A flat torus is the quotient of $\mathbb R^n$ by a lattice, which is a discrete subgroup of the additive group of $\mathbb R^n$ and can be shown to always be of the form $A\mathbb Z^n$ for some full-rank^[of course the lattice could have a smaller rank than $n$ but that would just give us a lower-dimensional lattice] $A$. Because the metric on these objects is flat, the Laplace-Beltrami operator on them is locally just the standard laplacian, and we do not have to do much to analyze it. This simplicity is perhaps why the first examples of isospectral non-isometric manifolds found were two sixteen dimensional flat, described in a [small note](https://www.pnas.org/doi/abs/10.1073/pnas.51.4.542) by John Milnor. Flat tori provide an interesting class of spaces to study spectral geometry, because they are one of the very few manifolds in which we can explicitly write down the eigenvalues, and thus prove isospectrality using direct calculations as opposed to indirect methods like Sunada's method. Because of this direct representation, many of the phenomena of the general theory can be seen here, but the spectral geometry of flat tori also has a special connection with topics from other fields of math like quadratic forms, modular forms, linear codes, etc. ## The spectra of flat tori What is the Laplace spectrum of a flat torus $\mathbb R^n/\Gamma$? We can use the [[Sunada's method#Upstairs and Downstairs|upstairs-downstairs correspondence]] and say that eigenfunctions $f$ satisfying $\Delta f = \lambda f$ on the torus are precisely those that satisfy this condition on $\mathbb R^n$ and also satisfy $f(x+\gamma) = f(x)$ for all $\gamma \in \Gamma$. The "eigenfunctions" upstairs are the *plane waves* $e^{i\langle k, x \rangle}$ for fixed $k \in \mathbb R^n$ having eigenvalue $\left\| k\right\|^2$, and for them to be invariant under the action of $\Gamma$ we need $k$ to satisfy $e^{i\langle k, \gamma\rangle} = 1$ which means $\langle k, \gamma \rangle = 2m\pi$ for some integer $m$. This equation means that $\frac k{2\pi}$ must be an element of the *dual lattice* $\Gamma^*$, which is the set of all vectors in $\mathbb R^n$ whose inner product with elements of $\Gamma$ is an integer. Thus the spectrum of the Laplacian is $4\pi^2\|\gamma^*\|^2$ where $\gamma^*$ ranges over elements of the dual lattice $\Gamma^*$. ^[This is not entirely valid just yet because the plane waves in $\mathbb R^n$ are *not* eigenfunctions of the $L^2$ laplacian as they're not square-integrable. However, their quotients in $\mathbb R^n/\Gamma$ *are* square integrable as tori have finite volume, and we can use Sturm-Liouville theory to see that these are the only eigenfunctions and that they form an orthonormal basis of $L^2(\mathbb R^n/\Gamma)$.] Thus, the spectrum of the Laplacian is precisely the set of lengths of *dual* vectors of the lattice $\Gamma.$ We can ease the pain of bringing in the dual lattice somewhat by using the *Poisson summation formula*. For this, some definitions are in order: 1. The *heat trace*, which we can define on any object with a Laplace spectrum, is the function $h_\Gamma(t) = \sum_{k\ge 0}e^{-\lambda_kt} = \sum_{\gamma^* \in \Gamma^*} e^{-4\pi^2\|\gamma^*\|^2t}$ 2. The following function, which is a sum over the *length spectrum* of $\mathbb R^n/\Gamma$, is defined as $l_\Gamma(t) = \sum_{\gamma \in \Gamma}e^{-\|\gamma\|^2/4t}$ 3. The *theta series*, which is defined for lattices, is $\theta_\Gamma(z) = \sum_{\gamma\in\Gamma}e^{i\pi z\|\gamma\|^2}$ In the above definitions $t\in\mathbb R$ and $\in \mathbb C$ $\operatorname{Im} z > 0$. Note that $l_\Gamma(t) = \theta_\Gamma(\frac1{4\pi i})$. The Poisson summation formula says that for $t\in(0, \infty)$, the heat trace and $l_\Gamma(t)$ converge and satisfy the relation $h_\Gamma(t) = \frac{\operatorname{Vol}(\Gamma)}{(4\pi t)^{\frac n2}}l_\Gamma(t)$We see an example of the [[The Laplace spectrum and the length spectrum|geodesic-spectrum correspondence]] quite clearly in this case, with the Poisson summation formula serving as the key result connecting the two. The length spectrum is precisely the set of lengths of closed geodesics. If we consider a point $x$ on the torus, then a closed geodesic from $x$ to itself on the torus lifts to a path from $x$ to $\gamma x$ for some $\gamma \in \Gamma$, which means that the lengths of closed geodesics correspond bijectively to the lengths of lattice vectors. ## Connections with other objects So far, we have seen some objects associated to a flat torus. The first is the lattice $\Gamma$ used to generate it, another is the matrix $A$ which forms the lattice via $\Gamma = A\mathbb Z^n$, and the series $h, l,$ and $\theta$ computed from the spectrum. We shall explore some of these objects, and more connected objects. Firstly, we comment on representing lattices using the matrix $A$: When do two such matrices lead to the same (or equivalent) lattices? Firstly, acting on the matrix $A$ by a *unimodular* matrix $U \in GL_n(\mathbb Z)$ does not change the lattice. Secondly, multiplying $A$ by an *orthogonal* matrix $O \in O(n)$ just rotates it and gives us a *congruent* matrix. Thus we can identify the set of lattices in $\mathbb R^n$ with the double-coset space $O(n)\textbackslash GL_n(\mathbb R) /GL_n(\mathbb Z)$. How, then, do we calculate whether two lattices represented by $A_1$ and $A_2$ are congruent? The connection with *quadratic forms* and their *representation numbers* helps us resolve this question. Every *positive-definite* quadratic form $q(x) = x^TQx$ admits a *Cholesky factorization* $Q = A^TA$, and thus we can associate to a quadratic form a lattice $A\mathbb Z^n$. Cholesky decomposition is unique up to multiplication by a unitary matrix, so the lattices associated to a quadratic form are congruent. The *representation numbers* of a quadratic form are the values it takes (with multiplicity) for vectors with integer coordinates; it can be shown that fore positive-definite quadratic forms these multiplicities are finite. The key connection between this theory and that of isospectral lattices and tori is that *quadratic forms with the same representation numbers have isospectral underlying lattices,* which we can show by using the length spectrum and taking $\|Ax\|$ for $x \in \mathbb Z^n$ (an element of the length spectrum) to $x^T(A^TA)x = \|Ax\|^2$ (a representation number). To check if two lattices are congruent, we can use some theory associated with quadratic forms described in [Nilsson, Rowlett, Rydell](https://arxiv.org/abs/2110.09457), section 3.1. Using the description above, we can compute when two lattices are congruent. What about isospectrality? In theory, we would need to compute the dual lattice and check an infinite number of vectors to see when two lattices are isospectral. In this task we are saved by the fact that the *theta series* $\theta_\Gamma$ of an even-dimensional lattice is a modular form. It turns out that there are a finite number of initial coefficients that completely determine a modular form, and this number can be computed. Using this theory, we can figure out if the theta series of two tori match, which would consequently imply that their heat traces match, [[The heat kernel and the heat trace|which determines the entire spectrum]]. For details on the modular form connection, we refer the reader to [Nilsson, Rowlett, Rydell](https://arxiv.org/abs/2110.09457), section 3.2. There is also a connection between lattices and *linear codes*, which is explained in section 3.3. ## The moduli space of flat tori and the choir numbers Flat tori also provide a case where there are easy proofs of results controlling the size of isospectral sets, such as *spectral rigidity* and *spectral isolation* results. A rigidity result says that one cannot deform an object smoothly while preserving its spectrum, and an isolation result says that an object is spectrally different from objects very similar to it in shape. Formally speaking, a rigidity result says that if $M_t$ is smooth curve in the moduli space of manifolds of a given sort such that $M_0 = M$ then there is a neighborhood $(-\epsilon, \epsilon)$ of $M$ such that no other manifold in $M_t$, $t \in (-\epsilon, \epsilon)$ has the same spectrum as $M_0$. An isolation result is slightly stronger and says that $M$ admits a neighborhood in the moduli space where it is the only manifold with its spectrum. Isolation implies rigidity, but not necessarily the other way around. First, we note Scott Wolpert's proof of the spectral rigidity of flat tori. Let $\Gamma_k$ be a sequence of flat tori for $k \in \mathbb N$ that converges to $\Gamma$. What this means is that $A(k)$ is s series of matrices that converge to $A$. If we can show that this sequence eventually becomes stationary then that would immediately imply that flat tori are spectrally rigid, because if $\Gamma_s$ is a continuous family of isospectral tori then $\Gamma_{\frac 1n}$ is such a isospectral sequence. In fact, this result would also imply that flat tori are spectrally isolated, because if not we could use the shrinking neighborhoods around a point to exhibit an isospectral convergent sequence. **Proposition:** If $\Gamma_k$ is a sequence of isospectral flat tori converging to $\Gamma$. Then, eventually the sequence $\Gamma_k$ stabilizes to $\Gamma$. **Proof:** Let $A(k)$ be the matrix associated to $\Gamma_k$ and let $S(k) = A(k)^TA(k)$ be the associated quadratic form. Because the tori are isospectral the image of $S(k)$ over $\mathbb Z^n$ is the same. By the continuity of the determinant we know that $\det(A) = \det(A(k))$, so $A(k)$ is invertible for all $k$. Now, fix $x \in \mathbb Z^n$ and by the triangle inequality, $\|A(k)x\|-\|Ax\| \le \|(A(k) - A)x\|$. The right hand side of this has to go to zero, but the left hand side can only take values in a discrete set, so we know that eventually $\|A(k)x\| = \|Ax\|$ for large enough $k$. We can now choose a large finite set of $x$, a $k$ big enough that this equality holds, and consider that $S(k)$ must be identical to $x$ on all those $x$, showing that $S(k)$ stabilizes. For a more detailed proof, see \[4\]. We can say something stronger: an isospectral set of tori, which we shall now a call a *choir* (following \[1\]), is finite. The proof uses an important result called *Mahler's compactness theorem*, that says that if we have an infinite sequence of lattices whose volume has an upper bound and whose shortest length has a lower bound, then that sequence contains a convergent subsequence. Let $\Gamma_k$ be a set of infinite distinct isospectral lattices. Isospectral lattices have the same volume and vector lengths, so the conditions hold, and we have a convergent subsequence $\Gamma_{k_i}$. By the proposition above this convergent subsequence stabilizes, contradicting our assumption that all $\Gamma_k$ were distinct. In fact we have an even stronger result. If $\Gamma_k$, $1\le k \le N$ is an isospectral choir, then the previous result just tells us that $N$ is a well-defined finite number, but gives no bounds on it. It could still be possible that for some dimension $d$, there are arbitrarily large choirs. However, Suwa-Bier's thesis^[which I have been unable to find, but which is cited by Rowlett et al] shows that in fact there is a bound on $N$ for a fixed dimension. This result lets us think of the dimension as giving a *global* bound on the number of isospectral flat tori regardless of the actual family of tori we are talking about. For any dimension $n$, we call the maximum size of an isospectral choir in that dimension the *choir number* $\flat_n$. Suwa-Bier's result shows that this number is indeed well defined for any given dimension. We shall soon see that isospectral flat tori in dimensions 1, 2, and 3 are isometric implying that $\flat_n = 1$ for $n=1,2,3$. The existence of pairs of isospectral tori in all dimensions greater than 3 implies $\flat_n \ge 2$ for $n \ge 4$. What other bounds are known on these? - In [this paper](https://arxiv.org/pdf/2412.16709), Rowlett, Rydell, and Mardby construct a triple of isospectral tori in dimension six, showing that $\flat_6 \ge 3$ - By adding on a one-dimensional lattice to flat tori in lower dimensions, we can extend an isospectral family to higher dimensions. This means that $\flat$ is a nondecreasing function of $n$ as there are at least as many flat tori in higher dimensions as in lower dimensions. - We can do a little better than this previous result: By combining $\flat_n$ lattices in dimension $n$ to $\flat_m$ lattices in dimension $m$, we can produce $\flat_n \times \flat_m$ tori in dimension $m+n$^[we have to scale the copies a little different to avoid two copies being the same if $m=n$]. Thus we get $\flat_{m+n} \ge \flat_m \flat_n$, and using this result with $\flat_6 = 3$ we get that $\flat_{6n} \ge 3^n$, which is the best known asymptotic result on choir numbers. These lower bounds that we do have come from simple counting arguments that don't use much about the structure of the problem; we do not know much about how many irreducible flat tori can be found in a dimension. We also do not know any upper bounds on the choir numbers beyond the fact that they are finite. ## Low dimensional inverse spectral results and Schiemann's algorithm We stated earlier that $\flat_1 = \flat_2 = \flat_3 = 1$. In one dimension, a flat torus is just a circle and it is easy to see that it is determined by its spectrum. Let us now consider the two-dimensional case. Let there be two lattices given by $A$ and $A'$. In two dimensions we can choose a basis for the lattices given by shortest vectors, and we can rotate the lattice so that the shortest is on the horizontal. Thus we can write the matrices as $A = \begin{bmatrix}a&b\\ 0&d\end{bmatrix},\;\;A'=\begin{bmatrix}a'&b'\\ 0&d'\end{bmatrix}$Since the length of vectors is a spectral invariant, $a=a'$, and because isospectral lattices have the same determinant, $ad = ad' \implies d=d'$. To make $b^2 + d^2 = b'^2=d'^2$, either $b=b'$ and we are done, or $b = -b'$ in which case we can multiply $A'$ with the unimodular matrix $\begin{bmatrix}-1&0\\0&1\end{bmatrix}$ and get equivalent lattices that have the same elements. Thus, two dimensional isospectral lattices are isometric. The proof that $\flat_3=1$ is much harder and accomplished using a technique called *Schiemann's algorithm* after Alexander Schiemann who proved this result in his thesis in 1994. A detailed exposition of this algorithm is given in Rydell's thesis \[2\], with a brief overview in \[3\]. The algorithm proceeds by identifying the space of 3-dimensional positive-definite quadratic forms with a subset $D$ of $\mathbb R^6$ and using algorithms using polyhedral cones to whittle down the set of isospectral quadratic forms in $D\times D$ until we are just left with the diagonal $(f, f) \in D \times D$. The proof boils down to using this algorithm for a large computer search to show that there is no pair of isospectral non-isometric 3-dimensional quadratic forms. We know by explicit counterexample that $\flat_n \ge 2$. ## References 1. Most of this note follows the excellent paper *The isospectral problem for flat tori from three perspectives* by Nilsson, Rowlett and Rydell, available [here](https://arxiv.org/abs/2110.09457) 2. Felix Rydell's thesis *Three perspectives on Schiemann's theorem* expands considerably on section 5 of the above paper and talks about Schiemann's algorithm. It is available freely on the Gothenburg university library website at [https://gupea.ub.gu.se/handle/2077/64923](https://gupea.ub.gu.se/handle/2077/64923) 3. Nilsson's thesis *Can one hear the shape of a flat torus?* gives much of the analytical and differential-geometric background of the results discussed hear, and is also available openly at [https://hdl.handle.net/20.500.12380/300211](https://hdl.handle.net/20.500.12380/300211) 4. Some of the information about finiteness results comes from Scott Wolpert's *The eigenvalue spectrum as moduli for flat tori*, available [here](https://www.jstor.org/stable/1997901)