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.

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