# Geometric modularity for algebraic and non-algebraic weights

Hanneke Wiersema<EMAIL_ADDRESS>University of Cambridge, Department of Pure
Mathematics and Mathematical Statistics, Centre for Mathematical Sciences,
Wilberforce Road, Cambridge CB3 0WB

###### Abstract.

In this short paper we generalise a result of Diamond–Sasaki connecting
geometric modularity of algebraic weights to geometric modularity of non-
algebraic weights and vice versa. In particular, we show that geometric
modularity of non-algebraic weights implies geometric modularity of multiple
algebraic weights, which is easy to see. More significantly, we show that
geometric modularity of multiple algebraic weights sometimes implies geometric
modularity of a non-algebraic weight. This connects work of Diamond–Sasaki to
earlier work on generalisations of the weight part of Serre’s conjecture.

Serre’s modularity conjecture [Ser87], now a theorem of Khare and
Wintenberger, asserts a connection between two-dimensional mod $p$ Galois
representations of the rationals and modular forms. The strong version
includes a prediction for the minimal weight $k\geq 2$ such that a modular
form of weight $k$ corresponds to a given Galois representation, this is the
_weight part_ of Serre’s conjecture. Edixhoven refined the weight part by
including modular forms of weight one by considering geometric mod $p$ modular
forms in [Edi92].

More generally, it is conjectured that such a connection should also exist
between mod $p$ representations of $G_{F}$, where $F$ is a totally real field,
and Hilbert modular forms. In this setting, Diamond and Sasaki give an
analogue of Edixhoven’s version of the weight part of Serre’s conjecture by
considering geometric mod $p$ Hilbert modular forms in [DS17], based on work
by Andreatta and Goren [AG05]. Crucially, they attach Galois representations
to mod $p$ Hilbert eigenforms of _arbitrary_ weights, extending work by
Goldring and Koskivirta [GK19] and Emerton, Reduzzi and Xiao [ERX17] for
paritious weights.

In the classical case, the geometric approach allowed for the inclusion of
weight one modular forms. In the more general setting we obtain many more new
weights: those of partial weight one. We wish to connect the partial weight
one setting to previous work on generalisations of Serre’s conjecture where
the weights are algebraic (i.e. each integer in the tuple of weights is at
least two), most notably the work of Buzzard, Diamond and Jarvis [BDJ10], and
of Gee et al. [GK14],[GLS14],[GLS15]. In the case where $F$ is a real
quadratic field in which $p$ is inert, Diamond and Sasaki proved a connection
between modularity of algebraic weights and modularity of non-algebraic
weights [DS17, Lemma 11.3.1].

The main theorem in this paper (Theorem 3.1) relates modularity of non-
algebraic weights to modularity of algebraic weights for arbitrary totally
real $F$ in which $p$ is unramified. As we shall see, the work of generalising
the result of Diamond–Sasaki mostly lies in dealing with the more complicated
combinatorics and in finding explicit formulas for the weights.

Therefore key to this work is, given any non-algebraic weight, finding a set
of corresponding algebraic weights whose modularity will imply modularity of
the non-algebraic weight (under some technical conditions). We describe this
process in Section 2, after first introducing the modular forms involved in
Section 1. In Section 3 we give the precise statement of our main result as
well as the proof. The proof uses methods introduced in [DS17] involving
partial Hasse invariants and partial Theta operators.

The results in this paper complement the work on the parallel weight one case
by Dimitrov–Wiese [DW20] and Gee–Kassaei [GK13]. The former proved that the
Galois representation attached to a mod $p$ Hilbert modular eigenform of
parallel weight 1 and level prime to $p$ is unramified above $p$, and the
latter proved, under some mild conditions, that if a modular representation
$\rho$ is unramified at every $v\mid p$, that then there exists a form $f$ of
parallel weight 1 such that $\rho\sim\rho_{f}$.

### Notation

We let $p$ be a prime number. Fix algebraic closures
${\overline{\mathbb{Q}}_{p}}$ and $\overline{{\mathbb{Q}}}$ of
${{\mathbb{Q}}_{p}}$ and ${\mathbb{Q}}$ respectively, and fix embeddings of
$\overline{{\mathbb{Q}}}$ into ${\overline{\mathbb{Q}}_{p}}$ and
${\mathbb{C}}$. Let $F$ be a totally real field in which $p$ is unramified. We
let $\mathcal{O}_{F}$ be the ring of integers of $F$ and
$\mathcal{O}_{F,p}=\mathcal{O}_{F}\otimes{\mathbb{Z}}_{p}$. We let
$\hat{\mathcal{O}}_{F}$ denote the profinite completion of $\mathcal{O}_{F}$.
Let $\Sigma$ be the set of embeddings $F\to\overline{{\mathbb{Q}}}$.

We let $L/{\mathbb{Q}}_{p}$ be a sufficiently large finite extension of
${{\mathbb{Q}}_{p}}$ containing the image of every embedding in $\Sigma$. Let
$E$ be the residue field. We identify $\Sigma$ with the set of embeddings of
$F$ into $L$ and with the set of embeddings of $\mathcal{O}_{F}/p$ into
${\overline{\mathbb{F}}_{p}}$. We let $\operatorname{Fr}$ be the absolute
Frobenius of ${\overline{\mathbb{F}}_{p}}$. For $v\mid p$ we denote
$\Sigma_{v}$ for the set of embeddings of $\mathcal{O}_{F}/v$ into
${\overline{\mathbb{F}}_{p}}$. Note that Frobenius has a cyclic action on
$\Sigma_{v}$. For any $\tau_{0}\in\Sigma_{v}$ we let $\tau_{i}=\tau_{i+1}^{p}$
so that $\tau_{i}=\operatorname{Fr}\circ\tau_{i+1}$. We often identify
$\Sigma$ with $\prod_{v\mid p}\Sigma_{v}$.

For any $\tau\in\Sigma$, we let $e_{\tau}$ denote the basis element of
${\mathbb{Z}}^{\Sigma}$ associated to $\tau$.

### Acknowledgements

I would like to thank Fred Diamond for suggesting the problem and for helpful
discussions. I would like to thank both Fred Diamond and James Newton for
comments on earlier versions of this paper. As the reader will notice, the
methods of this paper owe a debt to the work of Diamond–Sasaki and it is a
pleasure to thank them both for conversations about their work.

##  1\. Background

###  1.1. Mod $p$ Hilbert modular forms

In this paper mod $p$ Hilbert modular forms will be sections of automorphic
line bundles on Hilbert modular varieties of level prime to $p$ in
characteristic $p$ as defined in [DS17, Def 3.2.2]. We let $M_{k,l}(U;E)$ be
the space consisting of all mod $p$ Hilbert modular forms $f$ of weight
$(k,l)\in{\mathbb{Z}}^{\Sigma}\times{\mathbb{Z}}^{\Sigma}$ and of level $U$,
where $U$ is any open compact subgroup of $\mathop{\rm
GL}\nolimits_{2}(\hat{\mathcal{O}}_{F})$ containing $\mathop{\rm
GL}\nolimits_{2}(\mathcal{O}_{F,p})$, and with coefficients in $E$. Recall we
say a weight $(k,l)$ is _algebraic_ if $k\in{\mathbb{Z}}_{\geq 2}^{\Sigma}$.

We will need that the minimal weight of a mod $p$ Hilbert modular form as in
[DS17, Section 5.2] lies in a minimal cone $\Xi_{\min}$ [DK17, Corollary 1.2]:

$\Xi_{\min}=\\{k\in{\mathbb{Z}}^{\Sigma}\,|\,pk_{\tau}\geq
k_{\operatorname{Fr}^{-1}\circ\tau}\text{ for all }\tau\in\Sigma\\}.$

If $\mathfrak{n}$ is a prime ideal of $\mathcal{O}_{F}$ we define
$U_{1}(\mathfrak{n})$ as the open compact subgroup of $\mathop{\rm
GL}\nolimits_{2}(\hat{\mathcal{O}}_{F})$ given by

$U_{1}(\mathfrak{n})=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathop{\rm
GL}\nolimits_{2}(\hat{\mathcal{O}}_{F})\,|\,c,d-1\in\mathfrak{n}\hat{\mathcal{O}}_{F}\right\\}.$

In the following, we write $S_{v}$ and $T_{v}$ for the Hecke operators defined
in [DS17, Section 4.3] (for $v\nmid p$ and arbitrary weight $(k,l)$) and
[DS17, Section 9.7] (for $v\mid p$ and for algebraic weight $(k,0)$).

###  1.2. Partial Hasse invariants and partial $\Theta$ operators

For any $\tau\in\Sigma$, we let $\operatorname{Ha}_{\tau}$ denote the partial
Hasse invariant defined in [DS17, Section 5.1]. We write
$k_{\operatorname{Ha}_{\tau}}=pe_{\operatorname{Fr}^{-1}\circ\tau}-e_{\tau}$
for the weight of $\operatorname{Ha}_{\tau}$. Multiplication by
$\operatorname{Ha}_{\tau}$ gives a new mod $p$ Hilbert modular form
$\operatorname{Ha}_{\tau}f$ if $f$ has level $U$ where $U$ is a subgroup of
$\mathop{\rm GL}\nolimits_{2}(\mathbb{A}_{F}^{\infty})$ containing
$\mathop{\rm GL}\nolimits_{2}(\mathcal{O}_{F,p})$; this is Proposition 5.1.1
of [DS17]. In particular, we have an injective map

$M_{k,l}(U;E)\to M_{k+k_{\operatorname{Ha}_{\tau}},l}(U;E),$

commuting with $T_{v}$ and $S_{v}$ for all $v\nmid p$ such that $\mathop{\rm
GL}\nolimits_{2}(\mathcal{O}_{F,v})\subset U$.

For any $\tau\in\Sigma$, we let $\Theta_{\tau}$ denote the partial $\Theta$
operator defined in [DS17, Definition 8.2.1]. By Theorem 10.4.2 of [DS17], for
any $f\in M_{k,l}(U;E)$ and $\tau\in\Sigma$ we obtain a new Hilbert modular
form $\Theta_{\tau}(f)\in M_{\hat{k},\hat{l}}(U;E)$ where

  * •

if $\operatorname{Fr}\circ\tau=\tau$, then $\hat{k}_{\tau}=k_{\tau}+p+1$ and
$\hat{k}_{\tau^{\prime}}=k_{\tau^{\prime}}$ if $\tau^{\prime}\neq\tau$;

  * •

if $\operatorname{Fr}\circ\tau\neq\tau$, then
$\hat{k}_{\tau}=k_{\tau}+1,\hat{k}_{\operatorname{Fr}^{-1}\circ\tau}=k_{\operatorname{Fr}^{-1}\circ\tau}+p$
and $\hat{k}_{\tau^{\prime}}=k_{\tau^{\prime}}$ if
$\tau\not\in\\{\operatorname{Fr}^{-1}\circ\tau,\tau\\}$;

  * •

$\hat{l}_{\tau}=l_{\tau}-1$, and $\hat{l}_{\tau^{\prime}}=l_{\tau^{\prime}}$
if $\tau^{\prime}\neq\tau$.

###  1.3. Properties of Hilbert modular eigenforms

In this section we assume $(k,l)$ is algebraic. We introduce some properties
of eigenforms, details of each can be found in [DS17, Section 10].

####  1.3.1. Twisted forms

Let $\mathfrak{m}$ be an ideal of $\mathcal{O}_{F}$ such that
$\mathfrak{m}\mid\mathfrak{n}$. We let
$V_{\mathfrak{m}}\subset\hat{\mathcal{O}}^{\times}_{F}$ denote the kernel of
the natural projection to $(\mathcal{O}_{F}/\mathfrak{m})^{\times}$. We say a
character

$\xi:\\{a\in(\mathbb{A}_{F}^{\infty})^{\times}\mid
a_{p}\in\mathcal{O}_{F,p}^{\times}/V_{\mathfrak{m}}\\}\to E^{\times},$

is a character of weight $l^{\prime}\in{\mathbb{Z}}^{\Sigma}$ if
$\xi(\alpha)=\overline{\alpha}^{l^{\prime}}$ for all $\alpha\in
F_{+}^{\times}\cap\mathcal{O}_{F,p}^{\times}$.

For any $f\in M_{k,l}(U_{1}(\mathfrak{n});E)$ and any character of weight
$l^{\prime}$ and conductor $\mathfrak{m}$, we write $f_{\xi}\in
M_{k,l+l^{\prime}}(U_{1}(\mathfrak{n}\mathfrak{m}^{2});E)$ for the twisted
form defined in [DS17, Section 10.3].

####  1.3.2. Normalised eigenforms

A _normalised eigenform_ is a Hilbert modular form $f$ which is an eigenform
for Hecke operators $T_{v}$ for all $v\nmid p$ and $S_{v}$ for all
$v\nmid\mathfrak{n}p$, and where all $f_{\xi}$, twists of $f$ by characters
$\xi$ of weight $-l$ and conductor $\mathfrak{m}$ coprime to $p$, with values
in extensions $E^{\prime}$ of $E$, are eigenforms for $T_{v}$ for all $v\mid
p$. Finally, the coefficient of $q$ in the $q$-expansion should be $1$. If $f$
is a normalised eigenform in $M_{k,l}(U_{1}(\mathfrak{n}),E)$ and $\xi$ is a
character of weight $-l^{\prime}$ and conductor $\mathfrak{m}$ coprime to $p$,
then $f_{\xi}$ is a normalised eigenform in
$M_{k,l+l^{\prime}}(U_{1}(\mathfrak{n}\mathfrak{m}^{2});E)$.

####  1.3.3. Stabilised eigenform

A normalised eigenform $f\in M_{k,l}(U_{1}(\mathfrak{n});E)$ is _stabilised_
if $T_{v}f=0$ for all $v\mid\mathfrak{n}$. A stabilised eigenform is _strongly
stabilised_ if and only if $T_{v}f_{\xi}=0$ for all $v\mid p$ and characters
$\xi$ of weight $-l$.

Let us note that this property is crucial to the proof of the main result. We
need that there is at most one strongly stabilised eigenform of a given level
and weight giving rise to a given Galois representation, as we will see.

If $f$ is a normalised (resp. stabilised, strongly stabilised) eigenform, then
the same is true for both $\operatorname{Ha}_{\tau}f$ and $\Theta_{\tau}(f)$
for any $\tau$ (assuming $k_{\tau}\geq 3$ if
$\tau\neq\operatorname{Fr}\circ\tau$ in the case of
$\operatorname{Ha}_{\tau}f$). This is [DS17, Remark 10.6.6].

###  1.4. Geometric modularity

Let $U$ be an open compact subgroup $U\subset\mathop{\rm
GL}\nolimits_{2}(\hat{\mathcal{O}}_{F})$ containing $\mathop{\rm
GL}\nolimits_{2}(\mathcal{O}_{F,p})$. Let $Q$ be a finite set of primes
containing all $v\mid p$ and all $v$ such that $\mathop{\rm
GL}\nolimits_{2}(\mathcal{O}_{F,v})\not\subset U$. Suppose $f\in M_{k,l}(U;E)$
is an eigenform for $T_{v}$ and $S_{v}$ for all $v\not\in Q$, then we write
$\rho_{f}$ for the Galois representation attached to $f$ as in [DS17, Theorem
6.1.1].

######  Definition 1.1.

An irreducible, continuous and totally odd Galois representation
$\rho:G_{F}\to\mathop{\rm GL}\nolimits_{2}({\overline{\mathbb{F}}_{p}})$ is
_geometrically modular of weight $(k,l)$_ if $\rho$ is equivalent to the
extension of scalars of $\rho_{f}$ for some eigenform $f\in M_{k,l}(U;E)$ as
above.

This means $\rho$ is geometrically modular of weight $(k,l)$ if there is a
non-zero element $f\in M_{k,l}(U;E)$ for some $U\supset\mathop{\rm
GL}\nolimits_{2}(\mathcal{O}_{F,p})$ and $E\subset{\overline{\mathbb{F}}_{p}}$
such that

$T_{v}f=\mathop{\rm tr}\nolimits(\rho(\operatorname{Fr}_{v}))f\quad\text{ and
}\text{Nm}_{F/{\mathbb{Q}}}(v)S_{v}f=\operatorname{det}(\rho(\operatorname{Fr}_{v}))f,$

for all but finitely many primes $v$.

In the following theorem we collect some of the results we need for our proof,
which are results from [DS17].

######  Theorem 1.2.

Suppose $\rho:G_{F}\to\mathop{\rm
GL}\nolimits_{2}({\overline{\mathbb{F}}_{p}})$ is irreducible and
geometrically modular of weight $(k,l)$.

  1. (1)

Then $\rho$ arises from an eigenform $f$ of weight $(k,l)$ and level
$U_{1}(\mathfrak{n})$ for some $\mathfrak{n}$ prime to $p$.

  2. (2)

Moreover, we can take $f$ such that $\Theta_{\tau}(f)\neq 0$ for all
$\tau\in\Sigma$.

  3. (3)

If $k_{\tau}\geq 2$ for all $\tau$, then $f$ can moreover be taken to be a
normalised eigenform of level $U_{1}(\mathfrak{n})$ for some $\mathfrak{n}$
prime to $p$.

  4. (4)

Moreover, $\rho$ is also geometrically modular of weight
$(k+k_{\operatorname{Ha}_{\tau}},l)$ for any $\tau\in\Sigma$.

  5. (5)

Moreover, $\rho$ is also geometrically modular of weight $(\hat{k},\hat{l})$
as in Section 1.2 for any $\tau\in\Sigma$.

###### Proof.

  1. (1)

This is [DS17, Lemma 10.2.1].

  2. (2)

This is [DS17, Lemma 10.4.1].

  3. (3)

This is [DS17, Proposition 10.5.2].

  4. (4)

This follows from the commutativity of $\operatorname{Ha}_{\tau}$ with the
Hecke operators.

  5. (5)

This is [DS17, Theorem 10.4.2], which follows from $(2)$ and commutativity of
$\Theta_{\tau}$ with the Hecke operators. ∎

##  2\. Weight combinatorics

In this section we prepare to state the main result connecting modularity of
algebraic weights to non-algebraic weights and vice versa. We introduce the
Hilbert modular forms with algebraic weights we associate to Hilbert modular
forms of some non-algebraic weight.

###  2.1. The algebraic weights

Suppose $f$ is a non-zero mod $p$ Hilbert modular form with non-algebraic
weight $(k,0)$ such that $k_{\tau}>0$ for all $\tau\in\Sigma$. We associate
multiple Hilbert modular forms to it, all with algebraic weights. The first
form is obtained from multiplication by Hasse invariants, and the other ones
from multiplication by Hasse invariants and then acting on it with a partial
Theta operator.

######  Definition 2.1.

We let $M$ be the subset of $\Sigma$ consisting of $\tau\in\Sigma$ such that

$k_{\operatorname{Fr}^{-1}\circ\tau}=\dots=k_{\operatorname{Fr}^{(1-s)}\circ\tau}=2\text{
and }k_{\operatorname{Fr}^{-s}\circ\tau}=1,$

for some $s\geq 1$ (the first condition is vacuous if $s=1$).

#### The definition of $k^{\prime}$

We let $f^{\prime}$ be the modular form

(2.1) $\displaystyle f\cdot\prod_{\tau\in M}\operatorname{Ha}_{\tau}.$

We denote $(k^{\prime},l^{\prime})$ for the weight of $f^{\prime}$. We have
$l^{\prime}=l=(0,\dots,0)$. As the weight of $\operatorname{Ha}_{\tau}$ is
$pe_{\operatorname{Fr}^{-1}\circ\tau}-e_{\tau}$ the weight of $f^{\prime}$ can
easily be seen to be algebraic.

######  Example 2.2.

Say $k=(1,1,k_{2})$. If $k_{2}\geq 3$ we multiply $f$ by
$\operatorname{Ha}_{2}$ and $\operatorname{Ha}_{0}$ to obtain $f^{\prime}$ of
weight $(k^{\prime},0)$ with $k^{\prime}=(p,p+1,k_{2}-1)$. If $k_{2}=2$, then
we multiply $f$ by $\operatorname{Ha}_{2}$, $\operatorname{Ha}_{0}$ and
$\operatorname{Ha}_{1}$ to obtain $f^{\prime}$ of weight $(k^{\prime},0)$ with
$k^{\prime}=(p,p,p+1)$.

#### The definition of $k^{\mu}$

We define several modular forms $f^{\mu}$ obtained from $f$ by using partial
Hasse invariants and partial Theta operators.

######  Definition 2.3.

Let $\tilde{M}$ be the set of $\tau\in\Sigma$ such that one of the following
conditions holds:

  * •

$k_{\tau}\geq 3$ and $\tau\in M$,

  * •

$k_{\tau}=2,k_{\operatorname{Fr}^{-1}\circ\tau}=1,k_{\operatorname{Fr}^{-2}\circ\tau}=1$
or $2$ and if $k_{\operatorname{Fr}^{-2}\circ\tau}=2$ then
$\operatorname{Fr}^{-2}\circ\tau\in M$.

Note that if there is no case where $k_{\tau}=2$ and
$k_{\operatorname{Fr}^{-1}\circ\tau}=1$, that $\tilde{M}$ is just the set
$\tau\in\Sigma$ such that $k_{\tau}\neq
1,k_{\operatorname{Fr}^{-1}\circ\tau}=1$.

Let $\mu\in\tilde{M}$, then we define $f^{\mu}$ to be the form given by

(2.2)
$\displaystyle\Theta_{\mu}\left(f\cdot\prod_{\tau\in{M}\setminus\\{\mu\\}}\operatorname{Ha}_{\tau}\right).$

We denote $(k^{\mu},l^{\mu})$ for the weight of $f^{\mu}$. We have
$l^{\mu}=-e^{\mu}$. The Theta operator adds
$pe_{\operatorname{Fr}^{-1}\circ\tau}+e_{\tau}$ to the weight of the form it
acts on. It can easily be verified that the weight of $f^{\mu}$ is algebraic.

######  Example 2.4.

Say $k=(1,1,k_{2})$ with $k_{2}\geq 3$, then we multiply $f$ by
$\operatorname{Ha}_{0}$ and act on this with $\Theta_{2}$ to obtain $f^{\mu}$
of weight $(k^{\mu},l^{\mu})=((p,p+1,k_{2}+1),(0,0,-1))$. If $k_{2}=2$, then
we multiply $f$ by $\operatorname{Ha}_{0}$ and $\operatorname{Ha}_{1}$, act on
this with $\Theta_{2}$ to obtain $f^{\mu}$ of weight
$(k^{\mu},l^{\mu})=((p,p,p+3),(0,0,-1))$.

##  3\. Statement and proof of the theorem

######  Theorem 3.1.

Let $G_{F}\to\mathop{\rm GL}\nolimits_{2}({\overline{\mathbb{F}}_{p}})$ be an
irreducible representation. Suppose $\rho$ is geometrically modular of a non-
zero non-algebraic weight $(k,0)$ such that $k\in\Xi_{\min}$ and $k_{\tau}>0$
for all $\tau\in\Sigma$. Then

  1. (1)

$\rho$ is geometrically modular of weight $(k^{\prime},0)$,

  2. (2)

for each $\mu\in\tilde{M},$ $\rho$ is geometrically modular of weight
$(k^{\mu},l^{\mu})$.

Moroever, the converse holds if we assume in addition that

  1. (3)

for $\mu\in\tilde{M}$ let $f^{\mu}$ be the form as defined above. For each
$\mu\in\tilde{M}$ we have $T_{v}((f^{\mu})_{\xi})=0$ for all $v\mid p$ and
characters $\xi$ of weight $-l^{\mu}$,

  2. (4)

there is no place $v\mid p$ such that $k_{\tau}=1$ for all
$\tau\in\Sigma_{v}$,

  3. (5)

$k_{\mu}\not\equiv 1\mod p$ for all $\mu\in\tilde{M}.$

We note first that the theorem holds just as well for $(k,l)$ (with obvious
adjustments to Section 2.1), we take $l=0$ to ease exposition.

The condition (3) is implied by demanding that for all primes $v|p$ the local
representation $\rho|_{G_{F_{v}}}$ is not of the form
$\begin{pmatrix}\chi_{1}&\ast\\\ 0&\chi_{2}\end{pmatrix}$ where
$\chi_{1}|_{I_{F_{v}}}=\omega_{\mu}$ for any $\mu\in\tilde{M}$ and applying
[DS17, Corollary 10.7.2]. The reason for this condition is to ensure we obtain
strongly stabilised forms which we will need to show divisibility by certain
Hasse invariants. The key to the proof is that there is at most one strongly
stabilised eigenform $f\in M_{k,l}(U_{1}(\mathfrak{n};E))$ giving rise to
$\rho$.

We exclude the weights in (4) as these would require methods similar to the
aforementioned [DW20] and [GK13]. The weights in (5) are excluded as our
methods break down for that case.

###### Proof.

First assume $\rho$ is geometrically modular of $f\in M_{(k,0)}(U;E)$. By
Theorem 1.2(4) it is also geometrically modular of weight $k^{\prime}$ by
multiplying $f$ by the Hasse invariants as in (2.1). Now for each $f^{\mu}$ we
proceed similarly, we multiply by Hasse invariants and act on the result by
partial Theta operators as in (2.2) in which case it follows from Theorem
1.2(4-5).

For the converse, suppose we have (1)-(5). By (1) and Theorem 1.2(3) $\rho$
arises from a normalised eigenform in
$M_{{k^{\prime}},{0}}(U_{1}(\mathfrak{m});E)$ for an ideal $\mathfrak{m}$
prime to $p$ (and sufficiently large $E$).

From (2), and for any $\mu\in\tilde{M}$, Theorem 1.2(3) again implies that
$\rho$ arises from a normalised eigenform in
$M_{{k}^{\mu},{l}^{\mu}}(U_{1}(\mathfrak{m}_{\mu});E)$ for some ideal
$\mathfrak{m}_{\mu}$ prime to $p$.

Let
$\mathfrak{n}=\mathfrak{m}^{2}\prod_{\mu\in\tilde{M}}\mathfrak{m}_{\mu}^{2}$.
Then $\mathfrak{n}\subset\mathfrak{m}$ satisfies the conditions in [DS17,
Lemma 10.6.2], so this means $\rho$ arises from a stabilised eigenform
$f^{\prime}\in M_{k^{\prime},0}(U_{1}(\mathfrak{n});E)$. Similarly, for each
$\mu\in\tilde{M}$, the ideal $\mathfrak{n}\subset\mathfrak{m}_{\mu}$ satisfies
the conditions in [DS17, Lemma 10.6.2] with $\mathfrak{m}=\mathfrak{m}_{\mu}$,
so this means $\rho$ arises from a stabilised eigenform $f^{\mu}\in
M_{{k}^{\mu},{l}^{\mu}}(U_{1}(\mathfrak{n});E)$.

Fix $\mu\in\tilde{M}$. By [DS17, Proposition 9.4.1] $\Theta_{\mu}(f^{\prime})$
will be a strongly stabilised eigenform of weight
$(k^{\prime}+pe_{\mu}+e_{\operatorname{Fr}^{-1}\circ\mu},-e_{\mu})$ and level
$\mathfrak{n}$.

By (3) $\operatorname{Ha}_{\mu}(f^{\mu})$ is a strongly stabilised eigenform
of weight $(k^{\mu}+pe_{\mu}-e_{\operatorname{Fr}^{-1}\circ\mu},-e_{\mu})$ and
level $\mathfrak{n}$.

Now by [DS17, Lemma 10.6.5], the modular form that $\rho$ arises from with
these specifics is unique so that

$\Theta_{\mu}(f^{\prime})=\operatorname{Ha}_{\mu}(f^{\mu}).$

By Theorem 8.2.2 of [DS17] and (5) we know that $f^{\prime}$ is divisible by
$\operatorname{Ha}_{\mu}$.

We can do this step for every $\mu\in\tilde{M}$ to find that $\rho$ is
geometrically modular of a form $g$ such that

(3.1) $\displaystyle
f^{\prime}=g\prod_{\mu\in\tilde{M}}\operatorname{Ha}_{\mu}.$

The form $g$ is of level $\mathfrak{n}$ and of weight
$k^{\prime}-\sum_{\mu\in\tilde{M}}k_{\operatorname{Ha}_{\mu}}$. If this equals
$k$, the proof is done.

If not, this means that the set $M^{\prime}:=M\setminus\tilde{M}$ is non-
empty. We finish the proof by showing $g$ is divisible by
$\prod_{\tau\in{M^{\prime}}}\operatorname{Ha}_{\tau}$. Write
$k^{\prime\prime}$ for the weight of $g$. Suppose $\tau\in M^{\prime}$, then
$\tau$ falls into one of the following three cases:

  1. i

$k_{\tau}=1,k_{\operatorname{Fr}^{-1}\circ\tau}=1$,

  2. ii

$k_{\tau}=2,k_{\operatorname{Fr}^{-1}\circ\tau}=1,k_{\operatorname{Fr}^{-2}\circ\tau}\neq
1$, and if $k_{\operatorname{Fr}^{-2}\circ\tau}=2$ then
$\operatorname{Fr}^{-2}\circ\tau\not\in M$,

  3. iii

$k_{\tau}=2,k_{\operatorname{Fr}^{-1}\circ\tau}=2$ with $\tau\in M$.

In the first case, for some $s\geq 1$ there must be an embedding
$\operatorname{Fr}^{s}\circ\tau\in\tilde{M}$ such that
$k_{\operatorname{Fr}^{s-1}\circ\tau}=\dots=k_{\tau}=1$. We find
$(k^{\prime\prime}_{\operatorname{Fr}^{s-1}\circ\tau},\dots,k^{\prime\prime}_{\tau},k^{\prime\prime}_{\operatorname{Fr}^{-1}\circ\tau})=(0,p,p,\dots,p)$
or $(0,p,\dots,p,p+1)$. Theorem 1.1 of [DK17], which implies divisibility by
$\operatorname{Ha}_{\tau}$ if $pk_{\tau}<k_{\operatorname{Fr}^{-1}\circ\tau}$,
gives us divisibility of
$\operatorname{Ha}_{\operatorname{Fr}^{s-1}\circ\tau}$. The result now follows
from Lemma 3.2.

In the second case, there exist $s,t\geq 0$ such that
$\operatorname{Fr}^{s+t}\circ\tau\in\tilde{M}$ and

(3.2) $\displaystyle
k_{\operatorname{Fr}^{s+t-1}\circ\tau}=\dots=k_{\operatorname{Fr}^{s}\circ\tau}=1,k_{\operatorname{Fr}^{s-1}\circ\tau}=\dots=k_{\tau}=2,$

If $t=0$, we have
$(k^{\prime\prime}_{\operatorname{Fr}^{s+t-1}\circ\tau},\dots,k^{\prime\prime}_{\tau},k^{\prime\prime}_{\operatorname{Fr}^{-1}\circ\tau})=(1,p+1,p+1,\dots,p+1)$.
By [DK17, Theorem 1.1] and Lemma 3.2 we obtain divisibility by
$\operatorname{Ha}_{\operatorname{Fr}^{s+t-1}\circ\tau}\cdots\operatorname{Ha}_{\tau}$.
If $t\geq 1$, then we obtain $(0,p,\dots,p,p+1,\dots,p+1)$. We get
divisibility by
$\operatorname{Ha}_{\operatorname{Fr}^{s+t-1}\circ\tau}\cdots\operatorname{Ha}_{\operatorname{Fr}^{s}\circ\tau}$
by [DK17, Theorem 1.1] and Lemma 3.2. We obtain divisibility by
$\operatorname{Ha}_{\operatorname{Fr}^{s-1}\circ\tau}\cdots\operatorname{Ha}_{\tau}$
from again applying Lemma 3.2. The third case is covered in the second case
(note here $\tau\in M$ implies $\operatorname{Fr}^{-1}\circ\tau\in M$). ∎

######  Lemma 3.2.

Suppose $f$ is of weight $(k,l)$ with $k\in{\mathbb{Z}}^{\Sigma}_{\geq 0}$.
Suppose we know $f$ is divisible by $\operatorname{Ha}_{\tau}$ and suppose for
some $s\geq 1$ we have

$k_{\operatorname{Fr}^{-1}\circ\tau}=\dots=k_{\operatorname{Fr}^{-s}\circ\tau}=m,k_{\operatorname{Fr}^{-s-1}}=m+1,$

for some positive integer $m\leq p+1$. Then $f$ is divisible by
$\operatorname{Ha}_{\tau}\cdots\operatorname{Ha}_{\operatorname{Fr}^{-s}\circ\tau}$.

###### Proof.

We use Theorem 1.1 of [DK17], which implies divisibility by
$\operatorname{Ha}_{\tau}$ if $pk_{\tau}<k_{\operatorname{Fr}^{-1}\circ\tau}$.
After dividing by $\operatorname{Ha}_{\tau}$ the entry corresponding to
$\operatorname{Fr}^{-1}\circ\tau$ is $m-p$. We have $p(m-p)<m$ since $m\leq
p+1$. It is easy to see that
$p(k_{\operatorname{Fr}^{-(j-1)}\circ\tau}-p)<k_{{\operatorname{Fr}^{-j}\circ\tau}}$
for all $j\in\\{2,\dots,s+1\\}$. This concludes the proof. ∎

######  Example 3.3.

Assume $p$ is inert in $F$ and suppose $k=(1,1,k_{2},2,2,1,2,2)$ with
$k_{2}\geq 3$. We have

$\displaystyle k^{\prime}=$ $\displaystyle(p,p+1,k_{2}-1,p+1,p+1,p,p+1,p+1),$
$\displaystyle k^{\mu_{1}}=$ $\displaystyle(p,p+1,k_{2}+1,p+1,p+1,p,p+1,p+1),$
$\displaystyle k^{\mu_{2}}=$ $\displaystyle(p,p+1,k_{2}-1,p+1,p+3,p,p+1,p+1),$
$\displaystyle k^{\mu_{3}}=$ $\displaystyle(p,p+1,k_{2}-1,p+1,p+1,p,p+1,p+3).$

By the above matching procedure we find
$k^{\prime\prime}=(0,p+1,k_{2},1,p+2,0,p+1,p+2)$, the result follows from
Theorem 1.1 of [DK17] and Lemma 3.2.

## References

  * [AG05] F. Andreatta and E. Z. Goren.  Hilbert modular forms: mod $p$ and $p$-adic aspects.  Mem. Amer. Math. Soc., 173(819):vi+100, 2005.
  * [BDJ10] Kevin Buzzard, Fred Diamond, and Frazer Jarvis.  On Serre’s conjecture for mod $\ell$ Galois representations over totally real fields.  Duke Math. J., 155(1):105–161, 10 2010.
  * [DK17] Fred Diamond and Payman L. Kassaei.  Minimal weights of Hilbert modular forms in characteristic $p$.  Compos. Math., 153(9):1769–1778, 2017.
  * [DS17] Fred Diamond and Shu Sasaki.  A Serre weight conjecture for geometric Hilbert modular forms in characteristic p.  arXiv e-prints, page arXiv:1712.03775, Dec 2017.
  * [DW20] Mladen Dimitrov and Gabor Wiese.  Unramifiedness of Galois representations attached to Hilbert modular forms ${\rm mod}\,p$ of weight $1$.  J. Inst. Math. Jussieu, 19(2):281–306, 2020.
  * [Edi92] Bas Edixhoven.  The weight in Serre’s conjectures on modular forms.  Inventiones mathematicae, 109(3):563–594, 1992.
  * [ERX17] Matthew Emerton, Davide A. Reduzzi, and Liang Xiao.  Galois representations and torsion in the coherent cohomology of Hilbert modular varieties.  J. Reine Angew. Math., 726:93–127, 2017.
  * [GK13] Toby Gee and Payman Kassaei.  Companion forms in parallel weight one.  Compos. Math., 149(6):903–913, 2013.
  * [GK14] Toby Gee and Mark Kisin.  The Breuil-Mézard conjecture for potentially Barsotti-Tate representations.  Forum Math. Pi, 2:e1, 56, 2014.
  * [GK19] Wushi Goldring and Jean-Stefan Koskivirta.  Strata Hasse invariants, Hecke algebras and Galois representations.  Invent. Math., 217(3):887–984, 2019.
  * [GLS14] Toby Gee, Tong Liu, and David Savitt.  The Buzzard-Diamond-Jarvis conjecture for unitary groups.  J. Amer. Math. Soc., 27(2):389–435, 2014.
  * [GLS15] Toby Gee, Tong Liu, and David Savitt.  The weight part of Serre’s conjecture for $GL_{2}$.  Forum Math. Pi, 3:e2, 52, 2015.
  * [Ser87] Jean-Pierre Serre.  Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline{\bf Q}/{\bf Q})$.  Duke Math. J., 54(1):179–230, 1987.


# Embedding dimensions of simplicial complexes
on few vertices

Florian Frick  Dept. Math. Sciences, Carnegie Mellon University, Pittsburgh,
PA 15213, USA
Inst. Math., Freie Universität Berlin, Arnimallee 2, 14195 Berlin, Germany
<EMAIL_ADDRESS>,  Mirabel Hu  Dept. Math. Sciences, Carnegie Mellon University,
Pittsburgh, PA 15213, USA<EMAIL_ADDRESS>,  Nick Scheel  Dept. Math.,
Bard College, Annandale-on-Hudson, NY 12504, USA<EMAIL_ADDRESS>and  Steven
Simon  Dept. Math., Bard College, Annandale-on-Hudson, NY 12504, USA
<EMAIL_ADDRESS>
###### Abstract.

We provide a simple characterization of simplicial complexes on few vertices
that embed into the $d$-sphere. Namely, a simplicial complex on $d+3$ vertices
embeds into the $d$-sphere if and only if its non-faces do not form an
intersecting family. As immediate consequences, we recover the classical van
Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado
theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition
that such complexes satisfy the rigidity property that continuous and linear
embeddability are equivalent.

FF was supported by NSF grant DMS 1855591, NSF CAREER grant DMS 2042428, and a
Sloan Research Fellowship.

##  1\. Introduction and Statement of Results

Planar graphs are characterized as those without a $K_{5}$\- or
$K_{3,3}$-minor [18]. By a theorem of Fáry [7], a graph can be continuously
embedded into the plane if and only if there is an embedding where every edge
is a straight line segment. For higher-dimensional simplicial complexes and
embeddings into $\mathbb{R}^{d}$, the situation is much more intricate: No
efficient characterization of complexes that embed into $\mathbb{R}^{d}$
exists in general, algorithmically deciding the existence of an embedding can
be – depending on the dimensions of the complex and codomain – difficult or
even impossible (see, e.g., [11, 12]), and an analogue of Fáry’s theorem
asserting the equivalence of topological and linear embeddability fails even
in dimension three [4, 15].

Here we show that if one restricts attention to simplicial complexes on few
vertices, then both a simple characterization of complexes that embed into
$\mathbb{R}^{d}$ as well as the equivalence of topological and linear
embeddability can be salvaged. The only simplicial complex on $d+2$ vertices
that does not embed into the $d$-sphere $S^{d}$ is the $(d+1)$-dimensional
simplex $\Delta_{d+1}$. We thus focus on deciding whether simplicial complexes
on $d+3$ vertices embed into $S^{d}$. We note that a finite complex embeds
into $S^{d}$ but not into $\mathbb{R}^{d}$ if and only if it is homeomorphic
to $S^{d}$.

Fix a dimension $d$ and let $\mathcal{F}$ be a family of subsets of
$[d+3]=\\{1,2,\dots,d+3\\}$. Denote by $\Sigma(\mathcal{F})$ the simplicial
complex on $[d+3]$ with missing faces $\mathcal{F}$, that is,

$\Sigma(\mathcal{F})=\\{\sigma\subset[d+3]\ :\ \tau\notin\mathcal{F}\
\text{for all}\ \tau\subset\sigma\\}.$

Note that every simplicial complex is of this form by letting $\mathcal{F}$ be
the family of non-faces. A family $\mathcal{F}$ of subsets of $[d+3]$ is
called _intersecting_ if for all $\sigma,\tau\in\mathcal{F}$ we have that
$\sigma\cap\tau\neq\emptyset$. Our characterization of complexes on $d+3$
vertices that embed into $S^{d}$ is surprisingly simple:

######  Theorem 1.

Let $\mathcal{F}$ be a family of subsets of $[d+3]$. Then
$\Sigma(\mathcal{F})$ embeds into $S^{d}$ if and only if $\mathcal{F}$ is not
intersecting.

As with the Hanani–Tutte theorem [5, 16] for non-planar graphs, our proof of
Theorem 1 for non-embeddings shows there must exist disjoint simplices of
$\Sigma(\mathcal{F})$ whose images overlap (see Lemma 5). As an example,
suppose that $\mathcal{F}$ consists of all the $(d+2)$-subsets of $[2d+3]$, in
which case $\Sigma(\mathcal{F})$ is the $d$-skeleton $\Delta_{2d+2}^{(d)}$ of
the $(2d+2)$-simplex. The classical van Kampen–Flores theorem [8, 17] is then
recovered as an immediate corollary:

######  Theorem 2 (Van Kampen–Flores theorem).

For any continuous map $f\colon\Delta_{2d+2}^{(d)}\rightarrow\mathbb{R}^{2d}$,
there exist disjoint simplices $\sigma$ and $\tau$ of $\Delta_{2d+2}^{(d)}$
such that $f(\sigma)\cap f(\tau)\neq\emptyset$.

In a different direction, we show that Theorem 1 implies an extremal criterion
for embeddability of simplicial complexes with few vertices. Recall that the
Erdős–Ko–Rado theorem [6] states that an intersecting family of $k$-element
subsets of $[n]=\\{1,\ldots,n\\}$ has size at most $\binom{n-1}{k-1}$ whenever
$n\geq 2k$. As we will see, combining this with Theorem 1 quickly gives the
following:

######  Theorem 3.

Let $\Sigma$ be a simplicial complex on $d+3$ vertices with fewer than
$\binom{d+2}{k}$ faces of dimension ${k-1}$, where
$k\leq\lfloor\frac{d+3}{2}\rfloor$. Then $\Sigma$ embeds into $S^{d}$.

Observe that together with Theorem 1, Theorem 3 recovers the Erdős–Ko–Rado
theorem itself. Thus Theorem 3 provides a natural topological generalization
of the Erdős–Ko–Rado theorem. (The proof of the Erdős–Ko–Rado theorem is used
in the proof of Theorem 3, however.) We also note that the lower bound of
Theorem 3 is sharp, precisely because the the upper bound of the Erdős–Ko–Rado
theorem is.

Our final result shows that the analogue of Fáry’s theorem holds for
simplicial complexes on few vertices. To state this formally, we say that an
embedding of a simplicial complex $\Sigma$ into $\mathbb{R}^{d}$ is _linear_
if the image of each face of $\Sigma$ is the convex hull of the image of its
vertices, and likewise that an embedding of $\Sigma$ into $S^{d}$ is
_geodesic_ if the image of each face is geodesically convex in $S^{d}$, that
is, for any two points in the image any shortest path (in the isotropic round
metric) connecting them is also in the image. We then have the following
rigidity theorem:

######  Theorem 4.

Let $\Sigma$ be a simplicial complex on $d+3$ vertices. Then $\Sigma$ embeds
into $\mathbb{R}^{d}$ (respectively, $S^{d}$) if and only if it embeds
linearly into $\mathbb{R}^{d}$ (respectively, geodesically into $S^{d}$).

In fact, we will show (Lemma 7) that any simplicial complex on $d+3$ vertices
that embeds into $S^{d}$ is actually a subcomplex of a convex $(d+1)$-polytope
on $d+3$ vertices inscribed into $S^{d}$, that is, with all vertices on the
unit sphere. Mani [9] showed that any triangulation of $S^{d}$ on at most
$d+4$ vertices is the boundary complex of a convex polytope. However, our
construction in Lemma 7 is optimal in that there exist complexes on $d+4$
vertices that embed into $S^{d}$ but which are not contained in any simplicial
$(d+1)$-polytope on $d+4$ vertices; see Remark 10 for an example when $d=3$.
It remains open whether Theorem 4 holds for simplicial complexes on $d+4$ or
$d+5$ vertices; see Problem 9. Moreover, our result does not extend to linear
embeddings of polyhedra. For instance, Barnette [3] gives a simple example of
a polyhedral $2$-complex on six vertices that embeds into $\mathbb{R}^{3}$ but
for which no linear embedding exists.

We refer to Matoušek [10] for the basics about simplicial complexes and to
Ziegler [19] for the basics about polytopes.

##  2\. Proofs

###  2.1. Proof of Theorem 1

To prove Theorem 1, we first show that $\Sigma(\mathcal{F})$ on $d+3$ vertices
cannot embed in a $d$-sphere if $\mathcal{F}$ is intersecting.

######  Lemma 5.

Let $\mathcal{F}$ be an intersecting family of subsets of $[d+3]$. For any
continuous map $f\colon\Sigma(\mathcal{F})\rightarrow S^{d}$, there must exist
disjoint faces $\sigma$ and $\tau$ of $\Sigma(\mathcal{F})$ such that
$f(\sigma)\cap f(\tau)\neq\emptyset$.

While this result follows easily from a special case of Sarkaria’s lower bound
for dimensions of Euclidean embeddings via chromatic numbers of Kneser graphs
[14], we derive it instead as a direct consequence of the Topological Radon
theorem [1].

######  Theorem 6 (Topological Radon theorem).

For any continuous map $f\colon\Delta_{d+1}\rightarrow\mathbb{R}^{d}$, there
exist disjoint simplices $\sigma$ and $\tau$ of $\Delta_{d+1}$ such that
$f(\sigma)\cap f(\tau)\neq\emptyset$.

###### Proof of Lemma 5.

Given a continuous map $f\colon\Sigma(\mathcal{F})\rightarrow S^{d}$, we
extend it to a continuous map $F\colon\Delta_{d+2}\rightarrow\mathbb{R}^{d+1}$
as follows: Any point $x$ from the simplex $\Delta_{d+2}$ can be uniquely
expressed as the convex sum of a point $s$ from $\Sigma(\mathcal{F})$ and a
convex combination $b$ of barycenters of pairwise incident faces of
$\Delta_{d+2}$ not contained in $\Sigma(\mathcal{F})$. Letting $x=(1-t)b+ts$,
$0\leq t\leq 1$, we define $F(x)=tf(s)$. Here we think of $S^{d}$ as the unit
sphere in $\mathbb{R}^{d+1}$. All that we need is that $F$ continuously
extends $f$ and that $F^{-1}(S^{d})$ only consists of points in
$\Sigma(\mathcal{F})$.

Now suppose that $\sigma$ and $\tau$ are disjoint faces of $\Delta_{d+2}$ for
which $f(\sigma)\cap f(\tau)\neq\emptyset$. If $\mathcal{F}$ is intersecting,
then it is easy to see that both $\sigma$ and $\tau$ must lie in
$\Sigma(\mathcal{F})$. Indeed, any two simplices which are not contained in
$\Sigma(\mathcal{F})$ must intersect. On the other hand, suppose that $\sigma$
and $\tau$ are disjoint, where $\sigma$ lies in $\Sigma(\mathcal{F})$ and
$\tau$ does not. Then clearly $f(\sigma)\cap f(\tau)=\emptyset$ as well. ∎

To conclude the proof of Theorem 1, we show that $\Sigma(\mathcal{F})$ embeds
in $S^{d}$ whenever $\mathcal{F}$ is not intersecting, and moreover that this
embedding is geodesic. To that end, recall that the _matching number_
$\nu(\mathcal{F})$ of $\mathcal{F}$ is the maximum number of pairwise disjoint
sets of $\mathcal{F}$. As $\nu(\mathcal{F})=1$ if and only if $\mathcal{F}$ is
intersecting, the following lemma completes the proof of Theorem 1.

######  Lemma 7.

Let $\mathcal{F}$ be a family of subsets of $[d]=\\{1,\ldots,d\\}$. Then
$\Sigma(\mathcal{F})$ embeds geodesically into $S^{d-\nu(\mathcal{F})-1}$. In
fact, $\Sigma(\mathcal{F})$ is a subcomplex of a convex
$(d-\nu(\mathcal{F}))$-polytope with all $d$ vertices on the unit sphere.

###### Proof of Lemma 7.

Let $\nu=\nu(\mathcal{F})$ and let $\mathcal{M}=\\{S_{1},\ldots,S_{\nu}\\}$ be
a maximal collection of pairwise disjoint sets of $\mathcal{F}$. As
$\Sigma(\mathcal{F})$ is a subcomplex of $\Sigma(\mathcal{M})$, it suffices to
show that $\Sigma(\mathcal{M})$ embeds geodesically into the
$(d-\nu-1)$-sphere. To that end, let $\Delta_{i}$ denote the simplex
determined by $S_{i}$ for each $1\leq i\leq\nu$. As the $S_{i}$ are pairwise
disjoint,

$\Sigma(\mathcal{M})=\partial\Delta_{1}\ast\cdots\ast\partial\Delta_{\nu}\ast\Delta$

is the join of the boundaries of the $\Delta_{i}$ with the join of the simplex
$\Delta$ given by any vertices of $[d]$ (if any) which are not covered by
$\mathcal{M}$. Understanding $S^{-1}$ to be the empty set, each
$\partial\Delta_{i}$ is a $(|S_{i}|-2)$-dimensional sphere, so their join is a
sphere of dimension $\sum_{i}|S_{i}|-\nu-1$.

The boundary of any $k$-simplex is a triangulation of the $(k-1)$-dimensional
sphere and can be realized with geodesically convex faces, for example, by
radially projecting a $k$-simplex inscribed into the sphere. The join
$\partial\Delta_{1}\ast\cdots\ast\partial\Delta_{\nu}\ast\Delta$ may then be
realized with geodesically convex faces as well, and so $\Sigma(\mathcal{F})$
embeds geodesically into the sphere as a subcomplex of $\Sigma(\mathcal{M})$.
∎

###  2.2. Proof of Theorem 3

Using Theorem 1, we now prove our topological generalization of the
Erdős–Ko–Rado theorem. We recall the statement of the latter:

######  Theorem 8 (Erdős–Ko–Rado theorem).

Let $k\geq 2$ and $n\geq 2k$ be integers. If $\mathcal{F}$ is an intersecting
family of $k$-element subsets of $[n]$, then
$|\mathcal{F}|\leq\binom{n-1}{k-1}$.

###### Proof of Theorem 3.

Suppose that $\Sigma$ be a simplicial complex on $d+3$ vertices with fewer
than $\binom{d+2}{k}$ faces of dimension ${k-1}$, $d+3\geq 2k$. Let
$\mathcal{F}$ denote the family of non-faces of $\Sigma$, i.e.,
$\mathcal{F}=\\{\sigma\subseteq[d+3]\mid\sigma\notin\Sigma\\}$. Thus
$\Sigma=\Sigma(\mathcal{F})$. As $\Sigma$ has fewer than $\binom{d+2}{k}$
faces of dimension $k-1$, $\mathcal{F}$ has more than
$\binom{d+3}{k}-\binom{d+2}{k}=\binom{d+2}{k-1}$ subsets of order $k$. Thus
$\mathcal{F}$ cannot be intersecting by the Erdős–Ko–Rado theorem and so
$\Sigma$ must embed into $S^{d}$ by Theorem 1. ∎

###  2.3. Proof of Theorem 4

The proof of our Fáry-type result is again a direct consequence of the lemmas
above.

###### Proof of Theorem 4.

Let $f\colon\Sigma\rightarrow S^{d}$ be a continuous embedding, and let
$\mathcal{F}$ be the set of non-faces of $\Sigma$. Thus
$\Sigma=\Sigma(\mathcal{F})$. By Theorem 1, $\mathcal{F}$ is not intersecting,
and thus $\Sigma$ geodesically embeds into $S^{d}$ by Lemma 7. To finish the
proof, suppose that $f\colon\Sigma\rightarrow\mathbb{R}^{d}$ is an embedding.
Composing with the inverse of the stereographic projection then gives a
continuous embedding $f\colon\Sigma\rightarrow S^{d}$ into the punctured
$d$-sphere. By Lemma 7, $\Sigma$ is a proper subcomplex of the boundary of a
convex $(d+1)$-polytope $P$ and therefore linearly embeds into
$\mathbb{R}^{d}$, for example by considering the Schlegel diagram of $P$ with
respect to a facet not contained in $\Sigma$. ∎

We conclude with a problem and some comments concerning the optimality of
Theorem 4 and Lemma 7.

######  Problem 9.

Brehm [4] constructed a triangulation of the Möbius strip on nine vertices
that does not linearly embed into $\mathbb{R}^{3}$. Thus, embeddability and
linear embeddability into $\mathbb{R}^{d}$ differ for complexes on $d+6$
vertices. As far as we know, it remains open whether these notions of
embeddability coincide for simplicial complexes on $d+4$ or even $d+5$
vertices.

It is tempting to think that the two triangulations of $S^{3}$ on eight
vertices that are not boundary complexes of convex polytopes [2] are good
candidates to show that Theorem 4 cannot be extended to complexes on $d+5$
vertices. As shown by Mihalisin and Williams [13], these non-polytopal
$3$-spheres do linearly embed into $\mathbb{R}^{4}$, however. As an
application of Theorem 4, we provide a quick alternative proof of this fact in
Remark 11 below. Nonetheless, we can use one of these two exceptional spheres
to show that our construction in Lemma 7 cannot be extended to complexes on
$d+4$ vertices:

######  Remark 10.

We construct a simplicial complex on seven vertices that embeds into
$\mathbb{R}^{3}$ but which is not contained in any convex simplicial
$4$-polytope on seven vertices. Following the notation and facet list of [13],
the simplicial $3$-sphere $\mathcal{M}^{\prime}$ has a vertex $p$ whose vertex
link is an octahedron. Every edge not involving $p$ is present in
$\mathcal{M}^{\prime}$. The complex $\Sigma$ obtained from
$\mathcal{M}^{\prime}$ by deleting $p$ and all its incident faces is a
$3$-ball, and thus embeds into $\mathbb{R}^{3}$. Suppose $\Sigma$ were a
subcomplex of a convex simplicial $4$-polytope on seven vertices. To
triangulate the boundary octahedron of $\Sigma$, this polytope must have an
additional edge. This leads to a double edge, however, since all edges are
already present in $\Sigma$.

######  Remark 11.

Finally, we remark that we can simplify the constructions of Mihalisin and
Williams [13] for linear embeddings of the two non-polytopal $3$-spheres on
eight vertices. Indeed, let $\mathcal{M}^{\prime}$ be the non-polytopal
triangulation of $S^{3}$ on eight vertices whose $1$-skeleton is not complete
(i.e., “Barnette’s sphere”). Delete the two endpoints $v$ and $w$ of the
missing edge and all their incident faces from $\mathcal{M}^{\prime}$. The
resulting complex embeds into $\mathbb{R}^{3}$ and has six vertices; thus it
embeds linearly by Theorem 4. Thinking of $\mathbb{R}^{3}$ as a hyperplane in
$\mathbb{R}^{4}$, place $v$ above this hyperplane and $w$ below it. This
yields a linear embedding of $\mathcal{M}^{\prime}$ into $\mathbb{R}^{4}$. Now
consider $\mathcal{M}$, the non-polytopal $3$-sphere on eight vertices with
complete $1$-skeleton (i.e., “Grünbaum’s sphere”). Deleting any vertex yields
a stacked $3$-ball which linearly embeds into $\mathbb{R}^{3}$. One now only
need add the deleted vertex back by placing it above the
$\mathbb{R}^{3}$-hyperplane in $\mathbb{R}^{4}$.

## References

  * [1] E. Bajmóczy and I. Bárány. On a common generalization of Borsuk’s and Radon’s theorem. _Acta Math. Hung._ , 34: 347–350, 1979.
  * [2] D. Barnette. The triangulations of the 3-sphere with up to 8 vertices. _J. Combin. Theory, Ser. A_ , 14(1): 37–53, 1973.
  * [3] D. Barnette. Two “simple” 3-spheres. _Discrete Math._ , 67(1): 97–99, 1987.
  * [4] U. Brehm. A nonpolyhedral triangulated Möbius strip. _Proc. Amer. Math. Soc._ , 89(3): 519–522, 1983.
  * [5] Ch. Chojnacki. Über wesentlich unplättbare Kurven im dreidimensionalen Raume. _Fund. Math._ , 23(1): 135–142, 1934.
  * [6] P. Erdős, C. Ko, and R. Rado. Intersection theorems for systems of finite sets. _Q.J. Math._ , 12(1): 313–320, 1961.
  * [7] I. Fáry. On straight-line representation of planar graphs. _Acta Sci. Math. (Szeged)_ , 12: 229–233, 1948.
  * [8] A. Flores, Über $n$-dimensionale Komplexe die im $\mathbb{R}^{2n+1}$ absolut selbstverschlungen sind. _Ergeb. Math. Kolloq._ , 6: 4–7, 1933.
  * [9] P. Mani. Spheres with few vertices. _J. Comb. Theory, Ser. A_ , 13(3): 346–352, 1972.
  * [10] J. Matoušek. _Using the Borsuk–Ulam theorem: Lectures on topological methods in combinatorics and geometry._ Springer-Verlag, 2008.
  * [11] J. Matoušek, M. Tancer, and U. Wagner. Hardness of embedding simplicial complexes in $\mathbb{R}^{d}$. _J. Eur. Math. Soc._ , 13(2): 259–295, 2011.
  * [12] A. de Mesmay, Y. Rieck, E. Sedgwick, and M. Tancer. Embeddability in $\mathbb{R}^{3}$ is NP-hard. _J. ACM_ , 67(4): 1–29, 2020.
  * [13] J. Mihalisin and G. Williams. Nonconvex embeddings of the exceptional simplicial 3-spheres with 8 vertices. _J. Combin. Theory, Ser. A_ , 98(1): 74–86, 2002.
  * [14] K. S. Sarkaria. A generalized Kneser conjecture. _J. Comb. Theory, Ser. B_ , 49(2): 236–240, 1990.
  * [15] L. Schewe. Nonrealizable minimal vertex triangulations of surfaces: showing nonrealizability using oriented matroids and satisfiability solvers. _Discrete Comput. Geom._ , 43(2): 289–302, 2010.
  * [16] W. T. Tutte. Toward a theory of crossing numbers. _J. Comb. Theory_ , 8: 45–53, 1970.
  * [17] E. van Kampen. Komplexe in euklidischen Räumen. _Abh. Math. Semin. Univ. Hamb._ , 9: 72–78, 1932.
  * [18] K. Wagner. Über eine Eigenschaft der ebenen Komplexe. _Math. Ann._ , 114: 570–590, 1937.
  * [19] G. M. Ziegler. _Lectures on polytopes_. Grad. Texts Math. 152, Springer-Verlag, New York, 1995.


[1] [1]

Data curation, Writing - Original draft preparation

[cor1]Corresponding author [cor2]Principal corresponding author

# The formation of Pareto distribution in tracer systems on the water surface

S.V. Filatov<EMAIL_ADDRESS>142432, Institute of Solid State Physics, RAS
142432, Landau Institute for Theoretical Physics, RAS  A.V. Poplevin  A.A.
Levchenko  E.V. Lebedeva  L.P. Mezhov-Deglin

###### Abstract

We experimentally studied a clusterization process in a system of polyamide
tracers that are used for visualizing the flow of liquids on their surface. It
was shown that in a surface structure system appearing on the water surface a
Pareto distribution is formed for normalized cluster density and it is well
described by a $C/x^{-n}$ power-law function. One can suggest that the growth
of a surface structure is defined by the action of surface tension forces, and
the number of surface structures decreases exponentially, while maintaining
their total surface area. We show experimentally the significant role of
background liquid flows on the surface in the clusterization process.

###### keywords:

pareto distribution clusterization surface structure

The interaction of neutral tracer particles on the water surface is provided
by capillary forces.

The growth of large-scale structures is observed experimentally; a Pareto
distribution is formed in the cluster system - the power-law distribution of
the particle density over their areas.

The maximum cluster size is limited by random fluid flows.

##  1 Introduction

The experimental procedure of studying vortex currents created by waves on the
surface of liquids includes visualization using a decorative powder with the
density that is close to the density of the liquid Von Kameke et al. (2011);
Filatov et al. (2015); Francois et al. (2013); Mezhov-Deglin et al. (2019).
Normally a polyamide-12 powder with a mean size of $\sim$ 30 $\mu$m (Von
Kameke et al., 2011; Filatov et al., 2015; Francois et al., 2013) or hollow
glass spheres of about 50 $\mu$m in diameter [Mezhov-Deglin et al. (2019)] are
used. The liquid surface is photographed from above by a camera with a certain
shooting frequency. Compairing the sequence of photos allows computing the
liquid’s surface velocity by means of the PIVlab software package [Thielicke
and Stamhuis (2014)]. It requires the mean density of the decorative particles
to be no less than 10 per cm2 for the reliable operation of PIVlab.

In our study, the polyamide particles on the liquid surface gather into about
0.1 mm-large tracer-complexes that have a rigid structure and are constantly
moving under an external force. The tracer complexes form larger clusters, the
structure of the latter maintains to be flat, but is strongly indented. The
size of the clusters during the experiment gets larger, and at measurement
time of over 10 hours their size can increase up to several centimeters, their
quantity decreases by one order so that it is no more possible to use the PIV
method Thielicke and Stamhuis (2014).

The clustering of particles is observed in many systems, for example, in the
semiconductor bulk impurities segregation takes place [Käshammer and Sinno
(2015)], also at grain boundaries [Straumal et al. (2016)]. At the same time,
the motion of the dopants may be caused by an electric force or mechanical
tension. The capillary interaction between floating particles on the liquid
surface, which leads to their convergence, had been widely studied for many
years, see for example [Singh and Joseph (2005)]. In our studies, polyamide-12
particles form homogeneously structured complexes under Van der Waals forces
[Lifshitz and Hamermesh (1992)], while the clusters get formed by differently
structured complexes. The complexes are maintained within the clusters by
surface tension forces [Levchenko et al. (2019)]. The presence of background
fluid flows on the surface provides a random convergence of the complexes.

##  2 Experimental procedure

The experimental procedure of generating vortex current by waves and
visualizing liquid surface flows is described in detail in our paper [Filatov
et al. (2018, 2019)]. In our current experiments on the clusterization of
polyamide-12 particles on the water surface, a 70-by-70 cm bath was used. The
polyamide particles were sputtered above the water surface. Under the action
of gravity, they descended onto the surface, forming various structures.
Strong plunger pumping which can produce high-amplitude waves and vortices was
used for creation of uniform distribution of rigid poliamide complexes over
the surface and over the size. The velocity of the surface flow exceeded 10
cm/s. After the pumping was switched off, a series of photos of the water
surface was taken at various intervals for a period of up to 120 hours. During
the first 1.5 hours the shots were taken each 5 seconds, the rest of the time
each next shot was made in 30 seconds. To remove the impact of the bath walls
on measurement results, the number of the surface structures was calculated
within the area of 50x50 cm2 in size. The particles on the water surface were
illuminated by a LED strip mounted on the side panes of the bath. The area of
the structures was calculated from the number of pixels per surface structure.

Therefore, the experiments were carried out under different conditions, which
gave rise to different average values of the background velocity, namely
normal illumination (V $\sim$ 0.01 cm/s), switched off illumination (V $\sim$
0.005 cm/s), a thermally isolated container (V $\sim$ 0.002 cm/s), and
artificial operation of the plunger (0.006 to 0.25 cm/s).

##  3 Experimental results

As soon as the pumping is switched off, the surface flow velocity decreases
due to the wave decay because of viscous loses. Fig. 1 shows the flow velocity
reaching its minimum in several hours, slow random movements with the average
velocity of the order of 0.01 cm/s remain on the surface afterwards. Such
background movement on the surface maintains itself without plunger pumping,
being caused by volume convectional flows that occur due to vertical and
horizontal thermal gradients. The thermal gradients occur because of heat
generation by the LED strip on the bath walls.

Figure 1: Time dependence of an average modulus of the flow velocity on the
water surface.

Fig. 2 illustrates the photos of bath fragments with polyamide particle
complexes on the water surface after 10 minutes (a), 22 hours (b), and 60
hours (c) after switching-off pumping. Initially, after switching off pumping,
there are approximately 105 complexes with an area of the order of 10-4 \- 100
cm2 on the surface, formed by polyamide microparticles.

As it can be seen, most complexes are disc-shaped (2a). However, we can
observe the processes of the merging of two disc-complexes into one, as well
as of three discs into linear I- and T-shaped complexes. Over time, the
complexes formed at the first consolidation stage are collected into larger
formations with a linear size of up to 0.3 cm, and this process continues for
tens of hours, Fig. 2b. At this stage the complexes are characterized by their
surface structure homogeneity. In 60 hours, the complexes form clusters which
have a complex branched structure of several centimeters in size.

This can be seen more clearly in Fig. 2 (d) that shows complexes on the water
surface 20 hours after switching off the pumping and without continuous
illumination by the LED strip. The background convective flow velocity got
halved, reducing to 0.005 cm/s. The photo demonstrates clearly that the
clusters size is larger than that in Fig. 2(c) that was taken after 60 hours
of clusterization with the heat source activated. We can conclude that the
clustering process on the water surface has two stages. At the first stage,
the continuous complexes grow up to their characteristic size of less than 0.5
cm. At the second stage the continuous complexes form clusters with a complex
branched structure. The consolidation rate of the complexes and their maximum
size are determined by the surface convective flow.

a)

b)

c)

d)

Figure 2: Photographs of structures on the water surface 10 minutes (a), 22
hours (b), and 60 hours (c) after switching off pumping. (d) is the
photographs of the water surface 20 hours after switching off pumping without
continuous illumination.

a)

b)

c)

d)

e)

f)

Figure 3: Distribution of absolute velocity over the surface 20 hours (a), 60
hours (b) and 86 hours (c) after switching off pumping. Distribution of
particular velocity gradients (shear) $\sqrt{(\partial V_{x}/\partial
y)^{2}+(\partial V_{y}/\partial x)^{2}}$ over the surface 20 hours (d), 60
hours (e), and 86 hours (f) after switching off pumping.

Fig. 3 illustrates the distribution of absolute velocity and velocity
gradients (shear) over the surface 20, 60 and 86 hours after switching off
pumping. One can see in the images that velocity distribution over the surface
changes significantly with time. This is because heating conditions leading to
the emergence of convection are non-stationary and non-uniform. It may also be
seen from the figure 3 (a-c) that the characteristic correlation lengths for
the velocity are much larger than the size of the clusters.

Fig. 4 shows the distributions of the normalized number of structures on the
surface over their area $N(S)$, obtained by dividing the number of complexes
and clusters from the photos, the area of which fits within the specified
interval, by the number of photos and interval width. The interval increases
with a logarithmically increasing step from 10-4 cm2 to 50 cm2, the number of
photos is 103 units. Curve 1 was obtained 12 minutes, Curve 2 16 hours, Curve
3 49.3 hours, and Curve 4 116 hours after switching off pumping. It is seen
that the probability of finding large clusters on the photos is less than
unit.

Figure 4:  Distribution of the normalized number of structures on the water
surface over their area $N(S)$ after 12 minutes (1), 16 hours (2), 49.3 hours
(3), and 116 hours (4) after switching off pumping. The average surface flow
velocity is 0.01 cm/s.

Immediately after switching off pumping (see Curve 1), when the surface flow
velocity is still high, most surface structures have an area of about 10-2 cm2
or less. Only a few of them have an area of about 1 cm2. At 16 hours after
switching off pumping (see Curve 2), when the wave motion has totally decayed,
the photo shows many structures in wide range of areas of up to 2 cm2.
Nevertheless, in distribution 2 has not reached its equilibrium state,
therefore, it cannot be described by an universal function. At 49.3 hours (see
Curve 3), the large cluster growth continues by joining smaller complexes.
Clusters of about 10 cm2 in size appear. At 116 hours after switching off
pumping we observe that the distribution of the surface structures in a broad
range of areas from 0.001 cm2 up to 1 cm2 is getting close to the power-law
function. It is clearly seen that the obtained distribution $N(S)$ can be
described by the $N(S)\sim(S/S_{0})^{-n}$ function with an exponent close to n
= 4/3. However, we should notice that the $N(S)$ distribution at $S>1$ cm2
significantly deviates from the power-law function. The reason for this may be
too high velocity of the background surface water flow, or lack of the
observation time.

Thus, the distribution of the normalized number of surface structures over
their area $N(S)$ at long observation time becomes a power-law one, with its
characteristic size $S_{0}\approx 2\cdot 10^{-2}$ cm2.

As we mentioned above, the average value V of random background flow velocity
on the surface can define the characteristic time of the $N(S)$ function
reaching its equilibrium state, as well as the maximum size of clusters. To
test this assumption, the average velocity V was reduced to 0.002 cm/s by
setting additional 5-cm thick Styrofoam insulation on the bath walls and
changing the illumination mode. Fig. 5 shows the distributions $N(S)$ obtained
5.5 minutes, 25 minutes, 100.5 minutes, and 275.5 minutes after switching off
pumping at frequency of 3 Hz. The mean surface current velocity immediately
the pump at a frequency of 3 Hz was switched off getting close to 10 cm/s

Figure 5:  Distribution of the normalized particle cluster number the size
$N(S)$ in 5.5 minutes (1), 25 minutes (2), 100.5 minutes (3), and 275.5
minutes after switching off pumping. The average surface flow velocity is
0.002 cm/s.

It is clearly seen that the process of the establishment of a power-law
distribution proceeds much quicker than in the case of high values of
background flow velocity. The power-law distribution is already observed in
about 1.5 hours (see Curve 3).

In an area range from 10-1 cm2 to 101 cm2, the observed dependence $N(S)$ is
described by a power function with an exponent close to n = 9/4. It can be
seen that the number of clusters increased due to a decrease in the number of
smaller clusters. Curve 4 obtained in 4.6 hours in an area range from 10-2 to
101 cm2, can be well described by the power function $N(S)\sim(S/S_{0})^{-n}$
with an exponent close to n= 5/3 and $S_{0}=1.0\cdot 10^{-1}$ cm2. The process
of redistribution of areas between small and large structures leads to a
decrease in the absolute value of the exponent of the function describing the
experimental dependences $N(S)$. The probability of discovering a large
cluster on the water surface increases with time.

The maxima on the distributions $N(S)$ (see Curves 1, 2 and 3) are clearly
seen. One can assume that the concentration of complexes near $S\approx
10^{-2}$ cm2 within the first minutes of observation is caused by intensive
plunger pumping that makes large clusters break into smaller ones by intensive
fluid flows. Fig. 6(a) shows the time dependence of an average area of
structures in semi-logarithmic coordinates after switching off excitation at
background flow velocity of 0.01 cm/s. The average size of structures, defined
as the sum of the areas of all complexes and clusters $S$, divided by their
number $N$, increases with time, $S_{EV}=S/N$. One can see that, although 116
hours had elapsed since switching off excitation, the clustering process was
not completed. We assume that during the first 40 hours the average area
$S_{EV}$ of complexes and clusters increases exponentially within a
characteristic time of $\tau_{0}\sim 15$ hours. Starting from the 45th hour,
an average size of a cluster increases exponentially within a characteristic
time of $\tau_{0}\sim 40$ hours as a result of the linear structure fusion.

Fig. 6 (b) demonstrates the time dependence of the number of clusters, in
semi-logarithmic coordinates as well, after switching off pumping. It is
obvious that the number of clusters $N(t)$ decreases from $8\cdot 10^{4}$
units to $5\cdot 10^{3}$ with time. At times of less than 45 hours, the
decreasing process is approximately exponential, its characteristic time being
$\tau_{0}\approx 15$ hours. At times of over 45 hours, the process of a
decrease in the cluster number is also exponential, $N(t)\sim
exp(-t/\tau_{0}),\tau_{0}\approx 40$ hours. The difference between the
characteristic times is due to the qualitative change in the structures that
form complexes and clusters. In the beginning, the formation of linear
structure-complexes of large area, involving the disc-complexes, seems to be
the main process. But at long times, the growth of clusters is associated
mainly with the merging of linear surface structures.

The comparison of Fig. 6 (a) and 6 (b) shows clearly that an average size of
clusters increases due to a decrease in the total number of structures while
their total area $S$ remains the same.

a)

b)

Figure 6: Time dependence of an average cluster size (a) and the number of
clusters (b) on the water surface 50x50 cm2 on time after switching off
pumping. The background liquid flow is 0.01 cm/s.

##  4 Discussion

The power-law distribution of probability of a $C/x^{n}$-type event is called
the Pareto distribution [Manceau et al. (2019)] and is observed in economics
[Axtell (2001)], geophysics [Taubert et al. (2018)], astrophysics [Edward and
Russell (1991)], and condensed-matter physics [Stefani et al. (2009); Manceau
et al. (2019)]. It should be noted that the Pareto distribution is often
formed in random interactions in system of limited resources, such as the
income of money per citizen of a country or distribution of cities over the
number of inhabitants [Jones (2015); Ribeiro Justo (2014)]. In our
experiments, the total surface structure area is limited by the total number
of polyamide-12 particles sputtered on the water surface, and it cannot exceed
the area of our experimental bath. The randomness of interaction is defined by
a background flow on the surface water.

In our studies, the polyamide-12 particles had been already merged into the
complexes due to the action of Van der Waals forces. We assume that an average
density of the complexes turned out to be lower than that of water due to
hollow spaces that occur in their bulk. So, therefore, after mixing by
intensive pumping, they are immersed in water, and a part of them with the
height $h$, completely moistened with a liquid, protrudes above the surface.
By interacting with each other under the liquid surface by van der Waals
forces, the complexes are assembled into flat discs with a characteristic size
of about 0.2-0.3 cm and a thickness of about 0.01 cm. Such complexes include
up to 106 polyamide-12 particles. This is evidenced by the predominant number
of complexes in Figs. 4 and 5 with areas of less than 10-2 cm2.

The disc-complexes with a continuous surface structure seemingly cannot reach
larger sizes due to the rigidity of the structure, the directional action of
Van der Vaals forces, and the destructive effect of the flows spreading on the
surface. The total number of smaller complexes significantly exceeds the
number of larger ones, so the process of increase in the average area of the
surface structures is mainly related to the merging of the smaller complexes -
the first stage. One can assume that the process of structure merging has an
accidental character, so the velocity of changes in the number of clusters is
proportional to their quantity $N$:

$dN(t)/dt=1/\tau_{0}\cdot N$ (1)

At the second stage, decrease in the number of clusters is mainly due to the
addition of the smaller linear complexes to the larger clusters. Within very
long consolidation times, when the number of clusters drops to just few, the
exponential dependence will no longer make sence.

Let us take a qualitative look at the process of merging of two Van der Vaals
disc-complexes that have formed into one complex, that is, a linear structure,
at the first clusterization stage. The converging of the two discs to the
distance of the same order of their radius is performed by the background
surface liquid flow. The values of the disc diameters are close to the
capillary length $\sqrt{2\sigma/(\rho g)}$ , which is equal to 3.9 mm [Landau
and Lifshitz (1987)] for water, $\sigma$ is the surface tension coefficient,
$g$ is the free fall acceleration, and $\rho$ is the water density. Therefore,
we assume that the process of complex merging at this stage is defined by the
action of surface tension forces.

Let us assume that in the formed complex the distance between the centers of
the disks is equal to the sum of their radii, $R_{1}+R_{2}$. We also assume
that the thickness and the total complexes area do not change when they get
merged. So the energy balance between only the energy change in the side
surface of the complexes and the height $h$ should be taken into account. The
sum of side surface energies of two different discs can be written as:

$E_{S}=2\pi h\sigma(R_{1}+R_{2})$

After merging into a dumbbell shape, the energy of the side surface will be
written as:

$E_{L}\approx h\sigma(\pi+2)(R_{1}+R_{2})$

It can be seen that the energy ratio $E_{S}/E_{L}\approx 2\pi/(\pi+2)$ always
exceeds unit. To prove our suggestions, let us look at Fig. 2 (b) that shows a
large quantity of linear complexes which is the evidence of the initial
prevailing of the disc complexes that do not change after merging. The
formation of large disc-shaped complexes at this stage is rare.

At the next stage, large clusters with a complicated structure get formed.
Their "building material" is complexes of various shapes: discs, dumbbells, I-
and T-shaped structures. As a result of the merging, complicated structures
get formed that do not have any symmetry, with broken boundaries and holes in
their surface (See Fig. 2(c) and (d)). The clusters drift on the water surface
and retain their shape for a long time. Changes occur when they merge with a
smaller complex, or two larger clusters merge together.

We should note that the process of merging with a complex and merging of
clusters are possible only when there is a chaotic liquid flow with the
average velocity $V$. However, the presence of very high velocities of the
background fluid flow will prevent an increase in the cluster sizes.

It follows from simple considerations that the joining of a small complex of
the radius $R_{S}$ to a large cluster of the size $R_{B}$ is possible if a
decrease in the total energy of the large plus small clusters exceeds the
kinetic energy of the small cluster.

The energy change in the united larger cluster is approximately equal to:

$\Delta E\approx(\pi-2)h\sigma R_{S}$

The cluster kinetic energy equals to $M_{S}V^{2}/2$. The cluster mass equals
to $M_{S}=\pi R_{S}^{2}h\rho_{c}$, where $\rho_{c}$ is the cluster density.

Thus, the condition for joining a smaller cluster to a larger one is expressed
in the inequation:

$(\pi-2)h\sigma R_{S}>\pi R_{S}^{2}h\rho V^{2}$,

from which we get qualitative size restriction for the smaller complex that
can merge with a larger cluster at a given liquid flow velocity V.

$R_{S}<(\pi-2)/\pi\sigma/\rho V^{2}$

Thus, the growth of larger clusters is due to the joining of mainly smaller
complexes to them. The burnout of smaller complexes of less than 0.3 cm in
size occurs mainly on the surface (see Figs. 4 and 5).

However, the size $R_{B}$ of a larger cluster is also restricted by the action
of forces caused by shear liquid flows that break the bonds holding the
complexes together. Let us assume that the velocity gradient $\Delta V$ is
order to $V/R_{B}$ in order of magnitude, the cluster area is $\sim
R_{B}^{2}$, then for the estimation we can write:

$V/R_{B}\cdot R_{B}^{2}\cdot\eta\sim\sigma R_{S}$ (2)

From (2) we find that the maximum cluster area with an increase in the
velocity of the background shear flow should decrease according to the law:

$R_{B}^{2}\sim(\sigma R_{S}/V/\eta)^{2},$ (3)

where $\eta$ is the coefficient of dynamic viscosity.

To testify this dependence, we investigated the consolidation process at
various plunger operating regimes. The mean background flow velocity $V$ of
the liquid changes almost by two orders of magnitude from 0.006 to 0.25 cm/s.
Fig. 6 shows the dependencies of an average area of the structures at various
background velocities. It can be seen that at velocities of 0.01 and 0.006
cm/s an average cluster size continually and exponentially increases with time
throughout 25 hours of investigation. At a velocity of 0.04 cm/s and higher,
the value of $S_{EV}$ reaches its maximum value, moreover, the higher the
velocity $V$, the less the maximum value of $S_{EV}$. It is surprising that
the experimentally measured average cluster size decreases in an inverse ratio
to the squared velocity of the background flow according to formula (3) (See
the insert in Fig.7).

Note that it was shown earlier in [Higashitani and Iimura (1998)] that for
clusters with a fractal structure, the average size of a stable aggregate
decreases with an increase in the shear flow velocity by a power law with the
exponent less than unity in modulus. However, for clusters with closer packing
[Vassileva et al. (2006)], the average size of an aggregate slightly depends
on the shear flow velocity. The long-term dynamics of the shear-induced
breakage of individual colloidal clusters covering a wide range of fractal
dimensions were studied in [Harshe and Lattuada (2016)].

Figure 7: Time dependence of average cluster area on the water surface
$S_{EV}$ on time at various background flow velocity values in semi-
logarithmic coordinates. The insert demonstrates dependence of the maximum
average cluster area on the surface flow velocity in a log-log scale. 1 -
0.006 cm/s, 2 - 0.01 cm/s, 3 - 0.04 cm/s, 4 - 0.08 cm/s, 5 - 0.14 cm/s, 6 -
0.25 cm/s.

Fig.6 shows that the liquid flow with a velocity of 0.04 cm/s and higher
significantly restricts cluster growth. Note that at all values of the
background liquid flow velocity, power-law distribution of the normalized
number of clusters over the surface is observed. As an example, Fig. 8
illustrates the results obtained at a flow velocity of 0.08 cm/s.

Figure 8: Distribution of the normalized number of clusters over the surface
area $N(S)$ 25.2 minutes (1), 150 minutes (2), 375 minutes (3), 750 minutes
(4), and 1250 (5) minutes after switching off excitation.

At 25 minutes switching off pumping, the distribution $N(S)$ is far from being
a power-law type. However, starting from the 150th minute, on the
distributions $N(S)$ we can clearly distinguish two regions that are described
by the power functions $\sim S^{-n}$ with the power exponent $n$ being
approximately 7/10 and 5/2. The presence of two power-law distributions
indicates different conditions for the formation of surface structures and
proves our assumption that there are two consolidation stages, i.e., the
formation of large complexes and formation of branched clusters from them.

##  5 Conclusion

It has been experimentally determined that in the system of structures formed
on the water surface by polyamide particles, the Pareto distribution is
established in a wide range of their areas. Long-term studies have been
carried out (up to 120 hours) of the process of transformation of macroscopic
quasi-two-dimensional tracers that get formed from polyamide tracers used for
visualizing the water surface flows in a pool of a limited size. The tracers
consist of light neutral polyamide particles with a mean diameter of $\sim$ 30
$\mu km$. The finely dispersed polyamide-12 powder is sputtered onto the water
surface in a bath, so that the uncovered water area exceeds the area covered
with the dopant clusters by more than 100 times. During the experiment, the
total area of the dopant clusters on the water surface remained unchanged. It
observed that at long-term observation, the distribution of the normalized
density of quasi-two-dimensional polyamide clusters can be described by the
power function of $N(S/S_{0})\sim C(S/S_{0})^{-n}$ \- Pareto distribution,
where $S$ is the area of a single cluster surface, $S_{0}$ is the
characteristic cluster area. The formation of two-dimensional structures on
the water surface in a bath depends largely on the action of surface tension
forces. The number of surface structures decreases exponentially while their
total area remains constant. It has been also shown experimentally that the
background liquid flow on the surface plays the key role in the clusterization
process.

##  6 Acknolgement

We are grateful to V. Lebedev, I. Kolokolov, and V. Parfenyev for valuable
discussions. This work was supported by the Russian Ministry of Science and
Higher Education, project No. 075-15-2019-1893.

## References

  * Axtell (2001) Axtell, R.L., 2001.  Zipf distribution of U.S. firm sizes.  Science 293, 1818–1820.  doi:10.1126/science.1062081.
  * Edward and Russell (1991) Edward, T.L., Russell, J.H., 1991\.  Avalanches and the distribution of solar flares.  The Astrophysical Journal 380, L89–L92.  doi:10.1017/CBO9781107415324.004, arXiv:arXiv:1011.1669v3.
  * Filatov et al. (2015) Filatov, S.V., Brazhnikov, M.Y., Levchenko, A.A., 2015.  Generation of a vortex flow by waves on the surface of a liquid.  JETP Letters 102, 432--436.  URL: http://link.springer.com/10.1134/S0021364015190054, doi:10.1134/S0021364015190054.
  * Filatov et al. (2018) Filatov, S.V., Levchenko, A.A., Brazhnikov, M.Y., Mezhov-Deglin, L.P., 2018\.  A Technique for Registering Wave and Vortex Motions on a Liquid Surface.  Instruments and Experimental Techniques 61, 757--760.  URL: http://link.springer.com/10.1134/S0020441218040188, doi:10.1134/S0020441218040188.
  * Filatov et al. (2019) Filatov, S.V., Levchenko, A.A., Khramov, D.A., 2019.  Decay of a vortex lattice formed by gravity waves on the water surface.  Results in Physics 13, 102229\.  URL: https://doi.org/10.1016/j.rinp.2019.102229, doi:10.1016/j.rinp.2019.102229.
  * Francois et al. (2013) Francois, N., Xia, H., Punzmann, H., Shats, M., 2013\.  Inverse energy cascade and emergence of large coherent vortices in turbulence driven by Faraday waves.  Physical Review Letters 110, 194501\.  doi:10.1103/PhysRevLett.110.194501, arXiv:1302.2993.
  * Harshe and Lattuada (2016) Harshe, Y.M., Lattuada, M., 2016\.  Universal breakup of colloidal clusters in simple shear flow.  The Journal of Physical Chemistry B 120, 7244--7252.  URL: https://doi.org/10.1021/acs.jpcb.6b03220, doi:10.1021/acs.jpcb.6b03220, arXiv:https://doi.org/10.1021/acs.jpcb.6b03220. pMID: 27348206\.
  * Higashitani and Iimura (1998) Higashitani, K., Iimura, K., 1998\.  Two-dimensional simulation of the breakup process of aggregates in shear and elongational flows.  Journal of Colloid and Interface Science 204, 320 -- 327.  URL: http://www.sciencedirect.com/science/article/pii/S0021979798955618, doi:https://doi.org/10.1006/jcis.1998.5561.
  * Jones (2015) Jones, C.I., 2015.  Pareto and Piketty: The macroeconomics of top income and wealth inequality.  Journal of Economic Perspectives 29, 29--46.  doi:10.1257/jep.29.1.29.
  * Käshammer and Sinno (2015) Käshammer, P., Sinno, T., 2015\.  A mechanistic study of impurity segregation at silicon grain boundaries.  Journal of Applied Physics 118.  URL: http://dx.doi.org/10.1063/1.4929637, doi:10.1063/1.4929637.
  * Landau and Lifshitz (1987) Landau, L.D., Lifshitz, E.M., 1987\.  Fluid mechanics.  Pergamon Books Ltd.
  * Levchenko et al. (2019) Levchenko, A.A., Lebedeva, E.V., Mezhov-Deglin, L.P., Pelmenev, A.A., 2019\.  Self-organization of neutral particles on the surface of superfluid He II.  Low Temperature Physics 45.  URL: http://dx.doi.org/10.1063/1.5097354, doi:10.1063/1.5097354.
  * Lifshitz and Hamermesh (1992) Lifshitz, E., Hamermesh, M., 1992\.  The theory of molecular attractive forces between solids.  Perspectives in Theoretical Physics 2, 329--349.  doi:10.1016/b978-0-08-036364-6.50031-4.
  * Manceau et al. (2019) Manceau, M., Spasibko, K.Y., Leuchs, G., Filip, R., Chekhova, M.V., 2019.  Indefinite-Mean Pareto Photon Distribution from Amplified Quantum Noise.  Physical Review Letters 123, 123606\.  URL: https://doi.org/10.1103/PhysRevLett.123.123606, doi:10.1103/PhysRevLett.123.123606, arXiv:1807.11784.
  * Mezhov-Deglin et al. (2019) Mezhov-Deglin, L., Pel’menev, A., Levchenko, A., 2019.  Kelvin-Helmholtz instability forced by the intensive Faraday waves on the free surface of superfluid He-II.  Materials Letters 238, 226--228.  URL: https://doi.org/10.1016/j.matlet.2018.12.016, doi:10.1016/j.matlet.2018.12.016.
  * Ribeiro Justo (2014) Ribeiro Justo, W., 2014.  Zipf’s Law and the Gibrat’s Law: What Do the Facts Have to Say about the Brazilian Cities?  Journal of Finance and Economics 2, 136--144.  doi:10.12691/jfe-2-5-1.
  * Singh and Joseph (2005) Singh, P., Joseph, D.D., 2005\.  Fluid dynamics of floating particles.  Journal of Fluid Mechanics 530, 31–80.  doi:10.1017/S0022112005003575.
  * Stefani et al. (2009) Stefani, F.D., Hoogenboom, J.P., Barkai, E., 2009.  Beyond quantum jumps: Blinking nanoscale light emitters.  Physics Today 62, 34--39.  doi:10.1063/1.3086100.
  * Straumal et al. (2016) Straumal, B.B., Mazilkin, A.A., Baretzky, B., 2016.  Grain boundary complexions and pseudopartial wetting.  Current Opinion in Solid State and Materials Science 20, 247--256.  URL: http://dx.doi.org/10.1016/j.cossms.2016.05.006, doi:10.1016/j.cossms.2016.05.006.
  * Taubert et al. (2018) Taubert, F., Fischer, R., Groeneveld, J., Lehmann, S., Müller, M.S., Rödig, E., Wiegand, T., Huth, A., 2018\.  Global patterns of tropical forest fragmentation.  Nature 554, 519--522.  URL: http://dx.doi.org/10.1038/nature25508, doi:10.1038/nature25508.
  * Thielicke and Stamhuis (2014) Thielicke, W., Stamhuis, E.J., 2014\.  PIVlab – Towards User-friendly, Affordable and Accurate Digital Particle Image Velocimetry in MATLAB.  Journal of Open Research Software , 2(1):e30URL: http://openresearchsoftware.metajnl.com/articles/10.5334/jors.bl/, doi:10.5334/jors.bl.
  * Vassileva et al. (2006) Vassileva, N.D., van den Ende, D., Mugele, F., Mellema, J., 2006\.  Restructuring and break-up of two-dimensional aggregates in shear flow.  Langmuir 22, 4959--4967.  URL: https://doi.org/10.1021/la053460k, doi:10.1021/la053460k, arXiv:https://doi.org/10.1021/la053460k. pMID: 16700581.
  * Von Kameke et al. (2011) Von Kameke, A., Huhn, F., Fernández-García, G., Muñuzuri, A.P., Pérez-Muñuzuri, V., 2011.  Double Cascade Turbulence and Richardson Dispersion in a Horizontal Fluid Flow Induced by Faraday Waves.  Physical Review Letters 107, 074502\.  doi:10.1103/PhysRevLett.107.074502.