# 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