nLab Jacobi theta function

Idea

The fundamental example of theta functions is the Jacobi theta function given by

$$\vartheta(z,\tau) = \underoverset{n = - \infty}{\infty}{\sum} \exp(\pi i n^2 \tau + 2\pi i n z) \,.$$

As a variable of two arguments, this is actually a Jacobi form. These are the local coordinate expressions of the the covariantly constant sections of the Hitchin connection (for circle gauge group) on the moduli space of elliptic curves (Hitchin 90, remark 4.12). See there for more and see at theta function for the general idea.

At $z =0$ the function

$$\vartheta(0,\tau) = \underoverset{n = - \infty}{\infty}{\sum} \exp(\pi i n^2 \tau) \,.$$

is what in number theory is often just called “the theta function”. This is the one whose Mellin transform is the Riemann zeta function, see at Riemann zeta function – Relation to Jacobi theta function

Properties

Functional equation and Reciprocity

By the Poisson summation formula the number-theoretic theta function $\theta(0,z)$ satisfies the following functional equation:

$$\theta(0,\tau) = \frac{1}{\sqrt{\tau}} \theta(0,\frac{1}{\tau}) \,.$$

Under the Mellin transform this implies the functional equation of the Riemann zeta function, see at Riemann zeta function – Functional equation.

It also provides an analytic proof of the Landsberg-Schaar relation?

$$\frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi i n^2 q}{p}\right)=\frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi i n^2 p}{2q}\right)$$

where $p$ and $q$ are arbitrary positive integers. To prove it, apply theta reciprocity to $\tau=2iq/p+\epsilon$, $\epsilon \gt 0$, and then let $\epsilon\to 0$.

This reduces to the formula for the quadratic Gauss sum when $q=1$:

$$\sum_{n=0}^{p-1} e^{2 \pi i n^2 / p} = \left\{ \array{ \sqrt{p} & {if } \; p\equiv 1\mod 4 \\ i\sqrt{p} & {if } \; p\equiv 3\mod 4 } \right.$$

(where $p$ is an odd prime). From this, it’s not hard to deduce Gauss’s “golden theorem”.

quadratic reciprocity: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)(q-1)/4}$ for odd primes $p$ and $q$.

See e.g. (Karlsson).

some of this material from this MO discussion

Related concepts

• Poisson summation formula, functional equation

context/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)		class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$	Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

Due to Carl Jacobi.

Review includes

• Wikipedia, Jacobi theta-function

• section 9 in Analytic theory of modular forms (pdf)

• Anders Karlsson, Applications of heat kernels on abelian groups: $\zeta(2n)$, quadratic reciprocity, Bessel integrals (psd)

• Nigel Hitchin, Flat connections and geometric quantization, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (Euclid)