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 |
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