variables with half-integer characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407039.png" /> (cf. [[Theta-function|Theta-function]]).
+
As C.G.J. Jacobi already observed (1832), the inversion problem for  $  p> 1 $
 +
must be considered for the set of all Abelian integrals of the first kind,  $  \int \phi _ {1} \dots \int \phi _ {p} $,
 +
since one has to obtain functions with  $  2p $
 +
periods. In the general case when  $  p \geq  1 $,  
 +
the rational statement of the Jacobi inversion problem is as follows: Suppose one is given a system of equalities
  
The period matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407040.png" /> of the basis Abelian differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407041.png" /> has the form
+
$$ \tag{1 }
 +
\int\limits _ { c _ {1} } ^ { {w _ 1 } } \phi _ {j} + \dots +
 +
\int\limits _ { c _ {p} } ^ { {w _ p } } \phi _ {j}  = z _ {j} ,\ \
 +
j = 1 \dots p,
 +
$$
  
<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/j/j054/j054070/j05407042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
where  $  c _ {1} \dots c _ {p} $
 +
are fixed points on  $  F $,
 +
$  w _ {1} \dots w _ {p} $
 +
are variable points on  $  F $
 +
and  $  z = ( z _ {1} \dots z _ {p} ) $
 +
are arbitrary but given complex numbers. It is required to find how and under what conditions the system (1) can be inverted, that is, to obtain a representation of all possible symmetric rational functions of  $  w _ {k} $,
 +
$  k = 1 \dots p $,
 +
as functions of  $  z = ( z _ {1} \dots z _ {p} ) $.
  
<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/j/j054/j054070/j05407043.png" /></td> </tr></table>
+
Since the Abelian integrals in (1), as functions of the upper limit  $  w _ {k} $,
 +
depend on the form of the path on  $  F $
 +
that joins  $  c _ {k} $
 +
and  $  w _ {k} $,
 +
they are multiple-valued: When the path is varied they can increase by an integer linear combination of periods. Hence it follows that (1) is essentially a system of congruences modulo the periods of the differentials  $  \phi _ {1} \dots \phi _ {p} $.
 +
The values of the functions of  $  z= ( z _ {1} \dots z _ {p} ) $
 +
obtained by solving the Jacobi inversion problem must remain unchanged when the argument is increased by any integer combination of the periods of the differentials  $  \phi _ {1} \dots \phi _ {p} $.  
 +
Consequently, they are special Abelian functions with  $  2p $
 +
independent periods (cf. [[Abelian function|Abelian function]]).
  
and the Riemann relations (see [[Abelian function|Abelian function]]) between the periods ensure that the series representing the theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407044.png" />, constructed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407045.png" />, converge uniformly on compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407046.png" />. By using the theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407047.png" /> of zero characteristic one can construct the superposition
+
For the case  $  p = 1 $,
 +
that is, for an [[Elliptic integral|elliptic integral]], the elliptic functions solving the inversion problem can be constructed by using the comparatively-simple Jacobi theta-functions in a single complex variable  $  z $,  
 +
and meromorphic elliptic functions can be constructed as quotients of entire theta-functions. It is also possible to solve the general Jacobi inversion problem by using the theta-functions  $  \theta _ {H} ( z) = \theta _ {H} ( z _ {1} \dots z _ {p} ) $
 +
of the first order in  $  p $
 +
complex variables with half-integer characteristics  $  H $(
 +
cf. [[Theta-function|Theta-function]]).
  
<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/j/j054/j054070/j05407048.png" /></td> </tr></table>
+
The period matrix  $  W $
 +
of the basis Abelian differentials  $  \phi _ {j} $
 +
has the form
 +
 
 +
$$ \tag{2 }
 +
= \left \|
 +
\begin{array}{ccccccc}
 +
\pi i  & 0  & 0  & 0  &a _ {11}  &\dots  &a _ {1p}  \\
 +
0  &\dots  &\cdot  &\cdot  &\cdot  &\dots  &\cdot  \\
 +
\cdot  &\dots  &\pi i  & 0  &\cdot  &\dots  &\cdot  \\
 +
0  & 0  & 0  &\pi i  &a _ {p1}  &\dots  &a _ {pp}  \\
 +
\end{array}
 +
\
 +
\right \| ,
 +
$$
 +
 
 +
$$
 +
a _ {jk}  = a _ {kj} ,
 +
$$
 +
 
 +
and the Riemann relations (see [[Abelian function|Abelian function]]) between the periods ensure that the series representing the theta-functions  $  \theta _ {H} ( z) $,
 +
constructed in terms of  $  W $,
 +
converge uniformly on compact sets in  $  \mathbf C  ^ {p} $.
 +
By using the theta-function  $  \theta ( z) = \theta _ {0} ( z) $
 +
of zero characteristic one can construct the superposition
 +
 
 +
$$
 +
\Phi ( w)  = \theta ( u ( w) - z),
 +
$$
  
 
where
 
where
  
<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/j/j054/j054070/j05407049.png" /></td> </tr></table>
+
$$
 +
u ( w)  = \
 +
\left \{
 +
u _ {1} ( w _ {1} )  = \
 +
\int\limits _ { c _ {1} } ^ { {w _ 1 } } \phi _ {1} \dots
 +
u _ {p} ( w _ {p} )  = \int\limits _ { c _ {p} } ^ { {w _ p } }
 +
\phi _ {p} \right \}
 +
$$
  
is the vector of Abelian integrals and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407050.png" /> is a system of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407051.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407052.png" /> is called the [[Riemann theta-function|Riemann theta-function]]. For a given set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407053.png" />, either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407054.png" /> has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407055.png" /> a unique system of zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407056.png" /> (the normal case), or it vanishes identically (the exceptional case). The zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407057.png" /> give a solution to the Jacobi inversion problem. The exceptional points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407058.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407059.png" /> form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407060.png" /> a set of lower dimension.
+
is the vector of Abelian integrals and $  w = ( w _ {1} \dots w _ {p} ) $
 +
is a system of points of $  F $;  
 +
$  \Phi ( w) $
 +
is called the [[Riemann theta-function|Riemann theta-function]]. For a given set of numbers $  z \in \mathbf C  ^ {p} $,  
 +
either $  \Phi ( w) $
 +
has on $  F $
 +
a unique system of zeros $  \eta _ {1} \dots \eta _ {p} $(
 +
the normal case), or it vanishes identically (the exceptional case). The zeros $  \eta _ {1} \dots \eta _ {p} $
 +
give a solution to the Jacobi inversion problem. The exceptional points $  z $
 +
for which $  \Phi ( w) \equiv 0 $
 +
form in $  \mathbf C  ^ {p} $
 +
a set of lower dimension.
  
Explicit expressions for special Abelian functions that completely solve the Jacobi inversion problem can be constructed by using quotients of theta-functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407061.png" />, where the common denominator is the theta-function of zero characteristic. When periods are added to the argument, the theta-functions are multiplied by specific multipliers. For quotients of theta-functions, after cancellation the only non-trivial multiplier is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407062.png" />. Consequently, the squares of the quotients are unchanged when periods are added to the argument, and one obtains Abelian functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407063.png" /> periods.
+
Explicit expressions for special Abelian functions that completely solve the Jacobi inversion problem can be constructed by using quotients of theta-functions of the form $  \theta _ {H} ( z)/ \theta ( z) $,  
 +
where the common denominator is the theta-function of zero characteristic. When periods are added to the argument, the theta-functions are multiplied by specific multipliers. For quotients of theta-functions, after cancellation the only non-trivial multiplier is $  - 1 $.  
 +
Consequently, the squares of the quotients are unchanged when periods are added to the argument, and one obtains Abelian functions with $  2p $
 +
periods.
  
Closely related to the Jacobi inversion problem is the important problem of constructing for a given system of theta-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407064.png" /> with a common matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407065.png" /> satisfying the convergence conditions the corresponding algebraic function fields and the corresponding Riemann surface. For this construction to be possible, the distinct elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407067.png" /> (there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407068.png" /> of them) must satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407069.png" /> additional relations, the investigation of which when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054070/j05407070.png" /> is a very difficult problem (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]], [[#References|[5]]]).
+
Closely related to the Jacobi inversion problem is the important problem of constructing for a given system of theta-functions $  \theta _ {H} ( z) $
 +
with a common matrix $  W $
 +
satisfying the convergence conditions the corresponding algebraic function fields and the corresponding Riemann surface. For this construction to be possible, the distinct elements $  a _ {jk} $
 +
of $  W $(
 +
there are $  p ( p + 1)/2 $
 +
of them) must satisfy $  ( p - 2) ( p - 3)/2 $
 +
additional relations, the investigation of which when $  p > 3 $
 +
is a very difficult problem (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]], [[#References|[5]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.G. Chebotarev,  "The theory of algebraic functions" , Moscow-Leningrad  (1948)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Springer,  "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Clebsch,  P. Gordan,  "Theorie der Abelschen Funktionen" , Teubner  (1866)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Conforto,  "Abelsche Funktionen und algebraische Geometrie" , Springer  (1956)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Mumford,  "Structure of moduli spaces of curves and Abelian varieties" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 457–465</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.G. Chebotarev,  "The theory of algebraic functions" , Moscow-Leningrad  (1948)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Springer,  "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Clebsch,  P. Gordan,  "Theorie der Abelschen Funktionen" , Teubner  (1866)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Conforto,  "Abelsche Funktionen und algebraische Geometrie" , Springer  (1956)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Mumford,  "Structure of moduli spaces of curves and Abelian varieties" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 457–465</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.L. Siegel,  "Topics in complex function theory" , '''2''' , Interscience  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Mumford,  "Tata lectures on Theta" , '''1''' , Birkhäuser  (1983–1984)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.L. Siegel,  "Topics in complex function theory" , '''2''' , Interscience  (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Mumford,  "Tata lectures on Theta" , '''1''' , Birkhäuser  (1983–1984)</TD></TR></table>

Revision as of 22:14, 5 June 2020


The problem of inverting the Abelian integrals (cf. Abelian integral) of the first kind of an arbitrary algebraic function field (cf. Algebraic function). In other words, the problem of inverting the Abelian integrals of the first kind on the compact Riemann surface $ F $ of genus $ p \geq 1 $ that corresponds to a given algebraic equation $ F ( z, w) = 0 $.

Let $ \phi _ {1} \dots \phi _ {p} $ be a basis of the Abelian differentials of the first kind on $ F $( cf. Abelian differential). The inversion of a single Abelian integral, for example, $ \int _ {c _ {1} } ^ {w _ {1} } \phi _ {1} \equiv u _ {1} ( w _ {1} ) = z _ {1} $, that is, the representation of all possible rational functions of $ w _ {1} $, in particular, the representation of $ w _ {1} $ as a function of $ z _ {1} $, $ w _ {1} = w _ {1} ( z _ {1} ) $, is only meaningful when $ p = 1 $. In this case one speaks of the inversion of an elliptic integral, and this leads to doubly-periodic elliptic functions (cf. Elliptic function). For example, the inversion of an integral of the first kind in Legendre normal form

$$ u _ {1} ( w _ {1} ) = \ \int\limits _ { 0 } ^ { {w _ 1} } \frac{dt}{\sqrt {( 1 - t ^ {2} ) ( 1 - k ^ {2} t ^ {2} ) } } = z _ {1,} $$

leads to the Jacobi elliptic function $ w _ {1} = \mathop{\rm sn} z _ {1} $( cf. Jacobi elliptic functions).

As C.G.J. Jacobi already observed (1832), the inversion problem for $ p> 1 $ must be considered for the set of all Abelian integrals of the first kind, $ \int \phi _ {1} \dots \int \phi _ {p} $, since one has to obtain functions with $ 2p $ periods. In the general case when $ p \geq 1 $, the rational statement of the Jacobi inversion problem is as follows: Suppose one is given a system of equalities

$$ \tag{1 } \int\limits _ { c _ {1} } ^ { {w _ 1 } } \phi _ {j} + \dots + \int\limits _ { c _ {p} } ^ { {w _ p } } \phi _ {j} = z _ {j} ,\ \ j = 1 \dots p, $$

where $ c _ {1} \dots c _ {p} $ are fixed points on $ F $, $ w _ {1} \dots w _ {p} $ are variable points on $ F $ and $ z = ( z _ {1} \dots z _ {p} ) $ are arbitrary but given complex numbers. It is required to find how and under what conditions the system (1) can be inverted, that is, to obtain a representation of all possible symmetric rational functions of $ w _ {k} $, $ k = 1 \dots p $, as functions of $ z = ( z _ {1} \dots z _ {p} ) $.

Since the Abelian integrals in (1), as functions of the upper limit $ w _ {k} $, depend on the form of the path on $ F $ that joins $ c _ {k} $ and $ w _ {k} $, they are multiple-valued: When the path is varied they can increase by an integer linear combination of periods. Hence it follows that (1) is essentially a system of congruences modulo the periods of the differentials $ \phi _ {1} \dots \phi _ {p} $. The values of the functions of $ z= ( z _ {1} \dots z _ {p} ) $ obtained by solving the Jacobi inversion problem must remain unchanged when the argument is increased by any integer combination of the periods of the differentials $ \phi _ {1} \dots \phi _ {p} $. Consequently, they are special Abelian functions with $ 2p $ independent periods (cf. Abelian function).

For the case $ p = 1 $, that is, for an elliptic integral, the elliptic functions solving the inversion problem can be constructed by using the comparatively-simple Jacobi theta-functions in a single complex variable $ z $, and meromorphic elliptic functions can be constructed as quotients of entire theta-functions. It is also possible to solve the general Jacobi inversion problem by using the theta-functions $ \theta _ {H} ( z) = \theta _ {H} ( z _ {1} \dots z _ {p} ) $ of the first order in $ p $ complex variables with half-integer characteristics $ H $( cf. Theta-function).

The period matrix $ W $ of the basis Abelian differentials $ \phi _ {j} $ has the form

$$ \tag{2 } W = \left \| \begin{array}{ccccccc} \pi i & 0 & 0 & 0 &a _ {11} &\dots &a _ {1p} \\ 0 &\dots &\cdot &\cdot &\cdot &\dots &\cdot \\ \cdot &\dots &\pi i & 0 &\cdot &\dots &\cdot \\ 0 & 0 & 0 &\pi i &a _ {p1} &\dots &a _ {pp} \\ \end{array} \ \right \| , $$

$$ a _ {jk} = a _ {kj} , $$

and the Riemann relations (see Abelian function) between the periods ensure that the series representing the theta-functions $ \theta _ {H} ( z) $, constructed in terms of $ W $, converge uniformly on compact sets in $ \mathbf C ^ {p} $. By using the theta-function $ \theta ( z) = \theta _ {0} ( z) $ of zero characteristic one can construct the superposition

$$ \Phi ( w) = \theta ( u ( w) - z), $$

where

$$ u ( w) = \ \left \{ u _ {1} ( w _ {1} ) = \ \int\limits _ { c _ {1} } ^ { {w _ 1 } } \phi _ {1} \dots u _ {p} ( w _ {p} ) = \int\limits _ { c _ {p} } ^ { {w _ p } } \phi _ {p} \right \} $$

is the vector of Abelian integrals and $ w = ( w _ {1} \dots w _ {p} ) $ is a system of points of $ F $; $ \Phi ( w) $ is called the Riemann theta-function. For a given set of numbers $ z \in \mathbf C ^ {p} $, either $ \Phi ( w) $ has on $ F $ a unique system of zeros $ \eta _ {1} \dots \eta _ {p} $( the normal case), or it vanishes identically (the exceptional case). The zeros $ \eta _ {1} \dots \eta _ {p} $ give a solution to the Jacobi inversion problem. The exceptional points $ z $ for which $ \Phi ( w) \equiv 0 $ form in $ \mathbf C ^ {p} $ a set of lower dimension.

Explicit expressions for special Abelian functions that completely solve the Jacobi inversion problem can be constructed by using quotients of theta-functions of the form $ \theta _ {H} ( z)/ \theta ( z) $, where the common denominator is the theta-function of zero characteristic. When periods are added to the argument, the theta-functions are multiplied by specific multipliers. For quotients of theta-functions, after cancellation the only non-trivial multiplier is $ - 1 $. Consequently, the squares of the quotients are unchanged when periods are added to the argument, and one obtains Abelian functions with $ 2p $ periods.

Closely related to the Jacobi inversion problem is the important problem of constructing for a given system of theta-functions $ \theta _ {H} ( z) $ with a common matrix $ W $ satisfying the convergence conditions the corresponding algebraic function fields and the corresponding Riemann surface. For this construction to be possible, the distinct elements $ a _ {jk} $ of $ W $( there are $ p ( p + 1)/2 $ of them) must satisfy $ ( p - 2) ( p - 3)/2 $ additional relations, the investigation of which when $ p > 3 $ is a very difficult problem (see [1], [3], [4], [5]).

References

[1] N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)
[2] G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10
[3] A. Clebsch, P. Gordan, "Theorie der Abelschen Funktionen" , Teubner (1866)
[4] F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956)
[5] D. Mumford, "Structure of moduli spaces of curves and Abelian varieties" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 457–465

Comments

References

[a1] C.L. Siegel, "Topics in complex function theory" , 2 , Interscience (1971)
[a2] D. Mumford, "Tata lectures on Theta" , 1 , Birkhäuser (1983–1984)
How to Cite This Entry:
Jacobi inversion problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_inversion_problem&oldid=47457
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article