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I usually ask questions on math.stackexchange but I figure this one is more suited to being asked here. I should preface that I am a complete novice undergraduate, and unlikely to understand answers which refer to complicated differential geometry or number theory.

The Poisson summation formula (also called Jacobi Inversion Formula?) for a lattice in $\mathbb{R}^n$ is given for $t\in(0,+\infty)$ by \begin{align} \sum_{\gamma^*\in \Gamma^*} e^{-4\pi^2||\gamma^*||^2t} = (4\pi t)^{-n/2}\text{Vol}(\Gamma)\sum_{\gamma\in \Gamma} e^{-\frac{||\gamma||^2}{4t}}. \end{align} I am interested in the connection to the Laplace-eigenvalues of the flat torus $\mathbb{R}^n/\Gamma$ which we know are given in the spectrum as $\left\{ 4\pi^2||y||^2 : y\in \Gamma^* \right\}$. In particular I do not understand the following statement from Gordon:

"Next observe that the geodesic length spectrum of a torus $\Gamma\backslash \mathbb R^n$ coincides precisely with the length spectrum of the lattice $\Gamma$, if we define the multiplicity of a length in the geodesic length spectrum to be the number of free homotopy classes of loops containing a geodesic of the given length. The Jacobi inversion formula implies that two lattices $\Gamma_1$ and $\Gamma_2$ have the same length spectrum if and only if the dual lattices $\Gamma_1^*$ and $\Gamma_2^*$ have the same length spectrum. We conclude that two flat tori are isospectral if and only if they have the same geodesic length spectrum."

This is put in another way in the book Old and New Aspects in Spectral Geometry as simply saying that due to the Poisson summation formula, two flat tori $\mathbb{R}^n/\Gamma$ and $\mathbb{R}^n/\Gamma'$ are isospectral if and only if the share the same length spectrum $\{||y||:y\in\Gamma\}=\{||y'||:y'\in\Gamma'\}$.

I have tried playing with the formula by changing $t\mapsto 1/t$ and subsequently trying to show that the RHS determines the eigenvalues by taking away the zeroth eigenvalue from the series and multiplying with an exponential $e^{rs}$ ($r\in\mathbb{R}$) similarly to how one shows it for the LHS. But I can't get this approach to work due to the presence of $(4\pi)^{-n/2}s^{n/2}$.

I have also searched the web (and textbooks) to the best of my ability, but have come out empty. Any help would thus be greatly appreciated, thank you.

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The theta function (the left-hand side of the Jacobi identity) uniquely determines the values of ${||\gamma^*||:\gamma^*\in\Gamma^*}$ counted with multiplicities, just by looking inductively at its asymptotic expansion as $t\to +\infty$: the leading term will be $1$ since the multiplicity of $0$ is 1, the next term will be $Ne^{-4\pi^2\alpha t}$, where $\alpha$ is the smallest norm and $N$ its multiplicity, and so on. Alternatively, you can use the inverse Laplace transform formula.

Similarly, the RHS uniquely determines the values of ${||\gamma||:\gamma\in\Gamma}$ by looking at the asymptotic expansion as $t\to 0+$.

So the two tori are isospectral iff they have the same theta function iff they have the same length spectrum.

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  • $\begingroup$ Sorry, this might be a stupid objection, but when $t\to 0+$ doesn't the LHS go to $\infty$? I would expect the argument to be something like $t\to 0 \implies LHS=eigenvalue=RHS=length$. Is the equality of the two series not preserved when taking the limit? $\endgroup$
    – manifoldcurious
    Commented May 27, 2019 at 14:00
  • $\begingroup$ I do not understand what the the equalities in your comment mean. Yes, both sides go to infinity as $t \to 0+$. The point is that the asymptotic expansion of the RHS allows you to recover $||\gamma||$ (with mupliplicities), as follows: the first term of the expasion is the prefactor of the RHS, the second one is the prefactor times $Ne^{-\alpha/4t}$, where $\alpha$ is the minimum $||\gamma||$ and $N$ its mupliplicities, etc. $\endgroup$
    – Kostya_I
    Commented May 29, 2019 at 8:58
  • $\begingroup$ Sorry I was very sloppy in my explanation. Let $m_i$ be the multiplicity to $\lambda_1$. I was trying to say that I was expecting the following: for $r\in\mathbb{R}$ consider $E_r=e^{rt}(LHS-1)$. Then $r=\lambda_1$ is the unique value so that $E_{\lambda_1}\to m_1$ when $t\to\infty$. I was expecting $e^{rt}(RHS-1)$ to do something similar when $t\to\infty$. Another way I guess of stating my confusion is the following question: suppose I have both LHS and RHS series being true, but I do not have equality. Can I make the same conclusion about isospectrality? If not, why not? $\endgroup$
    – manifoldcurious
    Commented May 30, 2019 at 7:11
  • $\begingroup$ Sorry, what do you mean by "both LHS and RSH series being true"? For each $t$, they are numerical expressions (associated to a particular torus), not statements. The argument goes as follows: suppose two tori are isospectral. Then, they have the same RHS. By Poisson identity, they also have the same LHS, for all $t$. But since LHS uniquely determines the collection of $||\gamma||$, they have the same length spectrum. This works the same way in the other direction. $\endgroup$
    – Kostya_I
    Commented May 30, 2019 at 8:02
  • $\begingroup$ Again I apologise, I meant convergent when I wrote "true". Don't you mean that the RHS determines all $||\gamma||$? I think you might want to switch LHS and RHS in your latest comment. In that case I think I get it! I believe I needed the full argument, thank you! Anyway I will accept your answer. $\endgroup$
    – manifoldcurious
    Commented May 30, 2019 at 8:24

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