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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t0926002.png" />-function, of one complex variable''
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A quasi-doubly-periodic [[Entire function|entire function]] of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t0926003.png" />, that is, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t0926004.png" /> having, apart from a period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t0926005.png" />, also a quasi-period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t0926006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t0926007.png" />, the addition of which to the argument multiplies the value of the function by a certain factor. In other words, one has the identities (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t0926008.png" />):
+
{{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/t/t092/t092600/t0926009.png" /></td> </tr></table>
+
'' $  \theta $-
 +
function, of one complex variable''
 +
 
 +
A quasi-doubly-periodic [[Entire function|entire function]] of a complex variable  $  z $,
 +
that is, a function  $  \theta ( z) $
 +
having, apart from a period  $  \omega $,
 +
also a quasi-period  $  \omega \tau $,
 +
$  \mathop{\rm Im}  \tau > 0 $,
 +
the addition of which to the argument multiplies the value of the function by a certain factor. In other words, one has the identities (in  $  z $):
 +
 
 +
$$
 +
\theta ( z + \omega )  = \theta ( z),\ \
 +
\theta ( z + \omega \tau )  = \phi ( z) \theta ( z).
 +
$$
  
 
As a periodic entire function, a theta-function can always be represented by a series
 
As a periodic entire function, a theta-function can always be represented by a series
  
<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/t/t092/t092600/t09260010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\theta ( z)  = \
 +
\sum _ {n \in \mathbf Z }
 +
c _ {n}  \mathop{\rm exp} \left ( {
 +
\frac{2 \pi in } \omega
 +
} z \right ) ,
 +
$$
  
where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260011.png" /> must be chosen so as to ensure convergence. The series (1) is called a theta-series (because of the original notation). Other representations of theta-functions, for example as infinite products, are also possible.
+
where the coefficients $  c _ {n} $
 +
must be chosen so as to ensure convergence. The series (1) is called a theta-series (because of the original notation). Other representations of theta-functions, for example as infinite products, are also possible.
  
 
In applications one usually restricts oneself to multipliers of the form
 
In applications one usually restricts oneself to multipliers of the form
  
<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/t/t092/t092600/t09260012.png" /></td> </tr></table>
+
$$
 +
\phi ( z)  = q  \mathop{\rm exp} (- 2 \pi ikz),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260013.png" /> is a natural number, called the order or the weight of the theta-function, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260014.png" /> is a number. Convergence is ensured, for example, by using coefficients of the form
+
where $  k $
 +
is a natural number, called the order or the weight of the theta-function, and $  q $
 +
is a number. Convergence is ensured, for example, by using coefficients of the form
  
<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/t/t092/t092600/t09260015.png" /></td> </tr></table>
+
$$
 +
c _ {n}  =   \mathop{\rm exp} ( an  ^ {2} + 2bn + c),\ \
 +
\mathop{\rm Re}  a < 0.
 +
$$
  
 
In many problems it is convenient to take the theta-functions that satisfy the conditions
 
In many problems it is convenient to take the theta-functions that satisfy the conditions
  
<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/t/t092/t092600/t09260016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\theta ( z + 1)  = \theta ( z),
 +
$$
  
<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/t/t092/t092600/t09260017.png" /></td> </tr></table>
+
$$
 +
\theta ( z + \tau )  =   \mathop{\rm exp} (- 2 \pi ikz) \cdot \theta ( z).
 +
$$
  
All theta-functions of the form (2) of the same order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260018.png" /> form a vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260019.png" />. A basis for this vector space can be written in the form
+
All theta-functions of the form (2) of the same order $  k $
 +
form a vector space of dimension $  k $.  
 +
A basis for this vector space can be written in the form
  
<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/t/t092/t092600/t09260020.png" /></td> </tr></table>
+
$$
 +
\theta _ {r} ( z)  = \
 +
\sum _ {s \in \mathbf Z }
 +
\mathop{\rm exp} [ \pi i \tau s ( k ( s - 1) + 2r)
 +
+ 2 \pi i ( ks + r) z],
 +
$$
  
<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/t/t092/t092600/t09260021.png" /></td> </tr></table>
+
$$
 +
= 0 \dots k - 1.
 +
$$
  
 
Individual examples of theta-functions are already encountered in the work of J. Bernoulli (1713), L. Euler, and in the theory of heat conduction of J. Fourier. C.G.J. Jacobi subjected theta-functions to a systematic investigation, and picked out four special theta-functions, which formed the basis of his theory of elliptic functions (cf. [[Jacobi elliptic functions|Jacobi elliptic functions]]).
 
Individual examples of theta-functions are already encountered in the work of J. Bernoulli (1713), L. Euler, and in the theory of heat conduction of J. Fourier. C.G.J. Jacobi subjected theta-functions to a systematic investigation, and picked out four special theta-functions, which formed the basis of his theory of elliptic functions (cf. [[Jacobi elliptic functions|Jacobi elliptic functions]]).
  
Theta-functions of several complex variables arise as a natural generalization of theta-functions of one complex variable. They are constructed in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260022.png" /> be a row-matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260023.png" /> complex variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260024.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260025.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260026.png" />-th row of the identity matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260027.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260028.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260029.png" /> be an integer row-matrix, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260030.png" /> be a symmetric complex matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260031.png" /> such that the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260032.png" /> gives rise to a positive-definite quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260033.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260034.png" /> is the transpose of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260035.png" />.) The multiple theta-series
+
Theta-functions of several complex variables arise as a natural generalization of theta-functions of one complex variable. They are constructed in the following way. Let $  z = ( z _ {1} \dots z _ {p} ) $
 +
be a row-matrix of $  p $
 +
complex variables, $  p \geq  1 $,  
 +
let $  e _  \mu  $
 +
be the $  \mu $-
 +
th row of the identity matrix $  E $
 +
of order $  p $,  
 +
let $  n = ( n _ {1} \dots n _ {p} ) $
 +
be an integer row-matrix, and let $  A = \| a _ {\mu \nu }  \| $
 +
be a symmetric complex matrix of order $  p $
 +
such that the matrix $  \mathop{\rm Im}  A = \|  \mathop{\rm Im}  a _ {\mu \nu }  \| $
 +
gives rise to a positive-definite quadratic form $  n (  \mathop{\rm Im}  A) n  ^ {T} $.  
 +
(Here $  n  ^ {T} $
 +
is the transpose of the matrix $  n $.)
 +
The multiple theta-series
 +
 
 +
$$ \tag{3 }
 +
\theta ( z)  = \
 +
\sum _ {n \in \mathbf Z }
 +
\mathop{\rm exp} [ \pi ( nAn  ^ {T} + 2nz  ^ {T} ) ]
 +
$$
 +
 
 +
converges absolutely and uniformly on compacta in  $  \mathbf C  ^ {p} $,
 +
and hence defines an entire transcendental function of  $  p $
 +
complex variables  $  z _ {1} \dots z _ {p} $,
 +
called a theta-function of order  $  1 $.  
 +
The individual elements of the matrix  $  A $
 +
are called moduli, or parameters, of the theta-function  $  \theta ( z) $.  
 +
The number of moduli is equal to  $  p ( p + 1)/2 $.  
 +
A theta-function  $  \theta ( z) $
 +
of the first order satisfies the following basic identities (in  $  z $):
  
<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/t/t092/t092600/t09260036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{4 }
 +
\left .
  
converges absolutely and uniformly on compacta in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260037.png" />, and hence defines an entire transcendental function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260038.png" /> complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260039.png" />, called a theta-function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260041.png" />. The individual elements of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260042.png" /> are called moduli, or parameters, of the theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260043.png" />. The number of moduli is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260044.png" />. A theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260045.png" /> of the first order satisfies the following basic identities (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260046.png" />):
+
\begin{array}{c}
 +
\theta ( z + e _  \mu  )  = \theta ( z), \\
 +
\theta ( z + e _  \mu  A)  =   \mathop{\rm exp} [- \pi i ( a _ {\mu \mu }  + 2z _  \mu  )] \cdot \theta ( z), \\
 +
2 ( 1 + \delta _ {\mu \nu }  ) \pi
 +
\frac{\partial  \theta }{\partial  a _ {\mu \nu }  }
 +
  =
 +
\frac{\partial  ^ {2} \theta }{\partial  z _  \mu  \partial  z _  \nu  }
 +
,  \\
 +
\end{array}
  
<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/t/t092/t092600/t09260047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\right \}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260049.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260051.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260052.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260053.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260054.png" /> is the moduli system or system of periods and quasi-periods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260057.png" /> are arbitrary integer row-matrices, then the periodicity property of theta-functions can be written in its most general form as
+
where $  \mu , \nu = 1 \dots p $,  
 +
and $  \delta _ {\mu \nu }  = 1 $
 +
for $  \mu = \nu $
 +
and $  \delta _ {\mu \nu }  = 0 $
 +
for $  \mu \neq \nu $.  
 +
The $  ( p \times 2p) $-
 +
matrix $  S = ( E, A) $
 +
is the moduli system or system of periods and quasi-periods of $  \theta ( z) $.  
 +
If $  m = ( m _ {1} \dots m _ {p} ) $,  
 +
$  m  ^  \prime  = ( m _ {1}  ^  \prime  \dots m _ {p}  ^  \prime  ) $
 +
are arbitrary integer row-matrices, then the periodicity property of theta-functions can be written in its most general form as
  
<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/t/t092/t092600/t09260058.png" /></td> </tr></table>
+
$$
 +
\theta ( z + m  ^  \prime  + mA) =
 +
$$
  
<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/t/t092/t092600/t09260059.png" /></td> </tr></table>
+
$$
 +
= \
 +
\mathop{\rm exp} [- \pi ( mAm)  ^ {T} + 2m ( z + m  ^  \prime  )  ^ {T} ] \cdot \theta ( z).
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260061.png" /> be arbitrary complex row-matrices, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260062.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260063.png" />-matrix
+
Let $  \gamma = ( \gamma _ {1} \dots \gamma _ {p} ) $,  
 +
$  \gamma  ^  \prime  = ( \gamma _ {1}  ^  \prime  \dots \gamma _ {p}  ^  \prime  ) $
 +
be arbitrary complex row-matrices, and let $  \Gamma $
 +
be the $  ( 2 \times p) $-
 +
matrix
  
<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/t/t092/t092600/t09260064.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\begin{array}{c}
 +
\gamma \\
 +
\gamma  ^  \prime 
 +
\end{array}
 +
\
 +
\right \| .
 +
$$
  
 
Then the formula
 
Then the formula
  
<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/t/t092/t092600/t09260065.png" /></td> </tr></table>
+
$$
 +
\theta _  \Gamma  ( z)  = \
 +
\sum _ {n \in \mathbf Z  ^ {p} }
 +
\mathop{\rm exp} [ \pi ( n + \gamma ) A ( n + \gamma )  ^ {T} +
 +
2 ( n + \gamma ) ( z + \gamma  ^  \prime  )  ^ {T} ] =
 +
$$
 +
 
 +
$$
 +
= \
 +
\mathop{\rm exp} [ \pi i ( \gamma A \gamma  ^ {T} + 2 \gamma ( z + \gamma
 +
^  \prime  )  ^ {T} ) ] \cdot \theta ( z + \gamma  ^  \prime  + \gamma A)
 +
$$
 +
 
 +
defines a theta-function of order  $  1 $
 +
with characteristic (in general form)  $  \Gamma $.
 +
In this terminology the theta-function (3) has characteristic 0. The matrix  $  \Gamma $
 +
is also called the periodicity characteristic of the matrix  $  \gamma  ^  \prime  + \gamma A $.  
 +
One always has  $  \theta _ {- \Gamma }  (- z) = \theta _  \Gamma  ( z) $.  
 +
Property (4) generalizes to theta-functions of characteristic  $  \Gamma $:
  
<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/t/t092/t092600/t09260066.png" /></td> </tr></table>
+
$$ \tag{5 }
 +
\left .
  
defines a theta-function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260069.png" /> with characteristic (in general form) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260070.png" />. In this terminology the theta-function (3) has characteristic 0. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260071.png" /> is also called the periodicity characteristic of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260072.png" />. One always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260073.png" />. Property (4) generalizes to theta-functions of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260074.png" />:
+
\begin{array}{c}
 +
\theta _  \Gamma  ( z + e _  \mu  )  = \
 +
\mathop{\rm exp} ( 2 \pi i \gamma _  \mu  ) \cdot \theta _  \Gamma  ( z),  \\
 +
\theta _  \Gamma  ( z + e _  \mu  A) = \
 +
\mathop{\rm exp} [- \pi i
 +
( a _ {\mu \mu }  + 2 ( z _  \mu  - \gamma _  \mu  ^  \prime  ))]
 +
\cdot \theta _  \Gamma  ( z). \\
 +
\end{array}
  
<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/t/t092/t092600/t09260075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\right \}
 +
$$
  
The characteristic is said to be normal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260076.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260077.png" />.
+
The characteristic is said to be normal if $  0 \leq  \gamma _ {i} , \gamma _ {i}  ^  \prime  < 1 $
 +
for $  i = 1 \dots p $.
  
The most commonly used are fractional characteristics, where all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260079.png" /> are non-negative proper fractions with common denominator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260080.png" />. The simplest and most important case is of semi-integer or half characteristics, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260081.png" />. A semi-integer characteristic
+
The most commonly used are fractional characteristics, where all the $  \gamma _ {i} $
 +
and $  \gamma _ {i}  ^  \prime  $
 +
are non-negative proper fractions with common denominator $  \delta $.  
 +
The simplest and most important case is of semi-integer or half characteristics, where $  \delta = 2 $.  
 +
A semi-integer 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/t/t092/t092600/t09260082.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left \|
 +
\begin{array}{c}
 +
h \\
 +
h  ^  \prime 
 +
\end{array}
 +
\
 +
\right \|
 +
$$
  
can be thought of as being made up of the numbers 0 and 1 (usually a "theta-characteristictheta-characteristic" is used to mean just such a characteristic). For a theta-function with characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260083.png" /> equations (5) take the form
+
can be thought of as being made up of the numbers 0 and 1 (usually a "theta-characteristictheta-characteristic" is used to mean just such a characteristic). For a theta-function with characteristic $  H $
 +
equations (5) take the form
  
<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/t/t092/t092600/t09260084.png" /></td> </tr></table>
+
$$
 +
\theta _ {H} ( z + e _  \mu  )  = \
 +
(- 1) ^ {h _  \mu  } \cdot \theta _ {H} ( z),
 +
$$
  
<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/t/t092/t092600/t09260085.png" /></td> </tr></table>
+
$$
 +
\theta _ {H} ( z + e _  \mu  A)  = (- 1) ^ {h _  \mu  ^  \prime  }  \mathop{\rm exp}
 +
[- \pi i ( a _ {\mu \mu }  + 2z _  \mu  )] \cdot \theta _ {H} ( z).
 +
$$
  
A theta-characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260086.png" /> is called even or odd, depending on whether the theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260087.png" /> is even or odd. In other words, the theta-characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260088.png" /> is even or odd, depending on whether the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260089.png" /> is even or odd, since
+
A theta-characteristic $  H $
 +
is called even or odd, depending on whether the theta-function $  \theta _ {H} ( z) $
 +
is even or odd. In other words, the theta-characteristic $  H $
 +
is even or odd, depending on whether the number $  h  ^  \prime  h  ^ {T} $
 +
is even or odd, since
  
<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/t/t092/t092600/t09260090.png" /></td> </tr></table>
+
$$
 +
\theta _ {H} (- z)  = \
 +
(- 1) ^ {h  ^  \prime  h  ^ {T} }
 +
\cdot \theta _ {H} ( z).
 +
$$
  
There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260091.png" /> distinct theta-characteristics, of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260092.png" /> are even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260093.png" /> are odd. The theta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260094.png" /> takes the value zero at those points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260095.png" /> whose theta-characteristic
+
There are $  2  ^ {2p} $
 +
distinct theta-characteristics, of which $  2 ^ {p - 1 } ( 2  ^ {p} + 1) $
 +
are even and $  2 ^ {p - 1 } ( 2  ^ {p} - 1) $
 +
are odd. The theta-function $  \theta _ {H} ( z) $
 +
takes the value zero at those points $  ( g  ^  \prime  + gA)/2 $
 +
whose 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/t/t092/t092600/t09260096.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left \|
 +
\begin{array}{c}
 +
g \\
 +
g  ^  \prime 
 +
\end{array}
 +
\
 +
\right \|
 +
$$
  
yields an odd theta-characteristic when added to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260097.png" />. Jacobi used theta-functions with semi-integer characteristics in his theory of elliptic functions, except that his had period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260098.png" /> rather than 1.
+
yields an odd theta-characteristic when added to $  H $.  
 +
Jacobi used theta-functions with semi-integer characteristics in his theory of elliptic functions, except that his had period $  \pi i $
 +
rather than 1.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t09260099.png" /> be a natural number. An entire transcendental function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600100.png" /> is called a theta-function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600103.png" /> with characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600104.png" /> if it satisfies the identities
+
Let $  k $
 +
be a natural number. An entire transcendental function $  \theta _  \Gamma  ( z) $
 +
is called a theta-function of order $  k $
 +
with characteristic $  \Gamma $
 +
if it satisfies the identities
  
<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/t/t092/t092600/t092600105.png" /></td> </tr></table>
+
$$
 +
\theta _  \Gamma  ( z + e _  \mu  )  = \
 +
\mathop{\rm exp} ( 2 \pi i \gamma _  \mu  ) \cdot \theta _  \Gamma  ( z),
 +
$$
  
<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/t/t092/t092600/t092600106.png" /></td> </tr></table>
+
$$
 +
\theta _  \Gamma  ( z + e _  \mu  A)  =   \mathop{\rm exp} [- \pi i ( ka _ {\mu \mu }  + 2kz _  \mu  - 2 \gamma _  \mu  ^  \prime  )] \cdot \theta _  \Gamma  ( z).
 +
$$
  
For example, the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600107.png" /> theta-functions of order 1 is a theta-function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600108.png" />.
+
For example, the product of $  k $
 +
theta-functions of order 1 is a theta-function of order $  k $.
  
Using theta-functions of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600109.png" /> with semi-integer characteristics one can construct meromorphic Abelian functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600110.png" /> periods. The periods of an arbitrary Abelian function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600111.png" /> complex variables satisfy the Riemann–Frobenius relations, which yield convergence for the series defining the theta-functions with the corresponding system of moduli. According to a theorem formulated by K. Weierstrass and proved by H. Poincaré, an Abelian function can be represented as a quotient of entire theta-functions with corresponding moduli system. For the solution of the [[Jacobi inversion problem|Jacobi inversion problem]] on Abelian integrals, one constructs a special [[Riemann theta-function|Riemann theta-function]], whose argument is a system of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600112.png" /> on a Riemann surface.
+
Using theta-functions of order $  1 $
 +
with semi-integer characteristics one can construct meromorphic Abelian functions with $  2p $
 +
periods. The periods of an arbitrary Abelian function in $  p $
 +
complex variables satisfy the Riemann–Frobenius relations, which yield convergence for the series defining the theta-functions with the corresponding system of moduli. According to a theorem formulated by K. Weierstrass and proved by H. Poincaré, an Abelian function can be represented as a quotient of entire theta-functions with corresponding moduli system. For the solution of the [[Jacobi inversion problem|Jacobi inversion problem]] on Abelian integrals, one constructs a special [[Riemann theta-function|Riemann theta-function]], whose argument is a system of points $  w _ {1} \dots w _ {p} $
 +
on a Riemann surface.
  
 
See also [[Theta-series|Theta-series]].
 
See also [[Theta-series|Theta-series]].
Line 97: Line 298:
 
====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. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1964) pp. Chapt.8 {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Krazer, "Lehrbuch der Theta-Funktionen" , Chelsea, reprint (1970)</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>
 
<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. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1964) pp. Chapt.8 {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Krazer, "Lehrbuch der Theta-Funktionen" , Chelsea, reprint (1970)</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====
The conditions on the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600113.png" /> used in the construction of a theta-function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600114.png" /> variables (3) are precisely those needed in order that the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600115.png" /> defined by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600116.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600117.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600118.png" /> be an [[Abelian variety|Abelian variety]]. All Abelian varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600119.png" /> arise this way. Thus, there is a theta-function attached to any Abelian variety.
+
The conditions on the matrix $  A $
 +
used in the construction of a theta-function in $  p $
 +
variables (3) are precisely those needed in order that the lattice $  L $
 +
defined by the matrix $  ( I _ {p} A) $
 +
in $  \mathbf C  ^ {p} $
 +
be such that $  \mathbf C  ^ {p} / L $
 +
be an [[Abelian variety|Abelian variety]]. All Abelian varieties over $  \mathbf C $
 +
arise this way. Thus, there is a theta-function attached to any Abelian variety.
  
 
In particular, the conditions are satisfied by the canonical period matrix for Abelian differentials of the first kind on a Riemann surface (cf. [[Abelian differential|Abelian differential]]), thus determining the [[Jacobi variety|Jacobi variety]] of the Riemann surface and an associated theta-function.
 
In particular, the conditions are satisfied by the canonical period matrix for Abelian differentials of the first kind on a Riemann surface (cf. [[Abelian differential|Abelian differential]]), thus determining the [[Jacobi variety|Jacobi variety]] of the Riemann surface and an associated theta-function.
  
For a not necessarily canonical period matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600120.png" /> these relations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600121.png" /> (Riemann's equality, which becomes symmetry for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600122.png" /> in the canonical case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600123.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600124.png" /> is positive-definite Hermitean (Riemann's inequality, which becomes positive definiteness of the imaginary part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600125.png" /> in the canonical case (using the symmetry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092600/t092600126.png" />)), [[#References|[a8]]], p. 27. Together these two relations are sometimes known as the Riemann bilinear relations.
+
For a not necessarily canonical period matrix $  ( B, A) $
 +
these relations are $  A  ^ {T} B - B  ^ {T} A = 0 $(
 +
Riemann's equality, which becomes symmetry for $  A $
 +
in the canonical case when $  B = I _ {p} $)  
 +
and $  i B  ^ {T} \overline{A}\; - i A  ^ {T} \overline{B}\; $
 +
is positive-definite Hermitean (Riemann's inequality, which becomes positive definiteness of the imaginary part of $  A $
 +
in the canonical case (using the symmetry of $  A $)),  
 +
[[#References|[a8]]], p. 27. Together these two relations are sometimes known as the Riemann bilinear relations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.L. Siegel, "Topics in complex function theory" , '''2''' , Wiley (Interscience) (1971) {{MR|1013364}} {{MR|1008931}} {{MR|1008930}} {{MR|0476762}} {{MR|0257326}} {{ZBL|0719.11028}} {{ZBL|0635.30003}} {{ZBL|0635.30002}} {{ZBL|0257.32002}} {{ZBL|0184.11201}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </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><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Mumford, "On the equations defining abelian varieties I" ''Invent. Math.'' , '''1''' (1966) pp. 287–354 {{MR|0204427}} {{ZBL|0219.14024}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D. Mumford, "On the equations defining abelian varieties II-III" ''Invent. Math.'' , '''3''' (1967) pp. 71–135; 215–244</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press (1985) {{MR|2514037}} {{MR|1083353}} {{MR|0352106}} {{MR|0441983}} {{MR|0282985}} {{MR|0248146}} {{MR|0219542}} {{MR|0219541}} {{MR|0206003}} {{MR|0204427}} {{ZBL|0583.14015}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J.-i. Igusa, "Theta functions" , Springer (1972) {{MR|0325625}} {{ZBL|0251.14016}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R.C. Gunning, "Riemann surfaces and generalized theta functions" , Springer (1976) {{MR|0457787}} {{ZBL|0341.14013}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J.D. Fay, "Theta functions on Riemann surfaces" , Springer (1973) {{MR|0335789}} {{ZBL|0281.30013}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.L. Siegel, "Topics in complex function theory" , '''2''' , Wiley (Interscience) (1971) {{MR|1013364}} {{MR|1008931}} {{MR|1008930}} {{MR|0476762}} {{MR|0257326}} {{ZBL|0719.11028}} {{ZBL|0635.30003}} {{ZBL|0635.30002}} {{ZBL|0257.32002}} {{ZBL|0184.11201}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </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><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Mumford, "On the equations defining abelian varieties I" ''Invent. Math.'' , '''1''' (1966) pp. 287–354 {{MR|0204427}} {{ZBL|0219.14024}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> D. Mumford, "On the equations defining abelian varieties II-III" ''Invent. Math.'' , '''3''' (1967) pp. 71–135; 215–244</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press (1985) {{MR|2514037}} {{MR|1083353}} {{MR|0352106}} {{MR|0441983}} {{MR|0282985}} {{MR|0248146}} {{MR|0219542}} {{MR|0219541}} {{MR|0206003}} {{MR|0204427}} {{ZBL|0583.14015}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J.-i. Igusa, "Theta functions" , Springer (1972) {{MR|0325625}} {{ZBL|0251.14016}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R.C. Gunning, "Riemann surfaces and generalized theta functions" , Springer (1976) {{MR|0457787}} {{ZBL|0341.14013}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J.D. Fay, "Theta functions on Riemann surfaces" , Springer (1973) {{MR|0335789}} {{ZBL|0281.30013}} </TD></TR></table>

Revision as of 14:55, 7 June 2020


$ \theta $- function, of one complex variable

A quasi-doubly-periodic entire function of a complex variable $ z $, that is, a function $ \theta ( z) $ having, apart from a period $ \omega $, also a quasi-period $ \omega \tau $, $ \mathop{\rm Im} \tau > 0 $, the addition of which to the argument multiplies the value of the function by a certain factor. In other words, one has the identities (in $ z $):

$$ \theta ( z + \omega ) = \theta ( z),\ \ \theta ( z + \omega \tau ) = \phi ( z) \theta ( z). $$

As a periodic entire function, a theta-function can always be represented by a series

$$ \tag{1 } \theta ( z) = \ \sum _ {n \in \mathbf Z } c _ {n} \mathop{\rm exp} \left ( { \frac{2 \pi in } \omega } z \right ) , $$

where the coefficients $ c _ {n} $ must be chosen so as to ensure convergence. The series (1) is called a theta-series (because of the original notation). Other representations of theta-functions, for example as infinite products, are also possible.

In applications one usually restricts oneself to multipliers of the form

$$ \phi ( z) = q \mathop{\rm exp} (- 2 \pi ikz), $$

where $ k $ is a natural number, called the order or the weight of the theta-function, and $ q $ is a number. Convergence is ensured, for example, by using coefficients of the form

$$ c _ {n} = \mathop{\rm exp} ( an ^ {2} + 2bn + c),\ \ \mathop{\rm Re} a < 0. $$

In many problems it is convenient to take the theta-functions that satisfy the conditions

$$ \tag{2 } \theta ( z + 1) = \theta ( z), $$

$$ \theta ( z + \tau ) = \mathop{\rm exp} (- 2 \pi ikz) \cdot \theta ( z). $$

All theta-functions of the form (2) of the same order $ k $ form a vector space of dimension $ k $. A basis for this vector space can be written in the form

$$ \theta _ {r} ( z) = \ \sum _ {s \in \mathbf Z } \mathop{\rm exp} [ \pi i \tau s ( k ( s - 1) + 2r) + 2 \pi i ( ks + r) z], $$

$$ r = 0 \dots k - 1. $$

Individual examples of theta-functions are already encountered in the work of J. Bernoulli (1713), L. Euler, and in the theory of heat conduction of J. Fourier. C.G.J. Jacobi subjected theta-functions to a systematic investigation, and picked out four special theta-functions, which formed the basis of his theory of elliptic functions (cf. Jacobi elliptic functions).

Theta-functions of several complex variables arise as a natural generalization of theta-functions of one complex variable. They are constructed in the following way. Let $ z = ( z _ {1} \dots z _ {p} ) $ be a row-matrix of $ p $ complex variables, $ p \geq 1 $, let $ e _ \mu $ be the $ \mu $- th row of the identity matrix $ E $ of order $ p $, let $ n = ( n _ {1} \dots n _ {p} ) $ be an integer row-matrix, and let $ A = \| a _ {\mu \nu } \| $ be a symmetric complex matrix of order $ p $ such that the matrix $ \mathop{\rm Im} A = \| \mathop{\rm Im} a _ {\mu \nu } \| $ gives rise to a positive-definite quadratic form $ n ( \mathop{\rm Im} A) n ^ {T} $. (Here $ n ^ {T} $ is the transpose of the matrix $ n $.) The multiple theta-series

$$ \tag{3 } \theta ( z) = \ \sum _ {n \in \mathbf Z } \mathop{\rm exp} [ \pi ( nAn ^ {T} + 2nz ^ {T} ) ] $$

converges absolutely and uniformly on compacta in $ \mathbf C ^ {p} $, and hence defines an entire transcendental function of $ p $ complex variables $ z _ {1} \dots z _ {p} $, called a theta-function of order $ 1 $. The individual elements of the matrix $ A $ are called moduli, or parameters, of the theta-function $ \theta ( z) $. The number of moduli is equal to $ p ( p + 1)/2 $. A theta-function $ \theta ( z) $ of the first order satisfies the following basic identities (in $ z $):

$$ \tag{4 } \left . \begin{array}{c} \theta ( z + e _ \mu ) = \theta ( z), \\ \theta ( z + e _ \mu A) = \mathop{\rm exp} [- \pi i ( a _ {\mu \mu } + 2z _ \mu )] \cdot \theta ( z), \\ 2 ( 1 + \delta _ {\mu \nu } ) \pi \frac{\partial \theta }{\partial a _ {\mu \nu } } = \frac{\partial ^ {2} \theta }{\partial z _ \mu \partial z _ \nu } , \\ \end{array} \right \} $$

where $ \mu , \nu = 1 \dots p $, and $ \delta _ {\mu \nu } = 1 $ for $ \mu = \nu $ and $ \delta _ {\mu \nu } = 0 $ for $ \mu \neq \nu $. The $ ( p \times 2p) $- matrix $ S = ( E, A) $ is the moduli system or system of periods and quasi-periods of $ \theta ( z) $. If $ m = ( m _ {1} \dots m _ {p} ) $, $ m ^ \prime = ( m _ {1} ^ \prime \dots m _ {p} ^ \prime ) $ are arbitrary integer row-matrices, then the periodicity property of theta-functions can be written in its most general form as

$$ \theta ( z + m ^ \prime + mA) = $$

$$ = \ \mathop{\rm exp} [- \pi ( mAm) ^ {T} + 2m ( z + m ^ \prime ) ^ {T} ] \cdot \theta ( z). $$

Let $ \gamma = ( \gamma _ {1} \dots \gamma _ {p} ) $, $ \gamma ^ \prime = ( \gamma _ {1} ^ \prime \dots \gamma _ {p} ^ \prime ) $ be arbitrary complex row-matrices, and let $ \Gamma $ be the $ ( 2 \times p) $- matrix

$$ \left \| \begin{array}{c} \gamma \\ \gamma ^ \prime \end{array} \ \right \| . $$

Then the formula

$$ \theta _ \Gamma ( z) = \ \sum _ {n \in \mathbf Z ^ {p} } \mathop{\rm exp} [ \pi ( n + \gamma ) A ( n + \gamma ) ^ {T} + 2 ( n + \gamma ) ( z + \gamma ^ \prime ) ^ {T} ] = $$

$$ = \ \mathop{\rm exp} [ \pi i ( \gamma A \gamma ^ {T} + 2 \gamma ( z + \gamma ^ \prime ) ^ {T} ) ] \cdot \theta ( z + \gamma ^ \prime + \gamma A) $$

defines a theta-function of order $ 1 $ with characteristic (in general form) $ \Gamma $. In this terminology the theta-function (3) has characteristic 0. The matrix $ \Gamma $ is also called the periodicity characteristic of the matrix $ \gamma ^ \prime + \gamma A $. One always has $ \theta _ {- \Gamma } (- z) = \theta _ \Gamma ( z) $. Property (4) generalizes to theta-functions of characteristic $ \Gamma $:

$$ \tag{5 } \left . \begin{array}{c} \theta _ \Gamma ( z + e _ \mu ) = \ \mathop{\rm exp} ( 2 \pi i \gamma _ \mu ) \cdot \theta _ \Gamma ( z), \\ \theta _ \Gamma ( z + e _ \mu A) = \ \mathop{\rm exp} [- \pi i ( a _ {\mu \mu } + 2 ( z _ \mu - \gamma _ \mu ^ \prime ))] \cdot \theta _ \Gamma ( z). \\ \end{array} \right \} $$

The characteristic is said to be normal if $ 0 \leq \gamma _ {i} , \gamma _ {i} ^ \prime < 1 $ for $ i = 1 \dots p $.

The most commonly used are fractional characteristics, where all the $ \gamma _ {i} $ and $ \gamma _ {i} ^ \prime $ are non-negative proper fractions with common denominator $ \delta $. The simplest and most important case is of semi-integer or half characteristics, where $ \delta = 2 $. A semi-integer characteristic

$$ H = \ \left \| \begin{array}{c} h \\ h ^ \prime \end{array} \ \right \| $$

can be thought of as being made up of the numbers 0 and 1 (usually a "theta-characteristictheta-characteristic" is used to mean just such a characteristic). For a theta-function with characteristic $ H $ equations (5) take the form

$$ \theta _ {H} ( z + e _ \mu ) = \ (- 1) ^ {h _ \mu } \cdot \theta _ {H} ( z), $$

$$ \theta _ {H} ( z + e _ \mu A) = (- 1) ^ {h _ \mu ^ \prime } \mathop{\rm exp} [- \pi i ( a _ {\mu \mu } + 2z _ \mu )] \cdot \theta _ {H} ( z). $$

A theta-characteristic $ H $ is called even or odd, depending on whether the theta-function $ \theta _ {H} ( z) $ is even or odd. In other words, the theta-characteristic $ H $ is even or odd, depending on whether the number $ h ^ \prime h ^ {T} $ is even or odd, since

$$ \theta _ {H} (- z) = \ (- 1) ^ {h ^ \prime h ^ {T} } \cdot \theta _ {H} ( z). $$

There are $ 2 ^ {2p} $ distinct theta-characteristics, of which $ 2 ^ {p - 1 } ( 2 ^ {p} + 1) $ are even and $ 2 ^ {p - 1 } ( 2 ^ {p} - 1) $ are odd. The theta-function $ \theta _ {H} ( z) $ takes the value zero at those points $ ( g ^ \prime + gA)/2 $ whose theta-characteristic

$$ G = \ \left \| \begin{array}{c} g \\ g ^ \prime \end{array} \ \right \| $$

yields an odd theta-characteristic when added to $ H $. Jacobi used theta-functions with semi-integer characteristics in his theory of elliptic functions, except that his had period $ \pi i $ rather than 1.

Let $ k $ be a natural number. An entire transcendental function $ \theta _ \Gamma ( z) $ is called a theta-function of order $ k $ with characteristic $ \Gamma $ if it satisfies the identities

$$ \theta _ \Gamma ( z + e _ \mu ) = \ \mathop{\rm exp} ( 2 \pi i \gamma _ \mu ) \cdot \theta _ \Gamma ( z), $$

$$ \theta _ \Gamma ( z + e _ \mu A) = \mathop{\rm exp} [- \pi i ( ka _ {\mu \mu } + 2kz _ \mu - 2 \gamma _ \mu ^ \prime )] \cdot \theta _ \Gamma ( z). $$

For example, the product of $ k $ theta-functions of order 1 is a theta-function of order $ k $.

Using theta-functions of order $ 1 $ with semi-integer characteristics one can construct meromorphic Abelian functions with $ 2p $ periods. The periods of an arbitrary Abelian function in $ p $ complex variables satisfy the Riemann–Frobenius relations, which yield convergence for the series defining the theta-functions with the corresponding system of moduli. According to a theorem formulated by K. Weierstrass and proved by H. Poincaré, an Abelian function can be represented as a quotient of entire theta-functions with corresponding moduli system. For the solution of the Jacobi inversion problem on Abelian integrals, one constructs a special Riemann theta-function, whose argument is a system of points $ w _ {1} \dots w _ {p} $ on a Riemann surface.

See also Theta-series.

References

[1] N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)
[2] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) pp. Chapt.8 MR0173749 Zbl 0135.12101
[3] A. Krazer, "Lehrbuch der Theta-Funktionen" , Chelsea, reprint (1970)
[4] F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956) MR0079316 Zbl 0074.36601

Comments

The conditions on the matrix $ A $ used in the construction of a theta-function in $ p $ variables (3) are precisely those needed in order that the lattice $ L $ defined by the matrix $ ( I _ {p} A) $ in $ \mathbf C ^ {p} $ be such that $ \mathbf C ^ {p} / L $ be an Abelian variety. All Abelian varieties over $ \mathbf C $ arise this way. Thus, there is a theta-function attached to any Abelian variety.

In particular, the conditions are satisfied by the canonical period matrix for Abelian differentials of the first kind on a Riemann surface (cf. Abelian differential), thus determining the Jacobi variety of the Riemann surface and an associated theta-function.

For a not necessarily canonical period matrix $ ( B, A) $ these relations are $ A ^ {T} B - B ^ {T} A = 0 $( Riemann's equality, which becomes symmetry for $ A $ in the canonical case when $ B = I _ {p} $) and $ i B ^ {T} \overline{A}\; - i A ^ {T} \overline{B}\; $ is positive-definite Hermitean (Riemann's inequality, which becomes positive definiteness of the imaginary part of $ A $ in the canonical case (using the symmetry of $ A $)), [a8], p. 27. Together these two relations are sometimes known as the Riemann bilinear relations.

References

[a1] C.L. Siegel, "Topics in complex function theory" , 2 , Wiley (Interscience) (1971) MR1013364 MR1008931 MR1008930 MR0476762 MR0257326 Zbl 0719.11028 Zbl 0635.30003 Zbl 0635.30002 Zbl 0257.32002 Zbl 0184.11201
[a2] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[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
[a4] D. Mumford, "On the equations defining abelian varieties I" Invent. Math. , 1 (1966) pp. 287–354 MR0204427 Zbl 0219.14024
[a5] D. Mumford, "On the equations defining abelian varieties II-III" Invent. Math. , 3 (1967) pp. 71–135; 215–244
[a6] D. Mumford, "Abelian varieties" , Oxford Univ. Press (1985) MR2514037 MR1083353 MR0352106 MR0441983 MR0282985 MR0248146 MR0219542 MR0219541 MR0206003 MR0204427 Zbl 0583.14015
[a7] J.-i. Igusa, "Theta functions" , Springer (1972) MR0325625 Zbl 0251.14016
[a8] R.C. Gunning, "Riemann surfaces and generalized theta functions" , Springer (1976) MR0457787 Zbl 0341.14013
[a9] J.D. Fay, "Theta functions on Riemann surfaces" , Springer (1973) MR0335789 Zbl 0281.30013
How to Cite This Entry:
Theta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Theta-function&oldid=49468
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article