# Riemann theta-function

A superposition of theta-functions (cf. Theta-function) of the first order , , with half-integral characteristics , and of Abelian integrals (cf. Abelian integral) of the first order, used by B. Riemann in 1857 to solve the Jacobi inversion problem.

Let be an algebraic equation which defines a compact Riemann surface of genus ; let be a basis of the Abelian differentials (cf. Abelian differential) of the first kind on with -dimensional period matrix Let be the vector of basis Abelian integrals of the first kind, where is a fixed system of points in and is a varying system of points in . For any theta-characteristic where the integers take the values 0 or 1 only, it is possible to construct a theta-function with period matrix such that satisfies the fundamental relations (1)

Here is the -th row vector of the identity matrix , . If is a fixed vector in the complex space , then the Riemann theta-function can be represented as the superposition (2)

In the domain that is obtained from after removal of sections along the cycles of a homology basis of , 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 with zero characteristic . In particular, the zeros of the corresponding Riemann theta-function determine the solution to the Jacobi inversion problem.

Quotients of Riemann theta-functions of the type with a common denominator are used to construct analytic expressions solving the inversion problem. It can be seen from (1) that such quotients can have as non-trivial factors only , and the squares of these quotients are single-valued meromorphic functions on , i.e. rational point functions on the surface . The squares and other rational functions in quotients of theta-functions used in this case are special Abelian functions (cf. Abelian function) with periods. The specialization is expressed by the fact that different elements of the symmetric matrix , when , are connected by definite relations imposed by the conformal structure of , so that remain independent among them.

Riemann theta-functions constructed for a hyper-elliptic surface , when where is a polynomial of degree without multiple roots, are sometimes referred to as hyper-elliptic theta-functions.