THC Science

In mathematics, theta functions are special functions of several complex variables. They are important in several areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory, specifically string theory and D-branes.

The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this is shown to come from a line bundle condition of descent.

Jacobi theta function

The Jacobi theta function is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula

\vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi in^{2}\tau +2\pi inz).

If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity

\vartheta (z+1;\tau )=\vartheta (z;\tau ).

The function also behaves very regularly with respect to addition by τ and satisfies the functional equation

\vartheta (z+a+b\tau ;\tau )=\exp(-\pi ib^{2}\tau -2\pi ibz)\vartheta (z;\tau )

where a and b are integers.

Auxiliary functions

It is convenient to define three auxiliary theta functions, which we may write

\vartheta _{01}(z;\tau )=\vartheta (z+1/2;\tau )

\vartheta _{10}(z;\tau )=\exp(\pi i\tau /4+\pi iz)\vartheta (z+\tau /2;\tau )

\vartheta _{11}(z;\tau )=\exp(\pi i\tau /4+\pi i(z+1/2))\vartheta (z+(\tau +1)/2;\tau ).

This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome $q=\exp(\pi i\tau )$ rather than τ, and theta there is called $\theta _{3}$ , with $\vartheta _{01}$ termed $\theta _{0}$ , $\vartheta _{10}$ named $\theta _{2}$ , and $\vartheta _{11}$ called $-\theta _{1}$ .

If we set z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane (sometimes called theta constants.) These can be used to define a variety of modular forms, and to parametrize certain curves; in particular the Jacobi identity is

\vartheta (0;\tau )^{4}=\vartheta _{01}(0;\tau )^{4}+\vartheta _{10}(0;\tau )^{4}

which is the Fermat curve of degree four.

Jacobi identities

Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ+1 and τ ↦ -1/τ. We already have equations for the first transformation; for the second, let

\alpha =(-i\tau )^{1/2}\exp({\pi iz^{2}}{\tau }).

Then

\vartheta (z/\tau ;-1/\tau )=\alpha \vartheta (z;\tau )

\vartheta _{01}(z/\tau ;-1/\tau )=\alpha \vartheta _{10}(z;\tau )

\vartheta _{10}(z/\tau ;-1/\tau )=\alpha \vartheta _{01}(z;\tau )

\vartheta _{11}(z/\tau ;-1/\tau )=-\alpha \vartheta _{11}(z;\tau )

Theta functions in terms of the nome

Instead of ϑ in terms of z and τ, we may express it in terms of arguments w and the nome q, where $w=\exp(\pi iz)$ and $q=\exp(\pi i\tau )$ . In this form, ϑ becomes

\vartheta (w;q)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.

We may also express the auxiliary theta functions using these arguments, as

\vartheta _{01}(w;q)=\sum _{n=-\infty }^{\infty }(-1)^{n}w^{2n}q^{n^{2}},

\vartheta _{10}(w;q)=q^{1/4}\sum _{n=-\infty }^{\infty }w^{2n+1}q^{n^{2}+n},

\vartheta _{11}(w;q)=iq^{1/4}\sum _{n=-\infty }^{\infty }(-1)^{n}w^{2n+1}q^{n^{2}+n}.

It is worth noting that in the form given in terms of the variables w and q, the theta function does not require the exponential function for its definition, and can be used to define the function over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers.

Product representations

The Jacobi triple product tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have

\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+w^{2}q^{2m-1}\right)\left(1+w^{-2}q^{2m-1}\right)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.

It can be proven by elementary means, as for instance in Hardy and Wright's An Introduction to the Theory of Numbers.

If we express the theta function in terms of the nome $q=\exp(\pi i\tau )$ and $w=\exp(\pi iz)$ then

\vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi i\tau n^{2})\exp(\pi iz2n)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.

We therefore obtain a product formula for the theta function in the form

\vartheta (z;\tau )=\prod _{m=1}^{\infty }\left(1-\exp(2m\pi i\tau )\right)\left(1+\exp((2m-1)\pi i\tau +2\pi iz)\right)\left(1+\exp((2m-1)\pi i\tau -2\pi iz)\right)

Expanding terms out, the Jacobi triple product can also be written

\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+(w^{2}+w^{-2})q^{2m-1}+q^{4m-2}\right),

which we may also write as

\vartheta (z|q)=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right).

This form is valid in general but clearly is of particular interest when z is real. Similar product formulas for the auxiliary theta functions are

\vartheta _{01}(z|q)=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right).

\vartheta _{10}(z|q)=2q^{1/4}\cos(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right).

\vartheta _{11}(z|q)=-2q^{1/4}\sin(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).

Integral representations

The Jacobi theta functions have the following integral representations:

\vartheta (z;\tau )=-i\int _{i-\infty }^{i+\infty }{e^{i\pi \tau u^{2}}\cos(2uz+\pi u) \over \sin(\pi u)}du

\vartheta _{01}(z;\tau )=-i\int _{i-\infty }^{i+\infty }{e^{i\pi \tau u^{2}}\cos(2uz) \over \sin(\pi u)}du.

\vartheta _{10}(z;\tau )=-ie^{iz+i\pi \tau /4}\int _{i-\infty }^{i+\infty }{e^{i\pi \tau u^{2}}\cos(2uz+\pi u+\pi \tau u) \over \sin(\pi u)}du

\vartheta _{11}(z;\tau )=e^{iz+i\pi \tau /4}\int _{i-\infty }^{i+\infty }{e^{i\pi \tau u^{2}}\cos(2uz+\pi \tau u) \over \sin(\pi u)}du

Relation to the Riemann zeta function

The relation

\vartheta (0;-1/\tau )=(-i\tau )^{1/2}\vartheta (0;\tau )

was used by Riemann to prove the functional equation for Riemann's zeta function, by means of the integral

\Gamma \left({\frac {s}{2}}\right)\pi ^{-s/2}\zeta (s)={\frac {1}{2}}\int _{0}^{\infty }\left[\vartheta (0;it)-1\right]t^{s/2}{\frac {dt}{t}}

which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z not zero is given in the article on the Hurwitz zeta function.

Relation to the Weierstrass elliptic function

The theta function was used by Jacobi to construct (in a form adapted to easy calculation) his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since

\wp (z;\tau )=-(\log \vartheta _{11}(z;\tau ))''+c

where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of $\wp (z)$ at z = 0 has zero constant term.

Some relations to modular forms

Let η be the Dedekind eta function. Then

\vartheta (0;\tau )={\frac {\eta ^{2}\left(\tau +{\frac {1}{2}}\right)}{\eta (2\tau +1)}}

.

A solution to heat equation

The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z = x to be real, and taking τ = it with t real and positive. Then we can write

\vartheta (x,it)=1+2\sum _{n=1}^{\infty }\exp(-\pi n^{2}t)\cos(2\pi nx)

which solves the heat equation

{\frac {\partial }{\partial t}}\vartheta (x,it)={\frac {1}{4\pi }}{\frac {\partial ^{2}}{\partial x^{2}}}\vartheta (x,it).

That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb:

\lim _{t\rightarrow 0}\vartheta (x,it)=\sum _{n=-\infty }^{\infty }\delta (x-n)

where δ is the Dirac delta function. Thus, general solution can be specified by convolving the (periodic) boundary condition at t = 0 with the theta function.

Relation to the Heisenberg group

The Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. This invariance is presented in the article on the theta representation of the Heisenberg group.

Generalizations

If F is a quadratic form in n variables, then the theta function associated with F is

\theta _{F}(z)=\sum _{m\in Z^{n}}\exp(2\pi izF(m))

with the sum extending over the lattice of integers Zⁿ. This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,

\theta _{F}(z)=\sum _{k=0}^{\infty }R_{F}(k)\exp(2\pi ikz)

,

the numbers R_F(k) are called the representation numbers of the form.

Ramanujan theta function

See main articles Ramanujan theta function and mock theta function.

Riemann theta function

Let

\mathbb {H} _{n}=\{F\in M(n,\mathbb {C} )\;\mathrm {s.t.} \,F=F^{T}\;{\textrm {and}}\;{\mbox{Im}}F>0\}

be set of symmetric square matrices whose imaginary part is positive definite. H_n is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The n-dimensional analogue of the modular group is the symplectic group Sp(2n,Z); for n = 1, Sp(2,Z) = SL(2,Z). The n-dimensional analog of the congruence subgroups is played by ${\textrm {Ker}}\{{\textrm {Sp}}(2n,\mathbb {Z} )\rightarrow {\textrm {Sp}}(2n,\mathbb {Z} /k\mathbb {Z} )\}$ .

Then, given $\tau \in \mathbb {H} _{n}$ , the Riemann theta function is defined as

\theta (z,\tau )=\sum _{m\in Z^{n}}\exp \left(2\pi i\left({\frac {1}{2}}m^{T}\tau m+m^{T}z\right)\right).

Here, $z\in \mathbb {C} ^{n}$ is an n-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with n = 1 and $\tau \in \mathbb {H}$ where $\mathbb {H}$ is the upper half-plane.

The Riemann theta converges absolutely and uniformly on compact subsets of $\mathbb {C} ^{n}\times \mathbb {H} _{n}.$

The functional equation is

\theta (z+a+\tau b,\tau )=\exp 2\pi i\left(-b^{T}z-{\frac {1}{2}}b^{T}\tau b\right)\theta (z,\tau )

which holds for all vectors $a,b\in \mathbb {Z} ^{n}$ , and for all $z\in \mathbb {C} ^{n}$ and $\tau \in \mathbb {H} _{n}$ .

Q-theta function

See main article Q-theta function.

References

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See section 16.27ff.)
Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4 (See Chapter 6 for treatment of the Riemann theta)
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fourth edition (1959) , Oxford University Press
David Mumford, Tata Lectures on Theta I (1983), Birkhauser, Boston ISBN 3-7643-3109-7
James Pierpont Functions of a Complex Variable, Dover
Harry E. Rauch and Hershel M. Farkas, Theta Functions with Applications to Riemann Surfaces, (1974) Williams & Wilkins Co. Baltimore ISBN 683-07196-3.

Integral representations of Jacobi theta functions at PlanetMath.

@@ Line 152: / Line 152: @@
 ==Some relations to modular forms==
 Let η be the [[Dedekind eta function]]. Then
-:<math>\vartheta(0;\tau)=\frac{\eta^2\left(\frac{\tau+1}{2}\right)}{\eta(\tau+1)}</math>.
+:<math>\vartheta(0;\tau)=\frac{\eta^2\left(\tau+\frac{1}{2}\right)}{\eta(2\tau+1)}</math>.
 ==A solution to heat equation==