In which dimensions do there exist inequivalent lattices with the same theta function?

Equivalently, "Is a lattice determined by the distances (with multiplicities) of its points from the origin?"

By a lattice $$L$$ I mean a discrete additive subgroup of Euclidean space $$\mathbb{R}^{n}$$ which can be expressed as the integer span of $$n$$ basis vectors. The standard inner product on $$\mathbb{R}^{n}$$ induces an inner product on $$L$$. Two lattices are equivalent if one can be transformed into the other via rotations and reflections of $$\mathbb{R}^{n}$$ (so that the inner product is preserved). Note: to check whether two lattices are equivalent, it does not suffice to simply compare their basis vectors because when $$n>1$$ there are infinitely many different choices of basis vectors for the same lattice which are not related by rotations and reflections.

The theta function of $$L$$ is the generating function $$\theta_{L}$$ which counts the number of lattice points of a given squared-distance from the origin in $$L$$:

$$\theta_{L}=\sum_{x\in L}q^{x\cdot x}=\sum_{\alpha\in\left\{ x\cdot x\mid x\in L\right\} }N_{\alpha}q^{\alpha},\quad N_{\alpha}:=\#\left\{ x\in L\bigm|x\cdot x=\alpha\right\} .$$

(A more common convention includes a factor of $$\tfrac{1}{2}$$ in the exponent of $$q$$.)

Note that $$\theta_L$$ can be viewed either as a formal power series in $$q$$ (with possibly irrational exponents), or as a holomorphic function $$\theta_L(\tau)$$ of the variable $$\tau$$ in the upper-half-plane after replacing $$q^\alpha\mapsto \exp(\alpha \pi i \tau)$$.

It was shown by Alexander Schiemann that there exist inequivalent lattices with the same theta function if and only if $$n\geq4$$.

This question has a fascinating history. The first example of a pair of inequivalent lattices with the same theta function was discovered by Witt in 1941 in 16 dimensions [1]. (His argument is based on the theory of modular forms, and I have summarized the essential parts in this question.) In a short note [2], Milnor observed in 1964 that the eigenfunctions of the Laplacian on the dual torus $$\mathbb{R}^{n}/L^{*}$$ are in correspondence with the points of $$L$$, and the corresponding eigenvalues are equal to $$(2\pi)^{2}$$ times the squared-distances. Thus any pair of inequivalent lattices with the same theta function (in particular Witt's pair) yields a pair of isospectral manifolds which are not isometric. For this reason, such lattice pairs are often called isospectral and discussed under the heading of hearing the shape of a drum. Note that it's important that the spectrum be considered with multiplicities, since already in two dimensions, the hexagonal lattice has a rectangular sublattice of index 2 which represents all the same lengths from the full hexagonal lattice.

If there exists some example pair of equivalent lattices in dimension $$n$$, then there exist examples in all dimensions $$m\geq n$$. Namely, if $$L_{1}$$ and $$L_{2}$$ are an example pair, and if $$\Lambda$$ is any lattice, then the orthogonal direct sums $$\Lambda\oplus L_{1}$$ and $$\Lambda\oplus L_{2}$$ are also example pairs. Thanks to the identity $$\theta_{\Lambda\oplus L}(q)=\theta_{\Lambda}(q)\theta_{L}(q)$$ it follows that $$\Lambda\oplus L_{1}$$ and $$\Lambda\oplus L_{2}$$ have the same theta functions. Inequivalence of $$\Lambda\oplus L_{1}$$ and $$\Lambda\oplus L_{2}$$ follows from the theorem of Eichler and Kneser [3] that any lattice decomposes into the orthogonal direct sum of unique irreducible sublattices (see p. 6 of Nebe's talk for a summary of Kneser's construction).

In dimensions $$n\leq2$$ it is easy to show that a lattice is uniquely determined up to equivalence by its theta function. Between this and the example of Witt, it is clear that there is some minimal dimension $$n_{\mathrm{min}}$$ for which an example pair exists, and $$3\leq n_{\mathrm{min}}\leq16$$.

Schiemann published two papers [4,5] which, when taken together, prove that $$n_{\mathrm{min}}=4$$. In 1990 he showed $$n_{\mathrm{min}}\leq4$$ by discovering the first example pair in four dimensions, and in 1997 he proved that $$n_{\mathrm{min}}>3$$ by proving that 3D lattices are uniquely determined up to equivalence by their theta functions.

The two papers of Schiemann are highly technical. For the 1997 paper, the 2020 master's thesis of Felix Rydell does an excellent job of providing background and fleshing out the details. Below I explain the example 4D pair in elementary terms.

The 4-dimensional lattices $$L^{-}$$ and $$L^{+}$$ have the same theta series.

Via a computer search, Schiemann discovered a pair of inequivalent 4D lattices $$S_{1}$$ and $$S_{2}$$ having identical theta series. The squared-distances are even integers, and the theta series begins as $$1+0q^2+2q^{4}+0q^6+4q^{8}+\cdots.$$ To show that the theta series agree for all coefficients, Schiemann's original proof used a large space of rather complicated modular forms. It's easier to describe the work of Conway and Sloane [7], who constructed a four-parameter family of lattice pairs $$L^{\pm}(a,b,c,d)$$ and provided a simple geometric proof that their theta functions are equivalent (via coordinate reflections of certain subsets of points). The particular pair $$L^{\pm}:=L^{\pm}(1,7,13,19)$$ is equivalent to Schiemann's original example with $$S_{1}\sim L^{-}$$ and $$S_{2}\sim L^{+}$$. This provides a very elementary and concise proof that the theta functions are equal.

$$L^{-}$$ and $$L^{+}$$ are inequivalent lattices.

Schiemann proved inequivalence based on the theory of Minkowski reduction, while Conway and Sloane provided no details about how they verified that $$L^{-}$$ and $$L^{+}$$ are inequivalent. (Presumably they simply checked with a computer, since there are algorithms to determine whether or not two lattices are equivalent.) Here I provide a simple original proof that $$L^{-}$$ and $$L^{+}$$ are inequivalent.

Sketch: Assuming that the theta series stated above is correct, there are a total of exactly seven vectors of length $$\leq\sqrt{8}$$ in each of $$L^{-}$$ and $$L^{+}$$. Due to the symmetry $$x\to-x$$ of any lattice, these seven vectors consist of the zero-vector plus three pairs of antipodal vectors. One pair has length $$\sqrt{4}$$ and the other two pairs have length $$\sqrt{8}$$. We verify below that in $$L^{-}$$ these three pairs are coplanar, while in $$L^{+}$$ they span a three-dimensional subspace. Since distance and coplanarity are preserved under rotations and reflections, $$L^{-}$$ and $$L^{+}$$ cannot be equivalent.

Click the images for an interactive visualization of vectors of length $$\leq\sqrt8$$

in $$L^-$$:

in $$L^+$$:

Details: To perform explicit computations, we compute the Gram matrices of inner products with respect to Conway and Sloane's bases $$(v_{\infty}^{\pm},v_{0}^{\pm},v_{1}^{\pm},v_{2}^{\pm})$$: $$A_{L^{-}}=\left(\begin{array}{cccc} 4 & 2 & 2 & 5\\ 2 & 8 & -3 & 2\\ 2 & -3 & 12 & -2\\ 5 & 2 & -2 & 16 \end{array}\right),\qquad A_{L^{+}}=\left(\begin{array}{cccc} 4 & -1 & -4 & -4\\ -1 & 8 & 0 & -4\\ -4 & 0 & 12 & 1\\ -4 & -4 & 1 & 16 \end{array}\right).$$ For details, see this PARI/GP script. Assume for the moment that the theta series stated above is correct. Then it suffices to enumerate the three antipodal pairs of integer coefficient vectors $$c=(c_{\infty},c_{0},c_{1},c_{2})^{T}$$ such that $$x=c_\infty v_\infty+c_0 v^\pm_0 + c_1 v^\pm_1+c_2 v^\pm_2\in L^{\pm}$$ with $$\left\Vert x\right\Vert ^{2}=c^{T}A_{L^{\pm}}c\leq8$$ and check their rank. For both $$L^{\pm}$$, the nonzero pair of length $$\sqrt{4}$$ has components $$\pm\left(\begin{array}{cccc} 1 & 0 & 0 & 0\end{array}\right)^{T}$$. The two pairs of length $$\sqrt{8}$$ are: \begin{align*} L^{-}: & \pm\left(\begin{array}{cccc} 0 & 1 & 0 & 0\end{array}\right)^{T},\ \pm\left(\begin{array}{cccc} 1 & -1 & 0 & 0\end{array}\right)^{T},\\ \\ L^{+}: & \pm\left(\begin{array}{cccc} 0 & 1 & 0 & 0\end{array}\right)^{T},\ \pm\left(\begin{array}{cccc} 1 & 0 & 1 & 0\end{array}\right)^{T}. \end{align*} (See colvectors_ALm and colvectors_ALp in the PARI/GP script.) It is clear from the components that the three nonzero pairs for $$L^{+}$$ are linearly independent while those for $$L^{-}$$ are dependent.

To verify that the theta function stated above is correct so that these are the only vectors of length $$\leq\sqrt{8}$$, it suffices to use the Rayleigh quotient inequality $$8\geq c^{T}A_{L^{\pm}}c\geq\lambda_{\mathrm{min}}c^{T}c=\lambda_{\mathrm{min}}\left\Vert c\right\Vert ^{2}$$ and check the finitely many integer vectors with $$\left\Vert c\right\Vert \leq\sqrt{8/\lambda_{\mathrm{min}}}$$. (Both $$A_{L^{-}}$$ and $$A_{L^{+}}$$ have smallest eigenvalue $$\lambda_{\mathrm{min}}=1$$.) This check shows that the pairs found above are complete.

Q.E.D.

References

1. Witt, E. (1941), Eine Identität zwischen Modulformen zweiten Grades.

2. Milnor, J. (1964), Eigenvalues of the Laplace operator on certain manifolds. Proceedings of the National Academy of Sciences of the United States of America, 51 (4): 542ff

3. Kneser, M. (1954) Zur Theorie der Kristallgitter. Mathematische Annalen 127 : 105-106. Link

4. Schiemann, A. (1990) Ein Beispiel positiv definiter quadratischer Formen der Dimension 4 mit gleichen Darstellungszahlen. Archiv der Mathematik, 54(4), pp.372-375.

5. Schiemann A. (1997) Ternary positive definite quadratic forms are determined by their theta series. Mathematische Annalen, 308(3), pp.507-517.

6. Rydell, F. (2020) Three Perspectives of Schiemann’s Theorem. Master's thesis. Link

7. Conway, J.H. and Sloane, N.J. (1992) Four-dimensional lattices with the same theta series. International Mathematics Research Notices, 1992(4), pp.93-96. Link