$p$-adic variation of $L$ functions of exponential sums, I
Hui June Zhu
For a polynomial $f(x)$ in $(\Z_p\cap\Q)[x]$ of degree $d\geq 3$ let $L(f\otimes\F_p;T)$ be the $L$ function of the exponential sum of $f\bmod p$. Let $\NP(f\otimes\F_p)$ denote the Newton polygon of $L(f\otimes\F_p;T)$. Let $\HP(f)$ denote the Hodge polygon of $f$, which is the lower convex hull in $\R^2$ of the points $(n,\frac{n(n+1)}{2d})$ for $0\leq n\leq d-1$. We prove that there is a Zariski dense subset $\cU$ defined over $\Q$ in the space $\A^d$ of degree-$d$ monic polynomials over $\Q$ such that $\lim_{p\rightarrow\infty}\NP(f\otimes\F_p)=\HP(f)$ for all $f\in\cU(\Q)$. Moreover, we determine the $p$-adic valuation of every coefficient of $L(f\otimes\F_p;T)$ for $p$ large enough and $f$ in $\cU(\Q)$.
math0111194.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% This file is the AMS-LaTeX file for the paper        %%
%%							%%
%%   p-adic variation of L functions                    %%
%%       of exponential sums I   			%%
%%							%%
%%           by                                         %%
%%							%%
%%      Hui June Zhu                                    %%
%%                                                      %%
%%	Contact Information:				%%
%%	Hui June Zhu 					%%
%%      Department of mathematics			%%
%%	University of California			%%
%%	Berkeley, CA 94720-3840				%%
%%      Email: zhu@alum.calberkeley.org         	%%
%%	or     zhu@math.berkeley.edu                    %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[twoside]{amsart}
\usepackage{amsmath,amsthm,amssymb}

%% theorem definition
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{acknowledgments}{Acknowledgments}
\renewcommand{\theacknowledgments}{{\hspace{-.3em}}}

%\let\oldlabel=\label
%\renewcommand\label[1]{\oldlabel{#1}\marginpar{#1}}

%% preamble: define newcommand
%%bold characters
\renewcommand{\r}{{\bf r}}
\newcommand{\A}{{\mathbb A}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\llceil}{\left\lceil}
\newcommand{\rrceil}{\right\rceil}
\newcommand{\llfloor}{\left\lfloor}
\newcommand{\rrfloor}{\right\rfloor}
\def\bar{\overline}
\def\today{Nov 18, 2001}
%%mathcal
\newcommand{\cC}{ {\mathcal C} }
\newcommand{\cL}{ {\mathcal L} }
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cS}{ {\mathcal S} }
\newcommand{\cU}{ {\mathcal U} }
\newcommand{\cV}{ {\mathcal V} }
\newcommand{\cW}{ {\mathcal W} }

%%roman characters
\newcommand{\ord}{{\rm ord}}
\newcommand{\sgn}{{\rm sgn} }
\newcommand{\NP }{ {\rm  NP} }     %Newton polygon
\newcommand{\HP }{ {\rm  HP} }     %Hodge polygon
\newcommand{\Tr}{ {\rm Tr} }
\newcommand{\Zeta}{{\rm Zeta}}
\newcommand{\N}{{\rm N}}
%%pagestyle
\pagestyle{headings}

%%begin document
\begin{document}

%Topmatter
\title{
$p$-adic variation of $L$ functions of exponential sums, I
}
\author{Hui June Zhu}
\address{
Hui June Zhu,
Department of mathematics,
University of California,
Berkeley, CA 94720-3840.
The United States.
}
\email{zhu@alum.calberkeley.org}

\date{\today}
\keywords{
Newton polygons; Hodge polygons;
$L$ functions; Zeta functions;
exponential sums; Artin-Schreier curves;
$p$-adic variation; A conjecture of Wan.
}
\subjclass{11,14}

\begin{abstract}
For a polynomial $f(x)$ in $(\Z_p\cap\Q)[x]$ of degree $d\geq 3$ let
$L(f\otimes\F_p;T)$ be the $L$ function of the exponential sum of
$f\bmod p$.  Let $\NP(f\otimes\F_p)$ denote the Newton polygon of
$L(f\otimes\F_p;T)$.  Let $\HP(f)$ denote the Hodge polygon of $f$,
which is the lower convex hull in $\R^2$ of the points
$(n,\frac{n(n+1)}{2d})$ for $0\leq n\leq d-1$. We prove that there is
a Zariski dense subset $\cU$ defined over $\Q$ in the space $\A^d$ of
degree-$d$ monic polynomials over $\Q$ such that
$\lim_{p\rightarrow\infty}\NP(f\otimes\F_p)=\HP(f)$ for all
$f\in\cU(\Q)$.  Moreover, we determine the $p$-adic valuation of every
coefficient of $L(f\otimes\F_p;T)$
for $p$ large enough and $f$ in
$\cU(\Q)$.
\end{abstract}

\maketitle

\section{Introduction}\label{S:1}

In this paper $d$ is an integer $\geq 3$.  Let $\A^d$ be the
affine space of dimension $d$, identified with the space of
degree-$d$ monic polynomials parametered by their coefficients.
We always assume that $p$ is any prime
coprime to $d$.
Let  $f(x)$ a polynomial of one variable
in $\A^d(\Z_p\cap\Q)$.
Let
$E(x)=\exp(\sum_{j=0}^{\infty}\frac{x^{p^j}}{p^j})$ be the Artin-Hasse
exponential. Let $\gamma$ be a root of $\log(E(x))$ in $\bar\Q_p$ with
$\ord_p \gamma=\frac{1}{p-1}$. Then $E(\gamma)$ is a primitive $p$-th
root of unity. Denote it by $\zeta_p$. It is observed that
$\Z_p[\gamma]=\Z_p[\zeta_p]$.
For every $\ell\geq 1$, recall the exponential sums of
the reduction $f\otimes\F_p$ of $f$ modulo $p$
$$S_\ell(f\otimes\F_p):=
\sum_{x\in\F_{p^\ell}}\zeta_p^{\Tr_{\F_{p^\ell}/\F_p}(f(x)\otimes\F_p )}.$$
Then $L$-function of the exponential sum
of $f\otimes\F_p$ is defined by
\begin{eqnarray}\label{E:Lfunction}
L(f\otimes\F_p;T):
=\exp(\sum_{\ell=1}^{\infty}S_\ell(f\otimes\F_p)\frac{T^\ell}{\ell}).
\end{eqnarray}
It is a theorem of Dwork-Bombieri-Grothendieck that
\begin{eqnarray}\label{E:20}
L(f\otimes\F_p;T)&=&1+b_1T+b_2T^2+\ldots+b_{d-1}T^{d-1}\in\Z[\zeta_p][T].
\end{eqnarray}

Let $\ord_p (\cdot)$ denote the unique extension of the (additive)
$p$-adic valuation in $\Q_p$ to $\bar\Q_p$.  In this paper we also
denote by $\ord_p(\cdot)$ the $p$-adic valuation of the {\em content of a
power series over $\bar\Z_p$} (see \cite{Lang} for its standard definition).
Define the {\em Newton polygon} of the
$L$-function of $f\otimes\F_p$ as the lower convex hull of the points
$(n,\ord_p b_n)$ in $\R^2$ for $0\leq n\leq d-1$, where we set
$b_0=1$. We denote it by $\NP(f\otimes\F_p)$.  The {\em Hodge polygon}
of $f$ is the lower convex hull in $\R^2$ of the points
$(n,\frac{n(n+1)}{2d})$ for $0\leq n\leq d-1$.  It is known that
$\HP(f)$ is a lower bound of $\NP(f\otimes\F_p)$ and that if $p\equiv
1\bmod d$ then $\NP(f\otimes\F_p)=\HP(f)$ for every $f\in\A^d(\Q)$
(see
\cite{Adolphson-Sperber} and \cite{Bombieri}).

The main result of this paper is Theorem \ref{T:1}. It was a
conjecture of Daqing Wan, proposed in the following form in the number
theory seminar at Berkeley in the fall of 2000 (see also~\cite[Section
2.5]{Wan} for developments related to this topic).

\begin{theorem}\label{T:1}
There is a Zariski dense open subset $\cU$ defined over $\Q$
in $\A^d$ such that
for all $f(x)\in\cU(\Q)$ we have
$$\lim_{p\rightarrow\infty}\NP(f\otimes\F_p)=\HP(f).$$
\end{theorem}

The cases $d=3$ and $4$ are proved by \cite{Sperber} and \cite{Hong}
respectively.  The first
slope case is proved recently by an elementary method in \cite{SZ:3}
(see also \cite{SZ:2}). Further results concerning more
conjectures of Daqing Wan \cite{Wan:2} will be in \cite{Zhu:2}.

Theorem \ref{T:1} yields a one-variable answer towards questions
proposed in \cite{Katz:2}, there Katz asked how the Newton polygon of
the $L$ function varies with the prime $p$. See Katz's questions and
Sperber's example on page 151 of \cite[Chapter 5.1]{Katz:2}.  For
more developments in these directions see \cite{Sperber:3},
\cite{Sperber:4} and \cite{Adolphson-Sperber:2} and their
bibliographies.

It is well-known that the $L(f(x)\otimes\F_p;T)$ is intimately
connected to the study of the $p$-cover of the projective line.
Let
$X_f: y^p-y=f(x)\otimes\F_p$ be an Artin-Schreier curve over $\F_p$.
Recall that
$\NP(X_f\otimes\F_p)$ is the $p$-adic Newton polygon of the numerator
of the Zeta function $\Zeta(X_f\otimes\F_p;T)$ of $X_f$ over $\F_p$.
We have
\begin{eqnarray}\label{E:Zeta}
\Zeta(X_f\otimes\F_p;T)
=\frac{\N_{\Q(\zeta_p)/\Q}(L(f\otimes\F_p;T))}{(1-T)(1-pT)}
\end{eqnarray}
where the norm $\N_{\Q(\zeta_p)/\Q}$
being interpreted as the product of the conjugates
of $L(f\otimes\F_p;T)$ over $\Q(\zeta_p)$, the automorphism
acting trivially on the variable $T$.
Since we do not find suitable reference for this well-known
fact, we sketch its proof below.
For convenience let $X_f^\dagger$ be the curve $X_f$ minus the
point at infinity. Here $\Tr(\cdot)$ abbreviates the trace map from
$\F_{p^\ell}$ to $\F_p$.
It is easy to verify that
$$
\# X_f^\dagger(\F_{p^\ell})
=\sum_{a\in\F_p}\sum_{x\in\F_{p^\ell}}\zeta_p^{a\Tr(f(x)\otimes\F_p)}.
$$
Then
\begin{eqnarray*}
\Zeta(X_f\otimes\F_p;T) &=&\exp(\sum_{\ell=1}^\infty
(1+\#X_f^\dagger(\F_{p^\ell}))\frac{T^\ell}{\ell})
=\frac{1}{(1-T)}\exp(\sum_{\ell=1}^\infty
\#X_f^\dagger(\F_{p^\ell})\frac{T^\ell}{\ell})\\
&=&\frac{1}{(1-T)}\exp(\sum_{\ell=1}^\infty
(\sum_{a\in\F_p}\sum_{x\in\F_{p^\ell}}
\zeta_p^{a\Tr(f(x)\otimes\F_p)})\frac{T^\ell}{\ell})\\
&=&\frac{1}{(1-T)}\exp(\sum_{\ell=1}^\infty
(p^\ell+\sum_{a\in\F_p^*}\sum_{x\in\F_{p^\ell}}
\zeta_p^{a\Tr(f(x)\otimes\F_p)})\frac{T^\ell}{\ell})\\
&=&\frac{1}{(1-T)(1-pT)}\exp(\sum_{\ell=1}^\infty
(\sum_{a\in\F_p^*}\sum_{x\in\F_{p^\ell}}
\zeta_p^{a\Tr(f(x)\otimes\F_p)})\frac{T^\ell}{\ell})\\
&=&\frac{1}{(1-T)(1-pT)}\prod_{a\in{\F_p^*}}
\exp(\sum_{\ell=1}^\infty (\sum_{x\in\F_{p^\ell}}
\zeta_p^{a\Tr(f(x)\otimes\F_p)})\frac{T^\ell}{\ell})\\
&=&\frac{1}{(1-T)(1-pT)}\N_{\Q(\zeta_p)/\Q}(
\exp(\sum_{\ell=1}^\infty S_\ell(f\otimes\F_p)\frac{T^\ell}{\ell})),
\end{eqnarray*}
and this proves our claim.
Recall that the Newton polygon $\NP(X_f\otimes\F_p)$ is the $p$-adic Newton
polygon of the denominator of $\Zeta(X_f\otimes\F_p;T)$.
By (\ref{E:Zeta}) it is easily seen that
$\NP(f\otimes\F_p)$ is precisely
equal to $\NP(X_f\otimes\F_p)$ shrunk
by a factor of $\frac{1}{p-1}$
horizontally and vertically, which is denoted by
$\frac{\NP(X_f\otimes\F_p)}{p-1}$. Thus the following corollary follows
from Theorem \ref{T:1}.

\begin{corollary}
There exists a Zariski dense open subset $\cU$ defined over $\Q$
in $\A^d$ such that for
every $f\in \cU(\Q)$ we have
$$\lim_{p\rightarrow\infty}\frac{\NP(X_f\otimes\F_p)}{p-1} = \HP(f).$$
\end{corollary}

\begin{remark}
For every $d\geq 3$ the limit
$\lim_{p\rightarrow\infty}\NP(x^d\otimes\F_p)$ does not exist.
See Proposition \ref{P:2} for details.
\end{remark}


This paper is organized as follows.  We use Dwork p-adic theory in
section \ref{S:2} to derive a formula of the desired $L$ function of
exponential sums of $f(x)$ over $\F_p$ in terms of some Fredholm
determinants denoted by $C_n(\cdot)$.  Section \ref{S:3} is a key step
in the proof, it defines some Zariski dense subset
defined over $\Q$ in $\A^{d-1}$, which only depends on the
residue class of $p$ and not the size of $p$.
It is recommended that the reader continues with section
\ref{S:4}, where we immediately
use Dwork's $p$-adic theory to compute $p$-adic
valuations of the Fredholm matrix of the Frobenius map and
Fredholm determinant for $\vec{a}\in\cW(\Q)$
and for $p$ large enough.  The proof of Theorem \ref{T:1} is located
in section \ref{S:Proof}.  Finally we conclude the paper with a few
remarks of open questions and examples in section 6.

\begin{acknowledgments}
It is my great pleasure to thank Alan Adolphson and Steven Sperber who
exposed me to Dwork theory during the Dwork trimester in Italy. I
thank Hanfeng Li for careful reading and comments to an earlier
version. Most of all I thank Bjorn Poonen for
generously sharing ideas and offering support to this work.
\end{acknowledgments}




\section{Dwork p-adic theory}\label{S:2}

The fundamental material in our exposition follows
\cite[Sections II and III]{Bombieri}
(see also \cite{Dwork} \cite{Dwork2}
and \cite{Adolphson-Sperber}).
Recall that $p$ is any prime number coprime to $d$.
Let $\bar f(x)=x^d+\sum_{i=1}^{d-1}\bar a_ix^i$ be a polynomial
over $\F_p$. (See section \ref{S:Proof} to find out why
we confine ourselves to a polynomial with no constant term.

Write $\vec{\hat{a}}=(\hat{a}_1,\ldots,\hat{a}_{d-1})$ where
$\hat{a}_i$ is the Teichm\"uller lifting of $\bar a_i$, that is,
$\hat{a}_i\equiv a_i\bmod p$ and $\hat{a}_i^p=\hat{a}_i$.
Let $f(x)=x^d+\sum_{i=1}^{d-1}a_ix^i\in (\Z_p\cap\Q)[x]$
and $a_d=1$ such that reduction $f(x)\otimes\F_p$ of $f(x)$
at $p$ is equal to $\bar f(x)$.
Let $\theta(x)=E(\gamma x)$, where $E(\cdot)$ and $\gamma$ as
defined in section 1.
Then we may write
$\theta(x)=\sum_{m=0}^{\infty}\lambda_mx^m$
for $\lambda_m\in\Z_p[\zeta_p]$.
Note the following properties,
\begin{eqnarray}\label{E:22}
\ord_p \lambda_m&\geq&\frac{m}{p-1};
\end{eqnarray}
for $0\leq m\leq p-1$ we have,
\begin{eqnarray}\label{E:33}
\lambda_m=\frac{\gamma^m}{m!}\mbox{ and  } \ord_p \lambda_m=\frac{m}{p-1}.
\end{eqnarray}
Let $\vec{A}=(A_1,\ldots,A_{d-1})$ be a vector of variables and
$\vec{m}=(m_1,\ldots,m_{d-1})$.
Write $\vec{A}^{\vec{m}}$ for the monomial
$A_1^{m_1}\cdots A_{d-1}^{m_{d-1}}$.
Let $G_n(\vec{A})=0$ for $n<0$.
For every integer $n\geq 0$ let
\begin{eqnarray}\label{E:3}
G_n(\vec{A}):=\sum_{\substack{m_\ell\geq 0\\\sum_{\ell=1}^{d}\ell m_\ell=n}}
\lambda_{m_1}\cdots \lambda_{m_d}\vec{A}^{\vec{m}}.
\end{eqnarray}
Clearly we observe
that $G_n(\vec{A})\in \Z_p[\zeta_p][\vec{A}]$,
that is, $G_n(\vec{A})$ is a polynomial in variable $\vec{A}$
and with coefficients in $\Z_p[\zeta_p]$.
For all integers
$m_1,\ldots,m_d\geq 0$ such that $\sum_{\ell=1}^{d}\ell m_\ell=n$,
we have $d(m_1+\cdots +m_d)\geq \sum_{\ell=1}^{d}\ell m_\ell = n$
and so
$\min(m_1+\cdots +m_d)=\llceil\frac{n}{d}\rrceil$.
Therefore by (\ref{E:3}) we have
\begin{eqnarray}\label{E:Bound}
\ord_p G_n(\vec{A})
\geq
\frac{\min(m_1+\cdots +m_{d})}{p-1}
\geq
\frac{\llceil\frac{n}{d}\rrceil}{p-1}
\geq \frac{n}{d(p-1)}.
\end{eqnarray}
Let $G(X):=\prod_{i=1}^{d}\theta(\hat a_i X^i)\in\Z_p[\zeta_p][[X]]$.
We have
\begin{eqnarray*}
G(X)=
(\sum_{m_1=0}^{\infty}\lambda_{m_1}\hat a_1^{m_1}X^{m_1})
\cdots
(\sum_{m_d=0}^{\infty}\lambda_{m_d}\hat a_d^{m_d}X^{dm_d})
=\sum_{n=0}^{\infty}G_n(\vec{\hat a})X^n.
\end{eqnarray*}
Let $C_0(\vec{A})=1$, and for every $n\geq 1$ let
\begin{eqnarray}\label{E:1}
C_n(\vec{A}):=\sum_{
\substack{1\leq u_1<u_2<\ldots<u_n\\\sigma\in S_n}}
\sgn(\sigma)\prod_{i=1}^{n}G_{pu_i-u_{\sigma(i)}}(\vec{A}),
\end{eqnarray}
where $\sgn(\sigma)$ is the signature of the permutation $\sigma$
in the $n$-th symmetric group $S_n$.
It can be verified that
this definition makes sense and that $C_n(\vec{A})\in
\Z_p[\zeta_p][[\vec{A}]]$.

\begin{lemma}\label{L:Dwork}
Let $p$ be a prime coprime to $d$.
For $f(x)=x^d+\sum_{i=1}^{d-1}a_ix^i\in(\Z_p\cap\Q)[x]$,
write $\vec{a}=(a_1,\ldots,a_{d-1})$. Let
$\vec{\hat{a}}=(\hat a_1,\ldots,\hat a_{d-1})$
be Teichm\"uller lifting of $\vec{\bar a}=(\bar a_1,\ldots,\bar a_{d-1})$.
Then
\begin{eqnarray}
\label{E:Dwork}
L(f\otimes\F_p;T)&=&1+b_1(\vec{a})T+\cdots +b_{d-1}(\vec{a})T^{d-1}
\\
&=&\frac{1+\sum_{n=1}^{\infty}(-1)^nC_n(\vec{\hat a})T^n}
{(1-pT)(1+\sum_{n=1}^{\infty} (-1)^nC_n(\vec{\hat a})p^nT^n)},
\nonumber
\end{eqnarray}
where $b_1(\vec{a}),\ldots,b_{d-1}(\vec{a})\in\Z[\zeta_p]$.
\end{lemma}
\begin{proof}
The first equality is a rephrase of (\ref{E:20}).
For every positive integer $\ell$ let
$$S^*_\ell(f\otimes\F_p):=
\sum_{x\in\F_{p^\ell}^*}\zeta_p^{\Tr_{\F_{p^\ell}/\F_p}(f(x)\otimes\F_p)}.$$
Let
$$
L^*(f\otimes\F_p;T)
:=\exp(\sum_{\ell=1}^{\infty}S^*_\ell(f\otimes\F_p)\frac{T^\ell}{\ell}).
$$
Note that $S^*_\ell(f\otimes\F_p)=S_\ell(f\otimes\F_p)-1$ so
\begin{eqnarray}\label{E:Lstar}
L^*(f\otimes\F_p;T)
&=&\exp(\sum_{\ell=1}^\infty (S_\ell(f\otimes\F_p)-1))\nonumber\\
&=&(1-T)\exp(\sum_{\ell=1}^\infty S_\ell(f\otimes\F_p)\frac{T^\ell}{\ell})
\nonumber\\
&=&(1-T)L(f\otimes\F_p;T).
\end{eqnarray}
For any $c>0$ and $b\in \R$ let
$\cL(c,b)$ be the set of power series defined by
$$\cL(c,b):=\{
\sum_{n=0}^{\infty}A_nX^n\mid A_n\in \Q_p(\zeta_p),
\ord_p  A_n\geq \frac{cn}{d}+b\}.$$
Let $\cL(c):=\bigcup_{b\in\R}\cL(c,b)$.
{}From (\ref{E:Bound}) we have
$G(X)=\sum_{n=0}^{\infty}G_n(\vec{\hat a})X^n$ lies in $\cL(1/(p-1))$.
For any $\sum B_n X^n$ in $\cL(c)$,
let $\psi$ be the Hecke operator from $\cL(c)$ to $\cL(cp)$
given by
$\psi(\sum B_n X^n) = \sum B_{pn}X^n$.
Let $\alpha_1:=\psi\cdot G(X)$ be the endomorphism
$\cL(p/(p-1))$ defined by the composition of the multiplication map by
$G(X)$ then $\psi$, namely,
$$
\alpha_1\left(\sum_{i=0}^{\infty}B_iX^i\right)=\sum_{i=0}^{\infty}
\left(\sum_{j=0}^{\infty}G_{pi-j}(\vec{\hat a})B_j \right)X^i.
$$
Choose the standard monomial basis $\{1,x,x^2,\ldots\}$
for the $p$-adic Banach space $\cL(p/(p-1))$.
Then the $\Q_p(\zeta_p)$-endomorphism
$\alpha_1$ of $\cL(p/(p-1))$ has a matrix
representation by $\{G_{pi-j}(\vec{\hat a})\}_{i,j\geq 0}$.
We identify $\alpha_1$ with this matrix from now on.
By the Dwork trace formula (see \cite[Section III]{Bombieri})
we have
\begin{eqnarray*}
L^*(f\otimes\F_p;T)&=&\frac{\det(1-\alpha_1\cdot T)}{\det(1-\alpha_1\cdot pT)}.
\end{eqnarray*}
For the first row (i.e., $i=0$) of $\alpha_1$, we have
$G_{pi-j}(\vec{\hat a})=0$ for all $j\geq 1$
and $G_{0}(\vec{\hat a})=1$.
By (\ref{E:1}) we have
$$\det(1-\alpha_1\cdot T)
=(1-T)\det(1-\{G_{pi-j}(\vec{\hat a})T\}_{i,j\geq 1})
=(1-T)\sum_{n=0}^{\infty}(-1)^nC_n(\vec{\hat a})T^n.$$
Therefore, by (\ref{E:Lstar}) we have
\begin{eqnarray*}(1-T)L(f\otimes\F_p;T)
=L^*(f\otimes\F_p;T)=\frac{(1-T)\sum_{n=0}^{\infty}(-1)^nC_n(\vec{\hat
a})T^n}{ (1-pT)\sum_{n=0}^{\infty}(-1)^nC_n(\vec{\hat a}) p^nT^n}.
\end{eqnarray*}
By simplification of the above formula, our assertion follows.
\end{proof}


\section{Zariski dense subsets}\label{S:3}

This section is self-contained. We will construct a suitable Zariski
dense open subset $\cW$ defined over $\Q$ in $\A^{d-1}$ with suitable
property we shall utilize in the proof of Theorem \ref{T:1} in section
5. The introduction of many curiously defined invariants here will be
self-explained in section 4.

Fix an integer $r$ with $1\leq r\leq d-1$ and
$\gcd(d,r)=1$.
Let $1\leq n\leq d-1$.  For any
$\vec{m}=(m_1,\ldots,m_{d-1})\in\Z_{\geq 0}^{d-1}$ let
$|\vec{m}|=\sum_{\ell=1}^{d-1}m_\ell$ and $\vec{m}!=m_1!\cdots
m_{d-1}!$.  Recall that $d$ is an integer $\geq 3$.  Let $1\leq
i,j\leq d-1$. Let
\begin{eqnarray*}
r_{ij}&:=&d\llceil\frac{ri-j}{d}\rrceil-(ri-j).
\end{eqnarray*}

\begin{lemma}\label{L:r_n}
Let $\r_n:=\{r_{ij}\}_{1\leq i,j\leq n}$.
Then $0\leq r_{ij}\leq d-1$ and
there are no two identical entries in any row (column).
In $\r_{d-1}$ for every $1\leq i\leq d-1$ there is
a unique $j$ with $r_{ij}=0$.
\end{lemma}
\begin{proof}
By definition, $r_{ij}$ is
the least non-negative residue of $-(ri-j)\bmod d$ so we have $0\leq
r_{ij}\leq d-1$.
We prove for rows.
The argument for columns is almost identical.
Suppose we have $r_{ij}=r_{ij'}$ then
$ri-j\equiv ri-j'\bmod d$ by definition. Then $j\equiv j'\bmod d$.
Since $1\leq j,j'\leq n\leq d-1$ we have $j=j'$. So there are no
identical entries in any row of $\r_n$.
Note that $r$ is coprime to $d$ so for every
$1\leq i\leq d-1$ there is a unique $1\leq j\leq d-1$
such that $ri\equiv j\bmod d$. This is equivalent to
$r_{ij}=0$ by definition. This proves the last assertion.
\end{proof}

By Lemma \ref{L:r_n}
for every $1\leq i\leq d-1$ we have a unique
$1\leq j\leq d-1$ such that $r_{ij}=0$. We denote this $j$ by $J_i$.
Let
$$
\delta_{ij}=\left\{
\begin{array}{ll}
0&\mbox{for $j\leq J_i$}\\
1&\mbox{for $j>J_i$}.
\end{array}
\right.
$$
We want to point out the following elementary property
and leave the proof to the reader as an exercise.

\begin{lemma}\label{L:added}
For $1\leq i,j\leq d-1$ we have
\begin{eqnarray}
\label{E:added1}
\llceil\frac{pi-j}{d}\rrceil=\frac{pi+r_{ij}-j}{d}\equiv \frac{r_{ij}-j}{d}
\bmod p\\
\label{E:added2}
\llceil\frac{pi-j}{d}\rrceil=\llceil\frac{pi-1}{d}\rrceil-\delta_{ij}.
\end{eqnarray}
\end{lemma}



For any $0\leq s\leq n$ define a subset in $\Z_{\geq 0}^{d-1}$ by
\begin{eqnarray*}
\cM^s_{ij}&:=&\{\vec{m}=(m_1,m_2,\ldots,m_{d-1})\in\Z_{\geq 0}^{d-1}\mid
\sum_{\ell=1}^{d-1}\ell m_{d-\ell}=r_{ij}+ds\}.
\end{eqnarray*}
Note the important yet easy
fact that $\cM^s_{ij}$ is non-empty for any $0\leq
s\leq n$.
Recall $\vec{A}^{\vec{m}}=A_1^{m_1}\cdots A_{d-1}^{m_{d-1}}$.
For $1\leq i,j\leq n$
define an auxiliary polynomial
\begin{eqnarray}
\label{E:H}
&&H^s_{ij}(\vec{A})\\\nonumber
&&:=
\sum_{\vec{m}\in\cM^s_{ij}}
\frac{(\frac{r_{i1}-1}{d}+n)(\frac{r_{i1}-1}{d}+n-1)\cdots
(\frac{r_{i1}-1}{d}-\delta_{ij}+s+1-|\vec{m}|)}{\vec{m}!}
\vec{A}^{\vec{m}}.
\end{eqnarray}
This non-zero polynomial in $\Q[\vec{A}]$
plays a central role in the technique of this paper.

\begin{lemma}\label{L:H}
Let $0\leq s\leq n$ and $1\leq i,j\leq n$.

1). The polynomial $H^s_{ij}(\vec{A})$ in $\Q[\vec{A}]$
is supported on every $\vec{m}\in
\cM^{s}_{ij}$. The degree of its monomials ranges from
$s+\llceil\frac{r_{ij}+s}{d-1}\rrceil$ up to
$r_{ij}+ds$, where the maximal degree is attained at exactly one
monomial $A_{d-1}^{r_{ij}+ds}$ while the minimal degree is attained
at one or more monomials.

2). The polynomial $H^s_{ij}(\vec{A})$ has a
constant term if and only if $s=r_{ij}=0$;
it has a linear term
if and only if $s=0$ and $r_{ij}\neq 0$, in which
case this linear monomial is exactly $A_{d-r_{ij}}$. \\
\end{lemma}
\begin{proof}
1). Since $\gcd(r,d)=1$ we have $-(ir-1)\not\equiv 1\bmod d$.  Hence
$\frac{r_{i1}-1}{d}\not\in\Z$ and $h^s_{\vec{m},i,j}\neq 0$.
Therefore, in the polynomial $H^s_{ij}(\vec{A})$,
This proves the first assertion.
By the first assertion, it remains to prove
\begin{eqnarray}
\label{E:min}
\min_{\vec{m}\in \cM^s_{ij}}|\vec{m}|&=&
s+\llceil\frac{r_{ij}+s}{d-1}\rrceil\\
\label{E:max}
\max_{\vec{m}\in \cM^s_{ij}}|\vec{m}|&=&r_{ij}+ds.
\end{eqnarray}
For $\vec{m}\in \cM^{s}_{ij}$ we have
$|\vec{m}|\leq \sum_{\ell=1}^{d-1}\ell m_{d-\ell}=r_{ij}+ds$
and the equality holds if and only if $m_1=\cdots=m_{d-2}=0$ and
$m_{d-1}=r_{ij}+ds$. This verifies (\ref{E:max}). On the other
hand, for $\vec{m}\in\cM^{s}_{ij}$ clearly $(d-1)|\vec{m}|\geq
\sum_{\ell=1}^{d-1}\ell m_{d-\ell}=r_{ij}+ds$.
So $$|\vec{m}|\geq
\llceil\frac{r_{ij}+ds}{d-1}\rrceil =s+\llceil\frac{r_{ij}+s}{d-1}
\rrceil.$$
It is easy to see there are
$m_1,\ldots, m_{d-1}\geq 0$ satisfying
\begin{eqnarray}
\label{E:combine}
\sum_{\ell=1}^{d-1} m_{\ell}=\llceil\frac{r_{ij}+ds}{d-1}\rrceil
\quad \mbox{and}\quad
\sum_{\ell=1}^{d-1}(d-\ell)m_\ell=r_{ij}+ds.
\end{eqnarray}
For example, let $\kappa$ be the least non-negative
residue of $-(r_{ij}+ds)\bmod (d-1)$ then
let $m_1=\llceil\frac{r_{ij}+ds}{d-1}\rrceil-\kappa$,
$m_2=\kappa$ and let the rest $m_\ell=0$.
This says that there are $\vec{m}\in \cM^{s}_{ij}$
with $|\vec{m}|=\llceil\frac{r_{ij}+ds}{d-1}\rrceil$.
This proves (\ref{E:min}).

2). {}From 1) suppose $H^s_{ij}(\vec{A})$ has a linear term then
$s+\llceil\frac{r_{ij}+s}{d-1}\rrceil =1$, which implies $s=0$ and
$r_{ij}\neq 0$. In this case the only solution to (\ref{E:combine})
is $m_{d-r_{ij}}=1$ and $m_\ell=0$ for all
$\ell\neq r_{ij}$.  So the linear monomial is $A_{d-r_{ij}}$.  In the
same vein we obtain the assertion about constant term.
\end{proof}


Let
\begin{eqnarray}
\label{E11}
c_n&:=&\frac{1}{d}
\left(\max_{\sigma\in S_n}\sum_{i=1}^{n}r_{i,\sigma(i)}-
\min_{\sigma\in S_n}\sum_{i=1}^{n}r_{i,\sigma(i)}\right);
\end{eqnarray}
For any $0\leq t\leq c_n$, let
\begin{eqnarray}
\label{E10}
S_n^t &:=&\{\sigma\in S_n |
\sum_{i=1}^{n}r_{i,\sigma(i)}
      =\min_{\sigma\in S_n}
       \sum_{i=1}^{n}r_{i,\sigma(i)} +dt\}.
\end{eqnarray}
It is easy to note that $S_n=\bigcup_{t=0}^{c_n}S_n^t$.
For any $0\leq t\leq c_n$, let
\begin{eqnarray}
\label{E12}
f_n^t(\vec{A})&:=&
\sum_{\substack{s_0+s_1+\cdots+s_n=t\\s_\ell\geq 0}}
\sum_{\sigma\in S_n^{s_0}}\sgn(\sigma)
\prod_{i=1}^{n}H^{s_i}_{i,\sigma(i)}(\vec{A}).
\end{eqnarray}

\begin{lemma}\label{L:5}
Let $1\leq n\leq d-1$. Then
the polynomial $f_n^t(\vec{A})\neq 0$ for some $0\leq t\leq c_n$.
\end{lemma}
\begin{proof}
First step, we show that in the formal expansion of
$\sum_{t=0}^{c_n}f_n^t$ every lowest-degree-monomial
is in the formal expansion of $\prod_{i=1}^{n}H^0_{i,\sigma(i)}$
for some $\sigma\in S_n$.
In other words the lowest-degree-monomials
are in the summand where $s_1=\cdots=s_n=0$ and $s_0=t$.
Suppose $s_1>0$ say, by Lemma \ref{L:H} we have
for every $\sigma\in S_n$ the degree of a lowest-degree-monomial
in $\prod_{i=1}^{n}H^0_{i,\sigma(i)}$ is
$\sum_{i=1}^{n}\llceil\frac{r_{i,\sigma(i)}}{d-1}\rrceil$
which is strictly less than
$\sum_{i=1}^{n}(s_i+\llceil
\frac{r_{i,\sigma(i)}+s_i}{d-1}\rrceil)$, the degree of
a lowest-degree-monomial in $\prod_{i=1}^{n}H^{s_i}_{i,\sigma(i)}$.

By the first step, the lowest-degree-monomials lie
in the following sub-sum of $\sum_{t=0}^{c_n}f_n^t$:
$$\sum_{t=0}^{c_n}\sum_{\sigma\in S_n^t}
\sgn(\sigma)\prod_{i=1}^{n}H^0_{i,\sigma(i)}(\vec{A})
=\sum_{\sigma\in S_n}
\sgn(\sigma)\prod_{i=1}^{n}H^0_{i,\sigma(i)}(\vec{A})$$
since $S_n=\bigcup_{t=0}^{c_n}S_n^t$.

Now we proceed to our second step.
For every $i$ let $H^\dagger_{i,\sigma(i)}$ be
the lowest-degree-monomial of
$H^0_{i,\sigma(i)}$. (For example,
if $H^0_{i,\sigma(i)}=-3A_1^5A_2+\frac{2}{3}A_3$ then
$H^\dagger_{i,\sigma(i)}=A_3$ and
if $H^0_{i,\sigma(i)}=-3A_1^5A_4+\frac{2}{3}$ then
$H^\dagger_{i,\sigma(i)}=1$.)
Let
\begin{eqnarray}
D_n(\vec{A})&:=&\sum_{\sigma\in S_n}
        \sgn(\sigma)\prod_{i=1}^{n}H^\dagger_{i,\sigma(i)}(\vec{A})
     \in\Q[\vec{A}].
\end{eqnarray}
Recall from Lemma \ref{L:H} that $H^\dagger_{i,\sigma(i)}= 1$ or
$A_{d-r_{i,\sigma(i)}}$ depending on $r_{i,\sigma(i)}=0$ or not,
respectively. By adding in an auxiliary variable $A_d$ we let
$\tilde D_n$ be the determinant
of a matrix $\{A_{d-r_{ij}}\}_{1\leq i,j\leq n}$ and
\begin{eqnarray}\nonumber
\tilde{D}_n(A_1,\ldots,A_d)
   &=&\sum_{\sigma\in S_n}\sgn(\sigma)
                    \prod_{i=1}^{n}A_{d-r_{i,\sigma(i)}}\\
\label{E:D_n}
&=&\sum_{\sigma\in S_n}\sgn(\sigma)
A_d^{\#\{1\leq i\leq n|r_{i,\sigma(i)}=0\}}
\prod_{m=1}^{d-1}A_m^{\#\{1\leq i\leq n|r_{i,\sigma(i)}=d-m\}}.
\end{eqnarray}
It is clear that $\tilde D_n$ is homogeneous of degree $n$ in
$\Q[A_1,\ldots,A_d]$.  We claim there is a unique
highest-lexicographic-order-monomial in the formal expansion of
$\tilde D_n$.  Let $\cS_{-1}=S_n$. For every $0\leq k\leq d-1$ let
$\cS_k$ be the set of all $\sigma\in \cS_{k-1}$ such that
$\sigma(i)=j$ if $r_{ij}=k$ and if every entry in the $i$-th row and
the $j$-th column is $\geq k$.  Perusing the definition we note that
$\cS_k$ is not empty by Lemma \ref{L:r_n}. By Lemma \ref{L:r_n} again,
we note that $\cS_{d-1}$ contains exactly one element.  It is easy to
see that the set of permutations in $\cS_0$ gives monomials whose
$A_d$-exponents are highest in (\ref{E:D_n}).  Inductively, we note
that the unique permutation in $\cS_{d-1}$ gives the
highest-lexicographic-order-monomial, denoted by
$M(A_1,\ldots,A_{d})$, in $\tilde D_n$.  Since
$D_n(\vec{A})=
\tilde{D}_n(A_1,\ldots,A_{d-1},1)$, the monomial
$M(A_1,\ldots,A_{d-1},1)$ is a lowest-degree-monomial in $D_n$ and of
the highest-lexicographic-order among these lowest-degree-monomials.
Its uniqueness assures that
it can not be cancelled out among the lowest-degree-monomials.
Let $\sigma_n$ be the unique element in $\cS_{d-1}\subset S_n$.
Since $S_n=\bigcup_{t=0}^{c_n}S_n^t$, there is an $0\leq t\leq c_n$
such that $\sigma_n\in S_n^t$.
Then $f_n^t$ is not a zero polynomial.
\end{proof}

Let $t_n$ be the least $t$ with $0\leq t\leq c_n$ such that
$f_n^t(\vec{A})\neq 0$. (By Lemma \ref{L:5} this $t_n$ exists.)

\begin{corollary}\label{C:0}
Let $\cV_n$ be the complement in $\A^{d-1}$ of the
hypersurface defined by $f^{t_n}_n(\vec{A})=0$.
Let $\cW_r:=\bigcap_{n=1}^{d-1}\cV_n$.
Then $\cW_r$ is a Zariski dense open subset defined over $\Q$
in $\A^{d-1}$.
\end{corollary}
\begin{proof}
Lemma \ref{L:5} asserts that none of these hypersurfaces
is equal to $\A^{d-1}$. Hence $\cV_n$ is non-empty and
Zariski open in $\A^{d-1}$, but $\A^{d-1}$ is irreducible,
so $\cV_n$ is Zariski dense open in $\A^{d-1}$ for every $n$.
As a finite intersection, $\cW_r$ is hence also Zariski dense open
in $\A^{d-1}$.
\end{proof}


\section{Fredholm determinants and the $L$ function}
\label{S:4}

In this section, we consider the $p$-adic valuation of
$G_{pi-j}(\vec{A})$ in $\Z_p[\zeta_p][\vec{A}]$ and
that of the Fredholm determinant $C_n(\vec{A})$
in $\Z_p[\zeta_p][[\vec{A}]]$. We will consider
$G_{pi-j}(\vec{A})$ and  $C_n(\vec{A})$
as formal expressions in $\Z_p[\vec{A}][\gamma]$ and
$\Z_p[\vec{A}][[\gamma]]$, respectively.

Let notations be as in sections 2 and 3.
Recall $r$ is an integer with $1\leq r\leq d-1$ and $\gcd(r,d)=1$.
Let $n$ be an integer with $1\leq n\leq d-1$.
Let $p$ be a prime that $p\equiv r\bmod d$.

For any rational number $R$ let $\gamma^{>R}$ denote
the terms in $\Q_p(\zeta_p)[[\vec{A}]]$
whose coefficients have $p$-adic valuation $>R/(p-1)$.
We also use it to denote algebraic numbers in $\Q_p(\zeta_p)$
with $p$-adic valuation $>R/(p-1)$ and this shall not cause
any confusion.
Let
$$M_n:=\min_{\sigma\in S_n}\sum_{i=1}^{n}\llceil\frac{pi-\sigma(i)}{d}
\rrceil.
$$
The first lemma below derives properties from
(\ref{E11}) and (\ref{E10}).

\begin{lemma}\label{L:1}
For any $s\geq 0$  we have
\begin{eqnarray}
c_n&=&\max_{\sigma\in S_n}
\sum_{i=1}^{n}\llceil\frac{pi-\sigma(i)}{d}\rrceil-M_n\leq n;
\label{E2}\\
M_n &=&\frac{n(n+1)(p-1)}{2d}+
       \frac{1}{d}\min_{\sigma\in S_n}\sum_{i=1}^{n}r_{i,\sigma(i)};
\label{E1}\\
S^s_n&=&\{\sigma\in S_n|
      \sum_{i=1}^{n}\llceil\frac{pi-\sigma(i)}{d}\rrceil=M_n+s\}.\label{E3}
\end{eqnarray}
\end{lemma}
\begin{proof}
Suppose $\sigma_1,\sigma_2\in S_n$ are minimizer and
maximizer of $\sum_{i=1}^{n}\llceil\frac{pi-\sigma(i)}{d}\rrceil$,
respectively. Note that $\llceil\frac{pi-j}{d}\rrceil=\frac{pi-j+r_{ij}}{d}$
thus
$$
\max_{\sigma\in S_n}\sum_{i=1}^{n}\llceil\frac{pi-\sigma(i)}{d}\rrceil-M_n
=\frac{1}{d}\left(\max_{\sigma\in S_n}\sum_{i=1}^{n}r_{i,\sigma(i)}
 -\min_{\sigma\in S_n}\sum_{i=1}^{n}r_{i,\sigma(i)}\right)=c_n.
$$
For any $i$, since $1\leq \sigma(i)\leq d-1$, we have
$$
\llceil \frac{pi-\sigma_2(i)}{d} \rrceil
\leq\llceil\frac{pi-\sigma_1(i)}{d}\rrceil+1.
$$
Taking sum both sides and get
\begin{eqnarray*}
\max_{\sigma\in S_n}
\sum_{i=1}^{n}\llceil\frac{pi-\sigma(i)}{d}\rrceil\leq M_n+n.
\end{eqnarray*}
This proves (\ref{E2}).
Since
\begin{eqnarray*}
\sum_{i=1}^{n}\llceil\frac{pi-\sigma(i)}{d}\rrceil
=\frac{n(n+1)(p-1)}{2d}+\frac{1}{d}\sum_{i=1}^{n}r_{i,\sigma(i)},
\end{eqnarray*}
we see that (\ref{E1}) and (\ref{E3}) follows.
\end{proof}


For $0\leq s\leq c_n$, and $i,j\geq 1$ let
\begin{eqnarray}\label{E:K}
K^s_{ij}(\vec{A})&:=&
\sum_{\vec{m}\in\cM^s_{ij}}\frac{\vec{A}^{\vec{m}}}
{\vec{m}!(\llceil\frac{pi-j}{d}\rrceil+s-|\vec{m}|)!}.
\end{eqnarray}
Note that $K^s_{ij}(\vec{A})\in\Z_p[\vec{A}]$ for $p\geq d^2$.


\begin{lemma}\label{L:11}
Let $G_{n}(\vec{A})$ be defined as in (\ref{E:3}).
For any $i,j\geq 1$ and $pi-j\geq 0$ we have
\begin{eqnarray}\label{E:4}
\ord_p G_{pi-j}(\vec{A})\geq \frac{\llceil\frac{pi-j}{d}\rrceil}{p-1},
\end{eqnarray}
where $\ord_p(\cdot)$ denote $p$-adic valuation of the
content of $G_{pi-j}(\vec{A})$.
If, in addition, $p\geq (d^2+1)(d-1)$ then
\begin{eqnarray*}\label{E:5}
G_{pi-j}(\vec{A})
&=&\sum_{s=0}^{c_n}\gamma^{\llceil\frac{pi-j}{d}\rrceil+s}K^s_{ij}(\vec{A})
         +\gamma^{>\llceil\frac{pi-j}{d}\rrceil+c_n}.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Clearly, (\ref{E:4}) is a special case of (\ref{E:Bound}).
We shall prove the second equality.
For $0\leq s\leq c_n$, $m_\ell\geq 0$
and $m_1+\cdots+m_d=\llceil\frac{pi-j}{d}\rrceil+s$,
since $p\geq d^2-1$, we have
$m_\ell\leq \llceil\frac{pi-j}{d}\rrceil+c_n\leq p-1$.
And by (\ref{E:33}) and (\ref{E:3}), we have
\begin{eqnarray*}\label{E:39}
G_{pi-j}(\vec{A})
&=&\sum_{
\substack{m_1+\cdots+m_d\leq \llceil\frac{pi-j}{d}\rrceil+c_n\\
\sum_{\ell=1}^{d}\ell m_\ell=pi-j}}
\lambda_{m_1}\cdots \lambda_{m_d}\vec{A}^{\vec{m}} +\gamma^{>\llceil\frac{pi-j}{d}\rrceil+c_n}\\
&=&\sum_{s=0}^{c_n}\sum
\frac{\gamma^{m_1+\cdots+m_d}\vec{A}^{\vec{m}}}{m_1!\cdots m_d!}
 +\gamma^{>\llceil\frac{pi-j}{d}\rrceil+c_n}
\end{eqnarray*}
where the last sum ranges over all $m_\ell\geq 0$ such that $m_1+\cdots
+m_d=\llceil\frac{pi-j}{d}\rrceil+s$ and $\sum_{\ell=1}^{d}\ell
m_\ell=pi-j$.  It is easy to see that this is a subset of
$\cM^s_{ij}$.  Conversely, if $\vec{m}\in \cM^s_{ij}$ then
$$\sum_{\ell=1}^{d-1}m_\ell\leq \sum_{\ell=1}^{d-1}\ell
m_{d-\ell}=r_{ij}+ds\leq d-1+ds
\leq
\llceil\frac{pi-j}{d}\rrceil+s
$$
since $p\geq (d^2+1)(d-1)$.
Set $m_d=\llceil\frac{pi-j}{d}\rrceil+s-\sum_{\ell=1}^{d-1}m_\ell$,
then $m_1+\cdots+m_{d}=\llceil\frac{pi-j}{d}\rrceil+s$ and
$\sum_{\ell=1}^{d}\ell m_\ell=pi-j$ where $m_\ell\geq 0$.
Thus we have
\begin{eqnarray*}
G_{pi-j}(\vec{A})
&=&\sum_{s=0}^{c_n}
   \gamma^{\llceil\frac{pi-j}{d}\rrceil+s}
   \sum_{\vec{m}\in\cM^s_{ij}}\frac{\vec{A}^{\vec{m}}}
   {\vec{m}!(\llceil\frac{pi-j}{d}\rrceil+s-|\vec{m}|)!}
   +\gamma^{>\llceil\frac{pi-j}{d}\rrceil+c_n}.
\end{eqnarray*}
The lemma follows.
\end{proof}


\begin{lemma}\label{L:2}
Let $p\geq (d^2+1)(d-1)$ and let
$C_n(\vec{A})$ be defined as in (\ref{E:1}).
We have
\begin{eqnarray*}
C_n(\vec{A})
&=&\det\{G_{pi-j}(\vec{A})\}_{1\leq i,j\leq n} +\gamma^{>M_n+c_n}\\
&=&\sum_{t=0}^{c_n}
\gamma^{M_n+t}
\sum_{\substack{s_0+\cdots+s_n=t\\\sigma\in S_n^{s_0}}}\sgn(\sigma)
\prod_{i=1}^{n}K^{s_i}_{i,\sigma(i)}(\vec{A})
+\gamma^{>M_n+c_n}.
\end{eqnarray*}
Let $\vec{\hat{a}}\in (\Z_p\cap\Q)^{d-1}$.
Suppose there exists $0\leq t_n'\leq c_n$ such that
$$
\sum_{\substack{s_0+\cdots+s_n=t\\\sigma\in S_n^{s_0}}}\sgn(\sigma)
\prod_{i=1}^{n}K^{s_i}_{i,\sigma(i)}(\vec{\hat{a}})
\left\{
\begin{array}{ll}
\equiv 0 \bmod p&\mbox{for all $0\leq t<t_n'$}\\
\not\equiv 0\bmod p&\mbox{for $t=t_n'$}.
\end{array}
\right.
$$
Then
$$\ord_pC_n(\vec{\hat{a}})=\frac{M_n+t_n'}{p-1}.$$
\end{lemma}
\begin{proof}
Recall $C_n(\vec{A})$ from (\ref{E:1}).
Suppose there is a $t$ such that $u_t>n$,
we claim that
\begin{eqnarray}\label{E:contradiction}
\min_{\sigma\in S_n}\ord_p \prod_{t=1}^{n} G_{pu_t-u_{\sigma(t)}}(\vec{A})
&>& \frac{M_n+c_n}{p-1}.
\end{eqnarray}
Since $\sum_{t=1}^{n}u_t>\sum_{i=1}^{n}i$, we have
\begin{eqnarray}\label{E:twin1}
\frac{1}{p-1}\sum_{t=1}^{n}\llceil\frac{pu_t-u_{\sigma(t)}}{d}\rrceil
&\geq & \frac{1}{p-1}\sum_{t=1}^{n}\frac{pu_t-u_{\sigma(t)}}{d}\nonumber\\
&=&\frac{1}{d}\sum_{t=1}^{n}u_t\nonumber\\
&\geq&\frac{1}{d}\sum_{i=1}^{n}i+\frac{1}{d}
   \nonumber\\
&=& \frac{1}{p-1}\sum_{i=1}^{n}\frac{pi-\delta(i)}{d}+\frac{1}{d}
\end{eqnarray}
for any $\delta\in S_n$.
For $p\geq (d^2+1)(d-1)> d^2-d+1$ we have
\begin{eqnarray}\label{E:twin2}
\frac{1}{p-1}\sum_{i=1}^{n}\llceil\frac{pi-\delta(i)}{d}\rrceil
&\leq&\frac{1}{p-1}\sum_{i=1}^{n}\frac{pi-\delta(i)}{d}+\frac{n}{p-1}
  \nonumber\\
&\leq&\frac{1}{p-1}\sum_{i=1}^{n}\frac{pi-\delta(i)}{d}+\frac{d-1}{p-1}
  \nonumber\\
&<&\frac{1}{p-1}\sum_{i=1}^{n}\frac{pi-\delta(i)}{d}+\frac{1}{d}
\end{eqnarray}
Therefore, by (\ref{E:twin1}) and (\ref{E:twin2}),  we have
\begin{eqnarray*}
\min_{\sigma\in S_n}\ord_p \prod_{t=1}^{n}G_{pu_t-u_{\sigma(t)}}(\vec{A})
&>&\frac{1}{p-1}\max_{\delta\in S_n}
\sum_{i=1}^{n}\llceil\frac{pi-\delta(i)}{d}\rrceil
=\frac{M_n+c_n}{p-1}
\end{eqnarray*}
by (\ref{E2}).
Perusal of (\ref{E:1}) then shows the validity of
our first equality. Now we have by Lemmas \ref{L:1} and \ref{L:11}
\begin{eqnarray*}
C_n(\vec{A})
&=&\sum_{\sigma\in S_n}\sgn(\sigma)
\prod_{i=1}^{n}G_{pi-\sigma(i)}(\vec{A})+\gamma^{>M_n+c_n}\\
&=&\sum_{\sigma\in S_n}\sgn(\sigma)
\prod_{i=1}^{n}\sum_{s_i=0}^{c_n}\left(
\gamma^{\llceil\frac{pi-\sigma(i)}{d}\rrceil+s_i}K^{s_i}_{i,\sigma(i)}(\vec{A})
+\gamma^{>\llceil\frac{pi-\sigma(i)}{d}\rrceil+c_n}\right)
+\gamma^{>M_n+c_n}
\\
&=&\sum_{s_0=0}^{c_n}\sum_{\sigma\in S_n^{s_0}}\sgn(\sigma)
   \sum_{\ell=0}^{c_n-s_0}\gamma^{M_n+s_0+\ell}\sum_{s_1+\cdots+s_n=\ell}
   \prod_{i=1}^{n}K^{s_i}_{i,\sigma(i)}(\vec{A})+ \gamma^{>M_n+c_n}\\
&=&\sum_{t=0}^{c_n}\gamma^{M_n+t}
\sum_{s_0+\cdots+s_n=t}\sum_{\sigma\in S_n^{s_0}}\sgn(\sigma)
\prod_{i=1}^{n}K^{s_i}_{i,\sigma(i)}(\vec{A})+\gamma^{>M_n+c_n}.
\end{eqnarray*}
This finishes the proof of the second equality.

To show the last assertion, it is important to note that
\begin{eqnarray}\label{E:G}
\sum_{\substack{s_0+\cdots+s_n=t\\\sigma\in S_n^{s_0}}}\sgn(\sigma)
\prod_{i=1}^{n}K^{s_i}_{i,\sigma(i)}(\vec{\hat{a}})
 \end{eqnarray}
lies in $\Z_p$.
For $0\leq t<t_n'$
since (\ref{E:G}) vanishes $\bmod p$
its $p$-adic valuation is $\geq 1$.
So
\begin{eqnarray}\label{E:G2}
\ord_p
\left(\gamma^{M_n+t}\sum_{\substack{s_0+\cdots+s_n=t\\
\sigma\in S_n^{s_0}}}\sgn(\sigma)
\prod_{i=1}^{n}K^{s_i}_{i,\sigma(i)}(\vec{\hat{a}})\right)
&\geq& \frac{M_n+t+p-1}{p-1}\\
&>&\frac{M_n+c_n}{p-1}\nonumber\\
&\geq&\frac{M_n+t_n'}{p-1}.\nonumber
\end{eqnarray}
For $t=t_n'$ then (\ref{E:G}) does not vanish $\bmod p$
so it is a $p$-adic unit, denoted by $u$.  Combining with (\ref{E:G2})
we get
$$C_n(\vec{\hat{a}})=u\gamma^{M_n+t_n'}+\gamma^{>M_n+t_n'}.$$
Then our last assertion follows easily.
\end{proof}


\section{Proof of Theorem \ref{T:1}}\label{S:Proof}

Let notations be as in previous sections.
For $\vec{a}\in\cW(\Q)$ and for $p$ large enough
we give the exact $p$-adic
valuation of $C_n(\vec{a})$ (in Lemma \ref{L:14}) and then
$b_n(\vec{a})$ (in Corollary \ref{C:1}).  These accumulate to the
proof of Theorem \ref{T:1} at the end of this section.

Before presenting our main result
we start with a very elementary yet useful lemma.

\begin{lemma}\label{L:19}
Fix an integer $r$ with $1\leq r\leq d-1$ and $\gcd(r,d)=1$.
Let $k$ be a rational number. Then
$k\neq 0$ if and only if
there exists $N>0$ such that for all $p>N$ and
$p\equiv r\bmod d$ we have $k\not\equiv 0\bmod p$.
\end{lemma}
\begin{proof}
Suppose $k=0$.  Then of course $k\equiv 0\bmod p$ for all $p$ and of
course for all $p\equiv r\bmod d$. This proves the first direction.
Conversely, suppose $k\neq 0$. Then there are
only finitely many primes $p$ with $\ord_p k\neq 0$. So there exists
$N>0$ such that $k\not\equiv 0\bmod p$ for all $p>N$
and this certainly includes all $p\equiv r\bmod d$.
\end{proof}


For any $1\leq n\leq d-1$, we recall from section \ref{S:3} that $t_n$
is the least non-negative integer with $f_n^{t_n}(\vec{A})\neq 0$.
Recall that $\cV_n$ is the complement in $\A^{d-1}$ of the
hypersurface defined by $f_n^{t_n}(\vec{A})$.

\begin{lemma}\label{L:14}
Let $1\leq n\leq d-1$.
Let $\vec{a}\in\cV_n$.
If $p\equiv r\bmod d$ and $p$ is large enough
then we have
\begin{eqnarray}\label{E:C_n}
\ord_p C_n(\vec{\hat a})= \frac{M_n+t_n}{p-1}=
\frac{n(n+1)}{2d}+\frac{\min_{\sigma\in S_n}
     \sum_{i=1}^{n}r_{i,\sigma(i)}+dt_n}{d(p-1)}.
\end{eqnarray}
Moreover,
$$
\lim_{p\rightarrow\infty}\ord_p C_n(\vec{\hat a}) = \frac{n(n+1)}{2d}.
$$
\end{lemma}
\begin{proof}
Through out this proof assume $p\equiv r\bmod d$.
Let
\begin{eqnarray}\label{E:epsilon}
\epsilon_n&:=&
\frac{\min_{\sigma\in S_n}\sum_{i=1}^{n}r_{i,\sigma(i)}+dt_n}{d(p-1)}.
\end{eqnarray}
Note that $1\leq r_{ij}\leq d-1$ for all $1\leq i,j\leq d-1$,
and $t_n\leq c_n\leq n\leq d-1$ by (\ref{E2}),
we have $\epsilon_n\leq \frac{n(2d-1)}{d(p-1)}$. Thus
$\epsilon_n$ goes to $0$ as $p$ approaches $\infty$.
So it suffices to prove (\ref{E:C_n}).
Since $p\geq d^2$ we derive that
$H^s_{ij}(\vec{A}), K^s_{ij}(\vec{A})\in\Z_p[\vec{A}]$ from their
definitions in (\ref{E:H}) and (\ref{E:K}), respectively.
Since $p\geq d^2-1$ we always have
$1\leq \llceil\frac{pi-1}{d}\rrceil+n\leq p-1$.
Then
$$u_n:=\prod_{i=1}^{n}(\llceil\frac{pi-1}{d}\rrceil+n)!$$
is a $p$-adic unit in $\Z_p$.
Recall from Lemma \ref{L:added} that
\begin{eqnarray*}
\llceil\frac{pi-1}{d}\rrceil=\frac{pi+r_{i1}-1}{d}\equiv
\frac{r_{i1}-1}{d}\bmod p\\
\llceil\frac{pi-j}{d}\rrceil = \llceil\frac{pi-1}{d}\rrceil-\delta_{ij}
\equiv \frac{r_{i1}-1}{d}-\delta_{ij}\bmod p.
\end{eqnarray*}
Let $p\geq (d^2+1)(d-1)$. We then have
\begin{eqnarray*}
H^s_{ij}(\vec{A})
&&=
\sum_{\vec{m}\in\cM_{ij}^s}
\frac{(\llceil\frac{pi-1}{d}\rrceil+n)(\llceil\frac{pi-1}{d}\rrceil+n-1)
\cdots(\llceil\frac{pi-j}{d}\rrceil+s+1-|\vec{m}|)}{\vec{m}!}
\vec{A}^{\vec{m}}
\\
&&\equiv
\sum_{\vec{m}\in\cM_{ij}^s}
\frac{(\llceil\frac{pi-1}{d}\rrceil+n) !}
{\vec{m}!(\llceil\frac{pi-j}{d}\rrceil+s-|\vec{m}|)!}
\vec{A}^{\vec{m}}
\\
&&=
(\llceil\frac{pi-1}{d}\rrceil+n)!\ K^s_{ij}(\vec{A})\bmod p.
\end{eqnarray*}
Thus for $0\leq t\leq c_n$,
by definition of $f^{t}_n(\vec{A})$ in (\ref{E12}),
we have
\begin{eqnarray}
\nonumber
f^{t}_n(\vec{a})
&\equiv&u_n\sum_{
      \substack{s_0+\cdots+s_n=t\\s_\ell\geq 0}}
      \sum_{\sigma\in S_n^{s_0}}\sgn(\sigma)
    \prod_{i=1}^{n}K^{s_i}_{i,\sigma(i)}(\vec{a})\\
&\equiv& u_n
\sum_{\substack{s_0+\cdots+s_n=t\\s_\ell\geq 0}}
\sum_{\sigma\in S_n^{s_0}}\sgn(\sigma)
    \prod_{i=1}^{n} K^{s_i}_{i,\sigma(i)}(\vec{\hat a})\bmod p
\label{E:f_n}
\end{eqnarray}
since $K^s_{ij}(\vec{a})\equiv K^s_{ij}(\vec{\hat a})\bmod p$.
Recall that $f_n^t(\vec{A})=0$ for all $0\leq t<t_n$
and $f^{t_n}_n(\vec{A})\neq 0$.
Now suppose $\vec{a}\in\cV_n$.
By definition $f^{t_n}_n(\vec{a})\neq 0$.
Following Lemma~\ref{L:19} there exists $N>0$ such that for all
$p>N$ and $p\equiv r\bmod d$ we have
$f^{t_n}_n(\vec{a})\not\equiv 0\bmod p$.
By (\ref{E:f_n}) these above imply
$$
\sum_{\substack{s_0+\cdots+s_n=t\\s_\ell\geq 0}}
\sum_{\sigma\in S_n^{s_0}}\sgn(\sigma)
    \prod_{i=1}^{n} K^{s_i}_{i,\sigma(i)}(\vec{\hat a})
\left\{
\begin{array}{ll}
\equiv 0\bmod p&\mbox{for all $0\leq t< t_n$}\\
\not\equiv 0 \bmod p&\mbox{for $t=t_n$.}
\end{array}
\right.
$$
Apply Lemma \ref{L:2} it is now clear that
$\ord_p C_n(\vec{\hat a})=\frac{M_n+t_n}{p-1}.$
Now we conclude (\ref{E:C_n}) by
recalling the identity in (\ref{E1}).
Thus $\lim_{p\rightarrow\infty} \ord_p C_n(\vec{\hat a})=\frac{n(n+1)}{2d}$.
This finishes our proof.
\end{proof}

Write $\vec{a}=(a_1,\ldots,a_{d-1})$.
Let $f(x)=x^d+a_{d-1}x^{d-1}+\ldots+a_1x$ in $(\Z_p\cap\Q)[x]$.
Recall from Lemma \ref{L:Dwork}
that $L(f\otimes\F_p;T)=1+b_1(\vec{a})T+\cdots + b_{d-1}(\vec{a})
T^{d-1} \in\Z[\zeta_p][T]$.

%%proof of \lim ord b_n = n(n+1)/2d


\begin{corollary}\label{C:1}
Let $1\leq n\leq d-1$.
Let $\vec{a}\in\cW_r(\Q)$.
If $p\equiv r\bmod d$
and $p$ is large enough we have
\begin{eqnarray*}
\ord_p b_n(\vec{a})=\frac{M_n+t_n}{p-1}=\frac{n(n+1)}{2d}
+\frac{\min_{\sigma\in S_n}\sum_{i=1}^{n}r_{i,\sigma(i)}+dt_n}{d(p-1)}.
\end{eqnarray*}
Moreover,
\begin{eqnarray}\label{E:14}
\lim_{p\rightarrow \infty}\ord_p  b_n(\vec{a})=
\frac{n(n+1)}{2d}.
\end{eqnarray}
\end{corollary}
\begin{proof}
Write $\epsilon_n$ as in (\ref{E:epsilon}).
As discussed in Lemma \ref{L:14} it suffices to prove the first statement.
Let $p$ be large enough so that  statements in
Lemma \ref{L:14} hold and for any $1\leq n\leq d-1$ we have
$$\ord_p C_n(\vec{\hat a})=\frac{n(n+1)}{2d}+\epsilon_n.$$
Below we abbreviate $C_n$ for $C_n(\vec{\hat a})$ for all $n$.
Write
$$
 \frac{1}{1+\sum_{n=1}^{\infty}(-1)^nC_np^nT^n}
 = 1+ \sum_{n=1}^{\infty}\sum_{\substack{n_1+\cdots+n_k=n\\
n_1,\ldots,n_k\geq 1,k\geq 1}}
 C_{n_1}\cdots C_{n_k}p^nT^n=\sum_{n=0}^{\infty}W_nT^n.
$$
We claim that
$\ord_p W_n = n(\frac{d+1}{d}+\epsilon_1)$.
It is trivially true for $n=0$.
For $1\leq n\leq d-1$ we have
\begin{eqnarray}\label{E:Wn}
\ord_p W_n\geq n+ \min_{\substack{n_1+\cdots+n_k=n\\n_1,\ldots,n_k\geq
1,k\geq 1}} \left(\sum_{\ell=1}^{k} \frac{n_\ell(n_\ell+1)}{2d}
+\sum_{\ell=1}^{k}\epsilon_{n_\ell}\right),
\end{eqnarray}
where the equality holds if its minimum is unique.
But $\sum_{\ell=1}^{k} n_\ell(n_\ell+1)\geq \sum_{\ell=1}^{k}2n_\ell$,
so for $n_1+\cdots +n_k=n$, $n_\ell\geq 1$, we have
$\sum_{\ell=1}^{k}\frac{n_\ell(n_\ell+1)}{2d}\geq \frac{n}{d}$
where the equality certainly holds if $k=n$, that is, $n_1=\cdots=n_k=1$.
Since $\epsilon_1$ converges to $0$ as $p$ approaches $\infty$,
when $p$ is big enough we have
$$
\frac{n}{d}+n\epsilon_1
<
\min_{\substack{n_1+\cdots+n_k=n\\n_1,\ldots,n_k\geq 1, k<n}}
\sum_{i=1}^{k}\frac{n_\ell(n_\ell+1)}{2d}.
$$
Thus the minimum in (\ref{E:Wn}) is unique at $k=n$
in which case we have
$$\ord_p W_n = n+(\frac{n}{d}+n\epsilon_1)=n(\frac{d+1}{d}+\epsilon_1).$$
By (\ref{E:Dwork}),
\begin{eqnarray*}
L(f\otimes\F_p;T)
&=&1+b_1(\vec{a})T+\cdots+b_{d-1}(\vec{a})T^{d-1}\\
&=&\left(\sum_{n_1=0}^{\infty}p^{n_1}T^{n_1}\right)
\left(\sum_{n_2=0}^{\infty}(-1)^{n_2}C_{n_2}T^{n_2}\right)
\left(\sum_{n_3=0}^{\infty}W_{n_3}T^{n_3}\right)\\
&=&1+\sum_{n=1}^{\infty}\sum_{\substack{n_1+n_2+n_3
=n\\n_1,n_2,n_3\geq 0}}(-1)^{n_2}p^{n_1}C_{n_2}W_{n_3}T^n.
\end{eqnarray*}
So for large enough $p$
one may carry out a simple computation and
conclude that
$$\ord_p ((-1)^{n_2}p^{n_1}C_{n_2}W_{n_3})
=n_1+(\frac{n_2(n_2+1)}{2d}+\epsilon_{n_2})
+(\frac{d+1}{d}+\epsilon_1)n_3.$$
For $1\leq n\leq d-1$, $n_1+n_2+n_3=n$ and
$n_1, n_2, n_3\geq 0$ the minimal
$\ord_p ((-1)^{n_2}p^{n_1}C_{n_2}W_{n_3})$ is achieved at
$n_1=0, n_2=n, n_3=0$.
Thus for $1\leq n\leq d-1$ we have
$$\ord_p b_n(\vec{a}) = \min_{\substack{n_1+n_2+n_3=n\\n_1,n_2,n_3\geq 0}}
\ord_p ((-1)^{n_2}p^{n_1}C_{n_2}W_{n_3})
=\frac{n(n+1)}{2d}+\epsilon_{n}.$$
This finishes our proof.
\end{proof}


\begin{proof}[Proof of Theorem 1.1]
Let $f(x)=x^d+a_{d-1}x^{d-1}+\ldots+a_1x$ in $(\Z_p\cap\Q)[x]$.  Write
$\vec{a}=(a_1,\ldots,a_{d-1})$.  Let $r$ be coprime to $d$ and $1\leq
r\leq d-1$.  By Corollary \ref{C:1} we have a Zariski dense open
subset $\cW_r$ defined over $\Q$
in $\A^{d-1}$ such that for every $\vec{a}\in\cW_r(\Q)$ we have
\begin{eqnarray*}
\lim_{\substack{p\rightarrow\infty\\p\equiv r\bmod d}}
\ord_p b_n(\vec{a})
&=&\frac{n(n+1)}{2d}.
\end{eqnarray*}
Hence
\begin{eqnarray*}
\lim_{\substack{p\rightarrow\infty\\p\equiv r\bmod d}}
\NP(f\otimes\F_p)&=&\HP(f).
\end{eqnarray*}
Let $\cW:=\bigcap_{\substack{1\leq r\leq d-1\\\gcd(r,d)=1}} \cW_r$.
It is Zariski dense open in $\A^{d-1}$ by Corollary \ref{C:0},
For every $f$ with coefficient vector $\vec{a}\in \cW(\Q)$ we have
\begin{eqnarray}\label{E:00}
\lim_{p\rightarrow \infty}\NP(f\otimes\F_p)=\HP(f).
\end{eqnarray}
Consider the natural projection map $\mu:\A^d\rightarrow \A^{d-1}$
by $\mu(f+a_0)=\vec{a}$.
The pullback $\cU:=\mu^{-1}(\cW)$ is defined over $\Q$ and
is Zariski dense open in $\A^d$.
Now for any $a_0\in\Z_p\bigcap\Q$, by (\ref{E:Lfunction}),
we have $L((f+a_0)\otimes\F_p;T)=L(f\otimes\F_p;\zeta_p^{a_0}T)$.
Thus
\begin{eqnarray}\label{E:99}
\NP((f+a_0)\otimes\F_p)=\NP(f\otimes\F_p).
\end{eqnarray}
By (\ref{E:99}) and (\ref{E:00}) we have
$$\lim_{p\rightarrow \infty}\NP((f+a_0)\otimes\F_p)=\HP(f)=\HP(f+a_0)$$
for every $f+a_0\in\cU(\Q)$. This finishes the proof of the theorem.
\end{proof}




\section{More remarks}


Regarding Theorem
\ref{T:1} it is not yet established for a fixed $d\geq 3$ how large
the complementary set $(\A^d-\cU)(\Q)$ should be and what polynomial $f(x)$
should be in there. In Proposition \ref{P:2} we show that polynomial
$f(x)=x^d$ is in $(\A^d-\cU)(\Q)$ for every $d\geq 3$.  Since
$\NP(f\otimes\F_p)=\frac{\NP(X_f\otimes\F_p)}{p-1}$ for the
Artin-Schreier curve $X_f: y^p-y=f(x)$ (see discussion in section 1),
the eigenvalues of $L(f\otimes\F_p;T)$ are Gauss sums as worked out in
\cite[chapter III]{Koblitz:2}.  The prime factorization of the Gauss
sums are given by Stickelberger theorem (see \cite[Chapter
6]{Washington}). This yields the $p$-adic Newton polygon of
$L(f\otimes\F_p;T)$ and hence $\NP(f\otimes\F_p)$.  Let us describe
the resulted Newton polygon: Let $\sigma$ be a permutation in the
degree $d-1$ symmetric group $S_{d-1}$ such that for every $1\leq
n\leq d-1$ we let $\sigma(n)$ be the least positive residue of
$pn\bmod d$. Write $\sigma$ as a product of disjoint cycles (including
1-cycles).  Let $\sigma_i$ be a $\ell_i$-cycle in $\sigma$.  Let
$\lambda_i:=\sum n/(d\ell_i)$ where the sum ranges over all $n$ in the
standard representation of the $\ell_i$-cycle $\sigma_i$. Arrange
$\sigma_i$ in such an order that $\lambda_1\leq \lambda_2\leq \cdots$.
For every $\sigma_i$ in $\sigma$ let the pair $(\lambda_i,\ell_i)$ of
rational numbers represent the line segment of (horizontal) length
$\ell_i$ and of slope $\lambda_i$.  The {\em joint of line segments}
$(\lambda_i,\ell_i)$ is the lower convex hull consisting of line
segment $(\lambda_i,\ell_i)$'s connected at their endpoints. It
uniquely determines a Newton polygon.  One may show that
$\NP(x^d\otimes\F_p)$ is the joint of $(\lambda_i,\ell_i)$'s.
Note that $\NP(x^d\otimes\F_p)$ only depends on the
residue class of $p\bmod d$. I thank Kiran Kedlaya for
discussions which simplified my initial formulation.

\begin{proposition}\label{P:2}
The limit of $\NP(x^d\otimes\F_p)$ does not exist as $p$
approaches $\infty$.
\end{proposition}
\begin{proof}
Let notation be as above.
We claim that there exists $\sigma_i$ in $\sigma$ with
$\ell_i>1$. Support $\sigma$ is a product of $1$-cycles then $\sigma$
is identity and so $pn\equiv n\bmod d$ for every $1\leq n\leq
d-1$. Choose $n$ to be coprime to $d$ then we get $p\equiv 1\bmod d$.
This implies that for $p\not\equiv 1\bmod d$ there exists at least one
$\ell_i>1$. But for $p\equiv 1\bmod d$ we have $\ell_i=1$ for all
$i$. Thus $\lim_{p\rightarrow\infty}\NP(f\otimes\F_p)$ does not exist.
\end{proof}

\begin{remark}
For any $c_1,c_2\in\F_p^*$ and $c_3,c_4\in \F_p$ one observes that
$X_f:y^p-y=x^d$ is isomorphic over $\F_p$ to
$X_f':y^p-y=c_1(c_2x+c_3)^d+c_4$.  Thus $$\NP((c_1(c_2x+c_3)^d+c_4)
\otimes\F_p)=\NP(x^d\otimes\F_p).$$
\end{remark}

\begin{remark}
Permutations polynomials are potential elements in $(\A^d-\cU)(\Q)$.  (See
\cite[Chapter 7]{FiniteField}.)  Suppose that $f(x)\in \A^d(\Q)$ is a
permutation polynomial over $\F_p$ for all $p\equiv r\bmod d$ for some
$r$, then $S_1(f\otimes\F_p)=
\sum_{x\in\F_p}\zeta_p^{f(x)}=\sum_{x\in\F_p}\zeta_p^x=0$.  Then by
(\ref{E:1}) we have $b_1=S_1(f\otimes\F_p)=0$ for all $p\equiv r\bmod d$.
Consequently $\lim_{p\rightarrow\infty}\ord_p b_\ell \neq
\frac{\ell(\ell+1)}{2d}$.  Therefore,
$\lim_{p\rightarrow\infty}\NP(f\otimes\F_p)\neq \HP(f)$.
\end{remark}

\begin{remark}
Let $f_d(x)$ be a $d$-th Dickson polynomial (see more general
definition in \cite[Section 7.2]{FiniteField} and in particular (7.6)
here), then it is well-known that $f_d(x)$ is a permutations
polynomial over $\F_{p^\ell}$ (i.e., this polynomial function defines
an onto map from $\F_{p^\ell}$ to $\F_{p^\ell}$) if and only if
$\gcd(d,p^{2\ell}-1)=1$ (see proof in \cite[Theorem
7.16]{FiniteField}).  Then it is an exercise to show that for every
odd $d>3$, we have $b_1=S_1(f_d\otimes\F_p)=0$ for all $p\equiv 2\bmod
d$.  Thus the Newton polygon does not converge to the Hodge
one as $p$ goes to infinity.
\end{remark}



\begin{thebibliography}{99}

\bibitem{Adolphson-Sperber}
{\sc Allen Adolphson; Steven Sperber:}
Newton polyhedra and
the degree of the $L$-function associated to an exponential sum.
{\it Invent. Math.} {\bf 88} (1987), 555--569.

\bibitem{Adolphson-Sperber:2}
{\sc Allen Adolphson; Steven Sperber:}
Exponential sums and Newton polyhedra:
Cohomology and estimates,
{\it Annals of Mathematics} {\bf 130} (1989), 367--406.

\bibitem{Bombieri}
{\sc Enrico Bombieri:}
On exponential sums in finite fields,
{\it American J. Math.} {\bf 88} (1966), 71--105.

\bibitem{Dwork}
{\sc Bernard Dwork:}
On the zeta function of a hypersurface. II.
{\it Ann. of Math.}
{\bf 80} (1964), 227--299.

\bibitem{Dwork2}
{\sc Bernard Dwork:}
On the zeta function of a hypersurface.
{\it Publication Math. IHES,}
(1962), 5--68.

\bibitem{Hong}
{\sc Shao-Fang Hong:}
Newton polygons of $L$ functions associated with
exponential sums of polynomials of degree four over
finite fields.
{\it Finite Fields Appl.} {\bf 7} (2001), 205--237.

\bibitem{Katz:2}
{\sc Nicholas M. Katz:}
Sommes exponentielles,
{\it Ast\'erisque} {\bf 79},
Soci\'et\'e math\'ematique de France. 1980.

\bibitem{Koblitz:2}
{\sc Neal Koblitz:}
$p$-adic analysis: a short course on recent work,
{\it London mathematical society lecture note series} {\bf 46}.
Cambridge university press, 1980.

\bibitem{Koblitz}
{\sc Neal Koblitz:}
$p$-adic numbers, p-adic analysis, and Zeta-functions,
(Second edition),
{\it Graduate Texts in Mathematics} {\bf 58}.
Springer-Verlag, 1984.

\bibitem{Lang}
{\sc Serge Lang:}
{\it Algebra}
Third edition. Addison-Wesley, 1993.


\bibitem{FiniteField}
{\sc Rudolf Lidl; Harald Niederreiter:}
{\it Finite fields}.
{\it Second edition},
Encyclopedia of mathematics and its applications {\bf 20},
Edited by G.-C. Rota.
Cambridge University Press, 1997.

\bibitem{SZ:2}
{\sc Jasper Scholten; Hui June Zhu:}
Slope estimates of Artin-Schreier curves,
electronic preprint available online at
{\tt http://xxx.lanl.gov/abs/math.AG/0105005}.

\bibitem{SZ:3}
{\sc Jasper Scholten; Hui June Zhu:}
The first slope case of Wan's conjecture.
Preprint available at
{\tt http://www.math.berkeley.edu/zhu/preprints.html/}.

\bibitem{Sperber:3}
{\sc Steven Sperber:}
Congruence properties of the hyperkloosterman sum,
{\it Compositio Math.,} {\bf 40} (1980), 3--33.

\bibitem{Sperber:4}
{\sc Steven Sperber:}
Newton polygons for general hyperkloosterman sums,
in {\it $p$-adic cohomology},
{\it Ast\'erisque}, {\bf 119--120} (1984), 267--330.

\bibitem{Sperber}
{\sc Steven Sperber:}
On the $p$-adic theory of exponential sums,
{\it American J. Math.,}
{\bf 109} (1986), 255--296.

\bibitem{Wan}
{\sc Daqing Wan:}
An introduction to the theory of Newton polygons for L-functions
of exponential sums, To appear. Preprint available at
{\tt http://www.math.uci.edu/dwan/Overview.html}.

\bibitem{Wan:2}
{\sc Daqing Wan:}
{\it In preparation.}

\bibitem{Washington}
{\sc Lawrence Washington:}
Introduction to cyclotomic fields.
{\it Second edition.}
Graduate texts in mathematics {\bf 83}.
Springer, 1997.

\bibitem{Zhu:2}
{\sc Hui June Zhu:}
$p$-adic variation of $L$ functions
of exponential sums, II. (In preparation).

\end{thebibliography}

\end{document}





math0111194.extracted_bib
\begin{thebibliography}{99}

\bibitem{Adolphson-Sperber}
{\sc Allen Adolphson; Steven Sperber:}
Newton polyhedra and
the degree of the $L$-function associated to an exponential sum.
{\it Invent. Math.} {\bf 88} (1987), 555--569.

\bibitem{Adolphson-Sperber:2}
{\sc Allen Adolphson; Steven Sperber:}
Exponential sums and Newton polyhedra:
Cohomology and estimates,
{\it Annals of Mathematics} {\bf 130} (1989), 367--406.

\bibitem{Bombieri}
{\sc Enrico Bombieri:}
On exponential sums in finite fields,
{\it American J. Math.} {\bf 88} (1966), 71--105.

\bibitem{Dwork}
{\sc Bernard Dwork:}
On the zeta function of a hypersurface. II.
{\it Ann. of Math.}
{\bf 80} (1964), 227--299.

\bibitem{Dwork2}
{\sc Bernard Dwork:}
On the zeta function of a hypersurface.
{\it Publication Math. IHES,}
(1962), 5--68.

\bibitem{Hong}
{\sc Shao-Fang Hong:}
Newton polygons of $L$ functions associated with
exponential sums of polynomials of degree four over
finite fields.
{\it Finite Fields Appl.} {\bf 7} (2001), 205--237.

\bibitem{Katz:2}
{\sc Nicholas M. Katz:}
Sommes exponentielles,
{\it Ast\'erisque} {\bf 79},
Soci\'et\'e math\'ematique de France. 1980.

\bibitem{Koblitz:2}
{\sc Neal Koblitz:}
$p$-adic analysis: a short course on recent work,
{\it London mathematical society lecture note series} {\bf 46}.
Cambridge university press, 1980.

\bibitem{Koblitz}
{\sc Neal Koblitz:}
$p$-adic numbers, p-adic analysis, and Zeta-functions,
(Second edition),
{\it Graduate Texts in Mathematics} {\bf 58}.
Springer-Verlag, 1984.

\bibitem{Lang}
{\sc Serge Lang:}
{\it Algebra}
Third edition. Addison-Wesley, 1993.


\bibitem{FiniteField}
{\sc Rudolf Lidl; Harald Niederreiter:}
{\it Finite fields}.
{\it Second edition},
Encyclopedia of mathematics and its applications {\bf 20},
Edited by G.-C. Rota.
Cambridge University Press, 1997.

\bibitem{SZ:2}
{\sc Jasper Scholten; Hui June Zhu:}
Slope estimates of Artin-Schreier curves,
electronic preprint available online at
{\tt http://xxx.lanl.gov/abs/math.AG/0105005}.

\bibitem{SZ:3}
{\sc Jasper Scholten; Hui June Zhu:}
The first slope case of Wan's conjecture.
Preprint available at
{\tt http://www.math.berkeley.edu/zhu/preprints.html/}.

\bibitem{Sperber:3}
{\sc Steven Sperber:}
Congruence properties of the hyperkloosterman sum,
{\it Compositio Math.,} {\bf 40} (1980), 3--33.

\bibitem{Sperber:4}
{\sc Steven Sperber:}
Newton polygons for general hyperkloosterman sums,
in {\it $p$-adic cohomology},
{\it Ast\'erisque}, {\bf 119--120} (1984), 267--330.

\bibitem{Sperber}
{\sc Steven Sperber:}
On the $p$-adic theory of exponential sums,
{\it American J. Math.,}
{\bf 109} (1986), 255--296.

\bibitem{Wan}
{\sc Daqing Wan:}
An introduction to the theory of Newton polygons for L-functions
of exponential sums, To appear. Preprint available at
{\tt http://www.math.uci.edu/dwan/Overview.html}.

\bibitem{Wan:2}
{\sc Daqing Wan:}
{\it In preparation.}

\bibitem{Washington}
{\sc Lawrence Washington:}
Introduction to cyclotomic fields.
{\it Second edition.}
Graduate texts in mathematics {\bf 83}.
Springer, 1997.

\bibitem{Zhu:2}
{\sc Hui June Zhu:}
$p$-adic variation of $L$ functions
of exponential sums, II. (In preparation).

\end{thebibliography}