Theta function

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.

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 behaviour with respect to the addition of a period of the associated elliptic functions (sometimes called quasi-periodicity, though this is not related to the use of that term for dynamical systems). In the abstract theory this is shown to come from a line bundle condition of descent.

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

$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)$

If τ is fixed, this becomes a Fourier series for a periodic entire function of z, with period one; the theta function satisfying 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

$\vartheta(z+a+b\tau;\tau) = \exp(-\pi i b^2 \tau -2 \pi i b z)\vartheta(z;\tau)$

where a and b are integers.

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

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

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

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

This notation follows Riemann and David Mumford; Jacobi's original formulation was in terms of q = exp(πτ) rather than τ, and theta there is called θ₃, with $\vartheta_{01}$ termed θ₀, $\vartheta_{10}$ named θ₂, and $\vartheta_{11}$ called - θ₁.

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 parameterize 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.

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

$\mathcal{P}(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c$

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

References

Mumford, David, Tata Lectures on Theta I, Birkhauser

Pierpont, James Functions of a Complex Variable, Dover

Categories: Mathematics | Special functions | Modular forms

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