# Hearing Shapes via $p$-Adic Laplacians

Patrick Erik Bradley and Ángel Morán Ledezma

###### Abstract

For a finite graph, a spectral curve is constructed as the zero set of a two-
variate polynomial with integer coefficients coming from $p$-adic diffusion on
the graph. It is shown that certain spectral curves can distinguish non-
isomorphic pairs of isospectral graphs, and can even reconstruct the graph.
This allows the graph reconstruction from the spectrum of the associated
$p$-adic Laplacian operator. As an application to $p$-adic geometry, it is
shown that the reduction graph of a Mumford curve and the product reduction
graph of a $p$-adic analytic torus can be recovered from the spectrum of such
operators.

##  1 Introduction

The aim of spectral geometry is to describe the relationship between the
geometry of certain objects like surfaces, or more general Riemannian
manifolds, and the spectra of differential operators, like Laplacians, defined
on them. In other words, as stated by M. Kac in [11], ”Can one hear the shape
of a drum?” Ideally, one would like to be able to recover the geometric
object, up to isometry, from the spectra of one or several naturally defined
operators. Many counter-examples for Riemanian manifolds have appeared showing
that isospectral but non-isometric manifolds exist, which gives a negative
answer to the question. For example for the drum problem:

$\begin{cases}\nabla u=-\lambda u\\\ u|_{\partial D}=0\end{cases}\,$

Carolyn Gordon, David Webb, and Scott Wolpert in 1992, showed the existence of
a pair of drums with different shapes but which are isospectral [7]. On the
other hand, information about the geometry of the object can be extracted from
the spectrum. This kind of problems is known as “inverse problems”. Many
famous results towards this direction have been stablished, for example the
famous Weyl asymptotic law [26]. These problems extend to objects other than
Riemannian manifolds, like graphs, where for the adjacency matrix and the
Laplacian, non-isomorphic isospectral graphs have been found. This problem has
been intensively studied, cf. e.g. [21, 18, 24]. Recovering the structure of a
graph from the spectrum of an operator may lead to an invariant to describe
the topology of the graph. This could lead to new applications, like
recovering the structure of a graph from a diffusion process, which has many
potential applications, e.g. for topological access methods for spatial data
[10], to name only one.

Prime numbers play a fundamental role in many mathematical theories and
applications to sciences. From the realm of arithmetic as the fundamental
blocks or ”atoms” of integers, to applications in physics, from quantum
physics to the theory of complex disordered systems and geophysics,
information processing, biology, and cognitive science, see [20] and the
references therein. One powerful framework for the application of number
theory in sciences is the so called $p$-adic analysis or more general
ultrametric analysis [25, 14]. An important example is given in the theory of
disordered systems (spin glasses) where the $p$-adic structure is encoded in a
Parisi matrix which arises by the intrinsic hierarchical structure inside the
spin glasses [19]. This lead in the middle of the 80s to the idea of using
ultrametric spaces to describe the state of complex systems. A central idea in
physics of complex systems (like proteins) states that the dynamics on such
systems is generated by a random walk (diffusion equation) in the
corresponding energy landscape. By using interbasin kinetics methods, an
energy landscape is approximated by an ultrametric space and a function on
this space describing the distribution of the activation barriers, see e.g.
[15] and the references therein. Most of the applications towards this
direction recquire well-defined and natural pseuddifferential operators
constructed on ultrametric structures such as Non-Archimedean fields, where
the Taibleson-Vladimirov operator plays a fundamental role for diffusion on
the field $p$-adic numbers [25]. Differential operators and spectral geometry
on Riemannian manifolds have been extensively studied, nevertheless there is
no comparable theory of Non-Archimedean spectral geometry of
pseudodifferential operators over $p$-adic structures. Many other operators
have been developed, some of them with the aim of applications, and others as
generalisations to more general structures. For the former we have many
classes of $p$-adic operators from the work of W. Zúñiga, Kozyrev, Khrennikov,
where the relation of graph theory and $p$-adic integral and
pseudodifferential operators is explicitly stated, see [27, 14, 17], and the
reference there in.

For the latter one of the authors initiated the study of heat equations and
integral operators on the Non-Archimedean kin of Riemannian surfaces i.e.
Mumford Curves [3]. All those developments in the theory of pseudodifferential
equations over Non-Archimedean spaces clearly deal (indirectly) with one of
the main problems in spectral geometry, that is, direct problems in which a
description of the eigenvalues is needed.

In this article we initiate the study of inverse problems of spectral geometry
in the Non-Archimedean framework. Moreover, a new invariant for an arbitrary
combinatorial simple graph is introduced, showing that the spectra of certain
$p$-adic operators defined on the graph lead to a complete characterisation of
its isomorphisim class. The question ”Can you hear the shape of a graph?” has
already been answered in different contexts. In [8], the question was posed in
the context of quantum graphs, and was answered in the affirmative, that is,
they showed that the spectrum of the Schrödinger operator on a finite, metric
graph determines uniquely the connectivity matrix and the bond lengths under
certain conditions. In [28], a new spectral invariant in quantum graphs has
been introduced. In [9], it was proved that the spectral determinant of the
Laplace operator on a ﬁnite connected metric graph determines the number of
spanning trees under certain conditions. Understanding how the spectra of
certain operators in general graphs determine the geometry of a graph is an
important task for applications like graph comparison in graph analytics. For
example in [23], the Network Laplacian Spectral Descriptor, a graph
representation method that allows for straightforward comparisons of large
graphs, is proposed. Moreover, our results are applied to $p$-adic structures
like Mumford curves and $p$-adic analytic tori. Hence these results initiate
the study of inverse problems in spectral geometry in the Non-Archimedean
framework.

Given a graph $G$ and a matrix $\Delta\in\mathbb{N}^{|G|\times|G|}$, we study
a generalisation of a graph Laplacian $\Lambda_{G}^{\Delta}$ defined in
$L^{2}(G\times K)$, where $K$ is a non-archimedean local field. The space
$L^{2}(G\times K)$ can be decomposed as a direct sum of finite dimensional
spaces of dimension $|G|$, this leads of the following representation of
$\Lambda_{G}^{\Delta}$ ,

$\Lambda_{G}^{\Delta}=\bigoplus_{G\mathcal{K}}L(G_{r}^{\Delta}),$

where the matrices $L(G_{r}^{\Delta})$ are the Laplacian matrix of a weighted
version of the graph, and for $r=1$, we have that $G_{1}=G$. Therefore, this
operator can be understood as a direct sum of scaled replica of the original
graph. The spectrum of each copy belongs to a common plane algebraic curve
$V(P_{G}^{\Delta})$ called the spectral curve of the graph. For a suitable
choice of $\Delta$ we prove that $P_{G}^{\Delta}$ is an invariant of the graph
$G$. This leads to a reconstruction theorem which enable us to reconstruct the
graph through the spectra of the operator $\Lambda_{G}^{\Delta}$ (see Theorem
$4.7$ and Corollary $4.9$). Finally using these results we are able to
reconstruct the reduction graph of a Mumford curve and the product graph
coming from the reduction of a $p$-adic analytic torus using the spectrum of a
$p$-adic Laplacian.

##  2 Notation and Some Results from $p$-Adic Analysis

In this section we review some results from $p$-adic analysis, for a complete
exposition of the subject and proofs the reader may consult [1].

Let $K$ be a Non-Archimedean local field. Let $|\cdot|_{K}$ denote the
absolute value of the field $K$. Denote the local ring of $K$ by
$\mathcal{O}_{K}=\\{x\in K:|x|_{K}\leq 1\\}$ and its maximal ideal by
$\mathfrak{m}_{K}=\\{x\in K:|x|_{K}<1\\}$. Let $\chi$ be a fixed non-constant
complex-valued additive character on $K$. We denote by $dx$ the Haar measure
on the additive group of $K$, normalised such that the measure of
$\mathcal{O}_{K}$ is equal to $1$. The Fourier transform of an absolute
integrable complex-valued function $f\in L^{1}(K)$ will be written as

$\mathscr{F}(f)(\xi)=\int_{K}\chi(\xi x)f(x)dx,\ \xi\in K.$

If $\mathscr{F}(f)=\hat{f}\in L^{1}(K),$ we get the inversion formula

$f(x)=\int_{K}\chi(-x\xi)\hat{f}(\xi)d\xi.$

Since the mapping $\mathscr{F}:L^{1}(K)\cap L^{2}(K)\rightarrow L^{2}(K)$ is
an isometry, this mapping has an extension to an $L^{2}-$isometry from
$L^{2}(K)$ into $L^{2}(K)$, where the inverse Fourier transform will be
denoted as $\mathscr{F}^{-1}$.

Now we introduce the Vladimirov-Taibleson operator. Let $\mathcal{D}\subset
L^{2}(K)$ be its domain given by the set of those $f\in L^{2}(K)$, for which
$|\xi|^{\alpha}\hat{u}(\xi)\in L^{2}(K)$. The Vladimirov operator
$(\Delta^{\alpha},\mathcal{D})$, $\alpha>0$, for $f\in\mathcal{D}$ is defined
by

$\Delta^{\alpha}f(x)=\mathscr{F}^{-1}_{\xi\mapsto
x}(|\xi|_{K}^{\alpha}\mathscr{F}_{x\mapsto\xi}(f)(\xi))(x),\ x\in K.$

The operator $\Delta^{\alpha}$ is an unbonded operator in $L^{2}(K)$, and
since it is unitarily equivalent to the operator of multiplication by
$|\xi|_{K}^{\alpha}$, it is self-adjoint, its spectrum consists of the
eigenvalues $\lambda_{r}=q^{\alpha r}$, where $r\in\mathbb{Z}$ and $q$ is the
cardinality of the residue field $\mathcal{O}_{K}/\mathfrak{m}_{K}$. Moreover
we have the following result

######  Theorem 2.1 (Kozyrev).

There exist a complete orthonormal system of eigenfunctions of the operator
$\Delta^{\alpha}$ of the form $\psi_{r,n}(x)\in L^{2}(K)$ , where
$r\in\mathbb{Z}$ and $n\in\mathbb{N}$ such that

$\Delta^{\alpha}\psi_{r,n}(x)=q^{\alpha(1-r)}\psi_{r,n}.$

###### Proof.

For the case of $\mathds{Q}_{p}$, cf. [16]. The case of a general Non-
Archimedean local field $K$, cf. [1]. ∎

Henceforth this basis of $L^{2}(K)$ from Theorem 2.1 will be denoted by
$\mathcal{K}$.

##  3 Spectral Curves for Diffusion Pairs

In this section, we introduce the objects necessary for constructing the
spectral curve of a so-called _diffusion pair_ which is actually nothing but a
weighted graph, where the weights are integer powers of a fixed variable $Y$.
These objects are $p$-adic matrix-valued Laplacian operators reflecting the
adjacency structure of a graph.

###  3.1 $p$-Adic Laplacians for Graphs

Let $G\subset K/O_{K}$ be a finite set. Then we have isomorphisms

$L^{2}(G\times K)\cong L^{2}(G)\otimes L^{2}(K)\cong\bigoplus\limits_{a\in
G}L^{2}(K_{a})$

where $K_{a}$ is a copy of $K$ for each $a\in G$. We define maps:

(3)

where we write

$L^{2}(K_{a})=\bigoplus\limits_{\psi\in\mathcal{K}}\mathds{C}\psi_{a}$

using the set $\mathcal{K}$ of Kozyrev wavelets on $K$, and

$\psi_{a}(x)=\psi(x).$

The map $H_{G}$ is given as follows:

$\displaystyle H_{G}$ $\displaystyle\colon(u_{g})_{g\in G}\mapsto(f_{g})_{g\in
G},\;f_{g}=\sum\limits_{a\in G}C_{ga}\Delta_{ga}u_{a}$

where $\Delta_{ga}$ is the Vladimirov operator

$\Delta_{ga}\colon L^{2}(K_{a})\to
L^{2}(K_{g}),\psi_{a}\mapsto\Delta^{\alpha_{ga}}\psi_{g}$

where

$\Delta^{\alpha_{ga}}=\mathscr{F}^{-1}\left\lvert\cdot\right\rvert_{K}^{\alpha_{ga}}\mathscr{F}$

behaves like the usual Vladimirov operator, except for being applied to
different copies of Kozyrev wavelets indexed by vertices of $G$. In
particular, it simply multiplies the indexed Kozyrev wavelet $\psi_{g}$ by an
integer power of $q^{\alpha_{ga}}$.

Notice that in the basis of $L^{2}(G\times K)$ given by

$G\mathcal{K}={\left\\{\psi_{g}\colon g\in G,\;\psi\in\mathcal{K}\right\\}}$

we can represent $H_{G}$ by the $\left\lvert G\right\rvert\times\left\lvert
G\right\rvert$-matrix

$(C_{ag}\Delta_{ag})$

And the matrix $C=(C_{ag})$ can be viewed as an adjacency matrix of a simple
graph with vertex set $G$.

In order to obtain a graph Laplacian matrix, we consider instead of $H_{G}$
the operator

$\Lambda_{G}^{\Delta}\colon L^{2}(K)^{\left\lvert G\right\rvert}\to
L^{2}(K)^{\left\lvert G\right\rvert}$

represented by the matrix

$(L_{ab}^{\Delta})_{a,b\in G}$

with

$L_{ab}^{\Delta}=\begin{cases}-C_{ab}\Delta_{ab},&a\neq b\\\ \sum\limits_{g\in
G}C_{ag}\Delta_{ag},&a=b\end{cases}$

Here, $\Delta=(\Delta_{ga})$ can be viewed as a a matrix in
$\mathds{N}^{\left\lvert G\right\rvert\times\left\lvert G\right\rvert}$ having
entry $\alpha_{ga}$ whenever $ga$ represents an edge of the graph.

Later, we will show that there exist choices of diffusion parameters
$\alpha_{ag}\in\mathds{N}$ such that the spectrum of the operator
$\Lambda_{G}^{\Delta}$ determines the isomorphism class of the combinatorial
simple graph $G$.

######  Definition 3.1.

The operator $\Lambda_{G}^{\Delta}$ is called the _$p$ -adic Laplacian
associated with the diffusion pair $(G,\Delta)$_.

###  3.2 The Spectral Curve of a Diffusion Pair

Let $(G,\Delta)$ be a diffusion pair. Recall that $L=L_{1}$ is the Laplacian
of the graph $G$. We begin with the following observation:

######  Lemma 3.2.

It holds true that

$\operatorname{Spec}(L)\subset\operatorname{Spec}(\Lambda_{G}^{\Delta})$

as an inclusion of multi-sets.

###### Proof.

Observe first that in the basis $G\mathcal{K}$, the operator
$\Lambda_{G}^{\Delta}$ is represented by an
$\mathds{N}\times\mathds{N}$-matrix having a block-diagonal structure with
blocks of size $\left\lvert G\right\rvert\times\left\lvert G\right\rvert$
after a suitable linear ordering of the basis. Now, the non-zero entries of
each block away from the diagonal consist of Vladimirov eigenvalues. By
Theorem 2.1, they are of the form

$q^{\alpha_{ag}(1-r)}$

with $r\in\mathds{Z}$. For $r=1$ we identify the Laplacian matrix $L$ as one
of the blocks. Hence, the eigenvalues of $L$ are contained in the spectrum of
$\Lambda_{G}^{\Delta}$. ∎

Observe further that the block-diagonal structure found in the proof of the
above Lemma is in fact a replication of Laplacian matrices for the same
combinatorial graph structure on $G$, except that now the edge lengths are
powers of $p$ of the form $p^{\alpha_{ab}(1-r)}$ for fixed $r\in\mathds{Z}$.
This means that there is a family of graphs $G_{r}^{\Delta}$ parametrised by
$r\in\mathds{Z}$ having the same combinatorial Laplacian matrix $L$. And in
order to find all eigenvalues of $\Lambda_{G}^{\Delta}$, it is necessary and
sufficent to find the Laplacian eigenvalues for each graph in the family
$G_{r}^{\Delta}$ with $r\in\mathds{Z}$.

We will now examine the characteristic polynomial of each graph Laplacian
$L_{r}^{\Delta}$ associated with graph $G_{r}^{\Delta}$ from the family.
Notice that

$L_{1}^{\Delta}=L_{1}=L,\quad G_{1}^{\Delta}=G_{1}=G,$

are independent of the diffusion parameters symbolised by $\Delta$. We also
assume that the parameters $\alpha_{ab}$ are all pairwise different positive
natural numbers. The characteristic polynomial of $G_{r}^{\Delta}$ is

$P_{r}^{\Delta}(X)\in\mathds{Q}[X]$

and its degree is $\left\lvert G\right\rvert$. Again, we have

$P_{1}(X)=P_{1}^{\Delta}(X)$

is independent of $\Delta$, and coincides with the characteristic polynomial
of the graph Laplacian $L$.

The coefficients of $P_{r}^{\Delta}$ are given by the Leibniz formula for the
determinant as polynomials with integer coefficients in another variable $Y$
evaluated in $p^{1-r}$. Hence, we obtain a polynomial

$P_{G}^{\Delta}(X,Y)\in\mathds{Z}[X,Y]$

whose zero set in $\overline{\mathds{Q}}^{2}$ contains the spectrum of
$\Lambda_{G}^{\Delta}$ as the first coordinate of some of its points. Here, we
mean by $\overline{\mathds{Q}}$ the algebraic closure of $\mathds{Q}$.

######  Definition 3.3.

The completion of the plane algebraic curve $V(P_{G}^{\Delta})$ to a
projective algebraic curve is called the _spectral curve_ of the pair
$(G,\Delta)$.

###  3.3 Recovering Spectral Curves

Assume that we are given the $\operatorname{Spec}\Lambda_{G}^{\Delta}$ as a
multi-set, where $\Lambda_{G}^{\Delta}$ is the $p$-adic Laplacian associated
with diffusion pair $(G,\Delta)$. Assume also that the task is to recover the
graph $G$ from that spectrum. One way would be to try to recover the spectral
polynomial $P(X,Y)=P_{G}^{\Delta}(X,Y)$ and use the Reconstruction Theorem
(Theorem 4.7) proved below.

In this situation, an algorithm which terminates in finite time cannot be
expected, because each individual eigenvalue has to be associated with one of
the graphs $G_{r}$ ($r\in\mathds{Z}$) in the family induced by the spectral
pair. But from a purely existential standpoint, we can say that there exists a
classification of eigenvalues (including their multiplicities) such that each
class is $\operatorname{Spec}(L_{r})$ with $r\in\mathds{Z}$. Once this
classification is made, then each coefficient

$a_{i}(p^{1-r}),\quad i=1,\dots,n$

of the polynomial

$P(X,Y)=\sum\limits_{i=1}^{n}a_{i}(Y)X^{i}$

with $r\in\mathds{Z}$ can be calculated in each class of eigenvalues. All that
is then needed, is for each $r\in\mathds{Z}$ the value of

$P(X,p^{1-r})$

in finitely many places $x_{s}$. Then interpolation yields the coefficients of
$P(X,Y)$.

######  Definition 3.4.

A set of pairs $(x_{s},p^{1-r})$ with $s,r\in R\subset\mathds{N}$ is called a
_recovery datum_ , if $R$ is a finite set and $P(X,Y)$ can be interpolated
after evaluating the polynomial in that set of pairs.

######  Theorem 3.5.

Let $(G,\Delta)$ be a diffusion pair. Given $\operatorname{Spec}(L_{r})$ as
distinguished multi-sets for sufficiently but finitely many $r\in\mathds{Z}$,
it is possible to obtain recovery data for the spectral polynomial
$P_{(G,\Delta})$ with a terminating algorithm.

###### Proof.

Since all graphs $G_{r}$ are simple and have the same underlying combinatorial
graph with $n$ vertices and

$\left\lvert E\right\rvert\leq\frac{1}{2}n(n-1)=:b_{n}$

edges, it follows that the number of places to interpolate

$P(X,Y)=P_{G}^{\Delta}(X,Y)$

is bounded. As $n$ is given as the size of each multi-set
$\operatorname{Spec}(L_{r})$, it follows that $b_{n}$ such spectra are
sufficient in order to reproduce the characteristic polynomials

$P(X,p^{1-r})=P_{r}(X)$

of the graphs $G_{r}$. Evaluating the $b_{n}$ polynomials $P_{r}(X)$ at
$b_{n}$ places $x_{s}\in\mathds{R}$ yields pairs $(x_{s},p^{1-r})$ which form
a recovery datum, as now interpolation of $P(X,Y)$ is possible. Together, this
is an algorithm terminating after finitely many steps. ∎

The question is now, whether it is possible to extract distinguished multi-
sets $\operatorname{Spec}(L_{r})$ somehow by clustering spectral values. If it
is allowed to vary the prime number $p$, then this can be done in the
following game:

######  Game 1.

Assume that you are allowed to choose diffusion parameters $\Delta$ once, and
a prime number $p$ as many times as you wish. Then you will receive for each
$p$ the multi-set $\operatorname{Spec}(\Lambda_{G}^{\Delta})$ of an unknown
simple finite connected graph. If you manage to recover the spectral curve for
the diffusion pair $(G,\Delta)$ from these spectra, then you win, otherwise
you lose.

The following theorem states that there exist winning strategies for the Game
1:

######  Theorem 3.6.

For the $p$-adic Laplacian associated with any diffusion pair, there exists a
winning strategy in order to obtain distinguished multi-sets
$\operatorname{Spec}(L_{r})$ with $r\in\mathds{Z}$.

###### Proof.

In the case that the choice of diffusion parameters is to have them all equal
to $1$ (when beloning to an edge, otherwise, it is zero), a winning strategy
for connected graphs is to pick a large prime $p$. In this case, we have

$L_{r}=p^{1-r}L$

where $L$ is the Laplacian of the graph $G$.

Since the non-zero part of the spectrum of $L$ lies inside a compact interval
not containing $0$, as has been proven by [5], it then suffices to ask for
higher and higher prime numbers until there are increasing gaps between
clusters with relatively small inter-cluster distances between neigbouring
points. With increasing $p$, this phenomenon becomes more and more clearly
visible.

If not all parameters are chosen equal to $1$, we restrict to $r\leq 1$ and
again vary the prime $p$. From Matrix Perturbation Theory [2], we get that if
$p$ is sufficiently large, then in the range $r\leq 1$, again the inter-
cluster distances of neighbouring eigenvalues will be smaller than the intra-
cluster distances between neighbouring clusters. Hence, choosing $p$
sufficiently large, again removes overlaps between the clusters.

Since in both cases of diffusion parameter choices, the spectrum of $L$
remains fixed for any choice of varying the prime $p$, the reference cluster
for $r=1$ can be extracted, and then the spectra $\operatorname{Spec}(L_{r})$
for more values of $r\in\mathds{Z}$, or of $r\leq 1$ in the second case. After
having extracted sufficently many of these finite spectra, one can proceed to
the interpolation method and compute a recovery datum, as now the requirements
for Theorem 3.5 are met. ∎

##  4 Spectral Curves Which are Separating

In this section, we first separate pairs of non-isomrophic, but isospectral,
graphs via spectral curves for suitable diffusion pairs using analytic matrix
perturbation theory. After that, we prove for every finite graph the existence
of a diffusion pair such that the graph can be reconstructed from the spectral
polynomial. Although this generalises the first result, we believe that the
matrix perturbation method is of general interest, nevertheless.

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Figure 1: A pair of non-isomorphic, but isospectral graphs whose first Betti
number is $3$.

######  Example 4.1.

According to [18], the two graphs in Figure 1 are isospectral. Their first
Betti number equals $3$. This is the smallest example of an isospectral pair
of non-isomorphic simple graphs without bridges. The label ”2” on an edge
indicates a diffusion parameter value of $2$, i.e. an edge weight $Y^{2}$.
Unlabelled edges have diffusion parameter value $1$, i.e. edge weight $Y$.
Their respective spectral polynomials $P_{1}$ for the left, and $P_{2}$ for
the right graph of Figure 1 are:

$\displaystyle P_{1}(X,Y)$
$\displaystyle=\det\left(\begin{smallmatrix}X-3Y&Y&0&0&0&0&Y&Y\\\
Y&X-2Y&Y&0&0&0&0&0\\\ 0&Y&X-2Y&Y&0&0&0&0\\\ 0&0&Y&X-2Y&Y&0&0&0\\\
0&0&0&Y&X-2Y&Y&0&0\\\ 0&0&0&0&Y&X-3Y&Y&Y\\\ Y&0&0&0&0&Y&X-(2Y+Y^{2})&Y^{2}\\\
Y&0&0&0&0&Y&Y^{2}&X-(2Y+Y^{2})\end{smallmatrix}\right)$ $\displaystyle
P_{2}(X,Y)$ $\displaystyle=\det\left(\begin{smallmatrix}X-3Y&Y&Y&0&0&0&0&Y\\\
Y&X-2Y&Y&0&0&0&0&0\\\ Y&Y&X-3Y&Y&0&0&0&0\\\ 0&0&Y&X-2Y&Y&0&0&0\\\
0&0&0&Y&X-2Y&Y&0&0\\\ 0&0&0&0&Y&X-(2Y+Y^{2})&Y&Y^{2}\\\ 0&0&0&0&0&Y&X-2Y&Y\\\
Y&0&0&0&0&Y^{2}&Y&X-(2Y+Y^{2})\end{smallmatrix}\right)$

Their tangent cones $T_{1}$ of $P_{1}$ and $T_{2}$ of $P_{2}$ are:

$\displaystyle T_{1}(X,Y)$
$\displaystyle=\det\left(\begin{smallmatrix}X-3Y&Y&0&0&0&0&Y&Y\\\
Y&X-2Y&Y&0&0&0&0&0\\\ 0&Y&X-2Y&Y&0&0&0&0\\\ 0&0&Y&X-2Y&Y&0&0&0\\\
0&0&0&Y&X-2Y&Y&0&0\\\ 0&0&0&0&Y&X-3Y&Y&Y\\\ Y&0&0&0&0&Y&X-2Y&0\\\
Y&0&0&0&0&Y&0&X-2Y\end{smallmatrix}\right)$ $\displaystyle T_{2}(X,Y)$
$\displaystyle=\det\left(\begin{smallmatrix}X-3Y&Y&Y&0&0&0&0&Y\\\
Y&X-2Y&Y&0&0&0&0&0\\\ Y&Y&X-3Y&Y&0&0&0&0\\\ 0&0&Y&X-2Y&Y&0&0&0\\\
0&0&0&Y&X-2Y&Y&0&0\\\ 0&0&0&0&Y&X-2Y&Y&0\\\ 0&0&0&0&0&Y&X-2Y&Y\\\
Y&0&0&0&0&0&Y&X-2Y\end{smallmatrix}\right)$

These are spectral polynomials of the two bridgeless graphs of genus two,
where each edge has the same variable $Y$, shown in Figure 2. According to
[18, Thm. 3.1], these graphs are not isospectral, because they are not
isomorphic. It follows that the two tangent cones $T_{1}$ and $T_{2}$ are not
equal.

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---
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{8\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{7\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{6\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5}$
Figure 2: A pair of non-isomorphic bridgeless graphs whose first Betti number
is $2$. According to [18], it follows that they are not isospectral.

We saw that replacing one certain edge weight in each graph by $Y^{2}$ leads
to two polynomials $P_{1}(X,Y)$ and $P_{2}(X,Y)$ which are not the same. So,
in this case, the isospectral pair is separated by these two polynomials. It
only happens that

$P_{1}(X,1)=P_{2}(X,1)$

resulting in two identical characteristic polynomials for the two non-
isomorphic graphs. This example motivates the remainder of this article.

###  4.1 Separating Isospectral Pairs via Matrix Perturbation

An introduction to matrix perturbation theory can be found in [2]. We will use
this method in order to construct distinct spectral polynomials for non-
isomorphic, but isospectral graphs. All our calculations are explicit and most
of them reproduced here, even if many can also be found in that bibliographic
reference in a more general setting.

Let $k\neq\ell$ be natural numbers in ${\left\\{1,\dots,n\right\\}}$. We
define the matrix

$U(k,\ell)=(u_{ij})$

with

$u_{ij}=\begin{cases}1,&i=j=k,\;\text{or}\;i=k=\ell\\\
-1,&(i,j)=(k,\ell),\;\text{or}\;(i,j)=(\ell,k)\\\
0,&\text{otherwise}\end{cases}$

This is the Laplacian of the graph on $n$ vertices having precisely one
undirected edge $(k,\ell)$, since $U(k,\ell)=U(\ell,k)$.

Let $v=(v_{1},\dots,v_{n})\in\mathds{R}^{n}$. Then

$v^{\top}U(k,\ell)v=(v_{k}-v_{\ell})^{2}$

as can be verified by a simple calculation.

Now, let $E$ be a symmetric set of pairs $(i,j)$ of numbers $1,\dots,n$ with
$i\neq j$. Then

$U(E):=\sum\limits_{(k,\ell)\in E}U(k,\ell)$

and we have

$v^{\top}U(E)v=\sum\limits_{(k,\ell)\in E}(v_{k}-v_{\ell})^{2}$

We can now define

$\left\|v\right\|_{E}^{2}=v^{\top}U(E)v$

The function $\left\|\cdot\right\|_{E}$ is a semi-norm on $\mathds{R}^{n}$,
and we have

$\left\|v\right\|_{E}^{2}=0\quad\Leftrightarrow\quad\forall\;(i,j)\in E\colon
v_{i}=v_{j}$

######  Lemma 4.2.

Let $L$ be the Laplacian of a simple graph $G$ with $n$ vertices, and let
$E,E^{\prime}$ be two disjoint sets of pairs from
${\left\\{1,\dots,n\right\\}}$. Then there exists an eigenvector $v$ of $L$
such that

$\left\|v\right\|_{E}^{2}\neq\left\|v\right\|_{E^{\prime}}^{2}$

###### Proof.

This follows from the fact that $L$ is symmetric, i.e. from the Spectral
Theorem for symmetric real-valued matrices. Namely, if for all eigenvectors
$v$ of $L$ we had

$\left\|v\right\|_{E}=\left\|v\right\|_{E^{\prime}}$

then it would be impossible to generate via linear combination a vector
$x\in\mathds{R}^{n}$ having
$\left\|x\right\|_{E}\neq\left\|x\right\|_{E^{\prime}}$, because both
seminorms scale identically when an eigenvector is multiplied with a scalar. ∎

An equivalence of diffusion pairs $(G_{1},\Delta_{1})$ and
$(G_{2},\Delta_{2})$ where $G_{i}$ are graphs on the same vertex set $V$, is
given by a bijection $E_{1}\to E_{2}$ between the edge sets of $G_{1}$ and
$G_{2}$ which takes a weighted edge to an edge having the same weight. If two
graphs $G_{1},G_{2}$ are isospectral, then it is possible to define diffusion
parameters $\Delta_{1},\Delta_{2}$ such that there is an equivalence of
diffusion pairs $(G_{1},\Delta_{1})\sim(G,\Delta_{2})$. The following theorem
shows that this already allows to distinguish non-isomorphic graphs, if
suitable choices are taken.

######  Theorem 4.3.

Assume that $G_{1},G_{2}$ is a pair of non-isomorphic, but isospectral graphs.
Then there exist equivalent diffusion pairs $(G_{1},\Delta_{1})$,
$(G_{2},\Delta_{2})$ such that

$P_{G_{1}}^{\Delta_{1}}(X,Y)\neq P_{G_{2}}^{\Delta_{2}}(X,Y)$

as bivariate polynomials.

###### Proof.

Denote the edge set of graph $G_{i}$ as $E_{i}$ for $i=1,2$, and let
$C=E_{1}\cap E_{2}$, and $C_{1}=E_{1}\setminus E_{2}$, $C_{2}=E_{2}\setminus
E_{1}$. Since $G_{1}$ and $G_{2}$ are isospectral, it follows that $E_{1}$ and
$E_{2}$ have the same number of elements.

We are interested in the spectrum of the matrices

$L_{\varepsilon,i}=U(C)+\varepsilon U(C_{i})$

for $i=1,2$, and with $\varepsilon>0$. In first order w.r.t. the parameter
$\varepsilon$, we have

$\displaystyle\lambda_{\varepsilon,i}^{(1)}=\lambda+\varepsilon\left\|v\right\|_{C_{i}}^{2}+\text{terms
of higher order in $\varepsilon$}$ (4)

where $v\in\mathds{R}^{n}$ is an eigenvector of $U(C)$ associated with
eigenvalue $\lambda$, and $\lambda_{\varepsilon,i}^{(1)}$ approximates in
first order an eigenvalue $\lambda_{\varepsilon,i}$ of $L_{\varepsilon,i}$.
According to Lemma 4.2, one can find an eigenvector $v$ of $U(C)$ such that

$\left\|v\right\|_{C_{1}}\neq\left\|v\right\|_{C_{2}}$

This implies that the first order approximations of the eigenvalues
$\lambda_{\varepsilon,i}$ are different. It follows that the eigenvalues
$\lambda_{\varepsilon,1}$ and $\lambda_{\varepsilon,2}$ are different for the
range of $\varepsilon>0$ in which an analytic expansion in $\varepsilon$ is
possible and $\varepsilon>0$ is sufficiently small.

Now, we have varied one eigenvalue of $U(C)$ analytically in two different
ways. The other eigenvalues of $U(C)$ also vary analytically as in (4), except
that possibly it could be that some eigenvalues corresponding to
$L_{\varepsilon,1}$ and $L_{\varepsilon,2}$ are equal in this case. In any
case, the spectrum of $U(C)=L_{\varepsilon,0}$ is varied continuously in two
different manners. Hence, for $\varepsilon>0$ small, the discrete subsets
$\operatorname{Spec}(L_{\varepsilon,1})$ and
$\operatorname{Spec}(L_{\varepsilon,2})$ of $\mathds{R}$ are different. This
implies that the corresponding characteristic polynomials

$P_{i}(X,\varepsilon),\quad i=1,2$

are different. But these are evaluations of the spectral polynomials

$P_{i}(X,Y)$

evaluated at $Y=\varepsilon$ for all $\varepsilon>0$ sufficiently small, and
where these polynomials are given for edge weight maps:

$\Delta_{i}\colon C\cup E_{i}\to\mathds{N},\;e\mapsto\begin{cases}1,&e\in C\\\
0,&e\in E_{i}\end{cases}$

It follows that these two spectral polynomials are different.

In order to obtain positive diffusion constants, now look at the spectrum of
the matrix

$\varepsilon L_{\varepsilon,i}=\varepsilon U(C)+\varepsilon^{2}U(C_{i})$

which amounts to taking the diffusion parameters as

$\Delta_{i}\colon C\cup E_{i},\;e\mapsto\begin{cases}2,&e\in E_{i}\\\ 1,&e\in
C\end{cases}$

Hence, there exists a bijection $E_{1}\to E_{2}$ between sets of weighted
edges, i.e. an equivalence of diffusion pairs
$(G_{1},\Delta_{1})\sim(G_{2},\Delta_{2})$, such that
$P_{G_{1}}^{\Delta_{1}}(X,Y)\neq P_{G_{2}}^{\Delta_{2}}(X,Y)$ as asserted. ∎

###  4.2 A Reconstruction Theorem

Let $G=(V(G),E(G))$ be a graph with $n$ vertices. We begin with the following
observation:

######  Theorem 4.4 (Kel’mans, 1967).

Let

$P(X)=X^{n}-\sum\limits_{i=1}^{n-1}(-1)^{i}c_{i}X^{n-i}$

be the characteristic polynomial of the graph $G$. Then

$c_{i}=\sum\limits_{S\subset V\atop\left\lvert S\right\rvert=n-i}T(G_{S})$

where $T(H)$ is the number of spanning trees of graph $H$, and $G_{S}$ is the
quotient graph obtained by identifying all vertices in $S$ with a single
vertex.

###### Proof.

[12, 13] ∎

We recover these quantities from the spectral polynomial $P(X,Y)$ as follows:

$c_{i}(G)=a_{n-i}(1)$

for $i=1,\dots,n-1$.

We define

$\mathcal{F}(G)={\left\\{F\colon\text{$F$ is a forest with $V(F)=V(G)$ and
$E(F)\subset E(G)$}\right\\}}$

and call $\mathcal{F}(G)$ the _spanning forest set of $G$_. The following
subsets of $\mathcal{F}(G)$ are of interest:

$\mathcal{F}^{i}(G)={\left\\{F\in\mathcal{F}(G)\colon b_{0}(F)=i\right\\}}$

This allows us to formulate a generalisation of Theorem 4.4 which is also a
known result, but formulated here in the guise of spectral curves:

######  Theorem 4.5 (Buslov, 2014).

Let $(G,\Delta)$ be a diffusion pair. Then the coefficients of the spectral
polynomial

$P_{G}^{\Delta}(X,Y)=\sum\limits_{i=1}^{n}a_{i}(Y)X^{i}$

of a diffusion pair $(G,\Delta)$ are given as

$a_{i}(Y)=(-1)^{n-i}\sum\limits_{F\in\mathcal{F}^{i}(G)}\pi_{F},\quad\pi_{F}=\prod\limits_{e\in
E(F)}Y^{\alpha_{e}}$

for $i=1,\dots,n$.

###### Proof.

[4, Thm. 2]. ∎

Let $G^{\prime}$ be another graph. An isomorphism

$\mathcal{F}(G)\cong\mathcal{F}(G^{\prime})$

between spanning forest sets is given by a bijection
$f\colon\mathcal{F}(G)\to\mathcal{F}(G^{\prime})$ and an ismorphism $F\cong
F^{\prime}$ with $F\in\mathcal{F}(G)$ and $F^{\prime}=f(F)$ for all
$F\in\mathcal{F}(G)$.

######  Lemma 4.6.

Let $G,G^{\prime}$ be two graphs on $n$ vertices. Then $G$ and $G^{\prime}$
are isomorphic if and only if the strict forest subsets $\mathcal{F}(G)$ and
$\mathcal{F}(G^{\prime})$ are isomorphic.

###### Proof.

If the graphs are isomorphic, then clearly their strict forest sets are
isomorphic.

Assume now that $G=T$ and $G^{\prime}=T^{\prime}$ are trees. Since trees are
their own spanning forests, we clearly must have that $T\cong T^{\prime}$.

Now, assume that $G$ is not a tree. If $G$ is a forest, then we can apply the
result for trees to each individual connected component of $G$. So, we may
assume that $b_{1}(G)>0$, and that $G$ is connected. If
$\mathcal{F}(G)\cong\mathcal{F}(G^{\prime})$, then w.l.o.g. we may assume that
these two sets are equal, and that the vertex sets of the two graphs coincide.
Let $T$ be a spanning tree of $G$. Then

$\mathcal{F}(T)\subset\mathcal{F}(G)=\mathcal{F}(G^{\prime})$

Hence, by symmetry, each spanning tree of $G$ is a spanning tree of
$G^{\prime}$ and vice versa. This implies that $G=G^{\prime}$, as otherwise
there is an edge of $G$ not in $G^{\prime}$. But then a spanning tree of $G$
containing that edge is not a spanning tree of $G^{\prime}$, a contradiction.
Hence, $G\cong G^{\prime}$. ∎

######  Theorem 4.7 (Reconstruction Theorem).

For any finite graph $G$, there exists a $p$-adic Laplacian given by diffusion
parameters $\Delta$ such that the diffusion pair $(G,\Delta)$ has a spectral
polynomial $P(X,Y)$ such that for any diffusion pair $(G^{\prime},\Delta)$ its
spectral polynomial $P^{\prime}(X,Y)$ satisfies:

$P(X,Y)=P^{\prime}(X,Y)\quad\Leftrightarrow\quad(G,\Delta)\cong(G^{\prime},\Delta)$

In other words, the isomorphism class of $G$ is the unique family of graphs
having spectral polynomial $P(X,Y)$.

###### Proof.

We need only prove that if $P(X,Y)=P^{\prime}(X,Y)$, then
$(G,\Delta)\cong(G^{\prime},\Delta^{\prime})$.

Let

$\Delta\colon E(G)\to{\left\\{\alpha_{1},\dots,\alpha_{\left\lvert
E(G)\right\rvert}\right\\}},e\mapsto\alpha_{e}$

be a bijection. W.l.o.g. we may assume that the edges of $G$ are numbered as
$\alpha_{e}$, and that $\Delta$ is the identity map, so that we may write
$Y^{e}$ instead of $Y^{\alpha_{e}}$ in our polynomials. We further make the
following assumption:

We assume that no edge label equals the finite sum of (5) any other edge
labels (they are all positive integers).

Let $I\subset E(G)$. Then we define

$\displaystyle\mathcal{F}^{i}_{I}(G)$
$\displaystyle={\left\\{F\in\mathcal{F}^{i}(G)\colon E(F)=I\right\\}}$
$\displaystyle\mathcal{F}^{i}(I)$
$\displaystyle={\left\\{F\in\mathcal{F}^{i}(G)\colon E(F)\subseteq
I\right\\}}$

Then $\mathcal{F}^{i}(G)$ is the disjoint union of all the sets
$\mathcal{F}^{i}_{I}(G)$ for $I\subset E(G)$. Also, $\mathcal{F}^{i}_{I}(G)$
is either empty or consists of precisely one forest.

Let $E(G)=:E$, and let $H$ be a spanning subgraph of $G$. Then

$a_{H,i}(Y)=(-1)^{n-i}\sum\limits_{F\in\mathcal{F}^{i}(H)}\prod\limits_{e\in
E(F)}Y^{e}$

and for any spanning tree $T$ of $G$, we have

$\displaystyle a_{i}(Y)=a_{T,i}(Y)+a_{G\setminus T,i}(Y)$ (6)

where $G\setminus T$ is obtained from $G$ by removing the edges of $T$.

Now, given a spectral polynomial $P(X,Y)$, the polynomial

$a_{1}(Y)=\sum\limits_{k=1}^{M}a_{1k}Y^{k}$

with $M>>0$ recovers the set of spanning trees as follows: a monomial
$a_{1k}Y^{k}$ means that there are $\left\lvert
a_{1k}\right\rvert\in\mathds{N}$ spanning trees whose total sum of edge labels
equals $k$. Because of our assumption (5), we have that

$\left\lvert a_{1k}\right\rvert\leq 1$

for all $k=1,2,\dots$. Hence, each non-zero coefficient of $a_{1}(Y)$ encodes
precisely one spanning tree of $G$. Also, the two parts of the decomposition
(6) have no monomials in common.

In order to recover the diffusion constants, we look at

$a_{n-1}(Y)$

Again, assumption (5) ensures that all coefficients are either $1$ or zero.
So, the exponents of the non-zero monomials retrieve all the distinct labelled
edges of $G$.

We can go now further to extract for every spanning tree with a given set of
edge labels, the set of all spanning forests, thereby knowing their numbers of
connected components. Beginning with all pairs of distinct edges, we can also
extract the sets of edges in each connected component of any spanning forest.
The assumption (5) makes this possible. In this way, a unique spanning tree is
constructed. Doing this for all spanning trees, we obtain a unique set of
spanning forests $\mathcal{F}(G)$ for some graph $G$. By Lemma 4.6, the
isomorphism class of $G$ is now uniquely determined. ∎

######  Remark 4.8.

We remark that the spectral polynomial for any diffusion pair satisfying (5)
has coefficients $0,\pm 1$, as has been seen in the proof of the
Reconstruction Theorem (Theorem 4.7).

######  Corollary 4.9.

Playing Game 1 with choosing diffusion parameters which satisfy (5), allows
you to reconstruct the unknown graph $G$.

###### Proof.

This is an immediate consequence of Theorem 4.7, because Game 1 has a winning
strategy for any choice of diffusion parameters according to Theorem 3.6. ∎

######  Remark 4.10.

Notice that we are not solving the graph isomorphism problem in polynomial
time, as the computation of the coefficients of the spectral polynomial can be
expected to be far too time-consuming.

##  5 Hearing Shapes of $p$-adic geometric objects

Corollary 4.9 can also be applied to objects of $p$-adic geometry which have
an underlying graph sructure. The first kind of objects consists of Mumford
curves which have reduction graphs whose first Betti number equals the genus
of the curve. The second kind are analytic tori which after a base change look
like products of Mumford curves of genus $1$, so-called _Tate curves_. In
order to be able to do this, we will construct embeddings of the sets of
$K$-rational points of these objects into $K$. The details of this procedure
are presented in the following two subsections.

###  5.1 Hearing the Shape of a Mumford Curve

Mumford curves are explained in some detail in [6, Ch. 5]. However, we will
not need to much of their construction. All we need is that they are
projective algebraic curves admitting a finite cover by holed disks. Certain
types of coverings by holed disks in $K$, called _verticial coverings_ are
introduced in [3]. These produce certain types of reduction graphs, also
called verticial.

Let $X$ be a Mumford curve. In [3], the concept of verticial covering of the
set $X(K)$ of its $K$-rational points was developped. Let such a covering be
given, and let $G$ be the corresponding verticial reduction graph. It is a
connected graph whose vertices all have degree at least $2$, and its first
Betti number equals the genus $g$ of the curve. We require Mumford curves to
have positive genus.

The verticial covering of $X(K)$ consists of holed disks which are in one-to-
one correspondence with the vertices of $G$. The edges of $G$ correspond to
annuli (as rigid analytic spaces) with minimal positive thickness, i.e. they
do not contain $K$-rational points. These annuli connect two otherwise
disjoint holed disks in $X(K)$ without introducing extra $K$-rational points.

As a $p$-adic manifold, $X(K)$ is simply a disjoint union of finitely many
holed disks. This compact manifold can be embedded into $K$ as a closed-open
subset in such a way that each patch of the embedded verticial covering of
$X(K)$ contains a distinct point representing a class in $K/O_{K}$. Now, we
are in the setting of Section 3.1, and have a $p$-adic Laplacian operator
acting on the space $L^{2}(K)^{\left\lvert G\right\rvert}$.

######  Corollary 5.1.

Playing Game 1 with diffusion parameters satisfying (5) allows to reconstruct
a verticial reduction graph of a Mumford curve.

###### Proof.

This is immediate from the construction above and Corrolary 4.9. ∎

###  5.2 Hearing the Shape of a $p$-Adic Analytic Torus

The theory of $p$-adic analytic tori is outlined in [6, Ch. 6]. We will
collect the data in what follows, and then proceed with the application.

Let $A$ be an analytic torus

$A=\mathds{G}_{m}/\Lambda$

with multiplicative lattice $\Lambda$ generated by a basis

$\tilde{q}_{i}=q_{1}^{\alpha_{i1}}\cdots q_{g}^{\alpha_{ig}}$

for $i=1\dots,g$ with

$q_{i}=qe_{i}$

where $q=p^{f}$, $e_{i}$ is the unit vector of the $i$-th component in the
product space $K^{\times}\times\cdots\times K^{\times}$, and
$\alpha_{ij}\in\mathds{Z}$.

The lattice basis yields a decomposition

$K^{g}=\bigoplus\limits_{i=1}^{g}K\tilde{q}_{i}$

On each component $K\tilde{q}_{i}$, we can define an ultrametric norm as
follows: the generator $\tilde{q}_{i}$ is determined by an integer vector

$\alpha_{i}=(\alpha_{i1},\dots,\alpha_{ig})\in\mathds{Z}^{g}$

Its associated _primitive vector_ is a vector

$\lambda_{i}\in\mathds{N}^{g}$

such that

$\alpha_{i}=k\cdot\lambda_{i}$

with $k\in\mathds{Z}$, and $\left\lvert k\right\rvert$ is maximal with this
property. If we write

$\tilde{q}=q_{1}\cdots q_{g}$

and

$\tilde{q}^{\beta}=q_{1}^{\beta_{1}}\cdots q_{g}^{\beta_{g}}$

for

$\beta=(\beta_{1},\dots,\beta_{g})\in\mathds{Z}^{g}$

Then we have

$\tilde{q}_{i}=\tilde{q}^{\alpha_{i}}$

Its associated _primitive generator_ of the line $K\tilde{q}_{i}$ is defined
as

$\tilde{b}_{i}:=\tilde{q}^{\lambda_{i}}$

where $\lambda_{i}$ is the primitive vector associated with $\alpha_{i}$.

######  Definition 5.2.

The ultrametric norm associated with the line $K\tilde{q}_{i}$ is defined as

$\left\|x\right\|_{i}=\left\|\tilde{b}_{i}\right\|_{K^{g}}^{-\log_{q}\left\lvert\lambda\right\rvert_{K}}$

where $x=\lambda\tilde{b}_{i}\in K\tilde{q}_{i}$ with primitive generator
$\tilde{b}_{i}$, and $\lambda\in K$. The Haar measure $\mu_{i}$ on
$K\tilde{q}_{i}$ is normalised such that the unit ball w.r.t.
$\left\|\cdot\right\|_{i}$ has measure one.

The element $\tilde{q}_{i}$ defines a Tate curve, whowe reduction graph we
assume to be simplicial, as follows: as we have

$\alpha_{i}=n_{i}\lambda_{i}$

with natural $n_{i}>2$, it follows that

$\left\|\tilde{q}_{i}\right\|_{i}=\left\|\tilde{b}_{i}\right\|^{n_{i}}=(q^{-c_{i}})^{n_{i}}$

with integer $c_{i}>0$. Hence, the component Tate curve here is

$T_{i}=(K^{\times}\tilde{b}_{i})/\langle\tilde{q}_{i}\rangle\cong
K^{\times}/\left\langle(q^{c_{i}})^{n_{i}}\right\rangle$

However, there is a difference in the reduction graph structures on both sides
of the isomorphism: a verticial covering of the curve on the right has
$c_{i}n_{i}$ vertices, whereas anyone on the left has $n_{i}$ vertices.

We have a decomposition

$L^{2}(K^{g})=\bigotimes\limits_{i=1}^{g}L^{2}(K\tilde{q}_{i})$

On each factor, we repeat the construction of the previous subsection and
obtain an operator

$\Lambda\colon\bigotimes\limits_{i=1}^{g}L^{2}(K\tilde{q}_{i})^{\left\lvert
G_{i}\right\rvert}\to\bigotimes\limits_{i=1}^{g}L^{2}(K\tilde{q}_{i})^{\left\lvert
G_{i}\right\rvert}$

where $G_{i}$ is a verticial reduction graph of $T_{i}$. This operator
generalises the product space operator from [22] and decomposes as

$\Lambda_{G}=\Lambda_{1}+\dots+\Lambda_{g}$

where

$\Lambda_{i}=1\otimes\cdots\otimes 1\otimes\Lambda_{G_{i}}^{\Delta_{i}}\otimes
1\otimes\dots\otimes 1$

for $i=1,\dots,g$.

Instead of playing Game 1 for each component graph $G_{i}$, we recover the
product graph for the torus $A$:

######  Corollary 5.3.

Playing Game 1 is possible for $p$-adic analytic tori in a successful way in
order to recover the product graph composed of $G_{1},\dots,G_{g}$.

###### Proof.

Let $G$ be the product graph of $G_{1},\dots,G_{g}$. Then $\Lambda_{G}$ can be
viewed in fact as an operator

$\Lambda_{G}\colon L^{2}(K)^{\left\lvert G\right\rvert}\to
L^{2}(K)^{\left\lvert G\right\rvert}$

by taking suitable isomorphisms. Then play Game 1 using assumption (5). This
recovers $G$. ∎

## Acknowledgements

Evgeny Zelenov is thanked for posing this problem to one of the authors. David
Weisbart is thanked for valuable discussions. This research is partially
supported by the Deutsche Forschungsgemeinschaft under project number
469999674

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