# Riemann theta-function

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 |

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Riemann theta-function.

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