Theta-function

$ \theta $-function, of one complex variable

A quasi-doubly-periodic entire function of a complex variable $ z $, that is, a function $ \theta ( z) $ having, apart from a period $ \omega $, also a quasi-period $ \omega \tau $, $ \mathop{\rm Im} \tau > 0 $, 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 $ z $):

$$ \theta ( z + \omega ) = \theta ( z),\ \ \theta ( z + \omega \tau ) = \phi ( z) \theta ( z). $$

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

$$ \tag{1 } \theta ( z) = \ \sum _ {n \in \mathbf Z } c _ {n} \mathop{\rm exp} \left ( { \frac{2 \pi in } \omega } z \right ) , $$

where the coefficients $ c _ {n} $ 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

$$ \phi ( z) = q \mathop{\rm exp} (- 2 \pi ikz), $$

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

$$ c _ {n} = \mathop{\rm exp} ( an ^ {2} + 2bn + c),\ \ \mathop{\rm Re} a < 0. $$

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

$$ \tag{2 } \theta ( z + 1) = \theta ( z), $$

$$ \theta ( z + \tau ) = \mathop{\rm exp} (- 2 \pi ikz) \cdot \theta ( z). $$

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

$$ \theta _ {r} ( z) = \ \sum _ {s \in \mathbf Z } \mathop{\rm exp} [ \pi i \tau s ( k ( s - 1) + 2r) + 2 \pi i ( ks + r) z], $$

$$ r = 0, \dots, k - 1. $$

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 $ z = ( z _ {1}, \dots, z _ {p} ) $ be a row-matrix of $ p $ complex variables, $ p \geq 1 $, let $ e _ \mu $ be the $ \mu $-th row of the identity matrix $ E $ of order $ p $, let $ n = ( n _ {1}, \dots, n _ {p} ) $ be an integer row-matrix, and let $ A = \| a _ {\mu \nu } \| $ be a symmetric complex matrix of order $ p $ such that the matrix $ \mathop{\rm Im} A = \| \mathop{\rm Im} a _ {\mu \nu } \| $ gives rise to a positive-definite quadratic form $ n ( \mathop{\rm Im} A) n ^ {T} $. (Here $ n ^ {T} $ is the transpose of the matrix $ n $.) The multiple theta-series

$$ \tag{3 } \theta ( z) = \ \sum _ {n \in \mathbf Z } \mathop{\rm exp} [ \pi ( nAn ^ {T} + 2nz ^ {T} ) ] $$

converges absolutely and uniformly on compacta in $ \mathbf C ^ {p} $, and hence defines an entire transcendental function of $ p $ complex variables $ z _ {1}, \dots, z _ {p} $, called a theta-function of order $ 1 $. The individual elements of the matrix $ A $ are called moduli, or parameters, of the theta-function $ \theta ( z) $. The number of moduli is equal to $ p ( p + 1)/2 $. A theta-function $ \theta ( z) $ of the first order satisfies the following basic identities (in $ z $):

$$ \tag{4 } \left . \begin{array}{c} \theta ( z + e _ \mu ) = \theta ( z), \\ \theta ( z + e _ \mu A) = \mathop{\rm exp} [- \pi i ( a _ {\mu \mu } + 2z _ \mu )] \cdot \theta ( z), \\ 2 ( 1 + \delta _ {\mu \nu } ) \pi \frac{\partial \theta }{\partial a _ {\mu \nu } } = \frac{\partial ^ {2} \theta }{\partial z _ \mu \partial z _ \nu } , \\ \end{array} \right \} $$

where $ \mu , \nu = 1, \dots, p $, and $ \delta _ {\mu \nu } = 1 $ for $ \mu = \nu $ and $ \delta _ {\mu \nu } = 0 $ for $ \mu \neq \nu $. The $ ( p \times 2p) $-matrix $ S = ( E, A) $ is the moduli system or system of periods and quasi-periods of $ \theta ( z) $. If $ m = ( m _ {1}, \dots, m _ {p} ) $, $ m ^ \prime = ( m _ {1} ^ \prime , \dots, m _ {p} ^ \prime ) $ are arbitrary integer row-matrices, then the periodicity property of theta-functions can be written in its most general form as

$$ \theta ( z + m ^ \prime + mA) = $$

$$ = \ \mathop{\rm exp} [- \pi ( mAm) ^ {T} + 2m ( z + m ^ \prime ) ^ {T} ] \cdot \theta ( z). $$

Let $ \gamma = ( \gamma _ {1}, \dots, \gamma _ {p} ) $, $ \gamma ^ \prime = ( \gamma _ {1} ^ \prime, \dots, \gamma _ {p} ^ \prime ) $ be arbitrary complex row-matrices, and let $ \Gamma $ be the $ ( 2 \times p) $-matrix

$$ \left \| \begin{array}{c} \gamma \\ \gamma ^ \prime \end{array} \ \right \| . $$

Then the formula

$$ \theta _ \Gamma ( z) = \ \sum _ {n \in \mathbf Z ^ {p} } \mathop{\rm exp} [ \pi ( n + \gamma ) A ( n + \gamma ) ^ {T} + 2 ( n + \gamma ) ( z + \gamma ^ \prime ) ^ {T} ] = $$

$$ = \ \mathop{\rm exp} [ \pi i ( \gamma A \gamma ^ {T} + 2 \gamma ( z + \gamma ^ \prime ) ^ {T} ) ] \cdot \theta ( z + \gamma ^ \prime + \gamma A) $$

defines a theta-function of order $ 1 $ with characteristic (in general form) $ \Gamma $. In this terminology the theta-function (3) has characteristic 0. The matrix $ \Gamma $ is also called the periodicity characteristic of the matrix $ \gamma ^ \prime + \gamma A $. One always has $ \theta _ {- \Gamma } (- z) = \theta _ \Gamma ( z) $. Property (4) generalizes to theta-functions of characteristic $ \Gamma $:

$$ \tag{5 } \left . \begin{array}{c} \theta _ \Gamma ( z + e _ \mu ) = \ \mathop{\rm exp} ( 2 \pi i \gamma _ \mu ) \cdot \theta _ \Gamma ( z), \\ \theta _ \Gamma ( z + e _ \mu A) = \ \mathop{\rm exp} [- \pi i ( a _ {\mu \mu } + 2 ( z _ \mu - \gamma _ \mu ^ \prime ))] \cdot \theta _ \Gamma ( z). \\ \end{array} \right \} $$

The characteristic is said to be normal if $ 0 \leq \gamma _ {i} , \gamma _ {i} ^ \prime < 1 $ for $ i = 1, \dots, p $.

The most commonly used are fractional characteristics, where all the $ \gamma _ {i} $ and $ \gamma _ {i} ^ \prime $ are non-negative proper fractions with common denominator $ \delta $. The simplest and most important case is of semi-integer or half characteristics, where $ \delta = 2 $. A semi-integer characteristic

$$ H = \ \left \| \begin{array}{c} h \\ h ^ \prime \end{array} \ \right \| $$

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 $ H $ equations (5) take the form

$$ \theta _ {H} ( z + e _ \mu ) = \ (- 1) ^ {h _ \mu } \cdot \theta _ {H} ( z), $$

$$ \theta _ {H} ( z + e _ \mu A) = (- 1) ^ {h _ \mu ^ \prime } \mathop{\rm exp} [- \pi i ( a _ {\mu \mu } + 2z _ \mu )] \cdot \theta _ {H} ( z). $$

A theta-characteristic $ H $ is called even or odd, depending on whether the theta-function $ \theta _ {H} ( z) $ is even or odd. In other words, the theta-characteristic $ H $ is even or odd, depending on whether the number $ h ^ \prime h ^ {T} $ is even or odd, since

$$ \theta _ {H} (- z) = \ (- 1) ^ {h ^ \prime h ^ {T} } \cdot \theta _ {H} ( z). $$

There are $ 2 ^ {2p} $ distinct theta-characteristics, of which $ 2 ^ {p - 1 } ( 2 ^ {p} + 1) $ are even and $ 2 ^ {p - 1 } ( 2 ^ {p} - 1) $ are odd. The theta-function $ \theta _ {H} ( z) $ takes the value zero at those points $ ( g ^ \prime + gA)/2 $ whose theta-characteristic

$$ G = \ \left \| \begin{array}{c} g \\ g ^ \prime \end{array} \ \right \| $$

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

Let $ k $ be a natural number. An entire transcendental function $ \theta _ \Gamma ( z) $ is called a theta-function of order $ k $ with characteristic $ \Gamma $ if it satisfies the identities

$$ \theta _ \Gamma ( z + e _ \mu ) = \ \mathop{\rm exp} ( 2 \pi i \gamma _ \mu ) \cdot \theta _ \Gamma ( z), $$

$$ \theta _ \Gamma ( z + e _ \mu A) = \mathop{\rm exp} [- \pi i ( ka _ {\mu \mu } + 2kz _ \mu - 2 \gamma _ \mu ^ \prime )] \cdot \theta _ \Gamma ( z). $$

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

Using theta-functions of order $ 1 $ with semi-integer characteristics one can construct meromorphic Abelian functions with $ 2p $ periods. The periods of an arbitrary Abelian function in $ p $ 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 $ w _ {1}, \dots, w _ {p} $ 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 $ A $ used in the construction of a theta-function in $ p $ variables (3) are precisely those needed in order that the lattice $ L $ defined by the matrix $ ( I _ {p} A) $ in $ \mathbf C ^ {p} $ be such that $ \mathbf C ^ {p} / L $ be an Abelian variety. All Abelian varieties over $ \mathbf C $ 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 $ ( B, A) $ these relations are $ A ^ {T} B - B ^ {T} A = 0 $ (Riemann's equality, which becomes symmetry for $ A $ in the canonical case when $ B = I _ {p} $) and $ i B ^ {T} \overline{A} - i A ^ {T} \overline{B} $ is positive-definite Hermitean (Riemann's inequality, which becomes positive definiteness of the imaginary part of $ A $ in the canonical case (using the symmetry of $ A $)), [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