# Newton Polygons of Sums on Curves I: Local-to-Global Theorems

Joe Kramer-Miller and James Upton

###### Abstract

The purpose of this article is to study Newton polygons of certain abelian
$L$-functions on curves. Let $X$ be a smooth affine curve over a finite field
$\mathbb{F}_{q}$ and let $\rho:\pi_{1}(X)\to\mathbb{C}_{p}^{\times}$ be a
finite character of order $p^{n}$. By previous work of the first author, the
Newton polygon $\operatorname{\mathrm{NP}}(\rho)$ lies above a ‘Hodge polygon’
$\operatorname{\mathrm{HP}}(\rho)$, which is defined using local ramification
invariants of $\rho$. In this article we study the touching between these two
polygons. We prove that $\operatorname{\mathrm{NP}}(\rho)$ and
$\operatorname{\mathrm{HP}}(\rho)$ share a vertex if and only if a
corresponding vertex is shared between the Newton and Hodge polygons of
‘local’ $L$-functions associated to each ramified point of $\rho$. As a
consequence, we determine a necessary and sufficient condition for the
coincidence of $\operatorname{\mathrm{NP}}(\rho)$ and
$\operatorname{\mathrm{HP}}(\rho)$.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Main results
    3. 1.3 Outline of proof
    4. 1.4 Applications to $\mathbb{Z}_{p}$-towers of curves
      1. 1.4.1 $\mathbb{Z}_{p}$-Towers with Strictly Stable Monodromy
      2. 1.4.2 Overconvergent $\mathbb{Z}_{p}$-Towers
    5. 1.5 Acknowledgments
  2. 2 Growth Conditions
    1. 2.1 Weak Completions
    2. 2.2 Weak Base Change
  3. 3 Fredholm Theory
    1. 3.1 Spectral Theory of Completely Continuous Operators
      1. 3.1.1 Normed Vector Spaces
      2. 3.1.2 Completely Continuous Maps
      3. 3.1.3 Orthonormal and Integral Bases
    2. 3.2 Nuclear Operators
    3. 3.3 Newton and Hodge Polygons
    4. 3.4 Iteration
  4. 4 Geometric Setup
    1. 4.1 Mapping to $\mathbb{P}_{\mathbb{F}_{q}}^{1}$
    2. 4.2 The Global Lifting
    3. 4.3 Semi-Local Liftings
    4. 4.4 Semi-Local Decomposition
  5. 5 $\sigma$-Modules
    1. 5.1 Definitions
    2. 5.2 $\sigma$-Modules over the Global Lifting
    3. 5.3 Semi-Local Growth Conditions
    4. 5.4 Global growth conditions
    5. 5.5 Semi-Local Twisting
  6. 6 Estimating Dwork Operators
    1. 6.1 Local Estimates
      1. 6.1.1 First case: $\eta(P)=0$ or $\infty$
      2. 6.1.2 Second Case: $\eta(P)=1$
    2. 6.2 Global Estimates
  7. 7 Perturbing Operators and Main Results
    1. 7.1 Newton-Hodge Interaction and Perturbation Theory
    2. 7.2 Local-to-global extensions
    3. 7.3 Perturbations Coming from Local-to-Global Extensions
    4. 7.4 Main results
  8. 8 Glossary of notation
    1. 8.1 Basic definitions
    2. 8.2 The space $A$, its modifications, and some associated notation

##  1 Introduction

###  1.1 Motivation

Let $p$ be an odd prime and let $q$ be a power of $p$. Let $X/\mathbb{F}_{q}$
be a smooth affine variety with étale fundamental group $\pi_{1}(X)$. Consider
a continuous character $\rho:\pi_{1}(X)\to\mathbb{C}^{\times}$ of the étale
fundamental group of $X$. A fundamental problem in arithmetic geometry is to
understand the character sums

$\displaystyle S_{k}(X,\rho)=\sum_{x\in
X(\mathbb{F}_{q^{k}})}\rho(\text{Frob}_{x}),$

where $\text{Frob}_{x}$ denotes the Frobenius element at $x$. By the Weil
conjectures, there exist algebraic integers
$\alpha_{1},\dots,\alpha_{d_{1}},\beta_{1},\dots,\beta_{d_{2}}$ such that

$\displaystyle S_{k}(X,\rho)$
$\displaystyle=\sum_{i=1}^{d_{1}}\beta_{i}^{k}-\sum_{i=1}^{d_{2}}\alpha_{i}^{k}.$

What can we say about these algebraic integers? The Riemann Hypothesis states
that the $\alpha_{i}$ and $\beta_{i}$ are $q$-Weil numbers, i.e. that there
exist positive integers $a_{1},\dots,a_{d_{1}},b_{1},\dots,b_{d_{2}}$ such
that for any Archemedian place $|\cdot|_{\infty}$ we have
$|\alpha_{i}|_{\infty}=q^{\frac{a_{i}}{2}}$ and
$|\beta_{i}|_{\infty}=q^{\frac{b_{i}}{2}}$. We also understand what happens at
the finite places away from $p$: both the $\alpha_{i}$ and the $\beta_{i}$ are
$\ell$-adic units for any prime $\ell\neq p$.

The $p$-adic properties of these numbers are less definitive. There is no
exact formula as in the $\ell$-adic and Archimedean cases. In general, the
best one can hope for are lower bounds on the $p$-adic valuations, i.e., that
the Newton polygon of $\rho$ lies above some sort of Hodge polygon. Under
additional congruence conditions involving $p$ and the local monodromy of
$\rho$, one may hope to completely determine the $p$-adic valuations (i.e.
that the Newton and Hodge polygons coincide). The classical approach to
‘Newton over Hodge’ results for exponential sums has been restricted to the
case where $X$ is an algebraic torus. This wonderful theory was developed by
Robba, Adolphson-Sperber, Wan, and others, building off of ideas of Dwork. For
instance, Adolphson-Sperber prove a general ‘Newton over Hodge’ result for
exponential sums over $\mathbb{G}_{m}^{d}$ (see [1]) and Wan establishes a
criterion for equality of the two polygons (see [27]). Outside of the
$\mathbb{G}_{m}^{d}$ case nothing was known until recent work of the first
author (see [16] and [17]). In these articles Kramer-Miller establishes a
‘Newton over Hodge’ theorem for abelian $L$-functions on curves of any genus.
Unfortunately, this work ignores the question of touching or equality between
these two polygons.

The purpose of this article is to study touching between the Newton and Hodge
polygons of character sums on higher genus curves. Our main result establishes
a local criterion that is necessary and sufficient for the Newton and Hodge
polygons to touch at a vertex. It roughly states the following: Assume the
compactification of $X$ is an ordinary curve $\overline{X}$.111By the Deuring-
Shafarevich formula, if $\overline{X}$ is non-ordinary, we immediately know
that the Newton polygon will be strictly above the Hodge polygon For each
point at infinity $P\in\overline{X}\backslash X$, we may localize $\rho$ at
$P$ and construct a local-to-global extension $\rho_{P}^{\mathrm{ext}}$, in
the sense of Gabber-Katz. This $\rho_{P}^{\mathrm{ext}}$ is a representation
of $\pi_{1}(\mathbb{G}_{m})$, whose localization at $\infty$ agrees with the
localization of $\rho$ at $P$. When $\rho$ has order $p^{n}$, we prove that
the Newton and Hodge polygons of $\rho$ share a vertex if and only if the
Newton and Hodge polygons of $\rho_{P}^{\mathrm{ext}}$ share a corresponding
vertex for every $P\in\overline{X}\backslash X$. This allows us to reduce the
study of Newton-Hodge interaction to the classical $\mathbb{G}_{m}$ case. In
particular, we establish a necessary and sufficient local condition for the
Newton and Hodge polygons of $\rho$ to agree. In subsequent papers we will
apply these local-to-global theorems to study $\mathbb{Z}_{p}$-towers of
curves.

###  1.2 Main results

Suppose now that $X$ is a smooth affine curve over $\mathbb{F}_{q}$ with
smooth compactification $\overline{X}$. Let $S$ denote the complement of $X$
in $\overline{X}$. Let $\rho:\pi_{1}(X)\to\mathbb{C}^{\times}_{p}$ be a
character of order $p^{n}$, assumed to be totally ramified at each $P\in S$.
We will write

$L(\rho,s)=\prod_{x\in|X|}\frac{1}{1-\rho(\mathrm{Frob}_{x})s^{\deg(x)}}\in
1+s\mathbb{Z}_{p}[\rho][s]$

for the Artin $L$-function of $\rho$ and
$\operatorname{\mathrm{NP}}_{q}(\rho)$ for the $q$-adic Newton polygon of
$L(\rho,s)$.

For each $P\in S$, let $F_{P}$ denote the local field at $P$ and let $G_{P}$
denote the absolute Galois group of $F_{P}$. We localize $\rho$ to obtain a
continuous character

$\rho_{P}:G_{P}\to\mathbb{C}^{\times}.$

By a theorem of Katz-Gabber, the local character $\rho_{P}$ extends to a
character $\rho_{P}^{\mathrm{ext}}$ of the fundamental group
$\pi_{1}(\mathbb{A}_{\mathbb{F}_{q}}^{1})$. We may therefore speak of the
_local Newton polygon_
$\operatorname{\mathrm{NP}}_{q}(\rho_{P}^{\mathrm{ext}})$ of $\rho$ at $P$. If
$d_{P}$ denotes the Swan conductor of $\rho$ at $P$, then we define the _local
Hodge polygon_ $\operatorname{\mathrm{HP}}_{q}(\rho_{P}^{\mathrm{ext}})$ of
$\rho$ at $P$ to be the convex polygon with slope set

$\left\\{\frac{1}{d_{P}},\dots,\frac{d_{P}-1}{d_{P}}\right\\}.$

By a theorem of Liu-Wei [19], the Newton polygon
$\operatorname{\mathrm{NP}}_{q}(\rho_{P}^{\mathrm{ext}})$ lies above
$\operatorname{\mathrm{HP}}_{q}(\rho_{P}^{\mathrm{ext}})$ and both polygons
have the same terminal point.

Let $g$ denote the genus of $\overline{X}$. The _global Hodge polygon_
$\operatorname{\mathrm{HP}}_{q}(\rho)$ of $\rho$ is obtained by concatenating
the local Hodge polygons with $g-1+|S|$ segments of slope $0$ and slope $1$.
In other words, $\operatorname{\mathrm{HP}}_{q}(\rho)$ is the the convex
polygon with slope set:

$\big{\\{}\underbrace{0,\dots,0}_{g-1+|S|}\big{\\}}\sqcup\big{\\{}\underbrace{1,\dots,1}_{g-1+|S|}\big{\\}}\sqcup\bigsqcup_{P\in
S}\left\\{\frac{1}{d_{P}},\dots,\frac{d_{P}-1}{d_{P}}\right\\}.$

By a theorem of the first author [16], the Newton polygon
$\operatorname{\mathrm{NP}}_{q}(\rho)$ lies above
$\operatorname{\mathrm{HP}}_{q}(\rho)$ and both polygons have the same
terminal point.

To state our main theorem we introduce the following convention: For each
$r>0$, the _$r$ -truncated Newton polygon_
$\operatorname{\mathrm{NP}}_{q}^{<r}(\rho)$ is obtained from
$\operatorname{\mathrm{NP}}_{q}(\rho)$ by removing all segments of slope $\geq
r$. We will use similar notation for all Newton and Hodge polygons attached to
$\rho$.

######  Theorem 1.1.

Assume that $X$ is ordinary. Let $r\in[0,1]$. Then
$\operatorname{\mathrm{HP}}_{q}^{<r}(\rho)$ and
$\operatorname{\mathrm{NP}}_{q}^{<r}(\rho)$ have the same terminal point if
and only if $\operatorname{\mathrm{HP}}_{q}^{<r}(\rho_{P}^{\mathrm{ext}})$ and
$\operatorname{\mathrm{NP}}_{q}^{<r}(\rho_{P}^{\mathrm{ext}})$ have the same
terminal point for each $P\in S$.

######  Remark 1.2.

In Theorem 7.13 we prove the same result for a much wider class of characters,
which we call _$\bm{\delta}$ -Hodge_ characters. This general statement will
be used to study equicharacteristic $L$-functions and $T$-adic $L$-functions
in [18] and [25].

As a corollary to Theorem 1.1, we obtain a complete characterization of when
the Newton polygon $\operatorname{\mathrm{NP}}_{q}(\rho)$ coincides with
$\operatorname{\mathrm{HP}}_{q}(\rho)$. In this paper, we prove the following
theorem for $n=1$. The general case will follow from the $n=1$ case and from
our work on $\mathbb{Z}_{p}$-towers in the sequel [18]:

######  Theorem 1.3.

$\operatorname{\mathrm{NP}}_{q}(\rho)=\operatorname{\mathrm{HP}}_{q}(\rho)$ if
and only if

  1. 1.

$\overline{X}$ is ordinary.

  2. 2.

$\delta_{P}=d_{P}/p^{n-1}\in\mathbb{Z}$ for all $P\in S$.

  3. 3.

$p\equiv 1\pmod{\delta_{P}}$ for all $P\in S$.

######  Remark 1.4.

The “only if” part of Theorem 1.3 can be deduced from Theorem 1.1 for all $n$.
Indeed, assume
$\operatorname{\mathrm{NP}}_{q}(\rho)=\operatorname{\mathrm{HP}}_{q}(\rho)$.
Then by the Deuring-Shafarevich formula we know that $\overline{X}$ is
ordinary. From Theorem 1.1 we know that
$\operatorname{\mathrm{HP}}_{q}(\rho_{P}^{\mathrm{ext}})=\operatorname{\mathrm{NP}}_{q}(\rho_{P}^{\mathrm{ext}})$
for each $P\in S$. In particular, we know that
$\operatorname{\mathrm{NP}}_{q}(\rho_{P}^{\mathrm{ext}})$ contains the vertex
$(1,\frac{1}{d_{P}})$. However, the vertices of
$\operatorname{\mathrm{NP}}_{q}(\rho_{P}^{\mathrm{ext}})$ are contained in
$\frac{1}{p^{n-1}(p-1)}\mathbb{Z}$. This implies $d_{P}|p^{n-1}(p-1)$. To
prove 2 and 3 it suffices to show $p^{n-1}|d_{P}$. In particular, it is enough
to prove $d_{P}\geq p^{n-1}$. Let $F_{P,n}/F_{P}$ be the finite extension
corresponding to $\ker(\rho_{P})$, so that
$\mathrm{Gal}(F_{P,n}/F_{P})=\mathbb{Z}/p^{n}\mathbb{Z}$. We then let
$F_{P,i}\subset F_{P,n}$ be the field corresponding to
$p^{n-i}\mathbb{Z}/p^{n}\mathbb{Z}$. Let $d_{P,i}$ be the largest upper
numbering ramification break of $\mathrm{Gal}(F_{P,i}/F_{P})$ (so that
$d_{P,n}$ is $d_{P}$). By class field theory we know that $d_{P,i+1}\geq
pd_{P,i}$ (see e.g. [15, Lemma 5.2] for more details), which implies
$d_{P,n}\geq p^{n-1}$.

######  Remark 1.5.

We may apply Theorem 1.3 to determine the Newton polygon of certain covers of
an ordinary curve $\overline{X}$. Namely, let $X^{\prime}\to X$ be a finite
Galois $\mathbb{Z}/p^{n}\mathbb{Z}$-cover, which is totally ramified at each
$P\in S$. We will assume that for each $P$ the largest ramification break (in
upper numbering) of $X^{\prime}/X$ at $P$ is of the form $\delta_{P}p^{n-1}$,
where $\delta_{P}\in\mathbb{Z}$ and $p\equiv 1\mod\delta_{P}$. The zeta
function of $\overline{X}^{\prime}$ has a product decomposition

$\displaystyle
Z(\overline{X}^{\prime},s)=Z(\overline{X},s)\prod_{\rho}L(\rho,s),$ (1)

where the product is over all nontrivial characters $\rho$ of
$\mathrm{Gal}(\overline{X}/X)$. If $\rho$ has order $p^{j}$, then the Swan
conductor of $\rho$ at $P$ is exactly $\delta_{P}p^{j-1}$. In particular, we
may apply Theorem 1.3 to each $L$-function in (1) to completely determine the
Newton polygon of $\overline{X}^{\prime}$.

In the case of an Artin-Schreier cover, we are able to recover a result of
Booher and Pries (see [3, Corollary 4.3]). In fact, we obtain a much stronger
result: In [3], Booher and Pries show that there _exists_ an Artin-Schreier
cover $X^{\prime}\to X$ such that the $q$-adic Newton polygon of
$\overline{X}^{\prime}$ has slope set

$\big{\\{}\underbrace{0,\dots,0}_{\mathclap{pg+(p-1)(|S|-1)}}\big{\\}}~{}\sqcup~{}\big{\\{}\underbrace{1,\dots,1}_{\mathclap{pg+(p-1)(|S|-1)}}\big{\\}}\sqcup\bigsqcup_{P\in
S}\left\\{\frac{1}{d_{P}},\dots,\frac{d_{P}-1}{d_{P}}\right\\}^{\mathrlap{\times(p-1)}},$

where the $\times n$ superscript means each slope appears with multiplicity
$n$. By Grothendeick’s specialization theorem, this determines the Newton
polygon for a generic $\mathbb{Z}/p\mathbb{Z}$-cover of $X$ with these
ramification invariants. However, by the above paragraph we see that this
generic Newton polygon occurs for _every_ Artin-Schreier cover $X^{\prime}\to
X$ with the same ramification breaks.

###  1.3 Outline of proof

The starting point of our proof is the method developed by the first author in
[17] and [16]. However, the methods in this earlier work only produce lower
bounds on $\operatorname{\mathrm{NP}}_{q}(\rho)$, and are woefully inadequate
for pinpointing exact vertices. The necessary new ingredient is a perturbation
theory for $p$-adic operators, which allows us to relate the Newton polygons
$\operatorname{\mathrm{NP}}_{q}(\rho)$ and
$\operatorname{\mathrm{NP}}_{q}(\rho_{P}^{\mathrm{ext}})$.

##### Characters and Liftings

Our general approach is to study $L(\rho,s)$ via the Monsky trace formula,
which expresses $L(\rho,s)$ as a ratio of Fredholm determinants of operators.
This requires us to lift the Frobenius on $U$ to characteristic $0$. We use
the lifting construction developed by the first author in [17]. Choose a tame
Belyi map $\eta:\overline{X}\to\mathbb{P}_{\mathbb{F}_{q}}^{1}$ for which
$\eta(P)=0$ for all $P\in S$. Let $U\subseteq X$ denote the étale locus of
$\eta$. Let $A$ be a smooth lifting of the coordinate ring of $U$ to
characteristic $0$, and write $A^{\dagger}$ for the $p$-adic weak completion
of $A$ (see §2.1). Using $\eta$ we construct a lifting of Frobenius
$\sigma:A^{\dagger}\to A^{\dagger}$ whose local expression at each $P\in S$ is
particularly simple, see §4.3.

Let $\rho_{U}$ denote the restriction of $\rho$ to $\pi_{1}(U)$. Let
$\pi\in\mathbb{Z}_{p}[\rho]$ be a uniformizer, and consider the base change

$A_{\pi}^{\dagger}=\mathbb{Z}_{p}[\rho]\otimes_{\mathbb{Z}_{p}}A^{\dagger}.$

In this setting, the Monsky trace formula states that there is a
$\mathbb{Z}_{q}[\rho]$-linear endomorpshism $\Theta_{q}:A_{\pi}^{\dagger}\to
A_{\pi}^{\dagger}$ for which

$\displaystyle L(\rho_{U},s)$
$\displaystyle=\frac{\det(1-s\Theta_{q}|A_{\pi}^{\dagger}[\tfrac{1}{\pi}])}{\det(1-qs\Theta_{q}|A_{\pi}^{\dagger}[\tfrac{1}{\pi}])}.$

##### The Space $A_{\pi}^{\dagger}$

It follows from the trace formula that the truncated Newton polygon
$\operatorname{\mathrm{NP}}_{q}^{<1}(\rho_{U})$ agrees with that of the
characteristic series $\det(1-s\Theta_{q}|A_{\pi}^{\dagger}[\tfrac{1}{\pi}])$.
Thus, our goal in this paper is to estimate the Newton polygon of this series.
In the classical situation $X=\mathbb{G}_{m}$, the ring $A^{\dagger}$ may be
identified with the $p$-adic weak completion of the ring of Laurent
polynomials $\mathbb{Z}_{q}[t^{\pm}]$. The $\mathbb{Z}_{q}$-module structure
of $A^{\dagger}$ is quite explicit, as the elements $t^{k}$ for
$k\in\mathbb{Z}$ form a natural topological basis. Choosing a suitable basis
for the lifting $A^{\dagger}$ is a key point of the papers [17] and [16],
where a Riemann-Roch argument is used to construct a basis with prescribed
poles at each branch point $P$ of $\eta$.

For each such $P$, choose a local parameter $t_{P}$ at $P$ and consider the
ring $\mathcal{A}_{P}=\mathbb{Z}_{q}(\\!(t_{P})\\!)$. This ring is a lifting
of the local field $F_{P}$ to characteristic $0$, and there is a unique map
$A^{\dagger}\to\mathcal{A}_{P}^{\dagger}$ given by “expanding functions in the
parameter $t_{P}$”. The Riemann-Roch theorem implies that there is an exact
sequence of $\mathbb{Z}_{q}$-modules

$0\to L\to A^{\dagger}\to\bigoplus_{P}\mathcal{A}_{P}^{\dagger,\mathrm{tr}}\to
0,$

where $L$ is a free $\mathbb{Z}_{q}$-module of finite rank, and
$\mathcal{A}_{P}^{\dagger,\mathrm{tr}}$ is a space of truncated Laurent series
in $\mathcal{A}_{\pi,P}$ with prescribed poles. The space
$\mathcal{A}_{P}^{\dagger,\mathrm{tr}}$ admits a topological basis of the form
$B_{P}=\\{t_{P}^{-k}:k>\mu(P)\\}$, where $\mu(P)$ is a parameter depending on
the branching of the map $\eta$ at $P$. Is is then straightforward to “lift”
these bases to produce a global basis $B$ for $A^{\dagger}$.

##### Perturbing $\pi$-adic Operators

Let $P$ be a branch point of $\eta$, and let
$\mathcal{A}_{\pi,P}=\mathbb{Z}_{q}\otimes_{\mathbb{Z}_{p}}\mathcal{A}_{P}$.
The operator $\Theta_{q}$ extends in a natural way to the $\pi$-adic weak
completion $\mathcal{A}_{\pi,P}^{\dagger}$. If $P\in S$, then the truncated
space $\mathcal{A}_{\pi,P}^{\dagger,\mathrm{tr}}$ is invariant with respect to
the action of $\Theta_{q}$222Strictly speaking, this is only true after
replacing $\Theta_{q}$ with a suitable “twist.” This is a technical point
which we ignore in this introduction., and we may therefore speak of the local
Fredholm determinant

$\det(I-s\Theta_{q}|\mathcal{A}_{\pi,P}^{\dagger,\mathrm{tr}}[\tfrac{1}{\pi}])\in
1+s\mathbb{Z}_{q}[\rho]\llbracket s\rrbracket.$

In §7.2, we relate the Newton polygon of this series to that of the local
extension $\rho_{P}^{\mathrm{ext}}$. In particular, we are able to give a
lower bound for this Newton polygon in terms of the local Hodge polygon
$\operatorname{\mathrm{HP}}_{q}(\rho_{P}^{\mathrm{ext}})$.

The key insight of this paper is that the Newton polygon of $\Theta_{q}$
acting on $A_{\pi}^{\dagger}$ should be closely related to the local Newton
polygons at the ramified points $P\in S$. Let $\Psi=\Theta_{q}|B$ denote the
matrix of $\Theta_{q}$ with respect to our chosen global basis. In §7.3 we
construct a new matrix $\Psi^{\prime}$ in terms of the local matrices
$\Theta_{q}|B_{P}$ at each $P\in S$. To quantify the close relationship
between $\Psi$ and $\Psi^{\prime}$, we introduce in §7.1 the notion of a
_$\pi$ -adic $r$-perturbation_. The $r$-perturbation condition implies the
following relation between the Newton and Hodge polygons of both matrices:

$\displaystyle\begin{array}[]{l}\text{Touching between the $\pi$-adic
Hodge}\\\ \text{and Newton polygons of $\Psi$ for}\\\ \text{slopes smaller
than $r$}\end{array}$ $\displaystyle\iff\begin{array}[]{l}\text{Touching
between the $\pi$-adic Hodge}\\\ \text{and Newton polygons of $\Psi^{\prime}$
for}\\\ \text{slopes smaller than $r$}\end{array}$

In §7.3 we show using the estimates of Section 6 that $\Psi^{\prime}$ is an
$r$-perturbation of $\Psi$ for appropriate $r$. This allows us to deduce
Theorem 1.1.

###  1.4 Applications to $\mathbb{Z}_{p}$-towers of curves

Our primary motivation for this article is to study the $p$-adic variation of
$L$-functions along $\mathbb{Z}_{p}$-towers of curves. A _$\mathbb{Z}_{p}$
-tower_ $X_{\infty}/X$ is a sequence of finite Galois coverings

$\cdots\to X_{2}\to X_{1}\to X_{0}=X,$

together with compatible identifications
$\mathrm{Gal}(X_{n}/X)\cong\mathbb{Z}/p^{n}\mathbb{Z}$ for all $n$.
Equivalently, a $\mathbb{Z}_{p}$-tower over $X$ corresponds to a continuous
and surjective map $\rho:\pi_{1}(X)\to\mathbb{Z}_{p}$. We will typically
regard $X_{\infty}/X$ as a family of characters of $\pi_{1}(X)$ as follows:
For every continuous $p$-adic character
$\chi\in\mathrm{Hom}(\mathbb{Z}_{p},\mathbb{C}_{p}^{\times})$, we obtain a
composite character

$\rho_{\chi}=\chi\circ\rho:\pi_{1}(X)\to\mathbb{C}_{p}^{\times}.$

Daqing Wan has devised a program (see [28]) to understand how the Newton
polygon $\operatorname{\mathrm{NP}}_{q}(\rho_{\chi})$ varies as a function of
$\chi$. The general philosophy is that for well behaved towers we expect the
Newton polygons to exhibit a certain degree of regularity as $\chi$ varies.
Recently, there has been much progress surrounding the case $X=\mathbb{A}^{1}$
(see e.g. [6] for one of the first articles on the topic or [14] for results
on a general class of towers). However, essentially nothing is known for
higher genus curves or when multiple points along the tower are ramified. The
local-to-global results of this article allow us to approach these questions
when the base curve is ordinary. This is the content of the sequel articles
[18] and [25].

####  1.4.1 $\mathbb{Z}_{p}$-Towers with Strictly Stable Monodromy

Let $\bm{\delta}=(\delta_{P})_{P\in S}$ be a tuple of positive rational
numbers in $\mathbb{Z}[\tfrac{1}{p}]$. We say that the tower $X_{\infty}/X$
has _$\bm{\delta}$ -stable monodromy_ if for each $P\in S$, the highest
ramification break of $X_{n}/X$ at $P$ is of the form $p^{n-1}\delta_{P}$ for
all $n\gg 0$.

######  Theorem 1.6 ([18]).

Suppose that $\overline{X}$ is ordinary. Let $X_{\infty}/X$ be a
$\mathbb{Z}_{p}$-tower of curves with $\bm{\delta}$-stable monodromy. Then for
a finite character $\chi:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ of order
$p^{n}$:

  1. 1.

(Slope Uniformity) The $q$-adic Newton slopes of $\rho_{\chi}$ are
equidistributed in the interval $[0,1]$ as $n\to\infty$.

  2. 2.

(Slope Stability) Suppose that $\delta_{P}\in\mathbb{Z}$ and that
$\delta_{P}\equiv 1\pmod{p}$ for all $P\in S$. Then
$\operatorname{\mathrm{NP}}_{q}(\rho_{\chi})$ has slope set

$\\{\underbrace{0,...,0}_{g-1+|S|}\\}\sqcup\\{\underbrace{1,...,1}_{g-1+|S|}\\}\sqcup\bigsqcup_{P\in
S}\left\\{\frac{1}{\delta_{P}p^{n-1}},\dots\frac{\delta_{P}p^{n-1}-1}{\delta_{P}p^{n-1}}\right\\}.$

In particular, the invariants $\delta_{1},...,\delta_{P}$ completely determine
the Newton polygon of each curve $X_{n}$.

This theorem generalizes work of Kosters-Zhu (see [14]) for
$\bm{\delta}$-stable towers over $\mathbb{A}_{\mathbb{F}_{q}}^{1}$. To the
best of our knowledge, there were no prior examples of slope uniform or slope
stable towers beyond the $\mathbb{A}_{\mathbb{F}_{q}}^{1}$ case. Using the
methods of [16] we obtain similar theorems for $\bm{\delta}$-stable towers
twisted by a tame finite character
$\psi:\pi_{1}(X)\to\mathbb{Z}_{p}^{\times}$.

####  1.4.2 Overconvergent $\mathbb{Z}_{p}$-Towers

Any tower $X_{\infty}/X$ is determined by an Artin-Schreier-Witt equation
$F(x)-x=f$, where $f$ is an element of the $p$-typical Witt vectors of the
coordinate ring of $X$. We say that $X_{\infty}/X$ is _overconvergent_ if $f$
may be taken to be an overconvergent Witt vector in the sense of [5]. This is
a natural condition on the growth of the poles of the Witt coordinates of $f$,
closely related to the notion of overconvergence found in rigid cohomology. We
remark that any overconvergent tower is $\bm{\delta}$-stable for some
$\bm{\delta}$, but that most $\bm{\delta}$-stable towers are not
overconvergent.

A notable class of examples are the towers over
$X=\mathbb{A}_{\mathbb{F}_{q}}^{1}$ studied in work of Davis-Wan-Xiao [6].
Those authors prove that a special class of towers are slope-uniform and
slope-stable. In fact, they prove a beautiful _spectral halo theorem_ which
describes the $p$-adic variation of the zero locus of $L(\rho_{\chi},s)$ as
$\chi$ varies through the character space
$\mathrm{Hom}(\mathbb{Z}_{p},\mathbb{C}_{p}^{\times})$. In the final part of
this series of articles [25], the second author shows that the spectral halo
theorem holds for a general overconvergent tower
$X_{\infty}/\mathbb{A}_{\mathbb{F}_{q}}^{1}$. By combining this result with
our local-to-global theorems, we obtain the following stability result for
overconvergent towers over an ordinary curve:

######  Theorem 1.7.

Suppose that $X$ is ordinary. Let $X_{\infty}/X$ be an overconvergent
$\mathbb{Z}_{p}$-tower of curves. There exist a non-negative integer $m_{0}$
and postivie rational numbers $\alpha_{1},...,\alpha_{d}$ with the following
property: For every $\chi$ of order $p^{n}$ with $n>m_{0}$, the slope set of
$\operatorname{\mathrm{NP}}_{q}(\rho_{\chi})$ is

$\\{\underbrace{0,...,0}_{g-1}\\}\sqcup\\{\underbrace{1,...,1}_{g-1+|S|}\\}\sqcup\bigsqcup_{P\in
S}\bigsqcup_{i=0}^{p^{n-m_{0}}-1}\left\\{\frac{i}{p^{n-m_{0}-1}},\frac{\alpha_{1}+i}{p^{n-m_{0}-1}},...,\frac{\alpha_{d}+i}{p^{n-m_{0}-1}}\right\\}.$

As a byproduct of our approach, we obtain similar estimates for the Newton
polygon $\operatorname{\mathrm{NP}}_{q}^{<1}(\rho_{\chi})$ for _any_ $\chi$
(finite or infinite) near the boundary of the character space
$\mathrm{Hom}(\mathbb{Z}_{p},\mathbb{C}_{p}^{\times})$. This may be regarded
as a “small-slope” analogue of the spectral halo theorem of Davis-Wan-Xiao.
Unfortunately, our perturbation approach does not produce any information
regarding the zeroes of $L(\rho_{\chi},s)$ outside of the disk
$v_{p}(s)>-v_{p}(q)$. It seems likely that new techniques are necessary to
study zeros in this region.

###  1.5 Acknowledgments

We would like to thank Daqing Wan for sharing his insight on exponential sums
and towers. In addition, we acknowledge Kiran Kedlaya for discussing this work
with us.

##  2 Growth Conditions

Throughout this article, our basic coefficient ring will be a complete
discrete valuation ring $R$ with maximal ideal $\mathfrak{m}$ and residue
field $\mathbb{F}_{p}$ (we do not assume that $R$ has characteristic $0$). We
fix a non-zero topologically nilpotent element $\pi\in R$. We will frequently
work with algebras of functions over $R$, which are $\pi$-adically
overconvergent. In this section we spell out our conventions regarding these
“growth conditions” and introduce some basic constructions.

###  2.1 Weak Completions

Let $A$ be an $R$-algebra equipped with the $\pi$-adic topology. We will write
$A^{\infty}$ for the $\pi$-adic completion of $A$. Given real numbers $m$, $b$
with $m>0$, let $A^{m}(b)$ denote the set of $a\in A^{\infty}$ with the
following property: there exist $a_{1},...,a_{n}\in A$ and polynomials
$p_{j}\in\pi^{j}R[X_{1},...,X_{n}]$ such that

$a=\sum_{j}p_{j}(a_{1},...,a_{n})$ (2)

and $\deg(p_{j})<mj+b$. Then each $A^{m}(b)$ is a $\pi$-adically complete
submodule of $A^{\infty}$. The following basic properties are immediate:

######  Lemma 2.1.

For each $m>0$,

  1. 1.

$A^{m}(b_{1})\cdot A^{m}(b_{2})\subseteq A^{m}(b_{1}+b_{2})$.

  2. 2.

$\pi^{k}A^{m}(b)\subseteq A^{m}(b-km)$.

######  Definition 2.2.

The _weak completion_ $A^{\dagger}$ of $A$ is the union of the submodules
$A^{m}(b)$.

It is clear from Lemma 2.1 that $A^{\dagger}$ is a subring of $A^{\infty}$ and
is independent of the choice of $\pi\in R$. By [21, Theorem 1.4], the natural
map $A/\pi A\to A^{\dagger}/\pi A^{\dagger}$ is an isomorphism. Let us say
that a subset $S\subseteq A^{\dagger}$ is a set of _weak generators_ for
$A^{\dagger}$ if every $a\in A^{\dagger}$ has the form (2), where the $a_{i}$
are elements of $S$. For our purposes we will only consider weak completions
in two situations:

  1. 1.

Let $A$ be an $R$-algebra such that $A^{\dagger}$ admits a finite set of weak
generators. In this case a theorem of Fulton [8] guarantees that $A^{\dagger}$
is Noetherian.

  2. 2.

Let $A=R_{q}(\\!(t)\\!)$ be the ring of formal Laurent series over $R_{q}$.
Then

$A^{\dagger}\cong(R_{q}\llbracket
t\rrbracket\otimes_{R}R[X]^{\dagger})/(tX-1).$

This is once again a Noetherian ring. In this case we have:

$A^{m}(b)=\left\\{\sum_{k=-\infty}^{\infty}a_{k}t^{-k}:v_{\pi}(a_{k})\geq\frac{k-b}{m}\text{
for }k>0\right\\}.$

###  2.2 Weak Base Change

A typical situation for us will be as follows: Let $A$ be a
$\mathbb{Z}_{p}$-algebra, equipped with the $p$-adic topology. We define the
base change $A_{\pi}=R\otimes_{\mathbb{Z}_{p}}A$, which we equip with the
$\pi$-adic topology. We will always assume that the modules $A^{m}(b)$ are
given by $p$-adic growth conditions, whereas the $A_{\pi}^{m}(b)$ are given by
$\pi$-adic growth conditions (hence the choice of subscript).

Since many of our constructions will be given initially over $\mathbb{Z}_{p}$,
it is convenient to have a “comparison” between the weak completions
$A^{\dagger}$ and $A_{\pi}^{\dagger}$. First, we require a basic result from
commutative algebra:

######  Lemma 2.3.

Consider an exact sequence of $R$-modules

$0\to L\to M\xrightarrow{f}N.$ (3)

Let $\overline{f}:\overline{M}\to\overline{M}$ denote the reduction of $f$
modulo $\pi$. Then:

  1. 1.

If $M$ is $\pi$-adically complete, $N$ is $\pi$-adically separated, and
$\overline{f}$ is surjective, then $f$ is surjective.

  2. 2.

If $N$ is $\pi$-torsion free and $f$ is surjective, then
$\overline{L}=\ker(\overline{f})$.

  3. 3.

If $M$ and $N$ are $\pi$-torsion free, $M$ is $\pi$-adically separated, and
$\overline{f}$ is injective, then $f$ is injective.

###### Proof.

Let $n\in N$. Since $\overline{f}$ is surjective, there exists $m_{0}\in M$
such that $n=f(m_{0})+\pi n_{1}$, for some $n_{1}\in N$. Repeating this
procedure, we inductively construct $m_{i}\in M$ such that $n_{i}=f(m_{i})+\pi
n_{i+1}$. Since $M$ is $\pi$-adically complete, the sum

$m=\sum_{i=0}^{\infty}m_{i}\pi^{i}$

converges in $M$. The difference $f(m)-n$ vanishes because $N$ is
$\pi$-adically separated, proving (1). Since $R$ is a discrete valuation ring,
if $N$ is $\pi$-torsion free then $N$ is flat over $R$. Claim (2) then follows
by tensoring (3) with $R/\pi R$.

Suppose then that $M$ and $N$ are $\pi$-torsion free, $M$ is $\pi$-adically
separated, and that $\overline{f}$ is injective. Since $M$ and $N$ are flat
over $R$, by tensoring with the exact sequence

$0\to\pi^{j+1}R\to\pi^{j}R\to\pi^{j}R/\pi^{j+1}R\to 0,$

we have an identification
$\pi^{j}M/\pi^{j+1}M\cong\pi^{j}R/\pi^{j+1}R\otimes_{R/\pi R}\overline{M}$ and
similarly for $N$. Since $R$ is a discrete valuation ring,
$\pi^{j}R/\pi^{j+1}R$ is flat over $R/\pi R$ for all $j\geq 0$. By tensoring
$\overline{f}$ with $\pi^{j}R/\pi^{j+1}R$, and using the above identification,
we see that $f$ induces an injective map

$\pi^{j}M/\pi^{j+1}M\to\pi^{j}N/\pi^{j+1}N.$

It follows that the kernel of $f$ lies in $\bigcap_{j}\pi^{j}M=0$. Since $M$
is $\pi$-adically separated, this proves (3). ∎

By base change, the natural map $A\to A^{\dagger}$ induces a map
$A_{\pi}\to(A^{\dagger})_{\pi}$. Passing to weak completions, we obtain a map

$A_{\pi}^{\dagger}\to(A^{\dagger})_{\pi}^{\dagger}.$ (4)

Our “comparison” between $A^{\dagger}$ and $A_{\pi}^{\dagger}$ is provided by
the following:

######  Lemma 2.4 (Weak Base Change).

Suppose that $A$ is flat over $\mathbb{Z}_{p}$. Then the map (4) is an
isomorphism.

###### Proof.

The flatness condition guarantees that $A_{\pi}^{\dagger}$ and
$(A^{\dagger})_{\pi}^{\dagger}$ are $\pi$-torsion free and $\pi$-adically
separated. By Lemma 2.3, it suffices to prove that (4) is surjective. Since
$(A^{\dagger})_{\pi}$ is a set of weak generators for
$(A^{\dagger})_{\pi}^{\dagger}$, and since the image of (4) is weakly
complete, we only need to show that the image contains $(A^{\dagger})_{\pi}$.
Now $A$ constitutes a set of weak generators for $A^{\dagger}$, so every
element of $(A^{\dagger})_{\pi}$ is a finite sum of elements of the form

$r\otimes\sum_{j}p_{j}(a_{1},...,a_{n}),$

where $r\in R$, $a_{1},...,a_{n}\in A$ and $p_{j}\in
p^{j}\mathbb{Z}_{p}[X_{1},...,X_{n}]$ has degree $\leq mj+b$. Since $p$ is
topologically nilpotent in $R$, we see that this is exactly the image of the
convergent sum

$\sum_{j}(r\otimes p_{j})(a_{1},...,a_{n})\in A_{\pi}^{\dagger},$

where we regard $r\otimes p_{j}$ as a polynomial in $R[X_{1},...,X_{n}]$. ∎

##  3 Fredholm Theory

Let $K=R[\tfrac{1}{\pi}]$ denote the field of fractions of $R$. If $q$ is a
power of $p$, then we define the ring
$R_{q}=\mathbb{Z}_{q}\otimes_{\mathbb{Z}_{p}}R$ and its field of fractions
$K_{q}=K\otimes_{R}R_{q}$. We will now discuss the Fredholm theory of nuclear
operators acting on $K_{q}$-vector spacse and the corresponding theory of
Newton and Hodge polygons of such operators.

###  3.1 Spectral Theory of Completely Continuous Operators

####  3.1.1 Normed Vector Spaces

######  Definition 3.1.

A _normed vector space_ over $K_{q}$ is a $K_{q}$-vector space of the form
$V=K\otimes_{R}M$, where $M$ is a $\pi$-adically separated $R_{q}$-module. We
say that $V$ is a _Banach space_ over $K_{q}$ if $M$ is $\pi$-adically
complete.

Let $V=K\otimes_{R}M$ be a normed vector space over $K_{q}$. We topologize $V$
by taking the $R_{q}$-submodules $\mathfrak{m}^{n}M$ to be a fundamental
system of neighborhoods of $0$ in $V$. As the terminology suggests, the choice
of $M\subseteq V$ uniquely determines a norm $\|\cdot\|_{\pi}$ on $V$ defined
as follows: If $\pi R=\mathfrak{m}^{d}R$, then we define the _$\pi$ -adic
valuation_ on $V$ via

$v_{\pi}(x)=\frac{1}{d}\sup\\{n\in\mathbb{Z}:x\in\mathfrak{m}^{n}M\\}.$

The condition that $M$ is $\pi$-adically separated guarantees that
$\|x\|_{\pi}=p^{-v_{\pi}(x)}$ defines a norm on $V$. Since we can recover $M$
as the $R_{q}$-submodule of $x\in V$ with $\|x\|_{\pi}\leq 1$, the choice of
$M$ is equivlent to the norm $\|\cdot\|_{\pi}$ or the valuation $v_{\pi}$.

######  Example 3.2.

Let $I$ be a set. We define $b(I)$ to be the $K_{q}$-vector space of
$I$-families $x=(x_{i})_{i\in I}$ of elements of $K_{q}$ which are
$\pi$-adically bounded, in the sense that

$\|x\|_{\pi}=\sup_{i\in I}\|x_{i}\|<\infty.$

Then $b(I)$ is a Banach space over $K_{q}$. For each $i\in I$, let $e_{i}\in
b(I)$ denote the element whose $i$th component is 1, and whose other
components are all $0$. Then we may represent an element $x\in b(I)$ as a
formal sum

$x=\sum_{i\in I}x_{i}e_{i}.$ (5)

Let $c(I)$ be the subspace of $b(I)$ consisting of those $x$ for which
$x_{i}\to 0$ in the finite-complement topology on $I$. Then $c(I)$ is a Banach
space over $K_{q}$, and for each $x\in c(I)$ the formal sum (5) converges to
$x$ in the norm topology on $c(I)$.

Let $V=K\otimes_{R}M$ and $W=K\otimes_{R}N$ be normed vector spaces over
$K_{q}$. Let $M\hat{\otimes}_{R_{q}}N$ denote the $\pi$-adic completion of the
$R_{q}$-module $M\otimes_{R_{q}}N$. We define the _completed tensor product_
of $V$ and $W$ to be the $K_{q}$-Banach space
$V\hat{\otimes}_{K_{q}}W=K\otimes_{R}(M\hat{\otimes}_{R_{q}}N)$. Similarly,
for each $n\geq 0$ we let $\wedge^{n}M$ denote the $\pi$-adic completion of
the $n$th exterior power of $M$ (over $R_{q}$). We define the $n$th _completed
exterior power_ of $V$ to be the $K_{q}$-Banach space
$\wedge^{n}V=K\otimes_{R}\wedge^{n}M$.

We write $\mathrm{Hom}(V,W)$ for the $K_{q}$-vector space of _continuous_
$K_{q}$-linear maps $V\to W$, equipped with the usual norm

$\|\psi\|_{\pi}=\sup_{x\in M}\|\psi(x)\|_{\pi}<\infty.$

In particular, we will write $V^{*}=\mathrm{Hom}(V,K)$ for the _continuous_
dual space of $V$.

####  3.1.2 Completely Continuous Maps

######  Definition 3.3.

Let $V$ and $W$ be Banach spaces over $K_{q}$. We say that a continuous
$K_{q}$-linear map $\psi:W\to V$ is _completely continuous_ if $\psi$ is a
limit of operators of finite rank. Equivalently, $\psi$ is completely
continuous if and only if $\psi$ lies in the image of the natural map

$W^{*}\hat{\otimes}_{K_{q}}V\to\mathrm{Hom}(W,V).$

In particular, we identify the space of completely continuous operators $V\to
V$ with the Banach space $V^{*}\hat{\otimes}_{K_{q}}V$. The _trace map_ on
this space is defined in the usual way:

$\displaystyle\mathrm{Tr}:V^{*}\hat{\otimes}_{K_{q}}V$ $\displaystyle\to
K_{q}$ $\displaystyle f\otimes v$ $\displaystyle\mapsto f(v).$

If $\psi:V\to V$ is completely continuous, then for each $n\geq 0$ the $n$th
exterior power $\wedge^{n}\psi$ is a completely continuous operator on
$\wedge^{n}V$.

######  Definition 3.4.

Let $\psi:V\to V$ be a completely continuous operator. For each $n\geq 0$ we
define the $n$th _Fredholm coefficient_ to be

$c_{n}(\psi)=(-1)^{n}\mathrm{Tr}(\wedge^{n}\psi).$

The _Fredholm determinant_ of $\psi$ is the power series

$\det(I-s\psi)=\sum_{n=0}^{\infty}c_{n}(\psi)s^{n}\in 1+sK_{q}\llbracket
s\rrbracket.$

######  Lemma 3.5 ([10, Proposition 15]).

Let $\psi:V\to V$ be a completely continuous operator. Then the dual map
$\psi^{*}:V^{*}\to V^{*}$ is completely continuous, and there is an equality
of Fredholm series

$\det(I-s\psi)=\det(I-s\psi^{*}).$

####  3.1.3 Orthonormal and Integral Bases

######  Definition 3.6.

Let $V$ be a Banach space over $K_{q}$. Let $I$ be a set and let
$B=\\{e_{i}:i\in I\\}$ be a family of elements of $V$ indexed by $I$. We say
that $B$ is an _orthonormal basis_ for $V$ if, for every $x\in V$ there exists
a unique $(x_{i})\in c(I)$ such that $\|x\|_{\pi}=\sup_{i\in I}|x_{i}|_{\pi}$
and

$x=\sum_{i\in I}x_{i}e_{i}.$

Let $V=K_{q}\otimes_{R_{q}}M$ be a Banach space over $K_{q}$ and let
$B=\\{e_{i}:i\in I\\}$ be subset of $V$. By [10, Proposition 1], $B$ is an
orthonormal basis for $V$ if and only if $B\subseteq M$ and $B$ reduces to an
$\mathbb{F}_{q}$-basis for $\mathbb{F}_{q}\otimes_{R_{q}}M$. It follows that
every Banach space over $K_{q}$ admits an orthonormal basis. An orthonormal
basis for $V$ indexed by $I$ is equivalent to an isometric isomorphism of
Banach spaces $\iota:V\xrightarrow{\sim}c(I)$, sending each $x\in V$ to the
corresponding $I$-family $(x_{i})\in c(I)$.

######  Definition 3.7.

Let $V$ be a normed vector space over $K_{q}$. A _formal basis_ for $V$ is a
pair $(B,\iota)$, where $B=\\{e_{i}:i\in I\\}$ is a subset of $V$, and
$\iota:V\to b(I)$ is an isometric embedding sending $e_{i}\mapsto e_{i}$ for
all $i\in I$.

Typically we will abuse notation and refer to a formal basis $(B,\iota)$
simply by $B$, leaving the embedding into $b(I)$ implicit.

######  Example 3.8.

Let $V$ be a Banach space and let $B=\\{e_{i}:i\in I\\}$ be an orthonormal
basis for $V$. By the preceding discussion, $B$ is a formal basis for $V$ in a
natural way. For each $i\in I$, consider the continuous linear functional
$e_{i}^{*}\in V^{*}$ defined by $e_{i}^{*}(x)=x_{i}$. Let
$B^{*}=\\{e_{i}^{*}:i\in I\\}\subset V^{*}$. If $f\in V^{*}$ is any continuous
linear functional, then we define

$\iota(f)=(f(e_{i}))\in b(I).$

Then $\iota:V^{*}\cong b(I)$ is an isometric isomorphism of Banach spaces over
$K_{q}$, and the pair $(B^{*},\iota)$ is a formal basis for $V^{*}$.

Let $I$ be a set. Let $\Psi=(\psi_{i,j})$ be an $I\times I$ matrix with
entries in $K_{q}$. We say that $\Psi$ is of _trace class_ if $\psi_{i,i}\to
0$ in the finite-complement topology on $I$. In this case, we define the
_trace_ of $\Psi$ to be

$\mathrm{Tr}(\Psi)=\sum_{i\in I}\psi_{i,i}\in K_{q}.$

Choose any ordering on $I$. For each $n\geq 0$, the exterior power
$\wedge^{n}\Psi$ is a matrix with entries in $K_{q}$ indexed by the set

$\wedge^{n}I=\\{(i_{1},...,i_{n}):i_{1}<\cdots<i_{n}\\}.$

For each subset $J\subseteq I$, let $\Psi_{J}$ denote the $J\times J$
submatrix of $\Psi$ corresponding to $J$.

######  Definition 3.9.

Suppose that $\wedge^{n}\Psi$ is of trace class for all $n\geq 0$. We define
the $n$th _Fredholm coefficient_ of $\Psi$ to be

$c_{n}(\Psi)=(-1)^{n}\mathrm{Tr}(\wedge^{n}\Psi)=(-1)^{n}\sum_{|J|=n}\det(\Psi_{J}).$
(6)

The _Fredholm determinant_ of $\Psi$ is the power series

$\det(I-s\Psi)=\sum_{n=0}^{\infty}c_{n}(\Psi)s^{n}\in 1+sK_{q}\llbracket
s\rrbracket.$

######  Lemma 3.10 ([10, §5]).

Let $V$ be a Banach space over $K_{q}$ and let $\psi:V\to V$ be a completely
continuous operator. Let $B=\\{e_{i}:i\in I\\}$ be an orthonormal basis for
$V$, and let $\psi|B$ denote the $I\times I$ matrix of $\psi$ with respect to
$B$. Then there is an equality of Fredholm series

$\det(I-s\psi)=\det(I-s\psi|B).$

######  Definition 3.11.

We say that an $I\times I$ matrix $\Psi$ with entried in $K_{q}$ is _tight_
if:

  1. 1.

For each $i\in I$, the $i$th column $\Psi_{i}$ of $\Psi$ lies in $b(I)$.

  2. 2.

$\Psi_{i}\to 0$ in the finite-complement topology on $I$.

The definition ensures that if $\Psi$ is a tight $I\times I$ matrix and $x\in
b(I)$, then the sum

$\psi(x)=\sum_{i\in I}x_{i}\Psi_{i}.$

converges in $b(I)$. Thus every tight $I\times I$ matrix defines an associated
continuous $K_{q}$-linear operator $\psi:b(I)\to b(I)$.

######  Proposition 3.12.

Let $\Psi$ be a tight $I\times I$ matrix, and let $\psi:b(I)\to b(I)$ be the
corresponding $K_{q}$-linear operator. Then $\psi$ is completely continuous,
and there is an equality of Fredholm series

$\det(I-s\psi)=\det(I-s\Psi).$

###### Proof.

Let $\Psi^{\prime}$ denote the transpose of $\Psi$. From (6) we see that
$\det(I-s\Psi)=\det(I-s\Psi^{\prime})$. Since the rows of $\Psi^{\prime}$ tend
to $0$ in the finite-complement topology on $I$, $\Psi^{\prime}$ defines a
_completely continuous_ operator $\psi^{\prime}:c(I)\to c(I)$. If we identify
$c(I)^{*}\cong b(I)$ as in Example 3.8, then $\psi=(\psi^{\prime})^{*}$. By
Lemma 3.5, we have

$\det(I-s\psi)=\det(I-s\psi^{\prime})=\det(I-s\Psi^{\prime})=\det(I-s\Psi).$

∎

In light of the proposition, it will generally be convenient to identify the
tight $I\times I$ matrix $\Psi$ with the corresponding operator $\psi:b(I)\to
b(I)$.

######  Proposition 3.13.

Let $V^{\prime}\subseteq V$ be a containment of normed vector spaces over
$K_{q}$. Let $\psi:V\to V$ be a $K_{q}$-linear operator such that $V^{\prime}$
is $\psi$-invariant. Suppose that there exists a formal basis $B=\\{e_{i}:i\in
I\\}$ and elements $c_{i}\in K_{q}$ such that $B^{\prime}=\\{c_{i}e_{i}:i\in
I\\}$ is a formal basis for $V^{\prime}$. Then there is an equality of
Fredholm series

$\det(I-s\psi|B)=\det(I-s\psi|B^{\prime}),$

when either series exists.

###### Proof.

For each finite set $J\subseteq I$ the finite matrices $(\psi|B)_{J}$ and
$(\psi^{\prime}|B)_{J}$ are similar, and the result follows from the explicit
formula (6). ∎

###  3.2 Nuclear Operators

Let $V$ be a $K_{q}$-vector space and let $\psi:V\to V$ be a $K_{q}$-linear
operator. For every polynomial $g\in 1+sK_{q}[s]$, we define the subspace

$V_{g}=\bigcup_{n=1}^{\infty}\ker(g(\psi)^{n}).$

Let $\overline{K}_{q}$ be an algebraic closure of $K_{q}$. For any
$\lambda\in\overline{K}_{q}^{\times}$, let $g_{\lambda}=1+sK_{q}[s]$ be the
irreducible polynomial of $\lambda$ over $K_{q}$. We say that $\lambda$ is a
_non-zero eigenvalue_ if $V_{g_{\lambda}}\neq 0$.

######  Definition 3.14.

A $K_{q}$-linear operator $\psi:V\to V$ is _nuclear_ if:

  1. 1.

For every $g\in 1+sK_{q}[s]$ there is a $\psi$-equivariant decomposition
$V=F_{g}\oplus V_{g}$, where $g(\psi):F_{g}\to F_{g}$ is bijective and $V_{g}$
is finite-dimensional.

  2. 2.

The non-zero eigenvalues of $\psi$ tend to $\infty$ in the finite-complement
topology.

Let $\psi:V\to V$ be a nuclear operator. For each real number $r$ define the
subspace

$V^{<r}=\sum_{v_{\pi}(\lambda)>-r}V_{g_{\lambda}},$

Since $\psi$ is nuclear, each $V^{<r}$ is a finite-dimensional subspace of
$V$. The _characteristic series_ of $\psi$ is defined to be the power series

$C(\psi|V,s)=\lim_{r\to\infty}\det(I-s\psi|V^{<r})\in 1+K_{q}[[s]].$

Over $\overline{K}_{q}$, this series factors as a product
$C(\psi|V,s)=\prod_{\lambda}(1-s/\lambda)$, where $\lambda$ runs through the
non-zero eigenvalues of $\psi$ counted with multiplicity. It follows that
$C(\psi|V,s)$ is a $\pi$-adic entire function on $K_{q}$.

######  Example 3.15.

The following basic examples of nuclear operators, which we will refer to
frequently, are due to Monsky [20]:

  1. 1.

Let $V$ be a Banach space over $K_{q}$. Let $\psi:V\to V$ be a completely
continuous operator. Then by [20, Theorem 1.3], $\psi$ is nuclear and

$C(\psi|V,s)=\det(I-s\psi).$

  2. 2.

Let $V$ be a $K_{q}$-vector space, and let $\psi:V\to V$ be a nuclear
operator. Let $W\subseteq V$ be a $\psi$-invariant subspace such that $\psi$
restricts to a nuclear operator on $W$. Then by [20, Theorem 1.4] the action
of $\psi$ on $V/W$ is nuclear and

$C(\psi|V,s)=C(\psi|W,s)\cdot C(\psi|V/W,s).$

  3. 3.

Let $V$ be a $K_{q}$-vector space. Let $I$ be a linearly ordered set, and let
$\\{V_{i}\\}_{i\in I}$ be a family of $K_{q}$-subspaces of $V$ such that
$V_{j}\subseteq V_{j}$ whenever $i\leq j$, with $V=\bigcup_{i\in I}V_{i}$. Let
$\psi:V\to V$ be a $K_{q}$-linear operator. Suppose that for all $i\gg 0$,
$\psi$ restricts to a nuclear operator on $V_{i}$, and that the characteristic
series $C(\psi|V_{i},s)$ is independent of $i$. Then by [20, Theorem 1.6],
$\psi$ is a nuclear operator on $V$ and for such $i$ we have

$C(\psi|V,s)=C(\psi|V_{i},s).$

###  3.3 Newton and Hodge Polygons

######  Definition 3.16.

Let $S$ be a countable multiset of real numbers. We say that $S$ is a _slope
set_ if for all $r\in R$, the multiset $S^{<r}=\\{s\in S:s<r\\}$ is finite.

Let $S$ be a slope set. If $S$ is finite of cardinality $n$, then there is a
unique convex function $f:[0,n]\to\mathbb{R}\sqcup\\{\infty\\}$ such that:

  1. 1.

$f(0)=0$

  2. 2.

For each $0\leq i<n$, the restriction of $f$ to $[i,i+1]$ is a linear
function.

  3. 3.

$S$ is equal to the multiset $\\{f(i+1)-f(i):0\leq i<n\\}$.

In this case, we define the _Newton polygon_ $\operatorname{\mathrm{NP}}(S)$
of $S$ to be the graph of $f$ in $\mathbb{R}^{2}$. If $S$ is not finite, then
we define the _Newton polygon_ of $S$ to be the union

$\operatorname{\mathrm{NP}}(S)=\bigcup_{r}\operatorname{\mathrm{NP}}(S^{<r}).$

If $S^{\prime}$ is another slope set, we define the _concatenation_ of $S$ and
$S^{\prime}$ to be the slope set $S\sqcup S^{\prime}$. In particular, we will
write $S^{\times n}$ for the $n$-fold concatenation of $S$ with itself. The
basic operations on slope sets carry over to their Newton polygons as well. We
define a partial order on slope sets (or their Newton polygons) by writing
$S_{1}\succeq S_{2}$ whenever the Newton polygon of $S_{1}$ lies on or above
the Newton polygon of $S_{2}$.

######  Example 3.17.

  1. 1.

Let $P(s)=1+c_{1}s+\cdots+c_{d}s^{d}$ be a polynomial with coefficients in
$K_{q}$. We define the _$\pi$ -adic Newton polygon_
$\operatorname{\mathrm{NP}}_{\pi}P(s)$ of $P(s)$ to be the lower convex hull
in $\mathbb{R}^{2}$ of the points $(n,v_{\pi}(c_{n}))$ for $0\leq n\leq d$.
Then $\operatorname{\mathrm{NP}}_{\pi}P(s)$ is the Newton polygon of a unique
slope set which we call the $\pi$-adic _Newton slopes_ of $P(s)$. If we factor

$P(s)=\prod_{i=1}^{d}(1-\alpha_{i}s)$

over $\overline{K}_{q}$, then the $\pi$-adic Newton slopes of $P(s)$ are
precisely the $v_{\pi}(\alpha_{i})$ counted with multiplicity.

  2. 2.

Let $C(s)\in 1+sK_{q}\llbracket s\rrbracket$ be a power series converging in
the disk $v_{\pi}(s)>-r$. By the Weierstrass preparation theorem there is a
factorization

$C(s)=P(s)\cdot C^{\prime}(s),$

where $P(s)\in 1+sK_{q}[s]$ has all $\pi$-adic Newton slopes $<r$, and
$C^{\prime}(s)$ converges and is non-zero in the disk $v_{\pi}(s)>-r$. In this
case we define
$\operatorname{\mathrm{NP}}_{\pi}^{<r}C(s)=\operatorname{\mathrm{NP}}_{\pi}P(s)$.
In particular, if $C(s)$ is entire then we may write

$\operatorname{\mathrm{NP}}_{\pi}C(s)=\bigcup_{r}\operatorname{\mathrm{NP}}_{\pi}^{<r}C(s).$

  3. 3.

Let $V$ be a $K_{q}$-vector space and let $\psi:V\to V$ be a nuclear operator.
Then we define the _$\pi$ -adic Newton polygon_ of $\psi$ to be
$\operatorname{\mathrm{NP}}_{\pi}(\psi)=\operatorname{\mathrm{NP}}_{\pi}C(\psi,s)$.

######  Definition 3.18.

Let $V$ be a Banach space over $K_{q}$. Let $\psi:V\to V$ be a completely
continuous operator. The $\pi$-adic _Hodge polygon_
$\operatorname{\mathrm{HP}}_{\pi}(\psi)$ of $\psi$ is the lower convex hull in
$\mathbb{R}^{2}$ of the points $(n,v_{\pi}\wedge^{n}\psi)$.

Let $I$ be a set, and let $\Psi$ be a tight $I\times I$ matrix with entries in
$K_{q}$. Then we may write $\operatorname{\mathrm{HP}}_{\pi}(\Psi)$ for the
Hodge polygon of the induced operator $b(I)\to b(I)$. For tight matrices, the
following “naive” Hodge polygon is often useful for making explicit estimates:

######  Definition 3.19.

The _column Hodge polygon_ $\operatorname{\mathrm{cHP}}_{\pi}(\Psi)$ of $\Psi$
is the convex polygon with slope set

$\\{v_{\pi}\Psi(e_{i}):i\in I\\}.$

It is straightforward to show that $\operatorname{\mathrm{HP}}_{\pi}(\Psi)$
always lies on or above $\operatorname{\mathrm{cHP}}_{\pi}(\psi)$, and
therefore that $\operatorname{\mathrm{HP}}_{\pi}(\Psi)$ is the Newton polygon
of a unique slope set. We will frequently make use of the following much
stronger estimate:

######  Lemma 3.20.

Let $\Psi$ be a (finite) $n\times n$ matrix with entries in $K_{q}$. Then
every slope of $\operatorname{\mathrm{HP}}_{\pi}(\Psi)$ is greater than or
equal to the corresponding slope of $\operatorname{\mathrm{cHP}}_{\pi}(\Psi)$.
Consequently, if both polygons pass through a point $(n,m)$ then they agree on
the interval $[0,n]$.

###### Proof.

Recall that the “invariant factor theorem” states that $\Psi$ can be put in
diagonal form by applying finitely many of the following operations: (1)
Permuting the rows or columns of $\Psi$, (2) Adding an $R_{q}$-multiple of one
row of $\Psi$ to another, and (3) Given a column with one non-zero entry,
adding an $R_{q}$-multiple of this column to clear other entries in the same
row. It is well known that these operations do not affect the Hodge polygon of
$\Psi$. On the other hand, each of these operations can only increase the
column slopes $v_{\pi}\Psi(e_{i})$. Since
$\operatorname{\mathrm{HP}}_{\pi}(\Psi)=\operatorname{\mathrm{cHP}}_{\pi}(\Psi)$
when $\Psi$ is diagonal, this completes the proof. ∎

######  Lemma 3.21.

Let $V$ be a Banach space, and let $\psi:V\to V$ be a completely continuous
operator. Then

  1. 1.

$\operatorname{\mathrm{HP}}_{\pi}(\psi)$ is the Newton polygon of a unique
slope set.

  2. 2.

$\operatorname{\mathrm{NP}}_{\pi}(\psi)\succeq\operatorname{\mathrm{HP}}_{\pi}(\psi)$.

  3. 3.

If $V$ is finite-dimensional and $\det(\psi)\neq 0$, then
$\operatorname{\mathrm{NP}}_{\pi}(\psi)$ and
$\operatorname{\mathrm{HP}}_{\pi}(\psi)$ have the same terminal point.

###### Proof.

Let $B=\\{e_{i}:i\in I\\}$ be an orthonormal basis for $V$. Let $\Psi=\psi|B$
and let $\Psi^{\prime}$ denote the transpose of $\Psi$. Then $\Psi^{\prime}$
is tight, and
$\operatorname{\mathrm{HP}}_{\pi}(\psi)=\operatorname{\mathrm{HP}}_{\pi}(\Psi^{\prime})$.
Claim 1 then follows from the preceding discussion. Evidently
$v_{\pi}\mathrm{Tr}\wedge^{n}\psi\geq v_{\pi}\wedge^{n}\psi$ for all $n$,
proving 2. For 3, simply note that the terminal point of both polygons is
$(n,v_{\pi}\det(\psi))$. ∎

###  3.4 Iteration

Let $F$ denote the canonical Frobenius endomorphism of $\mathbb{Z}_{q}$. By
base change, $F$ induces an $R$-linear endomorphism $F:R_{q}\to R_{q}$.

######  Definition 3.22.

Let $V$ be a vector space over $K_{q}$. We say that a $K$-linear operator
$\psi:V\to V$ is _$F^{-1}$ -linear_ if

$\psi(F(r)x)=r\psi(x)$

for all $r\in R$ and $x\in V$. In this case, the iterate
$\psi_{q}=\psi^{v_{p}(q)}$ is a $K_{q}$-linear operator.

######  Definition 3.23.

Let $V$ be a normed vector space over $K_{q}$. Let $B_{q}$ be a formal basis
for $V$. An _associated $K$-Basis_ for $V$ is a formal basis for $V$ (as a
normed vector space over $K$) of the form

$B=\bigsqcup_{i=0}^{v_{p}(q)-1}F^{i}(\xi)\otimes B_{q},$ (7)

where $\xi\in\mathbb{Z}_{q}$ is any element such that the $F^{i}(\xi)$
generate $\mathbb{Z}_{q}$ as a $\mathbb{Z}_{p}$-module.

Let $\psi:V\to V$ be an $F^{-1}$-linear operator. Via the partition (7), the
matrix $\psi_{q}|B$ is block diagonal of the form

$\begin{bmatrix}\xi(\psi_{q}|B_{q})&&\\\ &\ddots&\\\
&&F^{v_{p}(q)-1}(\xi)(\psi_{q}|B_{q})\end{bmatrix}.$

Consider the parameter $\pi_{q}=\pi^{v_{p}(q)}$. We have the following
relation between the Newton polygons of the operators $\psi$ and $\psi_{q}$:

######  Lemma 3.24.

Let $V$ be a Banach space over $K_{q}$. Let $\psi:V\to V$ be a completely
continuous $F^{-1}$-linear operator. Then $\psi_{q}$ is completely continuous
as a $K_{q}$-linear operator and

$\operatorname{\mathrm{NP}}_{\pi_{q}}(\psi_{q})^{\times
v_{p}(q)}=\operatorname{\mathrm{NP}}_{\pi}(\psi).$

###### Proof.

Note that $\psi_{q}$ is completely continuous as a $K$-linear operator. Let
$B_{q}$ be an orthonormal $K_{q}$-basis for $V$, and let $B$ be an associated
$K$-basis for $V$. From the above block matrix, we see that the rows of
$\psi_{q}|B_{q}$ tend to $0$ in the finite-complement topology, thus
$\psi_{q}$ is completely continuous as a $K_{q}$-linear operator. We see
moreover that $\det(I-s\psi_{q}|B)=\det(I-s\psi_{q}|B_{q})^{v_{p}(q)}$. It
follows that

$\det(I-s^{v_{p}(q)}\psi_{q}|B_{q})^{v_{p}(q)}=\det(I-s^{v_{p}(q)}\psi_{q})=\prod_{\zeta^{v_{p}(q)}=1}\det(I-s\zeta\psi|B).$

The Newton polygons of the matrices $\zeta\psi|B$ do not depend on $\zeta$.
Thus,

$\operatorname{\mathrm{NP}}_{\pi_{q}}(\psi_{q}|B_{q})^{\times
v_{p}(q)}=\operatorname{\mathrm{NP}}_{\pi}\det(I-s^{v_{p}(q)}\psi_{q}|B_{q})=\operatorname{\mathrm{NP}}_{\pi}(\psi|B).$

The claim follows from Lemma 3.10. ∎

##  4 Geometric Setup

Fix a smooth projective curve $\overline{X}$ over $\mathbb{F}_{q}$ of genus
$g$. Let $S$ be a finite set of closed points of $\overline{X}$ and let
$X=\overline{X}\backslash S$. In this section we construct the global lifting
$(A^{\dagger},\sigma)$, which we will use to study abelian $L$-functions over
$X$. We will give our constructions over the ring $\mathbb{Z}_{p}$, with
$\pi=p$. Later, we will pass to general $R$ and $\pi$ by weak base change.

###  4.1 Mapping to $\mathbb{P}_{\mathbb{F}_{q}}^{1}$

The construction of our global lifting relies on an analogue of Belyi’s
theorem in characteristic $p$. For this we require:

######  Theorem 4.1 (Fulton [9]).

After extending the base field, there exists a finite, separable, tamely
ramified morphism $\eta_{0}:\overline{X}\to\mathbb{P}_{\mathbb{F}_{q}}^{1}$.

######  Remark 4.2.

In [23], Sugiyama and Yasuda extend Fulton’s result to the case $p=2$. We have
omitted this case for other reasons (see Remark 6.5). By a recent theorem of
Kedlaya-Litt-Witaszek, $\eta$ exists even without extending the base field
[13].

From now on we fix a choice of $\eta_{0}$ as in Theorem 4.1. As in [13,
Theorem 9.3], the existence of $\eta_{0}$ implies the existence of a tamely
ramified map $\overline{X}\to\mathbb{P}_{\mathbb{F}_{q}}^{1}$ with three
ramified points. We shall need some control over the branching of this map, so
we modify the construction slightly.

By extending the base field, we assume that all branch points of $\eta_{0}$
and every point in $S$ is $\mathbb{F}_{q}$-rational. We assume moreover that
$q$ is large enough so that there are two $\mathbb{F}_{q}$-rational points of
$\mathbb{P}_{\mathbb{F}_{q}}^{1}$ which are disjoint from the branch points of
$\eta_{0}$ and the image of $S$. Label these points as $0$ and $\infty$, and
let $1$ denote any other $\mathbb{F}_{q}$-rational point of
$\mathbb{P}_{\mathbb{F}_{q}}^{1}$. Consider the composition

$\eta_{q}:\mathbb{P}_{\mathbb{F}_{q}}^{1}\xrightarrow{q-1}\mathbb{P}_{\mathbb{F}_{q}}^{1}\to\mathbb{P}_{\mathbb{F}_{q}}^{1}\xrightarrow{p-1}\mathbb{P}_{\mathbb{F}_{q}}^{1}\to\mathbb{P}_{\mathbb{F}_{q}}^{1}.$

Here, the first and third maps denote the $(q-1)$\- and $(p-1)$-power maps,
respectively. The second map is a linear transformation fixing $1$ and
$\infty$ and sending $0$ to any other $\mathbb{F}_{q}$-rational point of
$\mathbb{P}_{\mathbb{F}_{q}}^{1}$. The final map is also a linear
transformation which fixes $\infty$ and swaps $0$ with $1$. Then $\eta_{q}$ is
ramified over $\\{0,1,\infty\\}$ and all branch points of $\eta_{q}$ are
$\mathbb{F}_{q}$-rational. We have:

######  Proposition 4.3.

The composite
$\eta=\eta_{q}\circ\eta_{0}:\overline{X}\to\mathbb{P}_{\mathbb{F}_{q}}^{1}$ is
a tame Belyi map such that

  1. 1.

$\eta(P)=0$ for each $P\in S$.

  2. 2.

If $\eta(P)=1$ then the ramification index of $\eta$ at $P$ is $p-1$.

For each $Q\in\\{0,1,\infty\\}$, let $r_{Q}$ denote the cardinality of the
fiber $\eta^{-1}(Q)$. If $\eta(P)=Q$, then we let $e_{P}$ denote the
ramification index of $P$ over $Q$. In particular, if $Q=1$ then $e_{P}=p-1$.
It follows that that $r_{1}(p-1)=\deg(\eta)$. The Riemann-Hurwitz formula
gives

$2(g-1)+r_{0}+r_{1}+r_{\infty}=\deg(\eta).$ (8)

###  4.2 The Global Lifting

Let $V=\mathbb{P}_{\mathbb{F}_{q}}^{1}\backslash\\{0,1,\infty\\}$. Let $u_{0}$
denote a parameter at $0$, and define $u_{1}=u_{0}-1$,
$u_{\infty}=u_{0}^{-1}$. The coordinate ring of $V$ is a generated by $u_{0}$,
$u_{\infty}$, and $u_{1}^{-1}$ over $\mathbb{F}_{q}$. We consider the lifting

$B=\mathbb{Z}_{q}[u_{0},u_{\infty},u_{1}^{-1}].$ (9)

We define $\sigma$ to be the unique Frobenius endomorphism of $B^{\dagger}$
satisfying $\sigma(u_{0})=u_{0}^{p}$.

Let $U=\eta^{-1}(V)$. Then $U$ is an affine open subset of $X$, which is
finite étale over $V$. We define $S_{\eta}=X-V$. We will lift $U$ to
characteristic $0$ rather than all of $X$, because the étale property of
$\eta$ is essential for constructing our Frobenius endomorphism: First, a
theorem of Elkik [7] states that there exists a smooth
$\mathbb{Z}_{q}$-algebra $A$ such that $U=\mathrm{Spec}(A/pA)$. By [26,
Theorem 2.4.4], there is a (necessarily unique) lifting of $\eta$ to an étale
map $B^{\dagger}\to A^{\dagger}$.

######  Definition 4.4.

The _global lifting_ is the ring $A^{\dagger}$ equipped with the unique
extension of $\sigma$ along $\eta$.

######  Lemma 4.5.

The Frobenius $\sigma:A^{\dagger}\to A^{\dagger}$ is injective, and
$A^{\dagger}$ is a finite projective $\sigma(A^{\dagger})$-module of rank $p$.

###### Proof.

The first statement is [21, Theorem 3.2(3)]. The second follows from that fact
that $A^{\dagger}$ is finite [21, Theorem 6.2] and flat [21, Theorem 2.1(2)]
over $\sigma(A^{\dagger})$. ∎

Let $\Omega_{A^{\dagger}}$ denote the $A^{\dagger}$-module of continuous
differentials of $A^{\dagger}/\mathbb{Z}_{q}$, and similarly for
$B^{\dagger}$. Then $\Omega_{B^{\dagger}}$ is a free $B^{\dagger}$-module of
rank $1$ generated by the differential $\tfrac{\mathrm{d}u_{0}}{u_{0}}$. Since
$\eta:B^{\dagger}\to A^{\dagger}$ is étale, we have

$\Omega_{A^{\dagger}}=A^{\dagger}\otimes_{B^{\dagger}}\Omega_{B^{\dagger}}=A^{\dagger}\frac{\mathrm{d}u_{0}}{u_{0}}.$

The Frobenius $\sigma$ induces an injective $\mathbb{Z}_{q}$-linear
endomorphism $\sigma_{1}$ of $\Omega_{A^{\dagger}}$. Explicitly,

$\sigma_{1}\left(a\frac{\mathrm{d}u_{0}}{u_{0}}\right)=\sigma(a)p\frac{\mathrm{d}u_{0}}{u_{0}}.$

Thus $\sigma_{1}(\Omega_{A^{\dagger}})=\Omega_{\sigma(A^{\dagger})}$ is a free
$\sigma(A^{\dagger})$-module of rank $1$, generated by
$p\tfrac{\mathrm{d}u_{0}}{u_{0}}$. By Lemma 4.5, there is a
$\sigma(A^{\dagger})$-linear _trace map_

$\mathrm{Tr}:A^{\dagger}\to\sigma(A^{\dagger}).$

Given $a\in A^{\dagger}$, the trace $\mathrm{Tr}(a)$ may be identified with
the coefficient of $X^{p-1}$ in the minimal polynomial $g_{a}(X)$ of $a$ over
$\sigma(A^{\dagger})$. Note that $g_{a}\equiv X^{p}-a^{p}\pmod{p}$, so that
$\mathrm{Tr}(a)$ in fact lies in $p\sigma(A^{\dagger})$. The trace map induces
a $\sigma(A^{\dagger})$-linear map
$\mathrm{Tr}_{1}:\Omega_{A^{\dagger}}\to\sigma_{1}(\Omega_{A^{\dagger}})$.
Explicitly,

$\mathrm{Tr}_{1}\left(a\frac{\mathrm{d}u_{0}}{u_{0}}\right)=\mathrm{Tr}(a)\frac{\mathrm{d}u_{0}}{u_{0}}=\frac{\mathrm{Tr}(a)}{p}p\frac{\mathrm{d}u_{0}}{u_{0}}.$

######  Definition 4.6.

The _global $U_{p}$ operator_ is the $\mathbb{Z}_{p}$-linear endomorphism

$U_{p}=\sigma_{1}^{-1}\circ\mathrm{Tr}_{1}:\Omega_{A^{\dagger}}\to\Omega_{A^{\dagger}}.$

The global $U_{p}$ operator is a $p$-Dwork operator on $\Omega_{A^{\dagger}}$
and the iterate $U_{q}=U_{p}^{v_{p}(q)}$ is a $q$-Dwork operator. If we
identify $\Omega_{A^{\dagger}}$ with $A^{\dagger}$ using the basis
$\tfrac{\mathrm{d}u_{0}}{u_{0}}$, then $U_{p}$ is identified with the operator

$U_{p}=\frac{1}{p}\sigma^{-1}\circ\mathrm{Tr}:A^{\dagger}\to A^{\dagger}.$

###  4.3 Semi-Local Liftings

For each $Q\in\\{0,1,\infty\\}$, the completion of $B/pB$ along $Q$ may be
identified with the local field $F_{Q}=\mathbb{F}_{q}(\\!(u_{Q})\\!)$. For
$P\in\overline{X}$ lying above $Q$, let $F_{P}$ denote the completion of
$A/pA$ along $P$. Then $F_{P}$ is a local field, and our assumptions on the
branching of $\eta$ allow us to choose a uniformizer $t_{P}\in F_{P}$ such
that $t_{P}^{e_{P}}=u_{Q}$. We have a Cartesian diagram:

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For each $Q\in\\{0,1,\infty\\}$ we define
$\mathcal{A}_{Q}=\mathbb{Z}_{q}(\\!(u_{Q})\\!)$. There is a natural map
$B^{\dagger}\to\mathcal{A}_{Q}^{\dagger}$ which is given by “expansion in the
parameter $u_{Q}$.” Similarly, if $P$ is a point of $\overline{X}$ lying above
$Q$, then we let $\mathcal{A}_{P}=\mathbb{Z}_{q}(\\!(t_{P})\\!)$. Since
$A^{\dagger}$ is étale over $B^{\dagger}$, the natural map $A^{\dagger}\to
F_{P}$ lifts uniquely to a map $A^{\dagger}\to\mathcal{A}_{P}^{\dagger}$,
which we regard as “expansion in the parameter $t_{P}$.” These maps fit into a
lifted Cartesian diagram:

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For each $Q\in\\{0,1,\infty\\}$, the endomorphism $\sigma:B\to B$ restricts to
an endomorphism of $\mathcal{A}_{Q}$. Explicitly,

$\displaystyle\sigma(u_{0})$ $\displaystyle=u_{0}^{p},$
$\displaystyle\sigma(u_{\infty})$ $\displaystyle=u_{\infty}^{p},$
$\displaystyle\sigma(u_{1})$ $\displaystyle=(u_{1}+1)^{p}-1.$

If $P$ is a point of $\overline{X}$ lying above $Q$, then $\sigma$ extends
uniquely along the étale map
$\mathcal{A}_{Q}^{\dagger}\to\mathcal{A}_{P}^{\dagger}$ to an endomorphism of
$\mathcal{A}_{P}^{\dagger}$. Evidently, if $\eta(P)=0$ or $\infty$ then
$\sigma(t_{P})=t_{P}^{p}$. The local Frobenius for $\eta(P)=1$ is more
complicated:

$\sigma(t_{P})=\sqrt[p-1]{\left(t_{P}^{p-1}+1\right)^{p}+1}.$

######  Definition 4.7.

Let $\mathcal{A}=\prod_{P\in S_{\eta}}\mathcal{A}_{P}$. The _semi-local
lifting_ is the ring $\mathcal{A}^{\dagger}$ equipped with the Frobenius
endomorphism $\sigma$, defined as the product of the local Frobenius
endomorphisms described above.

######  Lemma 4.8.

The Frobenius $\sigma:\mathcal{A}^{\dagger}\to\mathcal{A}^{\dagger}$ is
injective, and $\mathcal{A}^{\dagger}$ is a finite projective
$\sigma(\mathcal{A}^{\dagger})$-module of rank $p$.

###### Proof.

The first statement follows exactly as in Lemma 4.5, as does the fact that
$\mathcal{A}^{\dagger}$ is flat over $\sigma(\mathcal{A}^{\dagger})$. It
remains to show that $\mathcal{A}^{\dagger}$ is finite over
$\sigma(\mathcal{A}^{\dagger})$. To see this, note that if
$\eta(P)=Q\in\\{0,1,\infty\\}$, then we have a Cartesian diagram:

${\sigma(\mathcal{A}_{P}^{\dagger})}$${\mathcal{A}_{P}^{\dagger}}$${\sigma(\mathcal{A}_{Q}^{\dagger})}$${\mathcal{A}_{Q}^{\dagger}}$

The lower horizontal map is finite of degree $p$, as can be seen explicitly
from the local Frobenius structure at $Q$. ∎

As in the global situation, Lemma 4.8 implies there is a
$\sigma(\mathcal{A}^{\dagger})$-linear _trace map_

$\mathrm{Tr}:\mathcal{A}^{\dagger}\to\sigma(\mathcal{A}^{\dagger}).$

As before, image of $\mathrm{Tr}$ lies in $p\sigma(\mathcal{A}^{\dagger})$.
This allows us to extend $U_{p}$ to all of $\mathcal{A}^{\dagger}$:

######  Definition 4.9.

The _semi-local $U_{p}$-operator_ is defined to be the $\mathbb{Z}_{p}$-linear
operator

$U_{p}=\frac{1}{p}\sigma^{-1}\circ\mathrm{Tr}:\mathcal{A}^{\dagger}\to\mathcal{A}^{\dagger}.$

###  4.4 Semi-Local Decomposition

For local-to-global arguments, it will be convenient to have a description of
the underlying $\mathbb{Z}_{q}$-module structrue of the lifting $A^{\dagger}$,
analogous to the partial fraction decomposition of $B^{\dagger}$. For each
point $P$ of $\overline{X}$ lying above $Q\in\\{0,1,\infty\\}$, we define

$\mu(P)=\begin{cases}0&Q=0\text{ or }1\\\ p-1&Q=1\end{cases}.$ (11)

Consider the $\mathbb{Z}_{q}$-submodule of $\mathcal{A}_{P}$ consisting of
$\mu(P)$-truncated series

$\mathcal{A}_{P}^{\mathrm{tr}}=\left\\{\sum_{k}a_{k}t^{-k}\in\mathcal{A}_{P}:a_{k}=0\text{
for all }k\leq\mu(P)\right\\}.$ (12)

We define
$\mathcal{A}^{\mathrm{tr}}=\bigoplus_{P}\mathcal{A}_{P}^{\mathrm{tr}}$,
regarded as a $\mathbb{Z}_{q}$-submodule of $\mathcal{A}$. There is a natural
projection map $\mathrm{pr}:\mathcal{A}\to\mathcal{A}^{\mathrm{tr}}$. Since
$\mathcal{A}^{\mathrm{tr}}$ is a $p$-saturated submodule of $\mathcal{A}$, the
$p$-adic completion $\mathcal{A}^{\infty,\mathrm{tr}}$ is naturally a
$\mathbb{Z}_{q}$-submodule of $\mathcal{A}^{\infty}$. We define
$\mathcal{A}^{\dagger,\mathrm{tr}}=\mathcal{A}^{\dagger}\cap\mathcal{A}^{\infty,\mathrm{tr}}$.

######  Proposition 4.10.

There is an exact sequence of $\mathbb{Z}_{q}$-modules

$0\to L\to
A^{\dagger}\xrightarrow{\mathrm{pr}}\mathcal{A}^{\dagger,\mathrm{tr}}\to 0,$

where $L$ is a finite _free_ $\mathbb{Z}_{q}$-module of rank

$N=g-1+r_{0}+r_{1}+r_{\infty}.$ (13)

###### Proof.

First, consider the map
$\mathrm{pr}:A^{\infty}\to\mathcal{A}^{\infty,\mathrm{tr}}$. Any element of
the kernel $L$ has poles of finite order about each $P$, so that $L\subseteq
A^{\dagger}$. Consider the reduction modulo $p$:

$\overline{L}\to\overline{A}\xrightarrow{\overline{\mathrm{pr}}}\overline{\mathcal{A}}^{\mathrm{tr}}.$

By the Riemann-Roch theorem, $\overline{\mathrm{pr}}$ is surjective. The
kernel is precisely the global sections of the line bundle $\mathcal{L}(D)$,
where $D=\sum_{\eta(P)=1}(p-1)P$. Again by Riemann-Roch, this is a finite
$\mathbb{F}_{q}$-module of rank $N$. By Lemma 2.3(1)-(2), the map
$\mathrm{pr}:A^{\infty}\to\mathcal{A}^{\infty,\mathrm{tr}}$ is surjective and
$\ker(\overline{\mathrm{pr}})=\overline{L}$. Since $L$ is $p$-torsion free,
Nakayama’s lemma implies that $L$ is a finite free $\mathbb{Z}_{q}$-module of
rank $N$.

It remains to show that
$\mathrm{pr}:A^{\dagger}\to\mathcal{A}^{\dagger,\mathrm{tr}}$ is surjective.
Observe that $a\in\mathcal{A}^{\infty,\mathrm{tr}}$ lies in
$\mathcal{A}^{\dagger,\mathrm{tr}}$ if and only if there exists
$\hat{a}\in\mathcal{A}^{\dagger}$ such that $\mathrm{pr}(\hat{a})=a$. By the
preceding discussion, we may choose $\hat{a}$ to lie in the image of
$A^{\infty}$. Since $A^{\dagger}=\mathcal{A}^{\dagger}\cap A^{\infty}$, this
completes the proof. ∎

##  5 $\sigma$-Modules

###  5.1 Definitions

For this subsection, let $X$ denote a general affine $\mathbb{F}_{p}$-scheme.
By a _flat lifting_ of $X$ over $R$, we will mean a pair $(A,\sigma)$, where
$A$ is a lifting of the coordinate ring of $X$ to a flat $R$-algebra, and
$\sigma:A\to A$ is a lifting of the absolute Frobenius endomorphism of $X$.

######  Definition 5.1.

A _$\sigma$ -module_ over $A$ is a pair $(M,\phi)$, where $M$ is a finite
projective $A$-module and $\phi:\sigma^{*}M\to M$ is an $R$-linear map such
that $K\otimes\phi$ is an isomorphism. We say that $(M,\phi)$ is _unit-root_
if $\phi$ is an isomorphism.

Let $(M,\phi)$ be a $\sigma$-module over $A$. Let $(B,\tau)$ be a flat lifting
of another affine $\mathbb{F}_{p}$-scheme $Y$. Given a _Frobenius-compatible_
map $f:A\to B$ of $R$-algebras, we obtain by extension of scalars a
$\tau$-module $(B\otimes_{A}M,\tau\otimes\phi)$ over $B$.

######  Theorem 5.2 ([11], 4.1.1).

There is a rank-preserving equivalence between the category of unit-root
$\sigma$-modules over $A^{\infty}$ and the category of $R$-valued
representations of $\pi_{1}(X)$. Under this equivalence, the pullback of
representations along $Y\to X$ corresponds to extension of scalars along the
Frobenius-compatible map $f:A^{\infty}\to B^{\infty}$.

Let $(M,\phi)$ be a $\sigma$-module over $A$. By composition with the natural
map $M\to\sigma^{*}M$, we will usually regard $\phi$ as an endomorphism of $M$
which is $\sigma$-semilinear, in the sense that $\phi(am)=\sigma(a)\phi(m)$
for all $a\in A$ and all $m\in M$. We will typically restrict our attention to
the case that $M$ is a _free_ $A$-module of rank $1$. If $e\in M$ is any basis
for $M$ and $E=\phi(e)$ is the “matrix” of $\phi$ with respect to $E$, then we
have an isomorphism of $\sigma$-modules

$(M,\phi)\cong(A,E\circ\sigma),$

In this case, we refer to $E$ as a _Frobenius structure_ for $(M,\phi)$.

For each $x\in|X|$, let $\deg(x)=[k(x):\mathbb{F}_{q}]$. Consider the flat
lifting $(R(x),F)$ of $k(x)$ over $R$, where
$R(x)=R\otimes_{\mathbb{Z}_{p}}W(k(x))$ and $F$ is the canonical $R$-linear
lifting of Frobenius. By [20, Theorem 3.3] there is a unique Frobenius-
compatible map

$\hat{x}:A\to R(x),$

which we call the _Teichmüller lifting_ of $x$. Let $(M_{x},\phi_{x})$ denote
the $F$-module over $R(x)$ obtained by extension of scalars along $\hat{x}$.
The map $\phi_{x}$ is only $R$-linear, but the iterate
$\phi_{x}^{v_{p}(q)\deg(x)}$ is $R(x)$-linear. Since $R(x)$ is local, $M_{x}$
is free of finite rank and we may define:

######  Definition 5.3.

The _$L$ -function_ of $(M,\phi)$ is the power series

$L(\phi,s)=\prod_{x\in|U|}\frac{1}{\det(1-\phi_{x}^{v_{p}(q)\deg(x)}s^{\deg(x)})}\in
1+sR[[s]].$

###  5.2 $\sigma$-Modules over the Global Lifting

We will now return to the setting of §4. Let $(A^{\dagger},\sigma)$ denote the
global lifting of $U$ as defined in §4.2. Let
$A_{\pi}=R\otimes_{\mathbb{Z}_{p}}A$, which we equip with the $\pi$-adic
topology. By weak base change, we obtain a flat lifting
$(A_{\pi}^{\dagger},\sigma)$ of $U$ over $R$. Let $\rho:\pi_{1}(X)\to
R^{\times}$ be a continuous character. Let $\rho_{U}$ denote the restriction
of $\rho$ to $\pi_{1}(U)$, and let $(M,\phi)$ be the corresponding unit-root
$\sigma$-module over $A_{\pi}^{\infty}$. The $L$-function $L(\phi,s)$
coincides with the Artin $L$-function $L(\rho_{U},s)$ of $\rho_{U}$ over $U$.
In other words,

$\prod_{\begin{subarray}{c}x\in|X-U|\\\ x\notin
S\end{subarray}}(1-\rho(\mathrm{Frob}_{x})s)\cdot L(\rho,s)=L(\phi,s).$ (14)

######  Definition 5.4.

We say that $(M,\phi)$ is _overconvergent_ if $(M,\phi)$ is obtained by
extension of scalars from a $\sigma$-module over $A_{\pi}^{\dagger}$.

If $(M,\phi)$ is overconvergent, then $L(\phi,s)$ is known to be meromorphic
by the Monsky trace formula. Let us state the trace formula in the special
case that $M$ is free of rank $1$. First, observe that $(M,\phi)$ is
overconvergent if and only if $(M,\phi)$ admits an overconvergent Frobenius
structure $E\in A_{\pi}^{\dagger}$. By weak base change, the operator $U_{p}$
induces an $R$-linear endomorphism $U_{p}:A_{\pi}^{\dagger}\to
A_{\pi}^{\dagger}$. Upon fixing a choice of $E$, we define “Dwork operator”

$\Theta=U_{p}\circ E:A_{\pi}^{\dagger}\to A_{\pi}^{\dagger}.$

Let $V_{\pi}=K\otimes_{R}A_{\pi}$, regarded as a vector space over $K_{q}$. By
[20, Theorem 2.1], the action of $\Theta_{q}$ on $V_{\pi}^{\dagger}$ is
nuclear. We can now state Monsky’s trace formula:

######  Theorem 5.5 ([20], or [24] when $\mathrm{char}(R)=p$).

Suppose that $(M,\phi)$ is overconvergent and that $M$ is free of rank $1$.
Then

$L(\phi,s)=\frac{C(\Theta_{q}|V_{\pi}^{\dagger},s)}{C(\Theta_{q}|V_{\pi}^{\dagger},qs)}.$

######  Corollary 5.6.

Let $\rho:\pi_{1}(X)\to R^{\times}$ be a continuous character and let
$(M,\phi)$ be the unit-root $\sigma$-module over $A_{\pi}^{\dagger}$
corresponding to $\rho_{U}$. Assume that $(M,\phi)$ is overconvergent and that
$M$ is free of rank $1$. Then

$\prod_{\begin{subarray}{c}x\in|X-U|\\\ x\notin
S\end{subarray}}(1-\rho(\mathrm{Frob}_{x})s)\cdot
L(\rho,s)=\frac{C(\Theta_{q}|V_{\pi}^{\dagger},s)}{C(\Theta_{q}|V_{\pi}^{\dagger},qs)}.$

######  Corollary 5.7.

In the setting of Corollary 5.6, $L(\rho,s)$ is analytic in the disk
$v_{\pi}(s)>-v_{\pi}(q)$.

###### Proof.

Note that $L(\phi,s)$ is analytic in this disk by the Monsky trace formula. By
a theorem of Crew [4], $L(\rho,s)$ has no poles in the region $v_{\pi}(s)=0$.
The claim follows from Corollary 5.6. ∎

###  5.3 Semi-Local Growth Conditions

Let $P$ be a branch point of the covering $\eta$ defined in §4.1. Recall that
we have defined a flat lifting $(\mathcal{A}_{P}^{\dagger},\sigma)$ of the
local field $F_{P}$ over $\mathbb{Z}_{p}$ in §4.3. Let
$\mathcal{A}_{\pi,P}=R\otimes_{\mathbb{Z}_{p}}\mathcal{A}_{P}$, equipped with
the $\pi$-adic topology. Again by weak base change, we obtain a flat lifting
$(\mathcal{A}_{\pi,P}^{\dagger},\sigma)$ of $F_{P}$ over $R$. Let
$\rho:\pi_{1}(X)\to R^{\times}$ be a continuous character, and let $(M,\phi)$
be unit-root $\sigma$-module over $A_{\pi}^{\infty}$ corresponding to
$\rho_{U}$. By extension of scalars along the Frobenius-compatible map
$A_{\pi}^{\infty}\to\mathcal{A}_{\pi,P}^{\infty}$, we obtain a $\sigma$-module
$(M_{P},\phi_{P})$ over $\mathcal{A}_{\pi,P}^{\infty}$.

######  Lemma 5.8.

If $\rho$ is unramified at $P$, then $(M_{P},\phi_{P})$ admits a constant
Frobenius structure $\tilde{E}_{P}\in R_{q}$.

###### Proof.

Since $(M,\phi)$ is unramified, $\rho$ factors through a character
$\overline{\rho}:\pi_{1}(\mathrm{Spec}(\mathbb{F}_{q}))\to R^{\times}$. The
latter corresponds to a unit-root $F$-module $(\overline{M},\overline{\phi})$
over $R_{q}$. By Theorem 5.2, $(M,\phi)$ is obtained by extension of scalars
from $(\overline{M},\overline{\phi})$ along $R_{q}\to\mathcal{A}_{\pi,P}$, so
$(M,\phi)$ must admit a Frobenius structure in $R_{q}$. ∎

We define as before $\mathcal{A}_{\pi}=\prod_{P\in
S_{\eta}}\mathcal{A}_{\pi,P}$. By a _semi-local Frobenius structure_ for
$(M,\phi)$, we will mean an element
$\tilde{E}=(\tilde{E}_{P})\in\mathcal{A}_{\pi}^{\infty}$ such that
$\tilde{E}_{P}$ is a Frobenius structure for $(M_{P},\phi_{P})$ for all $P\in
S_{\eta}$.

######  Definition 5.9.

Let $\bm{\delta}=(\delta_{P})$ be a tuple of positive rational numbers indexed
by $S$. We say that $\rho$ is $\pi$-adically _$\bm{\delta}$ -overconvergent_
if $(M,\phi)$ admits a semi-local Frobenius structure $\tilde{E}$ such that:

  1. 1.

If $P\notin S$, then $\tilde{E}_{P}\in 1+\pi R_{q}$ is constant. In
particular, $(M,\phi)$ is unramified at $P$.

  2. 2.

If $P\in S$, then $\tilde{E}_{P}=\tilde{E}_{P}\in
R_{q}[t_{P}^{-1}]^{\delta_{P}}\cap(1+\pi R_{q}\langle t_{P}^{-1}\rangle)$.

In this case we refer to $\tilde{E}$ as a _$\bm{\delta}$ -Frobenius structure_
for $(M,\phi)$.

######  Example 5.10.

  1. 1.

Suppose that $\rho:\pi_{1}(X)\to R^{\times}$ is finite of order $p^{n}$. Let
$\pi\in\mathbb{Z}_{p}[\rho]$ be a uniformizer. For each $P\in S$, let $d_{P}$
denote the Swan conductor of $\rho$ at $P$, and let
$\delta_{P}=d_{P}/p^{n-1}$. By [17, Proposition 5.5], $\rho$ is $\pi$-adically
$\bm{\delta}$-overconvergent.

  2. 2.

Let $X_{\infty}/X$ be a $\mathbb{Z}_{p}$-tower of curves (§1.4.1), regarded as
a surjective map

$\rho:\pi_{1}(X)\to\mathbb{Z}_{p}.$

Let $\chi:\mathbb{Z}_{p}\to R^{\times}$ be a continuous character, and let
$\rho_{\chi}=\chi\circ\rho$. We define $\pi_{\chi}=\chi(1)-1$. In [18] we
prove that if $X_{\infty}/X$ has $\bm{\delta}$-stable monodromy and $R$ has
characteristic $p$, then $\rho_{\chi}$ is $\pi_{\chi}$-adically
$\bm{\delta}$-overconvergent.

  3. 3.

Continuing the previous example: Let $X_{\infty}/X$ be an overconvergent
$\mathbb{Z}_{p}$-tower of curves (1.4.2), so that $X_{\infty}/X$ has
$\bm{\delta}$-stable monodromy for some $\bm{\delta}=(\delta_{P})_{P\in S}$.
In [25], we show that for $v_{\pi_{\chi}}(p)\gg 0$, $\rho_{\chi}$ is
$\pi_{\chi}$-adically $\bm{\delta}$-overconvergent.

###  5.4 Global growth conditions

The Monsky trace formula requires that the $\sigma$-module $(M,\phi)$ in
question is overconvergent (i.e. it has an overconvergent Frobenius
structure). The following proposition says that one may check the
overconvergence of $M$ locally at each $P\in S$. In particular, the Monsky
trace formula may be applied to any $\sigma$-module that is
$\bm{\delta}$-overconvergent.

######  Proposition 5.11.

Let $(M,\phi)$ be a $\sigma$-module over $A_{\pi}^{\infty}$ with Frobenius
structure $E\in 1+\pi A_{\pi}^{\infty}$. Suppose that $(M,\phi)$ admits an
overconvergent semi-local Frobenius structure $\tilde{E}\in
1+\pi\mathcal{A}_{\pi}^{\dagger}$. Then there exists $a\in 1+\pi
A_{\pi}^{\infty}$ such that $E^{\prime}=\sigma(a)Ea^{-1}$ is contained in
$1+\pi A_{\pi}^{\dagger}$. In particular $(M,\phi)$ is overconvergent.

We define the $k$-th partial valuation on $\mathcal{A}_{\pi,P}^{\infty}$ as
follows: For $f\in\mathcal{A}_{\pi,P}^{\infty}$ write $f=\sum a_{n}t_{P}^{n}$.
We define $w_{k}^{P}(f)$ to be the $t_{P}$-adic order of $f$ reduced modulo
$\pi^{k+1}$. That is,

$\displaystyle w_{k}^{P}(f)$ $\displaystyle=\min_{v_{\pi}(a_{n})\leq k}n.$

Note that we have

$\displaystyle w_{k}^{P}(fg)$
$\displaystyle\geq\min_{i+j=k}w_{i}^{P}(f)+w_{j}^{P}(g)$ (15)

######  Lemma 5.12.

Let $m$ be an integer large enough so that
$\sigma(t_{P})\in\mathcal{A}_{\pi,P}^{pm}(p)$ (note that by our definition of
$\sigma$ and $t_{P}$, such an $m$ always exists). Let
$x\in\mathcal{A}_{\pi,P}^{\infty}$. Fix a natural number $k_{0}$. Assume that
$w^{P}_{k}(x)>-km$ for $k<k_{0}$ and $w^{P}_{k_{0}}(x)<-k_{0}m$. Then
$w^{P}_{k_{0}}(\sigma(x))=pw^{P}_{k_{0}}(x)$ and $w^{P}_{k}(\sigma(x))>-kpm$
for $k<k_{0}$.

###### Proof.

This is deduced from (15). ∎

######  Lemma 5.13.

Let $E_{P}^{\prime}$ be a Frobenius structure of $(M_{P},\phi_{P})$ with
$E_{P}^{\prime}\equiv 1\mod\pi$ and let $C>0$. Assume that for $k\gg 0$ we
have either $w_{k}^{P}(E_{P}^{\prime})\in\mathbb{Z}\backslash p\mathbb{Z}$ or
$w_{k}^{P}(E_{P}^{\prime})\geq-C$. Then $E_{P}^{\prime}$ is overconvergent.

###### Proof.

Let $\tilde{E}_{P}$ be the $P$-th coordinate of $\tilde{E}$. In particular,
$\tilde{E}_{P}$ is a Frobenius structure of $(M_{P},\phi_{P})$ contained in
$\mathcal{A}^{\dagger}_{\pi,P}$ with $\tilde{E}_{P}\equiv 1\mod\pi$. Then we
have $\tilde{E}_{P}\frac{\sigma(b)}{b}=E_{P}^{\prime}$ for some
$b\in\mathcal{A}_{\pi,P}^{\infty}$ with $b\equiv 1\mod\pi$. We need to prove
$b$ is overconvergent. Assume the contrary. In terms of partial valuations,
this means that for every $m>0$ there exists $k_{m}$ such that
$w_{k_{m}}^{P}(b)<-mk_{m}$. Assume that $k_{m}$ is the smallest integer such
that this inequality holds (so that $w_{k}^{P}(b)\geq-mk$ for $k\leq k_{m}$).
If $m$ is sufficiently large, we know from (15) and Lemma 5.12 that

$\displaystyle\begin{split}w_{k_{m}}^{P}\Big{(}\frac{\sigma(b)}{b}\Big{)}&=pw^{P}_{k_{m}}(b)<-pmk_{m},\\\
w_{k}^{P}\Big{(}\frac{\sigma(b)}{b}\Big{)}&>-pmk,\text{ for
$k<k_{m}$.}\end{split}$ (16)

Since $\tilde{E}_{P}$ is overconvergent, there exists $m_{0}>0$ such that
$w_{k}^{P}(\tilde{E}_{P})\geq-m_{0}k$ for all $k\geq 0$. Then from (15) and
(16) we have

$\displaystyle w_{k_{m}}^{P}(E_{P}^{\prime})$
$\displaystyle=pw^{P}_{k_{m}}(b),$

for $m$ sufficiently large. If we take $m$ to be larger than $C$, this
contradicts our assumption on the partial valuations of $E_{P}^{\prime}$. ∎

###### Proof.

(Of proposition 5.11) Recall that $E\in A_{\pi}^{\infty}$ be a Frobenius
structure of $(M,\phi)$ with $E\equiv 1\mod\pi$. For each $P$, let $A_{\pi,P}$
denote the subring of $A_{\pi}$ whose only poles are at $[P]$, the Teichmuller
lift of $P$. Then $\mathrm{Spec}(A_{\pi,P})$ is an affine curve over whose
special fiber is $\overline{X}-P$. We will show that there exists $a_{P}\in
A_{\pi,P}^{\infty}$ such that $E\frac{\sigma(a_{P})}{a_{P}}$ is overconvergent
at $P$ (i.e. it lies in $\mathcal{A}_{\pi,P}^{\dagger}$). Note that if $E$ is
overconvergent at $Q\neq P$, then so is $E\frac{\sigma(a_{P})}{a_{P}}$, since
$a_{P}$ does not have a pole at $Q$. The proposition will follow by repeating
for each $P$.

Let $C$ be sufficiently large so that
$H^{0}(\overline{X},\mathcal{O}_{\overline{X}}((C+n)P))/H^{0}(\overline{X},\mathcal{O}_{\overline{X}}(CP))$
has dimension $n$. We will find $a_{P}\in A_{\pi,P}^{\infty}$ such that
$E\frac{\sigma(a_{P})}{a_{P}}$ satisfies the hypothesis of Lemma 5.13. More
precisely, we will inductively construct $a_{P,n}\in A_{\pi,P}^{\infty}$
satisfying

  1. 1.

$a_{P,n+1}\equiv a_{P,n}\mod\pi^{n+1}$

  2. 2.

We can write $E\frac{\sigma(a_{P,n})}{a_{P,n}}\equiv
b_{P,n}+c_{P,n}\mod\pi^{n+1}$ where $c_{P,n}\in t_{P}^{-C}R\llbracket
t_{P}\rrbracket$ and $b_{P,n}$ is of the form

$\displaystyle b_{P,n}$ $\displaystyle=\sum_{p\nmid k}x_{P,n,k}t_{P}^{k}.$

We then take $a_{P}=\lim a_{P,n}$ and the result follows from Lemma 5.13.

For $n=0$ we take $a_{P,n}=1$. Let $n>0$ and assume such an $a_{P,n}$ exists.
We write

$\displaystyle E\frac{\sigma(a_{P,n})}{a_{P,n}}$
$\displaystyle=b_{P,n}+c_{P,n}+\pi^{n+1}r_{n}\mod\pi^{n+2}.$

Let $r_{n,0}$ denote the reduction of $r_{n}$ modulo $\pi$. Break up the
$t_{P}$-adic expansion of $r_{n,0}$:

$\displaystyle
r_{n,0}=\underbrace{\sum_{k=-pC}^{\infty}y_{0,k}t_{P}^{k}}_{\alpha_{0}}+\underbrace{\sum_{\stackrel{{\scriptstyle
k<-pC}}{{p\nmid
k}}}y_{0,k}t_{P}^{k}}_{\beta_{0}}+\underbrace{\sum_{pk<-pC}y_{0,pk}t_{P}^{pk}}_{\gamma_{0}}.$

By our assumption on $C$, there exists a regular function $c_{0}$ on
$\overline{X}-P$ with

$\displaystyle c_{0}^{p}$ $\displaystyle\equiv\gamma_{0}\mod
t_{P}^{-C}\mathbb{F}_{q}\llbracket t_{P}\rrbracket.$

We then set $r_{n,1}=r_{n,0}-c_{0}^{p}+c_{0}$. Break up $r_{n,1}$ into
$\alpha_{1}+\beta_{1}+\gamma_{1}$ as above and we find $c_{1}$ with
$c_{1}^{p}\equiv\gamma_{1}\mod t_{P}^{-C}\mathbb{F}_{q}\llbracket
t_{P}\rrbracket$. We define $r_{n,2}=c_{1}^{p}-c_{1}$. Repeat this process.
Note that the order of the pole of $\gamma_{i}$ decreases. In particular, for
$i$ large enough we have $\gamma_{i}=0$. Write $c=\sum c_{i}$ and let
$\tilde{c}$ be a lift of $c$ to $A_{\pi,P}$. By definition we have

$\displaystyle r_{n,0}-c^{p}+c$
$\displaystyle=\sum_{k=-pC}^{\infty}y_{k}t_{P}^{k}+\sum_{\stackrel{{\scriptstyle
k<-pC}}{{p\nmid k}}}y_{k}t_{P}^{k}.$

We define $a_{P,n+1}=a_{P,n}(1-\pi^{n+1}\tilde{c})$, which satisfies the
correct properties ∎

###  5.5 Semi-Local Twisting

Henceforth we will assume that $\rho:\pi_{1}(X)\to R^{\times}$ is
$\bm{\delta}$-overconvergent, and let $(M,\phi)$ denote the unit-root
$\sigma$-module over $A_{\pi}^{\dagger}$ corresponding to $\rho$. We fix a
$\bm{\delta}$-Frobenius structure $\tilde{E}$ for $(M,\phi)$ and a global
Frobenius structure $E=\sigma(a)\tilde{E}a^{-1}\in A_{\pi}^{\dagger}$ as in
Proposition 5.11. By weak base change, the semi-local $U_{p}$ operator (4.3)
induces a $p$-Dwork operator
$U_{p}:\mathcal{A}_{\pi}^{\dagger}\to\mathcal{A}_{\pi}^{\dagger}$. We define
$\tilde{\Theta}$ to be the $p$-Dwork operator

$\tilde{\Theta}=U_{p}\circ\tilde{E}:\mathcal{A}_{\pi}^{\dagger}\to\mathcal{A}_{\pi}^{\dagger}.$

Since $\Theta$ is $\sigma^{-1}$-linear, we have

$\displaystyle\Theta=U_{p}\circ E=U_{p}\circ(\sigma(a)\circ\tilde{E}\circ
a^{-1})=a\circ(U_{p}\circ\tilde{E})\circ a^{-1}.$

Let $\tilde{A}_{\pi}^{\dagger}=a^{-1}\cdot A_{\pi}^{\dagger}$ and
$\tilde{V}_{\pi}^{\dagger}=K\otimes_{R}\tilde{A}_{\pi}^{\dagger}$, regarded as
a $K_{q}$-vector space. The following is now immediate:

######  Proposition 5.14.

The operator $\tilde{\Theta}$ restricts to an endomorphism of
$\tilde{V}_{\pi}^{\dagger}$. Moreover, the action of $\tilde{\Theta}_{q}$ on
this space is nuclear, and there is an equality of Fredholm series

$C(\tilde{\Theta}_{q}|\tilde{V}_{\pi}^{\dagger},s)=C(\Theta_{q}|V_{\pi}^{\dagger},s).$

For each point $P$ of $\overline{X}$ with $\eta(P)\in\\{0,1,\infty\\}$, let
$\mathcal{A}_{\pi,P}^{\mathrm{tr}}=R\otimes_{\mathbb{Z}_{p}}\mathcal{A}_{P}^{\mathrm{tr}}$,
regarded as an $R_{q}$-submodule of $\mathcal{A}_{\pi,P}$. We define as before
$\mathcal{A}_{\pi}^{\mathrm{tr}}=\bigoplus_{P}\mathcal{A}_{\pi,P}^{\mathrm{tr}}$.
As in Proposition 4.10, we see that there is an exact sequence of
$R_{q}$-modules

$0\to L_{\pi}\to
A_{\pi}^{\dagger}\to\mathcal{A}_{\pi}^{\dagger,\mathrm{tr}}\to 0,$

where $L_{\pi}$ is a finite free $R_{q}$-module of rank $N$ (13). The
following lemma states that the “twisted” space $\tilde{A}_{\pi}^{\dagger}$
can be decomposed in a similar way:

######  Lemma 5.15.

There is an exact sequence of $R_{q}$-modules

$0\to\tilde{L}_{\pi}\to\tilde{A}_{\pi}^{\dagger}\to\mathcal{A}_{\pi}^{\dagger,\mathrm{tr}}\to
0,$

where $\tilde{L}_{\pi}=a^{-1}L_{\pi}$ is a finite free $R_{q}$-module of rank
$N$.

###### Proof.

Indeed, since $a\equiv 1\pmod{\pi}$ this sequence reduces modulo $\pi$ to the
exact sequence described above. The result follows from Lemma 2.3. ∎

##  6 Estimating Dwork Operators

In this section we give explicit estimates for the Newton polygon of
$\tilde{\Theta}_{q}$ acting on the twisted space $\tilde{V}_{\pi}^{\dagger}$.

###  6.1 Local Estimates

We begin by studying the action of $\tilde{\Theta}$ on the local spaces
$\mathcal{V}_{\pi,P}^{\dagger}=K\otimes_{R}\mathcal{A}_{\pi,P}^{\dagger}$. Our
approach will be to restrict $\tilde{\Theta}$ to a Banach subspace
$\mathcal{V}_{\pi,P}^{\mathbf{m}}\subset\mathcal{V}_{\pi,P}^{\dagger}$, which
is defined in terms of certain local growth conditions at $P$. In §6.1.1-6.1.2
below we estimate the action of $\tilde{\Theta}$ on this subspace.

First, let us explain our notational conventions. The superscript
$\mathbf{m}=(m_{P})$ will denote a tuple of positive rational numbers indexed
by $S_{\eta}$. Since the Newton polygon of $\tilde{\Theta}$ remains the same
under extension of scalars, it will be convenient to assume that $R$ contains
an $m_{P}$-th root of $\pi$ for each $P\in S_{\eta}$. We will define a
subspace
$\mathcal{A}_{\pi,P}^{\mathbf{m}}\subset\mathcal{A}_{\pi,P}^{\dagger}$ using
certain growth conditions depending on $m_{P}$ and $\eta(P)$. For $\eta(P)=0$
or $\infty$, the definition is simple:

$\mathcal{A}_{\pi,P}^{\mathbf{m}}=\mathcal{A}_{\pi,P}^{m_{P}}\subseteq\mathcal{A}_{\pi,P}^{\dagger}.$

When $\eta(P)=1$, the definition of $\mathcal{A}_{\pi,P}^{\mathbf{m}}$ is more
complicated and will be given in §6.1.2.

Our present goal is to estimate the columns of the matrix of $\tilde{\Theta}$
with respect to a formal basis of
$\mathcal{A}_{\pi,P}^{\mathbf{m},\mathrm{tr}}=\mathcal{A}_{\pi,P}^{\mathbf{m}}\cap\mathcal{A}_{\pi,P}^{\dagger,\mathrm{tr}}$,
at least for suitable $\mathbf{m}$. To explain the condition on $\mathbf{m}$,
consider the tuple of rational numbers $\mathbf{m}_{\pi}=(m_{\pi,P})_{P\in
S_{\eta}}$ defined by:

$m_{\pi,P}=\begin{cases}\frac{\delta_{P}}{p}&P\in S\\\ 0&P\not\in S\text{ and
}\eta(P)\in\\{0,\infty\\}\\\ \frac{1}{v_{\pi}(p)}&\eta(P)=1\end{cases}.$

We will write $\mathbf{m}\geq\mathbf{m}_{\pi}$ if $m_{P}\geq m_{\pi,P}$ for
all $P\in S_{\eta}$. We will give our estimates for all tuples $\mathbf{m}$ of
_positive_ rational numbers with $\mathbf{m}\geq\mathbf{m}_{\pi}$.

####  6.1.1 First case: $\eta(P)=0$ or $\infty$

We first consider the case $\eta(P)=0$ or $\infty$. In this case, we have
already defined $\mathcal{A}_{\pi,P}^{\mathbf{m}}$ to be
$\mathcal{A}_{\pi,P}^{m_{P}}$. We have $\sigma(t_{P})=t_{P}^{p}$, and the
operater $U_{p}$ is given explicitly as follows: Let $k=p\ell+r$ with $0\leq
r<p$. Then

$U_{p}(t_{P}^{k})=\begin{cases}t_{P}^{\ell}&r=0\\\ 0&r\neq 0\end{cases}.$

From the above formula we see that for every $m>0$,

$U_{p}(\mathcal{A}_{\pi,P}^{m})\subseteq\mathcal{A}_{\pi,P}^{m/p}.$ (17)

######  Proposition 6.1.

Suppose that $\eta(P)=0$ or $\infty$, and that
$\mathbf{m}\geq\mathbf{m}_{\pi}$. Then

$\tilde{\Theta}(\pi^{k/m_{P}}t_{P}^{-k})\in\pi^{\frac{k(p-1)}{pm_{P}}}\mathcal{A}_{\pi,P}^{\mathbf{m}}.$

###### Proof.

If $P\notin S$, then $\tilde{E}_{P}\in 1+\pi R_{q}$ is constant. If $P\in S$,
then $\tilde{E}_{P}\in\mathcal{A}_{\pi,P}^{\delta_{P}}$ since $(M,\phi)$ is
$\bm{\delta}$-overconvergent. In either case, we have
$\tilde{E}_{P}\in\mathcal{A}_{\pi,P}^{pm_{P}}$. By definition, we know that
$\pi^{k/m_{P}}t_{P}^{-k}$ is contained in
$\pi^{\frac{k(p-1)}{pm_{P}}}\mathcal{A}_{\pi,P}^{pm_{P}}$. The result follows
from (17). ∎

####  6.1.2 Second Case: $\eta(P)=1$

We now turn to the case $\eta(P)=1$. Since $P\notin S$, $\tilde{E}_{P}$ is
constant. Thus we will focus on estimating the action of the operator $U_{p}$.
Let $u_{P}=t_{P}^{p-1}$ and let $\mathcal{B}_{\pi,P}=R_{q}(\\!(u_{P})\\!)$.
Then $\mathcal{A}_{\pi,P}/\mathcal{B}_{\pi,P}$ is finite and Galois so we have
a decomposition

$\mathcal{A}_{\pi,P}=\bigoplus_{i=0}^{p-1}t_{P}^{-i}\mathcal{B}_{\pi,P}$

into eigenspaces for the distinct characters of the Galois group. The induced
splitting of $\mathcal{A}_{\pi,P}^{\dagger}$ is $\sigma$-equivariant. Consider
the subspace:

$\mathcal{B}_{\pi,P}^{\mathbf{m}}=\mathcal{B}_{\pi,P}^{m_{P}}=\left\\{\sum_{k=-\infty}^{\infty}b_{k}u^{-k}:v_{\pi}(b_{k})>\frac{k}{m_{P}}\text{
for all }k>0\right\\}.$

######  Lemma 6.2.

Suppose that $\mathbf{m}\geq\mathbf{m}_{\pi}$. Let $k=p\ell+r\in\mathbb{Z}$
with $0\leq r<p$. Then

$U_{p}(t^{-k})\in t^{-(\ell+r)}\mathcal{B}_{\pi,P}^{\mathbf{m}}.$

###### Proof.

For $R=\mathbb{Z}_{p}$ and $\pi=p$, this is [17, Corollary 4.7]. The condition
$\mathbf{m}\geq\mathbf{m}_{\pi}$ guarantees that
$m_{P}\geq\tfrac{1}{v_{\pi}(p)}$, so the general result follows by tensoring
up to $R$. ∎

######  Definition 6.3.

We define

$\mathcal{A}_{\pi,P}^{\mathbf{m}}=\bigoplus_{i=0}^{p-1}t_{P}^{-i}\mathcal{B}_{\pi,P}^{\mathbf{m}}.$

For each $k>p-1$, define the positive integer

$a(k)=\left\lfloor\frac{k-1}{p-1}\right\rfloor.$

Then the elements $\pi^{a(k)/m_{P}}t_{P}^{-k}$ constitute a formal basis for
$\mathcal{A}_{\pi,P}^{\mathbf{m}}$. We can now give our local estimate for
$\tilde{\Theta}$ in this case:

######  Proposition 6.4.

Suppose $\eta(P)=1$ and that $\mathbf{m}\geq\mathbf{m}_{\pi}$. Then for each
$k=p\ell+r>p-1$, we have

$\tilde{\Theta}(\pi^{a(k)/m_{P}}t_{P}^{-k})\in\pi^{\ell/m_{P}}\mathcal{A}_{\pi,P}^{\mathbf{m}}.$

###### Proof.

Observe that $a(k)-a(\ell+r)=\ell$. Thus, from Lemma 6.2 we have

$U_{p}(\pi^{a(k)/m_{P}}t_{P}^{-k})=\pi^{\ell/m_{P}}\pi^{a(\ell+r)/m_{P}}t^{-(\ell+r)}\mathcal{B}_{\pi,P}^{\mathbf{m}}\subseteq\pi^{\ell/m_{P}}\mathcal{A}_{\pi,P}^{\mathbf{m}}.$

The claim follows since $\tilde{E}_{P}\in 1+\pi R_{q}$. ∎

######  Remark 6.5.

Suppose that $p=2$. For $k\geq 3$, define $a(k)=\lfloor(k-1)/3\rfloor$. A
similar construction provides a submodule
$\mathcal{A}_{\pi,P}^{\mathbf{m}}\subseteq\mathcal{A}_{\pi,P}^{\dagger}$ with
the following property: Let $k=2\ell-r$ with $r=0$ or $1$. Then

$U_{p}(\pi^{a(k)/m_{P}}t_{P}^{-k})\in\pi^{(a(k)-a(\ell+r))/m_{P}}\mathcal{A}_{\pi,P}^{\mathbf{m}}.$

This estimate is too low for applications to the global setting. For example,
if $k=5=2\cdot 3-1$, then $a(k)-a(\ell+r)=0$, and this contributes an extra
segment of slope $0$ in the global Hodge bound below.

###  6.2 Global Estimates

Let $\mathbf{m}\geq\mathbf{m}_{\pi}$, and consider the “twisted” spaces
$\tilde{A}_{\pi}^{\mathbf{m}}=\mathcal{A}_{\pi}^{\mathbf{m}}\cap\tilde{A}_{\pi}^{\dagger}$
and $\tilde{V}_{\pi}^{\mathbf{m}}=K\otimes_{R}\tilde{A}_{\pi}^{\mathbf{m}}$.
We will now define a formal basis for $\tilde{V}_{\pi}^{\mathbf{m}}$ by
lifting the local bases used in §6.1. Let $\mathcal{A}_{\pi}^{\mathbf{m}}$
denote the product of the $\mathcal{A}_{\pi,P}^{\mathbf{m}}$ defined in §6.1.
Define
$\mathcal{A}_{\pi}^{{\mathbf{m}},\mathrm{tr}}=\mathcal{A}_{\pi}^{\mathbf{m}}\cap\mathcal{A}_{\pi}^{\dagger,\mathrm{tr}}$.
Note that we have an exact sequence

$0\to\tilde{L}_{\pi}\to\tilde{A}_{\pi}^{\mathbf{m}}\to\mathcal{A}_{\pi}^{{\mathbf{m}},\mathrm{tr}}\to
0$

and recall that $\tilde{L}_{\pi}$ is a finite free $R_{q}$-module of rank $N$.
Let ${B}_{q,0}^{\mathbf{m}}=\\{{e}_{0,k}^{\mathbf{m}}:1\leq k\leq N\\}$ be any
$R_{q}$-basis for $\tilde{L}_{\pi}$. For $P$ and each $k>n(P)$, choose a
lifting

${e}_{P,k}=t_{P}^{-k}+c_{P,k}\in\mathcal{A}_{\pi,P}^{\mathbf{m}},$

where $\mathrm{pr}(c_{P,k})=0$. Then we define
${B}_{q,P}^{\mathbf{m}}=\\{{e}_{P,k}^{\mathbf{m}}:k>\mu(P)\\}$, where

${e}_{P,k}^{\mathbf{m}}=\begin{cases}\pi^{k/m_{P}}{e}_{P,k}&\eta(P)=0\text{ or
}\infty\\\ \pi^{a(k)/m_{P}}{e}_{P,k}&\eta(P)=1\end{cases}.$

Our desired formal basis for $\tilde{V}_{\pi}^{\mathbf{m}}$ is:

${B}_{q}^{\mathbf{m}}={B}_{q,0}^{\mathbf{m}}\sqcup\bigsqcup_{P}{B}_{q,P}^{\mathbf{m}}.$
(18)

######  Proposition 6.6 (Global Column Estimate).

Suppose that $\mathbf{m}\geq\mathbf{m}_{\pi}$. Then:

  1. 1.

If $\eta(P)=0$ or $\infty$, then for all $k>0$

$\tilde{\Theta}(e_{P,k}^{\mathbf{m}})\in\pi^{\frac{k(p-1)}{pm_{P}}}\tilde{V}_{\pi,P}^{\mathbf{m}}.$

  2. 2.

If $\eta(P)=1$, then for all $k=p\ell+r>p-1$ with $0\leq r<p$

$\tilde{\Theta}(e_{P,k}^{\mathbf{m}})\in\pi^{\frac{\ell}{m_{P}}}\tilde{V}_{\pi,P}^{\mathbf{m}}.$

###### Proof.

We consider each case separately:

  1. 1.

If $\eta(P)=0$ or $\infty$ then we have

$\tilde{\Theta}({e}_{P,k}^{\mathbf{m}})=\tilde{\Theta}(\pi^{k/m_{P}}t_{P}^{-k})+\pi^{k/m_{P}}\tilde{\Theta}(c_{k}).$

The first term lies in
$\pi^{k(p-1)/pm_{P}}\mathcal{A}_{\pi,P}^{{\mathbf{m}},\mathrm{tr}}$ by
Proposition 6.1. Moreover, we know that $\tilde{\Theta}$ restricts to an
endomorphism of $\mathcal{A}_{\pi,P}^{\mathbf{m}}$, so that the second term
lies in $\pi^{k/m_{P}}\mathcal{A}_{\pi,P}^{\mathbf{m}}$. The claim follows
since
$\pi^{k}\mathcal{A}_{\pi}^{\mathbf{m}}\cap\tilde{A}_{\pi}^{\dagger}=\pi^{k}(\mathcal{A}_{\pi}^{\mathbf{m}}\cap\tilde{A}_{\pi}^{\dagger})$

  2. 2.

If $\eta(P)=1$, then

$\tilde{\Theta}({e}_{P,k}^{\mathbf{m}})=\tilde{\Theta}(\pi^{a(k)/m_{P}}t_{P}^{-k})+\pi^{a(k)/m_{P}}\tilde{\Theta}(c_{P,k}).$

By assumption, $\mathbf{m}\geq\mathbf{m}_{\pi}$ so that Proposition 6.4 holds.
Thus, the first term lies in $\pi^{\ell e}\mathcal{A}_{\pi}^{\mathbf{m}}$. The
$\pi$-adic valuation of the second term is at least
$a(k)/m_{P}\geq\ell/m_{P}$. The claim follows exactly as above.

∎

######  Corollary 6.7.

For all $\mathbf{m}\geq\mathbf{m_{\pi}}$, the characteristic series
$C(\Theta_{q}|V_{\pi}^{\dagger},s)$ agrees with the Fredholm determinant
$\det(I-s\tilde{\Theta}_{q}|B_{q}^{\mathbf{m}})$.

###### Proof.

First, observe that $\tilde{V}_{\pi}^{\dagger}$ is a union of Banach subspaces

$\tilde{V}_{\pi}^{\dagger}=\bigcup_{\mathbf{m}}\tilde{V}_{\pi}^{\mathbf{m}}.$

Let $B^{\mathbf{m}}$ be an associated $K$-basis to $B_{q}^{\mathbf{m}}$, as in
§3.4. By the estimates in Proposition 6.6, if $\mathbf{m}\geq\mathbf{m}_{\pi}$
then the matrix $\tilde{\Theta}|B^{\mathbf{m}}$ is tight. It follows that
$\tilde{\Theta}_{q}|B_{q}^{\mathbf{m}}$ is tight, and so $\tilde{\Theta}_{q}$
acts completely continuously on $\tilde{V}_{\pi}^{\mathbf{m}}$. By Proposition
3.13, the Fredholm series $\det(I-s\psi|\tilde{V}_{\pi}^{\mathbf{m}})$ is
_independent_ of $\mathbf{m}$. The claim follows as in Example 3.15(3). ∎

######  Corollary 6.8 (Global Hodge Bound).

The Newton polygon
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<v_{\pi}(p)}(\Theta_{q}|V_{\pi}^{\dagger})$
lies on or above the convex polygon with slope set:

$\\{\underbrace{0,...,0}_{r}\\}\sqcup\bigsqcup_{P\in
S}\left\\{\frac{k(p-1)}{\delta_{P}}:1\leq k<v_{\pi}(p)\delta_{P}\right\\}.$

###### Proof.

Fix a positive number $e\leq v_{\pi}(p)$, and consider the tuple
$\mathbf{m}_{e}=(m_{e,P})_{P\in S_{\eta}}$ defined by

$m_{e,P}=\begin{cases}\frac{\delta_{P}}{p}&P\in S\\\ \frac{1}{pe}&P\not\in
S\text{ and }\eta(P)\in\\{0,\infty\\}\\\ \frac{1}{e}&\eta(P)=1\end{cases}.$

Then $\mathbf{m}_{e}\geq\mathbf{m}_{\pi}$. Since the matrix of
$\tilde{\Theta}$ with respect to ${B}^{\mathbf{m}_{e}}$ has coefficients in
$R$, we see that $v_{\pi}(\tilde{\Theta}({e}_{0,k}^{\mathbf{m}_{e}}))\geq 0$
for all $1\leq k\leq N$. Now let $B^{\mathbf{m}_{e}}$ be an associated
$K$-basis to $B_{q}^{\mathbf{m}_{e}}$, as in §3.4. By Proposition 6.6, the
column Hodge polygon
$\operatorname{\mathrm{cHP}}_{\pi}^{<e}(\tilde{\Theta}|B^{\mathbf{m}_{e}})$
lies on or above the convex polygon with slope set:

$\\{\underbrace{0,...,0}_{rv_{p}(q)}\\}\sqcup\bigsqcup_{P\in
S}\left\\{\frac{k(p-1)}{\delta_{P}}:1\leq
k<e\delta_{P}\right\\}^{\times{v_{p}(q)}}.$

Upon taking the limit $e\to v_{\pi}(p)^{+}$ (or setting $e=v_{\pi}(p)$ if $R$
has characteristic $0$) the claim follows from Corollary 6.7 and Lemma 3.24. ∎

##  7 Perturbing Operators and Main Results

In this final section we will study the interaction of the Newton polygon
$\operatorname{\mathrm{NP}}_{\pi_{q}}(\tilde{\Theta}_{q}|\tilde{V}_{\pi}^{\dagger})$
with certain Hodge polygons attached to the action of $\tilde{\Theta}$ on the
Banach spaces $\tilde{V}_{\pi}^{\mathbf{m}}$. For this purpose we fix a
positive rational number $e\leq v_{\pi}(p)$, and let $\mathbf{m}_{e}$ be
defined as in the proof of Corollary 6.8 (when $R$ has characteristic $0$, we
may take $e=v_{\pi}(p)$).

###  7.1 Newton-Hodge Interaction and Perturbation Theory

Let $I$ be a countable set. Let $\Psi$ be a tight matrix with entries in $R$
indexed by $I$, regarded as a completely continuous operator $\psi:b(I)\to
b(I)$. We will now discuss the interaction between the Newton, Hodge, and
column Hodge polygons of $\Psi$. For every $r>0$, let

$I^{<r}(\Psi)=\\{i\in I:v_{\pi}\psi(e_{i})<r\\}.$

Let $\Psi^{<r}$ denote the finite diagonal submatrix of $\Psi$ indexed by
$I^{<r}(\Psi)$. We are primarily interested in the following strong type of
Newton-Hodge interaction:

######  Lemma 7.1.

The following are equivalent:

  1. 1.

$\operatorname{\mathrm{NP}}_{\pi}^{<r}(\Psi)$ and
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$ have the same terminal point.

  2. 2.

$\operatorname{\mathrm{NP}}_{\pi}(\Psi^{<r})$ and
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$ have the same terminal point.

###### Proof.

Choose an ordering on $I$ so that the column slopes $v_{\pi}\psi(e_{i})$ are
increasing. Label the elements $I^{<r}=\\{i_{1},...,i_{n}\\}$ in increasing
order, and let $(n,m)$ denote the terminal point of
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$ so that

$m=v_{\pi}\Psi(e_{i_{1}})+\cdots+v_{\pi}\Psi(e_{i_{n}}).$

Let $J=\\{j_{1},...,j_{n}\\}$ be any subset of $I$ of cardinality $n$, with
elements labelled in increasing order. Then for every $1\leq k\leq n$ we have

$v_{\pi}\Psi(e_{j_{k}})\geq v_{\pi}\Psi(e_{i_{k}}).$

If $J\neq I^{<r}(\Psi)$, then the inequality is _strict_ for $k=n$, since
$e_{j_{n}}\notin I^{<r}(\Psi)$. Since
$\operatorname{\mathrm{NP}}_{\pi}(\Psi_{J})\succeq\operatorname{\mathrm{cHP}}_{\pi}(\Psi_{J})$,
we see that every $J\times J$ minor of $\Psi$ has $\pi$-adic valuation $\geq
v_{\pi}\det(\Psi^{<r})$, with strict inequality when $J\neq I^{<r}(\Psi)$.
Thus $\operatorname{\mathrm{NP}}_{\pi}^{<r}(\Psi)$ and
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$ have the same terminal point if
and only if $v_{\pi}\det(\Psi^{<r})=m$, i.e. if and only if
$\operatorname{\mathrm{NP}}_{\pi}(\Psi^{<r})$ and
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$ have the same terminal point ∎

######  Lemma 7.2.

Suppose that $\operatorname{\mathrm{NP}}_{\pi}^{<r}(\Psi)$ and
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$ have the same terminal point.
Then the polygons

$\operatorname{\mathrm{HP}}_{\pi}(\Psi^{<r}),\
\operatorname{\mathrm{cHP}}_{\pi}(\Psi^{<r}),\
\operatorname{\mathrm{HP}}_{\pi}^{<r}(\Psi),\
\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$

are all equal.

###### Proof.

Observe that each of the listed polygons lies on or below
$\operatorname{\mathrm{HP}}_{\pi}(\Psi^{<r})$ and on or above
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$. By Lemma 7.1, all of these
polygons share the same terminal point. Since each column slope of $\Psi^{<r}$
is greater than or equal to the corresponsing column slope of $\Psi$, we must
have that
$\operatorname{\mathrm{cHP}}_{\pi}(\Psi^{<r})=\operatorname{\mathrm{cHP}}_{\pi}(\Psi)$.
It suffices then to consider the case that $\Psi=\Psi^{<r}$ is a finite matrix
whose $\pi$-adic column slopes are $<r$. But this is immediate from Lemma
3.20. ∎

######  Definition 7.3.

Let $\Psi^{\prime}=\Psi+\varepsilon$, where $\varepsilon$ is a matrix with
entries in $R$ indexed by $I$. We say that $\Psi^{\prime}$ is a $\pi$-adic
_$r$ -perturbation_ of $\Psi$ if:

  1. 1.

$v_{\pi}\varepsilon(e_{i})>v_{\pi}\Psi(e_{i})$ for all $i\in I^{<r}(\Psi)$.

  2. 2.

$v_{\pi}\varepsilon(e_{i})\geq r$ for all $i\notin I^{<r}(\Psi)$.

If $\Psi^{\prime}$ is a tight $\pi$-adic $r$-perturbation of $\Psi$, then the
truncated column Hodge polygons $\operatorname{\mathrm{cHP}}^{<r}_{\pi}(\Psi)$
and $\operatorname{\mathrm{cHP}}^{<r}_{\pi}(\Psi^{\prime})$ of Lemma 3.20
necessarily agree. The stronger perturbation condition guarantees the
following interaction between the Newton polygons of both matrices:

######  Lemma 7.4 (Perturbation Lemma).

Suppose that $\Psi^{\prime}=\Psi+\varepsilon$ is a tight $\pi$-adic
$r$-perturbation of $\Psi$. If $\operatorname{\mathrm{NP}}_{\pi}^{<r}(\Psi)$
has the same terminal point as
$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)=\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi)$,
then the same is true of
$\operatorname{\mathrm{NP}}_{\pi}^{<r}(\Psi^{\prime})$.

###### Proof.

By Lemmas 7.1 and 7.2, we immediately reduce to the case that $\Psi$ and
$\Psi^{\prime}$ are finite matrices whose column slopes are all $<r$. We must
show that $v_{\pi}\det(\Psi)=v_{\pi}\det(\Psi^{\prime})$. Choose an ordering
of $I$ so that the sequence $v_{\pi}\Psi(e_{i})$ is increasing. Given an index
$(i_{1},...,i_{k})\in\wedge^{n}I$, we will abbreviate
$e_{i_{1},...,i_{k}}=e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}$. Let us write

$\wedge^{k}\Psi^{\prime}=\wedge^{k}\Psi+\varepsilon_{k}$

for all $k\geq 0$. A straightforward computation shows that

$\varepsilon_{k+1}(e_{i_{1},...,i_{k+1}})=\wedge^{k}\Psi(e_{i_{1},...,i_{k}})\wedge\varepsilon(e_{i_{k+1}})+\varepsilon_{k}(e_{i_{1},...,i_{k}})\wedge\Psi^{\prime}(e_{i_{k+1}}).$

Since $\Psi^{\prime}$ is an $r$-perturbation of $\Psi$, by induction on $k$ we
see that

$v_{\pi}(\varepsilon_{n})>v_{\pi}\Psi(e_{1})+\cdots+v_{\pi}\Psi(e_{n}).$

By assumption, the latter is exactly $\det(\Psi)$. It follows that

$v_{\pi}\det(\Psi^{\prime})=v_{\pi}(\det(\Psi)+\varepsilon_{n})=v_{\pi}\det(\Psi).$

∎

###  7.2 Local-to-global extensions

Let $P\in S$. Let $G_{P}$ denote the absolute Galois group of $F_{P}$, and let
$\rho_{P}$ denote the restriction of $\rho$ to $G_{P}$. By a theorem of Katz-
Gabber [12], $\rho_{P}$ extends in a canonical way to a continuous character

$\rho_{P}^{\mathrm{ext}}:\pi_{1}(\mathbb{A}_{\mathbb{F}_{q}}^{1})\to
R^{\times}.$

Let us describe $\rho_{P}^{\mathrm{ext}}$ explicitly in terms of
$\sigma$-modules. We define a flat lifting $(A_{P},\sigma)$ of
$\mathbb{A}_{\mathbb{F}_{q}}^{1}$ over $\mathbb{Z}_{p}$, where
$A_{P}=\mathbb{Z}_{q}[t_{P}^{-1}]$ and $\sigma(t_{P}^{-1})=t_{P}^{-p}$. Let
$A_{\pi,P}=R\otimes_{\mathbb{Z}_{p}}A_{P}$. Since $\tilde{E}$ is a
$\bm{\delta}$-Frobenius structure for $(M,\phi)$, we may regard
$\tilde{E}_{P}$ as an element of $A_{\pi,P}^{\delta_{P}}$. Then
$\rho_{P}^{\mathrm{ext}}$ corresponds to the unit-root $\sigma$-module

$(M_{P}^{\mathrm{ext}},\phi_{P}^{\mathrm{ext}})=(A_{\pi,P}^{\dagger},\tilde{E}_{P}\circ\sigma).$

Since $\rho$ is $\bm{\delta}$-overconvergent, the Dwork trace formula (i.e.
the Monsky trace formula over $\mathbb{G}_{m}$) guarantees that
$L(\rho_{P}^{\mathrm{ext}},s)$ is analytic in the disk
$v_{\pi}(s)>-v_{\pi}(q)$. To state the trace formula, we define as before

$U_{p}=\frac{1}{p}\sigma^{-1}\circ\mathrm{Tr}:A_{P}^{\dagger}\to
A_{P}^{\dagger}.$

By weak base change, $U_{p}$ induces a $p$-Dwork operator on
$A_{\pi,P}^{\dagger}$. Consider the $p$-Dwork operator
$\tilde{\Theta}=U_{p}\circ\tilde{E}_{P}$. Let
$V_{\pi,P}^{\dagger}=K\otimes_{R}A_{\pi,P}^{\dagger}$. Then the action of
$\tilde{\Theta}_{q}$ on $V_{\pi,P}^{\dagger}$ is nuclear, and the Dwork trace
formula asserts that

$(1-\alpha
s)L(\rho_{P}^{\mathrm{ext}},s)=\frac{C(\tilde{\Theta}_{q}|V_{\pi,P}^{\dagger},s)}{C(\tilde{\Theta}_{q}|V_{\pi,P}^{\dagger},qs)},$

where $\alpha\in 1+\pi R_{q}$ denotes constant term of $\tilde{E}_{P}$.
Equivalently,

$C(\tilde{\Theta}_{q}|V_{\pi,P}^{\dagger},s)=\prod_{j=0}^{\infty}(1-\alpha
q^{j}s)L(\rho_{P}^{\mathrm{ext}},q^{j}s).$

######  Lemma 7.5.

There is a factorization

$C(\tilde{\Theta}_{q}|V_{\pi,P}^{\dagger},s)=(1-\alpha s)\cdot
C(\tilde{\Theta}_{q}|\mathcal{V}_{\pi,P}^{\dagger,\mathrm{tr}},s).$

###### Proof.

We have a decomposition
$A_{\pi,P}^{\dagger}=R_{q}\oplus\mathcal{A}_{\pi,P}^{\dagger,\mathrm{tr}}$. We
have already seen that the rightmost factor is $\tilde{\Theta}_{q}$-invariant
and that $\tilde{\Theta}_{q}$ acts as a nuclear operator on this space. The
desired factorization is obtained as in Example 3.15(2). ∎

######  Definition 7.6.

Let $P\in S$. The $\pi_{q}$-adic _local Newton polygon_ of $\rho_{P}$ is

$\operatorname{\mathrm{NP}}_{\pi_{q}}(\rho_{P})=\operatorname{\mathrm{NP}}_{\pi_{q}}(\tilde{\Theta}_{q}|\mathcal{V}_{\pi,P}^{\dagger,\mathrm{tr}}).$

By the Dwork trace formula, we see that
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<e}(\rho_{P})=\operatorname{\mathrm{NP}}_{\pi_{q}}^{<e}(\rho_{P}^{\mathrm{ext}})$,
and in particular this polygon does not depend on our local lifting
$(\mathcal{A}_{P},\sigma)$. We now turn our attention to the Hodge polygons of
local-to-global extensions. First, for each $P\in S$ we define the
_$\delta_{P}$ -Hodge polygon_ $\operatorname{\mathrm{HP}}(\delta_{P})$ to be
the convex polygon with slope set

$\left\\{\frac{p-1}{\delta_{P}},\frac{2(p-1)}{\delta_{P}},...\right\\}.$

Recall that since $P\in S$, the growth condition $m_{e,P}=\delta_{P}/p$ does
not depend on $e$. By Proposition 6.1, we know that

$\operatorname{\mathrm{HP}}_{\pi}(\tilde{\Theta}|\mathcal{V}_{\pi,P}^{\mathbf{m}_{e},\mathrm{tr}})\succeq\operatorname{\mathrm{HP}}(\delta_{P})^{\times
v_{p}(q)}.$ (19)

######  Definition 7.7.

We say that $\rho_{P}$ is $\pi$-adically _$\delta_{P}$ -Hodge_ if (19) is an
equality. In this case, we define the $\pi_{q}$-adic _local Hodge polygon_ of
$(M,\phi)$ at $P$ to be

$\operatorname{\mathrm{HP}}_{\pi_{q}}(\rho_{P})=\operatorname{\mathrm{HP}}(\delta_{P}).$

We say that $\rho$ is $\pi$-adically _$\bm{\delta}$ -Hodge_ if $\rho_{P}$ is
$\pi$-adically $\delta_{P}$-Hodge for all $P\in S$.

######  Remark 7.8.

In general, the local Hodge polygon
$\operatorname{\mathrm{HP}}_{\pi_{q}}(\rho_{P})$ _need not_ agree with the
Hodge polygon
$\operatorname{\mathrm{HP}}_{\pi_{q}}(\tilde{\Theta}_{q}|\mathcal{V}_{\pi,P}^{\mathbf{m}_{e},\mathrm{tr}})$.
However, by Lemma 3.24 we always have

$\operatorname{\mathrm{NP}}_{\pi_{q}}(\rho_{P})\succeq\operatorname{\mathrm{HP}}_{\pi_{q}}(\rho_{P}).$

######  Proposition 7.9.

Let $r_{i}$ be a sequence of positive rational numbers with $r_{i}\to\infty$.
Suppose that for each $i$, the polygons
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r_{i}}(\rho_{P})$ and
$\operatorname{\mathrm{HP}}^{<r_{i}}(\delta_{P})$ have the same terminal
point. Then $\rho_{P}$ is $\pi$-adically $\delta_{P}$-Hodge.

###### Proof.

By Lemma 3.24 we see that

$\operatorname{\mathrm{NP}}_{\pi}(\tilde{\Theta}|\mathcal{V}_{\pi,P}^{\dagger,\mathrm{tr}})\succeq\operatorname{\mathrm{HP}}_{\pi}(\tilde{\Theta}|\mathcal{V}_{\pi,P}^{\mathbf{m}_{e},\mathrm{tr}})\succeq\operatorname{\mathrm{cHP}}_{\pi}(\tilde{\Theta}|B_{P}^{\mathbf{m}_{e}})\succeq\operatorname{\mathrm{HP}}(\delta_{P})^{\times
v_{p}(q)}.$

By the estimates in §6.1.1, we know that every column slope of
$\tilde{\Theta}|B_{P}^{\mathbf{m}_{e}}$ is greater than or equal to the
corresponding column slope of $\operatorname{\mathrm{HP}}(\delta)^{\times
v_{p}(q)}$. By assumption, the $<r_{i}$-truncation of all four polygons have
the same terminal point for all $i$. Since $r_{i}\to\infty$, it follows from
Lemma 7.2 that

$\operatorname{\mathrm{HP}}_{\pi}(\tilde{\Theta}|V_{\pi,P}^{\mathbf{m}_{e},\mathrm{tr}})=\operatorname{\mathrm{cHP}}_{\pi_{q}}(\tilde{\Theta}|B_{P}^{\mathbf{m}_{e}})=\operatorname{\mathrm{HP}}(\delta_{P})^{\times
v_{p}(q)}.$

∎

######  Theorem 7.10.

Let $\rho:\pi_{1}(X)\to R^{\times}$ be a finite character of order $p^{n}$.
Let $\pi\in\mathbb{Z}_{p}[\rho]$ be a uniformizer. For each $P\in S$, let
$d_{P}$ denote the Swan conductor of $\rho$ at $P$, and let
$\delta_{P}=d_{P}/p^{n-1}$. Then:

  1. 1.

For all $P\in S$ and all $n>0$, the polygons
$\operatorname{\mathrm{NP}}_{\pi_{q}}(\rho_{P})$ and
$\operatorname{\mathrm{HP}}(\delta_{P})$ agree on the interval
$[nd_{P}-1,nd_{P}]$.

  2. 2.

$\rho$ is $\pi$-adically $\bm{\delta}$-Hodge.

###### Proof.

Since $L(\rho_{P}^{\mathrm{ext}},s)$ is a polynomial of degree $d_{P}-1$, we
see that $\operatorname{\mathrm{NP}}_{\pi_{q}}^{<e}(\rho_{P}^{\mathrm{ext}})$
and $\operatorname{\mathrm{HP}}^{<e}(\delta)$ share the same terminal point
$(d_{P}-1,(d_{P}-1)/\delta_{P})$. Claim 1 follows immediately from the Dwork
trace formula, and claim 2 follows from Proposition 7.9. ∎

In the follow-up papers [18], [25] we will verify that each of the
$\bm{\delta}$-overconvergent characters of Example 5.10 are also
$\bm{\delta}$-Hodge.

###  7.3 Perturbations Coming from Local-to-Global Extensions

We will now define a perturbation of the matrix
$\Psi=\tilde{\Theta}|{B}^{\mathbf{m}_{e}}$ using local-to-global extensions.
Using the partition (18), we may regard $\Psi=(\Psi_{\alpha,\beta})$ as a
block matrix, where $\alpha,\beta$ are either $0$ or one of the $P$. Let us
define a new block matrix $\Psi^{\prime}=(\Psi^{\prime}_{\alpha,\beta})$ as
follows: For any $\alpha$, we let $\Psi^{\prime}_{\alpha,0}=\Psi_{\alpha,0}$.
If $P\in S$, then we let
$\Psi^{\prime}_{P,P}=\tilde{\Theta}|B_{P}^{\mathbf{m}_{e}}$. We define all
other blocks to be zero.

<EMAIL_ADDRESS>50.00008pt}&&\makebox[0.0pt]{\smash{\raisebox{15.00002pt}{$\phantom{\Psi_{0,0}}0$}}}\Psi_{0,0}&&0&\makebox[0.0pt]{\smash{\raisebox{15.00002pt}{$\phantom{\cdots}P\in
S$}}}\cdots&0&0&\makebox[0.0pt]{\smash{\raisebox{15.00002pt}{$\phantom{\cdots}P\notin
S$}}}\cdots&0\\\
\hline\cr&&\Psi_{0,P}&&\tilde{\Theta}|B_{P}^{\mathbf{m}_{e}}&\cdots&0&0&\cdots&0\\\
\makebox[0.0pt]{$P\in S$\hskip
50.00008pt}&&\vdots&&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
&&\Psi_{0,P}&&0&\cdots&\tilde{\Theta}|B_{P}^{\mathbf{m}_{e}}&0&\cdots&0\\\
\hline\cr&&\Psi_{0,P}&&0&\cdots&0&0&\cdots&0\\\ \makebox[0.0pt]{$P\notin
S$\hskip 50.00008pt}&&\vdots&&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
&&\Psi_{0,P}&&0&\cdots&0&0&\cdots&0\\\ \end{array}\right]$ Figure 1: The shape
of the block matrix $\Psi^{\prime}$, with block indices labelled.

######  Lemma 7.11.

If $\rho$ is $\pi$-adically $\bm{\delta}$-Hodge, then $\Psi^{\prime}$ is an
$e$-perturbation of $\Psi$.

###### Proof.

First, recall from the global Hodge bound that the $\pi$-adic valuation of
$\tilde{\Theta}(e_{\alpha,k}^{\mathbf{m}_{e}})$ is $\geq e$ if either:
$\alpha=P\notin S$, or $\alpha=P\in S$ and $k\geq e\delta_{P}$. Thus we need
only consider the cases $\alpha=0$ and $\alpha=P\in S$ with $k<e\delta_{P}$.
Let $\Psi^{\prime}=\Psi+\varepsilon$, and regard
$\varepsilon=(\varepsilon_{\alpha,\beta})$ as a block matrix. Then by
construction we see that $\varepsilon_{\alpha,0}=0$ regardless of $\alpha$, so
for these columns $\Psi^{\prime}$ satisfies the perturbation condition.

It remains to consider the case $\alpha=P\in S$ and $k<e\delta_{P}$. Write
${e}_{P,k}=t_{P}^{-k}+c_{P,k}$, so that

$\tilde{\Theta}({e}_{P,k}^{\mathbf{m}_{e}})=\tilde{\Theta}(\pi^{kp/\delta_{P}}t^{-k})+\pi^{kp/\delta_{P}}\tilde{\Theta}(c_{k}).$

Regarding $\varepsilon$ as an endomorphism of
$\tilde{A}_{\pi}^{\mathbf{m}_{e}}$, we see that

$\varepsilon({e}_{P,k}^{\mathbf{m}_{e}})=-\pi^{kp/\delta_{P}}\tilde{\Theta}(c_{k}).$

It follows that

$v_{\pi}(\varepsilon({e}_{P,k}^{\mathbf{m}_{e}}))\geq\frac{kp}{\delta_{P}}.$

On the other hand, the assumption that $\rho$ is $\pi$-adically
$\bm{\delta}$-Hodge implies that

$v_{\pi}(\tilde{\Theta}({e}_{P,k}^{\mathbf{m}_{e}}))=\frac{k(p-1)}{\delta_{P}}.$

∎

###  7.4 Main results

We are now ready to state and prove the general form of Theorem 1.1. Let
$r\leq v_{\pi}(p)$, so that $L(\phi,s)$ is analytic in the region
$v_{\pi_{q}}(s)>-r$ by the Monsky trace formula. We define the $r$-truncated
_$\pi_{q}$ -adic Newton polygon_ of $(M,\phi)$ to be

$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\phi)=\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\Theta_{q}|V_{\pi}^{\dagger}).$

######  Lemma 7.12.

If $\overline{X}$ is ordinary, then
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\phi)$ has $N$ segments of slope
$0$.

###### Proof.

Observe that $(M,\phi)$ is congruent to the trivial $\sigma$-module
$(A_{\pi}^{\infty},\sigma)$ modulo $\pi$. In particular, if we write
$P(U,s)/(1-qs)$ for the zeta function of $U$, then

$L(\phi,s)\equiv P(U,s)\pmod{\pi}.$

The result follows from the Deuring-Shafarevich formula. ∎

Assume now that $\rho$ is $\pi$-adically $\bm{\delta}$-Hodge. We define the
$r$-truncated _$\pi_{q}$ -adic Hodge polygon_ of $(M,\phi)$ to be

$\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\phi)=\operatorname{\mathrm{NP}}\\{\underbrace{0,\dots,0}_{N}\\}\sqcup\bigsqcup_{P\in
S}\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\rho_{P}^{\mathrm{ext}}).$

######  Theorem 7.13.

Suppose that $\overline{X}$ is ordinary and that $\rho$ is $\pi$-adically
$\bm{\delta}$-Hodge. Then
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\rho_{P}^{\mathrm{ext}})$ and
$\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\rho_{P}^{\mathrm{ext}})$ have the
same terminal point for each $P\in S$ if and only if
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\phi)$ and
$\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\phi)$ have the same terminal
point.

###### Proof.

Since $r\leq v_{\pi}(p)$, we may enlarge $e$ as necessary and assume that
$r\leq e$. By Lemma 7.11 the matrix $\Psi^{\prime}$ of 7.3 is a $\pi$-adic
$r$-perturbation of $\Psi$. Since $\overline{X}$ is assumed to be ordinary, we
know that
$\operatorname{\mathrm{NP}}_{\pi}^{<r}(\Psi)=\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\phi)^{\times{v_{p}(q)}}$
must have $Nv_{p}(q)$ segments of slope $0$. By the perturbation lemma, the
same is true of $\Psi^{\prime}$. Thus

$\operatorname{\mathrm{NP}}_{\pi}^{<r}(\Psi^{\prime})=\operatorname{\mathrm{NP}}\\{\underbrace{0,\dots,0}_{Nv_{p}(q)}\\}\sqcup\bigsqcup_{P\in
S}\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\rho_{P}^{\mathrm{ext}})^{\times{v_{p}(q)}}.$

and

$\operatorname{\mathrm{cHP}}_{\pi}^{<r}(\Psi^{\prime})=\operatorname{\mathrm{NP}}\\{\underbrace{0,\dots,0}_{Nv_{p}(q)}\\}\sqcup\bigsqcup_{P\in
S}\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\rho_{P}^{\mathrm{ext}})^{\times
v_{p}(q)}.$

The claim is immediate from the perturbation lemma. ∎

######  Corollary 7.14.

Suppose that $\overline{X}$ is ordinary and that $\rho:\pi_{1}(X)\to
R^{\times}$ is $\pi$-adically $\bm{\delta}$-Hodge. Let $r\leq v_{\pi}(p)$. If
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\rho_{P}^{\mathrm{ext}})$ and