Theta function

A theta function is a holomorphic function formed by summing an exponential quadratic form over a lattice.

For a symmetric matrix $B$ with positive definite imaginary part, this theta function converges normally on compact subsets of ${\mathbb{C}}^g$. It obeys $\theta (z+m+Bn,B)=e^{-\pi i{n}^{t}Bn-2\pi i{n}^{t}z}\theta (z,B)$, so its zeros define a divisor on the associated complex torus. If $B$ is a curve's normalized period matrix, its zero divisor is a translate of the effective degree-$g-1$ locus, by the Riemann vanishing theorem. A general symmetric positive period matrix need not come from a curve.

For a full Euclidean lattice $\Lambda$ and a positive definite real symmetric matrix $A$, the Gaussian theta sum is ${\theta }_{\Lambda }(A)={\sum }_{\lambda \in \Lambda }{e}^{-\pi {\lambda }^{T}A\lambda }$. Its absolute convergence follows from Gaussian decay and the polynomial growth of the number of lattice points in a ball. The Poisson summation formula for a Euclidean lattice gives its anisotropic theta functional equation.

The lattice theta series of a full-rank Euclidean lattice records the multiset of squared lengths of lattice vectors. The Gaussian form shown here converges for $t>0$. For a flat torus ${\mathbb{R}}^d/\Lambda$, its heat trace is ${\Theta }_{{\Lambda }^{\ast }}(4\pi t)$, by the spectrum of a flat torus. Equal theta series do not in general imply orthogonal equivalence of lattices.

If every eigenvalue of a positive definite symmetric matrix $A$ tends to zero, then the anisotropic theta functional equation and the dominated convergence theorem show that $(\det A{)}^{1/2}{\theta }_{\Lambda }(A)\to \mathrm{covol}(\Lambda {)}^{-1}$. Indeed all eigenvalues of $A^{-1}$ tend to infinity, the zero term of the dual Gaussian theta sum is one, and its remaining terms tend to zero while being dominated by a fixed summable Gaussian. Merely requiring $\det A\to 0$ is insufficient.

Let $C=\sqrt{|D_K|}N(\mathfrak{b})$ and $n=[K:\mathbb{Q}]$. Normalize the Minkowski embedding of a number field to the covolume-one Euclidean lattice ${\Lambda }_{\mathfrak{b}}=C^{-1/n}j(\mathfrak{b})$, using the metric with complex coordinates weighted by $2$. Its theta function isWriting $\Vert y\Vert ={\prod }_v{y}_v^{[K_v:\mathbb{R}]}$ and using the trace dual of a fractional ideal gives $\Theta (y,\mathfrak{b})=\Vert y{\Vert }^{-1/2}\Theta (y^{-1},{\mathfrak{b}}^{\vee })$. Its small-parameter asymptotic of a lattice theta sum has leading coefficient one. For the unscaled ideal lattice, the leading coefficient is instead $C^{-1}$; one must specify the normalization when quoting this limit.

For a full-rank Euclidean lattice, ${\Theta }_{\Lambda }(\tau )={\sum }_{x\in \Lambda }{e}^{\pi i|x{|}^2\tau }$ obeys ${\Theta }_{\Lambda }(\tau )=(-i\tau {)}^{-n/2}m(\Lambda {)}^{-1}{\Theta }_{{\Lambda }^{\vee }}(-1/\tau )$. Apply the Poisson summation formula for a Euclidean lattice and the complex Gaussian Fourier transform. The identity holds without assuming an integral or self-dual lattice.

For a full Euclidean lattice and a positive definite real symmetric matrix $A$, the Gaussian theta sum obeysUse the Fourier transform kernel $e^{-2\pi ix\cdot z}$, the identity $\widehat{e^{-\pi x^{T}Ax}}(z)=(\det A{)}^{-1/2}{e}^{-\pi z^T{A}^{-1}z}$, and the Poisson summation formula for a Euclidean lattice. No integrality or self-duality of the Euclidean lattice is assumed.

The Jacobi theta functionsatisfies $\theta (\tau +2)=\theta (\tau )$ and $\theta (-1/\tau )=(-i\tau {)}^{1/2}\theta (\tau )$.

Use period-one spatial normalization and $Q=e^{\pi i\tau }$. The conventional odd function is ${\theta }_1=-\theta [1/2,1/2]$ with the characteristic convention of theta function with characteristics. Differentiating its Jacobi triple product gives ${\theta }_1^{\prime }(0)=2\pi Q^{1/4}{\prod }_{m\ge 1}(1-Q^{2m}{)}^3$. The three even constant products have product $2Q^{1/4}{\prod }_{m\ge 1}(1-Q^{2m}{)}^3$, proving the identity. The sign changes if one instead calls the characteristic series itself theta1.

For real characteristics $\alpha ,\beta$ and $\mathrm{Im}\tau >0$, Gaussian decay gives normal convergence, an entire function of $z$ and holomorphic dependence on $\tau$. Integer shifts of alpha leave the function unchanged; a unit shift of beta multiplies it by $e^{2\pi i\alpha }$. Its spatial quasi-periods are $\theta (z+1)=e^{2\pi i\alpha }\theta (z)$ and $\theta (z+\tau )=e^{-\pi i\tau -2\pi i(z+\beta )}\theta (z)$. Binary characteristic labels often mean alpha and beta divided by two, so the convention must be stated.

A theta function with characteristics evaluated at spatial argument zero. Its dependence on $\tau \in \mathbb{H}$ remains holomorphic. Among the four half-integer characteristic choices, the odd one vanishes identically, whereas the three even constants are nonzero on the half-plane. The zeros of theta functions with characteristics locate their spatial zero away from the origin. Ratios of equal powers of even constants yield the modular lambda function and other modular-invariant functions.

Use the radian spatial convention ${\theta }_3(z,\tau )={\sum }_n{e}^{\pi i{n}^2\tau +2inz}$ and ${\theta }_4(z,\tau )={\theta }_3(z+\pi /2,\tau )$. Their ratio is unchanged by $\pi$ and changes sign under $\pi \tau$. The numerator has simple zeros at $\pi (1+\tau )/2+\pi \mathbb{Z}+\pi \tau \mathbb{Z}$, while the denominator has simple zeros at $\pi \tau /2+\pi \mathbb{Z}+\pi \tau \mathbb{Z}$. The two sets are disjoint, so any period must lie in $\pi \mathbb{Z}+\pi \tau \mathbb{Z}$, by translating the zero set. The sign rule then gives the displayed exact lattice. Squaring the ratio removes that sign and has precisely the smaller lattice $\pi \mathbb{Z}+\pi \tau \mathbb{Z}$.

For the binary characteristic convention, each theta function with characteristics has one simple zero per lattice cell. The four zeros of the numerator, at the four half-periods, cancel those of ${\vartheta }_{11}(2z)$. Their quotient is an entire elliptic function, hence constant. Taking the limit at zero determines the factor one half. The product identity remains valid at the zeros, where the initial quotient requires removable extension.

Translation of tau by one gives $\theta [\alpha ,\beta ](z,\tau +1)=e^{-\pi i\alpha (\alpha -1)}\theta [\alpha ,\beta +\alpha -1/2](z,\tau )$. Gaussian Poisson summation gives $\theta [\alpha ,\beta ](z/\tau ,-1/\tau )=\sqrt{-i\tau }{e}^{\pi i{z}^2/\tau +2\pi i\alpha \beta }\theta [\beta ,-\alpha ](z,\tau )$. The square root is the holomorphic branch positive at tau=i. These formulas keep the characteristic phase, which is lost if one states only the transformation of unshifted theta constants.

Normal convergence allows termwise differentiation of the defining theta function with characteristics. A tau derivative multiplies a summand by $\pi i(n+\alpha {)}^2$, while its second spatial derivative multiplies it by $-4{\pi }^2(n+\alpha {)}^2$. This proves the displayed heat equation, linking its lattice-sum construction with diffusion and analytic continuation.

The logarithmic derivative is periodic under one and decreases by $2\pi i$ under tau. Integrating around a fundamental cell and applying the argument principle counts exactly one zero, with multiplicity. The displayed zero follows from translating the zero of $\theta [0,0]$ at $(1+\tau )/2$. It is consequently simple. For half-integer characteristics only the odd characteristic $(1/2,1/2)$ vanishes at the origin; the other three theta constants are nonzero.

The identity between Jacobi theta functions has the oscillator-product form ${\prod }_{r\in \mathbb{N}-1/2}(1+q^r{)}^8-{\prod }_{r\in \mathbb{N}-1/2}(1-q^r{)}^8=16q^{1/2}{\prod }_{n\ge 1}(1+q^n{)}^8$ for $|q|<1$. In a critical RNS string, dividing this identity by the eight-boson oscillator product proves equality of GSO projection-retained Neveu–Schwarz sector and Ramond sector state multiplicities at every mass level.

In the $e^{\pi iz}$ convention, Poisson summation gives ${\Theta }_3(-1/z)=\sqrt{-iz}{\Theta }_3(z)$ and interchanges ${\Theta }_2,{\Theta }_4$ with the same factor. Translation by one sends ${\Theta }_2$ to $e^{\pi i/4}{\Theta }_2$ and interchanges ${\Theta }_3,{\Theta }_4$. Eighth powers remove every phase and square-root ambiguity when proving integral-weight modularity.

This convention for the Jacobi theta function is the usual theta constant evaluated at twice the argument. The Poisson summation formula gives $\theta (-1/(4\tau ))=\sqrt{-2i\tau }\theta (\tau )$, with the square root holomorphic on the half-plane and positive on the imaginary axis after evaluating its positive real argument.

The number of ordered integer eight-tuples with square sum $n$ satisfies the displayed divisor formula for positive $n$, with $r_8(0)=1$ separately. The eighth power of the shifted theta series of integer squares is $(16E_4(2\tau )-E_4(\tau ))/15$. Its coefficient sign is $(-1{)}^n$, because squares and their integer roots have the same parity.

For $|q|<1$ and $z\ne 0$, the Jacobi triple product is

The odd Dirichlet character modulo four has ${\chi }_4(2r+1)=(-1{)}^r$. The series $B=\frac{1}{2}{\sum }_{n}n{\chi }_4(n)q^{n^2/8}$ is therefore $q^{1/8}$ times the displayed sum. The character theta proof of the eta product gives $B={\eta }^3$, proving the identity.

The identity holds for $|q|<1$, with both sides locally uniformly convergent. In the character theta proof of the eta product, pair the terms indexed by $n$ and $-n$ and then select $n=6r+1$; their Dirichlet character signs are $(-1{)}^r$. Removing $q^{1/24}$ gives the displayed product identity.

The theta group is ${\Gamma }_{\theta }=\Gamma (2)\cup \Gamma (2)S$. It is generated by $T^2$ and $S$ and has index three in the modular group.

The Jacobi triple product givesEvery factor is nonzero for $|q|<1$, and the product converges to a nonzero limit, so $\theta$ has no zero in the upper half-plane.