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Difference between revisions of "Riemann theta-function"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.G. Chebotarev,   "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich,   "Introduction to the classical theory of Abelian functions" , Moscow (1979) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Krazer,   "Lehrbuch der Thetafunktionen" , Chelsea, reprint (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Conforto,   "Abelsche Funktionen und algebraische Geometrie" , Springer (1956)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich, "Introduction to the classical theory of Abelian functions" , Moscow (1979) (In Russian) {{MR|0544988}} {{ZBL|0493.14023}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Krazer, "Lehrbuch der Thetafunktionen" , Chelsea, reprint (1970) {{MR|}} {{ZBL|0212.42901}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956) {{MR|0079316}} {{ZBL|0074.36601}} </TD></TR></table>
  
  
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====References====
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , '''1–2''' , Wiley (Interscience) (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford,   "Tata lectures on Theta" , '''1–2''' , Birkhäuser (1983–1984)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1–2''' , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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 {{MR|0934264}} {{ZBL|0696.14019}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford, "Tata lectures on Theta" , '''1–2''' , Birkhäuser (1983–1984) {{MR|2352717}} {{MR|2307769}} {{MR|2307768}} {{MR|1116553}} {{MR|0742776}} {{MR|0688651}} {{ZBL|1124.14043}} {{ZBL|1112.14003}} {{ZBL|1112.14002}} {{ZBL|0744.14033}} {{ZBL|0549.14014}} {{ZBL|0509.14049}} </TD></TR></table>

Revision as of 21:56, 30 March 2012

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
How to Cite This Entry:
Riemann theta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_theta-function&oldid=23964
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article