Theta-function

-function, of one complex variable

A quasi-doubly-periodic entire function of a complex variable , that is, a function having, apart from a period , also a quasi-period , , the addition of which to the argument multiplies the value of the function by a certain factor. In other words, one has the identities (in ):

As a periodic entire function, a theta-function can always be represented by a series

(1)

where the coefficients must be chosen so as to ensure convergence. The series (1) is called a theta-series (because of the original notation). Other representations of theta-functions, for example as infinite products, are also possible.

In applications one usually restricts oneself to multipliers of the form

where is a natural number, called the order or the weight of the theta-function, and is a number. Convergence is ensured, for example, by using coefficients of the form

In many problems it is convenient to take the theta-functions that satisfy the conditions

(2)

All theta-functions of the form (2) of the same order form a vector space of dimension . A basis for this vector space can be written in the form

Individual examples of theta-functions are already encountered in the work of J. Bernoulli (1713), L. Euler, and in the theory of heat conduction of J. Fourier. C.G.J. Jacobi subjected theta-functions to a systematic investigation, and picked out four special theta-functions, which formed the basis of his theory of elliptic functions (cf. Jacobi elliptic functions).

Theta-functions of several complex variables arise as a natural generalization of theta-functions of one complex variable. They are constructed in the following way. Let be a row-matrix of complex variables, , let be the -th row of the identity matrix of order , let be an integer row-matrix, and let be a symmetric complex matrix of order such that the matrix gives rise to a positive-definite quadratic form . (Here is the transpose of the matrix .) The multiple theta-series

(3)

converges absolutely and uniformly on compacta in , and hence defines an entire transcendental function of complex variables , called a theta-function of order . The individual elements of the matrix are called moduli, or parameters, of the theta-function . The number of moduli is equal to . A theta-function of the first order satisfies the following basic identities (in ):

(4)

where , and for and for . The -matrix is the moduli system or system of periods and quasi-periods of . If , are arbitrary integer row-matrices, then the periodicity property of theta-functions can be written in its most general form as

Let , be arbitrary complex row-matrices, and let be the -matrix

Then the formula

defines a theta-function of order with characteristic (in general form) . In this terminology the theta-function (3) has characteristic 0. The matrix is also called the periodicity characteristic of the matrix . One always has . Property (4) generalizes to theta-functions of characteristic :

(5)

The characteristic is said to be normal if for .

The most commonly used are fractional characteristics, where all the and are non-negative proper fractions with common denominator . The simplest and most important case is of semi-integer or half characteristics, where . A semi-integer characteristic

can be thought of as being made up of the numbers 0 and 1 (usually a "theta-characteristictheta-characteristic" is used to mean just such a characteristic). For a theta-function with characteristic equations (5) take the form

A theta-characteristic is called even or odd, depending on whether the theta-function is even or odd. In other words, the theta-characteristic is even or odd, depending on whether the number is even or odd, since

There are distinct theta-characteristics, of which are even and are odd. The theta-function takes the value zero at those points whose theta-characteristic

yields an odd theta-characteristic when added to . Jacobi used theta-functions with semi-integer characteristics in his theory of elliptic functions, except that his had period rather than 1.

Let be a natural number. An entire transcendental function is called a theta-function of order with characteristic if it satisfies the identities

For example, the product of theta-functions of order 1 is a theta-function of order .

Using theta-functions of order with semi-integer characteristics one can construct meromorphic Abelian functions with periods. The periods of an arbitrary Abelian function in complex variables satisfy the Riemann–Frobenius relations, which yield convergence for the series defining the theta-functions with the corresponding system of moduli. According to a theorem formulated by K. Weierstrass and proved by H. Poincaré, an Abelian function can be represented as a quotient of entire theta-functions with corresponding moduli system. For the solution of the Jacobi inversion problem on Abelian integrals, one constructs a special Riemann theta-function, whose argument is a system of points on a Riemann surface.

See also Theta-series.

References

[1]	N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) pp. Chapt. 9 (In Russian)
[2]	A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) pp. Chapt.8 MR0173749 Zbl 0135.12101
[3]	A. Krazer, "Lehrbuch der Theta-Funktionen" , Chelsea, reprint (1970)
[4]	F. Conforto, "Abelsche Funktionen und algebraische Geometrie" , Springer (1956) MR0079316 Zbl 0074.36601

Comments

The conditions on the matrix used in the construction of a theta-function in variables (3) are precisely those needed in order that the lattice defined by the matrix in be such that be an Abelian variety. All Abelian varieties over arise this way. Thus, there is a theta-function attached to any Abelian variety.

In particular, the conditions are satisfied by the canonical period matrix for Abelian differentials of the first kind on a Riemann surface (cf. Abelian differential), thus determining the Jacobi variety of the Riemann surface and an associated theta-function.

For a not necessarily canonical period matrix these relations are (Riemann's equality, which becomes symmetry for in the canonical case when ) and is positive-definite Hermitean (Riemann's inequality, which becomes positive definiteness of the imaginary part of in the canonical case (using the symmetry of )), [a8], p. 27. Together these two relations are sometimes known as the Riemann bilinear relations.

References

[a1]	C.L. Siegel, "Topics in complex function theory" , 2 , Wiley (Interscience) (1971) MR1013364 MR1008931 MR1008930 MR0476762 MR0257326 Zbl 0719.11028 Zbl 0635.30003 Zbl 0635.30002 Zbl 0257.32002 Zbl 0184.11201
[a2]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[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
[a4]	D. Mumford, "On the equations defining abelian varieties I" Invent. Math. , 1 (1966) pp. 287–354 MR0204427 Zbl 0219.14024
[a5]	D. Mumford, "On the equations defining abelian varieties II-III" Invent. Math. , 3 (1967) pp. 71–135; 215–244
[a6]	D. Mumford, "Abelian varieties" , Oxford Univ. Press (1985) MR2514037 MR1083353 MR0352106 MR0441983 MR0282985 MR0248146 MR0219542 MR0219541 MR0206003 MR0204427 Zbl 0583.14015
[a7]	J.-i. Igusa, "Theta functions" , Springer (1972) MR0325625 Zbl 0251.14016
[a8]	R.C. Gunning, "Riemann surfaces and generalized theta functions" , Springer (1976) MR0457787 Zbl 0341.14013
[a9]	J.D. Fay, "Theta functions on Riemann surfaces" , Springer (1973) MR0335789 Zbl 0281.30013