nLab Jacobi theta function



Theta functions

Complex geometry

Arithmetic geometry



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

ϑ(z,τ)=n=exp(πin 2τ+2πinz). \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=0z =0 the function

ϑ(0,τ)=n=exp(πin 2τ). \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


Functional equation and Reciprocity

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

θ(0,τ)=1τθ(0,1τ). \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?

1p n=0 p1exp(2πin 2qp)=e πi/42q n=0 2q1exp(πin 2p2q)\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 pp and qq are arbitrary positive integers. To prove it, apply theta reciprocity to τ=2iq/p+ϵ\tau=2iq/p+\epsilon, ϵ>0\epsilon \gt 0, and then let ϵ0\epsilon\to 0.

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

n=0 p1e 2πin 2/p={p ifp1mod4 ip ifp3mod4\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 pp is an odd prime). From this, it’s not hard to deduce Gauss’s “golden theorem”.

quadratic reciprocity: (pq)(qp)=(1) (p1)(q1)/4\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)(q-1)/4} for odd primes pp and qq.

See e.g. (Karlsson).

some of this material from this MO discussion

context/function field analogytheta function θ\thetazeta function ζ\zeta (= Mellin transform of θ(0,)\theta(0,-))L-function L zL_{\mathbf{z}} (= Mellin transform of θ(z,)\theta(\mathbf{z},-))eta function η\etaspecial values of L-functions
physics/2d CFTpartition function θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of complex structure τ\mathbf{\tau} of worldsheet Σ\Sigma (hence polarization of phase space) and background gauge field/source z\mathbf{z}analytically continued trace of Feynman propagator ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued trace of Feynman propagator in background gauge field z\mathbf{z}: L z(s)Tr reg(1(D z) 2) s= 0 τ s1θ(z,τ)dτL_{\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\tauanalytically continued trace of Dirac propagator in background gauge field z\mathbf{z} η z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s regularized 1-loop vacuum amplitude pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / regularized fermionic 1-loop vacuum amplitude pvη z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / vacuum energy 12L z (0)=Z H=12lndet reg(D z 2)-\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 operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of line bundle over Jacobian variety J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})zeta function of a Riemann surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function L zL_{\mathbf{z}} for z=0\mathbf{z} = 0 the trivial Galois representation)Artin L-function L zL_{\mathbf{z}} of a Galois representation z\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 (z=0\mathbf{z} = 0)/ Dirichlet theta function (z=χ\mathbf{z} = \chi a Dirichlet character)Riemann zeta function (being the Dirichlet L-function L zL_{\mathbf{z}} for Dirichlet character z=0\mathbf{z} = 0)Artin L-function of a Galois representation z\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


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: ζ(2n)\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)

Last revised on April 26, 2020 at 10:41:41. See the history of this page for a list of all contributions to it.