Newton slopes for twisted Artin--Schreier--Witt Towers

TL;DR

This paper investigates the $p$-adic Newton slopes of $L$-functions associated with twisted Artin--Schreier--Witt towers, showing they form arithmetic progressions when the character's conductor is sufficiently large.

Contribution

It establishes the pattern of Newton slopes as arithmetic progressions for a broad class of twisted $L$-functions in the context of Artin--Schreier--Witt towers.

Findings

Newton slopes form arithmetic progressions for large conductor

Results apply to twisted $L$-functions over finite fields

Provides new insights into $p$-adic properties of $L$-functions

Abstract

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto \omega(a)$ be the Teichm\"uller lift of $\mathbb F_q$, and let $\chi:\mathbb{Z}\to \mathbb C_p^\times$ be a finite character of $\mathbb Z_p$. The $L$-function associated to the polynomial $f$ and the so-called twisted character $\omega^u\times \chi$ is denoted by $L_f(\omega^u,\chi,s)$. We prove that, when the conductor of the character is large enough, the $p$-adic Newton slopes of this $L$-function form arithmetic progressions.

Equations

$$\omega :$$

$$\alpha \mapsto \omega (\alpha ):=\hat{\alpha }$$

$${\omega }^u:$$

$$\alpha \mapsto {\omega }^u(\alpha )={\hat{\alpha }}^u.$$

$$L_{f}(\chi,s)=\prod\limits_{x\in|\mathbb{G}_{m,\mathbb{F}_{q}}|}\frac{1}{1-\chi\Big{(}\mathrm{Tr}_{\mathbb{Q}_{q^{\deg(x)}}/\mathbb{Q}_{p}}\big{(}\hat{f}(\hat{x})\big{)}\Big{)}s^{\deg(x)}},$$

$$u=\sum _{i=0}^{a-1}u(i)p^i,$$

$$L_{f}(\omega^{u},\chi,s)=\prod\limits_{x\in|\mathbb{G}_{m,\mathbb{F}_{q}}|}\frac{1}{1-\omega^{u}\circ\mathrm{Norm}_{\mathbb{F}_{q^{\deg(x)}}/\mathbb{F}_{q}}(x)\cdot\chi\Big{(}\mathrm{Tr}_{\mathbb{Q}_{q^{\deg(x)}}/\mathbb{Q}_{p}}\big{(}\hat{f}(\hat{x})\big{)}\Big{)}s^{\deg(x)}},$$

$$y_u(k):=\frac{ak(k-1)(p-1)}{2d}+\frac{k}{d}\sum _{i=0}^{a-1}u(i).$$

$$\big{(}kd,\frac{y_{u}(kd)}{(p-1)p^{m_{\chi}-1}}\big{)}\quad\textrm{for any}\ 0\leq k\leq p^{m_{\chi}-1}.$$

$$\bigcup _{k=1}^{p^{m_{\chi }-1}}\{{\alpha }_{k1},{\alpha }_{k2},\dots ,{\alpha }_{kd}\},$$

$$\frac{a(k-1)}{p^{m_{\chi }-1}}+\frac{\sum _{i=0}^{a-1}u(i)}{d(p-1)p^{m_{\chi }-1}}\le {\alpha }_{kj}\le \frac{a(k-1)}{p^{m_{\chi }-1}}+\frac{\sum _{i=0}^{a-1}u(i)}{d(p-1)p^{m_{\chi }-1}}+\frac{a(d-1)}{d{p}^{m_{\chi }-1}}$$

$$\bigcup_{i=0}^{p^{m_{\chi}-m_{0}}-1}\Big{\{}\frac{\alpha_{1}+ai}{p^{m_{\chi}-m_{0}}},\frac{\alpha_{2}+ai}{p^{m_{\chi}-m_{0}}},\dots,\frac{\alpha_{dp^{m_{0}-1}}+ai}{p^{m_{\chi}-m_{0}}}\Big{\}}.$$

$$L_{f(x^{q-1})}(\chi ,s):=\prod _{u=0}^{q-2}{L}_f({\omega }^u,\chi ,s)$$

$${\mathcal{C}}_0\leftarrow {\mathcal{C}}_1\leftarrow \cdots \leftarrow {\mathcal{C}}_m\leftarrow \cdots$$

$$\{\begin{array}{ll}{L}_f({\omega }^u,\chi ,s)& \text{if}u\ne 0\\ \frac{L_f({\omega }^u,\chi ,s)}{1-\chi (\hat{f}(0))}& \text{if}u=0\end{array}$$

$$\operatorname{NP}_{T}(F):=\operatorname{NP}\Big{(}\Big{\{}(k,v_{T}(\mathbbm{u}_{k}))\;\big{|}\;0\leq k\leq n\Big{\}}\Big{)},$$

$${\text{SS}}_1\uplus {\text{SS}}_2$$

$${\mathrm{NP}}_1\oplus {\mathrm{NP}}_2$$

$$(t+\mathrm{NP})(k)=\mathrm{NP}(k)+t\text{for any}0\le k\le n,$$

$${\mathrm{NP}}_1(k)\ge N{P}_2(k)$$

$$\underset{i=1}{\overset{m}{⨁}}{\mathrm{NP}}_{1,i}\ge \underset{i=1}{\overset{m}{⨁}}{\mathrm{NP}}_{2,i}.$$

$$S_{f,\omega^{u},\chi}(\ell,s)=\sum_{x\in\mathbb{F}^{\times}_{q^{\ell}}}\omega^{u}\circ\mathrm{Norm}_{\mathbb{F}_{q^{\ell}}/\mathbb{F}_{q}}(x)\cdot(1+T)^{\Big{(}\mathrm{Tr}_{\mathbb{Q}_{q^{\ell}}/\mathbb{Q}_{p}}\big{(}\hat{f}(\hat{x})\big{)}\Big{)}}\in\mathbb{Z}_{q}[\![T]\!]$$

$$L_f({\omega }^u,T,s)=\exp (\sum _{\ell =1}^{\infty }{S}_{f,{\omega }^u,\chi }(\ell ,s)\frac{s^{\ell }}{\ell }).$$

$$L_{f}(\omega^{u},T,s)\big{|}_{T=\chi(1)-1}=L_{f}(\omega^{u},\chi,s).$$

$$L_g({\omega }^0,T,s)=\prod _{u=0}^{q-2}{L}_f({\omega }^u,T,s).$$

$$(1+T)^{\Big{(}\mathrm{Tr}_{\mathbb{Q}_{q^{\ell}}/\mathbb{Q}_{p}}\big{(}\hat{f}(\hat{x})\big{)}\Big{)}}:=\sum_{n=0}^{\infty}b_{\ell,n}(T)\hat{x}^{n}\in\mathbb{Z}_{p}[\![T]\!][\![\hat{x}]\!],$$

$$S_{f,{\omega }^u,\chi }(\ell ,s)=(q^{\ell }-1)\sum _{n=1}^{\infty }{b}_{\ell ,(q^{\ell }-1)(n-\frac{u}{q-1})}(T).$$

$$\sum _{u=0}^{q-2}{S}_{f,{\omega }^u,T}(\ell ,s)=(q^{\ell }-1)\sum _{u=0}^{q-2}\sum _{n=1}^{\infty }{b}_{\ell ,(q^{\ell }-1)(n-\frac{u}{q-1})}(T)=(q^{\ell }-1)\sum _{n=1}^{\infty }{b}_{\ell ,\frac{n(q^{\ell }-1)}{q-1}}(T).$$

$$S_{g,{\omega }^0,T}(\ell ,s)=(q^{\ell }-1)\sum _{n=1}^{\infty }{b}_{\ell ,\frac{n(q^{\ell }-1)}{q-1}}(T).$$

$$S_{g,{\omega }^0,T}(\ell ,s)=\sum _{u=0}^{q-2}{S}_{f,{\omega }^u,T}(\ell ,s),$$

$$L_g({\omega }^0,T,s)=\prod _{u=0}^{q-2}{L}_f({\omega }^u,T,s).\text{qed}$$

Full text

1. 1 Introduction
2. 2 Notation
3. 3 The T-adic Dwork’s Trace Formula
4. 4 Proof of Theorem 1.5 and Theorem 1.6

1. Introduction

Let $p$ be a prime number. Let $\mathbb{F}_{q}$ be the finite field of $q=p^{a}$ elements. Let

[TABLE]

be the Teichmüller lift of $\mathbb{F}_{q}$. For any $u\in\{0,1,\cdots,q-1\}$, we put

[TABLE]

We fix a monic polynomial $f(x)=x^{d}+a_{d-1}x^{d-1}+\cdots+a_{0}\in\mathbb{F}_{q}[x]$ of degree $d$ which is coprime to $p$. Set $a_{d}=1$ and put $\hat{a}_{i}:=\omega(a_{i})$ for $i=0,\dots,d$. The Teichmüller lift of the polynomial $f(x)$ is defined by $\hat{f}(x):=x^{d}+\hat{a}_{d-1}x^{d-1}+\cdots+\hat{a}_{0}\in\mathbb{Z}_{q}[x].$

The non-twisted $L$-function associated to $f(x)$ and a finite character $\chi:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ is defined as

[TABLE]

where $\mathbb{G}_{m,\mathbb{F}_{q}}$ is the one-dimensional torus over $\mathbb{F}_{q}$ and $\deg(x)$ stands for the degree of $x$.

In [DWX], Davis, Wan, and Xiao proved that

• •

If $\chi_{1}$ and $\chi_{2}$ are two finite characters with the same conductor $p^{m}\geq\frac{ap(d-1)^{2}}{8d}$, then $L_{f}(\chi_{1},s)$ and $L_{f}(\chi_{2},s)$ have the same Newton polygon.

• •

Let $\chi_{0}:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ be a fixed character with conductor $p^{\lceil\log_{p}\frac{ap(d-1)^{2}}{8d}\rceil}.$ Then the $p$-adic Newton slopes of $L_{f}(\chi_{1},s)$ (see Definition 1.4) form a disjoint union of arithmetic progressions determined by the $p$-adic Newton slopes of $L_{f}(\chi_{0},s)$.

In [BFZ], Blache, Ferard, and Zhu studied the so-called twisted $L$-functions (see Definition 1.2), whose $p$-adic Newton polygons satisfy a universal lower bound proved by C.Liu and W.Liu in [LL]. This lower bound is similar to the one given in [DWX]. Therefore, it is of interest to ask if the $p$-adic Newton slopes of the twisted $L$-functions also form arithmetic progressions. In this paper, we give an upper bound for the twisted $L$-function and prove that it coincides with the lower bound at $x=kd$ for any integer $k\geq 0$. As a consequence, we prove that its $p$-adic Newton slopes indeed form arithmetic progressions.

Notation 1.1**.**

For an integer $u$ in the set $\{0,1,\dots,q-2\}$, we put

[TABLE]

where $0\leq u(i)\leq p-1$ for any $0\leq i\leq a-1$.

Definition 1.2**.**

Let $\chi:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ be a finite character with conductor $p^{m_{\chi}}$. The twisted $L$-function associated to the characters $\chi$ and $\omega^{u}$ is defined by

[TABLE]

where $\mathbb{G}_{m,\mathbb{F}_{q}}$ is the one-dimensional torus over $\mathbb{F}_{q}$ and $\deg(x)$ stands for the degree of $x$. In [Liu-Wei], Liu and Wei prove that the $L$-function $L_{f}(\omega^{u},\chi,s)$ is a polynomial of degree $dp^{m_{\chi}-1}$.

Notation 1.3**.**

For simplicity of notations, we denote

[TABLE]

Definition 1.4**.**

We call the slopes of the line segments of the $p$-adic Newton polygon of $L_{f}(\omega^{u},\chi,s)$ the $p$-adic Newton slopes of $L_{f}(\omega^{u},\chi,s)$.

In this paper, we prove the following.

Theorem 1.5**.**

• (a)

The $p$-adic Newton polygon of $L_{f}(\omega^{u},\chi,s)$ passes through the points

[TABLE]

• (b)

The $p$-adic Newton polygon of $L_{f}(\omega^{u},\chi,s)$ has slopes (in increasing order)

[TABLE]

where

[TABLE]

for any $1\leq j\leq d$.

When the conductor $p^{m_{\chi}}$ of $L_{f}(\omega^{u},\chi,s)$ is large enough, the $p$-adic Newton slopes of $L_{f}(\omega^{u},\chi,s)$ have the following property.

Theorem 1.6** (Main theorem).**

Let $m_{0}$ be the minimal positive integer such that $p^{m_{0}}>\frac{adp}{8(p-1)}$ and let $0\leq\alpha_{1},\alpha_{2},\dots,\alpha_{dp^{m_{0}-1}}<a$ denote the slopes of the $p$-adic Newton polygon of $L_{f}(\omega^{u},\chi_{0},s)$ for a finite character $\chi_{0}:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ with $m_{\chi_{0}}=m_{0}$. Then for any finite character $\chi:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ with $m_{\chi}\geq m_{0}$, the $p$-adic Newton polygon of $L_{f}(\omega^{u},\chi,s)$ has slopes

[TABLE]

Theorem 1.6 says that when $m_{\chi}$ is large enough, the $p$-adic Newton slopes of $L_{f}(\omega^{u},\chi,s)$ form a disjoint union of arithmetic progressions determined by the $p$-adic Newton slopes of $L_{f}(\omega^{u},\chi_{0},s)$. A similar result is proved by Li in [Li] for the general Witt towers without twisting.

This paper is inspired by the $p$-adic Newton slopes of $L_{f}(\chi,s)$ in arithmetic progressions (proved in [DWX]), the twisted decomposition of

[TABLE]

in [BFZ], and the lower bound for the Newton polygon of $L_{f}(\omega^{u},\chi,s)$ given in [LL]. Let

[TABLE]

be the Artin–Schreier–Witt curve tower associated to the polynomial $f(x^{q-1})$, and let $Z(\mathcal{C}_{m},s)$ be the zeta function of the curve $\mathcal{C}_{m}$. It is known that

[TABLE]

are factors of $Z(\mathcal{C}_{m},s)$, and the degree of $L_{f}(\omega^{u},\chi_{0},s)$ is $\frac{1}{q-1}$ of the degree of $L_{f}(\chi_{0},s))$. Therefore, as a corollary of Theorem 1.6, we give a more precise description of zeros of $Z(\mathcal{C}_{m},s)$ than the one given in [DWX]. After we posted this paper on Arxiv, we are informed that there is a similar result obtained independently by Liu, Liu, and Niu.

Acknowledgments

The author thanks Dennis A. Eichhorn, Karl Rubin, Daqing Wan and Liang Xiao for many valuable discussions and suggestions.

2. Notation

In this section, we introduce some notations that we will use through out the paper.

Notation 2.1**.**

We write $v_{T}(-)$ for the $T$-adic valuation of elements in $\mathbb{C}_{p}[T]$ and $v_{p}(-)$ for the $p$-adic valuation of elements in $\mathbb{C}_{p}$.

Definition 2.2**.**

Given a set S:=\big{\{}(k,d_{k})\;\big{|}\;0\leq k\leq n\big{\}}. The Newton polygon of $S$, denoted by $\operatorname{NP}(S)$, is the lower convex hull of points in $S$. We call $n$ the length of $\operatorname{NP}(S)$.

For a power series $F(s)=\sum\limits_{i=0}^{n}\mathbbm{u}_{i}(T)s^{i}$, we put

[TABLE]

where $n\in\mathbb{Z}_{\geq 0}\cup\infty$.

Definition 2.3**.**

For a Newton polygon $\operatorname{NP}$, we write $R(\operatorname{NP})$ for the multiset of slopes in $\operatorname{NP}$.

It has an inverse, denoted by $R^{-1}$, mapping a multiset $\SS$ to the lower convex whose slopes coincide with this multiset.

Notation 2.4**.**

• (a)

Let $\SS_{1}$ and $\SS_{2}$ be two multisets in $\mathbb{Q}$. We denote by

[TABLE]

the union of $\SS_{1}$ and $\SS_{2}$ as multisets.

• (b)

For any two Newton polygons $\operatorname{NP}_{1}$ and $\operatorname{NP}_{2}$, we write

[TABLE]

for the Newton polygon whose slopes are the union of the slopes of $\operatorname{NP}_{1}$ and $\operatorname{NP}_{2}$.

• (c)

We denote by $\operatorname{NP}(i)$ the height of $\operatorname{NP}$ at $x=i$.

• (d)

For any $t\in\mathbb{Q}$, we denote $t+\operatorname{NP}$ be the Newton polygon such that

[TABLE]

where $n$ is the length of $\operatorname{NP}$.

Definition 2.5**.**

Let $\operatorname{NP}_{1}$ and $\operatorname{NP}_{2}$ be two polygons of same length $n$. If

[TABLE]

holds for any $0\leq k\leq n$, then we call that $\operatorname{NP}_{1}$ is above $\operatorname{NP}_{2}$ and denote this by $\operatorname{NP}_{1}\geq\operatorname{NP}_{2}$.

Lemma 2.6**.**

If \Big{\{}\operatorname{NP}_{1,i}\;\big{|}\;1\leq i\leq m\Big{\}} and \Big{\{}\operatorname{NP}_{2,i}\;\big{|}\;1\leq i\leq m\Big{\}} are two sets of Newton polygons such that for any $1\leq i\leq m$,

• •

$\operatorname{NP}_{1,i}$* and $\operatorname{NP}_{2,i}$ have the same length, and*

• •

$\operatorname{NP}_{1,i}\geq\operatorname{NP}_{2,i}$,

then

[TABLE]

Proof.

It follows directly from the definition of “$\oplus$”. ∎

Definition 2.7**.**

For any positive integer $\ell$, the sum

[TABLE]

is called a twisted $T$-adic exponential sum of $f(x)$.

The $L$-function $L_{f}(\omega^{u},T,s)$ (see Definition 1.2) satisfies

[TABLE]

It is easy to check

[TABLE]

Lemma 2.8**.**

If we put $g(x):=f(x^{q-1})$, then

[TABLE]

Proof.

Put

[TABLE]

where $b_{\ell,n}(T)\in\mathbb{Z}_{p}[\![T]\!]$ for all $n\geq 0$ and $\ell\geq 1$.

Notice that for each $u\in\{0,1,\dots,q-2\}$, we have

[TABLE]

Therefore, by taking the sum of $u$ over the set $\{0,1,\dots,q-2\}$, we get

[TABLE]

On the other hand, by definition, it is easy to check that

[TABLE]

Therefore, we have

[TABLE]

for all $\ell\geq 1$, which implies

[TABLE]

Definition 2.9**.**

The characteristic power series of $f$ is given by

[TABLE]

which is shown as a $p$-adic entire power series in [Liu].

By Lemma 2.8, we know that

[TABLE]

Notation 2.10**.**

We denote by $\operatorname{NP}_{f}(L,\omega^{u},T)$ (resp. $\operatorname{NP}_{f}(L,\omega^{u},\chi)$) the $T$-adic Newton polygon (resp. $p$-adic Newton polygon) of $L_{f}(\omega^{u},T,s)$ (resp. $L_{f}(\omega^{u},\chi,s)$).

Similarly, we write $\operatorname{NP}_{f}(C,\omega^{u},T)$ and $\operatorname{NP}_{f}(C,\omega^{u},\chi)$ for the $T$-adic Newton polygon (resp. $p$-adic Newton polygon) of $C_{f}(\omega^{u},T,s)$ and $C_{f}(\omega^{u},\chi,s)$ respectively.

3. The T-adic Dwork’s Trace Formula

In this section, we recall properties of the $L$-function associated to a $T$-adic exponential sum as considered by Liu and Wan in [LW]. Its specializations to appropriate values of $T$ interpolate the $L$-functions considered above.

Notation 3.1**.**

We first recall that the Artin–Hasse exponential series is defined by

[TABLE]

Setting $T=E(\pi)-1$ defines an isomorphism $\mathbb{Z}_{p}\llbracket\pi\rrbracket\cong\mathbb{Z}_{p}\llbracket T\rrbracket$.

Notation 3.2**.**

For our given polynomial $f(x)=\sum\limits_{i=0}^{d}a_{i}x^{i}\in\mathbb{Z}_{q}[x]$, we put

[TABLE]

We follow the notation of [LL]. Set

[TABLE]

and

[TABLE]

Notation 3.3**.**

• (a)

For two integers $n$ and $m$, we denote by $n\%m$ the residue class of $n$ modulo $m$ in $\{0,1,\dots,m-1\}$.

• (b)

Recall $u\in\{0,1,\dots,q-2\}$. We write $b_{u}|a$ for the minimal positive integer such that $u^{p^{b_{u}}}\equiv u\pmod{q}$.

• (c)

Denote

[TABLE]

and put

[TABLE]

to be the total Banach space associated to $u$.

• (d)

Choose a permutation $(i_{1},i_{2},\dots,i_{b_{u}})$ of $\{1,2,\dots,b_{u}\}$ such that the sequence $\{u_{i_{n}}\}$ is non-decreasing. Put

[TABLE]

to be a non-decreasing sequence.

It is easy to check that

[TABLE]

Let $\psi_{p}$ denote the operator on $\widetilde{\textbf{B}}_{u}$ given by

[TABLE]

and let $\psi$ be the composite linear operator

[TABLE]

where $\sigma$ is the Frobenius automorphism of $\mathbb{Z}_{q}$, and $E_{f}(x)$ acts on $g\in\widetilde{\textbf{B}}_{u}$ by

[TABLE]

By Dwork’s trace formula, we have

Lemma 3.4**.**

The characteristic power series $C_{f}(\omega^{u},T,s)$ satisfies

[TABLE]

Proof.

See [LL, Theorem 2.1]. ∎

By [LL, Lemma 4.2], we have

[TABLE]

We write

[TABLE]

for a basis of $\widetilde{\textbf{B}}_{u}$ over $\mathbb{Z}_{q}[\![T^{\frac{1}{d(q-1)}}]\!]$ and denote by $N$ the standard matrix of $\psi$ associated to the basis $B$.

It is not hard to check that $N$ is an infinite dimensional matrix of the form

[TABLE]

where all $*$ in this matrix are from $\mathbb{Z}_{q}[\![T^{\frac{1}{d(q-1)}}]\!]$.

Running an analogous argument to [RWXY, Corollary 3.9], we obtain

[TABLE]

Notation 3.5**.**

For a matrix $M$, we write

[TABLE]

for the $k\times k$-submatrix formed by elements whose row indices belong to $\{m_{1},\dots,m_{k}\}$ and whose column indices belong to $\{n_{1},\dots,n_{k}\}$.

Lemma 3.6**.**

Let $(t_{ij})_{j\in\mathbb{Z}_{\geq 0}}$ be $n$ non-decreasing sequences, and let $M_{1},M_{2},\dots,M_{n}$ be $n$ nuclear matrices such that

[TABLE]

where $M_{i}^{\prime}$ are infinite matrix whose entries belong to $\mathbb{Z}_{q}[\![T^{\frac{1}{d(q-1)}}]\!]$. Then the $T$-adic Newton polygon

[TABLE]

Proof.

Put

[TABLE]

From the definition of characteristic power series, we get

[TABLE]

Here and after, we set $m_{n+1,i}=m_{1,i}$ for all $1\leq i\leq k$. Since

[TABLE]

we complete the proof. ∎

Definition 3.7**.**

The Hodge polygon of $C_{f}(\omega^{u},T,s)$, denoted by $\mathrm{HP}(d,\omega^{u},T)$, is the lower convex hull of set

[TABLE]

Lemma 3.8**.**

Each point in \Big{\{}(k,\frac{a(p-1)}{db_{u}(q-1)}\sum\limits_{j=0}^{kb_{u}-1}c_{u,j})\Big{\}} is a vertex of $\mathrm{HP}(d,\omega^{u},T).$

Proof.

It follows that sequence \Big{(}\frac{a(p-1)}{db_{u}(q-1)}\sum\limits_{j=(k-1)b_{u}}^{kb_{u}-1}c_{u,j}\Big{)}_{k\in\mathbb{Z}_{\geq 0}} is strictly increasing in $k$. ∎

Recall

[TABLE]

Lemma 3.9**.**

We have

[TABLE]

Proof.

From (3.3.1), we know

[TABLE]

Since

[TABLE]

we know

[TABLE]

Corollary 3.10**.**

The Hodge polygon $\mathrm{HP}(d,\omega^{u},T)$ passes through the points

[TABLE]

Proposition 3.11**.**

The polygons $\operatorname{NP}_{f}(C,\omega^{u},T)$ and $\mathrm{HP}(d,\omega^{u},T)$ satisfy

[TABLE]

Proof.

Since the matrix $N$ is nuclear as in (3.4.3), its conjugates $\sigma^{i}(N)$ are also nuclear matrices of the same form. Therefore, applying Lemma 3.6 to the product of these matrices yields

[TABLE]

From (3.4.2) and (3.4.4), we have

[TABLE]

Corollary 3.12**.**

For any character $\chi:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ with conductor $p^{m_{\chi}}$, we have

[TABLE]

Proof.

It simply follows

[TABLE]

4. Proof of Theorem 1.5 and Theorem 1.6

In this section, we prove the main theorems.

Proposition 4.1**.**

• (a)

The Newton polygon $\operatorname{NP}_{f}(C,\omega^{u},T)$ passes through the points

[TABLE]

• (b)

If we write

[TABLE]

then for any $k\geq 0$ and $0\leq u\leq q-2$, the leading term of $r_{u,kd}$ is of the form

[TABLE]

where $*$ represents a $p$-adic unit.

Notation 4.2**.**

We denote by $\mathrm{UP}(d,\omega^{u},T)$ the lower convex hull of the points in

[TABLE]

Corollary 4.3**.**

The polygon $\mathrm{UP}(d,\omega^{u},T)$ forms an upper bound of $\operatorname{NP}_{f}(C,\omega^{u},T)$.

Proof.

This follows directly from Proposition 4.1 (a). ∎

Corollary 4.4**.**

Any finite character $\chi:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ with conductor $p^{m_{\chi}}$ satisfies

[TABLE]

Proof.

It follows from Theorem 4.1 (b). ∎

We will give the proof of Proposition 4.1 later.

Lemma 4.5**.**

Let $\operatorname{NP}_{1},\operatorname{NP}_{2},\dots,\operatorname{NP}_{n}$ be $n$ Newton polygons. Assume for each $1\leq i\leq n$ there is a rational number $c$ and a vertex $(k_{i},y_{i})$ of $\operatorname{NP}_{i}$ such that all segments of $\operatorname{NP}_{i}$ before this point have slopes strictly less than $c$, while all segments after that point have slopes greater than $c$. Then $\bigoplus\limits_{i=1}^{n}\operatorname{NP}_{i}$ passes though the point

[TABLE]

Proof.

The proof follows from the definition of direct sum “$\oplus$” of polygons. ∎

Lemma 4.6**.**

Any finite character $\chi$ with conductor $p^{m_{\chi}}$ satisfies

[TABLE]

Proof.

It is enough to show each monomial $aT^{i}\in\mathbb{Z}_{q}[T]$ satisfy

[TABLE]

which follows

[TABLE]

Proof of Proposition 4.1.

Proof of (a). Fix a finite character $\chi_{1}:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ with conductor $p$. By Lemma 4.6, we have

[TABLE]

By [DWX, Proposition 3.2], the $p$-adic Newton polygon $\operatorname{NP}_{g}(C,\omega^{0},\chi_{1})$ passes through the points

[TABLE]

Hence, we know that $\operatorname{NP}_{g}(C,\omega^{0},T)$ is not above point

[TABLE]

On the other hand, by Definition 3.7 and Lemma 3.9, we have

• (1)

For any $0\leq u\leq q-2$ and $k\geq 0$, the point

[TABLE]

is a vertex of $\mathrm{HP}(d,\omega^{u},T)$.

• (2)

All segments of $\mathrm{HP}(d,\omega^{u},T)$ before this point have slopes strictly less than $ak(p-1)$, while all segments after this point have slopes greater than $ak(p-1)$.

By checking the conditions in Lemma 4.5, we prove $\bigoplus\limits_{u=0}^{q-2}\mathrm{HP}(d,\omega^{u},T)$ passes through the points

[TABLE]

Combining it with Proposition 3.11 yields that $\operatorname{NP}_{g}(C,\omega^{0},T)$ is not above the points

[TABLE]

Thus,

[TABLE]

Now we show that (4.6.3) is actually an equality.

Consider

[TABLE]

Then we simplify the left-hand side of (4.6.3) by

[TABLE]

which is equal to its right-hand side. It implies for any $u\in\{0,1,\dots,q-2\}$, the Newton polygon $\operatorname{NP}_{f}(C,\omega^{u},T)$ passes through the points

[TABLE]

Proof of (b). From (a), we are able to write

[TABLE]

where $r_{u,kd,i}$ belongs to $\mathcal{O}_{\mathbb{C}_{p}}$.

Put $C_{g}(\omega^{0},T,s)=\sum\limits_{n=0}^{\infty}w_{n}(T)s^{n}.$ From [DWX], we know that the leading term of $w_{kd}(T)$ has the form

[TABLE]

where $*_{k}$ is a $p$-adic unit. It is easy to show that

[TABLE]

which implies that $r_{u,kd,i}$ are all $p$-adic units. ∎

Now we are ready to prove our main theorems of this paper.

Proof of Theorem 1.5.

(a) From (2.7.1), we obtain

[TABLE]

Therefore, by Proposition 4.1 (b), the Newton polygon $(p-1)p^{m_{\chi}-1}\operatorname{NP}_{f}(C,\omega^{u},\chi)$ is not above point

[TABLE]

On the other hand, the Hodge polygon $\mathrm{HP}(d,\omega^{u},T)$ forms a lower bound of $(p-1)p^{m_{\chi}-1}\operatorname{NP}_{f}(C,\omega^{u},\chi)$ and for all $k\geq 0$ the points

[TABLE]

are also vertices of $\mathrm{HP}(d,\omega^{u},T)$.

Therefore, the points

[TABLE]

are forced to be the vertices of $(p-1)p^{m_{\chi}-1}\operatorname{NP}_{f}(C,\omega^{u},\chi)$.

A simple argument about the relation between roots of a power series and its $p$-adic Newton polygon completes the proof.

(b) Since the slopes of segments of $\mathrm{HP}(d,\omega^{u},T)$ between $x=d(k-1)$ and $x=dk$ are in the interval

[TABLE]

by simply applying (a), we know that the slopes of segments of $(p-1)p^{m_{\chi}-1}\operatorname{NP}_{f}(C,\omega^{u},\chi)$ between $x=d(k-1)$ and $x=dk$ also in this interval, which completes the proof of (b). ∎

Recall $\mathrm{UP}(d,\omega^{u},T)$ is the upper bound of $\operatorname{NP}_{f}(C,\omega^{u},T)$ defined in Notation 4.2.

Lemma 4.7**.**

The vertical distance between points in $\mathrm{UP}(d,\omega^{u},T)$ and $\operatorname{NP}_{f}(C,\omega^{u},T)$ is bounded above by $\frac{ad(p-1)}{8}$.

Proof.

By Corollary 4.3 and Proposition 3.11, we know

[TABLE]

By Corollary 3.10, the polygon $\mathrm{HP}(d,\omega^{u},T)$ is above the parabola $P$ defined by

[TABLE]

Since all vertices $(kd,y_{u}(kd))$ of $\mathrm{UP}(d,\omega^{u},T)$ coincide with the parabola $P$, by simple calculation, the maximal vertical distance of $\mathrm{UP}(d,\omega^{u},T)$ and $P$ is equal to

[TABLE]

Proposition 4.8**.**

Let $\chi$ be a finite character with conductor $p^{m_{\chi}}>\frac{adp}{8}$. Then the Newton polygon $p^{m_{\chi}}\operatorname{NP}_{f}(C,\omega^{u},\chi)$ is independent of $\chi$.

Proof.

Recall in (4.1.1) we denote

[TABLE]

By Proposition 3.11 and Corollary 3.10, we are able to write $r_{u,k}(T)$ of the form

[TABLE]

Assume that $i(k)$ is the smallest integer such that

• •

$i(k)\leq\mathrm{UP}(d,\omega^{u},T)(k)$, where $\mathrm{UP}(d,\omega^{u},T)(k)$ is the height of $\mathrm{UP}(d,\omega^{u},T)$ at $x=k$.

• •

The corresponding coefficient $r_{u,k,i(k)}$ is a $p$-adic unit.

If such $i(k)$ does not exist, we simply put $i(k)=\infty$.

Then we will show that for any $\chi$ satisfying

[TABLE]

the Newton polygon $p^{m_{\chi}-1}(p-1)\operatorname{NP}_{f}(C,\omega^{u},\chi)$ is the same as \operatorname{NP}\Big{(}\Big{\{}(k,i(k))\;\big{|}\;k\geq 0\Big{\}}\Big{)}.

Since $C_{f}(\omega^{u},T,s)\in\mathbb{Z}_{p}[\![T]\!],$ for any $\ell<i(k)$ we have

[TABLE]

By Lemma 4.7 and the definition of $i(k)$, we know that

[TABLE]

It follows from the inequalities (4.8.3), (4.8.1) and (4.8.2) that

[TABLE]

The inequality above implies that v_{p}\big{(}u_{u,k}\cdot(\chi(1)-1)\big{)} is

• •

either equal to $\frac{i(k)}{{p^{m_{\chi}-1}(p-1)}}$

• •

or greater than $\frac{1}{(p-1)p^{m_{\chi}-1}}\mathrm{UP}(d,\omega^{u},T)(k)$.

Then this proposition follows directly from Corollary 4.4. ∎

For a Newton polygon $\operatorname{NP}$ and a rational number $t$ recall the definition of Newton polygon $t+\operatorname{NP}$ in Notation 2.4 (d).

Lemma 4.9**.**

Let $\chi:\mathbb{Z}_{p}\to\mathbb{C}_{p}^{\times}$ be a finite character with conductor $p^{m_{\chi}}$. Then we have

[TABLE]

where $R$ is defined in Definition 2.3.

Proof.

Since

[TABLE]

we know that

[TABLE]

Proof of Theorem 1.6.

By Proposition 4.8, we have

[TABLE]