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# On the variation of Frobenius eigenvalues in a skew-abelian Iwasawa tower

Asvin G111Department of Mathematics; University of Wisconsin, Madison. 222
email<EMAIL_ADDRESS>
###### Abstract

We study towers of varieties over a finite field such as
$y^{2}=f(x^{\ell^{n}})$ and prove that the characteristic polynomials of the
Frobenius on the étale cohomology show a surprising $\ell$-adic convergence.
We prove this by proving a more general statement about the convergence of
certain invariants related to a skew-abelian cohomology group. The key
ingredient is a generalization of Fermat’s Little Theorem to matrices. Along
the way, we will prove that many natural sequences of polynomials
$(p_{n}(x))_{n\geq 1}\in\mathbb{Z}_{\ell}[x]^{\mathbb{N}}$ converge
$\ell$-adically and give explicit rates of convergence.

###### Contents

  1. 1 Introduction
    1. 1.1 A congruence on characteristic polynomials
    2. 1.2 Some questions for future work
  2. 2 On the cohomology of a tower of curves
    1. 2.1 Two families of Iwasawa towers
    2. 2.2 On the distribution of Frobenius eigenvalues in towers of curves
  3. 3 On the convergence of a skew-abelian Iwasawa theoretic invariant
    1. 3.1 A cohomological interpretation
    2. 3.2 A generalization of Fermat’s little theorem to matrices
    3. 3.3 A proof of the main congruences
  4. 4 Explicit examples

######  Notation.

We will work throughout over a fixed finite field $\mathbb{F}_{q}$. A curve
$C$ over $\mathbb{F}_{q}$ refers to a smooth, projective, geometrically
connected scheme of dimension $1$. The base change to the algebraic closure
$\overline{\mathbb{F}}_{q}$ is denoted by $\overline{C}$. We denote its étale
cohomology with $\mathbb{Z}_{\ell}$ coefficients by
$H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell})$. By standard
functoriality arguments, it comes endowed with a linear action of the
geometric Frobenius $\sigma_{q}$. We fix an auxiliary prime $\ell$ throughout
and for simplicity assume that $\ell>2$ and $q\equiv 1\pmod{\ell}$. 333As
usual, the theorems go through if $\ell=2$ with appropriately stronger
hypothesis. For instance, if $\ell=2$ then we need $q\equiv 1\pmod{\ell^{2}}$.

##  1 Introduction

The eigenvalues of the Frobenius on the étale cohomology of a smooth,
projective variety over a finite field carry significant arithmetic
information. By the Weil conjectures, these eigenvalues are algebraic integers
and their absolute values under any complex embedding are understood.

We draw inspiration from Iwasawa theory to study the asymptotic behaviour of
these eigenvalues in an "Iwasawa tower" and in particular, we show that there
is a strong $\ell$-adic convergence statement to be made in many natural
examples. The Iwasawa algebras arising in this study are non-commutative due
to the non trivial action of the Frobenius on this monodromy group and we hope
that this perspective is interesting too. Let us begin with an example.

######  Example 1.

Consider the smooth projective curves $C_{n}$ corresponding to the equations

$Y^{2}=X^{2^{n}}+1\text{ over }\mathbb{F}_{5}.$

They define a tower $\dots\to C_{2}\to C_{1}$ with maps $C_{n+1}\to C_{n}$
defined by $(X,Y)\to(X^{2},Y)$. The characteristic polynomial of $\sigma_{5}$
on $H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})$ is

$f_{n}(x)\coloneqq\det\left(1-\sigma_{2}x|H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})\right)=(1-2x+5x^{2})\prod_{i=1}^{n-2}(1+x^{2^{i}}5^{2^{i-1}})^{2}.$

Note that $f_{n-1}(x)$ divides $f_{n}(x)$ and the inverse of the roots of
$g_{n}(x)=f_{n}(x)/f_{n-1}(x)$ are of the form $\sqrt{5}\zeta$ for $\zeta$ a
root of unity of order $2^{n-1}$ for $n\geq 3$. In Section 4, we show that for
$n$ sufficiently large, the normalized (by $\alpha\to\alpha/|\alpha|$ so that
the complex absolute value is $1$444we note that the complex norm $|\alpha|$
is independent of the embedding to $\mathbb{C}$ by the Weil conjectures) roots
of $g_{n+1}(x)$ are exactly all possible $\ell$-th roots of the normalized
roots of $g_{n}(x)$.

In fact, we prove the same statement for towers of Fermat curves (from which
the above follows) and Artin-Schreier curves. The proof of this statement
follows from realizing the roots of $g_{n}(x)$ as Jacobi sums and using
results of Coleman [C+87] on identities for Gauss sums (coming from the Gross-
Kubota p-adic Gamma function [GK79]).

###  1.1 A congruence on characteristic polynomials

This prompts the question of what happens in a more general context. For
instance, we could take a map $f:C\to\mathbb{P}^{1}$ or $f:C\to A$ for $A$ an
abelian variety of dimension $d$ and pull back by the following diagrams:

${C_{n}}$${\mathbb{P}^{1}}$${or}$${C_{n}}$${A}$${C}$${\mathbb{P}^{1}}$${C}$${A}$$\scriptstyle{\pi_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{t\to
t^{\ell^{n}}}$$\scriptstyle{\pi_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{[\ell^{n}]}$$\scriptstyle{f}$$\scriptstyle{f}$

We denote the first family of examples by Case A and the second family by Case
B. Note that in both the families, the $C_{n}\to C$ are geometrically
(branched) Galois extensions with an abelian Galois group
$G_{n}\cong(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}$ for $b=1$ or $2d$ in Case A
and Case B respectively. Note that the $G_{n}$ themselves have an action of
$\sigma_{q}$ and this will be crucial.

We define
$M_{n}=H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})/H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell})$,
$f_{n}(x)$ to be the characteristic polynomial of $\sigma_{q}$ on $M_{n}$ and
$g_{n}$ to be the characteristic polynomial
$\det\left(1-\sigma_{q}x|M_{n}/M_{n-1}\right)$. It does not seem to be true
that $g_{n}$ determines $g_{n+1}$ as in Example 1. Nonetheless, the following
weaker convergence statement is true.

Let $k_{n}$ be the order of $\sigma_{q}$ acting on
$\mu_{\ell^{n}}=\mathbb{G}_{m}[\ell^{n}]$ in Case A while in Case B, $k_{n}$
is a close relative of the order of $\sigma_{q}$ acting on $A[\ell^{n}]$. In
particular, it is independent of $C$ and can be made completely explicit. In
either case $k_{n}$ is of the form $\max\\{1,\ell^{n-n_{0}}\\}$ with $n_{0}$
depending on which case we are considering.

######  Theorem (Theorem 6).

In the above set up (with some mild assumptions on $f$ and $q$),

  1. 1.

We have a factorization into monic polynomials:

$f_{m}(x)=\prod_{n\leq m}g_{n}(x)$

where the $g_{n}$ are independent of $m$.

  2. 2.

There exist polynomials $h_{n}(y),\tilde{h}_{n}(y)\in\mathbb{Z}[y]$ such that,
in Case A

$g_{n}(x)=h_{n}(x^{k_{n}}).$

while in Case B

$g_{n}(x)=\tilde{h}_{n}(x^{k_{n}}).$

  3. 3.

In Case A: For $n$ sufficiently large so that $k_{n+1}=\ell k_{n}$ (Lemma 7),
we have the $\ell$-adic convergence

$h_{n+1}(y)\equiv h_{n}(y)\pmod{\ell^{n}}.$

In particular, the following $\ell$-adic limit exists in
$\mathbb{Z}_{\ell}[y]$:

$h_{\infty}(y)=\lim_{n\to\infty}h_{n}(y).$

In Case B: For $n\geq n_{0}$ sufficiently large so that $k_{n+1}=\ell k_{n}$,
we have the congruence

$\tilde{h}_{n+1}(y)\equiv\tilde{h}_{n}^{\ell^{(b-1)}}(y)\pmod{\ell^{n}}.$

In particular, the following $\ell$-adic limit exists in
$\mathbb{Z}_{\ell}[y]$ with $\exp,\log$ defined formally as power series:

$\tilde{h}_{\infty}(y)=\exp\left(\lim_{n\to\infty}\frac{1}{\ell^{(n-n_{0})(b-1)}}\log(\tilde{h}_{n}(y))\right).$

The first two properties of the theorem are fairly standard 555As a reviewer
pointed out, Part 2 has been ”known for a long time and rediscovered several
times”, for instance see [Gor79, Lemma 1.1]. For completeness, we give our own
proof too. and follow from understanding the structure of $M_{n}$ as a module
over $\mathbb{Z}_{\ell}[G_{n},\sigma_{q}]$ and in particular, depends on
$\sigma_{q}$ having "large" orbits when acting upon the characters of $G_{n}$.
The main body of the paper proves a more abstract statement (Theorem 9) about
the convergence of certain invariants of a non abelian cohomology group which
implies the third part of the above theorem on the towers of curves.

We note that this more abstract statement can be applied to many more
geometric contexts than just our two examples of towers of curves above
although we do not pursue this in our paper. It applies to any tower of
varieties with an action of an abelian group such that the Frobenius action on
the cohomology has a "large" orbit. For instance, we could take hypersurfaces
of the form

$f(x_{0}^{\ell^{n}},\dots,x_{n}^{\ell^{n}})=0\subset\mathbb{P}^{n}_{\mathbb{F}_{q}}.$

All the interesting cohomology is concentrated in the middle dimensional
cohomology and the above theorem holds for the characteristic polynomial of
the Frobenius action on this middle dimensional cohomology group.

We will see in Section 3 that $M_{\infty}=\varprojlim_{n}M_{n}$ is a free
module for a certain skew-abelian Iwasawa algebra and in particular, the
characteristic polynomials we study are all determined by a Galois cohomology
class with coefficients in matrices over a ring of power series. The bulk of
this paper consists in studying the $\ell$-adic properties of these power
series.

A key role in the study of the study of these algebraic properties is played
by the following generalization of Fermat’s little theorem (conjectured by
Arnold in [Arn06] and proven by Zarelua in [Zar08]):

######  Theorem (Arnold - Zarelua, Theorem 10).

Let $A$ be a $r\times r$ matrix over $\mathbb{Z}_{\ell}$. Then, the congruence

$\operatorname{\mathrm{tr}}(A^{\ell^{n+1}})\equiv\operatorname{\mathrm{tr}}(A^{\ell^{n}})\pmod{\ell^{n+1}}$

holds for any prime $\ell$ and any $n\in\mathbb{N}$.

Arnold’s conjecture goes back to before Arnold (Jänichen [Jan21] and Schur
[Sch37]). For a more recent expository survey and applications to topology and
dynamics, see Zarelua [Zar08]. Arnold’s conjecture has since been proven many
times in the literature (for instance, see [MP10]). We give a new proof666In
the course of writing this paper, we found essentially the same proof by
Qiaochu Yuan in a blog post from 2009. of a slightly refined statement since
we will use a similar technique in proving our main theorem.

To keep notation simple, we state a special (yet non-trivial) case of our
general $\ell$-adic convergence theorem.

######  Theorem (Theorems 8,9).

Let $F(t)$ be a $r\times r$ matrix with entries in $\mathbb{Z}_{\ell}[t]$.
Suppose that $q$ is a prime such that $q-1$ is divisible by $\ell$ but not
$\ell^{2}$. For each $n\geq 1$, we define the matrix

$A_{n}=\prod_{i=1}^{\ell^{n-1}}F(\zeta_{\ell^{n}}^{q^{i}})$

with characteristic polynomial $p_{n}(x)$. Then, the limit
$p_{\infty}(x)=\lim_{n}p_{n}(x)$ exists and we have the congruence

$p_{n+1}(x)\equiv p_{n}(x)\pmod{\ell^{n}}.$

We note that even in the simplest case where $r=1$, the above theorem is not
obvious.

###  1.2 Some questions for future work

We pose a few questions suggested by this work.

######  Question 1.

Our main theorem establishes the existence of $\ell$-adic limits
$h_{\infty}(x),\tilde{h}_{\infty}(x)$ in the two cases. In some simple cases,
the $h_{n}(x)$ are independent of $n$ for $n$ large enough and by the proof of
the Weil conjectures, are known to in fact be polynomials over $\mathbb{Z}$
while a-priori $h_{\infty}(x)$ is only defined over $\mathbb{Z}_{\ell}$.

Are the roots of $h_{\infty}(x)$ always transcendental numbers (except in the
cases where $h_{n}$ is eventually constant)?

######  Question 2.

Even if the roots of $h_{\infty}(x)$ are transcendental, is it possible to
describe them using simple $\ell$-adic transcendental functions?

######  Question 3.

What information about the original morphism $f:C\to\mathbb{P}^{1}$ does
$h_{\infty}(x)$ remember? In the classical Iwasawa theory set up, the limiting
characteristic polynomials turn out to be equal to various $\ell$-adic L
functions up to a unit (by the Main Conjecture of Iwasawa Theory), can we hope
for something similar in this case?

######  Question 4.

Let $(\Lambda,\sigma_{q})$ be as in Section 3 and $M$ a finite, free $\Lambda$
module with a $\sigma_{q}$ semi-linear endomorphism $\Phi:M\to M$. It might be
possible and interesting to completely classify such endomorphisms $\Phi$ in
the hope of a more conceptual proof of the main results. This question is
reminiscent of Manin’s classification of Dieudonne modules ([Man63]). Indeed,
the $(M\otimes_{\Lambda}\Lambda_{n}(v),\Phi)$ form a "compatible" system of an
"$\ell$-adic analogue of Dieudonne modules" over the "compatible" system of
cyclotomic local rings with endomorphism $(\Lambda_{n}(v),\sigma_{q})$ as $n$
varies - this final sentence is purely impressionistic!

######  Question 5.

Let $Q:\mathbb{Z}_{\ell}^{n}\to\mathbb{Z}_{\ell}^{n}$ be a linear automorphism
and for $v\in\mathbb{Z}_{\ell}^{n}$, let $k_{n}(v)$ be the smallest positive
integer so that $Q^{k_{n}(v)}v\equiv v\pmod{\ell^{n}}$. Let
$\lambda:\mathbb{Z}_{\ell}^{n}\to\mathbb{Z}_{\ell}$ be an arbitrary linear
form. Does the sequence

$S_{n}(\lambda,v)\coloneqq\sum_{j=1}^{k_{n}(v)}\zeta_{\ell^{n}}^{\lambda(Q^{-j}v)}$

defined in Remark 9 converge to $0$ as $n\to\infty$? If so, what is the rate
of convergence and is it uniform as $v$ ranges over primitive vectors?

Acknowledgements: I would like to thank my advisor Jordan Ellenberg for posing
a question that led to this paper, feedback on the writing of this paper and
many other useful discussions, Douglas Ulmer for many helpful discussions and
useful feedback on the writing of the paper, John Yin for helping with some
computer calculations.

I am also very grateful to the anonymous referee for helpful expository
suggestions and spotting an error in an earlier version of the proof of Lemma
2 and to Yifan Wei for helping me fix the error.

Outline of the paper: For expository reasons, the paper is not presented in
strictly logical order. Section 3 is independent of the rest of the paper and
its main results (Theorems 8 and 9) are used in proving our main geometric
result (Theorem 6). The reader interested in the geometry and willing to take
the $\ell$-adic analysis on faith can skip Section 3. The reader interested
only in the $\ell$-adic convergence results can skip Section 2.

##  2 On the cohomology of a tower of curves

In this section, we reduce Theorem 6 to an abstract statement about the
convergence of characteristic polynomials of a sequence of matrices.

We fix an odd prime $\ell$ and a finite field $\mathbb{F}_{q}$ with $q$ large
enough to be specified soon. For a variety $X/\mathbb{F}_{q}$, the notation
$H^{i}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell})$ denotes as usual the
étale cohomology of the variety
$X\times_{\mathbb{F}_{q}}\overline{\mathbb{F}}_{q}$ with $\mathbb{Z}_{\ell}$
coefficients. The Frobenius $\sigma_{q}$ acts on it through a linear
automorphism.

###  2.1 Two families of Iwasawa towers

######  Definition 1.

Let $C/\mathbb{F}_{q}$ be a curve. We will be interested in the following two
classes of towers:

  * •

Case A: Given a non-constant map $f:C\to\mathbb{P}^{1}$, we can construct
extensions $\pi_{n}:C_{n}\to C$ by the pull back diagram

${C_{n}}$${\mathbb{P}^{1}}$${C}$${\mathbb{P}^{1}}$$\scriptstyle{\pi_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{t\to
t^{\ell^{n}}}$$\scriptstyle{f}$

The $C_{n}$ form an inverse system with an action by the group

$\Gamma_{n}=\mathbb{Z}/\ell^{n}\mathbb{Z}\rtimes\mathbb{Z}$

where we denote a generator for the first factor by $\theta$ (corresponding to
$\theta(t)=\zeta_{\ell^{n}}t$) and a generator for the second factor by
$\sigma_{q}$ corresponding to the Frobenius operation. They satisfy the
commutation identity

$\sigma_{q}\theta=\theta^{q}\sigma_{q}.$

We require the $\pi_{n}:C_{n}\to C$ to be totally ramified over the preimage
$f^{-1}\left(\\{0,\infty\\}\right)$ \- for instance, this is satisfied if $f$
is unramified over $0,\infty$ or more generally, if the ramification indices
of $f$ over $0,\infty$ are co-prime to $\ell$. This guarantees that the
$C_{n}$ are geometrically irreducible.

  * •

Case B: Given an Abelian variety $A/\mathbb{F}_{q}$ of dimension $d$ and a map
$f:C\to A$, we construct $\pi_{n}:C_{n}\to C$ by the pullback diagram

${C_{n}}$${A}$${C}$${A}$$\scriptstyle{\pi_{n}}$$\scriptstyle{f_{n}}$$\scriptstyle{[\ell^{n}]}$$\scriptstyle{f}$

We require the $C_{n}$ to be geometrically irreducible, this is achieved for
instance if the induced map
$\pi^{\text{\'{e}t}}_{1}(f):\pi^{\text{\'{e}t}}_{1}(C)\to\pi^{\text{\'{e}t}}_{1}(A)$
on the étale fundamental groups is surjective. In this case, the $C_{n}$ are
acted upon by (with $b=2d$)

$\Gamma_{n}=(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}\rtimes\mathbb{Z}.$

The first factor can be identified with $A[\ell^{n}]$ and we denote a basis of
it by $\alpha_{1},\dots,\alpha_{d},\beta_{1},\dots,\beta_{d}$ so that the
Frobenius $\sigma_{q}$ (corresponding to the second factor) acts by a $b\times
b$ matrix $Q$ as

$\sigma_{q}v=Qv$

for $v\in A[\ell^{n}].$ The congruence $Q\equiv\operatorname{I}\pmod{\ell}$ is
equivalent to $\sigma_{q}$ acting as the identity on $A[\ell]$. This can
always be achieved by a finite extension of the base field $\mathbb{F}_{q}$
and we suppose that $q$ is large enough so that
$Q\equiv\operatorname{I}\pmod{\ell}$.777If $\ell=2$, we would need
$Q\equiv\operatorname{I}\pmod{\ell^{2}}$. Note that $1$ is not an eigenvalue
of $Q$ since $\sigma_{q}-1:A\to A$ has finite degree equal to
$A(\mathbb{F}_{q})$.

######  Remark 1.

These aren’t the only cases our main theorem applies to and in fact, we can
even generalize to higher dimensions. What is important is that our tower of
varieties has an action by a pro-$\ell$ abelian group as above and that the
growth in cohomology is "regular" in the tower so that as a module over the
group algebra, the rank of the cohomology groups are constant. For example, we
could take Fermat hypersurfaces of the form

$X_{n}:\sum_{j=0}^{d}x_{j}^{\ell^{n}}=0\subset\mathbb{P}^{b+1}$

with action by $G_{n}=(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}$. The only
interesting cohomology group is in degree $i=b$, in which case it is a rank
$1$ module over $\mathbb{Z}_{\ell}[G_{n}]$ ([And87, Theorem 6] for instance)
and a straightforward variant of Theorem 3 shows that growth in cohomology is
regular.

######  Remark 2.

Note that the automorphism groups $\Gamma_{n}$ aren’t abelian but they are
very close to being abelian, being the extension of an abelian group by the
Frobenius action. Therefore one could view this as an example of skew-abelian
Iwasawa theory.

In the remainder of this subsection we prove some basic results about the
cohomology of these towers (ignoring the Frobenius action initially).

######  Lemma 1.

For a finite extension of curves $f:X\to Y$,
$H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell})$ is a direct factor of
$H^{1}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell})$.

###### Proof.

Since $H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell})$ is dual to the
Tate module $T_{\ell}(Y)$, it suffices to show the corresponding fact for the
Tate modules of $X$ and $Y$, i.e., we need to show that the natural map
$f:T_{\ell}(X)\to T_{\ell}(Y)$ is surjective and that the kernel is torsion
free.

One easily checks the following composite map

$\mathrm{Jac}(Y)\xrightarrow{f^{*}}\mathrm{Jac}(X)\xrightarrow{f_{*}}\mathrm{Jac}(Y)$

is simply multiplication by the degree of $f$, for instance by using an
isomorphism $\mathrm{Jac}(X)\cong\mathrm{Pic}(X)$ and computing the map
explicitly in terms of divisors supported away from the ramification locus.
This shows that the second map is surjective which in turn implies that the
map on Tate modules $T_{\ell}(f):T_{\ell}(X)\to T_{\ell}(Y)$ is surjective.

Moreover, the kernel of $T_{\ell}(f)$ is torsion free since if $[P_{n}]_{n\geq
1}\in T_{\ell}(X)$ mapped to zero, then $P_{n}\in\mathrm{ker}(f)$ which would
imply that $\deg(f)\geq\ell^{n}$ for all $n$ which is a contradiction. ∎

When the extension is generically Galois, we can say more.

######  Lemma 2.

Suppose $f:X\to Y$ is a generically Galois (branched) extension of (smooth,
proper) curves with Galois group $G$. Then,
$H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell})$ is exactly the
submodule of $H^{1}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell})$ fixed by
the $G$ action.

###### Proof.

Let us first suppose that $X,Y$ are not necessarily proper but that $f:X\to Y$
is unramified. By the Hochschild-Serre spectral sequence,

$H^{r}(G,H^{s}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell}))\Longrightarrow
H^{r+s}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell}).$

If we want to let $r+s=1$, then we have either $r=0,s=1$ or $r=1,s=0$. But,
$H^{0}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell})=\mathbb{Z}_{\ell}$ with
the trivial $G$ action and therefore

$H^{1}(G,H^{0}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell}))=\mathrm{Hom}(G,\mathbb{Z}_{\ell})=0$

since $G$ is torsion and $\mathbb{Z}_{\ell}$ is torsion free. This causes the
spectral sequence to degenerate at the $(1,1)$ term and we have the required
isomorphism

$H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell})\cong
H^{0}(G,H^{1}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell})).$

Now, for a general (branched) $f:X\to Y$, let $T\subset Y$ be the ramification
divisor on $Y$ and $f^{-1}(T)=S\subset X$ its preimage in $X$ with
$U=X-S,V=Y-T$. With this set-up, we have following commutative diagram

${{H^{1}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell})}}$${{H^{s}_{\text{\'{e}t}}(\overline{U},\mathbb{Z}_{\ell})}}$${{H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell})}}$${{H^{1}_{\text{\'{e}t}}(\overline{V},\mathbb{Z}_{\ell})}.}$

Note that the cokernels along the horizontal rows have weight $2$ (i.e.,
$\sigma_{q}$ acts by $q$ on the cokernel) as can be seen either from the
excision long exact sequence or from the Lefschetz fixed point theorem for
compactly supported cohomology along with Poincaré duality. On the other hand
$H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell}),H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell})$
are both of weight $1$ (by the Weil conjectures, for instance).

The above diagram is $G$-equivariant since $S,T$ are. Therefore, the
$G$-invariants of $H^{1}_{\text{\'{e}t}}(\overline{X},\mathbb{Z}_{\ell})$ are
contained in $H^{1}_{\text{\'{e}t}}(\overline{V},\mathbb{Z}_{\ell})$ but the
above weight argument shows that it is in fact contained in
$H^{1}_{\text{\'{e}t}}(\overline{Y},\mathbb{Z}_{\ell})$ as required. ∎

Let us return to our specific towers above.

######  Definition 2.

In Case A, let $G_{n}=\mathbb{Z}/\ell^{n}\mathbb{Z}$ with generator $\theta$
while in Case B, let $G_{n}=(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}$ with
generators $\alpha_{i},\beta_{j}$ as discussed before. We also define the
group algebra $R_{n}=\mathbb{Z}_{\ell}[G_{n}]$.

By Lemma 1 and 2,
$M_{n}=H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})/H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell})$
is a free $\mathbb{Z}_{\ell}$ module with an action of $R_{n}$ described by
the following theorem with $g_{0}$ the genus of $C$.

######  Theorem 3.

Let us define

$r=\begin{cases}2g_{0}+s-2&\text{ in {Case A}}\\\ 2g_{0}-2&\text{ in {Case
B}}\end{cases}$

where in Case A, $s$ is the number of preimages of $0,\infty$ for the defining
map $f:C\to\mathbb{P}^{1}$.

As $R_{n}$ modules, we have an exact sequence

$0\to\mathbb{Z}_{\ell}^{r}\to R_{n}^{r}\to M_{n}\to 0.$

where $G_{n}$ acts trivially on the first term.

As a preliminary to the above theorem, we use Riemann-Hurwitz to compute the
dimensions of $M_{n}$.

######  Lemma 4.

Let $g_{n}$ be the genus of $C_{n}$. $M_{n}$ is a free $\mathbb{Z}_{\ell}$
module of rank $2(g_{n}-g_{0})$ and in Case A, we have

$\dim_{\mathbb{Z}_{\ell}}M_{n}=(\ell^{n}-1)(2g_{0}+s-2)$

while in Case B, we have

$\dim_{\mathbb{Z}_{\ell}}M_{n}=(\ell^{bn}-1)(2g_{0}-2)$

###### Proof.

By Lemma 1 and 2, $M_{n}$ is a free $\mathbb{Z}_{\ell}$ module. It remains to
compute its $\mathbb{Z}_{\ell}$ rank ($=2(g_{n}-g_{0})$). In Case A,
let$S_{0},S_{\infty}\subset C(\overline{\mathbb{F}}_{q})$ be the preimages of
$0,\infty$ under $f$ so that $s=|S_{0}|+|S_{\infty}|$. Note that $\pi_{n}$ is
only ramified over $S_{0},S_{\infty}$ and by assumption, it is totally
ramified to order $\ell^{n}$ over these points. By Riemann-Hurwitz, we then
have

$2g_{n}-2=\ell^{n}(2g_{0}-2)+s(\ell^{n}-1)\implies
2(g_{n}-g_{0})=(\ell^{n}-1)(2g_{0}+s-2).$

In Case B, $\pi_{n}$ is unramified and of degree $\ell^{bn}$ and therefore, we
simply have

$2g_{n}-2=\ell^{bn}(2g_{0}-2)\implies 2(g_{n}-g_{0})=(\ell^{bn}-1)(2g_{0}-2).$

∎

We finish the proof of Theorem 3 by using the Lefschetz fixed point theorem to
compute the character of $M$ in terms of fixed points.

###### Proof.

In Case A, let $g\in G_{n}$ be non trivial. Since $g$ is not the identity, the
only points it fixes on $C_{n}$ are the points lying over $0,\infty$ under the
map $C_{n}\to\mathbb{P}^{1}$. In the notation of the previous lemma, there are
$s$ such points in total and the local index at each point is $+1$. Moreover,
$g$ acts trivially on the degree $0,2$ cohomology groups. Therefore, by the
Lefschetz fixed point formula

$\operatorname{\mathrm{tr}}(g|H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell}))=2-s$

and since $G$ acts trivially on $C_{0}$,

$\operatorname{\mathrm{tr}}(g|H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell}))-\operatorname{\mathrm{tr}}(g|H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell}))=-(2g_{0}+s-2)=-r.$

On the other hand, the identity $\mathrm{id}\in G_{n}$ of course acts
trivially so that

$\operatorname{\mathrm{tr}}(\mathrm{id}|H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell}))-\operatorname{\mathrm{tr}}(\mathrm{id}|H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell}))=2(g_{n}-g_{0})=r(\ell^{n}-1)$

where the final equality is by the previous lemma.

In Case B, any $g\neq\mathrm{id}\in G_{n}$ acts on the abelian variety $A$ by
a non trivial translation and hence has no fixed points on either $C_{n}$ or
$A$. As before, by the Lefschetz fixed point theorem

$\operatorname{\mathrm{tr}}(g|H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell}))-\operatorname{\mathrm{tr}}(g|H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell}))=2-2g_{0}=-r.$

The identity element has trace equal to

$\dim H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})-\dim
H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell})=2(g_{n}-g_{0})=r(\ell^{bn}-1).$

If we then examine the exact sequence

$0\to\mathbb{Z}_{\ell}^{r}\to R_{n}^{r}\to X\to 0,$

we see that $X$ has the character we computed above in both cases for $M_{n}$
proving that $X\cong M_{n}$ as $G_{n}$ representations.

∎

######  Remark 3.

As an immediate corollary of the above theorem, we notice that in both Case A
and B, for every non trivial character
$\chi:G_{n}\to\overline{\mathbb{Z}}^{\times}$, the corresponding eigenspace
$M_{n}(\chi)$ of $M_{n}\otimes\mathbb{Z}_{\ell}[\zeta_{\ell^{n}}]$ is of
dimension $2g_{0}+s-2$ and $2g_{0}-2$ in the two cases respectively. In
particular it is independent of $n$ and we call the characters appearing in
$P_{n}=M_{n}/M_{n-1}$ "new" or "primitive" characters of level $n$.

We fix a set of generators $t_{1},\dots,t_{b}$ for
$G_{n}\cong(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}$ and identify characters $\chi$
of $G_{n}$ by vectors
$v=(v_{1},\dots,v_{b})\in(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}$ by defining
$\chi_{v}(t_{i})=t_{i}^{v_{i}}.$ Under this identification, primitive
characters correspond exactly to primitive vectors as defined below in
Definition 3. We denote the eigenspace of $\chi_{v}$ by $M_{n}(v)$.

The exact sequence in the above theorem implies that $M_{n}$ is not a free
$R_{n}$ module but nevertheless, the inverse limit
$M_{\infty}\coloneqq\lim_{n}M_{n}$ is a free module over
$\Lambda=\mathbb{Z}_{\ell}[[T_{1},\dots,T_{b}]]=\lim_{n}R_{n}$.

######  Lemma 5.

Let $\theta_{1},\dots,\theta_{b}$ be the generators of
$G_{n}=\left(\mathbb{Z}/\ell^{n}\mathbb{Z}\right)^{b}$ as above. Then the
projective limit $M_{\infty}\coloneqq\varprojlim_{n}M_{n}$ is a free module of
rank $r$ over $\Lambda=\mathbb{Z}_{\ell}[\\![T_{1},\dots,T_{b}]\\!]$. The
Frobenius $\sigma_{q}$ acts _semi_ -linearly on $M_{\infty}$, i.e.,
$\sigma_{q}$ is $\mathbb{Z}_{\ell}$ linear and satisfies

$\sigma_{q}\circ(1+T_{i})={\sigma_{q}}(1+T_{i})\circ\sigma_{q}$

where we identify $1+T_{i}$ with $\theta_{i}$ so that $\sigma_{q}$ acts on
$1+T_{i}$ through its action on $\varprojlim_{n}G_{n}$.

###### Proof.

By the above theorem, we have the following identification as
$\mathbb{Z}_{\ell}[G_{n}]$-modules

$M_{n}\cong\left(\frac{\mathbb{Z}_{\ell}[\theta_{1},\dots,\theta_{b}]}{(\theta_{1}^{\ell^{n}}=1,\dots,\theta_{b}^{\ell^{n}}=1,\prod_{i=1}^{b}\left(\sum_{j=0}^{\ell^{n}-1}\theta_{i}^{j}\right)}\right)^{r}$

since $\prod_{i=1}^{b}\left(\sum_{j=0}^{\ell^{n}-1}\theta_{i}^{j}\right)$
generates the unique $1$-dimensional $\mathbb{Z}_{\ell}$ submodule of
$\mathbb{Z}_{\ell}[G_{n}]$ with trivial $G_{n}$ action. Using this explicit
presentation, we define a map

$\Lambda^{r}=\left(\mathbb{Z}_{\ell}[\\![T_{1},\dots,T_{b}]\\!]\right)^{r}\to
M_{\infty}$

by mapping, for each factor, the $T_{i}\to\theta_{i}-1$ in each term in the
projective limit. We will prove that this map is an isomorphism. Since the map
is defined on each factor, we can assume henceforth that $r=1$. The kernel of
the induced map to $M_{n}$ is generated by the elements

$(1+T_{i})^{\ell^{n}}-1=\sum_{j=1}^{\ell^{n}}\binom{\ell^{n}}{j}T_{i}^{j}\text{
for }i=1,\dots,b$

and

$\prod_{i=1}^{b}\left(\frac{(1+T_{i})^{\ell^{n}}-1}{T_{i}}\right)=\prod_{i=1}^{b}\left(\sum_{j=1}^{\ell^{n}}\binom{\ell^{n}}{j}T_{i}^{j}\right).$

As $n\to\infty$, these elements tend to $0$ in the
$(\ell,T_{1},\dots,T_{b})$-adic topology of $\Lambda$ so that the map
$\Lambda\to M_{\infty}$ is injective. On the other hand, surjectivity is also
clear since the $\theta_{i}$ generate $G_{n}$, and consequently the
$\theta_{i}-1$ generate $\mathbb{Z}[G_{n}]$. The Frobenius action is induced
through this morphism, thus completing the proof. ∎

###  2.2 On the distribution of Frobenius eigenvalues in towers of curves

In this subsection, we prove that the characteristic polynomials $f_{n}(x)$ of
$\sigma_{q}$ on $H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})$ in
our two cases satisfy some striking congruences. We will treat the cases
uniformly by letting $Q=q,b=1$ in Case A.

######  Definition 3.

For $R$ a discretely valued ring (DVR) or a quotient of a DVR, we call $v\in
R^{b}$ primitive if at least one of its co-ordinates is a unit. We denote the
space of primitive vectors by $\mathcal{P}(R^{b})$.

For a primitive vector $v\in
H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})$, we define
$k_{n}(v)$ to be the smallest postive integer such that $Q^{k_{n}(v)}v\equiv
v\pmod{\ell^{n}}$. We define $k_{n}$ to be the minimum of $k_{n}(v)$ as $v$
ranges over primitive vectors. Lemma 7 shows the existence of a positive
integer $\beta_{v}$ such that $k_{n}(v)=\ell^{n-\beta_{v}}$ for
$n\geq\beta_{v}$. Moreover, $n_{0}=\max_{v\text{ primitive}}\beta_{v}$ is
finite so that $k_{n}=\ell^{n-n_{0}}$ for $n\geq n_{0}$.

######  Theorem 6.

Let $C_{n}$ be as in Case A or B of Definition 1 and

$f_{n}(x)=\det\left(1-\sigma_{q}x|M_{n}\right)$

be the characteristic polynomial of the Frobenius $\sigma_{q}$ acting on
$M_{n}=H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})/H^{1}_{\text{\'{e}t}}(\overline{C},\mathbb{Z}_{\ell})$.
It satisfies the following properties:

  1. 1.

We have a factorization into monic polynomials:

$f_{m}(x)=\prod_{n\leq m}g_{n}(x)$ (1)

where the $g_{n}$ are independent of $m$.

  2. 2.

There exist polynomials $h_{n}(y),\tilde{h}_{n}(y)$ such that, in Case A

$g_{n}(x)=h_{n}(x^{k_{n}}).$ (2)

while in Case B

$g_{n}(x)=\tilde{h}_{n}(x^{k_{n}}).$ (3)

  3. 3.

In Case A: For $n\geq n_{0}$ (Lemma 7), we have the $\ell$-adic convergence

$h_{n+1}(y)\equiv h_{n}(y)\pmod{\ell^{n}}.$ (4)

In particular, the following $\ell$-adic limit exists in
$\mathbb{Z}_{\ell}[y]$:

$h_{\infty}(y)=\lim_{n\to\infty}h_{n}(y).$

In Case B: For $n\geq n_{0}$, we have the congruence

$\tilde{h}_{n+1}(y)\equiv\left(\tilde{h}_{n}(y)\right)^{\ell^{(b-1)}}\pmod{\ell^{n}}.$
(5)

In particular, the following $\ell$-adic limit exists in
$\mathbb{Z}_{\ell}[y]$ with $\exp,\log$ defined formally as power series:

$\tilde{h}_{\infty}(y)=\exp\left(\lim_{n\to\infty}\frac{1}{\ell^{(n-n_{0})(b-1)}}\log(\tilde{h}_{n}(y))\right).$

######  Remark 4.

The Frobenius $\sigma_{q}$ is known to act semi-simply on the étale cohomology
of a curve888Semi-simplicity for abelian varieties and conjectured to act
semi-simply with rational coefficients on any variety over $\mathbb{F}_{q}$.
While the following proof simplifies slightly if we use the semi-simplicity of
$\sigma_{q}$ on $M_{n}$, we do not assume this so that the following proof can
be adapted more easily to cases where semi-simplicity is not known.

###### Proof.

Part $1$, i.e., Equation 1 is an immediate consequence of Lemma 1 once we
define $g_{n}(x)$ to be the characteristic polynomial of $\sigma_{q}$ on
$P_{n}=M_{n}/M_{n-1}$.

To prove Part $2$, i.e., Equations 2 and 3, we treat the two cases
simultaneously by taking $b=1,Q=q$ in Case A. Recall the notation that, for
$v\in\mathbb{Z}_{\ell}^{b}$, $M_{n}(v)$ is the eigenspace of $G_{n}$ for the
character $\chi_{v}(t_{i})=t_{i}^{v_{i}}$. The eigenspaces $M_{n}(v)$ get
permuted by $\sigma_{q}$ in the following manner:

$\sigma_{q}:M_{n}(v)\to M_{n}(Q^{-1}v)$

and therefore $\sigma_{q}^{k_{n}(v)}$ is an automorphism of $M_{n}(v)$. We
will prove that a Jordan block of $\sigma_{q}^{k_{n}(v)}$ acting on
$M_{n}(v)\subset P_{n}$ (with eigenvalue $\lambda\neq 0$) corresponds to
$k_{n}(v)$ distinct Jordan blocks of $\sigma_{q}$ acting on $P_{n}$ (with
eigenvalues $\mu^{1/k_{n}(v)}$). Since this claim is independent of passing to
an extension, we replace $P_{n}$ by
$P_{n}\otimes_{\mathbb{Z}_{\ell}}\overline{Q}_{\ell}$.

To that end, let $m_{1},\dots,m_{s}\in M_{n}(v)$ be some generalized
eigenvectors of $\sigma_{q}^{k_{n}}$ corresponding to a pure Jordan block of
eigenvalue $\lambda$ (possibly defined over an extension $\mathbb{Z}_{\ell}$)
so that

$\sigma_{q}^{k_{n}}(m_{i+1})=\lambda m_{i+1}+m_{i}$

(with the convention that $m_{0}=0$.) We will first show that the eigenvector
$m_{1}$ for $\sigma_{q}^{k_{n}(v)}$ corresponds to $k_{n}(v)$ distinct
eigenvectors for $\sigma_{q}$. For $m_{i}\in M_{n}(v)$, let
$m_{i,j}=\sigma_{q}^{j-1}(m_{i})$ for $j=1,\dots,k_{n}(v)$. Note that

$\sigma_{q}(m_{i+1,k_{n}(v)})=\sigma_{q}^{k_{n}(v)}(m_{i+1})=\lambda
m_{i+1,1}+m_{i,1}.$

For each $\mu$ a $k_{n}$ root of $\lambda$,
$n_{\mu}=\sum_{j=1}^{k_{n}(v)}\mu^{-j}m_{1,j}$ is an eigenvector of
$\sigma_{q}$. Indeed, we have

$\sigma_{q}(n_{\mu})=\sum_{j=1}^{k_{n}(v)-1}\mu^{-j}m_{1,j+1}+\lambda\mu^{-k_{n}(v)}m_{1,1}=\mu
n_{\mu}.$

Therefore, the $n_{\mu}$ are each an eigenvector of $\sigma_{q}$ and the
subspace $N=\mathrm{span}(n_{\mu}:\mu^{k_{n}(v)}=\lambda)$ is stable under
$\sigma_{q}$ and contains

$m_{1,j}=\frac{1}{k_{n}(v)}\sum_{\mu^{k_{n}(v)}=\lambda}\mu^{j}n_{\mu}\ \ \ \
\ \ \text{ for }j=1,\dots,k_{n}(v).$

Passing to the quotient $P_{n}/N$ therefore corresponds to replacing the
$m_{1},\dots,m_{s}$ by $m_{2},\dots,m_{s}$ (with $m_{2}$ now an eigenvalue of
$\sigma_{q}^{k_{n}(v)}$) and we continue inductively to show that each $m_{i}$
corresponds to $k_{n}(v)$ distinct generalized eigenvectors $n_{i,\mu}$ with
eigenvalue $\mu$.

Let $g_{n,v}(x)=\det(\operatorname{I}-\sigma_{q}x)$ be the characteristic
polynomial of $\sigma_{q}$ on
$N_{n}(v)=\bigoplus_{i=0}^{k_{n}(v)-1}M_{n}(Q^{i}(v))$. This module has
dimension exactly $k_{n}(v)$ times the dimension of $M_{n}(v)$ and since for
each generalized eigenvector $m_{i}$ of $M_{n}(v)$, we have constructed
$k_{n}(v)$ distinct generalized eigenvectors $n_{i,\mu}$ of $N_{n}(v)$
corresponding to the $k_{n}(v)$ distinct roots of $\lambda$, the $n_{i,\mu}$
together in fact span $N_{n}(v)$.

The identity

$\prod_{j=1}^{k_{n}(v)}(1-x\mu\zeta_{k_{n}(v)}^{j})=1-x^{k_{n}(v)}\mu^{k_{n}(v)}$

then shows that $g_{n,v}(x)=h_{n,v}(x^{k_{n}(v)})$ for some polynomial
$h_{n,v}(y)$ with roots $y=\lambda=\mu^{k_{n}(v)}$. We note that the above
proof in fact computes the $h_{n,v}(x)$ to be exactly the characteristic
polynomial of $\sigma_{q}^{k_{n}(v)}$ on $M_{n}(v)$. Since

$g_{n}(x)=\prod_{v\in\mathcal{P}(\mathbb{Z}/\ell^{n}\mathbb{Z})/\sim}g_{n,v}(x)$

where the product is over a set of representatives for the $\sigma_{q}$ action
on primitive vectors, the proof of Part (2) in Case A is completed by defining

$h_{n}(y)=\prod_{v\in\mathcal{P}(\mathbb{Z}/\ell^{n}\mathbb{Z})/\sim}h_{n,v}(y)$

and setting $y=x^{k_{n}}$.

For Case B, we define (again as a product over a similar set of
representatives for the $\sigma_{q}$ action on primitive vectors)

$\tilde{h}_{n}(y)=\prod_{v\in\mathcal{P}(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}/\sim}h_{n,v}(y^{k_{n}(v)/k_{n}})$

so that (with $y=x^{k_{n}}$)

$g_{n}(x)=\prod_{v\in\mathcal{P}(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}/\sim}g_{n,v}(x)=\prod_{v\in\mathcal{P}(\mathbb{Z}/\ell^{n}\mathbb{Z})^{b}/\sim}h_{n,v}(x^{k_{n}(v)})=\tilde{h}_{n}(x^{k_{n}}).$

Finally, we prove Part (3), i.e., equations 4 and 5. Let us fix a generating
set $m_{1},\dots,m_{r}$ for $M_{n}$ over
$\mathbb{Z}_{\ell}[G_{n}]\cong\mathbb{Z}_{\ell}[t_{1},\dots,t_{b}]/(t_{i}^{\ell^{n}}-1:i=1,\dots,b)$.
Since $M_{n}$ is not a free $\mathbb{Z}_{\ell}[G_{n}]$ module, it might not be
completely clear what a generating set should mean. For our purposes, it
suffices to choose $m_{1},\dots,m_{r}$ so that under any specialization that
maps the $t_{i}$ to $\ell^{n}$ roots of unity, the $m_{i}$ specialize to a
genuine basis over the induced specialization of $M_{n}$. That this is indeed
possible follows from the explicit description of the $M_{n}$ as $G_{n}$
modules in Lemma 5. Such a specialization corresponds to a representation
$\chi_{v}:G_{n}\to\overline{\mathbb{Q}}$ for $v\in\mathbb{Z}_{\ell}^{b}$ and
we denote the induced specialization also by $\chi_{v}:M_{n}\to M_{n}(v)$

In terms of the $m_{1},\dots,m_{r}$, $\sigma_{q}$ acting on $M_{n}$ can be
represented by some invertible matrix $F(t_{1},\dots,t_{b})$. From this point
on, we will be concerned only with this matrix $F(t_{1},\dots,t_{b})$. Since
$\sigma_{q}$ skew commutes with the $t_{i}$, we have

$\sigma_{q}^{k_{n}(v)}=\prod_{i=1}^{k_{n}(v)}F(t^{Q^{k_{n}(v)-i}v}).$

Therefore, with respect to the basis $\chi_{v}(m_{1}),\dots,\chi_{v}(m_{r})$
of $M_{n}(v)$, the action of $\sigma_{q}^{k_{n}(v)}$ corresponds to evaluating
the above product using the character $\chi_{v}$ and is represented by the
matrix

$A_{n}(v)=\prod_{i=1}^{k_{n}(v)}F(\zeta_{\ell^{n}}^{Q^{k_{n}(v)-i}v})$

of Section 3 (and we note that $A_{n}(v)$ is independent of our choice of $F$
or the $m_{1},\dots,m_{r}$). As noted above, the $h_{n,v}(y)$ are the
characteristic polynomials of $\sigma_{q}^{k_{n}(v)}$ on $M_{n}(v)$ and
therefore, correspond to the $p_{n,v}(y)$ in Section 3. We further see that
the $\tilde{h}_{n}(y)$ correspond to the polynomials $r_{n}(y)$ of Theorem 9
and by this theorem, we have the required congruence:

$\tilde{h}_{n+1}(y)\equiv\tilde{h}_{n}(y)\pmod{\ell^{n}}.$

∎

##  3 On the convergence of a skew-abelian Iwasawa theoretic invariant

In this section, we prove a general, abstract result about the convergence of
a certain cohomological invariant defined for a skew commutative Iwasawa
algebra. The set up is as follows.

We fix an odd999As usual, the arguments of this paper go through if $\ell=2$
with minor, standard modifications. prime $\ell$ and positive integers $b,r$
throughout this section. All cohomology groups in this section represent group
cohomology unless indicated otherwise. All congruences in this paper are in
$\mathbb{Z}_{\ell}$ (and hence only concerned with the $\ell$-adic valuation)
unless explicitly mentioned otherwise.

Let $\Lambda=\mathbb{Z}_{\ell}[[T_{1},\dots,T_{b}]]$ be the $b$ dimensional
Iwasawa algebra and set $t_{i}=1+T_{i}$. It is a local ring with maximal ideal
$\mathfrak{m}=(\ell,T_{1},\dots,T_{b})$. Note that for
$\lambda\in\mathbb{Z}_{\ell}$, the expression

$t_{i}^{\lambda}=(1+T_{i})^{\lambda}=\sum_{k\geq
0}\binom{\lambda}{k}T_{i}^{k}$

converges in $\Lambda$. For $v=(v_{1},\dots,v_{b})\in(\mathbb{Z}_{\ell})^{b}$,
we define $t^{v}=(t_{1}^{v_{1}},\dots,t_{b}^{v_{b}}).$ We suppose that
$\Lambda$ has an endomorphism $\sigma_{q}$ acting through a matrix
$Q=Q_{ij}\in\operatorname{\mathrm{GL}}_{b}(\mathbb{Z}_{\ell})$ in the
following way:

$\sigma_{q}(t^{v})=t^{Qv}\iff\sigma_{q}(T_{i})=\left[\prod_{j}(1+T_{j})^{Q_{ji}}\right]-1\text{
for all }i.$

We note that the action is well defined since
$\sigma_{q}(T_{i})\in\mathfrak{m}$. For $v\in\mathbb{Z}_{\ell}^{b}$, we denote
the size of the orbit of $v$ under $Q$ in
$(\mathbb{Z}_{\ell}/\ell^{n}\mathbb{Z}_{\ell})^{b}$ by $k_{n}(v)$.101010i.e.,
$Q^{k_{n}(v)}v\equiv v\pmod{\ell^{n}}$ and $k_{n}(v)$ is the least such
positive integer. We also define

$k_{n}=\min_{v\text{ primitive}}k_{n}(v).$

Assumption: We suppose henceforth that $Q\equiv\operatorname{I}\pmod{\ell}$
and that $Q$ fixes no vectors111111If $\ell=2$, then we would need to assume
that $Q\equiv 1\pmod{4}$..

######  Lemma 7.

Let $v$ be a primitive vector. Then there exist integers $\alpha\geq
1,\beta_{v}\geq 0$ so that

$k_{n}(v)=\begin{cases}\ell^{n-\alpha-\beta_{v}}&\text{ if
}n\geq\alpha+\beta_{v}\\\ 1&\text{ otherwise.}\end{cases}$

Moreover, there is some (minimal) $\beta_{0}$ such that
$\beta_{v}\leq\beta_{0}$ for all primitive $v$.

In particular, we have

$k_{n}=\begin{cases}\ell^{n-\alpha-\beta_{0}}&\text{ if }n\geq
n_{0}\coloneqq\alpha+\beta_{0}\\\ 1&\text{ otherwise.}\end{cases}$

###### Proof.

Since $Q\equiv\operatorname{I}\pmod{\ell}$, we have $\log Q=\ell^{\alpha}X$
for $\alpha\geq 1$ with $X\in M_{b}(\mathbb{Z}_{\ell})$ not divisible by
$\ell$. Since $\ell\geq 3$,

$(Q^{m}-\operatorname{I})v=\exp(m\log
Q)v-v=m\ell^{\alpha}Xv+\frac{(m\ell^{\alpha})^{2}}{2}X^{2}v+\dots.$

Since $Q$ does not fix any vectors, $Xv\neq 0$ so let $\beta_{v}$ be the
largest value such that $Xv\equiv 0\pmod{\ell^{\beta_{v}}}$. We see that
$k_{n}(v)$ is the smallest $m$ so that $(Q^{m}-\operatorname{I})v$ is
divisible by $\ell^{n}$. Since $X^{k}v\equiv 0\pmod{\ell^{\beta_{v}}}$ for any
$k\geq 1$ too, the $\ell$-adic valuation of $(Q^{m}-\operatorname{I})v$ is
determined by the leading term $m\ell^{\alpha}Xv$ so that

$k_{n}(v)=\begin{cases}\ell^{n-\alpha-\beta_{v}}&\text{ if
}n\geq\alpha+\beta_{v}\\\ 1&\text{ otherwise.}\end{cases}$

It remains to show that there is a uniform upper bound on $\beta_{v}$.

Let $\pi:\mathbb{Z}_{\ell}^{b}\to\mathbb{F}_{\ell}^{b}$ be the reduction map.
The primitive vectors correspond to the subspace
$\mathcal{P}=\pi^{-1}(\mathbb{F}_{\ell}^{b}-\\{0\\})$ which is a closed (and
open) subset of $\mathbb{Z}_{\ell}^{b}$. Therefore $\mathcal{P}$ is compact
and by continuity of multiplication by $X$,

$X\mathcal{P}=\\{Xv:v\in\mathcal{P}\\}\subset\mathbb{Z}_{\ell}^{b}$

is compact and closed too. By assumption on $Q$, $X\mathcal{P}$ does not
contain $0$ (since this would correspond to a fixed point of $Q$). This
implies that $X\mathcal{P}$ is in fact bounded away from $0$, i.e, there is
some minimal $\beta_{0}$ so that the image of $X\mathcal{P}$ in
$(\mathbb{Z}/\ell^{\beta_{0}+1}\mathbb{Z}_{\ell})^{b}$ does not contain $0$ so
that $\beta_{v}\leq\beta_{0}$ for every primitive $v$ (and
$\beta_{0}=\beta_{v}$ for some primitive $v$).

∎

######  Remark 5.

It is easy to see why we need to restrict to $v$ primitive and to $Q$ not
having any fixed vectors. If $Qv=v$, then $k_{n}(v)=1$ and if
$v=\ell^{s}v_{0}$, then $k_{n}(v)=1$ for $n\leq s$ which is an obstruction to
a uniform bound on $n$.

###  3.1 A cohomological interpretation

Let $M$ be a free $\Lambda$ module of rank $r$ with a $\Lambda$-linear
endomorphism $\Phi:M\to M$. Upon picking a basis $m_{1},\dots,m_{r}$ for $M$,
we express $\Phi$ as a matrix $F(T_{1},\dots,T_{b})$ with entries in
$\Lambda$. We suppose that $\Phi$ skew commutes with $\sigma_{q}$ in the
following sense:

$\sigma_{q}\circ
F=F(\sigma_{q}(T_{1}),\dots,\sigma_{q}(T_{b}))\circ\sigma_{q}.$

Note that $\sigma_{q}$ acts on $\operatorname{\mathrm{GL}}_{r}(\Lambda)$
through its action on $\Lambda$. This data of $M$ and the endomorphism $\Phi$
as above gives rise to an element $\eta$ in the non abelian cohomology group
$H^{1}(\mathbb{Z}\sigma_{q},\operatorname{\mathrm{GL}}_{r}(\Lambda))$ in the
following way:

Given a $F$ as above, we can define a cocycle representative by
$\eta(\sigma_{q})=F\in\operatorname{\mathrm{GL}}_{r}(\Lambda).$ A change of
basis by a matrix $P\in\operatorname{\mathrm{GL}}_{r}(\Lambda)$ corresponds to
$F\to P(\sigma_{q}(T_{1}),\dots,\sigma_{q}(T_{b}))FP^{-1}$ which is exactly
the boundary action. Therefore, the cohomology class $\eta\in
H^{1}(\mathbb{Z}\sigma_{q},\operatorname{\mathrm{GL}}_{r}(\Lambda))$ depends
only on $(M,\Phi)$.

For a positive integer $n$ and
$v=(v_{1},\dots,v_{b})\in\mathbb{Z}_{\ell}^{b}$, note that since
$T_{i}=\zeta_{\ell^{n}}^{v_{i}}-1$ is in the maximal ideal of
$\mathbb{Z}_{\ell}[\zeta_{\ell^{n}}]$, we can define the quotient

$\Lambda_{n}(v)=\frac{\mathbb{Z}_{\ell}[[T_{1},\dots,T_{b}]]}{(t_{1}=\zeta_{\ell^{n}}^{v_{1}},\dots,t_{b}=\zeta_{\ell^{n}}^{v_{b}})}.$

We note that $\sigma_{q}^{k_{n}(v)}$ fixes the ideal
$(t_{1}-\zeta_{\ell^{n}}^{v_{1}},\dots,t_{b}-\zeta_{\ell^{n}}^{v_{b}})\subset\mathbb{Z}_{\ell}[[T_{1},\dots,T_{b}]]$
and thus descends to an endomorphism of $\Lambda_{n}(v)$.

Henceforth, we fix $\eta\in
H^{1}(\mathbb{Z}\sigma_{q},\operatorname{\mathrm{GL}}_{r}(\Lambda)),v\in\mathbb{Z}_{\ell}^{b}$
and define the following sequence of invariants (implicitly depending on
$\eta$) taking values in polynomials in one variable:

$p_{n,-}(y):v\in
H^{1}(\mathbb{Z}\sigma_{q},\operatorname{\mathrm{GL}}_{r}(\Lambda))\xrightarrow{\mathrm{restriction}}H^{1}(\mathbb{Z}\sigma_{q}^{k_{n}(v)},\operatorname{\mathrm{GL}}_{r}(\Lambda_{n}(v)))\xrightarrow{\text{char
poly}}\Lambda_{n}(v)[y]\ni p_{n,v}(y)$

where for the first map, we restrict along
$\mathbb{Z}\sigma_{q}^{k_{n}(v)}\subset\mathbb{Z}\sigma_{q}$ and push forward
along the quotient
$\operatorname{\mathrm{GL}}_{r}(\Lambda)\to\operatorname{\mathrm{GL}}_{r}(\Lambda_{n}(v))$
and for the second map, since $\sigma_{q}^{k_{n}(v)}$ acts trivially on
$\operatorname{\mathrm{GL}}_{r}(\Lambda_{n}(v))$, we have

$H^{1}(\mathbb{Z}\sigma_{q}^{k_{n}(v)},\operatorname{\mathrm{GL}}_{r}(\Lambda_{n}(v)))=\mathrm{Hom}(\mathbb{Z}\sigma_{q}^{k_{n}(v)},\operatorname{\mathrm{GL}}_{r}(\Lambda_{n}(v)))/\text{conjugacy}=\operatorname{\mathrm{GL}}_{r}(\Lambda_{n}(v))/\text{conjugacy}$

which shows that the characteristic polynomial is a well defined invariant.
Tracing through the definition in terms of the value of $F=\eta(\sigma_{q})$
for $\eta\in
H^{1}(\mathbb{Z}\sigma_{q},\operatorname{\mathrm{GL}}_{r}(\Lambda))$,
$p_{n,v}(y)$ has the following explicit formula. For
$v\in\mathbb{Z}_{\ell}^{b}$, we denote
$F(t_{1}=\zeta_{\ell^{n}}^{v_{1}},\dots,t_{b}=\zeta_{\ell^{n}}^{v_{b}})$ by
$F(\zeta_{\ell^{n}}^{v})$ and define

$A_{n}(v)\coloneqq F(\zeta_{\ell^{n}}^{Q^{k_{n}(v)-1}v})\dots
F(\zeta_{\ell^{n}}^{v})=\prod_{i=1}^{k_{n}}F(\zeta_{\ell^{n}}^{Q^{-i}v})$ (6)

where we implicitly use that $Q^{k_{n}(v)}v\equiv v\pmod{\ell^{n}}$ for the
second equality. The characteristic polynomial of $A_{n}(v)$ is exactly

$p_{n,v}(y)=\det(\operatorname{I}-yA_{n}(v)).$

Equivalently, it is the characteristic polynomial of $\sigma_{q}^{k_{n}(v)}$
acting on $\Lambda_{n}(v)$.

As the main results of this section, we will prove two $\ell$-adic convergence
results regarding the sequence of polynomials $p_{n,v}(y)$ as $n\to\infty$.

######  Theorem 8.

Suppose that $Q=q\operatorname{I}$ is a scalar matrix. For $n$ sufficiently
large so that $k_{n+1}=\ell k_{n}$, the characteristic polynomials satisfy the
congruence

$p_{n+1,v}(y)\equiv p_{n,v}(y)\pmod{k_{n+1}}.$

######  Remark 6.

Unfortunately, this strong congruence is not true in general if $Q$ is not
scalar (even if $r=1$) as the following example shows. Take
$\ell=5,q_{1}=6,q_{2}=11$ and let $Q$ be the diagonal matrix with entries
$q_{1},q_{2}$. Take $F(t_{1},t_{2})=1+t_{1}^{3}t_{2}$ and
$v=(1,1)\in\mathbb{Z}_{\ell}^{2}$. Computation shows that
$A_{3}(v)=49,A_{2}(v)=7$ so that the difference is only divisible by $7$ and
not $k_{3}=49$ as the above theorem would suggest. Nevertheless, the
computational evidence also suggests that the $A_{n}(v)$ still converge, just
with a slower rate of convergence. As we will see in Remark 9, this will be
related to the vanishing of certain sums of roots of unity.

For our geometric applications, the following statement is sufficient. Recall
that $\mathcal{P}((\mathbb{Z}/\ell^{n}\mathbb{Z})^{b})$ denotes the space of
primitive vectors. It is acted upon by $Q$ and we denote a set of
representatives for the orbits of $Q^{\mathbb{Z}}$ acting on
$\mathcal{P}((\mathbb{Z}/\ell^{n}\mathbb{Z})^{b})$ by
$\mathcal{P}((\mathbb{Z}/\ell^{n}\mathbb{Z})^{b})/\sim$. For $v^{\prime}=Qv$,
we note that $p_{n,v}\equiv p_{n,v^{\prime}}$ so that $p_{n,v}$ is independent
of the choice of representative. The following polynomial depends only on the
class $\eta$.

######  Definition 4.

With $A_{n}(v)$ and $p_{n,v}$ as before, define

$r_{n}(y)\coloneqq\prod_{v\in\mathcal{P}((\mathbb{Z}/\ell^{n}\mathbb{Z})^{b})/\sim}p_{n,v}(y^{k_{n}(v)/k_{n}}).$

######  Theorem 9.

Let $Q$ be any matrix in the kernel of
$\operatorname{\mathrm{GL}}_{b}(\mathbb{Z}_{\ell})\to\operatorname{\mathrm{GL}}_{b}(\mathbb{F}_{\ell})$.
For $n\geq n_{0}$ so that $k_{n+1}=\ell k_{n}$, We have

$r_{n+1}(y)\equiv r_{n}^{\ell^{b-1}}(y)\pmod{\ell^{n}}.$

If $Q=q\operatorname{I}$, we have the stronger congruence

$r_{n+1}(y)\equiv r_{n}^{\ell^{b-1}}(y)\pmod{\ell^{nb}}.$

######  Remark 7.

When $b=1$, the two bounds agree since all matrices are scalar! Note that
Theorem 8 only implies the following weaker congruence for $b=1$:

$r_{n+1}(y)\equiv r_{n}(y)\pmod{k_{n+1}}.$

######  Remark 8.

Numerical evidence shows that these congruences are in fact sharp and the
bounds in Theorems 8 and 9 are realized in most cases (but not always!). For
instance, with $r=1,b=2,\ell=3$ and $Q=(1+\ell^{2})\operatorname{I}$ a scalar
matrix, the computation

$A_{3}\left(\frac{1}{1-\ell},\frac{1}{1-\ell}\right)-A_{2}\left(\frac{1}{1-\ell},\frac{1}{1-\ell}\right)=70\ell$

shows the sharpness of Theorem 8. The same example also shows the sharpness of
part $2$ of Theorem 9. Let $d\geq 1$ and
$\tau_{3},\tau_{2}\in\mathbb{Z}_{\ell}$ so that $r_{3}(y)=1-\tau_{3}y+\dots$
and $r_{2}^{\ell}(y)=1-\tau_{2}y+\dots$. Then

$\tau_{3}-\ell\tau_{2}=560\ell^{4}.$

Both the theorems above will depend on the following generalization of
Fermat’s little theorem to matrices to deal with the case when $r\geq 1$. This
generalization of Fermat’s little theorem can be seen as the degenerate case
of Theorem 8 when $F(T_{1},\dots,T_{b})=F_{0}$ is constant in the $T_{i}$.

###  3.2 A generalization of Fermat’s little theorem to matrices

In this subsection we state and prove a generalization of Fermat’s little
theorem to the case of matrices. As noted in the introduction, this
generalization was conjectured by Arnold [Arn06] and proved by Zarelua [Zar08]
(and many other following works). Our proof is short and apparently
new121212In the course of writing this paper, we found essentially the same
proof by Qiaochu Yuan in a blog post from 2009. and therefore we present it
here.

######  Theorem 10 (Arnold-Zarelua).

Let $A\in M_{r}(\mathbb{Z}_{\ell})$. Then:

$\operatorname{\mathrm{tr}}A^{\ell^{n+1}}\equiv\operatorname{\mathrm{tr}}A^{\ell^{n}}\pmod{\ell^{n+1}}.$

In fact, we also have

$\det(1-xA^{\ell^{n+1}})\equiv\det(1-xA^{\ell^{n}})\pmod{\ell^{n+1}}.$

###### Proof.

We fix a $n$. Since we are proving a congruence modulo $\ell^{n+1}$, we can
replace $A$ by a $r\times r$ matrix with non negative integer entries. Let $G$
be the directed multigraph with adjacency matrix $A$, i.e it has $r$ vertices
labelled from $1$ to $r$ and there are $a_{ij}$ many edges from $i$ to $j$.

A closed path of length $n$ on the graph corresponds to a sequence of edges
$e_{1},\dots,e_{n-1}$ such that the in-vertex of $e_{i+1}$ is the out-vertex
of $e_{i}$ and the path starts and ends at the same vertex. The quantity
$\operatorname{\mathrm{tr}}A^{n}$ has the graph theoretic interpretation of
being the number of closed paths of length $n$ on $G$.

Now, consider a closed path $P$ of length $\ell^{n+1}$. The cyclic group of
order $\ell^{n+1}$ acts on the path by permuting

$(e_{1},\dots,e_{n-1})\to(e_{2},\dots,e_{n-1},e_{1}).$

Since we are working modulo $\ell^{n+1}$, we can ignore those paths $P$ where
the orbit by this action has size $\ell^{n+1}$. The remaining paths $P$ are
exactly those which are concatenations of $\ell$ copies of a path of length
$\ell^{n}$. These are exactly counted by
$\operatorname{\mathrm{tr}}(A^{\ell^{n}})$ and therefore we have shown the
required congruence

$\operatorname{\mathrm{tr}}(A^{\ell^{n+1}})\equiv\operatorname{\mathrm{tr}}(A^{\ell^{n}})\pmod{\ell^{n+1}}.$

To prove the corresponding congruence for characteristic polynomials, we use
the well known determinant to trace exponential identity (as formal power
series in $x$)

$\det(1-xB)=\exp\left(-\sum_{d\geq
1}\frac{\operatorname{\mathrm{tr}}(B^{d})x^{d}}{d}\right).$ (7)

Let $d=d_{0}\ell^{e}$ for $d_{0}$ co-prime to $\ell$. The congruence above on
powers of $A^{d_{0}}$ then implies that

$\operatorname{\mathrm{tr}}(A^{d\ell^{n+1}})\equiv\operatorname{\mathrm{tr}}(A^{d\ell^{n}})\pmod{d\ell^{n+1}}.$

Since $\ell>2$, $\alpha\equiv\beta\pmod{\ell^{n}}$ for $n\geq 1$ implies that
$\exp(\alpha)\equiv\exp(\beta)\pmod{\ell^{n}}$:

To see this, let $t\in\mathbb{Z}_{\ell}$ such that $\ell^{n}|t$. We will show
that $e^{t}\equiv 1\pmod{\ell^{n}}$. Supposing this, we see that

$\alpha\equiv\beta\pmod{\ell^{n}}\implies e^{\alpha-\beta}\equiv
1\pmod{\ell^{n}}\implies e^{\alpha}\equiv e^{\beta}\pmod{\ell^{n}}$

since $e^{\beta}\in\mathbb{Z}_{\ell}[\\![x]\\!]$ in our case.

To show that $e^{t}\equiv 1\pmod{\ell^{n}}$, we argue by cases. The terms
appearing in the Taylor expansion of $\exp(t)$ are of the form $t^{r}/r!$. If
$r=1$, then $\ell^{n}|t$. In general, Legendre’s formula shows that $t^{r}/r$
is divisible by $\ell^{\delta_{n,r}}$ for $\delta_{n,r}\coloneqq
nr-r/(\ell-1)$. For $r\geq 2$, note that

$\delta_{n,r}\geq n\impliedby nr-r/2-n\geq 0\iff 2n\geq\frac{r}{r-1}\text{
which is always true for }n\geq 1.$

We finish our proof now by noting that the congruences on the traces implies
(by the exponential identity)

$\det(1-xA^{\ell^{n+1}})\equiv\det(1-xA^{\ell^{n}})\pmod{\ell^{n+1}}.$

∎

###  3.3 A proof of the main congruences

In this subsection, we prove Theorems 8 and 9. It will help to set up some
notation and make some easy reductions first.

Recall that $F(T_{1},\dots,T_{b})$ is a power series in the $T_{i}$ and to
define $A_{n+1}(v)$, we are required to evaluate $F$ at
$T_{i}=\zeta_{\ell^{n+1}}^{v_{i}}-1$ (for $i=1,\dots,b$) which is in the
maximal ideal for the local ring $\mathbb{Z}_{\ell}[\zeta_{\ell^{n+1}}]$.
Since we are interested in a congruence modulo $k_{n+1}(v)$ (or $k_{n+1}$), we
can truncate the $F$ at some finite degree $d$ so that
$(\zeta_{\ell^{n+1}}-1)^{d}\equiv 0\pmod{k_{n+1}(v)}$ and suppose that it is a
polynomial in the $t_{i}=T_{i}+1$ of the form

$F=\sum_{I\in\mathbb{N}^{b}}F_{I}t_{1}^{i_{1}}\dots t_{b}^{t_{b}}$

where the $F_{I}$ are $r\times r$ matrices over $\mathbb{Z}_{\ell}$.

Let $\rho\geq 1$ and for a tuple
$J=(I_{1},\dots,I_{\rho})\in\mathbb{(}N^{b})^{\rho}$, we define
$F_{J}=\prod_{j=1}^{\rho}F_{I_{j}}.$ Using the standard notation
$\langle-,-\rangle$ for inner products (and considering
$\mathbb{N}^{b}\subset\mathbb{Z}_{\ell}^{b}$), we also define the linear form

$\lambda_{J}(v)=\sum_{j=1}^{\rho}\langle I_{j},Q^{-j}v\rangle.$

In terms of this notation, we see that

$A_{n+1}^{d}(v)=\sum_{J\in\mathbb{(}N^{b})^{dk_{n+1}(v)}}F_{J}\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}$

where we have implicitly used that $Q^{k_{n+1}(v)}v\equiv v\pmod{\ell^{n+1}}$.
We denote cyclic permutations by

$\tau(J)=(I_{2},I_{3},\dots,I_{\rho},I_{1})$

and if $k_{n}(v)|\rho$, we note that

$\lambda_{\tau J}(v)\equiv\lambda_{J}(Qv)\pmod{\ell^{n}}.$ (8)

######  Notation.

We will argue by considering each tuple along with its cyclic permutations. To
that end, we fix some notation that we will use repeatedly. Let
$K=(I_{1},\dots,I_{\rho})$ be a tuple of length $\rho$ such that it is non
periodic.131313i.e., the tuples $\tau^{i}J$ are pairwise distinct for $1\leq
i<\rho$. For any $\delta=r\rho\in\mathbb{N}$, we define
$J_{K}(\delta)=(I_{1},\dots,I_{\delta})\coloneqq(K,\dots,K)$ to be the tuple
of length $\delta$ where $K$ is concatenated to itself $r$ times. We suppose
that $r=r_{0}\ell^{s}$ with $r_{0}$ co-prime to $\ell$.

We need one more lemma (which will in fact control the rate of congruence)
before the proof of Theorem 8.

######  Lemma 11.

For $n\geq 0$, suppose $\rho$ is an integer multiple of $k_{n}$. For any
$w\in\mathbb{Z}_{\ell}$,

$S_{\rho,n}(w)\coloneqq\sum_{i=1}^{\rho}\zeta_{\ell^{n}}^{q^{i}w}\equiv
0\pmod{\rho}.$

###### Proof.

Let $w=\ell^{m}w_{0}$ with $w_{0}$ a unit. Since $q^{k_{n}}\equiv
1\pmod{\ell^{n}}$ and $\rho/k_{n}\in\mathbb{Z}$, we see that

$S_{\rho,n}(w)=\sum_{i=1}^{\rho}\zeta_{\ell^{n-m}}^{q^{i}w_{0}}=\frac{\rho}{k_{n-m}}S_{k_{n-m},n-m}(w_{0})$

where we use the convention that $\zeta_{-m}=1$ if $m\geq 0$. Therefore, we
can suppose that $w$ is a unit and $\rho=k_{n}$ without loss of generality.
Let $\log q=\ell^{\alpha}x$ with $x$ a unit so that
$q^{i}-1=i\ell^{\alpha}x\pmod{\ell^{\alpha+1}}$. We now have two cases to
consider. Either $\alpha\geq n$ in which case
$\zeta_{\ell^{n}}^{q^{i}w}=\zeta_{\ell^{n}}^{w}$ and

$S_{\rho,n}(w)=\rho\zeta_{\ell^{n}}^{w}\equiv 0\pmod{\rho}$

or $\alpha<n$. In this second case, note that the $\zeta_{\ell^{n}}^{q^{i}w}$
are all pairwise distinct for $i\leq k_{n}=\ell^{n-\alpha}$:

If $1\leq j<i\leq\ell^{n-\alpha}$, then

$i-j<\ell^{n-\alpha}\implies\zeta_{\ell^{n}}^{(q^{i}-q^{j})w}=\zeta_{\ell^{n}}^{(i-j)w\ell^{\alpha}x+\dots}\neq
1.$

In fact, the $\zeta_{\ell^{n}}^{q^{i}w}$ are a complete set of roots for the
polynomial $z^{\ell^{n-\alpha}}=\zeta_{\ell^{\alpha}}^{w}$ and $S_{\rho,n}(w)$
is equal to the linear term of this polynomial which is $0$ thus completing
the proof.

∎

###### Proof of Theorem 8.

For this proof, recall that $Q=q\operatorname{I}$ is a scalar matrix so that
$k_{n}(v)=k_{n}$ for all primitive $v$. We reduce the congruence on the
characteristic polynomials $p_{n,v}$ to a congruence on traces using the
exponential identity (equation 7)

$p_{n,v}(y)=\exp\left(-\sum_{d\geq
0}\operatorname{\mathrm{tr}}(A_{n}^{d}(v))\frac{y^{d}}{d}\right)$

as in the proof of Theorem 10. Upon fixing $n$ such that $k_{n+1}=\ell
k_{n}$,it suffices to show the congruence

$t_{n}\coloneqq\operatorname{\mathrm{tr}}(A_{n+1}^{d}(v))-\operatorname{\mathrm{tr}}(A_{n}^{d}(v))\equiv
0\pmod{dk_{n+1}}.$

We will consider the contributions to $t_{n}$ from each tuple and its cyclic
permutations. In the notation of Notation, we take $\delta=dk_{n+1}$ and
$J=J_{K}(dk_{n+1})$ and if $\ell|r$, $J_{0}=J_{K}(dk_{n})$. Note that

$\lambda_{J}(v)=\sum_{i=1}^{\rho}\langle
I_{i},q^{-i}\left(1+q^{-\rho}+\dots+q^{-(r-1)\rho}\right)v\rangle=\frac{q^{-r\rho}-1}{q^{-\rho}-1}\lambda_{K}(v).$

Since $q^{i}-1=i\log(q)+\frac{1}{2}(i\log(q))^{2}+\dots$ is exactly divisible
by $i\log(q)$, there exists some $w\in\mathbb{Z}_{\ell}$ so that

$\frac{q^{-r\rho}-1}{q^{-\rho}-1}=\frac{\ell^{s}\rho}{\rho}w=\ell^{s}w\implies\lambda_{J}(v)=\ell^{s}w\lambda_{K}(v).$
(9)

Moreover, there exists some $y\in\mathbb{Z}_{\ell}$ so that $q^{-dk_{n}}\equiv
1+\ell^{n}y\pmod{\ell^{n+1}}$ and therefore

$\sum_{i=1}^{\ell-1}q^{-idk_{n}}\equiv\ell+\ell^{n}y\sum_{i=0}^{\ell-1}i\equiv\ell+\ell^{n+1}y\frac{\ell-1}{2}\pmod{\ell^{n+1}}\implies\lambda_{J}(v)\equiv\ell\lambda_{J_{0}}(v)\pmod{\ell^{n+1}}.$
(10)

We now have to consider two cases.

First, suppose $s=0$. In this case, the only contributions from tuples that
are repetitions of $K$ and its cyclic permutations comes from
$\operatorname{\mathrm{tr}}(A_{n+1}^{d}(v))$ and is of the form

$\sum_{i=1}^{\rho}\operatorname{\mathrm{tr}}(F_{\tau^{i}J})\zeta_{\ell^{n+1}}^{\lambda_{\tau^{i}J}(v)}=\operatorname{\mathrm{tr}}(F_{J})\sum_{i=1}^{\rho}\zeta_{\ell^{n+1}}^{\lambda_{J}(q^{i}v)}=\operatorname{\mathrm{tr}}(F_{J})\sum_{i=1}^{\rho}\zeta_{\ell^{n+1}}^{q^{i}w\lambda_{K}(v)}$

where for the first equality, we use that the trace is invariant under cyclic
permutations and equation 8 while for the second equality, we use equation 9
above and that $s=0$ by assumption. Now, $r\rho=dk_{n+1}$ and since $r$ is co-
prime to $\ell$, $k_{n+1}$ being a $\ell$-power necessarily divides $\rho$. In
fact, $\rho$ and $dk_{n+1}$ have the same $\ell$-adic valuation. Thus, we can
apply Lemma 11 to conclude

$\sum_{i=1}^{\rho}\operatorname{\mathrm{tr}}(F_{\tau^{i}J})\zeta_{\ell^{n+1}}^{\lambda_{\tau^{i}J}(v)}\equiv
0\pmod{\rho}\iff\sum_{i=1}^{\rho}\operatorname{\mathrm{tr}}(F_{\tau^{i}J})\zeta_{\ell^{n+1}}^{\lambda_{\tau^{i}J}(v)}\equiv
0\pmod{dk_{n+1}}.$

Next, suppose $s>0$. In this case, we will have contributions from both
$\operatorname{\mathrm{tr}}(A_{n+1}^{d}(v))$ and
$\operatorname{\mathrm{tr}}(A_{n}^{d}(v))$ and they are of the form

$\displaystyle\sum_{i=1}^{\rho}\operatorname{\mathrm{tr}}(F_{\tau^{i}J})\zeta_{\ell^{n+1}}^{\lambda_{\tau^{i}J}(v)}-\sum_{i=1}^{\rho}\operatorname{\mathrm{tr}}(F_{\tau^{i}J_{0}})\zeta_{\ell^{n}}^{\lambda_{\tau^{i}J_{0}}(v)}$
$\displaystyle=\left(\operatorname{\mathrm{tr}}(F_{K}^{r})-\operatorname{\mathrm{tr}}(F_{K}^{r/\ell})\right)\sum_{i=1}^{\rho}\zeta_{\ell^{n}}^{\lambda_{J}(q^{i}v)}$
$\displaystyle=\left(\operatorname{\mathrm{tr}}(F_{K}^{r})-\operatorname{\mathrm{tr}}(F_{K}^{r/\ell})\right)\sum_{i=1}^{\rho}\zeta_{\ell^{n+1}}^{q^{i}\ell^{s}w\lambda_{K}(v)}$
$\displaystyle\equiv 0\pmod{r\rho=dk_{n+1}}$

where the first equality follows from invariance of trace under cyclic
permutations and equation 10 while the second equation follows from equation
9. For the last congruence, Theorem 10 implies that

$\operatorname{\mathrm{tr}}(F_{K}^{r})-\operatorname{\mathrm{tr}}(F_{K}^{r/\ell})\equiv
0\pmod{r}.$

Moreover, since $dk_{n+1}=r\rho$, we see that $\rho$ is divisible by
$dk_{n+1}\ell^{-s}$ and in particular by $k_{n+1-s}$. Therefore, we can apply
Lemma 11 to conclude
$\sum_{i=1}^{\rho}\zeta_{\ell^{n+1-s}}^{q^{i}w\lambda_{K}(v)}\equiv
0\pmod{\rho}.$

∎

######  Remark 9.

We remark that the failure of this proof for the general case (see Remark 6)
happens exactly at Lemma 11. If $Q$ is not scalar, it is no longer true that
$\lambda_{J}(Qv)=Q\lambda_{J}(v)$ and consequently, there exist examples (with
$\lambda$ a linear form) such that

$S_{n}(\lambda;v)\coloneqq\sum_{j=1}^{k_{n}(v)}\zeta_{\ell^{n}}^{\lambda(Q^{-j}v)}\not\equiv
0\pmod{k_{n}(v)}.$

Nevertheless, the above proof shows that if the $S_{n}(\lambda_{J};v)\to 0$ as
$n\to\infty$, then the characteristic polynomials $p_{n,v}(y)$ will also
converge as $n\to\infty$. If $\lambda_{J}(\log(Q)v)\neq 0$, a variation of
Lemma 11 still applies to $S_{n}(\lambda_{J};v)$. In fact, numerical evidence
supports the vanishing of the limit (for $\lambda_{J}$ an arbitrary linear
form) but we do not know how to prove it.

From now on, we again let $Q\equiv\operatorname{I}\pmod{\ell}$ be a general
matrix. We recall some notation before the proof of Theorem 9. We let
$V=\mathbb{Z}_{\ell}^{b}$ be a free $\mathbb{Z}_{\ell}$ module,
$V_{n}=V/\ell^{n}V$, $\mathcal{P}(V_{n})$ to be the primitive vectors in
$V_{n}$ and $\mathcal{P}(V_{n})/\sim$ to be a set of representatives under the
action by $Q$. The characteristic polynomials we are interested in are

$r_{n}(y)=\prod_{v\in\mathcal{P}((\mathbb{Z}/\ell^{n}\mathbb{Z})^{b})/\sim}p_{n,v}(y^{k_{n}(v)/k_{n}}).$

We also fix $n$ sufficiently large and define (in the notation of Lemma 7)

$V_{e}=\\{v\in
V:\frac{k_{n}(v)}{k_{n}}|\ell^{e}\iff\beta_{v}\geq\beta_{0}-e\iff Xv\equiv
0\pmod{\ell^{\beta_{0}-e}}\\}\subset V.$

By the last equivalent condition, we see that $V_{e}$ is a (non-empty)
submodule of $V$. Since $Q$ commutes with $\log(Q)$ and hence also $X$, we see
that $Q$ preserves $V_{e}$. When $Q=q\operatorname{I}$, $V_{e}=V$ since
$\beta_{v}=\beta_{0}$ for all primitive $v$. Also define $V_{e,n}$ to be the
image of $V_{e}$ in $V/\ell^{n}V$ under the reduction map. Note that, in
general, $V_{e}\not\cong(\mathbb{Z}/\ell^{n}\mathbb{Z})^{c}$ for some $c$ and
is only a-priori a finite $\mathbb{Z}/\ell^{n}\mathbb{Z}$ module.

So, let $M$ be an arbitrary finite $\mathbb{Z}_{\ell}$ module and $n\geq 0$ be
the smallest value such that $\ell^{n}M=0$. An element $v\in M$ is said to be
primitive (generalizing our usual notion) when $\ell^{n-1}v\neq 0$ and the set
of primitive elements is denoted $\mathcal{P}(M)$. Our two definitions of
primitive are compatible in the sense that

$\mathcal{P}(V_{e,n})=\mathcal{P}(V/\ell^{n}V)\cap V_{e,n}.$

We need one more lemma (analogous to Lemma 11 and also the determining factor
for the rate of convergence) before the proof of Theorem 9.

######  Lemma 12.

Let $M$ be as above with

$\chi:M\to\overline{\mathbb{Z}}_{\ell}^{\times}$

a character. Then, we have the congruence

$\sum_{v\in\mathcal{P}(M)}\chi(v)\equiv 0\pmod{\ell^{n-1}}.$

If $M=(\mathbb{Z}_{\ell}/\ell^{n}\mathbb{Z}_{\ell})^{b}$, we have the stronger
congruence

$\sum_{v\in\mathcal{P}(M)}\chi(v)\equiv 0\pmod{\ell^{(n-1)b}}.$

###### Proof.

Let $|M|=m$, note that $S_{M}\coloneqq\sum_{v\in M}\chi(v)\equiv 0\pmod{m}:$
There are two cases to consider. First, if $\chi$ is the trivial character,
then $S_{M}=m$ and the congruence is clear. Second, if $\chi$ is not trivial,
we can find some $m_{0}\in M$ so that $\chi(m_{0})\neq 1$ and
$S_{M}=\chi(m_{0})S_{M}\implies S_{M}=0\equiv 0\pmod{m}.$

Define

$N=\\{v\in M:\ell^{n-1}v=0\\}\subset M$

so that $\mathcal{P}(M)=M-N$. The module $M$ has size at least $\ell^{n}$ and
the module $N$ has size at least $\ell^{n-1}$. Therefore, we have

$\sum_{v\in\mathcal{P}(M)}\chi(v)=\sum_{v\in M}\chi(v)-\sum_{w\in
N}\chi(w)=S_{M}-S_{N}\equiv 0\pmod{\ell^{n-1}}$

since $|M|\equiv|N|\equiv 0\pmod{\ell^{n-1}}$.

If $M=(\mathbb{Z}_{\ell}/\ell^{n}\mathbb{Z}_{\ell})^{b}$ so that
$N=(\mathbb{Z}_{\ell}/\ell^{n-1}\mathbb{Z}_{\ell})^{b}$, then the above
argument shows the stronger congruence

$\sum_{v\in\mathcal{P}(M)}\chi(v)=\sum_{v\in M}\chi(v)-\sum_{w\in
N}\chi(w)=S_{M}-S_{N}\equiv 0\pmod{\ell^{(n-1)b}}.$

∎

We now prove Theorem 9, along the same general lines as the proof of Theorem
8.

###### Proof of Theorem 9.

By the exponential identity 7, we have

$r_{n}(y)=\exp\left(-\sum_{v\in\mathcal{P}(V/\ell^{n}V)/\sim}\sum_{f\geq
0}\frac{\operatorname{\mathrm{tr}}A_{n}^{f}(v)}{f}y^{\frac{fk_{n}(v)}{k_{n}}}\right).$

Let us fix some $d=d_{0}\ell^{e}$ (with $d_{0}$ co-prime to $\ell$) and
collect the terms corresponding to $y^{d}$ so that with

$C_{d,n}=\sum_{v\in\mathcal{P}(V_{e,n})/\sim}\frac{k_{n}(v)}{dk_{n}}\operatorname{\mathrm{tr}}A_{n}^{\frac{dk_{n}}{k_{n}(v)}}(v)\text{,
we have }r_{n}(y)=\exp\left(-\sum_{d\geq 0}C_{d,n}y^{d}\right).$

As in the proof of Theorem 10, the congruence

$r_{n+1}(y)\equiv r_{n}^{\ell^{b-1}}(y)\pmod{\ell^{n}},$

is reduced to the congruence

$C_{d,n+1}\equiv\ell^{b-1}C_{d,n}\pmod{\ell^{n}}.$

Since a representative $v\in\mathcal{P}(V_{e,n})/\sim$ represents $k_{n}(v)$
many vectors in $\mathcal{P}(V_{e,n})$ and $A_{n}(Qv)$ is conjugate to
$A_{n}(v)$ so that their powers have the same traces, we can express $C_{d,n}$
as a sum over all primitive vectors by

$C_{d,n}=\sum_{v\in\mathcal{P}(V_{e,n})}\frac{1}{dk_{n}}\operatorname{\mathrm{tr}}A_{n}^{\frac{dk_{n}}{k_{n}(v)}}(v).$

Therefore we are reduced to proving the congruence

$t_{n}\coloneqq\sum_{v\in\mathcal{P}(V_{e,n+1})}\frac{1}{dk_{n+1}}\operatorname{\mathrm{tr}}A_{n+1}^{\frac{dk_{n+1}}{k_{n+1}(v)}}(v)-\sum_{v\in\mathcal{P}(V_{e,n})}\frac{\ell^{b}}{dk_{n+1}}\operatorname{\mathrm{tr}}A_{n}^{\frac{dk_{n}}{k_{n}(v)}}(v)\equiv
0\pmod{\ell^{n}}$

where we have implicitly used the assumption that $n$ is sufficiently large so
that $k_{n+1}=\ell k_{n}$. Since every vector in $\mathcal{P}(V_{e,n})$ has
$\ell^{b}$ many lifts to $\mathcal{P}(V_{e,n+1})$, we also have

$t_{n}=\frac{1}{dk_{n+1}}\sum_{v\in\mathcal{P}(V_{e,n+1})}\left(\operatorname{\mathrm{tr}}A_{n+1}^{\frac{dk_{n+1}}{k_{n+1}(v)}}(v)-\operatorname{\mathrm{tr}}A_{n}^{\frac{dk_{n}}{k_{n}(v)}}(v)\right).$

Note that in the expansion

$\operatorname{\mathrm{tr}}A_{n}^{\frac{dk_{n}}{k_{n}(v)}}(v)=\sum_{J\in(\mathbb{N}^{b})^{dk_{n}}}\operatorname{\mathrm{tr}}(F_{J})\zeta_{\ell^{n}}^{\lambda_{J}(v)},$

the tuples all have size $dk_{n}$ independent of $v$. As before, we will argue
by fixing a tuple $K$ and considering the contributions from tuples that are
multiples of $K$ and their cyclic permutations. In the notation of Notation,
let $J=J_{K}(dk_{n+1})$ and when $\ell|r$, $J_{0}=J_{K}(dk_{n})$.

First, we suppose that $s=0$. In this case, the only contribution to $t_{n}$
from $K$ will be through $J$ and will be of the form

$\frac{1}{dk_{n+1}}\sum_{v\in\mathcal{P}(V_{e,n+1})}\operatorname{\mathrm{tr}}(F_{J})\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}.$

We note that $\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}$ is a character on
$V_{e,n+1}$ and therefore, by Lemma 12, there exists some
$T_{\lambda_{J}}\in\mathbb{Z}_{\ell}$ such that

$\frac{1}{dk_{n+1}}\sum_{v\in\mathcal{P}(V_{e,n+1})}\operatorname{\mathrm{tr}}(F_{J})\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}=\frac{\ell^{n}}{dk_{n+1}}T_{\lambda_{J}}.$

Moreover, for any cyclic permutation $\tau^{i}J$ of $J$, the corresponding
contribution is of the same form as before since $Q^{i}$ permutes
$\mathcal{P}(V_{e,n+1})$:

$\frac{1}{dk_{n+1}}\sum_{v\in\mathcal{P}(V_{e,n+1})}\operatorname{\mathrm{tr}}(F_{\tau^{i}J})\zeta_{\ell^{n+1}}^{\lambda_{J}(Q^{i}v)}=\frac{1}{dk_{n+1}}\sum_{v\in\mathcal{P}(V_{e,n+1})}\operatorname{\mathrm{tr}}(F_{J})\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}=\frac{\ell^{n}}{dk_{n+1}}T_{\lambda_{J}}.$

Therefore, the contribution from all the cyclic permutations of $J$ is
together equal to

$\frac{\rho\ell^{n}}{dk_{n+1}}\operatorname{\mathrm{tr}}(F_{J})T_{\lambda_{J}}\equiv
0\pmod{\ell^{n}}$

since the $\ell$-adic valuation of $\rho$ is equal to the $\ell$-adic
valuation of $dk_{n+1}$.

Next, suppose $s>0$. In this case, the contribution from $K$ will be through
$J$ and $J_{0}$. Since $v\in V_{e}$, $dk_{n+1}$ is divisible by $k_{n+1}(v)$
so that $Q^{-idk_{n}}v=v+i\ell^{n}Yv$ for some $Y\in M_{b}(\mathbb{Z}_{\ell})$
and

$\left(\operatorname{I}+Q^{-dk_{n}}+\dots+Q^{-(\ell-1)dk_{n}}\right)v=\ell
v+i\ell^{n}\sum_{i=0}^{\ell-1}Yv=\ell v+\ell^{n+1}\frac{\ell-1}{2}Yv\equiv\ell
v\pmod{\ell^{n+1}}.$

This implies that

$\lambda_{J}(v)=\sum_{i=1}^{dk_{n}}\langle
I_{i},q^{-i}\left(1+Q^{-dk_{n}}+\dots+Q^{-(\ell-1)dk_{n}}\right)v\rangle\equiv\ell\lambda_{J_{0}}(v)\pmod{\ell^{n+1}}$

which is equivalent to
$\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}=\zeta_{\ell^{n}}^{\lambda_{J_{0}}(v)}$.
Therefore, the contribution from $J,J_{0}$ in $t_{n}$ is of the form

$\frac{1}{dk_{n+1}}\left(\operatorname{\mathrm{tr}}(F_{k}^{r})-\operatorname{\mathrm{tr}}(F_{k}^{r/\ell})\right)\sum_{v\in\mathcal{P}(V_{e,n+1})}\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}=\frac{\ell^{n}}{dk_{n+1}}\left(\operatorname{\mathrm{tr}}(F_{k}^{r})-\operatorname{\mathrm{tr}}(F_{k}^{r/\ell})\right)T_{\lambda_{J}}.$

As above, the cyclic permutations of $K$ give rise to exactly the same
contribution so that the total contribution from all cyclic permutations of
$K$ is

$\frac{\rho\ell^{n}}{dk_{n+1}}\left(\operatorname{\mathrm{tr}}(F_{k}^{r})-\operatorname{\mathrm{tr}}(F_{k}^{r/\ell})\right)T_{\lambda_{J}}\equiv
0\pmod{\ell^{n}}$

since
$\left(\operatorname{\mathrm{tr}}(F_{k}^{r})-\operatorname{\mathrm{tr}}(F_{k}^{r/\ell})\right)$
is divisible by $r$ by Theorem 10 and $r\rho=dk_{n+1}$.

When $Q=q\operatorname{I}$, the proof is exactly the same as above except that
we have the stronger congruence

$\sum_{v\in\mathcal{P}(V_{e,n+1})}\operatorname{\mathrm{tr}}(F_{J})\zeta_{\ell^{n+1}}^{\lambda_{J}(v)}\equiv
0\pmod{\ell^{nb}}.$

This follows from the second part of Lemma 12 since
$V_{e}=V=\mathbb{Z}_{\ell}^{b}$ in this case and
$V_{e,n+1}=(\mathbb{Z}_{\ell}/\ell^{n+1}\mathbb{Z}_{\ell})^{b}$. ∎

######  Remark 10.

As one sees from the proof, the modulus of the congruence in Theorem 9 depends
on the structure of $V_{e,n+1}$.

##  4 Explicit examples

In this section, we prove that the normalized eigenvalues of the
characteristic polynomials $h_{n,v}(x)$ defined in the proof of Theorem 6 are
independent of $n$ for $n$ sufficiently large in the following two examples:

  * •

Fermat Curves: This is the family of curves defined by the equation

$C_{n}:x^{\ell^{n}}+y^{\ell^{n}}+z^{\ell^{n}}=0\subset\mathbb{P}^{2}.$

We have maps

$\dots\to C_{n}\to C_{n-1}\to\dots\to C_{1}\cong\mathbb{P}^{1}$

with $G_{n}=\operatorname{\mathrm{Aut}}(C_{n}/C_{1})=(\mu_{\ell^{n}})^{2}$ and
the element $(\zeta_{1},\zeta_{2})$ acts by
$[x:y:z]\to[x\zeta_{1}:y\zeta_{2}:z]$.

  * •

Artin-Schreier Curves: This is the family of curves defined by the projective
closure of the equation

$C_{n}:y^{q}-y=x^{\ell^{n}}\subset\mathbb{P}^{2}/\mathbb{F}_{q}.$

The automorphism group in this case is
$G_{n}=\mathbb{F}_{q}\times\mu_{\ell^{n}}$. An element $(a,\zeta)$ in this
group acts on the curve by $(x,y)\to(\zeta x,y+a)$.

######  Remark 11.

The results of this section work in somewhat greater generality, for instance
we don’t need to restrict to Fermat or Artin-Schreier curves of degree a power
of $\ell$. The results also work for various quotients of these curves such as
the superelliptic curves $y^{m}=x^{\ell^{n}}+a$.

Since the computations in other cases are exactly analogous, we only deal with
the above two cases.

Throughout this section, we identify characters
$\chi:\mu_{\ell^{n}}\to\overline{\mathbb{Z}}_{\ell}$ with vectors
$v\in\mathbb{Z}_{\ell}$ by $\chi(v):\zeta_{\ell^{n}}\to\zeta_{\ell^{n}}^{v}$.
We also fix a compatible family of additive characters
$\psi_{n}:\mathbb{F}_{q^{n}}\to\overline{\mathbb{Z}}_{\ell}$ that satisfy
$\psi_{n}=\operatorname{\mathrm{tr}}(\mathbb{F}_{q^{n}}/\mathbb{F}_{q})\circ\psi_{1}$.

In both of the above families of curves, we can decompose
$M_{n}=H^{1}_{\text{\'{e}t}}(\overline{C}_{n},\mathbb{Z}_{\ell})$ into one
dimensional eigenspaces $M_{n}(\chi)$ indexed by characters $\chi$ of $G_{n}$.
In the Fermat curve case, the characters are naturally indexed by
$v\in(\mathbb{Z}/\ell^{n}\mathbb{Z})^{2}$ while in the second case, the
characters are indexed by $(\psi,v)$ where $\psi$ is an additive character of
$\mathbb{F}_{q}$ and $v\in\mathbb{Z}/\ell^{n}\mathbb{Z}$.

Given a character $\chi:\mu_{\ell^{n}}\to\overline{\mathbb{Z}}_{\ell}$ and
$q\equiv 1\pmod{\ell^{n}}$, we can define a multiplicative character of
$\mathbb{F}_{q}^{\times}$ since the map $x\to x^{\frac{q-1}{\ell^{n}}}$
induces a surjection

$\mathbb{F}_{q}^{\times}\to\mu_{\ell^{n}}(\mathbb{F}_{q})\cong\mu_{\ell^{n}}$

and we compose this surjection with $\chi$. By a slight abuse of notation, we
also denote this character by $\chi$.

The following well-known theorem ([Kat81, Corollary 2.2 and Lemma 2.3])
identifies the eigenvalues of the Frobenius $\sigma_{q}$ on $M(\chi)$ with
Gauss and Jacobi sums respectively.

######  Theorem 13.

We assume that $q\equiv 1\pmod{\ell^{n}}$.

  * •

For the Fermat curves $C_{n}$, let $\eta=(\chi,\chi_{2})$ be a character of
$G_{n}=(\mu_{\ell^{n}})^{2}$. The eigenvalues of $\sigma_{q}$ on the
eigenspace $M_{n}(\eta)$ are given by the Jacobi sum

$-J_{q}(\chi_{1},\chi_{2})=-\sum_{x\in\mathbb{F}_{q}}\chi_{1}(x)\chi_{2}(1-x).$

  * •

For the Artin-Schreier curves, let $\eta=(\psi,\chi)$ be a character of
$G_{n}=\mathbb{F}_{q}\times\mu_{\ell^{n}}$. The eigenvalues of $\sigma_{q}$ on
the eigenspace $M_{n}(\eta)$ are given by the Gauss sums

$-g_{q}(\psi,\chi)=-\sum_{x\in\mathbb{F}_{q}}\psi(x)\chi(x).$

###### Proof.

We sketch the proof for completeness. In the case of Fermat curves, we would
like to count points on the affine curve $x^{\ell^{n}}+y^{\ell^{n}}=-1$ while
in the case of Artin-Schreier curves, we would like to count points on
$y^{q}-y=x^{\ell^{n}}$.

We have the identities

$\sum_{\chi:\mathbb{F}_{q}^{\times}\to\mu_{\ell^{n}}}\chi(x)=\begin{cases}\ell^{n}&\text{
if }x=y^{\ell^{n}}\\\ 0&\text{ otherwise}\end{cases}$

and

$\sum_{\psi:\mathbb{F}_{q}\to\mu_{q}}\psi(x)=\begin{cases}q&\text{ if
}x=y^{q}-y\\\ 0&\text{ otherwise}\end{cases}.$

We can use these identities to test if an element $x\in\mathbb{F}_{q}$ is a
$\ell^{n}$-th power or of the form $y^{q}-y$ and therefore use it to count
points:

For the Fermat Curve, we have

$C_{n}(F_{q})=\sum_{z+w=-1}\sum_{\chi_{1},\chi_{2}:\mathbb{F}_{q}^{\times}\to\mu_{\ell^{n}}}\chi_{1}(x)\chi_{2}(y)$

while for Artin-Schreier curves:

$C_{n}(\mathbb{F}_{q})=\sum_{z\in\mathbb{F}_{q}}\sum_{\psi,\chi}\psi(z)\chi(z).$

Exchanging the summation, this shows that the point counts on the two curves
can be expressed in terms of Jacobi and Gauss sums respectively. Finally, we
use the Weil-conjectures to identify eigenvalues of the Frobenius action with
Jacobi/Gauss sums by varying over all powers of $q$.

∎

Let us return to the set-up of Theorem 6. The roots of the characteristic
polynomial $h_{n,v}(x)$ therefore correspond to
$\left(-J_{q}(\chi_{1},\chi_{2})\right)^{k_{n}}=-J_{q^{k_{n}}}(\chi_{1},\chi_{2})$
with $v$ corresponding to the character $\chi_{1},\chi_{2}$ and similarly for
the Gauss sum in the two cases we are interested in. Put another way, we
choose the minimal $q$ so that $q-1$ is exactly divisible by $\ell^{n}$ and we
are looking for a relation between these values for varying $n$.

Luckily, the exact statement we need is a result of Coleman [C+87] proved
using the p-adic Gamma function of Gross-Koblitz [GK79]. Stated in our
notation and specialized to our needs, [C+87, Theorem 11] takes the following
form:

######  Theorem 14 (Coleman).

Let $v\in\mathbb{Z}_{\ell}$, $q=p^{f}$ be such that $\ell^{n}$ exactly divides
$q-1$. In the notation of the previous theorem, we have

$g_{q^{\ell}}(\psi,\chi_{q^{\ell}}(v))=g_{q}(\psi,\chi_{q}(v))\chi_{q}(v)(\ell)c_{q}$

for $c_{q}=c_{p}^{f}$ and $c_{p}=(-1)^{r}p^{\frac{\ell-1}{2}}$ where $r$
depends only on $\ell$.

###### Proof.

In Theorem $11$ of loc. cit., take $b=v/\ell^{n+1}$, $d=\ell$. Note that there
is exactly one orbit of size $\ell$ and
$c=\left(\sqrt{-p}^{\ell-1}\phi_{d}(0)\right)^{f}$, $r=r_{\ell}+(\ell-1)/2$ in
the notation of that paper. ∎

The following theorem is an immediate consequence of Coleman’s theorem and is
the required relation.

######  Theorem 15.

Suppose that $q$ is such that $\ell^{n}$ exactly divides $q-1$. Let
$v_{1},v_{2}\in\mathbb{Z}_{\ell}$, $\chi_{q^{m}}(v_{i})$ multiplicative
characters of $\mu_{\ell^{\infty}}\left(\mathbb{F}_{q^{m}}\right)$
corresponding to $v_{i}$ and
$\psi_{n}:\mathbb{F}_{q^{n}}\to\overline{\mathbb{Z}}_{\ell}$ a compatible
series of additive characters as above.

Then, we have the following identities:

$\frac{J_{q}(\chi_{q}(v_{1}),\chi_{q}(v_{2}))}{q^{1/2}}=\frac{J_{q^{\ell}}(\chi_{q^{\ell}}(v_{1}),\chi_{q^{\ell}}(v_{2}))}{q^{\ell/2}}$
(11)

and

$\frac{g_{q}(\psi,\chi_{q}(v))\chi_{q}(v)(\ell)}{q^{1/2}}=\frac{g_{q^{\ell}}(\psi,\chi_{q^{\ell}}(v))}{q^{\ell/2}}$
(12)

###### Proof.

We first prove equation 11. We have the well known identity

$J_{q}(\chi_{1},\chi_{2})g_{q}(\psi,\chi_{1}\chi_{2})=g_{q}(\psi,\chi_{1})g_{q}(\psi,\chi_{2}).$

By Theorem 14, we then have

$\displaystyle J_{q^{\ell}}(\chi_{q^{\ell}}(v_{1}),\chi_{q^{\ell}}(v_{2}))$
$\displaystyle=\frac{g_{q^{\ell}}(\psi,\chi_{q^{\ell}}(v_{1}))g_{q^{\ell}}(\psi,\chi_{q^{\ell}}(v_{2}))}{g_{q^{\ell}}(\psi,\chi_{1}\chi_{2})}$
$\displaystyle=\frac{g_{q}(\psi,\chi_{q}(v_{1}))g_{q}(\psi,\chi_{q}(v_{2}))c_{q}}{g_{q}(\psi,\chi_{1}\chi_{2})}$
$\displaystyle=J_{q}(\chi_{q}(v_{1}),\chi_{q}(v_{2}))c_{q}$

where $q=p^{f}$. Since $c_{q}=\pm q^{\frac{\ell-1}{2}}$, we recover equation
11 up to a sign by dividing by $q^{\ell/2}$. Finally, upon reducing Theorem 8
$\pmod{\ell}$, we note that the normalized eigenvalues are all congruent
$\pmod{\ell}$ and therefore the sign has to be $+1$.

Equation 12 follows in exactly the same manner from Theorem 14. ∎

######  Remark 12.

We note that the above theorem is in exact accord with Case A, Theorem 6 since
in the notation of that theorem, it shows that the roots of $h_{n+1}(y)$ are
equal to the roots of $h_{n}(y)$. In other words, we not only have a
congruence $h_{n+1}(y)\equiv h_{n}(y)\pmod{\ell^{n}}$, we have an equality
$h_{n+1}(y)=h_{n}(y)$ in the two cases considered in this section.

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