# 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.

#### 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

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.
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.