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A superposition of theta-functions (cf. [[Theta-function|Theta-function]]) of the first order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820902.png" />, with half-integral characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820903.png" />, and of Abelian integrals (cf. [[Abelian integral|Abelian integral]]) of the first order, used by B. Riemann in 1857 to solve the [[Jacobi inversion problem|Jacobi inversion problem]].
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820904.png" /> be an algebraic equation which defines a compact [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820905.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820906.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820907.png" /> be a basis of the Abelian differentials (cf. [[Abelian differential|Abelian differential]]) of the first kind on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820908.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r0820909.png" />-dimensional period matrix
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{{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/r/r082/r082090/r08209010.png" /></td> </tr></table>
+
A superposition of theta-functions (cf. [[Theta-function|Theta-function]]) of the first order  $  \Theta _ {H} ( u) $,
 +
$  u = ( u _ {1} \dots u _ {p} ) $,
 +
with half-integral characteristics  $  H $,
 +
and of Abelian integrals (cf. [[Abelian integral|Abelian integral]]) of the first order, used by B. Riemann in 1857 to solve the [[Jacobi inversion problem|Jacobi inversion problem]].
 +
 
 +
Let  $  F( u, w) = 0 $
 +
be an algebraic equation which defines a compact [[Riemann surface|Riemann surface]]  $  F $
 +
of genus  $  p $;  
 +
let  $  \phi _ {1} \dots \phi _ {p} $
 +
be a basis of the Abelian differentials (cf. [[Abelian differential|Abelian differential]]) of the first kind on  $  F $
 +
with  $  ( p \times 2p) $-
 +
dimensional period matrix
 +
 
 +
$$
 +
= \| \pi i E, A \|  = \left \|
 +
 
 +
\begin{array}{cccccc}
 +
\pi i  &\dots  & 0  &a _ {11}  &\dots  &a _ {1p}  \\
 +
0  &\dots  & 0 &a _ {21}  &\dots  &a _ {2p}  \\
 +
\cdot  &\dots  &\cdot  &\cdot  &\dots  &\cdot  \\
 +
0  &\dots  &\pi i  &a _ {p1}  &\dots  &a _ {pp}  \\
 +
\end{array}
 +
 
 +
\right \| .
 +
$$
  
 
Let
 
Let
  
<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/r/r082/r082090/r08209011.png" /></td> </tr></table>
+
$$
 +
u( w)  = \left ( u _ {1} ( w _ {1} ) =
 +
\int\limits _ { c _ {1} } ^ { {w _ 1 } }
 +
\phi _ {1} \dots u _ {p} ( w _ {p} ) =
 +
\int\limits _ { c _ {p} } ^ { {w _ p } } \phi _ {p} \right )
 +
$$
  
be the vector of basis Abelian integrals of the first kind, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209012.png" /> is a fixed system of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209014.png" /> is a varying system of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209015.png" />. For any theta-characteristic
+
be the vector of basis Abelian integrals of the first kind, where $  ( c _ {1} \dots c _ {p} ) $
 +
is a fixed system of points in $  F $
 +
and $  w = ( w _ {1} \dots w _ {p} ) $
 +
is a varying system of points in $  F $.  
 +
For any theta-characteristic
  
<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/r/r082/r082090/r08209016.png" /></td> </tr></table>
+
$$
 +
= \left \|
  
where the integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209017.png" /> take the values 0 or 1 only, it is possible to construct a theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209018.png" /> with period matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209020.png" /> satisfies the fundamental relations
+
\begin{array}{c}
 +
h  \\
 +
h  ^  \prime  \\
 +
\end{array}
 +
\right \|  = \
 +
\left \|
 +
\begin{array}{ccc}
 +
h _ {1}  &\dots  &h _ {p}  \\
 +
h _ {1}  ^  \prime  &\dots  &h _ {p}  ^  \prime  \\
 +
\end{array}
 +
\right \| ,
 +
$$
  
<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/r/r082/r082090/r08209021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
where the integers  $  h _ {i} , h _ {i}  ^  \prime  $
 +
take the values 0 or 1 only, it is possible to construct a theta-function  $  \Theta _ {H} ( u) $
 +
with period matrix  $  W $
 +
such that  $  \Theta _ {H} ( u) $
 +
satisfies the fundamental relations
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209022.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209023.png" />-th row vector of the identity matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209026.png" /> is a fixed vector in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209027.png" />, then the Riemann theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209028.png" /> can be represented as the superposition
+
$$ \tag{1 }
 +
\left .  
 +
\begin{array}{c}
 +
\Theta _ {H} ( u + \pi ie _  \mu  )  = (- 1) ^ {h _  \mu  } \Theta _ {H} ( u), \\
 +
\Theta _ {H} ( u + e _  \mu  A)  = (- 1) ^ {h _  \mu  ^  \prime  }
 +
\mathop{\rm exp} (- a _ {\mu \mu }  - 2u _  \mu  ) \cdot \Theta _ {H} ( u). \\
 +
\end{array}
 +
\right \}
 +
$$
  
<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/r/r082/r082090/r08209029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
Here  $  e _  \mu  $
 +
is the  $  \mu $-
 +
th row vector of the identity matrix  $  E $,
 +
$  \mu = 1 \dots p $.  
 +
If  $  z = ( z _ {1} \dots z _ {p} ) $
 +
is a fixed vector in the complex space  $  \mathbf C  ^ {p} $,
 +
then the Riemann theta-function  $  \Phi _ {H} ( w) $
 +
can be represented as the superposition
  
In the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209030.png" /> that is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209031.png" /> after removal of sections along the cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209032.png" /> of a homology basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209033.png" />, the Riemann theta-functions (2) are everywhere defined and analytic. When crossing through sections the Riemann theta-functions, as a rule, are multiplied by factors whose values are determined from the fundamental relations (1). In this case, a special role is played by the theta-function of the first order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209034.png" /> with zero characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209035.png" />. In particular, the zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209036.png" /> of the corresponding Riemann theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209037.png" /> determine the solution to the Jacobi inversion problem.
+
$$ \tag{2 }
 +
\Phi _ {H} ( w)  = \Theta _ {H} ( u( w) - z).
 +
$$
  
Quotients of Riemann theta-functions of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209038.png" /> with a common denominator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209039.png" /> are used to construct analytic expressions solving the inversion problem. It can be seen from (1) that such quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209040.png" /> can have as non-trivial factors only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209041.png" />, and the squares of these quotients are single-valued meromorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209042.png" />, i.e. rational point functions on the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209043.png" />. The squares and other rational functions in quotients of theta-functions used in this case are special Abelian functions (cf. [[Abelian function|Abelian function]]) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209044.png" /> periods. The specialization is expressed by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209045.png" /> different elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209046.png" /> of the symmetric matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209047.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209048.png" />, are connected by definite relations imposed by the conformal structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209049.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209050.png" /> remain independent among them.
+
In the domain  $  F ^ { \star } $
 +
that is obtained from  $  F $
 +
after removal of sections along the cycles  $  a _ {1} , b _ {1} \dots a _ {p} , b _ {p} $
 +
of a homology basis of $  F $,
 +
the Riemann theta-functions (2) are everywhere defined and analytic. When crossing through sections the Riemann theta-functions, as a rule, are multiplied by factors whose values are determined from the fundamental relations (1). In this case, a special role is played by the theta-function of the first order  $  \Phi ( u) = \Theta _ {0} ( u) $
 +
with zero characteristic  $  H = 0 $.  
 +
In particular, the zeros  $  \eta _ {1} \dots \eta _ {p} $
 +
of the corresponding Riemann theta-function  $  \Phi ( w) = \Phi _ {0} ( w) $
 +
determine the solution to the Jacobi inversion problem.
  
Riemann theta-functions constructed for a hyper-elliptic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209051.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209052.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209053.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082090/r08209054.png" /> without multiple roots, are sometimes referred to as hyper-elliptic theta-functions.
+
Quotients of Riemann theta-functions of the type  $  \Psi _ {H} ( w) = \Theta _ {H} ( u( w)) $
 +
with a common denominator  $  \Psi ( w) = \Theta ( u( w)) = \Theta _ {0} ( u( w)) $
 +
are used to construct analytic expressions solving the inversion problem. It can be seen from (1) that such quotients  $  \Psi _ {H} ( w)/ \Psi ( w) $
 +
can have as non-trivial factors only  $  - 1 $,
 +
and the squares of these quotients are single-valued meromorphic functions on  $  F $,
 +
i.e. rational point functions on the surface  $  F $.  
 +
The squares and other rational functions in quotients of theta-functions used in this case are special Abelian functions (cf. [[Abelian function|Abelian function]]) with  $  2p $
 +
periods. The specialization is expressed by the fact that  $  p( p+ 1)/2 $
 +
different elements  $  a _ {\mu \nu }  $
 +
of the symmetric matrix  $  A $,
 +
when  $  p > 3 $,
 +
are connected by definite relations imposed by the conformal structure of  $  F $,
 +
so that  $  3( p- 1) $
 +
remain independent among them.
 +
 
 +
Riemann theta-functions constructed for a hyper-elliptic surface  $  F $,  
 +
when $  F( u, w) = w  ^ {2} - P( u) $
 +
where $  P( u) $
 +
is a polynomial of degree $  n \geq  5 $
 +
without multiple roots, are sometimes referred to as hyper-elliptic theta-functions.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.G. Chebotarev,   "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich,   "Introduction to the classical theory of Abelian functions" , Moscow (1979) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Krazer,   "Lehrbuch der Thetafunktionen" , Chelsea, reprint (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Conforto,   "Abelsche Funktionen und algebraische Geometrie" , Springer (1956)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich, "Introduction to the classical theory of Abelian functions" , Moscow (1979) (In Russian) {{MR|0544988}} {{ZBL|0493.14023}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Krazer, "Lehrbuch der Thetafunktionen" , Chelsea, reprint (1970) {{MR|}} {{ZBL|0212.42901}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956) {{MR|0079316}} {{ZBL|0074.36601}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
Line 36: Line 135:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , '''1–2''' , Wiley (Interscience) (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello,   "Periods of Abelian integrals, theta functions, and differential equations of KdV type" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''I''' , Amer. Math. Soc. (1987) pp. 623–627</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford,   "Tata lectures on Theta" , '''1–2''' , Birkhäuser (1983–1984)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1–2''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Arbarello, "Periods of Abelian integrals, theta functions, and differential equations of KdV type" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''I''' , Amer. Math. Soc. (1987) pp. 623–627 {{MR|0934264}} {{ZBL|0696.14019}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford, "Tata lectures on Theta" , '''1–2''' , Birkhäuser (1983–1984) {{MR|2352717}} {{MR|2307769}} {{MR|2307768}} {{MR|1116553}} {{MR|0742776}} {{MR|0688651}} {{ZBL|1124.14043}} {{ZBL|1112.14003}} {{ZBL|1112.14002}} {{ZBL|0744.14033}} {{ZBL|0549.14014}} {{ZBL|0509.14049}} </TD></TR></table>

Latest revision as of 14:55, 7 June 2020


A superposition of theta-functions (cf. Theta-function) of the first order $ \Theta _ {H} ( u) $, $ u = ( u _ {1} \dots u _ {p} ) $, with half-integral characteristics $ H $, and of Abelian integrals (cf. Abelian integral) of the first order, used by B. Riemann in 1857 to solve the Jacobi inversion problem.

Let $ F( u, w) = 0 $ be an algebraic equation which defines a compact Riemann surface $ F $ of genus $ p $; let $ \phi _ {1} \dots \phi _ {p} $ be a basis of the Abelian differentials (cf. Abelian differential) of the first kind on $ F $ with $ ( p \times 2p) $- dimensional period matrix

$$ W = \| \pi i E, A \| = \left \| \begin{array}{cccccc} \pi i &\dots & 0 &a _ {11} &\dots &a _ {1p} \\ 0 &\dots & 0 &a _ {21} &\dots &a _ {2p} \\ \cdot &\dots &\cdot &\cdot &\dots &\cdot \\ 0 &\dots &\pi i &a _ {p1} &\dots &a _ {pp} \\ \end{array} \right \| . $$

Let

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

be the vector of basis Abelian integrals of the first kind, where $ ( c _ {1} \dots c _ {p} ) $ is a fixed system of points in $ F $ and $ w = ( w _ {1} \dots w _ {p} ) $ is a varying system of points in $ F $. For any theta-characteristic

$$ H = \left \| \begin{array}{c} h \\ h ^ \prime \\ \end{array} \right \| = \ \left \| \begin{array}{ccc} h _ {1} &\dots &h _ {p} \\ h _ {1} ^ \prime &\dots &h _ {p} ^ \prime \\ \end{array} \right \| , $$

where the integers $ h _ {i} , h _ {i} ^ \prime $ take the values 0 or 1 only, it is possible to construct a theta-function $ \Theta _ {H} ( u) $ with period matrix $ W $ such that $ \Theta _ {H} ( u) $ satisfies the fundamental relations

$$ \tag{1 } \left . \begin{array}{c} \Theta _ {H} ( u + \pi ie _ \mu ) = (- 1) ^ {h _ \mu } \Theta _ {H} ( u), \\ \Theta _ {H} ( u + e _ \mu A) = (- 1) ^ {h _ \mu ^ \prime } \mathop{\rm exp} (- a _ {\mu \mu } - 2u _ \mu ) \cdot \Theta _ {H} ( u). \\ \end{array} \right \} $$

Here $ e _ \mu $ is the $ \mu $- th row vector of the identity matrix $ E $, $ \mu = 1 \dots p $. If $ z = ( z _ {1} \dots z _ {p} ) $ is a fixed vector in the complex space $ \mathbf C ^ {p} $, then the Riemann theta-function $ \Phi _ {H} ( w) $ can be represented as the superposition

$$ \tag{2 } \Phi _ {H} ( w) = \Theta _ {H} ( u( w) - z). $$

In the domain $ F ^ { \star } $ that is obtained from $ F $ after removal of sections along the cycles $ a _ {1} , b _ {1} \dots a _ {p} , b _ {p} $ of a homology basis of $ F $, the Riemann theta-functions (2) are everywhere defined and analytic. When crossing through sections the Riemann theta-functions, as a rule, are multiplied by factors whose values are determined from the fundamental relations (1). In this case, a special role is played by the theta-function of the first order $ \Phi ( u) = \Theta _ {0} ( u) $ with zero characteristic $ H = 0 $. In particular, the zeros $ \eta _ {1} \dots \eta _ {p} $ of the corresponding Riemann theta-function $ \Phi ( w) = \Phi _ {0} ( w) $ determine the solution to the Jacobi inversion problem.

Quotients of Riemann theta-functions of the type $ \Psi _ {H} ( w) = \Theta _ {H} ( u( w)) $ with a common denominator $ \Psi ( w) = \Theta ( u( w)) = \Theta _ {0} ( u( w)) $ are used to construct analytic expressions solving the inversion problem. It can be seen from (1) that such quotients $ \Psi _ {H} ( w)/ \Psi ( w) $ can have as non-trivial factors only $ - 1 $, and the squares of these quotients are single-valued meromorphic functions on $ F $, i.e. rational point functions on the surface $ F $. The squares and other rational functions in quotients of theta-functions used in this case are special Abelian functions (cf. Abelian function) with $ 2p $ periods. The specialization is expressed by the fact that $ p( p+ 1)/2 $ different elements $ a _ {\mu \nu } $ of the symmetric matrix $ A $, when $ p > 3 $, are connected by definite relations imposed by the conformal structure of $ F $, so that $ 3( p- 1) $ remain independent among them.

Riemann theta-functions constructed for a hyper-elliptic surface $ F $, when $ F( u, w) = w ^ {2} - P( u) $ where $ P( u) $ is a polynomial of degree $ n \geq 5 $ without multiple roots, are sometimes referred to as hyper-elliptic theta-functions.

References

[1] N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)
[2] A.I. Markushevich, "Introduction to the classical theory of Abelian functions" , Moscow (1979) (In Russian) MR0544988 Zbl 0493.14023
[3] A. Krazer, "Lehrbuch der Thetafunktionen" , Chelsea, reprint (1970) Zbl 0212.42901
[4] F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956) MR0079316 Zbl 0074.36601

Comments

Nowadays a Riemann theta-function is defined as a theta-function of the first order with half-integral characteristic corresponding to the Jacobi variety of an algebraic curve (or a compact Riemann surface). A general theta-function corresponds to an arbitrary Abelian variety. The problem of distinguishing the Riemann theta-functions among the general theta-functions is called the Schottky problem. It has been solved (see Schottky problem).

References

[a1] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1–2 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a2] E. Arbarello, "Periods of Abelian integrals, theta functions, and differential equations of KdV type" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , I , Amer. Math. Soc. (1987) pp. 623–627 MR0934264 Zbl 0696.14019
[a3] D. Mumford, "Tata lectures on Theta" , 1–2 , Birkhäuser (1983–1984) MR2352717 MR2307769 MR2307768 MR1116553 MR0742776 MR0688651 Zbl 1124.14043 Zbl 1112.14003 Zbl 1112.14002 Zbl 0744.14033 Zbl 0549.14014 Zbl 0509.14049
How to Cite This Entry:
Riemann theta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_theta-function&oldid=16988
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article