\section{Introduction}
Let $s\geq 1$ be an integer. 
For a $s$-tuple $\ul a=(a_1,\ldots,a_s)\in\Z^s_p$ of $p$-adic integers,
let
\[
F_{\ul a}(t)={}_sF_{s-1}\left({a_1,\ldots,a_s\atop 1,\ldots, 1}:t\right)
=\sum_{n=0}^\infty\frac{(a_1)_n}{n!}\cdots\frac{(a_s)_n}{n!}t^n
\]
be the hypergeometric power series where
$(\alpha)_n=\alpha(\alpha+1)\cdots(\alpha+n-1)$ denotes the Pochhammer symbol.
In his seminal paper \cite{Dwork-p-cycle}, B. Dwork discovered that
certain ratios of the hypergeometric power series are the uniform limit of rational functions.
We call his functions {\it Dwork's $p$-adic hypergeometric
functions}.
Let $\alpha'$ denote the Dwork prime, which is defined to be $(\alpha+l)/p$ where $l\in
\{0,1,\ldots,p-1\}$ is the unique integer such that $\alpha+l\equiv0$ mod $p$.
The $i$-th Dwork prime $a^{(i)}$ is defined by $a^{(i+1)}=(a^{(i)})'$ and $a^{(0)}=a$.
Write $\ul a'=(a'_1,\ldots,a'_s)$ and $\ul a^{(i)}=(a^{(i)}_1,\ldots,a^{(i)}_s)$. 
Then Dwork's $p$-adic hypergeometric function is defined
to be
\begin{equation}\label{Dwork}
\cF^{\mathrm{Dw}}_{\ul a}(t)=F_{\ul a}(t)/F_{\ul a'}(t^p).
\end{equation}
This is a convergent function in the sense of Krasner.
More precisely 
Dwork proved the congruence relations (\cite[p.41, Theorem 3]{Dwork-p-cycle})
\begin{equation}\label{Dwork-congruence}
\cF^{\mathrm{Dw}}_{\ul a}(t)
\equiv \frac{[F_{\ul a}(t)]_{<p^n}}{[F_{\ul a'}(t^p)]_{<p^n}}\mod p^n\Z_p[[t]]
\end{equation}
where for a power series $f(t)=\sum c_nt^n$, we write 
$[f(t)]_{<m}:=\sum_{n<m}c_nt^n$ the truncated polynomial.
This implies that 
$\cF^{\mathrm{Dw}}_{\ul a}(t)$ is a convergent function.
More precisely, for $f(t)\in\Z_p[t]$,
let $\ol{f(t)}\in \F_p[t]$ denote the reduction modulo $p$. 
Let $I\subset \Z_{\geq0}$ be a finite subset 
such that $\{\ol{[F_{\ul a^{(i)}}(t)]}_{<p}\}_{i\in \Z_{\geq0}}
=\{\ol{[F_{\ul a^{(i)}}(t)]}_{<p}\}_{i\in I}$
as sets. Put $h(t)=\prod_{i\in I}[F_{\ul a^{(i)}}(t)]_{<p}$.
Then \eqref{Dwork-congruence} implies
\[
\cF^{\mathrm{Dw}}_{\ul a}(t)
\in\Z_p\langle t,h(t)^{-1}\rangle:=\varprojlim_{n\geq1}(\Z_p/p^n\Z_p[t,h(t)^{-1}]),
\]
and hence that $\cF^{\mathrm{Dw}}_{\ul a}(t)$ is a convergent function on a domain
$\{z\in\C_p\mid |h(z)|_p=1\}$.

Dwork showed a geometric aspect of his $p$-adic hypergeometric functions.
Let $E$ be the elliptic curve over $\F_p$ defined by a Weierstrass equation
$y^2=x(1-x)(1- a x)$ with $a\in\F_p\setminus\{0,1\}$. 
Suppose that $E$ is ordinary, which means that $p\not{\hspace{-0.5mm}|}\,a_p$ 
where 
$T^2-a_pT+p$ is the characteristic polynomial of the Frobenius on $E$.
Let $\alpha_E$ be the root of $T^2-a_pT+p$ in $\Z_p$ such that $|\alpha_E|_p=1$,
which is often referred to as the unit root.
Then Dwork proved a formula
\[
\alpha_E=(-1)^{\frac{p-1}{2}}\cF^{\mathrm{Dw}}_{\frac{1}{2},\frac{1}{2}}(\wh a)
\]
where $\wh a\in \Z_p^\times$ is the Teichm\"uller lift of $a\in \F_p^\times$.
This is now called Dwork's unit root formula (cf. \cite[\S 7]{Put})


\medskip

In this paper, we introduce new $p$-adic hypergeometric functions, which we call
the {\it $p$-adic hypergeometric functions of logarithmic type}.
We shall first introduce the $p$-adic polygamma functions $\psi^{(r)}_p(z)$ 
in \S \ref{poly-sect}, which are slight modifications of Diamond's $p$-adic
polygamma functions \cite{Diamond}.
Let $W=W(\ol\F_p)$ be the Witt ring of $\ol\F_p$, and $K=\Frac W$ the fractional field.
Let $\sigma$ be a $p$-th Frobenius on $W[[t]]$
given by $\sigma(t)=ct^p$
with $c\in 1+pW$. Let $\log:\C^\times_p\to\C_p$ be the Iwasawa logarithmic function.
Let
\[
G_{\ul a}(t):=\psi_p(a_1)+\cdots+\psi_p(a_s)+s\gamma_p
-p^{-1}\log(c)+
\int_0^t(F_{\underline{a}}(t)-F_{\ul a'}(t^\sigma))\frac{dt}{t}
\]
be a power series
where 
$\int_0^t(-)\frac{dt}{t}$ means the 
operator which sends $\sum_{n\geq 1}a_nt^n$ to $\sum_{n\geq 1}\frac{a_n}{n}t^n$.
It is not hard to show $G_{\ul a}(t)\in W[[t]]$ (Lemma \ref{G-int}).
Then our new function is defined to be a ratio
\[
\cF^{(\sigma)}_{\underline{a}}(t):=
G_{\ul a}(t)/F_{\ul a}(t).
\]
Notice that $\cF^{(\sigma)}_{\ul a}(t)$ is also $p$-adically continuous with respect to
$\ul a$.
In case $a_1=\cdots=a_s=c=1$, one has $\cF^{(\sigma)}_{\ul a}(t)
=(1-t)\ln^{(p)}_1(t)$ the $p$-adic logarithm. In this sense, we can regard 
$\cF^{(\sigma)}_{\ul a}(t)$ as a deformation of the $p$-adic logarithm.

The first main result of this paper 
is the congruence relations for $\cF^{(\sigma)}_{\ul a}(t)$ that are 
similar to Dwork's. 
\begin{thm}[Theorem \ref{cong-thm}]
Suppose that $a_i\not\in\Z_{\leq 0}$ for all $i$. Then
\[
\cF^{(\sigma)}_{\underline{a}}(t)\equiv \frac{[G_{\ul a}(t)]_{<p^n}}{[F_{\ul a}(t)]_{<p^n}}
\mod p^nW[[t]]
\]
if $c\in 1+2pW$. If $p=2$ and $c\in 1+2W$, then the congruence holds modulo $p^{n-1}$.
\end{thm}
Thanks to this, 
$\cF^{(\sigma)}_{\ul a}(t)$ is a convergent function, 
\[
\cF^{\mathrm{Dw}}_{\ul a}(t)
\in\Z_p\langle t,h(t)^{-1}\rangle
\]
and then the special value at $t=\alpha$ is defined for $\alpha\in\C_p$ such that 
$|h(\alpha)|_p=1$.

\medskip

The second main result 
is to give a geometric aspect of our $\cF^{(\sigma)}_{\ul a}(t)$,
which concerns with the {\it syntomic regulator map}.
Let $X$ be a smooth variety over $W$.
We denote by $H^\bullet_{\syn}(X,\Q_p(j))$ the rigid syntomic cohomology groups by
Besser \cite{Be1} (see also \cite[1B]{NN}), which agree with the syntomic cohomology
groups by Fontaine-Messing \cite{FM} (see also \cite{KaV}) when $X$ is projective. 
Let $K_i(X)$ be Quillen's algebraic $K$-groups.
Then there is the syntomic regulator map
\[
\reg_\syn^{i,j}:K_i(X)\lra H^{2j-i}_\syn(X,\Q_p(j))
\]
for each $i,j\geq0$ (\cite[Theorem 7.5]{Be1}, \cite[Theorem A]{NN}).
We shall concern ourselves with only $\reg_\syn^{2,2}$, which we 
abbreviate to $\reg_\syn$ in this paper.
Note that there is the natural isomorphism
$H^2_\syn(X,\Q_p(2))\cong H^1_\dR(X_K/K)$ where $K=\Frac W$ is the fractional
field and $X_K:=X\times_WK$.
Our second main result is to relate $\cF^{(\sigma)}_{a,b}(t)$ with
the syntomic regulator of a certain element of $K_2$ of a 
{\it hypergeometric curve}, which is defined 
in the following way.
Let $N,M\geq 2$ be integers.
Let $p$ be a prime such that $p\not\hspace{-0.5mm}|NM$.
Let $\alpha\in W$ satisfy that $\alpha\not\equiv 0,1$ mod $p$.
Then we define a hypergeometric curve $X_\alpha$ to
be a projective smooth scheme over $W$ defined by
a bihomogeneous equation 
\[
(X_0^N-X_1^N)(Y_0^M-Y_1^M)=\alpha X^N_0Y_0^M
\]
in $\P^1_W(X_0,X_1)\times\P^1_W(Y_0,Y_1)$ (\S \ref{fermat-sect}).
We put $x=X_1/X_0$ and $y=Y_1/Y_0$.

\begin{thm}[Corollary \ref{main-thm4}]\label{intro-thm1}
Let $p>2$ be a prime such that $p\not|NM$.
Let
\[
\xi=\left\{\frac{x-1}{x-\nu_1},\frac{y-1}{y-\nu_2}\right\}\in K_2(X_\alpha)\ot\Q
\]
for $(\nu_1,\nu_2)\in\mu_N(K)\times\mu_M(K)$ where $\mu_m(K)$ denotes
the group of $m$-th roots of unity in $K$ (cf. \eqref{m-fermat-eq1}).
Let
\[
Q: H^1_\dR(X_{\alpha,K}/K)\ot H^1_\dR(X_{\alpha,K}/K)\lra 
H^2_\dR(X_{\alpha,K}/K)\cong K
\]
be the cup-product pairing.
Suppose that
$h(\alpha)\not\equiv0$ mod $p$ where $h(t)$ is as above.
For a pair of integers $(i,j)$ such that $0<i<N$ and $0<j<M$, we put
$\omega_{i,j}:=Nx^{i-1}y^{j-M}/(1-x^N)dx$ a regular $1$-form (cf. \eqref{fermat-form}),
and
$e_{i,j}^{\text{\rm unit}}$ the unit root vectors which are explicitly given in Theorem \ref{uroot-thm}.
Then we have
\[
Q(\reg_\syn(\xi), e_{N-i,M-j}^{\text{\rm unit}})
=N^{-1}M^{-1}
(1-\nu^{-i}_1)(1-\nu^{-j}_2)
\cF_{a_i,b_j}^{(\sigma_\alpha)}(\alpha)
Q(\omega_{i,j}, e_{N-i,M-j}^{\text{\rm unit}}).
\]
\end{thm}
Similar results hold for other curves (see \S \ref{gauss-sect} -- \S \ref{elliptic-sect}).
A more striking application of our new function $\cF^{(\sigma)}_{\ul a}(t)$
is that one can describe
the syntomic regulators of the Ross symbols in $K_2$ of the Fermat curves.
\begin{thm}[Theorem \ref{fermat-main2}]
Let $F$ be the Fermat curve over $K$ defined by an affine equation $z^N+w^M=1$
with $N|(p-1)$ and $M|(p-1)$.
Let $\{1-z,1-w\}\in K_2(F)$ be the Ross symbol \cite{ross2}.
Let 
\[
\reg_\syn(\{1-z,1-w\})=
\sum_{(i,j)\in I} A_{i,j}M^{-1}z^{i-1}w^{j-M}dz\in H^1_\dR(F/K)
\]
where the notation be as in the beginning of \S \ref{fermatcurve-sect}.
Then \[A_{i,j}=\cF_{a_i,b_j}^{(\sigma)}(1)\] for $(i,j)$ such that $i/N+j/M<1$.
\end{thm}
As long as the author knows, this is the first explicit description of the Ross symbol
in $p$-adic cohomology.


\medskip

We hope that our results will provide
a new direction of the study of the $p$-adic Beilinson conjecture
by Perrin-Riou \cite[4.2.2]{Perrin-Riou} (see also \cite[Conj.2.7]{Colmez}),
especially for $K_2$ of elliptic curves.
Let $E$ be an elliptic curve over $\Q$.
Let $p$ be a prime at which $E$
has a good ordinary reduction.
Let $\alpha_{E,p}$ be the unit root of the reduction $\ol E$ at $p$, 
and $e_\unit\in H^1_\dR(E/\Q_p)$ the eigenvector with eigenvalue $\alpha_{E,p}$. 
Let $\omega_E\in \vg(E,\Omega^1_{E/\Q})$ be a regular $1$-form.
Let $L_p(E,\chi,s)$ be the $p$-adic $L$-function
defined by Mazur and Swinnerton-Dyer \cite{MS}.
Then as a consequence of the $p$-adic Beilinson conjecture 
for elliptic curves,
one can expect
that there is an element $\xi\in K_2(E)$ which is integral in the sense of \cite{Scholl}
such that
\[
(1-p\alpha^{-1}_{E,p})\frac{Q(\reg_\syn(\xi),e_\unit)}{
Q(\omega_E,e_\unit)}\sim_{\Q^\times}L_p(E,\omega^{-1},0)
\]
where $x\sim_{\Q^\times}y$ means $xy\ne0$ and $x/y\in \Q^\times$.
We also refer to \cite[Conjecture 3.3]{ABC} for more precise statement.
According to our main results, we can replace the left hand side with the special values
of the $p$-adic hypergeometric functions of logarithmic type.
For example, let $E_a$ be the elliptic curve defined by $y^2=x(1-x)(1-(1-a)x)$ with $a\in \Q\setminus\{0,1\}$ and $p>3$ a prime where $E_a$ has a good ordinary reduction.
Then one predicts
\[
(1-p\alpha^{-1}_{E_a,p})
\cF_{\frac{1}{2},\frac{1}{2}}^{(\sigma_a)}(a)\sim_{\Q^\times}
L_p(E_a,\omega^{-1},0)
\]
if $a=-1,\pm2,\pm4,\pm8,\pm16,
\pm\frac{1}{2},  
\pm\frac{1}{8},\pm\frac{1}{4}, \pm\frac{1}{16}$ (Conjecture \ref{ell1-conj}).
See \S \ref{RZ-sect} for other cases.
The author has no idea how to attack the question in general, while
we have one example (the proof relies on Brunault's appendix in \cite{ABC}).
\begin{thm}[Theorem \ref{brunault}]\label{F. Brunault}
$(1-p\alpha^{-1}_{E_4,p})\cF_{\frac{1}{2},\frac{1}{2}}^{(\sigma_4)}(4)=-
L_p(E_4,\omega^{-1},0).$
\end{thm}
We note that this is a $p$-adic counterpart of 
a formula of Rogers and Zudilin (\cite[Theorem 2, p.399 and (6), p.386]{RZ})
\[
2L'(E_4,0)=
\mathrm{Re}\left[\log 4-
{}_4F_3\left({\frac{3}{2},\frac{3}{2},1,1\atop 2,2,2};4\right)\right]
\left(={}_3F_2\left({\frac{1}{2},\frac{1}{2},\frac{1}{2}\atop \frac{3}{2},1};
\frac{1}{4}\right)\right).
\]
The conjectures in \S \ref{RZ-sect} give the first formulation of 
the $p$-adic counterparts
of Rogers-Zudilin type formulas.


\medskip

This paper is organized as follows.
\S \ref{poly-sect} is the preliminary section on 
Diamond's $p$-adic polygamma functions.
More precisely we shall give a slight modification of Diamond's polygamma
(though it might be known to the experts).
We give a self-contained exposition, because
the author does not find a suitable reference, especially 
concerning with our modified functions.
In \S \ref{pHGlog-sect}, we introduce the $p$-adic hypergeometric functions of
logarithmic type, and prove the congruence relations.
In \S \ref{reg-sect}, we show that our new $p$-adic hypergeometric functions appear in
the syntomic regulators of the hypergeometric curves.
Finally we discuss the $p$-adic Beilinson conjecture for $K_2$ of elliptic curves
in \S \ref{weakB-sect}.

\medskip

\noindent{\bf Acknowledgement.}
The author would like to express sincere gratitude
to Professor Masataka Chida for the stimulating discussion on 
the $p$-adic Beilinson conjecture. The discussion with him is the origin of this work.
He very much appreciate Professor Fran\c{c}ois Brunault 
for giving him lots of comments on his paper \cite{regexp}, and for the help of the proof
of Theorem \ref{F. Brunault}.

\medskip

\noindent{\bf Notation.}
We denote by $\mu_n(K)$ the group of $n$-th roots of unity in a field $K$. We write $\mu_\infty(K)=\cup_{n\geq 1}\mu_n(K)$.
If there is no fear of confusion, we drop ``$K$'' and simply write $\mu_n$.
For a power series $f(t)=\sum_{i=0}^\infty a_it^i\in R[[t]]$ with coefficients in a 
commutative ring $R$, 
we denote
$f(t)_{<n}:=\sum_{i=0}^{n-1}a_it^i$ the truncated polynomial.

\section{$p$-adic polygamma functions}\label{poly-sect}
The complex analytic polygamma functions are the $r$-th derivative
\[
\psi^{(r)}(z):=\frac{d^r}{dz^r}\left(\frac{\Gamma'(z)}{\Gamma(z)}\right), \quad r\in\Z_{\geq0}.
\]
In his paper \cite{Diamond}, Jack Diamond gave a $p$-adic counterpart of the polygamma functions $\psi^{(r)}_{D,p}(z)$ which are given in the following way.
\begin{equation}\label{diamond0}
\psi^{(0)}_{D,p}(z)
=\lim_{s\to\infty}\frac{1}{p^s}\sum_{n=0}^{p^s-1}\log(z+n),
\end{equation}
\begin{equation}\label{diamond1}
\psi^{(r)}_{D,p}(z)=(-1)^{r+1}r!\lim_{s\to\infty}\frac{1}{p^s}\sum_{n=0}^{p^s-1}\frac{1}{(z+n)^r},
\quad r\geq 1,
\end{equation}
where $\log:\C_p^\times\to\C_p$ is the Iwasawa logarithmic function which is characterized as a continuous 
homomorphism satisfying
$\log(p)=0$ and
\[
\log(z)=-\sum_{n=1}^\infty\frac{(1-z)^n}{n},\quad |z-1|_p<1.
\]
It should be noticed that the series \eqref{diamond0} and \eqref{diamond1}
converge only when $z\not\in \Z_p$, and hence $\psi^{(r)}_{D,p}(z)$
turn out to be locally analytic functions on $\C_p\setminus\Z_p$.
This causes inconvenience in our discussion.
In this section we give a continuous function $\psi_p^{(r)}(z)$ on $\Z_p$
which is a slight {\it modification} of $\psi^{(r)}_{D,p}(z)$.
See \S \ref{polygamma-sect} for the definition and also \S \ref{measure-sect} 
for an alternative definition in terms of $p$-adic measure.

\subsection{$p$-adic polylogarithmic functions}
Let $x$ be an indeterminate.
For an integer $r\in\Z$, the $r$-th $p$-adic polylogarithmic function $\ln_r^{(p)}(x)$
is defined as a formal power series
\[
\ln_r^{(p)}(x):=\sum_{k\geq1,p\not{\hspace{0.7mm}|}\,k}\frac{x^k}{k^r}=\lim_{s\to\infty}
\left(\frac{1}{1-x^{p^s}}\sum_{1\leq k<p^s,\, p\not{\hspace{0.7mm}|}\,k}\frac{x^k}{k^r}\right)
\in \Z_p[[x]]
\]
which belongs to the ring 
\[
\Z_p\langle x,(1-x)^{-1}\rangle:=\varprojlim_{s}(\Z/p^s\Z[x,(1-x)^{-1}])
\]
of convergent power series.
If $r\leq 0$, this is a rational function, more precisely
\[
\ln_0^{(p)}(x)=\frac{1}{1-x}-\frac{1}{1-x^p},\quad 
\ln_{-r}^{(p)}(x)=\left(x\frac{d}{dx}\right)^r\ln_0^{(p)}(x).
\]
If $r>0$, this is known to be an {\it overconvergent function}, more precisely
it has a (unique) analytic continuation to the domain 
$|x-1|>|1-\zeta_p|$ where $\zeta_p\in \ol\Q_p$ 
is a primitive $p$-th root of unity.

Let $W(\ol\F_p)$ be the Witt ring of $\ol\F_p$ and $F$ the $p$-th Frobenius endomorphism.
Define the {\it $p$-adic logarithmic function}
\begin{equation}\label{log(p)}
\log^{(p)}(z):=\frac{1}{p}\log\left(\frac{z^p}{F(z)}\right):=-\sum_{n=1}^\infty
\frac{p^{-1}}{n}\left(1-\frac{z^p}{F(z)}\right)^n
\end{equation}
on $W(\ol\F_p)^\times$.
This is different from the Iwasawa $\log(z)$ in general, but
one can show 
$\log^{(p)}(1-z)=-\ln^{(p)}_1(z)$ for $z\in W(\ol\F_p)^\times$ such that $F(z)=z^p$
and $z\not\equiv 1$ mod $p$.
\begin{prop}[cf. \cite{C-dlog} IV Prop.6.1, 6.2]
Let $r\in\Z$ be an integer. Then
\begin{equation}\label{polylog-diffeq}
\ln^{(p)}_r(x)=x\frac{d}{dx}\ln^{(p)}_{r+1}(x),
\end{equation}
\begin{equation}\label{polylog-refl}
\ln^{(p)}_r(x)=(-1)^{r+1}\ln^{(p)}_r(x^{-1}),
\end{equation}
\begin{equation}\label{polylog-dstr}
\sum_{\zeta\in\mu_N}\ln_r^{(p)}(\zeta x)=\frac{1}{N^{r-1}}\ln^{(p)}_r(x^N)
\quad \mbox{\rm(distribution formula)}.
\end{equation}
\end{prop}
\begin{pf}
\eqref{polylog-diffeq} and \eqref{polylog-dstr} are immediate from the power series
expansion $\ln_r^{(p)}(x)=\sum_{k\geq 1,p\not{\hspace{0.7mm}|}\,k} x^k/k^r$.
On the other hand \eqref{polylog-refl} follows from the fact
\[
\frac{1}{1-x^{-p^s}}\sum_{1\leq k<p^s,\, p\not{\hspace{0.7mm}|}\,k}\frac{x^{-k}}{k^r}
=
\frac{-1}{1-x^{p^s}}\sum_{1\leq k<p^s,\, p\not{\hspace{0.7mm}|}\,k}\frac{x^{p^s-k}}{k^r}
\equiv
\frac{(-1)^{r+1}}{1-x^{p^s}}\sum_{1\leq k<p^s,\, p\not{\hspace{0.7mm}|}\,k}
\frac{x^{p^s-k}}{(p^s-k)^r}
\]
modulo $p^s\Z[x,(1-x)^{-1}]$.
\end{pf}
\begin{lem}\label{ln-formula1}
Let $m, N\geq 2$ be integers prime to $p$.
Let $\ve\in\mu_m\setminus\{1\}$.
Then for any $n\in \{0,1,\ldots,N-1\}$, we have
\[
N^r\sum_{\nu^N=\ve}\nu^{-n}\ln^{(p)}_{r+1}(\nu)=\lim_{s\to\infty}
\frac{1}{1-\ve^{p^s}}\sum_{\us{k+n/N\not\equiv0\text{ \rm mod }p}{0\leq k<p^s}}
\frac{\ve^k}{(k+n/N)^{r+1}}.
\]  
\end{lem}
\begin{pf}
Note $\sum_{\nu^N=\ve}\nu^i=N\ve^{i/N}$ if $N|i$ and $=0$ otherwise.
We have 
\begin{align*}
N^r\sum_{\nu^N=\ve}\nu^{-n}\ln^{(p)}_{r+1}(\nu x)
&=N^r\sum_{k\geq1,p\not{\hspace{0.7mm}|}\,k}\sum_{\nu^N=\ve}\frac{\nu^{k-n}x^k}{k^{r+1}}\\
&=N^{r+1}\sum_{N|(k-n),p\not{\hspace{0.7mm}|}\,k}\frac{\ve^{(k-n)/N}x^k}{k^{r+1}}\\
&=\sum_{n+\ell N\not\equiv 0\text{ mod }p,\,\ell\geq 0}\frac{(\ve x^N)^\ell x^n}{(\ell+n/N)^{r+1}}
\quad(\ell=(k-n)/N)\\
&\equiv \frac{1}{1-(\ve x^N)^{p^s}}\sum_{\us{n+\ell N\not\equiv 0
\text{ mod }p}{0\leq \ell<p^s}}\frac{(\ve x^N)^\ell x^n}{(\ell+n/N)^{r+1}}\\
\end{align*}
modulo $p^s\Z[x,(1-\ve x^N)^{-1}]$.
Since $\ve\not\equiv 1$ mod $p$, the evaluation at $x=1$ makes sense, and then we have the desired equation.
\end{pf}
\begin{lem}\label{p-zeta-def}
Let $r\ne1$ be an integer.
Then for any integer $N\geq 2$ prime to $p$,
\begin{equation}\label{p-zeta-def-eq1}
\sum_{\ve\in \mu_N\setminus\{1\}}\ln_r^{(p)}(\ve)
=-(1-N^{1-r})L_p(r,\omega^{1-r})
\end{equation}
where $L_p(s,\chi)$ is the $p$-adic $L$-function 
and $\omega$ is the Teichm\"uller character.
\end{lem}
\begin{pf}
This
is well-known to experts as {\it Coleman's formula}.
We give a self-contained and straightforward proof for convenience of the reader,
because the author does not find a suitable literature (note that \eqref{p-zeta-def-eq1} 
is not included in \cite[I, (3)]{C-dlog}).

\medskip


We first show \eqref{p-zeta-def-eq1} in case $r=-m$ with $m\in \Z_{\geq1}$.
Note that $\ln_{-m}^{(p)}(x)$ is a rational function.
More precisely let
\[
\ln_0(x):=\frac{x}{1-x},\quad 
\ln_{-m}(x):=\left(x\frac{d}{dx}\right)^m\ln_0(x),
\]
then $\ln_{-m}^{(p)}(x)=\ln_{-m}(x)-p^m\ln_{-m}(x^p)$.
Therefore
\[
\sum_{\ve\in \mu_N\setminus\{1\}}\ln_{-m}^{(p)}(\ve)
=(1-p^{m})
\sum_{\ve\in \mu_N\setminus\{1\}}\ln_{-m}(\ve).
\]
Since $L_p(-m,\omega^{1+m})=-(1-p^{m})B_{m+1}/(m+1)$ 
where $B_n$ are the Bernoulli numbers,
\eqref{p-zeta-def-eq1} for $r=-m$ is equivalent to
\begin{equation}\label{p-zeta-def-eq4}
(1-N^{m+1})\frac{B_{m+1}}{m+1}=
\sum_{\ve\in \mu_N\setminus\{1\}}\ln_{-m}(\ve).
\end{equation}
Put $\ell_r(x):=\ln_r(x)-N^{1-r}\ln_r(x^N)$.
By the distribution property
\[
\sum_{\ve\in \mu_N}\ln_r(\ve x)=
N^{1-r}\ln_r(x^N)
\]
which can be easily shown by a computation of power series expansions,
the right hand side of \eqref{p-zeta-def-eq4} equals to
the evaluation $-\ell_{-m}(x)|_{x=1}$ at $x=1$, and hence
\begin{equation}\label{p-zeta-def-eq5}
\sum_{\ve\in \mu_N\setminus\{1\}}\ln_{-m}(\ve)
=-\left(x\frac{d}{dx}\right)^m\ell_0(x)\bigg|_{x=1}.
\end{equation}
On the other hand, letting $x=e^z$, one has
\begin{align*}
\ell_0(e^z)=\frac{e^z}{1-e^z}-\frac{Ne^{Nz}}{1-e^{Nz}}
=-\sum_{n=1}^\infty \left(B_n\frac{z^{n-1}}{n!}-B_n\frac{N^nz^{n-1}}{n!}
\right)
=-\sum_{n=1}^\infty (1-N^n)B_n\frac{z^{n-1}}{n!}
\end{align*}
and hence
\begin{equation}\label{p-zeta-def-eq6}
\left(x\frac{d}{dx}\right)^m\ell_0(x)\bigg|_{x=1}=
\frac{d^m}{dz^m}\ell_0(e^z)\bigg|_{z=0}=
-(1-N^{m+1})\frac{B_{m+1}}{m+1}.
\end{equation}
Now \eqref{p-zeta-def-eq4} follows from \eqref{p-zeta-def-eq5} and
\eqref{p-zeta-def-eq6}.

\medskip

We have shown \eqref{p-zeta-def-eq1} for negative $r$.
Let $r\ne1$ be an arbitrary integer.
Since $\ln_r^{(p)}(x)=\sum_{p\not{\hspace{0.7mm}|}\, k}x^k/k^r$,
one has that
for any integers $r,r'$ such that $r\equiv r'$ mod $(p-1)p^{s-1}$,
\[\ln_r^{(p)}(x)\equiv \ln_{r'}^{(p)}(x)
\mod p^s\Z_p[[x]]
\]
and hence modulo $p^s\Z_p\langle x,(1-x)^{-1}\rangle$.
This implies
\[\ln_r^{(p)}(\ve)\equiv \ln_{r'}^{(p)}(\ve)
\mod p^sW(\ol\F_p).
\]
Take $r'=r-p^{s+a-1}(p-1)$ with $a\gg 0$.
It follows that
\[
(1-N^{1-r'})L_p(r',\omega^{1-r'})
=(1-N^{1-r'})L_p(r',\omega^{1-r})\to
(1-N^{1-r})L_p(r,\omega^{1-r})
\]
as $a\to\infty$ by the continuity of the $p$-adic $L$-functions.
Since $r'<0$, one can apply \eqref{p-zeta-def-eq1} and then 
\[
-(1-N^{1-r})L_p(r,\omega^{1-r})
\equiv \sum_{\ve\in \mu_N\setminus\{1\}}\ln_r(\ve)\mod p^sW(\ol\F_p)
\]
for any $s>0$. This completes the proof.

\end{pf}

\subsection{$p$-adic polygamma functions}\label{polygamma-sect}
Let $r\in\Z$ be an integer. For $z\in \Z_p$, define
\begin{equation}\label{wt-polygamma-def}
\wt{\psi}_p^{(r)}(z):=\lim_{n\in\Z_{>0},n\to z}\sum_{1\leq k<n,p\not{\hspace{0.7mm}|}\, k}
\frac{1}{k^{r+1}}.
\end{equation}
The existence of the limit follows from the fact that
\begin{equation}\label{equiv}
\sum_{1\leq k<p^s,p\not{\hspace{0.7mm}|}\,k}k^m\equiv 
\begin{cases}
-p^{s-1}&p\geq 3\mbox{ and }(p-1)|m\\
2^{s-1}&p=2\mbox{ and }2 |m\\
1&p=2\mbox{ and }s=1\\
0&\mbox{otherwise}
\end{cases}
\end{equation}
modulo $p^s$.
Thus $\wt\psi^{(r)}_p(z)$ is a $p$-adic continuous function on $\Z_p$.
More precisely
\begin{equation}\label{equiv-psi}
z\equiv z'\hspace{-0.3cm}\mod p^s\Longrightarro
\wt\psi^{(r)}_p(z)-\wt\psi^{(r)}_p(z') \equiv
\begin{cases}
0\hspace{-0.3cm}\mod p^s&p\geq 3\mbox{ and }(p-1)\not|(r+1)\\
0\hspace{-0.3cm}\mod p^s&p=2,\, s\geq 2\mbox{ and }2\not|(r+1)\\
0\hspace{-0.3cm}\mod p^{s-1}&\mbox{othewise.}
\end{cases}
\end{equation}
Define the {\it $p$-adic Euler constant} \footnote{This is different from
Diamond's $p$-adic Euler constant. His constant
is $p/(p-1)\gamma_p$, \cite[\S 7]{Diamond}.} by
\[
\gamma_p:=-\lim_{s\to\infty}\frac{1}{p^s}\sum_{0\leq j<p^s,p\not{\hspace{0.7mm}|}\,j}\log(j),\quad
(\log=\mbox{Iwasawa log}).
\]
where the convergence follows by
\begin{align}
\sum_{0\leq j<p^{s+1},p\not{\hspace{0.7mm}|}\,j}\log(j)-p
\sum_{0\leq j<p^s,p\not{\hspace{0.7mm}|}\,j}\log(j)
&=\sum_{k=0}^{p-1}\sum_{0\leq j<p^s,p\not{\hspace{0.7mm}|}\,j}
\log\left(1+\frac{kp^s}{j}\right)\notag\\
&\equiv\sum_{k=0}^{p-1}\sum_{0\leq j<p^s,p\not{\hspace{0.7mm}|}\,j}
\frac{kp^s}{j}\mod p^{2s-1}\notag\\
&\os{\eqref{equiv}}{\equiv} 0\mod p^{2s-1}.\label{equiv-log}
\end{align}
We define the $r$-th {\it $p$-adic polygamma function} to be
\begin{equation}\label{polygamma-def}
\psi_p^{(r)}(z):=\begin{cases}
-\gamma_p+\wt{\psi}^{(0)}_p(z)&r=0\\
-L_p(1+r,\omega^{-r})+\wt{\psi}^{(r)}_p(z)&r\ne0.
\end{cases}
\end{equation}
If $r=0$, we also write $\psi_p(z)=\psi_p^{(0)}(z)$ and call it the {\it $p$-adic digamma function}.
\subsection{Formulas on $p$-adic polygamma functions}\label{formula-sect}
\begin{thm}\label{polygamma-thm1}
\begin{enumerate}
\item[\rm(1)] $\wt\psi^{(r)}_p(0)=\wt\psi^{(r)}_p(1)=0$ or equivalently
$\psi^{(r)}_p(0)=\psi^{(r)}_p(1)=-\gamma_p$ or $=-\zeta_p(r+1)$.
\item[\rm(2)]
$\wt\psi^{(r)}_p(z)=(-1)^r\wt\psi^{(r)}_p(1-z)$ or equivalently
$\psi^{(r)}_p(z)=(-1)^r\psi^{(r)}_p(1-z)$ (note $L_p(1+r,\omega^{-r})=0$ for odd $r$).
\item[\rm(3)]
\[
\wt\psi^{(r)}_p(z+1)-\wt\psi^{(r)}_p(z)=\psi^{(r)}_p(z+1)-\psi^{(r)}_p(z)=\begin{cases}
z^{-r-1}&z\in \Z_p^\times\\
0&z\in p\Z_p.
\end{cases}
\]
\end{enumerate}
\end{thm}
Compare the above with \cite{NIST} p.144, 5.15.2, 5.15.5 and 5.15.6.
\begin{pf}
(1) and (3) are immediate from definition on noting \eqref{equiv}.
We show (2). 
Since $\Z_{>0}$ is a dense subset in $\Z_p$, it is enough to show in case $z=n>0$ an integer.
Let $s>0$ be arbitrary such that $p^s>n$.
Then
\begin{align*}
\wt\psi^{(r)}_p(n)&\equiv
\sum_{1\leq k<n,p\not{\hspace{0.7mm}|}\,k}\frac{1}{k^{r+1}}\equiv 
(-1)^{r+1}\sum_{-n< k\leq -1,p\not{\hspace{0.7mm}|}\,k}\frac{1}{k^{r+1}}
\equiv 
(-1)^{r+1}\sum_{p^s-n+1\leq k< p^s,p\not{\hspace{0.7mm}|}\,k}\frac{1}{k^{r+1}}\\
&\equiv 
(-1)^{r+1}\sum_{0\leq k< p^s,p\not{\hspace{0.7mm}|}\,k}\frac{1}{k^{r+1}}-
(-1)^{r+1}\sum_{0\leq k< p^s-n+1,p\not{\hspace{0.7mm}|}\,k}\frac{1}{k^{r+1}}\\
&\equiv(-1)^{r}\sum_{0\leq k< p^s-n+1,p\not{\hspace{0.7mm}|}\,k}\frac{1}{k^{r+1}}\\
&\equiv (-1)^r\wt\psi^{(r)}_p(1-n)
\end{align*}
modulo $p^s$ or $p^{s-1}$. Since $s$ is an arbitrary large integer, this means 
$\wt\psi^{(r)}_p(n)=(-1)^r\wt\psi^{(r)}_p(1-n)$ as required.
\end{pf}
\begin{thm}\label{polygamma-thm2}
Let $0\leq n<N$ be integers and suppose $p\not|N$. Then
\begin{equation}\label{polygamma-thm2-eq}
\wt{\psi}_p^{(r)}\left(\frac{n}{N}\right)=N^r
\sum_{\ve\in \mu_N\setminus\{1\}}(1-\ve^{-n})\ln_{r+1}^{(p)}(\ve).
\end{equation}
\end{thm}
For example 
\[
{\psi}_p^{(r)}\left(\frac{1}{2}\right)=
-L_p(1+r,\omega^{-r})+2^{r+1}\ln_{r+1}^{(p)}(-1)=(1-2^{r+1})L_p(1+r,\omega^{-r}).
\]
Compare this with \cite{NIST} p.144, 5.15.3.
\begin{pf}
We may assume $n>0$.
Let $s>0$ be an integer such that $p^s\equiv 1$ mod $N$.
Write $p^s-1=lN$.
\begin{align*}
S:=\sum_{\ve\in \mu_N\setminus\{1\}}(1-\ve^{-n})\ln_{r+1}^{(p)}(\ve)
&\equiv
\sum_{1\leq k<p^s,\, p\not{\hspace{0.7mm}|}\,k}\left(
\sum_{\ve\in \mu_N\setminus\{1\}}\frac{1-\ve^{-n}}{1-\ve^{p^s}}\frac{\ve^k}{k^{r+1}}\right)
\\
&\equiv
\sum_{1\leq k<p^s,\, p\not{\hspace{0.7mm}|}\,k}\left(
\sum_{\ve\in \mu_N\setminus\{1\}}\frac{\ve^k+\cdots+\ve^{k+N-n-1}}{k^{r+1}}\right)
\end{align*}
modulo $p^s$.
Note $\sum_{\ve\in \mu_N\setminus\{1\}}\ve^i=N-1$ if $N|i$ and $=-1$ otherwise.
By \eqref{equiv}, we have
\[
S\equiv 
\sum_{k}
\frac{N}{k^{r+1}}\mod p^{s-1}
\]
where $k$ runs over the integers such that
$0\leq k<p^s$, $p\not{\hspace{-0.4mm}|}\,k$ and there is an integer $0\leq  i<N-n$ such 
that
$k+i\equiv 0$ mod $N$.
Hence 
\begin{align*}
&N^rS
\equiv 
\sum_{k}\frac{1}{(k/N)^{r+1}}=
\sum_{k\equiv 0\text{ mod } N}
+\sum_{k\equiv -1\text{ mod } N}+\cdots+\sum_{k\equiv n-N+1\text{ mod }N}\\
&=
\sum_{\us{j\not\equiv 0\text{ mod }p}{1\leq j<p^s/N}}\frac{1}{j^{r+1}}
+\sum_{\us{j-1/N\not\equiv 0\text{ mod }p}{1\leq j<(p^s+1)/N}}\frac{1}{(j-1/N)^{r+1}}
+\cdots+
\sum_{\us{j-(N-n-1)/N\not\equiv 0\text{ mod }p}{1\leq j<(p^s+N-n-1)/N}}
\frac{1}{(j-(N-n-1)/N)^{r+1}}\\
&\equiv
\sum_{\us{j\not\equiv 0\text{ mod }p}{1\leq j\leq l}}\frac{1}{j^{r+1}}
+\sum_{\us{j+l\not\equiv 0\text{ mod }p}{1\leq j\leq l}}\frac{1}{(j+l)^{r+1}}
+\cdots+
\sum_{\us{j+l(N-n-1)\not\equiv 0\text{ mod }p}{1\leq j\leq l}}\frac{1}{(j+l(N-n-1))^{r+1}}\\
&=\sum_{\us{j\not\equiv 0\text{ mod }p}{1\leq j\leq l(N-n)}}\frac{1}{j^{r+1}}
=\sum_{\us{j\not\equiv 0\text{ mod }p}{0\leq j<l(N-n)+1}}\frac{1}{j^{r+1}}.
\end{align*}
Since $l(N-n)+1\equiv n/N$ mod $p^s$, the last summation is equivalent to $\wt\psi^{(r)}(n/N)$
modulo $p^{s-1}$ by definition. 
\end{pf}
\begin{rem}\label{definition-rem}
The complex analytic analogy of 
Theorem \ref{polygamma-thm2} is the following.
Let $\ln_r(z)=\ln^{an}_{r}(z)=\sum_{n=1}^\infty z^n/n^r$ be the analytic polylog.
Then
\begin{align*}
N^r\sum_{k=1}^{N-1}(1-e^{-2\pi ikn/N})\ln_{r+1}(e^{2\pi ik/N})&=
\sum_{m=1}^\infty\sum_{k=1}^{N-1}\frac{N^r}{m^{r+1}}(e^{2\pi ikm/N}-e^{2\pi ik(m-n)/N})\\
&=
\sum_{k=1}^\infty\frac{N^{r+1}}{(kN)^{r+1}}-\frac{N^{r+1}}{(kN-N+n)^{r+1}}\\
&=
\sum_{k=1}^\infty\frac{1}{k^{r+1}}-\frac{1}{(k-1+n/N)^{r+1}}.
\end{align*}
If $r=0$, then this is equal to $\psi(z)-\psi(1)$ (\cite{NIST} p.139, 5.7.6).
If $r\geq 1$, then this is equal to $\zeta(r+1)+(-1)^r/r!\psi^{(r)}(n/N)$
(\cite{NIST} p.144, 5.15.1).
\end{rem}
\begin{thm}\label{polygamma-thm4}
Let $m\geq 1$ be an positive integer prime to $p$. 
\begin{enumerate}
\item[\rm(1)]
Let $\psi_p(z)=\psi_p^{(0)}(z)$ be the $p$-adic digamma function.
Then
\[
\psi_p(mz)-\log^{(p)}(m)=\frac{1}{m}\sum_{i=0}^{m-1}\psi_p(z+\frac{i}{m}),
\]
(see \eqref{log(p)} for the definition of $\log^{(p)}(z)$).
\item[\rm(2)] If $r\ne 0$, we have
\[
\psi^{(r)}_p(mz)=\frac{1}{m^{r+1}}\sum_{i=0}^{m-1}\psi^{(r)}_p(z+\frac{i}{m}).
\]
\end{enumerate}
\end{thm}
Compare the above with \cite{NIST} p.144, 5.15.7.
\begin{pf}
By Lemma \ref{p-zeta-def}, the assertions are equivalent to
\begin{equation}\label{polygamma-thm4-eq}
\frac{1}{m^{r+1}}\sum_{i=0}^{m-1}\wt\psi^{(r)}_p(z+\frac{i}{m})
=\wt\psi^{(r)}_p(mz)
+\sum_{\ve\in\mu_N\setminus\{1\}}\ln^{(p)}_{r+1}(\ve)
\end{equation}
for all $r\in\Z$.
Since $\Z_{(p)}\cap[0,1)$ is a dense subset in $\Z_p$, it is enough to show the above
in case $z=n/N$ with $0\leq n<N$, $p\not|N$. 
By Theorem \ref{polygamma-thm2},
\begin{align*}
\frac{1}{m^{r+1}}\sum_{i=0}^{m-1}\wt\psi^{(r)}_p(z+\frac{i}{m})
&=\frac{1}{m^{r+1}}\sum_{i=0}^{m-1}\wt\psi^{(r)}_p(\frac{nm+iN}{mN})\\
&=\frac{N^r}{m}\sum_{i=0}^{m-1}\sum_{\nu\in\mu_{mN}\setminus\{1\}}
(1-\nu^{-nm-iN})\ln^{(p)}_{r+1}(\nu).
\end{align*}
The last summation is divided into the following two terms
\[
\sum_{i=0}^{m-1}
\sum_{\nu\in\mu_{N}\setminus\{1\}}
(1-\nu^{-nm})\ln^{(p)}_{r+1}(\nu)=
m\sum_{\nu\in\mu_{N}\setminus\{1\}}
(1-\nu^{-nm})\ln^{(p)}_{r+1}(\nu),
\]
\begin{align*}
\sum_{i=0}^{m-1}\sum_{\ve\in\mu_{m}\setminus\{1\}}\sum_{\nu^N=\ve}
(1-\nu^{-nm}\ve^{-i})\ln^{(p)}_{r+1}(\nu)
&=m\sum_{\ve\in\mu_{m}\setminus\{1\}}\sum_{\nu^N=\ve}
\ln^{(p)}_{r+1}(\nu)\\
&=\frac{m}{N^r}\sum_{\ve\in\mu_{m}\setminus\{1\}}
\ln^{(p)}_{r+1}(\ve)
\end{align*}
where the last equality follows from the distribution formula \eqref{polylog-dstr}.
Since the former is equal to $\wt\psi^{(r)}_p(nm/N)$ by Theorem \ref{polygamma-thm2},
the equality \eqref{polygamma-thm4-eq} follows.
\end{pf}
\subsection{$p$-adic measure}\label{measure-sect}
For a function $g:\Z_p\to \C_p$, the Volkenborn integral is defined by
\[
\int_{\Z_p}g(t)dt=\lim_{s\to\infty}\frac{1}{p^s}\sum_{0\leq j<p^s}g(j)
\]
if the limit exists.
\begin{thm}\label{meas-thm2}
Let $\log:\C_p^\times\to \C_p$ be the Iwasawa logarithmic function.
Let 
\[
{\mathbf 1}_{\Z_p^\times}(z):=\begin{cases}
1&z\in\Z_p^\times\\
0&z\in p\Z_p
\end{cases}
\]
be the characteristic function.
Then
\[
\psi_p(z)=\int_{\Z_p}\log(z+t){\mathbf 1}_{\Z_p^\times}(z+t)dt.
\]
\end{thm}
\begin{pf}
Using \eqref{equiv-log}, one can easily show that
the Volkenborn integral 
$Q(z):=\int_{\Z_p}{\mathbf 1}_{\Z_p^\times}(z+t)\log(z+t)dt$ is defined. 
Moreover we have
\[
Q(z+1)-Q(z)\equiv\begin{cases}
p^{-s}(\log(z)-\log(z+p^s))& z\in \Z_p^\times\\
0&z\in p\Z_p
\end{cases}\mod p^{s'}
\]
where $s'=s-1$ if $p=2$ and $s'=s$ if $p\geq 3$.
For $z\in \Z_p^\times$, since
\[
p^{-s}(\log(z)-\log(z+p^s))=-p^{-s}\log(1+z^{-1}p^s)
\equiv z^{-1}\mod p^{s'},
\]
it follows from Theorem \ref{polygamma-thm1} (3) that
 $Q(z)$ differs from $\psi_p(z)$ by a constant.
Since
\[
Q(0)=\lim_{s\to\infty} \frac{1}{p^s}\sum_{0\leq j<p^s,p\not{\hspace{0.7mm}|}\,j}\log(j)= -\gamma_p=\psi_p(0),
\]
we obtain $Q(z)=\psi_p(z)$.
\end{pf}

\begin{thm}\label{meas-thm1}
If $r\ne0$, then
\[
\psi^{(r)}_p(z)=-\frac{1}{r}\int_{\Z_p}(z+t)^{-r}{\mathbf 1}_{\Z_p^\times}(z+t)dt
\]
where ${\mathbf 1}_{\Z_p^\times}(z)$ denotes the characteristic function as in Theorem \ref{meas-thm2}.
\end{thm}
\begin{pf}
Using \eqref{equiv}, one sees that
the Volkenborn integral 
$Q(z)=-\frac{1}{r}\int_{\Z_p}(z+t)^{-r}{\mathbf 1}_{\Z_p^\times}(z+t)dt$ is defined.
Moreover 
if $z\in\Z_p^\times$, then
\[
Q(z+1)-Q(z)\equiv\frac{-1}{rp^s}\left(
\frac{1}{(z+p^s)^r}-\frac{1}{z^r}\right)\equiv z^{-1-r}\mod p^{s-\ord_p(r)},
\]
and if $z\in p\Z_p$, then $Q(z+1)\equiv Q(z)$.
This shows that $Q(z)-\psi^{(r)}_p(z)$ is a constant by Theorem \ref{polygamma-thm1} (3). 
We show $Q(0)=\psi^{(r)}_p(0)$.
By definition 
\[
Q(0)= \lim_{n\to\infty}\frac{-1}{rp^n}\sum_{0\leq k<p^n,p\not{\hspace{0.7mm}|}\,k}
\frac{1}{k^r}.
\]
Recall the original definition of
the $p$-adic $L$-function by Kubota-Leopoldt \cite{KL}
\[
L_p(s,\chi)=\frac{1}{s-1}\lim_{n\to\infty}\frac{1}{fp^n}
\sum_{0\leq k<fp^{n},p\not{\hspace{0.7mm}|}\,k}\chi(k)\langle k\rangle^{1-s},
\quad \langle k\rangle:=k/\omega(k)
\]
for a primitive Dirichlet character $\chi$ with conductor $f\geq1$.
This immediately implies $Q(0)=-L_p(1+r,\omega^{-r})=\psi^{(r)}_p(0)$.
\end{pf}
\section{$p$-adic hypergeometric functions of logarithmic type}\label{pHGlog-sect}
For an integer $n\geq 0$, we denote by $(a)_n$ the Pochhammer symbol,
\[
(a)_0:=1,\quad (a)_n:=a(a+1)\cdots(a+n-1),\,n\geq 1.
\]
For $a\in \Z_p$, we denote by $a':=(a+l)/p$ the {\it Dwork prime} 
where $l\in \{0,1,\ldots,p-1\}$ is the unique integer such that $a+l\equiv 0$ mod $p$.
We denote the $i$-th Dwork prime by 
$a^{(i)}$ which is defined to be $(a^{(i-1)})'$ with $a^{(0)}=a$.
\subsection{Definition}\label{pHGlog-defn}
Let $s\geq 1$ be a positive integer.
Let $a_i,b_j\in\Q_p$ with $b_j\not\in \Z_{\leq0}$. Let
\[
{}_sF_{s-1}\left({a_1,\ldots,a_s\atop b_1,\ldots b_{s-1}}:t\right)
=\sum_{n=0}^\infty\frac{(a_1)_n\cdots(a_s)_n}{(b_1)_n\cdots(b_{s-1})_n}\frac{t^n}{n!}.
\]
be the {\it hypergeometric power series} with coefficients.
In what follows we only consider the cases $a_i\in \Z_p$ and $b_j=1$, and then 
we write
\[
F_{\ul a}(t):={}_sF_{s-1}\left({a_1,\ldots,a_s\atop 1,\ldots 1}:t\right)\in \Z_p[[t]]
\]
for 
$\ul a=(a_1,\ldots,a_s)\in \Z_p^s$.
\begin{defn}[$p$-adic hypergeometric functions of logarithmic type]
Let $\ul a'=(a'_1,\ldots,a'_s)$ where $a_i'$ denotes the Dwork prime.
Let $W=W(\ol\F_p)$ be the Witt ring of $\ol\F_p$.
Let $\sigma:W[[t]]\to W[[t]]$ be the $p$-th Frobenius endomorphism given by  
$\sigma(t)=ct^p$ with $c\in 1+pW$,
compatible with the Frobenius on $W$.
Put a power series
\[
G_{\ul a}(t):=\psi_p(a_1)+\cdots+\psi_p(a_s)+s\gamma_p
-p^{-1}\log(c)+
\int_0^t(F_{\underline{a}}(t)-F_{\ul a'}(t^\sigma))\frac{dt}{t}
\]
where $\psi_p(z)$ is the $p$-adic digamma function defined in \S \ref{polygamma-sect},
and $\log(z)$ is the Iwasawa logarithmic function.
Then we define
\[
\cF^{(\sigma)}_{\underline{a}}(t)=G_{\ul a}(t)/F_{\underline{a}}(t),
\]
and call the {\rm $p$-adic hypergeometric functions of logarithmic type}.
\end{defn}
\begin{lem}\label{G-int}
$G_{\ul a}(t)\in W[[t]]$. Hence it follows $\cF^{(\sigma)}_{\underline{a}}(t)\in W[[t]]$.
\end{lem}
\begin{pf}
Let $G_{\underline{a}}(t)=\sum B_it^i$.
Let $F_{\underline{a}}(t)=\sum A_it^i$ and
$F_{\ul a'}(t)=\sum A^{(1)}_it^i$.
If $p{\not|}i$, then
$B_i=A_i/i$ is obviously a $p$-adic integer.
For $i=mp^k$ with $k\geq 1$ and $p\not| m$, one has
\[
B_i=B_{mp^k}=\frac{A_{mp^k}-c^{mp^{k-1}}A^{(1)}_{mp^{k-1}}}{mp^k}.
\]
Since $c^{mp^{k-1}}\equiv 1$ mod $p^k$, it is enough to see 
$A_{mp^k}\equiv A^{(1)}_{mp^{k-1}}$ mod $p^k$. 
However this follows from \cite[p.36, Cor. 1]{Dwork-p-cycle}. 
\end{pf}
\subsection{Congruence relations}\label{cong-pf-sect1}
For a power series $f(t)=\sum_{n=0}^\infty A_nt^n$, we denote $f(t)_{<m}:=\sum_{n<m}A_nt^n$
the truncated polynomial.
\begin{thm}\label{cong-thm}
Suppose that $a_i\not\in \Z_{\leq0}$ for all $i$.
Let us write $\cF^{(\sigma)}_{\underline{a}}(t)=G_{\ul a}(t)/F_{\ul a}(t)$.
If $c\in 1+2pW$, then for all $n\geq 1$
\begin{equation}\label{cong-thm-eq0}
\cF^{(\sigma)}_{\underline{a}}(t)\equiv
\frac{G_{\underline{a}}(t)_{<p^n}}{F_{\underline{a}}(t)_{<p^n}}\mod p^nW[[t]].
\end{equation}
If $p=2$ and $c\in 1+2W$ (not necessarily $c\in 1+4W$), 
then the above holds modulo $p^{n-1}$.
\end{thm}
\begin{cor}\label{cong-cor}
Suppose that there exists an integer $r\geq 0$ such that $a_i^{(r+1)}=a_i$ for all $i$
where $(-)^{(r)}$ denotes the $r$-th Dwork prime.
Then
\[
\cF_{\ul a}^{(\sigma)}(t)\in 
W\langle t,F_{\ul a}(t)_{<p}^{-1}, \ldots,F_{\ul a^{(r)}}(t)_{<p}^{-1}\rangle:=
\varprojlim_n (W/p^n[t,F_{\ul a}(t)_{<p}^{-1}, \ldots,F_{\ul a^{(r)}}(t)_{<p}^{-1}])
\]
is a convergent function.
For $\alpha\in W$ such that $F_{\ul a^{(i)}}(\alpha)_{<p}\not
\equiv 0$ mod $p$ for all $i$, the special value of $\cF_{\ul a}^{(\sigma)}(t)$ at $t=\alpha$
is defined, and it is explicitly given by
\[
\cF_{\ul a}^{(\sigma)}(\alpha)=\lim_{n\to\infty}
\frac{G_{\underline{a}}(\alpha)_{<p^n}}{F_{\underline{a}}(\alpha)_{<p^n}}.
\]
\end{cor}
\subsection{Proof of Congruence relations : Reduction to the case $c=1$}\label{cong-sect2}
Throughout the sections \ref{cong-sect2}, \ref{cong-sect3} and \ref{cong-sect4},
we use the following notation. Fix $s\geq 1$ and $\ul a=(a_1,\ldots,a_s)$ with $a_i\not\in \Z_{\leq0}$. Let $\sigma(t)=ct^p$ be the Frobenius. Put
\begin{equation}\label{cong-A}
F_{\ul a}^{(i)}(t):=\sum_{n=0}^\infty A^{(i)}_nt^n,\quad 
A^{(i)}_n:=\frac{(a_1^{(i)})_n}{n!}\cdots\frac{(a_s^{(i)})_n}{n!}
\end{equation}
where $a_k^{(i)}$ denotes the $i$-th Dwork prime.
Letting $\cF^{(\sigma)}_{\underline{a}}(t)=G_{\ul a}(t)/F_{\ul a}(t)$, we put
\[
G_{\ul a}(t)=\sum_{n=0}^\infty B_nt^n
\]
or explicitly 
\begin{equation}\label{cong-sect1-eq1-0}
B_0=\psi_p(a_1)+\cdots+\psi_p(a_s)+s\gamma_p,
\end{equation}
\begin{equation}\label{cong-sect1-eq1}
B_n=\frac{A_n}{n},\,(p\not|n),\quad
B_{mp^k}=\frac{A_{mp^k}-c^{mp^{k-1}}A^{(1)}_{mp^{k-1}}}{mp^k},\,(m,k\geq1).
\end{equation}

\begin{lem}\label{cong-lem0}
The proof of Theorem \ref{cong-thm} is reduced to the case $\sigma(t)=t^p$ (i.e. $c=1$).
\end{lem}
\begin{pf}
Write $f(t)_{\geq m}:=f(t)-f(t)_{<m}$. 
Put $n^*:=n$ if $c\in 1+2pW$ and $n^*=n-1$ if $p=2$ and $c\not
\in 1+4W$.
Theorem \ref{cong-thm}
is equivalent to saying
\[
F_{\ul a}(t)G_{\ul a}(t)_{\geq p^n}
\equiv F_{\ul a}(t)_{\geq p^n}G_{\ul a}(t)\mod p^{n^*}W[[t]],
\]
namely
\[
\sum_{i+j=m,\,i,j\geq0}A_{i+p^n}B_j-A_iB_{j+p^n}\equiv 0\mod p^{n^*}
\]
for all $m\geq 0$.
Suppose that this is true when $c=1$, namely
\begin{equation}\label{cong-lem0-eq1}
\sum_{i+j=m}A_{i+p^n}B^\circ_j-A_iB^\circ_{j+p^n}\equiv 0\mod p^{n^*}
\end{equation}
where $B^\circ_i$ are the coefficients \eqref{cong-sect1-eq1-0}
or \eqref{cong-sect1-eq1} when $c=1$.
We denote by $B_i$ the coefficients for an arbitrary $c\in 1+pW$.
We then want to show
\begin{equation}\label{cong-lem0-eq2}
\sum_{i+j=m}A_{i+p^n}(B^\circ_j-B_j)-A_i(B^\circ_{j+p^n}-B_{j+p^n})\equiv 0\mod p^{n^*}.
\end{equation}
Let $c=1+pe$ with $e\ne0$ (if $e=0$, there is nothing to prove). 
Then
\begin{align*}
\sum_{i+j=m}A_{i+p^n}(B^\circ_j-B_j)
&=A_{m+p^n}p^{-1}\log(c)+\sum_{1\leq j\leq m} p^{-1}\frac{(c^{j/p}-1)A_{m+p^n-j}A^{(1)}_{j/p}}{j/p}\\
&=A_{m+p^n}\sum_{i=1}^\infty\frac{(-1)^{i+1}}{i}p^{i-1}e^i+\sum_{1\leq j\leq m}(j/p)^{-1}
\sum_{i=1}^\infty\binom{j/p}{i}p^{i-1}e^iA_{m+p^n-j}A^{(1)}_{j/p}\\
&=\sum_{i=1}^\infty\left(A_{m+p^n}\frac{(-1)^{i+1}}{i}+\sum_{1\leq j\leq m}(j/p)^{-1}
\binom{j/p}{i}A_{m+p^n-j}A^{(1)}_{j/p}\right)p^{i-1}e^i\\
&=\sum_{i=1}^\infty\left(A_{m+p^n}\frac{(-1)^{i+1}}{i}+\sum_{1\leq j\leq m}i^{-1}
\binom{j/p-1}{i-1}A_{m+p^n-j}A^{(1)}_{j/p}\right)p^{i-1}e^i\\
&=\sum_{i=1}^\infty\left(\sum_{0\leq j\leq m}i^{-1}
\binom{j/p-1}{i-1}A_{m+p^n-j}A^{(1)}_{j/p}\right)p^{i-1}e^i
\end{align*}
where $A^{(k)}_{j/p}:=0$ if $p{\not|}j$.
Similarly
\[
\sum_{i+j=m}A_i(B^\circ_{j+p^n}-B_{j+p^n})
=\sum_{i=1}^\infty\left(\sum_{0\leq j\leq m}i^{-1}
\binom{(m+p^n-j)/p-1}{i-1}A_{j}A^{(1)}_{(m+p^n-j)/p}\right)p^{i-1}e^i.
\]
Therefore it is enough to show that
\[
\frac{p^{i-1}e^i}{i}\sum_{0\leq j\leq m}
\binom{j/p-1}{i-1}A_{m+p^n-j}A^{(1)}_{j/p}
\equiv\frac{p^{i-1}e^i}{i}\sum_{0\leq j\leq m}
\binom{(m+p^n-j)/p-1}{i-1}A_{j}A^{(1)}_{(m+p^n-j)/p}\mod p^{n^*}
\]
equivalently
\begin{equation}\label{cong-lem0-eq3}
\sum_{0\leq j\leq m}
(1-j/p)_{i-1}
A_{m+p^n-j}A^{(1)}_{j/p}
\equiv\sum_{0\leq j\leq m}
(1-(m+p^n-j)/p)_{i-1}A_{j}A^{(1)}_{(m+p^n-j)/p}\mod p^{n^*-i+1}i!e^{-i}
\end{equation}
for all $i\geq 1$ and $m\geq 0$. 
Recall the Dwork congruence 
\[
\frac{F(t^p)}{F(t)}\equiv
\frac{[F(t^p)]_{<p^m}}{F(t)_{<p^m}}\mod p^l\Z_p[[t]],\quad m\geq l
\]
from \cite[p.37, Thm. 2, p.45]{Dwork-p-cycle}.
This immediately imples \eqref{cong-lem0-eq3} in case $i=1$.
Suppose $i\geq 2$.
To show \eqref{cong-lem0-eq3}, it is enough to show 
\begin{equation}\label{cong-lem0-eq4}
\sum_{0\leq j\leq m}
(j/p)^k
A_{m+p^n-j}A^{(1)}_{j/p}
\equiv\sum_{0\leq j\leq m}
((m+p^n-j)/p)^kA_{j}A^{(1)}_{(m+p^n-j)/p}\mod p^{n^*-i+1}i!e^{-i}
\end{equation}
for each $k\geq 0$.
We write $A^*_j:=j^kA^{(1)}_j$, and put $F^*(t):=\sum_{j=0}^\infty A^*_jt^j$.
Then \eqref{cong-lem0-eq4} is equivalent to saying 
\begin{equation}\label{cong-lem0-eq5}
F(t)_{<p^n}F^*(t^p)\equiv
F(t)[F^*(t^p)]_{<p^n}\mod p^{n^*-i+1}i!e^{-i}\Z_p[[t]].
\end{equation}
We show \eqref{cong-lem0-eq5}, which finishes the proof of Lemma \ref{cong-lem0}.
It follows from \cite[p.45, Lem. 3.4 ]{Dwork-p-cycle}
that we have
\[
\frac{F^*(t)}{F(t)}\equiv
\frac{F^*(t)_{<p^m}}{F(t)_{<p^m}}\mod p^l\Z_p[[t]],\quad m\geq l.
\]
This implies
\[
\frac{F^*(t^p)}{F(t^p)}\equiv
\frac{F^*(t^p)_{<p^n}}{[F(t^p)]_{<p^n}}\mod p^{n-1}\Z_p[[t]].
\]
Therefore we have
\[
\frac{F^*(t^p)}{F(t)}=\frac{F(t^p)}{F(t)}\frac{F^*(t^p)}{F(t^p)}\equiv
\frac{[F(t^p)]_{<p^n}}{F(t)_{<p^n}}
\frac{[F^*(t^p)]_{<p^n}}{F(t^p)_{<p^n}}=\frac{[F^*(t^p)]_{<p^n}}{F(t)_{<p^n}}
\mod p^{n-1}\Z_p[[t]].
\]
If $p\geq 3$, then $\ord_p(p^{n^*-i+1}i!)=\ord_p(p^{n-i+1}i!)\leq n-1$ for any $i\geq2$, and hence
\eqref{cong-lem0-eq5} follows.
If $p=2$, then $\ord_p(p^{n-i+1}i!)\leq n$ but not necessarily $\ord_p(p^{n-i+1}i!)\leq n-1$.
If $e\in 2W\setminus\{0\}$, then $\ord_p(p^{n^*-i+1}i!e^{-i})=\ord_p(p^{n-i+1}i!e^{-i})\leq n-i<n-1$,
and hence
\eqref{cong-lem0-eq5} follows.
If $e$ is a unit, then 
$\ord_p(p^{n^*-i+1}i!e^{-i})=\ord_p(p^{n-i}i!)\leq n-1$ for any $i\geq2$, as required again.
This completes the proof.
\end{pf}
\subsection{Proof of Congruence relations : Preliminary lemmas}\label{cong-sect3}
Until the end of \S \ref{cong-sect4}, let $\sigma$ be the Frobenius given by $\sigma(t)=t^p$
(i.e. $c=1$).
Therefore
\begin{equation}\label{cong-B}
B_0=\psi_p(a_1)+\cdots+\psi_p(a_s)+s\gamma_p,\quad
B_i=\frac{A_i-A^{(1)}_{i/p}}{i},\quad i\in\Z_{\geq1}
\end{equation}
where $A^{(k)}_i$ are as in \eqref{cong-A}, and
we mean $A^{(k)}_{i/p}=0$ unless $i/p\in\Z_{\geq0}$.
\begin{lem}\label{prod-lem}
For an $p$-adic integer $\alpha\in \Z_p$ and $n\in\Z_{\geq1}$, we define 
\[
\{\alpha\}_n:=\prod_{\underset{p\not{\hspace{0.7mm}|}\,(a+i-1)}{1\leq i\leq n}}(\alpha+i-1),
\]
and $\{\alpha\}_0:=1$.
Let $a\in\Z_p\setminus\Z_{\leq 0}$, and 
$l\in \{0,1,\ldots,p-1\}$ the integer such that $a\equiv -l$ mod $p$.
Then for any $m\in \Z_{\geq0}$, we have
\[
m\equiv 0,1,\ldots,l\text{ \rm mod }p\quad\Longrightarrow\quad
\frac{(a)_m}{m!}\left(\frac{(a')_{\lfloor m/p\rfloor}}
{\lfloor m/p\rfloor !}\right)^{-1}=\frac{\{a\}_{m}}{\{1\}_{m}}\in\Z_p^\times,
\]
\[
m\equiv l+1,\ldots,p-1\text{ \rm mod }p\quad\Longrightarrow\quad
\frac{(a)_m}{m!}\left(\frac{(a')_{\lfloor m/p\rfloor}}
{\lfloor m/p\rfloor !}\right)^{-1}=\left(a+l+p\lfloor\frac{m}{p}\rfloor\right)
\frac{\{a\}_m}{\{1\}_m}
\]
where $a'=a^{(1)}$ is the Dwork prime.
\end{lem}
\begin{pf}
Straightforward.
\end{pf}
\begin{lem}[Dwork]\label{cong-lem5}
For any $m\in\Z_{\geq0}$, $A_m/A^{(1)}_{\lfloor m/p\rfloor}$ are $p$-adic integers, and
\[
m\equiv m'\mod p^n\quad\Longrightarrow\quad
\frac{A_m}{A^{(1)}_{\lfloor m/p\rfloor}}\equiv
\frac{A_{m'}}{A^{(1)}_{\lfloor m'/p\rfloor}}\mod p^n.
\]
\end{lem}
\begin{pf}
This is \cite[p.36, Cor. 1]{Dwork-p-cycle}, or one can easily show this
by using Lemma \ref{prod-lem} on noticing the fact that
\[
\{\alpha\}_{p^n}
\equiv \prod_{i\in (\Z/p^n\Z)^\times}i
\equiv\begin{cases}
1&p=2,\,n\ne 2\\
-1&\text{otherwise}
\end{cases}
\mod p^n.
\]
\end{pf}

\begin{lem}\label{b0-lem}
Let $a\in\Z_p\setminus\Z_{\leq 0}$ and $m,n\in \Z_{\geq1}$. Then
\begin{equation}\label{b0-lem-eq1}
1-\frac{(a')_{mp^{n-1}}}{(mp^{n-1})!}
\left(\frac{(a)_{mp^n}}{(mp^n)!}\right)^{-1}\equiv mp^n
(\psi_p(a)+\gamma_p)\mod p^{2n}.
\end{equation}
Moreover $A^{(1)}_{mp^{n-1}}/A_{mp^n}$
and $B_k/A_k$ 
are $p$-adic integers for all $k,m\geq 0$, $n\geq 1$, and
\begin{equation}\label{b0-lem-eq2}
\frac{A^{(1)}_{mp^{n-1}}}{A_{mp^n}}\equiv1-mp^n(\psi_p(a_1)+\cdots+\psi_p(a_s)+s\gamma_p)
\mod p^{2n},
\end{equation}
\begin{equation}\label{b0-lem-eq3}
p\not|m\quad \Longrightarrow\quad\frac{B_{mp^n}}{A_{mp^n}}=
\frac{1-A^{(1)}_{mp^{n-1}}/A_{mp^n}}{mp^n}\equiv B_0\mod p^n.
\end{equation}
\end{lem}
\begin{pf}
We already see that $A^{(1)}_{mp^{n-1}}/A_{mp^n}\in \Z_p$ in Lemma \ref{prod-lem}.
It is enough to show \eqref{b0-lem-eq1}.
Indeed \eqref{b0-lem-eq2} is immediate from \eqref{b0-lem-eq1}, and
\eqref{b0-lem-eq3} is immediate from \eqref{b0-lem-eq2}.
Moreover \eqref{b0-lem-eq2} also implies that $B_k/A_k\in \Z_p$ for any $k\in\Z_{\geq0}$.

Let us show \eqref{b0-lem-eq1}.
Let $a=-l+p^nb$ with $l\in \{0,1,\ldots,p^n-1\}$.
Then
\[
\frac{(a')_{mp^{n-1}}}{(mp^{n-1})!}
\left(\frac{(a)_{mp^n}}{(mp^n)!}\right)^{-1}
=\frac{\{1\}_{mp^n}}{\{a\}_{mp^n}}=\prod_{\underset{k-l\not\equiv0\text{ mod }p}{l<k<mp^n}} \frac{k-l}{k-l+p^nb}
\times\prod_{\underset{k-l\not\equiv0\text{ mod }p}{0\leq k<l}} \frac{k-l+mp^n}{k-l+p^nb}
\]
by Lemma \ref{prod-lem}.
If $(p,n)\ne(2,1)$, we have
\begin{align*}
\frac{\{1\}_{mp^n}}{\{a\}_{mp^n}}
&\equiv
\prod_{\underset{p\not{\hspace{0.7mm}|}\,k-l}{l<k<mp^n}}\left( 1-\frac{p^nb}{k-l}\right)
\prod_{\underset{p\not{\hspace{0.7mm}|}\,k-l}{0\leq k<l}}\left( 1-\frac{p^n(b-m)}{k-l}\right)\\
&\equiv
1-p^n\left(\sum_{\underset{p\not{\hspace{0.7mm}|}\,k-l}{l<k<mp^n}}\frac{b}{k-l}
+\sum_{\underset{p\not{\hspace{0.7mm}|}\,k-l}{0\leq k<l}}\frac{b-m}{k-l}\right)\\
&\os{\eqref{equiv}}{\equiv}
1-mp^n\sum_{\underset{p\not{\hspace{0.7mm}|}\,k-l}{l<k<mp^n}}\frac{1}{k-l}\\
&=
1-mp^n\sum_{1\leq k<mp^n-l,p \not{\hspace{0.7mm}|}\,k}\frac{1}{k}\\
&\os{\eqref{equiv-psi}}{\equiv}1-mp^n(\psi_p(a)+\gamma_p)
\end{align*}
modulo $p^{2n}$, which completes the proof of \eqref{b0-lem-eq1}.
In case $(p,n)=(2,1)$, we need another observation
(since the 3rd equivalence does not hold in general).
In this case
we have
\begin{align*}
\frac{\{1\}_{2m}}{\{a\}_{2m}}
&\equiv
1-2
\left(\sum_{\underset{2\not{\hspace{0.7mm}|}\, k-l}{l<k<2m}}\frac{b}{k-l}
+\sum_{\underset{2\not{\hspace{0.7mm}|}\,k-l}{0\leq k<l}}\frac{b-m}{k-l}\right)\mod 4\\
&=
1-2
\left(\sum_{\underset{2\not{\hspace{0.7mm}|}\, k-l}{l<k<2m}}\frac{m}{k-l}
+\sum_{\underset{2\not{\hspace{0.7mm}|}\,k-l}{0\leq k<2m}}\frac{b-m}{k-l}\right)\\
&
\equiv
1-2m
\left(\sum_{0<k<2m-l,\,2\not{\hspace{0.7mm}|}\,k}\frac{1}{k}
+b-m\right)\mod 4
\end{align*}
and
\[
\psi_2(a)+\gamma_2\equiv \sum_{0<k<L,\,2 \not{\hspace{0.7mm}|}\,k}\frac{1}{k}\mod 2
\]
where $L\in\{0,1,2,3\}$ such that $a=-l+2b\equiv L$ mod $4$.
Therefore \eqref{b0-lem-eq1} is equivalent to 
\[
m
\left(\sum_{0<k<2m-l,\,2 \not{\hspace{0.7mm}|}\,k}\frac{1}{k}
-\sum_{0<k<L,\,2 \not{\hspace{0.7mm}|}\,k}\frac{1}{k}
+b-m\right)\equiv 0\mod 2.
\]
We may assume that $m>0$ is odd and $b=0,1$ (hence $a=0,\pm 1,2$). 
Then one can check this
on a case-by-case analysis.
\end{pf}
\begin{lem}\label{cong-lem3}
For any $m, m'\in\Z_{\geq 0}$ and $n\in \Z_{\geq 1}$, we have \[
m\equiv m'\mod p^n\quad\Longrightarrow\quad
\frac{B_m}{A_m}\equiv
\frac{B_{m'}}{A_{m'}}\mod p^n.
\]
\end{lem}
\begin{pf}
If $p{\not|} m$, then $B_m/A_m=1/m$ and hence the assertion is obvious.
Let $m=kp^i$ with $i\geq 1$ and $p{\not|}k$. 
It is enough to show the assertion in case $m'=m+p^n$.
If $n\leq i$, then
\[
\frac{B_m}{A_m}\equiv \frac{B_{m'}}{A_{m'}}\equiv B_0\mod p^n
\]
by \eqref{b0-lem-eq3}. Suppose $n\geq i$.
Notice that
\[
1-m\frac{B_m}{A_m}=\frac{A^{(1)}_{m/p}}{A_m}=\prod_{r=1}^s \frac{\{1\}_m}{\{a_r\}_m}
\]
by \eqref{cong-B} and Lemma \ref{prod-lem}.
We have
\begin{align*}
1-m'\frac{B_{m'}}{A_{m'}}
&=\prod_r \frac{\{1\}_{kp^i+p^n}}{\{a_r\}_{kp^i+p^n}}\\
&=\prod_r \frac{\{1\}_{kp^i}}{\{a_r\}_{kp^i}}\frac{\{1+kp^i\}_{p^n}}{\{a_r+kp^i\}_{p^n}}\\
&=\left(1-m\frac{B_m}{A_m}\right)\prod_r 
\frac{\{1+kp^i\}_{p^n}}{\{a_r+kp^i\}_{p^n}}\\
&=\left(1-m\frac{B_m}{A_m}\right)\prod_r 
\frac{\{1\}_{p^n}}{\{a_r+kp^i\}_{p^n}}
\frac{\{1+kp^i\}_{p^n}}{\{1\}_{p^n}}\\
&\os{(\ast)}{\equiv}\left(1-m\frac{B_m}{A_m}\right)\prod_r 
(1-p^n(\psi_p(a_r+kp^i)-\psi_p(1+kp^i)))\mod p^{2n}\\
&\os{(\ast\ast)}{\equiv}\left(1-m\frac{B_m}{A_m}\right)
(1-p^nB_0)\mod p^{n+i}
\end{align*}
where 
$(\ast)$ follows from Lemmas \ref{prod-lem} and \ref{b0-lem}.
The equivalence 
$(\ast\ast)$ follows from \eqref{equiv-psi} in case $(p,i)\ne(2,1)$, and in case $(p,i)=(2,1)$,
it does from the fact that
\[
\psi_2(z+2)-\psi_2(z)\equiv1\mod 2,\quad z\in \Z_2.
\]
Therefore we have
\[
kp^i\left(\frac{B_{m'}}{A_{m'}}-\frac{B_{m}}{A_{m}}\right)\equiv -p^n\frac{B_{m'}}{A_{m'}}+p^nB_0
\mod p^{i+n}.
\]
By \eqref{b0-lem-eq3}, the right hand side vanishes.
This is the desired assertion.
\end{pf}
\begin{lem}\label{cong-lem4}
Put $S_m:=\sum_{i+j=m}A_{i+p^n}B_{j}-A_iB_{j+p^n}$ for $m\in \Z_{\geq0}$.
Then \[
S_m\equiv\sum_{i+j=m}(A_{i+p^n}A_{j}-A_iA_{j+p^n})\frac{B_{j}}{A_{j}}\mod p^n.
\]
\end{lem}
\begin{pf}
\begin{align*}
S_m
&=\sum_{i+j=m}A_{i+p^n}B_{j}-A_iA_{j+p^n}\frac{B_{j+p^n}}{A_{j+p^n}}\\
&\equiv\sum_{i+j=m}A_{i+p^n}B_{j}-A_iA_{j+p^n}\frac{B_{j}}{A_{j}}\mod p^n\quad(\mbox{Lemma \ref{cong-lem3}})\\
&=\sum_{i+j=m}(A_{i+p^n}A_{j}-A_iA_{j+p^n})\frac{B_{j}}{A_{j}}
\end{align*}
as required.
\end{pf}
\begin{lem}\label{cong-lem6}
\[
S_m\equiv\sum_{i+j=m}
(A^{(1)}_{\lfloor j/p\rfloor}A^{(1)}_{\lfloor i/p\rfloor+p^{n-1}}
-A^{(1)}_{\lfloor i/p\rfloor}A^{(1)}_{\lfloor j/p\rfloor+p^{n-1}}
)\frac{A_i}{A^{(1)}_{\lfloor i/p\rfloor}}\frac{A_j}{A^{(1)}_{\lfloor j/p\rfloor}}
\frac{B_j}{A_j}\mod p^n.
\]
\end{lem}
\begin{pf}
This follows from Lemma \ref{cong-lem4} and Lemma \ref{cong-lem5}.
\end{pf}


\begin{lem}\label{cong-lem7}
For all $m,k,s\in \Z_{\geq0}$ and $0\leq l\leq n$, 
we have
\begin{equation}\label{cong-lem7-eq1}
\sum_{\underset{i\equiv k\text{ mod }p^{n-l}}{i+j=m}}A_iA_{j+p^{n-1}}-A_jA_{i+p^{n-1}}
\equiv0
\mod p^l.
\end{equation}
\end{lem}
\begin{pf}
There is nothing to prove in case $l=0$. If $l=n$, then \eqref{cong-lem7-eq1} is obvious as
\[
\mbox{LHS}=\sum_{i+j=m}A_iA_{j+p^{n-1}}-A_jA_{i+p^{n-1}}=0.
\]
Suppose that $1\leq l\leq n-1$.
Let $A^{(r)}_i$ be as in \eqref{cong-A}.
For $r,k\in \Z_{\geq0}$ we put
\[
F^{(r)}(t):=\sum_{i=0}^\infty A^{(r)}_it^i,
\]
\begin{equation}\label{cong-lem7-eq3}
F^{(r)}_k(t):=\sum_{i\equiv k\text{ mod }p^{n-l}} A^{(r)}_it^i
=p^{-n+l}\sum_{s=0}^{p^{n-l}-1}\zeta^{-sk}F(\zeta^st)
\end{equation}
where $\zeta$ is a primitive $p^{n-l}$-th root of unity.
Then \eqref{cong-lem7-eq1} is equivalent to
\begin{equation}\label{cong-lem7-eq2}
F_k(t)F_{m-k}(t)_{<p^{n-1}}\equiv F_k(t)_{<p^{n-1}}F_{m-k}(t) \mod p^l
\end{equation}
where $F_k(t)=F^{(0)}_k(t)$.
It follows from the Dwork congruence
\cite[p.37, Thm. 2]{Dwork-p-cycle} that one has
\[
\frac{F^{(i)}(t)}{F^{(i+1)}(t^p)}\equiv 
\frac{F^{(i)}(t)_{<p^m}}{[F^{(i+1)}(t^p)]_{<p^m}}\mod p^n
\]
for any $m\geq n\geq1$.
This implies
\[
\frac{F^{(i)}(t^p)}{F^{(i+1)}(t^{p^2})}\equiv 
\frac{F^{(i)}(t^p)_{<p^{n+1}}}{[F^{(i+1)}(t^{p^2})]_{<p^{n+1}}}\mod p^n,\quad
\frac{F^{(i)}(t^{p^2})}{F^{(i+1)}(t^{p^3})}\equiv 
\frac{F^{(i)}(t^{p^2})_{<p^{n+2}}}{[F^{(i+1)}(t^{p^3})]_{<p^{n+2}}}\mod p^n,\ldots.
\]
Hence we have 
\begin{align*}
\frac{F(t)}{F^{(n-l)}(t^{p^{n-l}})}
&=\frac{F(t)}{F^{(1)}(t^p)}
\frac{F^{(1)}(t^p)}{F^{(2)}(t^{p^2})}\cdots
\frac{F^{(n-l-1)}(t^{p^{n-l-1}})}{F^{(n-l)}(t^{p^{n-l}})}\\
&\equiv\frac{[F(t)]_{<p^d}}{[F^{(1)}(t^p)]_{<p^d}}\frac{[F(t^p)]_{<p^d}}{[F^{(1)}(t^{p^2})]_{<p^d}}\cdots
\frac{[F^{(n-l-1)}(t^{p^{n-l-1}})]_{<p^d}}{[F^{(n-l)}(t^{p^{n-l}})]_{<p^d}}
\mod p^{d-n+l+1} \Z_p[[t]]\\
&=\frac{[F(t)]_{<p^d}}{[F^{(n-l)}(t^{p^{n-l}})]_{<p^d}},
\end{align*}
namely there are $a_i\in \Z_p$ such that
\[
\frac{F(t)}{F^{(n-l)}(t^{p^{n-l}})}=
\frac{F(t)_{<p^d}}{[F^{(n-l)}(t^{p^{n-l}})]_{<p^d}}
+p^{d-n+l+1}\sum_{i}a_it^i.
\]
Substitute $t$ for $\zeta^s t$ in the above and 
multiply it by
\[
\left(\frac{F(t)}{F^{(n-l)}(t^{p^{n-l}})}\right)^{-1}=
\left(\frac{F(t)_{<p^d}}{[F^{(n-l)}(t^{p^{n-l}})]_{<p^d}}
+p^{d-n+l+1}\sum_{i}a_it^i\right)^{-1}.
\]
Then we have
\[
F(\zeta^st)F(t)_{<p^d}-F(\zeta^st)_{<p^d}F(t)=p^{d-n+l+1}\sum_{i=0}^\infty b_i(\zeta^s)t^i
\]
where $b_i(x)\in \Z_p[x]$ are polynomials which do not depend on $s$.
Applying $\sum_{s=0}^{p^{n-l}-1}\zeta^{-sk}(-)$ on both side,
one has
\[
p^{n-l}F_k(t)F(t)_{<p^d}-p^{n-l}F_k(t)_{<p^d}F(t)=p^{d-n+l+1}\sum_{i=0}^\infty 
\sum_{s=0}^{p^{n-l}-1}\zeta^{-sk}b_i(\zeta^s)t^i
\]
by \eqref{cong-lem7-eq3}. Since $\sum_{s=0}^{p^{n-l}-1}\zeta^{sj}=0$ or $p^{n-l}$,
the right hand side is zero modulo $p^{d+1}$.
Therefore
\[
\frac{F_k(t)}{F(t)}\equiv 
\frac{F_k(t)_{<p^d}}{F(t)_{<p^d}}
\mod p^{d-n+l+1}\Z_p[[t]].
\]
This implies
\[
\frac{F_k(t)F_j(t)_{<p^d}-F_k(t)_{<p^d}F_j(t)}{F(t)}\equiv
\frac{F_k(t)_{<p^d}F_j(t)_{<p^d}-F_k(t)_{<p^d}F_j(t)_{<p^d}}{F(t)_{<p^d}}=0\mod p^{d-n+l+1}.
\]
Now \eqref{cong-lem7-eq2} is the case $(d,j)=(n-1,s-k)$.
\end{pf}

\subsection{Proof of Congruence relations : End of proof}\label{cong-sect4}
We finish the proof of Theorem \ref{cong-thm}.
Let $S_m$ be as in Lemma \ref{cong-lem4}.
The goal is to show
\[
S_m\equiv 0\mod p^n,\quad \forall\, m\geq 0.
\]
Let us put
\[
q_i:=\frac{A_i}{A^{(1)}_{\lfloor i/p\rfloor}},\quad
A(i,j):=A^{(1)}_iA^{(1)}_j,\quad A^*(i,j):=A(j,i+p^{n-1})-A(i,j+p^{n-1})
\]
\[
B(i,j):=A^*(\lfloor i/p\rfloor,\lfloor j/p\rfloor).
\]
Then 
\[
S_m\equiv\sum_{i+j=m}B(i,j)q_iq_j\frac{B_j}{A_j}\mod p^n
\]
by Lemma \ref{cong-lem6}.
It follows from Lemma \ref{cong-lem3} and Lemma \ref{cong-lem5} that we have
\begin{equation}\label{pf-eq1}
k\equiv k'\mod p^i\quad\Longrightarrow\quad \frac{B_k}{A_k}\equiv\frac{B_{k'}}{A_{k'}},
\,q_k\equiv q_{k'}
\mod p^{i}.
\end{equation}
By Lemma \ref{cong-lem7}, we have
\begin{equation}\label{pf-eq2}
\sum_{\underset{i\equiv k\text{ mod }p^{n-l}}{i+j=s}}A^*(i,j)\equiv0 \mod p^l,\quad 0\leq l\leq n
\end{equation}
for all $s\geq0$.
Let
$m=l+sp$ with $l\in\{0,1,\ldots,p-1\}$.
Note
\[
B(i,m-i)=\begin{cases}
A^*(k,s-k)&kp\leq i\leq kp+l\\
A^*(k,s-k-1)&kp+l< i\leq (k+1)p-1.
\end{cases}
\]
Therefore
\begin{align*}
S_{m}&\equiv \sum_{i+j=m}B(i,j)q_iq_j\frac{B_j}{A_j}\mod p^n\\
&=
\sum_{i=0}^{p-1}\sum_{k=0}^{\lfloor(m-i)/p\rfloor}B(i+kp,m-(i+kp))q_{i+kp}q_{m-(i+kp)}
\frac{B_{m-(i+kp)}}{A_{m-(i+kp)}}
\\
&=
\sum_{k=0}^s
B(i+kp,m-(i+kp))
\sum_{i=0}^{l}q_{i+kp}q_{m-(i+kp)}
\frac{B_{m-(i+kp)}}{A_{m-(i+kp)}}
\\
&\quad+\sum_{k=0}^{s-1
B(i+kp,m-(i+kp))
\sum_{i=l+1}^{p-1}q_{i+kp}q_{m-(i+kp)}
\frac{B_{m-(i+kp)}}{A_{m-(i+kp)}}
\\
&=
\sum_{k=0}^s
A^*(k,s-k)
\overbrace{\left(\sum_{i=0}^{l}q_{i+kp}q_{m-(i+kp)}
\frac{B_{m-(i+kp)}}{A_{m-(i+kp)}}\right)}^{P_k}
\\
&\quad+\sum_{k=0}^{s-1}
A^*(k,s-k-1)
\underbrace{\left(\sum_{i=l+1}^{p-1}q_{i+kp}q_{m-(i+kp)}
\frac{B_{m-(i+kp)}}{A_{m-(i+kp)}}\right)}_{Q_k}.
\end{align*}
We show that the first term vanishes modulo $p^n$.
It follows from \eqref{pf-eq1} that we have
\begin{equation}\label{pf-eq3}
k\equiv k'\mod p^i\quad\Longrightarrow\quad 
P_k\equiv P_{k'}
\mod p^{i+1}.
\end{equation}
Therefore one can write
\[
\sum_{k=0}^{s}A^*(k,s-k)P_k\equiv\sum_{i=0}^{p^{n-1}-1}P_i
\overbrace{\left(\sum_{k\equiv i\text{ mod } p^{n-1}}A^*(k,s-k)
\right)}^{(\ast)}\mod p^n.
\]
It follows from \eqref{pf-eq2} that $(\ast)$ is zero modulo $p$.
Therefore, again by \eqref{pf-eq3}, one can rewrite
\[
\sum_{k=0}^{s}A^*(k,s-k)P_k\equiv\sum_{i=0}^{p^{n-2}-1}P_i
\overbrace{\left(\sum_{k\equiv i\text{ mod } p^{n-2}}A^*(k,s-k)
\right)}^{(\ast\ast)}\mod p^n.
\]
It follows from \eqref{pf-eq2} that $(\ast\ast)$ is zero modulo $p^2$, so that one has
\[
\sum_{k=0}^{s}A^*(k,s-k)P_k\equiv\sum_{i=0}^{p^{n-3}-1}P_i
\left(\sum_{k\equiv i\text{ mod } p^{n-3}}A^*(k,s-k)
\right)\mod p^n
\]
by \eqref{pf-eq3}.
Continuing the same discussion, one finally obtains
\[
\sum_{k=0}^{s}A^*(k,s-k)P_k\equiv
\sum_{k=0}^sA^*(k,s-k)=0\mod p^n
\]
the vanishing of the first term.
In the same way one can show the vanishing of the second term,
\[
\sum_{k=0}^{s}A^*(k,s-1-k)Q_k\equiv0\mod p^n.
\]
We thus have $S_m\equiv 0$ mod $p^n$.
This completes the proof of Theorem \ref{cong-thm}.
\section{Geometric aspect of $p$-adic hypergeometric functions of logarithmic type}
\label{reg-sect}
We mean by a {\it fibration} over a ring $R$ a projective flat morphism 
of quasi-projective smooth $R$-schemes.
Let $X$ be a smooth $R$-scheme. We mean by a relative {\it normal crossing divisor}
(abbreviated to NCD) in $X$ over $R$
a divisor in $X$ which is locally defined by an equation $x^{r_1}_1\cdots x_s^{r_s}$
where $r_i>0$ are integers and $(x_1,\ldots,x_n)$ is a local coordinate of $X/R$.
We say a divisor $D$ simple if $D$ is a union of $R$-smooth divisors.
\subsection{Hypergeometric Curves}
\label{fermat-sect}
Let $A$ be a commutative ring.
We denote by $\P^1_A(Z_0,Z_1)$ the projective line over $A$ with homogeneous coordinate
$(Z_0:Z_1)$.
Let $N,M\geq 2$ be integers which are invertible in $A$.
Let $t\in A$ such that $t(1-t)\in A^\times$.
Define $X$ to be a projective scheme over $A$ defined by a bihomogeneous equation
\begin{equation}\label{HGC}
(X_0^N-X_1^N)(Y_0^M-Y_1^M)=tX^N_0Y_0^M
\end{equation}
in $\P^1_A(X_0,X_1)\times\P^1_A(Y_0,Y_1)$
We call it
a {\it hypergeometric curve} over $A$.
The morphism $X\to\Spec A$ is smooth projective
with connected fibers of relative dimension one, and 
the genus of a geometric fiber is $(N-1)(M-1)$ (e.g. Hurwitz formula).
We put $x:=X_1/X_0$ and $y:=Y_1/Y_0$, and often refer to an affine equation
\[
(1-x^N)(1-y^M)=t.
\]
In what follows, we only consider the case $A=W[t,(t-t^2)^{-1}]$ where
$W$ is a commutative ring in which $NM$ is invertible.
Then the morphism $X\to \Spec W[t,(t-t^2)^{-1}]$ extends to a projective flat morphism
\[
f:Y\lra \P^1_W=\P^1_W(T_0,T_1)
\]
of smooth $W$-schemes with $t:=T_1/T_0$ in the following way.
Let 
\[
Y_s:=\{(X_0:X_1)\times(Y_0:Y_1)\times(T_0:T_1)\mid
T_0(X_0^N-X_1^N)(Y_0^M-Y_1^M)=T_1X^N_0Y_0^M\}
\]
and let $f_s:Y_s\to \P^1_W$ be the 3rd projection, which is smooth over $S=\Spec W[t,(t-t^2)^{-1}]\subset \P^1_W$.
In a neighborhood of the fiber $f_s^{-1}(t=0)$, $Y_s$ is smooth over $W$ where
$t=0$ denotes the closed subscheme $\Spec W[t]/(t)$.
Moreover
the fiber $f_s^{-1}(t=0)$ is a simple relative NCD over $W$,
and all multiplicities of components of $f_s^{-1}(t=0)$ are one.
In a neighborhood of the fiber $f_s^{-1}(t=1)$, $Y_s$ is also smooth over $W$.
However we note that
the fiber $f_s^{-1}(t=1)$ is irreducible but not a NCD.
Finally, in a neighborhood of the fiber $f_s^{-1}(t=\infty)$ there are singular loci in $Y_s$.
Put $z=1/x$, $w=1/y$ and $s=1/t$.
Then the singular loci are $\{s=1-z^N=w=0\}$ and $\{s=z=1-w^M=0\}$, and 
the neighborhood of each locus
is locally isomorphic to
\[\Spec W[[u_1,u_2,u_3]]/(u_1u_2-u_3^k),\quad
k=N,M.
\]
Then, there is the standard way to resolve the singularities so that we have
a smooth projective $W$-scheme $Y$.

\medskip


Summing up the above construction, we have
\begin{lem}\label{int.model-lem}
Suppose that $NM$ is invertible in $W$.
Then there is a morphism
\[
f:Y\lra \P^1_W=\P^1_W(T_0,T_1)
\]
of smooth projective $W$-schemes satisfying the following. 
\begin{enumerate}
\item[$(1)$]
Let $S:=\Spec W[t,(t-t^2)^{-1}]\subset \P^1_W$ with $t:=T_1/T_0$.
Then $X=f^{-1}(S)\to S$ is the hypergeometric curve \eqref{HGC}.
\item[$(2)$]
$f$ has a semistable reduction at $t=0$.
The fiber $D:=f^{-1}(t=0)$ is a relative simple NCD, and 
the multiplicities of the components are one.
\end{enumerate}
\end{lem}
\begin{rem}
One can further resolve $Y$ so that we have $\wt f:\wt Y\to \P^1_W$ such that
$\wt f^{-1}(t=1)$ and $\wt f^{-1}(t=\infty)$ are relative simple NCD's, while
the multiplicities are not necessarily one.
\end{rem}
\subsection{Gauss-Manin connection}
In this section we assume that $W$ is an integral domain of characteristic zero.
Let $K=\Frac W$ be the fractional field.
For a $W$-scheme $Z$ and a $W$-algebra $R$, we write $Z_R=Z\times_W R$.
The group $\mu_N\times\mu_M=\mu_N(\ol K)\times \mu_M(\ol K)$ acts on $\ol X$ in the following way
\begin{equation}\label{fermat-ss}
[\zeta, \nu]\cdot(x,y,t)=(\zeta x,\nu y,t),\quad(\zeta ,\nu )\in \mu_N\times \mu_M.
\end{equation}
We denote by $V(i,j)$ the subspace on which $(\zeta ,\nu )$ acts by multiplication
by $\zeta^i\nu^j$ for all $(\zeta ,\nu )$.
Then one has the eigen decomposition
\[
H^1_\dR(X_{\ol K}/ S_{\ol K})=\bigoplus_{i=1}^{N-1}\bigoplus_{j=1}^{M-1}
H^1_\dR(X_{\ol K}/S_{\ol K})(i,j),
\]
and each eigenspace $H^1_\dR(X_{\ol K}/ S_{\ol K})(i,j)$ is free of rank $2$ over 
$\O(S_{\ol K})$
(\cite[Lemma 2.2]{A}).
Put
\begin{equation}\label{fermat-form-ab}
a_i:=1-\frac{i}{N},\quad 
b_j:=1-\frac{j}{M}.
\end{equation}
Let
\begin{equation}\label{fermat-form}
\omega_{i,j}:=N\,\frac{x^{i-1}y^{j-M}}{1-x^N}dx
=-M\,\frac{x^{i-N}y^{j-1}}{1-y^M}dy,
\end{equation}
\begin{equation}\label{fermat-form-2}
\eta_{i,j}:
=\frac{1}{x^N-1+t}\omega_{i,j}=Mt^{-1}x^{i-N}y^{j-M-1}dy
\end{equation}
for integers $i,j$ such that $1\leq i\leq N-1,\,1\leq j\leq M-1$.
Then $\omega_{i,j}$ is the 1st kind, and $\eta_{i,j}$ is the 2nd kind.
They form a $\O(S_{\ol K})$-free basis of $H^1_\dR(X_{\ol K}/S_{\ol K})(i,j)$.
According to this, we put
\[H_\dR^1(X_K/S_K)(i,j):=\O(S_K)\omega_{i,j}+\O(S_K)\eta_{i,j}\subset H^1_\dR(X_K/S_K).
\]
Let 
\[
F_{a,b}(t):={}_2F_1\left({a,b\atop 1};t\right)=\sum_{i=0}^\infty\frac{(a)_i}{i!}\frac{(b)_i}{i!}
t^i\in K[[t]]
\]
be the hypergeometric series.
Put
\begin{equation}\label{fermat-form-wt}
\wt\omega_{i,j}:=\frac{1}{F_{a_i,b_j}(t)}\omega_{i,j},\quad
\wt\eta_{i,j}:=-t(1-t)^{a_i+b_j}(F'_{a_i,b_j}(t)\omega_{i,j}+b_jF_{a_i,b_j}(t)\eta_{i,j}).
\end{equation}
For the later use, we give notation $V(i,j),\omega_{i,j},\eta_{i,j},\wt\omega_{i,j},\wt\eta_{i,j}$
for $(i,j)$ not necessarily a pair of integers.
Let $(i,j)=(q/r,q'/r')\in \Q^2$ such that $\gcd(r,N)=\gcd(r',M)=1$ and
$N\not| q$ and $M\not| q'$.
Let $i_0$, $j_0$ be the unique integers such that $i_0\equiv i$ mod $N$,
$j_0\equiv j$ mod $M$ and $1\leq i_0<N$, $1\leq j_0<M$. Then we define
\begin{equation}\label{rule}
V(i,j)=V(i_0,j_0),\quad \omega_{i,j}=\omega_{i_0,j_0},\quad\ldots\quad
\wt\eta_{i,j}=\wt\eta_{i_0,j_0}.
\end{equation}
\begin{prop}\label{fermat-GM}
Let $\nabla:H^1_\dR(X_K/S_K)\to\O(S_K)dt\ot H^1_\dR(X_K/S_K)$
be the Gauss-Manin connection. It naturally extends on $K((t))\ot_{\O(S)}H^1_\dR(X_K/S_K)$
which we also write by $\nabla$. Then
\[
\begin{pmatrix}
\nabla(\omega_{i,j})&\nabla(\eta_{i,j})
\end{pmatrix}
=dt\ot\begin{pmatrix}
\omega_{i,j}&\eta_{i,j}
\end{pmatrix}
\begin{pmatrix}
0&-a_i(t-t^2)^{-1}\\
-b_j&(-1+(1+a_i+b_j)t)(t-t^2)^{-1}
\end{pmatrix},
\]
\[
\begin{pmatrix}
\nabla(\wt\omega_{i,j})&\nabla(\wt\eta_{i,j})
\end{pmatrix}
=dt\ot\begin{pmatrix}
\wt\omega_{i,j}&\wt\eta_{i,j}
\end{pmatrix}
\begin{pmatrix}
0&0\\
t^{-1}(1-t)^{-a_i-b_j}F_{a_i,b_j}(t)^{-2}&0
\end{pmatrix}.
\]
\end{prop}
\begin{pf}
We may replace the base field with $\C$.
Since $\nabla$ is commutative with the action of $\mu_N(\C)\times\mu_M(\C)$, 
$\nabla$ preserves the eigen components $H^1_\dR(X/S)(i,j)$.
We think $X$ and $S$ of being complex manifolds.
For $\alpha\in \C\setminus\{0,1\}$ we write $X_\alpha=f^{-1}(t=\alpha)$.
Then there is a homology cycle $\delta_\alpha\in H_1(X_\alpha,\Q)$
such that
\[
\int_{\delta_\alpha}\omega_{i,j}=2\pi\sqrt{-1}\,{}_2F_1\left({a_i,b_j\atop 1};\alpha\right)
\]
(\cite[Lemma 2.3]{A}).
Let $\partial_t=\nabla_{\frac{d}{dt}}$ be the differential operator on 
$\O(S)^{an}\ot_{\O(S)} H^1_\dR(X/S)$.
Put $D=t\partial_t$ and $P_{\mathrm{HG}}=D^2-t(D+a_i)(D+b_j)$ the hypergeometric
differential operator. Therefore
\[
\int_{\delta_\alpha}P_{\mathrm{HG}}(\omega_{i,j})=0.
\]
Since $H_1(X_\alpha,\C)(i,j)$ is a 2-dimensional irreducible $\pi_1(S,\alpha)$-module,
we have $\int_{\gamma}P_{\mathrm{HG}}(\omega_{i,j})=0$ for all
$\gamma\in\pi_1(S,\alpha)$ which means
\begin{equation}\label{fermat-GM-eq1}
P_{\mathrm{HG}}(\omega_{i,j})=(D^2-t(D+a_i)(D+b_j))(\omega_{i,j})=0.
\end{equation}
Next we show
\begin{equation}\label{fermat-GM-eq2}
\partial_t(\omega_{i,j})=-b_j\eta_{i,j}.
\end{equation}
Write $\omega_{i,j}=y^{j-M}\phi$, and think of it being an element of 
$\vg(U,\Omega^1_{X/\C})$ with $U=\{x\ne\infty,y\ne\infty\}\subset X$. Since $\phi$ is a linear combination of $dx/(x-\nu)$'s,
one has $d(\phi)=0$.
Therefore
\[
d(\omega_{i,j})=
d(y^{j-M})\wedge\phi=(j-M)y^{j-M-1}dy\wedge\phi\in \vg(U,\Omega^2_{X/\C}).
\]
Apply
\[
\frac{dt}{t}=\frac{Nx^{N-1}dx}{x^N-1}+\frac{My^{M-1}dy}{y^M-1}
\]
to the above, we have
\[
d(\omega_{i,j})=(j-M)y^{j-M-1}\frac{y^M-1}{Mty^{M-1}}dt\wedge\phi
=-(1-j/M)\frac{y^M-1}{ty^{M}}dt\wedge\omega_{i,j}
=-b_jdt\wedge\eta_{i,j},
\]
which means $\nabla(\omega_{i,j})=-b_jdt\ot\eta_{i,j}$. This completes the proof of 
\eqref{fermat-GM-eq2}.
Now all the formulas on $\nabla$ follow from \eqref{fermat-GM-eq1} and
\eqref{fermat-GM-eq2}.
\end{pf}
The following is straightforward from Proposition \ref{fermat-GM}.
\begin{cor}\label{fermat-GM-cor1}
Let $\nabla_{i,j}$ be the connection on the eigen component
$H_{i,j}:=K((t))\ot_{\O(S)}H^1_\dR(X_K/S_K)(i,j)$. Then
$\ker\nabla_{i,j}=K\wt{\eta}_{i,j}$.
Moreover let $\ol\nabla_{i,j}$ be the connection on $M_{i,j}:=H_{i,j}/K((t))\wt\eta_{i,j}$
induced from $\nabla_{i,j}$.
Then $\ker\ol\nabla_{i,j}=K\wt{\omega}_{i,j}$.
\end{cor}
We mean by
a semistable family
$g:\cX\to \Spec R[[t]]$ over a commutative ring $R$ 
that $g$ is a proper flat morphism, smooth
over $\Spec R((t))$ and it is locally described by
\[
g:\Spec R[[x_1,\ldots,x_n]]\lra \Spec R[[t],\quad g^*(t)=x_1\cdots x_r
\]
in each formal neighborhood.
Let $D$ be the fiber at $\Spec R[[t]]/(t)$, which is a relative NCD in $\cX$ over $R$
with no multiplicities.
Let ${\mathscr U}:=\cX\setminus D$.
We define the log de Rham complex $\omega^\bullet_{\cX/R[[t]]}$
to be the subcomplex of $\Omega^\bullet_{{\mathscr U}/R((t))}$ generated by $dx_i/x_i$
($1\leq i\leq r$) and $dx_j$ ($j>r$) over $\O_\cX$.
Equivalently,
\begin{equation}\label{log-dR-cpx}
\omega^\bullet_{\cX/R[[t]]}:=\Coker\left[
\frac{dt}{t}\ot\Omega^{\bullet-1}_{\cX/R}(\log D)
\to \Omega^{\bullet}_{\cX/R}(\log D)\right]
\end{equation}
where $\Omega^{\bullet}_{\cX/R}(\log D)$ denotes the $t$-adic completion of
the complex of the 
algebraic K\"ahler differentials, 
\[
\Omega^{\bullet}_{\cX/R}(\log D):=
\varprojlim_{n\geq1}\big(\Omega^{\bullet,\text{alg}}_{\cX/R}(\log D)/t^n\Omega^{\bullet,\text{alg}}_{\cX/R}(\log D)\big)
\]
\begin{cor}\label{fermat-GM-cor2}
Let $f:Y\to\P^1_W$ be the morphism of projective smooth $W$-schemes in Lemma \ref{int.model-lem}.
Let $Y_K:=Y\times_WK$.
Let $\Delta_K:=\Spec K[[t]]\hra \P^1_K$ be the formal neighborhood, and put
$\cY_K:=f^{-1}(\Delta_K)$. Let $D_K\subset\cY_K$ be the fiber at $t=0$.
Let
\[
\xymatrix{
D_K\ar[r]\ar[d]&\cY_K\ar[d]\ar[r]&Y_K\ar[d]\\
\{0\}\ar[r]&\Delta_K\ar[r]&\P^1_K
}\]
Put $H_K:=H^1_\zar(\cY_K,\omega^\bullet_{\cY_K/K[[t]]})$.
It follows from \cite[(2.18)--(2.20)]{steenbrink} or
\cite[(17)]{zucker} that $H_K\to K((t))\ot_{\O(S)} H^1_\dR(X/S)$ is injective.
We identify $H_K$ with its image.
Then the eigen component
$H_K(i,j)$ is a free $K[[t]]$-module with basis
$\{\wt\omega_{i,j},\wt\eta_{i,j}\}$.
\end{cor}
\begin{pf}
$H_K$ is called {\it Deligne's canonical extension}, and 
is characterized by the following conditions (\cite[(17)]{zucker}).
\begin{description}
\item[(D1)] $H_K$ is a free $K[[\l]]$-module such that $K((t))\ot H_K=K((t))\ot_{\O(S)} H^1_\dR(X/S)$,
\item[(D2)]
the connection extends to have log pole, 
$\nabla:H_K\to\frac{dt}{t}\ot H_K$,
\item[(D3)]
each eigenvalue $\alpha$ of $\Res(\nabla)$ satisfies $0\leq \mathrm{Re}(\alpha)<1$, 
where $\Res(\nabla)$ is the $K$-linear endomorphism defined by a commutative diagram
\[
\xymatrix{
H_K\ar[r]^\nabla\ar[d]&
\frac{dt}{t}\ot H_K\ar[d]^{\Res\ot1}\\
H_K/tH_K\ar[r]^{\Res(\nabla)}&H_K/tH_K.
}
\]
\end{description}
Put $H^0_K:=\bigoplus_{i,j}K[[t]]\wt\omega_{i,j}+K[[t]]\wt\eta_{i,j}$.
We can directly check
that $H^0_K$ satisfies {\bf(D1)}--{\bf(D3)} by Propositioon \ref{fermat-GM}.
We then conclude $H_K=H^0_K$ thanks to the uniqueness of Deligne's canonical extension.
\end{pf}
\subsection{Rigid cohomology and a category $\FilFMIC(S)$}
In what follows the base ring $W$ is the Witt ring $W(\ol\F_p)$ with $p$ prime to $NM$.
Put $K:=\Frac W$ the fractional field.
We denote the $p$-th Frobenius on $W$ by $F$.


\medskip

Let $X\to S$ be the hypergeometric curve as before.
Write $X_{\ol\F_p}:=X\times_{W}\ol\F_p$ and $S_{\ol\F_p}:=S\times_{W}\ol\F_p$.
Let $\sigma$ be a $F$-linear $p$-th Frobenius on $W[t,(t-t^2)^{-1}]^\dag$ the ring of
overconvergent power series, which naturally extends on $K[t,(t-t^2)^{-1}]^\dag:=
K\ot W[t,(t-t^2)^{-1}]^\dag$.
Then the {\it rigid cohomology} groups
\[
H^\bullet_\rig(X_{\ol\F_p}/S_{\ol\F_p})
\]
are defined. We refer the book \cite{LS} for the general theory of rigid cohomology.
The required properties in below is the following.
\begin{itemize}
\item
$H^\bullet_\rig(X_{\ol\F_p}/S_{\ol\F_p})$ is a finitely generated 
$\O(S)^\dag=K[t,(t-t^2)^{-1}]^\dag$-module.
\item(Frobenius)
The $p$-th Frobenius $\Phi_{X/S}$ on $H^\bullet_\rig(X_{\ol\F_p}/S_{\ol\F_p})$ (depending on $\sigma$)
is defined in a natural way. This is a $\sigma$-linear endomorphism :
\[
\Phi_{X/S}(f(t)x)=\sigma(f(t))\Phi_{X/S}(x),\quad \mbox{for }x\in 
H^\bullet_\rig(X_{\ol\F_p}/S_{\ol\F_p}),\, 
f(t)\in \O(S)^\dag.
\]
\item(Comparison)
There is the comparison isomorphism with the algebraic de Rham cohomology,
\[
c:H^\bullet_\rig(X_{\ol\F_p}/S_{\ol\F_p})\cong H^\bullet_\dR(X/S)
\ot_{\O(S)} \O(S)^\dag.
\]
\end{itemize}
In \cite[2,1]{AM} we introduce
a category $\FilFMIC(S)=\FilFMIC(S,\sigma)$.
It consists of collections of datum $(H_{\dR}, H_{\rig}, c, \Phi, \nabla, \Fil^{\bullet})$ such that
    \begin{itemize}
            \setlength{\itemsep}{0pt}
        \item $H_{\dR}$ is a finitely generated $\O(S)$-module,
        \item $H_{\rig}$ is a finitely generated $\O(S)^\dag$-module,
        \item $c\colon H_\rig\cong H_{\dR}\otimes_{\O(S)}
        \O(S)^\dag$, the comparison 
        \item $\Phi\colon \sigma^{\ast}H_{\rig}\xrightarrow{\,\,\cong\,\,} H_{\rig}$ is an isomorphism of $\O(S)^\dag$-module,
        \item $\nabla\colon H_{\dR}\to \Omega_{S/\Q_p}^1\otimes H_{\dR}$ is an integrable connection
    that satisfies $\Phi\nabla=\nabla\Phi$.
        \item $\Fil^{\bullet}$ is a finite descending filtration on $H_{\dR}$ of locally free 
        $\O(S)$-module (i.e. each graded piece is locally free),
    that satisfies $\nabla(\Fil^i)\subset \Omega^1\ot \Fil^{i-1}$.
    \end{itemize}


Let $\mathrm{Fil}^\bullet$ denote the Hodge filtration on the de Rham cohomology,
and $\nabla$ the Gauss-Manin connection.
Let
\[
H^i(X/S):=(H^i_\dR(X/S),H^i_\rig(X_{\F_p}/S_{\F_p}),c,\Phi_{X/S},\nabla,\mathrm{Fil}^\bullet)
\]
be an object of $\FilFMIC(S)$. 

For an integer $r$, the Tate object $\O_S(r)\in \FilFMIC(S)$ 
is defined in a customary way (loc.cit.).
We simply write
\[
M(r)=M\ot \O_S(r)
\]
for an object $M\in \FilFMIC(S)$.



\begin{lem}\label{frobenius-lem}
Suppose that $\sigma$ is given by $\sigma(t)=ct^p$ with some $c\in 1+pW$.
Then, with the notation in Corollary \ref{fermat-GM-cor2},
the Frobenius $\Phi_{X/S}$ induces the action on 
$H_K$ in a natural way.
\end{lem}
\begin{pf}
Let $W((t))^\wedge$ be the $p$-adic completion and write $K((t))^\wedge:=K\ot_WW((t))^\wedge$ on which $\sigma$ extends as $\sigma(t)=ct^p$.
The Frobenius $\Phi_{X/S}$ on $H^1_\dR(X/S)\ot_{\O(S)}\O(S)^\dag$ naturally extends
on $H^1_\dR(X/S)\ot_{\O(S)}K((t))^\wedge$ via the homomorphism
$\O(S)^\dag\to K((t))^\wedge$. We show that the action of $\Phi_{X/S}$ preserves
the subspace $H_K$

Let $f:Y\to\P^1_W$ be the morphism of projective smooth $W$-schemes in Lemma \ref{int.model-lem}.
Let
\[
\xymatrix{
D_W\ar[r]\ar[d]&\cY\ar[d]\ar[r]&Y\ar[d]\\
0\ar[r]&\Delta_W\ar[r]&\P^1_W
}\]
where $\Delta_W:=\Spec W[[t]]\hra \P^1_W$ is the formal neighborhood, and 
$0=\Spec W[[t]]/(t)$.
Note that $D_W$ is reduced, namely $\cY/\Delta_W$ has a semistable reduction.
Write $\cY_{\ol\F_p}:=\cY\times_W\ol\F_p$ etc.
We employ the log-crystalline cohomology
\begin{equation}\label{log-crys}
H^\bullet_{\text{log-crys}}((\cY_{\ol\F_p},D_{\ol\F_p})/(\Delta_W,0))
\end{equation}
where $(\cX,\cD)$ denotes the log scheme with log structure
induced by the divisor $\cD$.
There is the comparison theorem by Kato \cite[Theorem 6.4]{Ka-log},
\begin{equation}\label{log-crys-isom}
H^\bullet_{\text{log-crys}}((\cY_{\ol\F_p},D_{\ol\F_p})/(\Delta_W,0))
\cong 
H^\bullet(\cY,\omega^\bullet_{\cY/W[[t]]})
\end{equation}
(see \eqref{log-dR-cpx} for the complex $\omega^\bullet_{\cY/W[[t]]}$).
The log-crystalline cohomology is endowed with
the $p$-th Frobenius $\Phi_{(\cY,D_W)}$
which is compatible with $\Phi_{X/S}$ under the map
\begin{align*}
H^1(\cY,\omega^\bullet_{\cY/W[[t]]})&\to
H^1(\cY_K,\omega^\bullet_{\cY_K/K[[t]]}))\\
&\hra
H^1_\dR(X/S)\ot_{\O(S)} K((t))\\
&\hra H^1_\dR(X/S)\ot_{\O(S)} K((t))^\wedge
\end{align*}
where $\cY_K:=\cY\times_{W[[t]]}K[[t]]$ and $D_K:=D_W\times_WK$.
Thus the assertion follows.
\end{pf}
\begin{prop}\label{frobenius-thm}
Let $\wt\omega_{i,j},\wt\eta_{i,j}$ be as in \eqref{fermat-form} and \eqref{fermat-form-2}.
Suppose that $\sigma$ is given by $\sigma(t)=ct^p$ with some $c\in 1+pW$.
Then
\
\Phi_{X/S}(\wt\eta_{p^{-1}i,p^{-1}j})\in K\wt\eta_{i,j},\quad
\Phi_{X/S}(\wt\omega_{p^{-1}i,p^{-1}j})\equiv p\wt\omega_{i,j}\mod K((t))\wt\eta_{i,j}
\
where we use the notation \eqref{rule}.
\end{prop}
\begin{pf}
Let $\nabla$ be the Gauss-Manin connection on $H^1_\dR(X/S)\ot_{\O(S)}K((t))$.
Since $\Phi_{X/S}\nabla=\nabla\Phi_{X/S}$, we have 
$\Phi_{X/S}\ker(\nabla)\subset \ker(\nabla)$.
Moreover,
$\Phi_{X/S}$ sends the eigencomponents $H_{i,j}:=H^1_\dR(X/S)(i,j)\ot_{\O(S)}K((t))$
onto the component $H_{pi,pj}$ as 
$\Phi_{X/S}[\zeta,\nu]=[\zeta,\nu]\Phi_{X/S}$.
Therefore we have
\[
\Phi_{X/S}(\wt\eta_{p^{-1}i,p^{-1}j})\in K\wt\eta_{i,j}
\]
by Corollary \ref{fermat-GM-cor1}.
We show the latter.
By Lemma \ref{frobenius-lem} together with Corollary \ref{fermat-GM-cor2}, 
there are $f_{i,j}(t), g_{i,j}(t)\in K[[t]]$ such that
$\Phi_{X/S}(\wt\omega_{p^{-1}i,p^{-1}j})=f_{i,j}(t)\wt\omega_{i,j}+g_{i,j}(t)\wt\eta_{i,j}$.
Put $M_{i,j}:=H_{i,j}/K((t))\wt\eta_{i,j}$.
Then $\Phi_{X/S}(M_{p^{-1}i,p^{-1}j})\subset M_{i,j}$ and $\Phi_{X/S}$ is
commutative with the connection $\ol\nabla$ on $M_{i,j}$.
Therefore $f_{i,j}(t)=C_{i,j}$ is a constant as $\ker(\ol\nabla)=K\wt\omega_{i,j}$
by Corollary \ref{fermat-GM-cor1},
\begin{equation}\label{frob-lem-eq1}
\Phi_{X/S}(\wt\omega_{p^{-1}i,p^{-1}j})=C_{i,j}\wt\omega_{i,j}+g_{i,j}(t)\wt\eta_{i,j}.
\end{equation}
We want to show $C_{i,j}=p$.
To do this, we recall the log-crystalline cohomology \eqref{log-crys}
\[
H^\bullet_{\text{log-crys}}((\cY_{\ol\F_p},D_{\ol\F_p})/(\Delta_W,0))
\cong H^\bullet(\cY,\omega^\bullet_{\cY/W[[t]]}).
\]
where we keep the notation in the proof of Lemma \ref{frobenius-lem}.
Let $Z_W$ be the intersection locus of $D_W$.
This is a disjoint union of  $NM$-copies of $\Spec W$.
More precisely, let $P_{\zeta,\nu}$ be the point of $Z_W$ defined by $x=\zeta$ and
$y=\nu$.
Then $Z_W=\{P_{\zeta,\nu}\mid(\zeta,\nu)\in\mu_N\times\mu_M\}$.
We consider the composition of morphisms 
\[
\omega^\bullet_{\cY/W[[t]]}\os{\wedge\frac{dt}{t}}{\lra}
\Omega^{\bullet+1}_{\cY/W}(\log D)\os{\Res}{\lra}
\O_{Z_W}[-1
\]
of complexes where $\Res$ is the Poincare residue.
This gives rise to the map
\[
R:H^1(\cY_K,\Omega^\bullet_{\cY_K/K[[t]]}(\log D_K))\lra 
H^0(Z_K,\O_{Z_K})=\bigoplus_{\zeta,\nu}K\cdot P_{\zeta,\nu}
\]
which is compatible with respect to the Frobenius $\Phi_{X/S}$ 
on the left
and the Frobenius $\Phi_Z$ on the right in the sense that
\begin{equation}\label{frobenius-thm-eq0}
R\circ \Phi_{X/S}=p\Phi_Z\circ R.
\end{equation}
Notice that $\Phi_Z$ is a $F$-linear map such that
$\Phi_Z(P_{\zeta,\nu})=P_{\zeta,\nu}$ where $F$ is the Frobenius
on $W$.
We claim
\begin{equation}\label{frobenius-thm-eq1}
R(\wt\eta_{i,j})=0,
\end{equation}
and
\begin{equation}\label{frobenius-thm-eq2}
R(\wt\omega_{i,j})=\sum_{\zeta,\nu}\zeta^i\nu^jP_{\zeta,\nu}.
\end{equation}
To show \eqref{frobenius-thm-eq1}, we recall the definition \eqref{fermat-form-wt}.
Since $R(t\wt\omega_{i,j})=0$ obviously, 
it is enough to show $R(t\eta_{i,j})=0$.
However since $t\eta_{i,j}=Mx^{i-N}y^{j-M}dy$ by
\eqref{fermat-form-2}, we have
\[
R(t\eta_{i,j})=\Res\left(Mx^{i-N}y^{j-M}dy\frac{dt}{t}\right)=0
\]
as required. One can show \eqref{frobenius-thm-eq2} as follows.
\begin{align*}
R(\wt\omega_{i,j})=R(\omega_{i,j})
&=\Res\left(
M\,\frac{x^{i-N}y^{j-1}}{y^M-1}dy\wedge\frac{dt}{t}\right)\\
&=\Res\left(
\frac{Mx^{i-N}y^{j-1}}{y^M-1}dy\wedge\frac{Nx^{N-1}}{x^N-1}dx\right)\\
&=\sum_{\zeta,\nu}\zeta^i\nu^jP_{\zeta,\nu}.
\end{align*}
We turn to the proof of $C_{i,j}=p$ in \eqref{frob-lem-eq1}.
Apply $R$ on the both side of \eqref{frob-lem-eq1}.
By \eqref{frobenius-thm-eq1}, the right hand side is $\alpha_{i,j}R(\wt\omega_{i,j})$,
and 
the left hand side is $p\Phi_Z\circ R(\wt\omega_{p^{-1}i,p^{-1}j})$ 
by \eqref{frobenius-thm-eq0},
\[
C_{i,j}R(\wt\omega_{i,j})=p\Phi_Z\circ R(\wt\omega_{p^{-1}i,p^{-1}j}).
\]
Apply \eqref{frobenius-thm-eq1} to the above. We have
\[
C_{i,j}\left(\sum_{\zeta,\nu}\zeta^i\nu^jP_{\zeta,\nu}\right)=
p\Phi_Z\left(\sum_{\zeta,\nu}\zeta^{p^{-1}i}\nu^{p^{-1}j}\cdot P_{\zeta,\nu}\right)
=p\left(\sum_{\zeta,\nu}\zeta^{i}\nu^{j}P_{\zeta,\nu}\right)
\]
and hence $C_{i,j}=p$ as required.
\end{pf}


\begin{thm}[Unit root formula]\label{uroot-thm}
Suppose $\sigma(t)=t^p$.
For a pair of integers $(i,j)$ with $1\leq i<N$ and $1\leq j<M$, we put
\[
e^{\mathrm{unit}}_{i,j}:=(1-t)^{-a_i-b_j}F_{a_i,b_j}(t)^{-1}\wt\eta_{i,j}.
\]
Let $s\geq 0$ be the minimal
integer such that $a_i^{(s+1)}=a_i$ and $b_j^{(s+1)}=b_j$ and
put $h(t):=\prod_{m=0}^s F_{a_i^{(m)},b_j^{(m)}}(t)_{<p}$. Then
\begin{equation}\label{uroot-thm-1}
e^{\mathrm{unit}}_{i,j}\in 
H^1_\dR(X/S)\ot_{\O(S)}K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle
\end{equation}
and
\begin{equation}\label{uroot-thm-2}
\Phi_{X/S}(e^{\mathrm{unit}}_{p^{-1}i,p^{-1}j})=
\frac{(1-t)^{a_i+b_j}}{(1-t^p)^{a^{(1)}_i+b^{(1)}_j}}
\cF^{\mathrm{Dw}}_{a_i,b_j}(t)e^{\mathrm{unit}}_{i,j}
\end{equation}
where $\cF^{\mathrm{Dw}}_{a_i,b_j}(t)$ is the Dwork $p$-adic hypergeometric function
\eqref{Dwork},
and we apply the convention \eqref{rule} to the notation
$e^{\mathrm{unit}}_{p^{-1}i,p^{-1}j}$.
In particular
$e^{\mathrm{unit}}_{p^{-1}i,p^{-1}j}$ is the eigen vector of 
$\Phi_{X/S}^{s+1}=\overbrace{\Phi_{X/S}\circ\cdots\circ\Phi_{X/S}}^{s+1}$, and
\begin{equation}\label{uroot-thm-3}
\Phi_{X/S}^{s+1}(e^{\mathrm{unit}}_{i,j})
=\left(\prod_{m=0}^s
\frac{(1-t^{p^m})^{a^{(m)}_i+b^{(m)}_j}}{(1-t^{p^{m+1}})^{a^{(m+1)}_i+b^{(m+1)}_j}}
\cF^{\mathrm{Dw}}_{a^{(m)}_i,b^{(m)}_j}(t^{p^m})\right)e^{\mathrm{unit}}_{i,j}.
\end{equation}
\end{thm}
Notice that $(1-t)^{a_i+b_j}\not\in \Z_p\langle t,(1-t)^{-1}\rangle$
but $(1-t)^{a_i+b_j}/(1-t^p)^{a_i^{(1)}+b_j^{(1)}}\in \Z_p\langle t,(1-t)^{-1}\rangle$.
\begin{pf}
Since
\begin{equation}\label{uroot-thm-eq1}
\frac{F'_{a_i,b_j}(t)}{F_{a_i,b_j}(t)}\in \Z_p\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle
\end{equation}
by \cite[p.45, Lem. 3.4 ]{Dwork-p-cycle}, \eqref{uroot-thm-1} follows.
We show \eqref{uroot-thm-2}, which is equivalent to
\begin{equation}\label{uroot-thm-eq2}
\Phi_{X/S}(\wt\eta_{p^{-1}i,p^{-1}j})=\wt\eta_{i,j}\in H^1_\dR(X/S)\ot_{\O(S)}K((t)).
\end{equation}
Let
\[
Q:H^1_\dR(X_K/S_K)\ot H^1_\dR(X_K/S_K)\lra H^2_\dR(X_K/S_K)\cong \O(S_K)
\]
be the cup-product pairing which is anti-symmetric and non-degenerate.
This extends on $H^1_\dR(X/S)\ot_{\O(S)}K((t))$ which we also write by $Q$.
Then the following is satisfied.
\begin{description}
\item[(Q1)]
$Q(\Phi_{X/S}(x),\Phi_{X/S}(y))=pQ(x,y)^\sigma$ for $x,y\in H^1_\dR(X/S)\ot_{\O(S)}K((t))$,
\item[(Q2)]
$Q(gx,gy)=Q(x,y)$ for $g=(\zeta,\nu)\in \mu_N\times \mu_M$,
\item[(Q3)]
$Q(F^1,F^1)=0$ where $F^1=\vg(X_K,\Omega^1_{X_K/S_K})$ is the Hodge filtration, 
\item[(Q4)]
$Q(\nabla(x),y)+Q(x,\nabla(y))=dQ(x,y)$.
\end{description
Put $H_{i,j}:=H^1_\dR(X/S)(i,j)\ot_{\O(S)}K((t))$ eigen components.
By {\bf(Q2)}, $Q$ induces a perfect pairing $H_{i,j}\ot H_{N-i,M-j}\to\O(S)$.
Therefore $Q(\wt\omega_{i,j},\wt\eta_{N-i,M-j})\ne0$ by {\bf(Q3)}.
We claim 
\begin{equation}\label{uroot-thm-eq3}
Q(\wt\eta_{i,j},\wt\eta_{N-i,M-j})=0,
\end{equation}
\begin{equation}\label{uroot-thm-eq4}
Q(\wt\omega_{i,j},\wt\eta_{N-i,M-j})\in \Q^\times.
\end{equation}
To show \eqref{uroot-thm-eq3}, we recall $H_K$ in Corollary \ref{fermat-GM-cor2}.
Since $Q$ is the cup-product pairing, this induces a pairing $H_K\ot H_K\to K[[t]]$, and hence 
\[
\ol Q:H_K/tH_K\ot_K H_K/tH_K\lra K.
\]
Since $\nabla(\wt\eta_{i,j})=0$, $Q(\wt\eta_{i,j},\wt\eta_{N-i,M-j})$ is a constant by {\bf(Q4)}.
Therefore if one can show $\ol Q(\wt\eta_{i,j},\wt\eta_{N-i,M-j})=0$, 
then \eqref{uroot-thm-eq3} follows. It follows from {\bf(Q4)} that
\[
\ol Q(\Res(\nabla)(x),y)+\ol Q(x,\Res(\nabla)(y))=0,\quad \forall\,x,y\in H_K/tH_K
\]
where $\Res(\nabla)$ is as in the proof of Corollary \ref{fermat-GM-cor2}.
Since $\Res(\nabla)(\wt\omega_{i,j})=\wt\eta_{i,j}$ 
and $\Res(\nabla)\wt\eta_{i,j}=0$ by Proposition \ref{fermat-GM},
one has
\[
\ol Q(\wt\eta_{i,j},\wt\eta_{N-i,M-j})
=\ol Q(\Res(\nabla)\wt\omega_{i,j},\wt\eta_{N-i,M-j})
=-\ol Q(\wt\omega_{i,j},\Res(\nabla)\wt\eta_{N-i,M-j})=0
\]
as required. We show \eqref{uroot-thm-eq4}.
Since $\nabla(\wt\omega_{i,j})
\in K((t))\wt\eta_{i,j}$, we have
\[
dQ(\wt\omega_{i,j},\wt\eta_{N-i,M-j})=
Q(\nabla(\wt\omega_{i,j}),\wt\eta_{N-i,M-j})=0
\]
by \eqref{uroot-thm-eq3} which means that $Q(\wt\omega_{i,j},\wt\eta_{N-i,M-j})$ is a constant.
Since $X/S$, $Q$ and $\wt\omega_{i,j},\wt\eta_{i,j}$ are defined over $\Q((t))$, 
the constant 
should belong to $\Q^\times$.
This completes the proof of \eqref{uroot-thm-eq4}.

\medskip
We turn to the proof of \eqref{uroot-thm-eq2}.
By Proposition \ref{frobenius-thm}, there is a constant $\alpha\in K$ such that
$\Phi_{X/S}(\wt\eta_{p^{-1}i,p^{-1}j})=\alpha\wt\eta_{i,j}$.
Put $c:=Q(\wt\omega_{i,j},\wt\eta_{N-i,M-j})$ which belongs to $\Q^\times$
by \eqref{uroot-thm-eq4}.
By {\bf(Q1)}, we have
$Q(\Phi_{X/S}(\wt\omega_{i,j}),\Phi_{X/S}(\wt\eta_{N-i,M-j}))=pQ(\wt\eta_{i,j},\wt\eta_{N-i,M-j})^\sigma=pc$,
and hence \[\alpha Q(\Phi_{X/S}(\wt\omega_{i,j}),\wt\eta_{N-i,M-j})=pc.\]
It follows from \eqref{uroot-thm-eq3} and Proposition \ref{frobenius-thm} 
that the left hand side is 
\[
p\alpha Q(\wt\omega_{i,j},\wt\eta_{N-i,M-j})=p\alpha c.
\]
Therefore $\alpha=1$. This completes the proof.
\end{pf}


\subsection{Syntomic Regulators of hypergeometric curves} \label{syn-reg-sect}
Let $W=W(\ol\F_p)$ and $K:=\Frac W$ be as before.
Let $f_R:Y_R\to \P^1_R$ be the fibration of hypergeometric curves 
in Lemma \ref{int.model-lem}
over a ring $R:=\Z[\zeta_N,\zeta_M,(NM)^{-1}]\subset W$ 
where $\zeta_n$ is a primitve $n$-th root of unity in $W$.
Let $\ol X_R:=f_R^{-1}(\Spec R[t,t^{-1}])$
and $U_R:=\Spec R[x,y,t,t^{-1}]/((1-x^N)(1-y^M)-t)$.
Put $Z_R:=\ol X_R\setminus U_R$ 
a disjoint union of copies of $\bG_{m,R}=\Spec R[t,t^{-1}]$.

\medskip

For $(\nu_1,\nu_2)\in\mu_N(R)\times \mu_M(R)$, let
\begin{equation}\label{m-fermat-eq1}
\xi=\xi(\nu_1,\nu_2)=\left\{
\frac{x-1}{x-\nu_1},\frac{y-1}{y-\nu_2}
\right\}\in K^M_2(\O(U_R))
\end{equation}
be a symbol in Milnor's $K_2$.
Let $K_2(U_R)^{(2)}\os{\partial}{\lra} K_1(Z_R)^{(1)}=(R[t,t^{-1}]^\times)^\op\ot\Q$
be the boundary map where $K_i(-)^{(j)}$ denotes the Adams weight piece, 
which is explicitly described by
\[
\{f,g\}\longmapsto (-1)^{\ord_P(f)\ord_P(g)}\frac{g^{\ord_P(f)}}{f^{\ord_P(g)}}\bigg|_P,\quad 
P\in Z_R.
\]
It is a simple exercise to show that
$\partial(\xi)=0$ and hence $\xi$ lies in the image of $K_2(\ol X_R)^{(2)}$.
Since $K_2(\ol X_R)^{(2)}\to K_2(U_R)^{(2)}$ is injective as $K_2(\bG_{m,R})^{(1)}=0$, 
we have the element in $K_2(\ol X_R)^{(2)}$ which we also write by $\xi$.
Let $\ol X:=\ol X_R\times _RW$ and $i:\ol X\to \ol X_R$ the projection. 
We then have an element $i^*(\xi) \in K_2(\ol X)$, which we also write by $\xi$.

Let $\mathrm{dlog}:K_2(\ol X)^{(2)}\to \vg(\ol X,\Omega^2_{\ol X/W})\ot\Q$
be the dlog map which is given by $\{f,g\}\mapsto\frac{df}{f}\wedge\frac{dg}{g}$.
One immediately has
\begin{equation}\label{m-fermat-eq2}
\mathrm{dlog}(\xi)=N^{-1}M^{-1}
\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)\frac{dt}{t}\omega_{i,j}.
\end{equation}
Let $\sigma$ be a $p$-th Frobenius on $W[[t]]$ given by $\sigma(t)=ct^p$ with
$c\in 1+pW$.
According to \cite[\S 4.1]{AM},
one can associate a $1$-extension
\begin{equation}\label{syn-reg-ext}
0\lra H^1(X/S)(2)\lra M_\xi(X/S)\lra \O_S\lra 0
\end{equation}
in the exact category $\FilFMIC(S)$ (loc.cit. Prop.4.2).
Let $e_\xi\in \mathrm{Fil}^0M_\xi(X/S)_\dR$ be the unique lifting of $1\in \O_S(S)$.
Let $\ve_k^{(i,j)}(t)$ and $E_k^{(i,j)}(t)$ be defined by
\begin{align}
e_\xi-\Phi(e_\xi)
&=N^{-1}M^{-1}
\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)[\ve^{(i,j)}_1(t)\omega_{i,j}
+\ve^{(i,j)}_2(t)\eta_{i,j}]\label{fermat-e-eq1}\\
&=N^{-1}M^{-1}
\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)[E^{(i,j)}_1(t)\wt\omega_{i,j}
+E^{(i,j)}_2(t)\wt\eta_{i,j}].\label{fermat-e-eq2}
\end{align}
Notice that $\ve_k^{(i,j)}(t)$ and $E^{(i,j)}_k(t)$ 
depend on the choice of the Frobenius $\sigma$.
The relation between $\ve_k^{(i,j)}(t)$ and $E^{(i,j)}_k(t)$ is explicitly given by
\begin{align}
\ve^{(i,j)}_1(t)&=E^{(i,j)}_1(t)F_{a_i,b_j}(t)^{-1}-t(1-t)^{a_i+b_j}F'_{a_i,b_j}(t)E_2^{(i,j)}(t)
\label{syn-reg-eq3}
\\
\ve^{(i,j)}_2(t)&=-b_jt(1-t)^{a_i+b_j}F_{a_i,b_j}(t)E_2^{(i,j)}(t)\label{syn-reg-eq4}.
\end{align}
By the definition $\ve^{(i,j)}_k(t)$ are automatically overconvergent functions,
\[\ve^{(i,j)}_k(t)\in K[t,(t-t^2)^{-1}]^\dag.\]
On the other hand, since $F'_{a_i,b_j}(t)/F_{a_i,b_j}(t)$ is a convergent function 
(cf. \eqref{uroot-thm-eq1}), so is
$E^{(i,j)}_1(t)/F_{a_i,b_j}(t)$,
\begin{equation}\label{syn-reg-eq5}
\frac{E^{(i,j)}_1(t)}{F_{a_i,b_j}(t)}\in K\langle
t,(t-t^2)^{-1},h(t)^{-1}\rangle,\quad h(t):=\prod_{m=0}^s F_{a_i^{(m)},b_j^{(m)}}(t)_{<p}
\end{equation}
where $s\geq 0$ is the minimal
integer such that $a_i^{(s+1)}=a_i$ and $b_j^{(s+1)}=b_j$.

\medskip

The following is the main theorem in this paper, which provides
a geometric aspect of 
$\cF^{(\sigma)}_{a,b}(t)$ the $p$-adic hypergeometric function
of logarithmic type defined in \S \ref{pHGlog-defn}.

\begin{thm}\label{fermat-main1}
Suppose that $\sigma$ is given by $\sigma(t)=ct^p$ with $c\in 1+pW$.
Then 
\begin{equation}\label{main-thm1-eq1}
\frac{E_1^{(i,j)}(t)}{F_{a_i,b_j}(t)}=-\cF^{(\sigma)}_{a_i,b_j}(t).
\end{equation}
Hence
\[
e_\xi-\Phi(e_\xi)\equiv 
\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}\frac{(1-\nu^{-i}_1)(1-\nu^{-j}_2)}{NM}\cF^{(\sigma)}_{a_i,b_j}(t)
\omega_{i,j}
\mod \sum_{i,j}K\langle t,(t-t^2),h(t)^{-1}\rangle e^{\mathrm{unit}}_{i,j}.\]
\end{thm}
\begin{pf}
The Frobenius  $\sigma$ extends on $K((t))$, and 
$\Phi$ also extends on $K((t))\ot H^1_\dR(X/S)\cong
H^1_{\text{log-crys}}((\cY_{\ol\F_p},D_{\ol\F_p})/(\Delta_W,0))\ot_{W[[t]]}K((t))
$ in the natural way where the isomorphism follows from \eqref{log-crys-isom}.
Apply the Gauss-Manin connection $\nabla$ on \eqref{fermat-e-eq2}.
Since $\nabla\Phi=\Phi\nabla$ and $\nabla(e_\xi)=-\mathrm{dlog}\xi$ (\cite[(2.30)]{AM}),
we have
\begin{align}\label{main-thm1-eq2-left}
&(1-\Phi)\left(-N^{-1}M^{-1}\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)
\frac{dt}{t}\omega_{i,j}\right)\\
=&N^{-1}M^{-1}\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)
\nabla(E^{(i,j)}_1(t)\wt\omega_{i,j}+E^{(i,j)}_2(t)\wt\eta_{i,j})\label{main-thm1-eq2-right}
\end{align}
by \eqref{m-fermat-eq2}.
Let $\Phi_{X/S}$ denote the $p$-th Frobenius on $H^1_\rig(X_{\ol\F_p}/S_{\ol\F_p})$.
Then the $\Phi$ on $H^1_\rig(X/S)(2)$ agrees with $p^{-2}\Phi_{X/S}$ by definition of Tate twists. 
It follows from Proposition \ref{frobenius-thm} that we have
\[
\Phi_{X/S}(\wt\omega_{p^{-1}i,p^{-1}j})\equiv p\wt\omega_{i,j}\mod K((t))\wt\eta_{i,j}
\]
where $m\in \{1,\ldots,N-1\}$ with $pm\equiv n$ mod $N$.
Therefore
\[
\eqref{main-thm1-eq2-left}\equiv-
N^{-1}M^{-1}\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)
(F_{a_i,b_j}(t)-F_{a_i^{(1)},b_j^{(1)}}(t^\sigma))\frac{dt}{t}\wt\omega_{i,j}
\]
modulo $\sum_{i,j} K((t))\wt\eta_{i,j}$.
On the other hand, 
\[
\eqref{main-thm1-eq2-right}\equiv
N^{-1}M^{-1}\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)
t\frac{d}{dt}(E^{(i,j)}_1(t))\cdot\frac{dt}{t}\wt\omega_{i,j}
\mod \sum_{i,j}K((t))\wt\eta_{i,j}
\]
by Proposition \ref{fermat-GM}.
We thus have
\begin{equation}\label{main-thm1-eq3}
t\frac{d}{dt}E_1^{(i,j)}(t)=-F_{a_i,b_j}(t)+F_{a_i^{(1)},b_j^{(1)}}(t^\sigma),
\end{equation}
and hence 
\[
E_1^{(i,j)}(t)=-\left(C+\int_0^tF_{a_i,b_j}(t)-F_{a_i^{(1)},b_j^{(1)}}(t^\sigma)\frac{dt}{t}\right)
\]
for some constant $C\in K$.
We determine the constant $C$ in the following way.
Firstly $E_1^{(i,j)}(t)/F_{a_i,b_j}(t)$ is a convergent function by \eqref{syn-reg-eq5}.
If $C=\psi_p(a_i)+\psi_p(b_j)+2\gamma_p$, then 
$E_1^{(i,j)}(t)/F_{a_i,b_j}(t)=\cF^{(\sigma)}_{a_i,b_j}(t)$ belongs to
$K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle$ by Corollary \ref{cong-cor}.
If there is another $C'$ such that 
$E_1^{(i,j)}(t)/F_{a_i,b_j}(t)\in K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle$,
then it follows
\[
\frac{C-C'}{F_{a_i,b_j}(t)}\in K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle.
\]
This is impossible, and hence
there is no possibility other than
$C=\psi_p(a_i)+\psi_p(b_j)+2\gamma_p$. 
To see this, 
we recall from Theorem \ref{uroot-thm}
the formula \eqref{uroot-thm-3}
\begin{align*}
\Phi_{X/S}^{s+1}(e^{\mathrm{unit}}_{i,j})
&=\left(\prod_{m=0}^s
\frac{(1-t^{p^m})^{a^{(m)}_i+b^{(m)}_j}}{(1-t^{p^{m+1}})^{a^{(m+1)}_i+b^{(m+1)}_j}}
\cF^{\mathrm{Dw}}_{a^{(m)}_i,b^{(m)}_j}(t^{p^m})\right)e^{\mathrm{unit}}_{i,j}\\
&=\left(\frac{(1-t)^{a_i+b_j}}{(1-t^{p^{s+1}})^{a_i+b_j}}\right)
\frac{F_{a_i,b_j}(t)}{F_{a_i,b_j}(t^{p^{s+1}})}e^{\mathrm{unit}}_{i,j}.
\end{align*}
Iterating $\Phi_{X/S}^{s+1}$ to the above, we have
\[
(\Phi_{X/S}^{s+1})^n(e^{\mathrm{unit}}_{i,j})
=\overbrace{\left(\frac{(1-t)^{a_i+b_j}}{(1-t^{p^{n(s+1)}})^{a_i+b_j}}\right)}^{\mu(t)}
\frac{F_{a_i,b_j}(t)}{F_{a_i,b_j}(t^{p^{n(s+1)}})}e^{\mathrm{unit}}_{i,j}.
\]
Put $q:=p^{n(s+1)}$.
Let $\alpha\in W$ satisfy $\alpha^q=\alpha$ and 
$(\alpha-\alpha^2)h(\alpha)\not\equiv0$ mod $p$.
Then the evaluation $\mu(\alpha)$ is a root of unity. 
Suppose $g(t):=F_{a_i,b_j}(t)^{-1}\in K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle$.
Let $g(t)=(t-\alpha)^kg_0(t)$ with $g_0(t)\in 
K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle$ such that $g_0(\alpha)\ne0$.
Then we have 
\[
\frac{F_{a_i,b_j}(t)}{F_{a_i,b_j}(t^q)}\bigg|_{t=\alpha}
=\frac{(t^q-\alpha)^kg_0(t^q)}{(t-\alpha)^kg_0(t)}\bigg|_{t=\alpha}
=(q\alpha^{q-1})^k=q^k.
\]
Since the first evaluation is a unit in $W$, we have $k=0$.
Thus
an eigen value of $(\Phi_{X/S}^{s+1})^n|_{t=\alpha}$ is a root of unity.
This contradicts with the Weil-Riemann hypothesis.
\end{pf}

\begin{thm}[Syntomic Regulator Formula]\label{main-thm3}
Suppose that $p>2$ is prime to $NM$.
Let $\alpha\in W$ such that $\alpha\not\equiv0,1$ mod $p$.
Let $\sigma_{\alpha}$ be the Frobenius given by $t^\sigma=F(\alpha)\alpha^{-p}t^p$
where $F$ is the Frobenius on $W$.
Let $X_\alpha$ be the fiber at $t=\alpha$ ($\Leftrightarrow$ $\l=1-\alpha$), 
which is a smooth projective variety over $W$ of relative dimension one.
Let
\[
\reg_\syn:K_2(X_\alpha)\lra H^2_\syn(X_\alpha,\Q_p(2))\cong H^1_\dR(X_\alpha/K)
\]
be the syntomic regulator map.
Then
\[
\reg_\syn(\xi|_{X_\alpha})=-
N^{-1}M^{-1}
\sum_{i=1}^{N-1}\sum_{j=1}^{M-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)
[\ve^{(i,j)}_1(\alpha)\omega_{i,j}
+\ve^{(i,j)}_2(\alpha)\eta_{i,j}].
\]
\end{thm}
\begin{pf}
This follows from \cite[Theorem 4.4]{AM}.
\end{pf}
\begin{cor}\label{main-thm4}
Let the notation and assumption be as in Theorem \ref{main-thm3}. Suppose further that
$h(\alpha)\not\equiv0$ mod $p$ where $h(t)$ is as in \eqref{syn-reg-eq5}.
Let $e_{N-i,M-j}^{\text{\rm unit}}$ be as in Theorem \ref{uroot-thm}, and
$Q: H^1_\dR(X_\alpha/K)\ot H^1_\dR(X_\alpha/K)\to H^2_\dR(X_\alpha/K)\cong K$ 
the cup-product pairing.
Then we have
\[
Q(\reg_\syn(\xi|_{X_\alpha}), e_{N-i,M-j}^{\text{\rm unit}})
=-N^{-1}M^{-1}
(1-\nu^{-i}_1)(1-\nu^{-j}_2)
\cF_{a_i,b_j}^{(\sigma_\alpha)}(\alpha)
Q(\omega_{i,j}, e_{N-i,M-j}^{\text{\rm unit}}).
\]
\end{cor}
\begin{pf}
Noticing $Q(e_{i,j}^{\text{\rm unit}},e_{N-i,M-j}^{\text{\rm unit}})=0$ by \eqref{uroot-thm-eq3},
this is immediate from Theorem \ref{main-thm3}.
\end{pf}

\subsection{Syntomic regulator of the Ross symbols of Fermat curves}
\label{fermatcurve-sect}
We apply Theorem \ref{fermat-main1} to the study of the syntomic regulator 
of the {\it Ross symbol} \cite{ross2}
\[
\{1-z,1-w\}\in K_2(F)\ot\Q
\]
of the Fermat curve
\[
F:z^N+w^M=1,\quad p{\not|}NM.
\] 
The group $\mu_N\times \mu_M$ acts on $F$ by $(\ve_1,\ve_2)\cdot (z,w)=(\ve_1z,\ve_2w)$.
Let $H^1_\dR(F/K)(i,j)$ denote the subspace on which $(\ve_1,\ve_2)$ acts by multiplication
by $\ve_1^i\ve_2^j$.
Let
\[
I=\left\{(i,j)\in\Z^2\mid1\leq i\leq N-1,1\leq j\leq M-1,\,\frac{i}{N}+\frac{j}{M}\ne1\right\},
\]
then 
\[
H^1_\dR(F/K)=\bigoplus_{(i,j)\in I}H^1_\dR(F/K)(i,j).
\]
Each eigen space $H^1_\dR(F/K)(i,j)$ 
is one-dimensional with basis 
$z^{i-1}w^{j-M}dz=-N^{-1}Mz^{i-N}w^{j-1}dw$, and 
\[
H^1_\dR(F/K)(i,j)\subset \vg(F,\Omega^1_{F/K})\quad\Longleftrightarrow\quad
\frac{i}{N}+\frac{j}{M}<1
\]
(e.g. \cite[\S 2]{gross}).
In particular, the genus of $F$ is $1+\frac{1}{2}(NM-N-M-\gcd(N,M))$.


\begin{thm}\label{fermat-main2}
Suppose that $p>2$ is prime to $NM$.
Let $F:z^N+w^M=1$ be the Fermat curve over $W$.  
Let
\[
\reg_\syn:K_2(F)\ot\Q\lra H^2_\syn(F,\Q_p(2))\cong H^1_\dR(F/K)
\]
be the syntomic regulator map and let $A^{(i,j)}\in K$ be defined by
\[
\reg_\syn(\{1-z,1-w\})=
\sum_{(i,j)\in I} A^{(i,j)}M^{-1}z^{i-1}w^{j-M}dz.
\]
Suppose that $(i,j)\in I$ satisfies
\begin{equation}\label{fermat-main2-eq1}
\mbox{\rm(i) }
\frac{i}{N}+\frac{j}{M}<1,\quad
\mbox{\rm(ii) }
 F_{\frac{i}{N},\frac{j}{M}}(1)_{<p^n}\not\equiv0\mod p,\,\forall n\geq1.
\end{equation}
Then we have
\begin{equation}\label{fermat-main2-eq2}
A^{(i,j)}=\cF^{(\sigma)}_{\frac{i}{N},\frac{j}{M}}(1)
\end{equation}
where $\sigma=\sigma_1$ (i.e. $\sigma(t)=t^p$).
\end{thm}
The following lemma gives a sufficient condition for that 
the conditions \eqref{fermat-main2-eq1} are satisfied.
\begin{lem}\label{fermat-main2-lem1}
\begin{enumerate}
\item[$(1)$]
Let $a,b\in \Z_p$. Then
$F_{a,b}(1)_{<p^n}\not\equiv0$ mod $p$ for all $n\geq1$ if and only if
$F_{a^{(k)},b^{(k)}}(1)_{<p}\not\equiv0$ mod $p$ for all $k\geq0$ where $a^{(k)}$ denotes the
Dwork $k$-th prime.
\item[$(2)$]
Let $a_0,b_0\in\{0,1,,\ldots,p-1\}$ satisfy $a\equiv -a_0$ and $b\equiv -b_0$ mod $p$.
Then 
\[
F_{a,b}(1)_{<p}\equiv \frac{\Gamma(1+a_0+b_0)}{\Gamma(1+a_0)\Gamma(1+b_0)}
=\frac{(a_0+b_0)!}{a_0!b_0!}
\mod p.
\]
In particular
\[
F_{a,b}(1)_{<p}\not\equiv 0
\quad\Longleftrightarrow\quad 
a_0+b_0\leq  p-1.
\]
\item[$(3)$]
Suppose that $N|(p-1)$ and $M|(p-1)$. Then for any 
$(i,j)$ such that $0<i<N$ and $0<j<M$ and $i/N+j/M<1$, 
the conditions \eqref{fermat-main2-eq1} hold.
\end{enumerate}
\end{lem}
\begin{pf}
(1) is a consequence of the Dwork congruence \eqref{Dwork-congruence}.
We show (2). Obviously $F_{a,b}(t)_{<p}\equiv F_{-a_0,-b_0}(t)_{<p}$ mod $p\Z_p[t]$, and
$F_{-a_0,-b_0}(t)_{<p}=F_{-a_0,-b_0}(t)$ as $a_0$ and
$b_0$ are non-positive integers greater than $-p$.
Then apply Gauss' formula (e.g. \cite{NIST} 15.4.20)
\[
{}_2F_1\left({a,b\atop c};1\right)=
\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},\quad \mathrm{Re}(c-a-b)>0.
\]
To see (3), letting $a=i/N$ and $b=j/M$, we note that 
$a^{(k)}=a$, $b^{(k)}=b$ and $a_0=i(p-1)/N$, $b_0=j(p-1)/M$.
Then the condition \eqref{fermat-main2-eq1} (ii) follows by (1) and (2).
\end{pf}

\bigskip

\noindent{\it Proof of Theorem \ref{fermat-main2}}.
We show that the theorem is reduced to the case $M=N$. Let $L$ be the least common multiple of $M,N$,
and $F_1$ the Fermat curve defined by an affine equation $F_1:z^L_1+w_1^L=1$.
There is a finite surjective map $\rho:F_1\to F$ given by $
\rho^*(z)=z_1^A$ $\rho^*(w)=w_1^B$ where $AN=BM=L$.
There is a commutative diagram
\[
\xymatrix{
K_2(F_1)\ot\Q\ar[r]\ar[d]_{\rho_*}&H^2_\syn(F_1,\Q_p(2))\ar[d]^{\rho_*}\\
K_2(F)\ot\Q\ar[r]&H^2_\syn(F,\Q_p(2))
}
\]
with surjective vertical arrows. It is a simple exercise to show that $\rho_*\{1-z_1,
1-w_1\}=\{1-z.1-w\}$ and $\rho_*(L^{-1}z_1^{i-1}w_1^{j-L}dz_1)
=M^{-1}z^{i'-1}w^{j'-M}dz$ if $(i,j)=(i'A,j'B)$ and $=0$ otherwise.
Thus the theorem for $F$ can be deduced from the theorem for $F_1$.

\medskip

We assume $N=M$ until the end of the proof.
Let $f:Y\to\P^1_W$ be as in Lemma \ref{int.model-lem}, which
has a bad reduction at $t=1$.
Let $\l:=1-t$ be a new parameter, and let $\l_0^N=\l$.
Let $\ol S_0:=\Spec W[\l_0,(1-\l^N_0)^{-1}]\to \P^1_W$
and $S_0:=\Spec W[\l_0,\l_0^{-1},(1-\l^N_0)^{-1}]\subset \ol S_0$.
Let $\ol X_s:=Y\times_{\P^1_W}\ol S$. Then
$\ol X_s$ has a unique singular point $(x,y,\l_0)=(0,0,0)$ in an affine open set
\[U_s=\Spec W[x,y,\l_0,(1-\l^N_0)^{-1}]/(x^Ny^N-x^N-y^N-\l_0^N)\subset \ol X_s.\]
Let $\ol X_0\to \ol X_s$ be the blow-up at $(x,y,\l_0)=(0,0,0)$. Then $\ol X_0\to \Spec W$ is smooth, and the morphism
\begin{equation}\label{fermat-main2-lem2}
\xymatrix{
f_0:\ol X_0\ar[r]& \ol S_0
}
\end{equation}
is projective flat
such that $X_0:=f_0^{-1}(S_0)\to S_0$ smooth and $f_0$ has a 
semistable reduction at $\l_0=0$.
The fiber $Z:=f^{-1}(\l_0=0)$ is a reduced divisor with two irreducible components
$F$ and $E$ where $F$ is the proper transform of the curve $x^Ny^N-x^N-y^N=0$
$\Leftrightarrow$ $z^N+w^N=1$ $(z:=x^{-1},w:=y^{-1})$, and $E$ is the exceptional curve. Both curves are
isomorphic to the Fermat curve $u^N+v^N=1$. 
Moreover $E$ and $F$ intersects transversally at $N$-points.


We recall the $K_2$-symbols $\xi(\nu_1,\nu_2)$ in \eqref{m-fermat-eq1}.
We think them to be elements of $K_2(\ol X_0)\ot\Q$, and
put
\[
\Xi:=\sum_{(\nu_1,\nu_2)\in \mu_N\times \mu_N}\xi(\nu_1,\nu_2)
=
\left\{\frac{(x-1)^N}{x^N-1},\frac{(y-1)^N}{y^N-1}\right\}\in K_2(\ol X_0)\ot\Q.
\]
Then the restriction of $\Xi$ on $F$ is 
\begin{align*}
\Xi|_F
&=\{(1-z)^N,(1-w)^N\}-\{(1-z)^N,1-w^N\}-\{1-z^N,(1-w)^N\}+\{1-z^N,1-w^N\}\notag\\
&=\{(1-z)^N,(1-w)^N\}-\{(1-z)^N,z^N\}-\{w^N,(1-w)^N\}+\{w^N,1-w^N\}\notag\\
&=N^2\{1-z,1-w\}
\end{align*}
This is the Ross symbol.
We thus have
\begin{equation}\label{Xi-eq1}
N^2\reg_\syn(\{1-z,1-w\})
\sum_{(\nu_1,\nu_2)\in \mu_N\times \mu_N}\reg_\syn(\xi(\nu_1,\nu_2)|_F).
\end{equation}
Write $\xi=\xi(\nu_1,\nu_2)$ simply.
Let $\sigma$ be the $p$-th Frobenius on $W[\l_0,\l_0^{-1}]$ such that $\sigma(t)=t^p$
$\Leftrightarrow$ $\sigma(\l_0)=(1-(1-\l_0^N)^p)^{\frac1N}$.
Recall \eqref{fermat-e-eq1} and \eqref{fermat-e-eq2},
\begin{align*}
e_\xi-\Phi_\sigma(e_\xi)
&=N^{-2}\sum_{1\leq i,j\leq N-1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)[
\ve^{(i,j)}_{1,\sigma}(t)\omega_{i,j}
+\ve^{(i,j)}_{2,\sigma}(t)\eta_{i,j}]\\
&=N^{-2}\sum_{1\leq i,j\leq N-1}
(1-\nu^{-i}_1)(1-\nu^{-j}_2)[
E^{(i,j)}_{1,\sigma}(t)\wt\omega_{i,j}
+E^{(i,j)}_{2,\sigma}(t)\wt\eta_{i,j}]
\end{align*}
where we write ``$(-)_\sigma$''
to emphasize that they depend on $\sigma$.
Let $\tau$ be the $p$-th Frobenius on $W[[\l_0]]$ such that $\tau(\l_0)=\l_0^p$.
Let
\begin{equation}\label{fermat-main2-eq5}
e_\xi-\Phi_\tau(e_\xi)
=N^{-2}\sum_{1\leq i,j\leq N-1}
(1-\nu^{-i}_1)(1-\nu^{-j}_2)[\ve^{(i,j)}_{1,\tau}(\l)\omega_{i,j}
+\ve^{(i,j)}_{2,\tau}(\l)\eta_{i,j}]
\end{equation}
be defined in the same way.
This is related to \eqref{Xi-eq1} in the following way.
Let
$\Delta:=\Spec W[[\l_0]]\to \ol S_0$, and  $\cX:=f^{-1}(\Delta)$.
We have the syntomic regulator
\[
\reg_\syn(\xi)\in H^2_\syn(\cX,\Z_p(2))
\]
in the syntomic cohomology group. 
We endow the log structure on $\Delta$ (resp. $\cX$)
defined by the divisor $O=\Spec W[[\l_0]]/(\l_0)$ (resp. $E+F$)
which we denote by the same notation $O$ (resp. $E+F$).
Let $\omega_{\cX/\Delta}$ be the log de Rham complex for $(\cX,E+F)/(\Delta,O)$.
Recall the log syntomic cohomology groups (e.g. \cite[\S 2]{T})
\[
H^i_\syn((X,M),\Z_p(j))
\]
of a log scheme $(X,M)$ satisfying several conditions 
(all log schemes appearing in this proof satisfy them).
Moreover one can further define the syntomic cohomology groups
$H^i_\syn((\cX,E+F)/(\Delta,O,\tau),\Z_p(j))$ following the construction in \cite[\S 3.1]{AM}, 
where we note that 
$\tau$ induces the $p$-th Frobenius on $(\Delta,O)$ (while so does not $\sigma$).
Let 
\[
\rho_\tau:H^2_\syn(\cX,\Z_p(2))
\lra H^2_\syn((\cX,E+F)/(\Delta,O,\tau),\Z_p(2))
\os{\cong}{\longleftarrow}H^1_\zar(\cX,\omega^\bullet_{\cX/\Delta})
\]
be the composition of natural maps.
We endow the log structure on $F$ defined by the divisor $T:=E\cap F$ which we denote by $T$. Put $U:=F\setminus T$.
Let $\omega^\bullet_F:=\Omega^\bullet_{F/W}(\log T)$ 
the log de Rham complex for $(F,T)/W$.
Let $\iota:F\hra\cX$ be the closed immersion.
Then there is a commutative diagram
\[
\xymatrix{
H^2_\syn(\cX,\Z_p(2))\ar[d]_{\rho_\tau}\ar[r]& H^2_\syn((F,T),\Z_p(2))\\
H^1_\zar(\cX,\omega^\bullet_{\cX/\Delta})\ar[r]^-{\iota^*}\ar[d]_\pi
&H^1_\zar(F,\omega^\bullet_{F})\ar[u]_\cong\ar[r]^-{\subset}&
H^1_\dR(U/K)\\
W((\l_0))\ot H^1_\dR(X_0/S_0)
}
\]
and we have
\begin{equation}\label{1-ext-thm1-eq1}
(\iota^*\circ\rho_\tau)(\reg_\syn(\xi))=\reg_\syn(\xi|_F)\in H^1_\dR(U/K).
\end{equation}
Moreover it follows from \cite[Theorem 3.8]{AM} that
\begin{equation}\label{1-ext-thm1-eq2}
(\pi\circ\rho_\tau)(\reg_\syn(\xi))=\Phi(e_\xi)-e_\xi
\in 
W((\l_0))\ot H^1_\dR(X_0/S_0).
\end{equation}
Note that $H^1_\dR(F/K)\to H^1_\dR(U/K)$ is injective, and the above element belongs to
the image of $H^1_\dR(F/K)$, so that we may replace $H^1_\dR(U/K)$ with $H^1_\dR(F/K)$
in \eqref{1-ext-thm1-eq1}.

\medskip

Write $X_{0,K}:=X_0\times_WK$ etc.
Let $\Delta_K:=\Spec K[[\l_0]]$ and $\cX_K:=\ol X_0\times_{\ol S_0} \Delta_K$.
Put 
\[
H_K:=H^1(\cX_K,\omega^\bullet_{\cX_K/\Delta_K})\hra K((\l_0))\ot H^1_\dR(X_0/S_0).
\]

\begin{lem}\label{fermat-main2-lem3}
Put $s:=(a_i+b_j)N$ which is a positive integer.
If $a_i+b_j<1$, then the eigencomponent $H_K(i,j)$ is a free $K[[\l_0]]$-module
of rank two
with a basis $\{\omega_{i,j},\l_0^{s}\eta_{i,j}\}$.
\end{lem}
\begin{pf}
This is proven in the same way as the proof of
Corollary \ref{fermat-GM-cor2}.
\end{pf}


\begin{lem}\label{fermat-main2-lem0}
Let $1\leq i,j\leq N-1$ be integers, and put
$a_i:=1-i/N$ and $b_j:=1-j/N$.
Put
\[
f_n(t)=f_{n,i,j}(t):=-\frac{(1-\nu_1^{-i})(1-\nu_2^{-j})}{N^2}\frac{1}{F_{a_i,b_j}(t)}
\left(\frac{d^{n-1}}{dt^{n-1}}\left(\frac{F_{a^{(1)}_i,b^{(1)}_j}(t)}{t}\right)\right)^\sigma
\]
for $n\in\Z_{\geq 1}$. 
Then
\[
\ve^{(i,j)}_{1,\tau}(\l)-\cF^{(\sigma)}_{a_i,b_j}(t)
=\sum_{n=1}^\infty\frac{(t^\tau-t^\sigma)^n}{n!}
p^{-1}f_n(t)
+b_j^{-1}\frac{F'_{a_i,b_j}(t)}{F_{a_i,b_j}(t)}
\ve^{(i,j)}_{2,\tau}(\l).
\]
\end{lem}
Notice that $f_n(t)$ is a convergent function on the rigion 
$\{F_{a_i,b_j}(t)_{<p^n}\not\equiv 0\}$ by
\cite[p.37, Thm. 2, p.45 Lem. 3.4]{Dwork-p-cycle}
\begin{pf}
The relation between $\ve^{(i,j)}_{k,\sigma}(t)$ and $\ve^{(i,j)}_{k,\tau}(t)$ is the following (e.g. \cite[6.1]{EK}, 
\cite[17.3.1]{Ke})
\begin{equation}\label{fermat-main2-eq6}
\Phi_\tau(e_\xi)-\Phi_\sigma(e_\xi)=
\sum_{n=1}^\infty\frac{(t^\tau-t^\sigma)^n}{n!}
\Phi_\sigma\partial^n_t e_\xi
\end{equation}
where $\partial_t=\nabla_{\frac{d}{dt}}$ is the differential operator on $M_\xi(X/S)_\dR$.
By \eqref{m-fermat-eq2},
\[
\partial_t(e_\xi)
=-\sum_{1\leq i,j\leq N-1}\frac{(1-\nu^{-i}_1)(1-\nu^{-j}_2)}{N^2}\frac{1}{t}\omega_{i,j}
=-\sum_{1\leq i,j\leq N-1}\frac{(1-\nu^{-i}_1)(1-\nu^{-j}_2)}{N^2}\frac{F_{a_i,b_j}(t)}{t}\wt\omega_{i,j}.
\] 
Let $\eta_{i,j}^*:=(1-t)^{-a_i-b_j}F_{a_i,b_j}(t)^{-1}\wt\eta_{i,j}\in H^1_\dR(X/S)\ot
K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle$ where $h(t)=\prod_{m=0}^N
F_{a^{(m)}_i,b^{(m)}_j}(t)$ with $N\gg0$.
By Proposition \ref{fermat-GM}, 
\[
\partial^n_t(e_\xi)
=-\sum_{1\leq i,j\leq N-1}\frac{(1-\nu^{-i}_1)(1-\nu^{-j}_2)}{N^2}
\frac{d^{n-1}}{dt^{n-1}}\left(\frac{F_{a_i,b_j}(t)}{t}\right)\wt\omega_{i,j}+(\cdots)\wt\eta_{i,j}
\]
and hence
\[
\Phi_\sigma\partial^n_t(e_\xi)
\equiv \sum_{1\leq i,j\leq N-1}
p^{-1}f_{n,i,j}(t)\mod K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle\eta^*_{i,j}\]
by Proposition \ref{frobenius-thm}.
Take the reduction of the both side of \eqref{fermat-main2-eq6} modulo 
$K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle\eta^*_{i,j}$.
We then have
\[
\ve^{(i,j)}_{1,\tau}(\l)-\ve^{(i,j)}_{1,\sigma}(t)
-b_j^{-1}\frac{F'_{a_i,b_j}(t)}{F_{a_i,b_j}(t)}
(\ve^{(i,j)}_{2,\tau}(\l)-\ve^{(i,j)}_{2,\sigma}(t))
=\sum_{n=1}^\infty\frac{(t^\tau-t^\sigma)^n}{n!}
p^{-1}f_n(t).
\]
On the other hand,
\[
\ve^{(i,j)}_{1,\sigma}(t)=\cF^{(\sigma)}_{a_i,b_j}(t)+b_j^{-1}\frac{F'_{a_i,b_j}(t)}{F_{a_i,b_j}(t)}
\ve^{(i,j)}_{2,\sigma}(t)
\]
by \eqref{syn-reg-eq3}, \eqref{syn-reg-eq4} and Theorem \ref{fermat-main1}.
Hence
\[
\ve^{(i,j)}_{1,\tau}(\l)-\cF^{(\sigma)}_{a_i,b_j}(t)
=\sum_{n=1}^\infty\frac{(t^\tau-t^\sigma)^n}{n!}
p^{-1}f_n(t)
+b_j^{-1}\frac{F'_{a_i,b_j}(t)}{F_{a_i,b_j}(t)}
\ve^{(i,j)}_{2,\tau}(\l)
\]
as required.
\end{pf}
\begin{lem}\label{fermat-main2-lem4}
If $a_i+b_j<1$, then $\ord_{\l=0}(\ve_{1,\tau}^{(i,j)}(\l))\geq 0$ and 
$\ord_{\l=0}(\ve_{2,\tau}^{(i,j)}(\l))\geq 1$.
\end{lem}
\begin{pf}
Since $e_\xi-\Phi_\tau(e_\xi)\in H_K$,
we have
\[
\ve_{1,\tau}^{(i,j)}(\l)\omega_{i,j}+\ve_{2,\tau}^{(i,j)}(\l)\eta_{i,j}\in H_K(i,j).
\]
If $a_i+b_j<1$, then this means
\[
\ve_{1,\tau}^{(i,j)}(\l_0^N)\, , \l_0^{-s}\ve_{2,\tau}^{(i,j)}(\l_0^N)\in K[[\l_0]].
\]
by Lemma \ref{fermat-main2-lem3}. Since $s=(a_i+b_j)N<N$, the assertion follows.
\end{pf}
\begin{lem}\label{fermat-main2-lem5}
If $a_i+b_j<1$ and $F_{a_i,b_j}(1)_{<p^n}\not\equiv 0$ mod $p$ for all $n\geq 1$, then 
\[
\ve_{1,\tau}^{(i,j)}(0)=\cF^{(\sigma)}_{a_i,b_j}(1)
\]
where the left hand side denotes the evaluation at $\l=0$ ($\Leftrightarrow$ $t=1$)
and the right hand side denotes the evaluation at $t=1$.
Note that the left value is defined by Lemma \ref{fermat-main2-lem4}.
\end{lem}
\begin{pf}
This is straightforward from Lemma \ref{fermat-main2-lem0} on noticing that 
$F'_{a_i,b_j}(t)/F_{a_i,b_j}(t)$ and $f_n(t)$ are convergent at $t=1$ by 
\cite[p.45, Lem. 3.4 ]{Dwork-p-cycle} under the condition that
$F_{a_i,b_j}(1)_{<p^n}\not\equiv 0$ mod $p$ for all $n\geq 1$.
\end{pf}

\medskip

We finish the proof of Theorem \ref{fermat-main2}.
Let $(i,j)$ satisfy $a_i+b_j<1$.
Let 
\[\rho_\tau(\reg_\syn(\xi))(i,j)\in H_K(i,j)\] be the eigencomponent of
$\psi_\tau(\reg_\syn(\xi))$, which agrees with
\[
-N^{-2}(1-\nu^{-i}_1)(1-\nu^{-j}_2)[
\ve^{(i,j)}_{1,\tau}(t)\omega_{i,j}
+\ve^{(i,j)}_{2,\tau}(t)\eta_{i,j}]
\]
by \eqref{fermat-main2-eq5} and \eqref{1-ext-thm1-eq2}.
It is straightforward to see $\iota^*(\omega_{i,j})=Nz^{N-i-1}w^{-j}dz.$
Then 
\begin{align*}
&\iota^*[\ve^{(i,j)}_{1,\tau}(\l)\omega_{i,j}+\ve^{(i,j)}_{2,\tau}(\l)\eta_{i,j}]\\
=&
\ve_{1,\tau}^{(i,j)}(\l_0^L)|_{\l_0=0}\cdot \iota^*(\omega_{i,j})+
(\l_0^{-s}\ve_{2,\tau}^{(i,j)}(\l_0^L))|_{\l_0=0}\cdot \iota^*(\l_0^s\eta_{i,j})]
&\text{(Lemma \ref{fermat-main2-lem3})}\\
=&
\ve_{1,\tau}^{(i,j)}(0)\cdot Nz^{N-i-1}w^{-j}dz
&\text{(Lemma \ref{fermat-main2-lem4} and $s<L$)}\\
=&
\cF_{a_i,b_j}^{(\sigma)}(1)\cdot Nz^{N-i-1}w^{-j}dz
& \text{(Lemma \ref{fermat-main2-lem5})}.
\end{align*}
Therefore 
\[\reg_\syn(\xi|_F)
=-N^{-2}
\sum_{a_i+b_j<1}(1-\nu^{-i}_1)(1-\nu^{-j}_2)
\cF_{a_i,b_j}^{(\sigma)}(1)Nz^{N-i-1}w^{-j}dz+\sum_{a_i+a_j> 1}(-)
\]
by \eqref{1-ext-thm1-eq1}.
Taking the summation over $(\nu_1,\nu_2)\in \mu_N\times \mu_M$, we have
\[
N^2\reg_{\syn}(\{1-z,1-w\})=-\sum_{a_i+a_j< 1}\cF_{a_i,b_j}^{(\sigma)}(1)Nz^{N-i-1}w^{-j}dz
+\sum_{a_i+a_j> 1}(-)
\]
by \eqref{Xi-eq1}. This finishes the proof of Theorem \ref{fermat-main2}.
 
 \bigskip
 
    
In \cite{ross2}, Ross showed the non-vanishing of the Beilinson regulator
\[
\reg_B\{1-z,1-w\}\in H^2_\cD(F,\R(2))\cong H^1_B(F,\R)^{F_\infty=-1}
\]          
of his element in the Deligne-Beilinson cohomology group.
We expect the non-vanishing also in the $p$-adic situation.
\begin{conj}
Under the condition \eqref{fermat-main2-eq1},
$\cF_{\frac{i}{N},\frac{j}{M}}^{(\sigma)}(1)\ne0$.
\end{conj}
By the congruence relation for $\cF^{(\sigma)}_{\ul a}(t)$ (Theorem \ref{cong-thm}), 
the non-vanishing $\cF_{\frac{i}{N},\frac{j}{M}}^{(\sigma)}(1)\ne0$ is equivalent to 
\[
G_{\frac{i}{N},\frac{j}{M}}^{(\sigma)}(1)_{<p^n}\not\equiv 0\mod p^n
\]
for some $n\geq 1$.
A number of computations by computer indicate that this holds
(possibly $n\ne1$). 
Moreover if the Fermat curve has a quotient to an elliptic curve over $\Q$, one can expect
that the syntomic regulator agrees with the special value of the $p$-adic $L$-function
according to the $p$-adic Beilinson conjecture by Perrin-Riou \cite[4.2.2]{Perrin-Riou}.
See Conjecture \ref{ell6-conj} below for detail.
\subsection{Syntomic Regulators of Hypergeometric curves of Gauss type}
\label{gauss-sect}
Let $W=W(\ol\F_p)$ and $K=\Frac W$. 
Let $N\geq 2$, $A,B>0$ be integers such that $0<A,B<N$ and $\gcd(N,A)=\gcd(N,B)=1$.
Let $X_{\gauss,K}\to\Spec K[\l,(\l-\l^2)^{-1}]$ a smooth projective morphism
of relative dimension one whose generic fiber is 
defined from an affine equation
\[
v^N=u^A(1-u)^B(1-\l u)^{N-B}.
\]
We call $X_{\gauss,K}$ the {\it hypergeometric curve of Gauss type} (\cite[\S 2.3]{A}).
The genus of a smooth fiber is $N-1$.
Let $X$ be the hypergeometric curve in \S \ref{fermat-sect} defined by an
affine equation $(1-x^N)(1-y^N)=t$.
Then there is a finite cyclic covering 
\begin{equation}
\rho:X\times_WK\lra X_{\gauss,K},\quad
\begin{cases}
\rho^*(u)=x^{-N}\\
\rho^*(v)=x^{-A}(1-x^{-N})y^{N-B}\\
\rho^*(\l)=1-t
\end{cases}
\end{equation}
of degree $N$ whose Galois group is generated by an automorphism 
$g_{A,B}:=[\zeta_N^B,\zeta_N^{-A}]\in \mu_N(K)\times \mu_N(K)$ 
(see \eqref{fermat-ss} for the notation) where $\zeta_N$ is
a fixed primitive $N$-th root of unity
($\rho$ is a generalization of the Fermat quotient, e.g. \cite[p.211]{gross}).

\medskip

Suppose that $N$ is prime to $p$.
We construct an integral model $X_\gauss$ over $W$ in the following way.
The cyclic group $\langle g_{A,B}\rangle$ generated by $g_{A,B}$ acts on $X$,
and it is a free action.
We define
$X_\gauss$ to be the quotient
\[
f:X_\gauss\os{\text{def}}{=}X/\langle g_{A,B}\rangle\lra S=\Spec W[t,(t-t^2)^{-1}].
\]
of $X$ by the cyclic group $\langle g_{A,B}\rangle$. Since it is a free action, $X_\gauss$
is smooth over $W$.
The cyclic group $\mu_N(K)\times \mu_N(K)/\langle g_{A,B}\rangle$ acts on $X_{\gauss,K}$,
and it is generated by an automorphism $h$ given by $(u,v)\mapsto (u,\zeta^{-1}_N v)$.
For an integer $n$, we denote by $V(n)$ the eigenspace on which $h$ acts by
multiplication by $\zeta_N^n$ for all $\zeta_N\in \mu_N(K)$.
Then the pull-back $\rho^*$ satisfies
\begin{equation}\label{gauss-eq1}
\rho^*(H^1_\dR(X_{\gauss,K}/S_K)(n))=H^1_\dR(X_K/S_K)(-nA,-nB),\quad
0< n<N
\end{equation}
and the push-forward $\rho_*$ satisfies
\begin{equation}\label{gauss-eq2}
\rho_*(H^1_\dR(X_K/S_K)(i,j))=\begin{cases}
H^1_\dR(X_{\gauss,K}/S_K)(n)&(i,j)\equiv (-nA,-nB)\text{ mod }N\\
0&\text{otherwise}
\end{cases}
\end{equation}
for $0<i,j<N$.
Put
\begin{equation}\label{gauss-eq3}
\omega_n:=\rho_*(\omega_{-nA,-nB}),\quad 
\eta_n:=\rho_*(\eta_{-nA,-nB})
\end{equation}
a basis of $H^1_\dR(X_{\gauss,K}/S_K)(n)$ (see \eqref{rule} for the notation $(-)_{-nA,-nB}$).
Recall $e^{\mathrm{unit}}_{i,j}$ from 
Theorem \ref{uroot-thm}.
Put
\[
e^{\mathrm{unit}}_n:=\rho_*e^{\mathrm{unit}}_{-nA,-nB}\in 
H^1_\dR(X_{\gauss,K}/S_K)(n)
\ot K\langle t,(t-t^2)^{-1},h(t)^{-1}\rangle
\] 
for $0<n<N$. 
Notice that
$\rho^*(\omega_{-nA,-nB})=N\omega_n$, 
$\rho^*(\eta_{-nA,-nB})=N\eta_n$ and 
$\rho^*e^{\mathrm{unit}}_{-nA,-nB}=Ne^{\mathrm{unit}}_n$ by \eqref{gauss-eq1} and
\eqref{gauss-eq2} together with the fact that $\rho_*\rho^*=N$.
We put
\begin{equation}\label{gauss-eq4}
\xi_\gauss=\xi_\gauss(\nu_1,\nu_2):=\rho_*\xi(\nu_1,\nu_2)\in K_2(X_\gauss)^{(2)}\subset
K_2(X_\gauss)\ot\Q
\end{equation}
where $\xi(\nu_1,\nu_2)$ is as in the beginning of \S \ref{syn-reg-sect}.
Let $\sigma_{\alpha}$ be the Frobenius given by $t^\sigma=ct^p$ with $c\in 1+pW$.
Taking the fixed part of \eqref{syn-reg-ext} by $\langle g_{A,B}\rangle$,
we have a $1$-extension
\[
0\lra H^1(X_\gauss/S)(2)\lra M_{\xi_\gauss}(X_\gauss/S)\lra \O_S\lra 0
\]
in the exact category $\FilFMIC(S)$.
Let $e_{\xi_\gauss}\in \Fil^0M_{\xi_\gauss}(X_\gauss/S)_\dR$ 
be the unique lifting of $1\in \O(S)$. 
\begin{thm}\label{gauss-main1}
Put $a_n:=nA/N-\lfloor nA/N\rfloor$ and $b_n:=nB/N-\lfloor nB/N\rfloor$.
Let $h(t)=\prod_{m=0}^sF_{a_n^{(m)},b_n^{(m)}}(t)_{<p}$ where $s$ is the minimal
integer such that $(a_n^{(s+1)},b_n^{(s+1)})=(a_n,b_n)$ for all $n\in\{1,2,\ldots,N-1\}$.
Then 
\[
e_{\xi_\gauss}-\Phi(e_{\xi_\gauss})\equiv -
\sum_{n=1}^{N-1}\frac{(1-\nu^{nA}_1)(1-\nu^{nB}_2)}{N^2}\cF^{(\sigma)}_{a_n,b_n}(t)
\omega_n\]
modulo $\sum_{n=1}^{N-1}K\langle t,(t-t^2),h(t)^{-1}\rangle e^{\mathrm{unit}}_n.$
\end{thm}
\begin{pf}
This is immediate by applying $\rho_*$ on the formula in Theorem \ref{fermat-main1}.
\end{pf}
\begin{cor}\label{gauss-main2}
Suppose that $p>2$ is prime to $N$.
Let $a\in W$ such that $a\not\equiv0,1$ mod $p$.
Let $\sigma_a$ be the Frobenius given by $t^\sigma=F(a)a^{-p}t^p$
where $F$ is the Frobenius on $W$.
Let $X_{\gauss,a}$ be the fiber at $t=a$ ($\Leftrightarrow$ $\l=1-a$), 
which is a smooth projective variety over $W$ of relative dimension one.
Let
\[
\reg_\syn:K_2(X_{\gauss,a})\lra 
H^2_\syn(X_{\gauss,a},\Q_p(2))\cong H^1_\dR(X_{\gauss,a}/K)
\]
be the syntomic regulator map. 
Let $Q$ be the cup-product pairing on $H^1_\dR(X_{\gauss,a}/K)$.
Then
\[
Q(\reg_\syn(\xi_\gauss|_{X_a}),e^{\mathrm{unit}}_{N-n})=
N^{-2}
(1-\nu^{nA}_1)(1-\nu^{nB}_2)
\cF_{a_n,b_n}^{(\sigma_a)}(a)
Q(\omega_n,e^{\mathrm{unit}}_{N-n}).
\]
\end{cor}
\begin{pf}
This is immediate from Theorem \ref{gauss-main1}
on noticing $Q(e^{\mathrm{unit}}_n,e^{\mathrm{unit}}_m)=0$, cf. Corollary \ref{main-thm4}.
\end{pf}





\subsection{Syntomic Regulators of elliptic curves}\label{elliptic-sect}
The methods in \S \ref{syn-reg-sect} work not only for the hypergeometric curves but
also for the elliptic fibrations listed in \cite[\S 5]{A}.
We here give the results together with a sketch of the proof because the discussion
is similar to before.
\begin{thm}\label{elliptic-thm1}
Let $p\geq 5$ be a prime number. Let $W=W(\ol\F_p)$ be the Witt ring and $F:=\Frac(W)$
the fractional field.
Let $f:Y\to\P^1$ be the elliptic fibration defined by a Weierstrass equation
$3y^2=2x^3-3x^2+1-t$ over $W$. Put $\omega=dx/y$.
Let 
\[\xi:=\left\{
\frac{y-x+1}{y+x-1},
\frac{t}{2(x-1)^3}
\right\}\in K_2(X),\quad X:=Y\setminus f^{-1}(0,1,\infty).
\]
Let $a\in W$ satisfy that
$a\not\equiv0,1$ mod $p$ and $X_a$
has a good ordinary reduction where $X_a$ is the fiber at 
$\Spec W[t]/(t-a)$.
Let $e_{\text{\rm unit}}\in H^1_\dR(X_a/K)$ be the eigen vector of
the unit root with respect to the $p$-th Frobenius $\Phi$.
Let $\sigma_a$ denote the $p$-th Frobenius given by $\sigma_a(t)=F(a)
a^{-p}t^p$.
Then, we have
\[
Q(\reg_\syn(\xi|_{X_a}), e_{\text{\rm unit}})
=\cF_{\frac{1}{6},\frac{5}{6}}^{(\sigma_a)}(a)
Q(\omega,e_{\text{\rm unit}})
\]
\end{thm}
\begin{pf} (sketch).
We first note that
\[
\dlog(\xi)=\frac{dx}{y}\frac{dt}{t}=\omega\wedge\frac{dt}{t}.
\]
Let $\cE$ be the fiber over the formal neighborhood $\Spec \Z_p[[t]]\hra \P^1_{\Z_p}$.
Let $\rho:\bG_m\to \cE$ be the uniformization, and $u$ the uniformizer of $\bG_m$.
Then we have 
\[
\rho^*\omega=F(t)\frac{du}{u}
\]
with $F(t)$ a formal power series in $\Z_p[[t]]$ which
satisfies the Picard-Fuchs equation. Namely $F(t)$ is a solution 
of the differential equation
\[
(t-t^2)\frac{d^2y}{dt^2}+(1-2t)\frac{dy}{dt}-\frac{5}{36}y=0.
\]
Therefore $F(t)$ coincides with the hypergeometric power series 
\[
F_{\frac{1}{6},\frac{5}{6}}(t)={}_2F_1\left({\frac{1}{6},\frac{5}{6}\atop 1};t\right)
\]
up to scalar.
Looking at the residue of $\omega$ at the point $(x,y,t)=(1,0,0)$, one finds that
the constant is $1$.
Hence we have
\[
\rho^*\omega=F_{\frac{1}{6},\frac{5}{6}}(t)\frac{du}{u}.
\]
Then the rest of the proof goes in the same way as Theorem \ref{fermat-main1}.
\end{pf}

\begin{thm}\label{elliptic-thm2}
Let $f:Y\to\P^1$ be the elliptic fibration defined by a Weierstrass equation
$y^2=x^3+(3x+4t)^2$, and 
\[\xi:=\left\{
\frac{y-3x-4t}{-8t},
\frac{y+3x+4t}{8t}
\right\}.
\]
Then, under the same notation and assumption in Theorem \ref{elliptic-thm1}, we have
\[
Q(\reg_\syn(\xi|_{X_a}), e_{\text{\rm unit}})
=3\cF_{\frac{1}{3},\frac{2}{3}}^{(\sigma_a)}(a)Q(
\omega,e_{\text{\rm unit}}).
\]
\end{thm}
\begin{pf}
Let $\cE$ be the fiber over the formal neighborhood $\Spec \Z_p[[t]]\hra \P^1_{\Z_p}$,
and let $\rho:\bG_m\to \cE$ be the uniformization.
Then one finds
\[
\dlog(\xi)=-3\frac{dx}{y}\frac{dt}{t}=-3\omega\wedge\frac{dt}{t}
\]
and 
\[
\rho^*\omega=\frac{1}{3}F_{\frac{1}{3},\frac{2}{3}}(t)\frac{du}{u}.
\]
The rest is the same as before.
\end{pf}

\begin{thm}\label{elliptic-thm3}
Let $f:Y\to\P^1$ be the elliptic fibration defined by a Weierstrass equation
$y^2=x^3-2x^2+(1-t)x$, and 
\[\xi:=\left\{\frac{y-(x-1)}{y+(x-1)},\frac{-t x}{(x-1)^3}\right\}.
\]
Then, under the same notation and assumption in Theorem \ref{elliptic-thm1}, we have
\[
Q(\reg_\syn(\xi|_{X_a}), e_{\text{\rm unit}})
=\cF_{\frac{1}{4},\frac{3}{4}}^{(\sigma_a)}(a)
Q(\omega,e_{\text{\rm unit}}).
\]
\end{thm}
\begin{pf}
One finds
\[
\dlog(\xi)=\frac{dx}{y}\frac{dt}{t}=\omega\wedge\frac{dt}{t}
\]
and
\[
\rho^*\omega=F_{\frac{1}{4},\frac{3}{4}}(t)\frac{du}{u}.
\]
The rest is the same as before.
\end{pf}
\section{$p$-adic Beilinson conjecture for elliptic curves over $\Q$}\label{weakB-sect}
\subsection{Statement}
The Beilinson regulator is a generalization of Dirichlet's regulators of number fields.
in higher $K$-groups of varieties.
He conjectured the formulas on the regulators and special values of motivic $L$-functions
which generalize the analytic class number formula.
For an elliptic curve $E$ over $\Q$, Beilinson proved 
that there is an integral symbol $\xi\in K_2(E)$
such that the real regulator $\reg_\R(\xi)\in H^2_\cD(E,\R(2))\cong \R$ 
agrees with the special value of the $L$-function $L(E,s)$ of $E$,
\[
\reg_\R(\xi)\sim_{\Q^\times}L'(E,0)\quad
(\Longleftrightarrow\,
\reg_\R(\xi)/L'(E,0)\in \Q^\times)
\]
where
$L'(E,s)=\frac{d}{ds}L(E,s)$ (\cite[Theorem 1.3]{B-reg}).

The $p$-adic counterpart of the Beilinson conjecture
was formulated by Perrin-Riou \cite[4.2.2]{Perrin-Riou},
which we call the {\it $p$-adic Beilinson conjecture}.
See also \cite{Colmez} for a general survey.
Her conjecture is formulated in terms of the $p$-adic etale regulators (which
are compatible with the syntomic regulators thanks to Besser's theorem) and the conjectural
$p$-adic measures which provide the $p$-adic $L$-functions of motives.
However
there are only a few of literatures
due to the extremal difficulty of the statement.
In a joint paper \cite{ABC} with Chida, 
we give a concise statement of the $p$-adic Beilinson
conjecture by restricting ourselves to the case of elliptic curves over $\Q$.

\begin{conj}[Weak $p$-adic Beilinson conjecture]\label{weakB-conj}
Let $E$ be an elliptuic curve over $\Q$.
Let $p>2$ be a prime at which $E$ has a good ordinary reduction.
Let $E_{\Q_p}:=E\times_\Q\Q_p$ and 
let $e_{\text{\rm unit}}\in H^1_\dR(X_\alpha/K)$ be the eigen vector for
the unit root $\alpha_{E,p}$ of the $p$-th Frobenius $\Phi$.
Let $L_p(E,\chi,s)$ be the $p$-adic $L$-function by Mazur and Swinnerton-Dyer
\cite{MS}.
Let $Q:H^1_\dR(E_{\Q_p}/\Q_p)^{\ot2}\to H^2_\dR(E_{\Q_p}/\Q_p)\cong\Q_p$ be the cup-product pairing.
Let 
\[
\reg_\syn:K_2(E)\to H^2_\syn(E,\Q_p(2))\cong H^1_\dR(E_{\Q_p}/\Q_p)
\]
be the syntomic regulator map. 
Fix a regular 1-form $\omega_E\in \vg(E,\Omega^1_{E/\Q})$.
Let $\omega:\Z_p^\times\to\mu_{p-1}$ be the Teichm\"uller character.
Then there is a constant $C\in \Q^\times$ which does not depend on $p$ such that
\[
(1-p\alpha_{E,p}^{-1})
\frac{Q(\reg_\syn(\xi),e_\unit)}{Q(\omega_E,e_\unit)}
=CL_p(E,\omega^{-1},0).
\]
\end{conj}
In \cite[Conjecture 3.3]{ABC}, we give a precise description of the constant $C$
in terms of the real regulator.




\subsection{Conjecture on Rogers-Zudilin type formulas}\label{RZ-sect}
In their paper \cite{RZ}, Rogers and Zudilin give descriptions of special values of
$L$-functions of elliptic curves in terms of
the hypergeometric functions ${}_3F_2$ or ${}_4F_3$. 
Apply theorems in \S \ref{elliptic-sect} to Conjecture \ref{weakB-conj}, we have
a statement of the $p$-adic counterpart of the Rogers-Zudilin type formulas.


\medskip

The following is obtained from Conjecture \ref{weakB-conj} and 
Corollary \ref{gauss-main2}
(note that the symbol \eqref{RZ-eq2} below
agrees with $\xi_\gauss|_{X_a}$ in \S \ref{gauss-sect} up to a constant). 
\begin{conj}\label{ell1-conj}
Let $X\to \P^1$ be the elliptic fibration defined by a Weierstrass equation
$y^2=x(1-x)(1-(1-t)x)$ (i.e. the hypergeometric curve of Gauss type, cf. \S \ref{gauss-sect}).
Let $X_a$ be the fiber at $t=a\in\Q\setminus\{0,1\}$.
Suppose that the symbol
\begin{equation}\label{RZ-eq2}
\xi_a=\left\{\frac{y-1+x}{y+1-x},\frac{a x^2}{(1-x)^2}\right\}\in K_2(X_a)
\end{equation}
is integral in the sense of Scholl \cite{Scholl}.
Let $p>2$ be a prime such that
$X_a$ has a good ordinary reduction at $p$. 
Let $\sigma_a:\Z_p[[t]]\to\Z_p[[t]]$ be the $p$-th Frobenius given by
$\sigma_a(t)=a^{1-p}t^p$.
Let $\alpha_{X_a,p}$ be the unit root.
Then there is a rational number $C_a\in \Q^\times$ not depending on $p$ such that
\[
(1-p\alpha^{-1}_{X_a,p})\cF_{\frac{1}{2},\frac{1}{2}}^{(\sigma_a)}(a)=C_a
L_p(X_a,\omega^{-1},0)
\]
where $\omega$ is the Teichm\"uller character.
\end{conj}
Here are examples of $a$ such that the symbol \eqref{RZ-eq2} is integral 
(cf. \cite[5.4]{A})
\[
a=-1,\pm2,\pm4,\pm8,\pm16,
\pm\frac{1}{2},  
\pm\frac{1}{8},\pm\frac{1}{4}, \pm\frac{1}{16}.
\]
F. Brunault compared
the symbol \eqref{RZ-eq2} with the Beilinson-Kato element in case $a=4$
(\cite[Appendix B]{ABC}).
Then, thanks to the main result of his paper \cite{regexp}, 
it follows that $\reg_\syn(\xi_a)$ gives the $p$-adic $L$-value
of $X_4$ (see \cite[Theorem 5.2]{ABC} for the precise statement). 
We thus have
\begin{thm
\label{brunault}
When $a=4$, Conjecture \ref{ell1-conj} is true and $C_4=-1$.
\end{thm}



We obtain the following statements from
Theorems \ref{elliptic-thm1}, \ref{elliptic-thm2} and \ref{elliptic-thm3}.
\begin{conj}\label{ell2-conj}
Let $a\in \Q\setminus\{0,1\}$ and 
let $X_a$ be the ellptic curve over $\Q$ defined by an affine equation
$3y^2=2x^3-3x^2+1-a$. 
Suppose that the symbol
\begin{equation}\label{RZ-eq3}
\left\{
\frac{y-x+1}{y+x-1},
\frac{a}{2(x-1)^3}
\right\}\in K_2(X_a)
\end{equation}
is integral.
Let $p>3$ be a prime such that 
$X_a$ has a good ordinary reduction at $p$. 
Then there is a rational number $C_a\in \Q^\times$ not depending on $p$ such that
\[
(1-p\alpha^{-1}_{X_a,p})\cF_{\frac{1}{6},\frac{5}{6}}^{(\sigma_a)}(a)=C_a
L_p(X_a,\omega^{-1},0).
\]
\end{conj}
There are infinitely many $a$ such that
the symbol \eqref{RZ-eq3} is integral.
For example, if $a=1/n$ with
$n\in\Z_{\geq 2}$ and $n\equiv 0,2$ mod $6$,
then 
the symbol \eqref{RZ-eq3} is integral (cf. \cite[5.4]{A}).

\begin{conj}\label{ell3-conj}
Let $a\in \Q\setminus\{0,1\}$ and 
let $X_a$ be the ellptic curve over $\Q$ defined by an affine equation
$y^2=x^3+(3x+4a)^2$.
Suppose that the symbol
\begin{equation}\label{RZ-eq4}
\left\{
\frac{y-3x-4a}{-8a},
\frac{y+3x+4a}{8a}
\right\}
\in K_2(X_a)
\end{equation}
is integral.
Let $p>3$ be a prime such that 
$X_a$ has a good ordinary reduction at $p$. 
Then there is a rational number $C_a\in \Q^\times$ not depending on $p$ such that
\[
(1-p\alpha^{-1}_{X_a,p})\cF_{\frac{1}{3},\frac{2}{3}}^{(\sigma_a)}(a)=C_a
L_p(X_a,\omega^{-1},0).
\]
\end{conj}
If $a=\frac{1}{6n}$ with
$n\in\Z_{\geq 1}$ arbitrary,
then 
the symbol \eqref{RZ-eq4} is integral (cf. \cite[5.4]{A}).
\begin{conj}\label{ell4-conj}
Let $a\in \Q\setminus\{0,1\}$ and 
let $X_a$ be the elliptic curve over $\Q$ defined by an affine equation
$y^2=x^3-2x^2+(1-a)x$.
Suppose that the symbol
\begin{equation}\label{RZ-eq5}
\left\{\frac{y-(x-1)}{y+(x-1)},\frac{-a x}{(x-1)^3}\right\}
\in K_2(X_a)
\end{equation}
is integral.
Let $p>2$ be a prime such that
$X_a$ has a good ordinary reduction at $p$. 
Then there is a rational number $C_a\in \Q^\times$ not depending on $p$ such that
\[
(1-p\alpha^{-1}_{X_a,p})\cF_{\frac{1}{4},\frac{3}{4}}^{(\sigma_a)}(a)=C_a
L_p(X_a,\omega^{-1},0).
\]
\end{conj}
If the denominator of $j(X_a)=64(1+3a)^3/(a(1-a)^2)$ is prime to
$a$ (e.g. $a=1/n$, $n\in \Z_{\geq 2}$), then 
the symbol \eqref{RZ-eq5} is integral.

\medskip

From Corollary \ref{main-thm4} and Theorem \ref{fermat-main2}, 
we have the following conjectures.

\begin{conj}\label{ell6-conj}
Let $a\in \Q\setminus\{0,1\}$ and 
let $X_a$ be the ellptic curve over $\Q$ defined by an affine equation
$(x^2-1)(y^2-1)=a$.
Suppose that the symbol
\begin{equation}\label{RZ-eq6}
\left\{\frac{x-1}{x+1},\frac{y-1}{y+1}\right\}\in K_2(X_a)
\end{equation}
is integral.
Let $p>2$ be a prime such that 
$X_a$ has a good ordinary reduction at $p$. 
Then there is a rational number $C_a\in \Q^\times$ not depending on $p$ such that
\[
(1-p\alpha^{-1}_{X_a,p})\cF_{\frac{1}{2},\frac{1}{2}}^{(\sigma_a)}(1)=C_a
L_p(X_a,\omega^{-1},0).
\]
\end{conj}
If the denominator of $j(X_a)=16(a^2-16a+16)^3/((1-a)a^4)$
is prime to $a$ (e.g. $a=\pm 2^n$, $n\in \{\pm 1,\pm 2,\pm 3\}$), then 
the symbol \eqref{RZ-eq6} is integral.
\begin{conj}\label{ell7-conj}
Let $F_{N,M}$ be the Fermat curve defined by an affine equation 
$z^N+w^M=1$, and $F^*_{2,4}$ the curve $z^2=w^4+1$. 
Let $\sigma=\sigma_1$ (i.e. $\sigma(t)=t^p$).
Then there are rational numbers $C,C^\prime,
C^{\prime\prime}\in \Q^\times$ not depending on $p$ such that
\[
(1-p\alpha^{-1}_{F_{3,3},p})\cF_{\frac{1}{3},\frac{1}{3}}^{(\sigma)}(1)=C
L_p(F_{3,3},\omega^{-1},0),
\]
\[
(1-p\alpha^{-1}_{F_{2,4},p})\cF_{\frac{1}{2},\frac{1}{4}}^{(\sigma)}(1)=C^\prime
L_p(F_{2,4},\omega^{-1},0),
\]
\[
(1-p\alpha^{-1}_{F_{2,4}^*,p})\cF_{\frac{1}{4},\frac{1}{4}}^{(\sigma)}(1)=C^{\prime\prime}
L_p(F_{2,4}^*,\omega^{-1},0).
\]
\end{conj}

