# Algebraic relations of special $v$-adic arithmetic gamma values and their
period interpretations

Chieh-Yu Chang  Department of Mathematics, National Tsing Hua University,
Hsinchu City 30042, Taiwan R.O.C<EMAIL_ADDRESS>,  Fu-Tsun Wei
Department of Mathematics, National Tsing Hua University, Hsinchu City 30042,
Taiwan R.O.C<EMAIL_ADDRESS>and  Jing Yu  Department of Mathematics,
National Taiwan University, Taipei City 10617, Taiwan R.O.C.
<EMAIL_ADDRESS>
###### Abstract.

Let $v$ be a finite place of $\mathbb{F}_{q}(\theta)$. We show that algebraic
relations of $v$-adic arithmetic gamma values over $\mathbb{F}_{q}(\theta)$
are explained by the standard functional equations together with Thakur’s
analogue of the Gross–Koblitz formula. A key step is working out a formula
expressing $v$-adic crystalline–de Rham periods of Carlitz motives with
Complex Multiplication as products of these special gamma values (a $v$-adic
Chowla–Selberg formula). The algebraic independence of those periods in
question results from determining the dimension of the motivic Galois groups,
adapting and further developing existing tools.

###### 2010 Mathematics Subject Classification:

Primary 11R58, 11J93

##  1\. Introduction

###  1.1. Motivation

This study is inspired by the Gross–Koblitz formula [GK79] and its function
field analogue established by Thakur [T88]. For a prime number $p$, Morita
[Mo75] defined a continuous function
$\Gamma_{p}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}^{\times}$ via a $p$-adic
interpolation of the factorials $n!$ (up to a sign). The special $p$-adic
gamma values we are interested in, are the values $\Gamma_{p}(z)$ for
$z\in\mathbb{Q}\cap\mathbb{Z}_{p}$. Since the very unique $\pi$ is missing in
the $p$-adic domains, $\Gamma_{p}(1/2)$ for odd prime $p$, becomes algebraic.
For an odd prime $p$, Gross–Koblitz managed to express a Gauss sum explicitly
in terms of a product of specific special Morita $p$-adic gamma values. As a
consequence, they derived the algebraicity of the special $p$-adic gamma value
$\Gamma_{p}(z)$ whenever the denominator of $z\in\mathbb{Q}\cap\mathbb{Z}_{p}$
is a factor of $p-1$.

Classically, one interpolates the factorials $n!$ to obtain the Euler gamma
function, and a folklore conjecture asserts that all the special gamma values
$\Gamma(z)$, for $z\in\mathbb{Q}\setminus\mathbb{Z}$, are transcendental. This
is only known when the denominator is $2$ as $\Gamma(\frac{1}{2})=\sqrt{\pi}$,
or when the denominator is $4$ or $6$. The latter case is due to Chudnovsky’s
work [Chud84, p. 6-8] (see also [BW07, p. 30]) on the algebraic independence
of a nonzero period and its corresponding quasi-period of a CM elliptic curve
defined over $\overline{\mathbb{Q}}$. The Lang-Rohrlich conjecture [La90]
asserts that all $\overline{\mathbb{Q}}$-algebraic relations among special
gamma values and $2\pi\sqrt{-1}$ are explained by the standard functional
equations satisfied by the Euler gamma function.

Back to the $p$-adic case, Gross-Koblitz’s algebraicity result clearly reveals
a deviation of the $p$-adic interpolations from the classical $\infty$-adic
interpolation. It is a natural question whether for any odd prime $p$, all the
special Morita gamma values are transcendental except those explained in
Gross-Koblitz’s paper. One may ask further whether there is a ‘$p$-adic
Lang–Rohrlich phenomenon’ (namely, the known existing algebraic relations
arising from the standard functional equations that Morita’s $p$-adic gamma
function satisfies together with the Gross-Koblitz formula) accounting for all
$\overline{\mathbb{Q}}$-algebraic relations among the special $p$-adic gamma
values. Our first main result (Theorem 1.2.1) is to give an affirmative answer
for this question in the positive characteristic world.

In this article, we investigate the $v$-adic arithmetic gamma function for the
rational function field $k$ of one variable over a finite field, after L.
Carlitz [Car35], D. Goss [Go80] and D. Thakur [T88], [T91]. In this case, the
transcendence of $\infty$-adic special arithmetic gamma values was well-known
before the 21 century [T96, All96]. For any finite place $v$ of $k$, we are
able to show that algebraicity of special arithmetic $v$-adic gamma values
comes only from Thakur’s analogue of Gross-Koblitz. The case of special
$v$-adic geometric gamma values will be treated in a forthcoming work.

Another motivation of our study on special $v$-adic arithmetic gamma values
comes from the classical Chowla-Selberg formula, which relates a nonzero
period of a CM elliptic curve over $\overline{\mathbb{Q}}$ to a specific
product of special gamma values. Inspired by the $p$-adic Chowla-Selberg
formula established by Oguz [O89], we prove in Theorem 1.3.1 that certain
crystalline–de Rham periods of Carlitz motives with complex multiplication can
be expressed as products of special $v$-adic arithmetic gamma values. It turns
out that such a period interpretation plays a crucial role in the process of
proving algebraic independence result of Theorem 1.2.1.

###  1.2. Algebraic relations among $v$-adic arithmetic gamma values

Let $A:=\mathbb{F}_{q}[\theta]$ be the polynomial ring in one variable
$\theta$ over a finite field $\mathbb{F}_{q}$ of characteristic $p$ with $q$
elements, and $k:=\mathbb{F}_{q}(\theta)$ be its fraction field. Put
$D_{0}:=1$, and for $i\geq 1$,
$D_{i}:=\prod_{j=0}^{i-1}(\theta^{q^{i}}-\theta^{q^{j}})\in A$. For each
$i\in\mathbb{N}$, $D_{i}$ equals the product of all monic polynomials of
degree $i$ in $A$. Carlitz [Car35] introduced his factorial for $A$ as
follows: given a non-negative integer $n$, write $n=\sum_{i\geq 0}n_{i}q^{i}$
for $0\leq n_{i}<q$ and put

$\Gamma_{\mathrm{ari},\infty}(n+1):=\prod_{i\geq 0}D_{i}^{n_{i}}\in A,$

which becomes the positive characteristic analogue of the factorial $n!$, see
(2.1.1). Taking the unit part of these factorial values, Goss [Go80]
discovered that one may do interpolation in Morita style, arriving at a
continuous function:

$\Gamma_{\mathrm{ari},\infty}:\mathbb{Z}_{p}\rightarrow\mathbb{F}_{q}(\\!(1/\theta)\\!)^{\times}.$

Goss successfully applied his strategy to all places of $k$. Given a monic
irreducible polynomial $v\in\mathbb{F}_{q}[\theta]$, let $k_{v}$ be the
completion of $k$ at $v$. Let $\bar{k}_{v}$ be a fixed algebraic closure of
$k_{v}$, and $\bar{k}$ be the algebraic closure of $k$ inside $\bar{k}_{v}$.
Using $v$-adic interpolation for the Carlitz factorials, Goss [Go88] and
Thakur [T91] defined a $v$-adic arithmetic gamma function
$\Gamma_{\mathrm{ari},v}:\mathbb{Z}_{p}\rightarrow k_{v}^{\times}$, whose
precise definition is given in Section 2.2.

In the $\infty$-adic case, the special values
$\Gamma_{\mathrm{ari},\infty}(z)$, for
$z\in\mathbb{Q}\cap\mathbb{Z}_{p}\setminus\mathbb{Z}$, are all transcendental
over $k$ by Thakur [T96] and Allouche [All96]. Furthermore, using
transcendence theory in terms of $t$-motives developed by Anderson-Brownawell-
Papanikolas [ABP04] and Papanikolas [P08], all algebraic relations among these
special values are shown to follow from the standard monomial relations
[CPTY10]. These relations come directly from the functional equations
satisfied by $\Gamma_{\mathrm{ari},\infty}$. Note that
$\mathbb{Q}\cap\mathbb{Z}_{p}=\bigcup_{\ell\in\mathbb{N}}\frac{1}{q^{\ell}-1}\cdot\mathbb{Z}$.
In particular, one has:

$\operatorname{tr.deg}_{k}k\big{(}\Gamma_{\mathrm{ari},\infty}(z)\ \big{|}\
z\in\mathbb{Q}\text{ with }(q^{\ell}-1)z\in\mathbb{Z}\big{)}=\ell.$

In the $v$-adic case, the functional equations of $\Gamma_{\text{ari},v}$
derived in [Go88] and [T91] also give rise to standard monomial relations
(2.2.3) among $v$-adic arithmetic gamma values at rational $p$-adic integers.
However for the finite place $v$, one naturally encounters the Gross–Koblitz
phenomenon, as shown by Thakur [T88]. This gives additional relations among
the special values in question (see Theorem 2.2.5 and Corollary 2.2.7). The
first main result of this paper is to prove the following theorem which
illustrates the $v$-adic Lang-Rohrlich phenomenon.

######  Theorem 1.2.1.

Given a positive integer $\ell$, we have that

$\operatorname{tr.deg}_{\bar{k}}\bar{k}\big{(}\Gamma_{\mathrm{ari},v}(z)\
\big{|}\ z\in\mathbb{Q}\text{ with
}(q^{\ell}-1)z\in\mathbb{Z}\big{)}=\ell-\gcd(\ell,\deg v).$

Consequently, all the $\bar{k}$-algebraic relations among $v$-adic arithmetic
gamma values at rational $p$-adic integers are explained by the functional
equations in Proposition 2.2.1, the standard monomial relations (2.2.3) and
Thakur’s analogue of Gross–Koblitz formula.

The decrease of transcendence degree for $v$-adic special values resulting
from the analogue of Gross-Koblitz formula is precisely given by
$\gcd(\ell,\deg v)$. Also, the above theorem complements Thakur’s theorem (cf.
Remark 4.1.10):

######  Corollary 1.2.2.

(See Corollary 4.1.9.) Given $a,b\in\mathbb{Z}$ with $\gcd(a,b)=1$ and $p\nmid
b$, we have that

$\Gamma_{\mathrm{ari},v}(\frac{a}{b})\text{ is transcendental over
$k$}\quad\text{if and only if}\quad q^{\deg v}\not\equiv 1\bmod b.$

######  Remark 1.2.3.

In the case of $\deg v=1$, Thakur [T98] initially derived the transcendence of
$\Gamma_{\mathrm{ari},v}(\frac{a}{b})$ in the aforementioned corollary using
methods from automata theory for special cases. Thakur’s approach was later
generalized to encompass general $\frac{a}{b}$ in [WY02]. Also, we point out
that a $v$-adic Anderson–Brownawell–Papanikolas criterion in [Chen23] would
lead to an alternative proof of the above corollary when $\deg v=1$.

Note that the naturally existing monomial relations indicate that the
transcendence degree specified in Theorem 1.2.1 is bounded above by
$\ell-\gcd(\ell,\deg v)$, see Theorem 2.2.9. To derive the opposite
inequality, we shall express the $v$-adic arithmetic gamma values in terms of
the $v$-adic periods of the “crystalline–de Rham comparison isomorphism” for
Carlitz $t$-motives with complex multiplication by constant field extensions.

###  1.3. A $v$-adic Chowla–Selberg formula

Let $\mathbb{F}_{q}[t]$ be the polynomial ring in another variable $t$ over
$\mathbb{F}_{q}$. Given a positive integer $\ell$, we let ${\bf{C}}_{\ell}$ be
the Carlitz $\mathbb{F}_{q^{\ell}}[t]$-module, then ${\bf{C}}_{\ell}$ is a CM
Drinfeld $\mathbb{F}_{q}[t]$-module of rank $\ell$ with CM field
$\mathbb{F}_{q^{\ell}}(t)$. In [CPTY10, Theorems 3.3.1 and 3.3.2], it is shown
that periods of the dual $t$-motive associated to ${\bf{C}}_{\ell}$ are
expressed as products of special $\infty$-adic gamma values (up to algebraic
multiples). By [Pe08, Sec. 4.2] (see also [CP12, (3.4.5) and (3.4.8)]),
periods and quasi-periods of ${\bf{C}}_{\ell}$ are explicitly related to the
periods of the dual $t$-motive in question, and hence combining these gives a
Chowla-Selberg formula for special $\infty$-adic arithmetic gamma values.

Here we shall present a $v$-adic counterpart of the phenomenon above. Let
$\matheur{C}_{\ell}$ be the Carlitz $t$-motive of degree $\ell$ over
$\mathbb{F}_{q}[\theta]$, and denote by
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ the ‘restriction of scalars’ of
$\matheur{C}_{\ell}$ (see Section 3.1). We note that the terminalogy ‘degree
$\ell$’ indicates that $\matheur{C}_{\ell}$ has complex multiplication by the
degree $\ell$ constant field extension over $\mathbb{F}_{q}(t)$. As
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ is defined ‘integrally’ over
$\mathbb{F}_{q}[\theta]$ and has ‘good reduction’ at $v$, one may associate
its crystalline module
$H_{\textrm{cris}}(\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v})$ and de Rham
module $H_{\textrm{dR}}(\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v})$ over
$k_{v}$ (see Remark 3.5.12). Moreover, the theory of Hartl–Kim in [HaKi20]
provides a crystalline–de Rham comparison isomorphism (see (3.5.13))

$\phi_{v}:H_{\textrm{cris}}(\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{\textrm{dR}}(\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}).$

Therefore the natural “isocrystal action” $\tau_{v}$ on
$H_{\textrm{cris}}(\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v})$ transforms
to a $k_{v}$-linear automorphism $\varrho_{v}$ on
$H_{\textrm{dR}}(\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v})$ through
$\phi_{v}$ (see (3.7.1) for the precise definition). We show in the following
theorem that ${\varrho}_{v}$ can be expressed explicitly in terms of special
$v$-adic arithmetic gamma values.

######  Theorem 1.3.1.

(See Theorem 3.7.7.) Choose the ‘global’ basis
$\\{\omega_{0},...,\omega_{\ell-1}\\}$ of the de Rham module
$H_{\textrm{dR}}(\matheur{R}_{\tau}(\matheur{C}_{\ell}),k)$ as in (3.7.4). For
each integer $s$ with $0\leq s<\ell$, we write $s+d=s_{0}+n_{s}^{\prime}\ell$,
where $s_{0}\in\mathbb{Z}$ with $0\leq s_{0}<\ell$ and
$n_{s}^{\prime}\in\mathbb{Z}_{\geq 0}$. Then we have

$\varrho_{v}\cdot\omega_{s}=(-1)^{n_{s}^{\prime}}\cdot
C_{s}\cdot\displaystyle\frac{\displaystyle\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\frac{q^{s+d}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}}{\displaystyle\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\frac{q^{s+d-1}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}^{q}}\cdot\omega_{s_{0}},\quad\text{
where }\quad C_{s}:=\begin{cases}-v,&\text{ if $s=0$,}\\\ 1,&\text{ if
$s>0$,}\end{cases}$

and $\langle z\rangle_{\operatorname{ari}}$ is the fractional part of $z$ for
every $z\in\mathbb{Q}$, i.e. $0\leq\langle z\rangle_{\operatorname{ari}}<1$
and $z-\langle z\rangle_{\operatorname{ari}}\in\mathbb{Z}$.

######  Remark 1.3.2.

We may regard Theorem 1.3.1 as a function field analogue of Ogus’ $p$-adic
Chowla–Selberg formula in [O89, Theorem 3.15].

The proof of Theorem 1.3.1 is based on an explicit description of the
crystalline–de Rham comparison isomorphism $\phi_{v}$ and a product expansion
formula for the special values of $\Gamma_{\mathrm{ari},v}$ in question (see
Corollary 2.3.5). By choosing a specific ‘global’
$\mathbb{F}_{q}[\theta][t]$-base $\mathfrak{B}$ of
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ as in (3.6.2), which induces a basis
$\Lambda_{v}=\\{\lambda_{v,0},...,\lambda_{v,\ell-1}\\}$ of
$H_{\textrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}$
(in (3.6.5)) and a basis $\beta_{v}=\\{\omega_{v,0},...,\omega_{v,\ell-1}\\}$
of $H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}$
(in (3.6.4)), we particularly have

$\omega_{v,s}=\begin{cases}\left(\displaystyle\prod_{n=1}^{n_{s}-1}(\theta-\theta^{q^{\ell
n+s}})\right)\cdot\omega_{s},&\text{ if $s=0$,}\\\
\left(\displaystyle\prod_{n=0}^{n_{s}-1}(\theta-\theta^{q^{\ell
n+s}})\right)\cdot\omega_{s},&\text{ if $s>0$,}\end{cases}$

given in (3.7.5), and the “period matrix” of $\phi_{v}$ with respect to
$\Lambda_{v}$ and $\beta_{v}$ is determined by the following result:

######  Theorem 1.3.3.

(See Theorem 3.6.6.) For every integer $s$ with $0\leq s<\ell$, we have

$\phi_{v}(\lambda_{v,s})=\prod_{n=n_{s}}^{\infty}(1-\frac{(\theta-\varepsilon)^{q^{\ell
n+s}}}{\theta-\varepsilon^{q^{\ell n+s}}})\cdot\omega_{v,s}.$

Here $\varepsilon=\lim_{i\rightarrow\infty}\theta^{q^{i\deg v}}$ $\in k_{v}$
is the Teichmüller image of $\theta\bmod v\in A/(v)$, and $n_{s}$ is the
smallest non-negative integer so that $s+n_{s}\ell\geq\deg v$. Consequently,
the period matrix of the isomorphism $\phi_{v}$ in (3.5.13) with respect to
$\beta_{v}$ and $\Lambda_{v}$ is

$\begin{pmatrix}\displaystyle\prod_{n=n_{0}}^{\infty}(1-\frac{(\theta-\varepsilon)^{q^{\ell
n}}}{\theta-\varepsilon^{q^{\ell n}}})&&0\\\ &\ddots&\\\
0&&\displaystyle\prod_{n=n_{\ell-1}}^{\infty}(1-\frac{(\theta-\varepsilon)^{q^{\ell
n+\ell-1}}}{\theta-\varepsilon^{q^{\ell n+\ell-1}}})\end{pmatrix}.$

Having this period matrix at hand, we are able to realize the action of
$\varrho_{v}$ on the basis $\beta_{v}$ as in (3.7.3). Combining the product
formula of $v$-adic arithmetic gamma values in Corollary 2.3.5 enables us to
derive Theorem 1.3.1.

###  1.4. Algebraic independence of the $v$-adic crystalline–de Rham periods

From the $v$-adic Chowla–Selberg formula in Theorem 1.3.1 and the concrete
description of the period matrix in Theorem 1.3.3, our proof of Theorem 1.2.1
relies on the algebraic independence of the following $v$-adic periods:

######  Theorem 1.4.1.

Given a positive integer $\ell$, the infinite products

$\Omega_{\ell,v,s}(v):=\prod_{n=0}^{\infty}(1-\frac{(\theta-\varepsilon)^{q^{\ell
n+s}}}{\theta-\varepsilon^{q^{\ell n+s}}})\quad\in k_{v},\quad 0<s\leq\ell,$

are algebraically independent over $\bar{k}$.

Indeed, by Theorem 1.4.1, the $v$-adic Chowla–Selberg formula enables us to
construct sufficiently many algebraically independent elements in the field

$\bar{k}\big{(}\Gamma_{\mathrm{ari},v}(z)\ \big{|}\ z\in\mathbb{Q}\text{ with
}(q^{\ell}-1)z\in\mathbb{Z}\big{)},$

for which one ensures that its transcendence degree over $\bar{k}$ is bounded
below by $\ell-\gcd(\ell,\deg v)$ (see Theorem 4.1.4), and hence proves
Theorem 1.2.1.

1.4.2. Strategy of deriving Theorem 1.4.1. In order to apply the powerful
machinery of Papanikolas’ theory on Grothendieck’s period conjecture in [P08],
our initial step is to regard $v$ as the “infinite place” of the projective
$\frac{1}{v}$-line over $\mathbb{F}_{q}$. More precisely, put
$\tilde{v}:=\frac{1}{v}$. Then $k$ is a finite extension of
$\mathbb{F}_{q}(\tilde{v})$, and $k_{v}$ is a finite extension of the Laurent
series field $\mathbb{F}_{q}(\\!(\frac{1}{\tilde{v}})\\!)$ (which is the
completion of $\mathbb{F}_{q}(\tilde{v})$ at the place $\infty_{\tilde{v}}$
corresponding to $\frac{1}{\tilde{v}}$). So $k_{v}$ is an
‘$\infty_{\tilde{v}}$-adic’ field. Let
$\iota:\mathbb{F}_{q}[t]\cong\mathbb{F}_{q}[\theta]$ be the
$\mathbb{F}_{q}$-algebra isomorphism sending $t$ to $\theta$, which extends to
a field isomorphism $\iota:\mathbb{F}_{q}(t)\cong\mathbb{F}_{q}(\theta)$. Put
$\tilde{T}_{v}:=\iota^{-1}(\tilde{v})\in\mathbb{F}_{q}(t)$. Then $\iota$
induces an $\mathbb{F}_{q}$-algebra isomorphism
$\mathbb{F}_{q}[\tilde{T}_{v}]\cong\mathbb{F}_{q}[\tilde{v}]$. To apply
Papanikolas’s theory [P08] for those special values in our
$\infty_{\tilde{v}}$-adic fields, we make the following replacements

$\theta\leftarrow\tilde{v},\quad{}t\leftarrow\tilde{T}_{v}$

to switch $\overline{k}[t]$-modules (resp. dual $t$-motives) to
$\overline{k}[\tilde{T}_{v}]$-modules (resp. dual $\tilde{T}_{v}$-motives). A
detailed description of how to view $v$-adic fields as
$\infty_{\tilde{v}}$-adic fields is given in Sec. 5.1.

Let $d:=\deg v$. Our proof of Theorem 1.4.1 is outlined in the following:

  * (1)

Define a “Carlitz-type” Hartl–Juschka module
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ (see Definition 6.1.3),
where the infinite products $\Omega_{d\ell,v,s}(v)$ in Theorem 1.4.1
(replacing $\ell$ by $d\ell$), for $0<s\leq d\ell$, show up as periods of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ (see Proposition 6.3.4).

  * (2)

By Papanikolas’ theorem, determining the transcendence degree of the field
generated by the periods of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ is equivalent to finding
the dimension of its $\tilde{T}_{v}$-motivic Galois group
$\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}$ (see (6.3.6)).

  * (3)

The “CM structure” of $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$
provides an upper bound of
$\dim\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}$ (cf.
(5.2.6)). On the other hand, from the analysis of the Hodge–Pink weights of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$, we obtain sufficiently
many “Hodge–Pink cocharacters” of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$, from which we get

$\dim\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}=d^{2}\ell\quad(=[\mathbb{F}_{q^{d\ell}}(t):\mathbb{F}_{q}(\tilde{T}_{v})]),$

see Theorem 6.4.4.

  * (4)

Therefore, the period interpretation of $\Omega_{d\ell,v,s}(v)$ in (1) above
for $0<s\leq d\ell$ shows their algebraic independence over $k$ (see Section
6.4). Finally, the result follows from the identity (see (6.4.7)):

$\Omega_{\ell,v,s}(v)=\prod_{j=0}^{d-1}\Omega_{d\ell,v,j\ell+s}(v),\quad
0<s\leq\ell.$

######  Remark 1.4.3.

Put $\tilde{\theta}_{v}:=(\theta-\varepsilon)^{-1}$, and set

$\tilde{\Omega}_{\ell}(x):=\prod_{n=0}^{\infty}(1-\frac{x}{\tilde{\theta}_{v}^{q^{\ell
n}}})\quad\in k_{v}[\\![x]\\!],\quad\forall\ell\in\mathbb{N}.$

When $\deg v=1$, we get $v=\theta-\varepsilon$, for which
$\varepsilon\in\mathbb{F}_{q}$. Therefore

$\Omega_{\ell,v,s}(v)=\tilde{\Omega}_{\ell}^{(s)}(x)\big{|}_{x=\tilde{\theta}_{v}}=\prod_{n=0}^{\infty}(1-\frac{\tilde{\theta}_{v}}{\tilde{\theta}_{v}^{q^{\ell
n+s}}}),$

and Theorem 1.4.1 can be simply derived from the algebraic independence of
periods and the quasi-periods of the Carlitz module of degree $\ell$ in
[CPTY10] (when replacing $\theta$ to $\tilde{\theta}_{v}$). However, when
$\deg v=d>1$, the infinite product $\Omega_{d\ell,v,s}(v)$, for $0<s\leq
d\ell$, is actually the $s$th Frobenius twist of $\tilde{\Omega}_{d\ell}(x)$
evaluated at $x=\tilde{\theta}_{v,s}:=(\theta-\varepsilon^{q^{s}})^{-1}$:

$\Omega_{d\ell,v,s}(v)=\tilde{\Omega}_{d\ell}^{(s)}(x)\big{|}_{x=\tilde{\theta}_{v,s}}=\prod_{n=0}^{\infty}(1-\frac{\tilde{\theta}_{v,s}}{\tilde{\theta}_{v}^{q^{d\ell
n+s}}}).$

Thus, to show the algebraic independence of $\Omega_{d\ell,v,s}(v)$ is
equivalent to determine the algebraic relations among the evaluations of the
Frobenius twists of $\tilde{\Omega}_{d\ell}(x)$ at $d$ distinct points
$x=\tilde{\theta}_{v,1},...,\tilde{\theta}_{v,d}$. In order to overcome this
issue, we consider the “restriction of scalars” down to
$\mathbb{F}_{q}[\tilde{T}_{v}]$, so that the periods of the Hartl–Juschka
$\bar{k}[\tilde{T}_{v}]$-module
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ do include all of these
evaluations.

######  Remark 1.4.4.

Theorem 1.4.1 implies that

$\operatorname{tr.deg}_{\bar{k}}\bar{k}\Big{(}\Omega_{\ell,v,s}(v)\ \Big{|}\
0<s\leq\ell\Big{)}=\ell=\dim\Gamma_{\matheur{R}_{\tau}(\matheur{C}_{\ell})},$

where $\Gamma_{\matheur{R}_{\tau}(\matheur{C}_{\ell})}$ is the “$t$-motivic
Galois group of the $t$-motive $\matheur{R}_{\tau}(\matheur{C}_{\ell})$” (see
[CPTY10, Lemma 3.2.1]). This coincidence appears to provide evidence for a
suitable analogue of Papanikolas’ theorem in [P08] (Grothendieck period
conjecture) associated with the $v$-adic periods that arise from
crystalline–de Rham comparison isomorphisms.

###  1.5. Organization of this paper

In Section 2, we start with fixing basic notation and recalling the Carlitz
factorials in Section 2.1. The definition of Goss’ $v$-adic arithmetic gamma
function and its functional equations is given in Section 2.2, as well as
Thakur’s analogue of Gross–Koblitz formula. In Section 2.3, we derive a
product expansion of $v$-adic arithmetic gamma values, a pivotal observation
that underscores their connection with the ”$v$-adic periods” Carlitz
$t$-motives with complex multiplication by constant field extensions.

In Section 3, we first introduce the “integral” Carlitz $t$-motive
$\matheur{C}_{\ell}$ of arbitrary degree $\ell$, along with its restriction of
scalars $\matheur{R}_{\tau}(\matheur{C}_{\ell})$ in Section 3.1. After setting
up the power series rings to be used and clarifying their relations in Section
3.2, we revisit the crystalline module and de Rham module of
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ in Section 3.3 and 3.4, respectively.
The crystalline–de Rham comparison isomorphism is explicitly constructed in
Section 3.5, leading to a detailed portrayal of the crystalline–de Rham period
matrix in Section 3.6. The proof of Theorem 1.3.1 is provided in Section 3.7.

In Section 4, we prove Theorem 1.2.1 on the algebraic independence of $v$-adic
arithmetic gamma values in Section 4.1, assuming the transcendence result of
crystalline periods in Theorem 1.4.1. Also, the distribution associated with
$v$-adic arithmetic gamma values is discussed in Section 4.2.

In Section 5, we lay the groundwork for the proof of Theorem 1.4.1. Section
5.1 introduces additional notation specifically concerning the treatment of
$v$ as the infinite place of the projective $\frac{1}{v}$-line. For
completeness, Section 5.2 offers the definition of Hartl–Juschka modules, the
associated motivic Galos groups, and the de Rham pairing. Also stated in
Section 5.2 is Papanikolas’s theorem, which serves as the function field
analogue of Grothedieck period conjecture. The Hodge–Pink cocharacters of a
Hartl–Juschka module is defined in Section 5.3.

In Section 6, we introduce Carlitz-type Hartl–Juschka modules and examine
their corresponding Betti modules. The definition of the module
$\tilde{\mathscr{C}}_{d\ell}$, as well as its restriction of scalars
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$, is provided in Section
6.1. Section 6.2 gives a detailed account of the uniformizability of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$, after writing down
explicitly the structure of the corresponding Betti modules. We articulate how
the crystalline–de Rham periods of $\matheur{R}_{\tau}(\matheur{C}_{d\ell})$
can be expressed as “abelian integrals” in Section 6.3. Finally, the motivic
Galois group of $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ is
determined in Section 6.4, culminating with the proof of Theorem 1.4.1.

##  2\. $v$-adic arithmetic gamma values

###  2.1. Notation

Throughout this paper, we fix the following notation. Let $p$ be a prime
number, and let $q$ be a power of $p$. Denote by $\mathbb{F}_{q}$ the finite
field of $q$ elements. Let $\theta$ be a variable and let
$A:=\mathbb{F}_{q}[\theta]$ be the polynomial ring in $\theta$ over
$\mathbb{F}_{q}$ with quotient field $k:=\mathbb{F}_{q}(\theta)$. For a monic
irreducible polynomial $v\in A$, we still denote by $v$ the corresponding
finite place of $k$ when it is clear from the context. The completion of $k$
at $v$ is denoted by $k_{v}$, and the residue field is denoted by
$\mathbb{F}_{v}$. In particular, we may regard $\mathbb{F}_{v}$ as a subfield
of $k_{v}$ via the Teichmüller embedding, which sends $\theta\bmod v$ to
$\varepsilon:=\lim_{i\rightarrow\infty}\theta^{q^{i\deg v}}$ (see [La02, Chap.
XII Ex. 16]), and $k_{v}$ is then identified with the Laurant series field
$\mathbb{F}_{v}(\\!(\theta-\varepsilon)\\!)$. Note that in the case $v=\theta$
we have $\varepsilon=0$, otherwise $\varepsilon$ generates all the roots of
unity in the local field $k_{v}$.

Choose an algebraic closure $\bar{k}_{v}$ of $k_{v}$, and denote by
$\mathbb{C}_{v}$ the completion of $\bar{k}_{v}$. Let $\bar{k}$ be the
algebraic closure of $k$ in $\mathbb{C}_{v}$. Let $A_{+}$ be the set of all
monic polynomials in $A$. Put $D_{0}:=1$, and for $i\geq 1$,
$D_{i}:=\prod_{j=0}^{i-1}(\theta^{q^{i}}-\theta^{q^{j}})\in A$. For each
$i\in\mathbb{N}$, we denote by $A_{+,i}$ the set of all monic polynomials of
degree $i$, and note that $D_{i}=\prod_{a\in A_{+,i}}a$ (see [Car35, p. 140]).
Carlitz [Car35] introduced the so-called Caritz factorials as the following.
For a non-negative integer $n$, we write $n=\sum_{i\geq 0}n_{i}q^{i}$ for
$0\leq n_{i}<q$ and define

$\Gamma_{\mathrm{ari}}(n+1):=\prod_{i\geq 0}D_{i}^{n_{i}}\in A.$

Note that $\Gamma_{\mathrm{ari}}(n+1)$ is analogous to the classical Euler
$\Gamma$-values $\Gamma(n+1)=n!$ for $n\geq 0$ as one has the property that
(due to W. Sinnott, see [Go88, Proposition 2.1.4])

(2.1.1) ${\rm{ord}}_{v}\left(\Gamma_{\mathrm{ari}}(n+1)\right)=\sum_{e\geq
1}\lfloor\frac{n}{q^{e\deg v}}\rfloor,$

where $q^{\deg v}$ is the cardinality of $A/(v)$.

###  2.2. Definition of $v$-adic arithmetic gamma function

Following Goss [Go80, Go88], and Thakur [T91], we consider the $v$-adic
arithmetic gamma function that interpolates the Carlitz factorials
$v$-adically as follows. For each $i\geq 0$, we define

$D_{i,v}:=\prod a,$

where $a$ in the product above runs over all monic polynomial of degree $i$
with $v\nmid a$. For any $z\in\mathbb{Z}_{p}$, the ring of $p$-adic integers,
we express $z$ as $z=\sum_{i\geq 0}z_{i}q^{i}$ for $0\leq z_{i}<q$ and define

$\Gamma_{\mathrm{ari},v}(z+1):=\prod_{i\geq 0}(-D_{i,v})^{z_{i}}.$

It is shown in [Go80, Appendix] that the infinite product $\prod_{i\geq
0}(-D_{i,v})^{z_{i}}$ converges in $k_{v}^{\times}$, and
$\Gamma_{\mathrm{ari},v}:\mathbb{Z}_{p}\rightarrow k_{v}^{\times}$ is a
continuous function satisfying the following functional equations:

######  Proposition 2.2.1.

(See [T04, Thm. 4.6.1, Lem. 4.6.2, and Thm. 4.6.3]) Let $v\in A_{+}$ be
irreducible. For each $z\in\mathbb{Z}_{p}$, we have

  * (1)

(Reflection)

$\Gamma_{\mathrm{ari},v}(z)\cdot\Gamma_{\mathrm{ari},v}(1-z)=\Gamma_{\mathrm{ari},v}(0).$

  * (2)

(Translation) When $z\neq 0$, put

$r=\text{\rm ord}_{q}(z):=\max\big{\\{}m\in\mathbb{Z}_{\geq 0}\ \big{|}\
q^{m}|z\big{\\}}.$

Then

$\frac{\Gamma_{\mathrm{ari},v}(z+1)}{\Gamma_{\mathrm{ari},v}(z)}=-\frac{D_{r,v}}{\prod_{i=0}^{r-1}D_{i,v}^{q-1}}.$

  * (3)

(Multiplication) For every $n\in\mathbb{N}$ with $p\nmid n$, we have

$\prod_{i=0}^{n-1}\Gamma_{\mathrm{ari},v}(z+\frac{i}{n})=\Gamma_{\mathrm{ari},v}(nz)\cdot\Gamma_{\mathrm{ari},v}(0)^{\frac{n-1}{2}}.$

As it is known that (see [T91, Theorem 4.4])

$\Gamma_{\mathrm{ari},v}(1-\frac{1}{q-1})^{q-1}=\Gamma_{\mathrm{ari},v}(0)=(-1)^{\deg
v-1},$

the square root of $\Gamma_{\mathrm{ari},v}(0)$ on the right hand side of the
equality in Proposition 2.2.1 (3) is chosen to be

$\Gamma_{\mathrm{ari},v}(0)^{\frac{1}{2}}:=\Gamma_{\mathrm{ari},v}(1-\frac{1}{q-1})^{\frac{q-1}{2}}.$

When taking values at a rational $p$-adic integer $z$, one has the following
standard monimial relations which directly come from the $q$-adic expansion of
$z$:

######  Proposition 2.2.2.

Let $v\in A_{+}$ be irreducible. Given $\ell\in\mathbb{N}$ and
$c_{0},...,c_{\ell-1}\in\mathbb{Z}$ with $0\leq c_{0},...,c_{\ell-1}<q$, we
have

(2.2.3)
$\Gamma_{\mathrm{ari},v}(1-\sum_{i=0}^{\ell-1}\frac{c_{i}q^{i}}{q^{\ell}-1})=\prod_{i=0}^{\ell-1}\Gamma_{\mathrm{ari},v}(1-\frac{q^{i}}{q^{\ell}-1})^{c_{i}}.$

######  Remark 2.2.4.

Regarding $\bar{k}$-algebraic relations among $v$-adic arimetic gamma values
at rational $p$-adic integers, in [T91, Sec. 2] Thakur showed that once we
have the monomial relations (2.2.3) together with translation (2) of
Proposition 2.2.1 (for $z\in\mathbb{Z}_{p}\cap\mathbb{Q}$), then the
reflection (1) and multiplication (3) of Proposition 2.2.1 are valid for
$z\in\mathbb{Z}_{p}\cap\mathbb{Q}$.

Thakur’s analogue of the Gross–Koblitz formula in [T88, Theorem VI] implies:

######  Theorem 2.2.5.

(See [T88, Corollary of Thm. V].) Let $v\in A_{+}$ be irreducible. For each
$z\in\mathbb{Q}$ with $(q^{\deg v}-1)\cdot z\in\mathbb{Z}$, we have

$\Gamma_{\mathrm{ari},v}(z)\in\bar{k}^{\times}.$

######  Remark 2.2.6.

In fact, Thakur’s formula in [T88, Theorem VI] shows that when
$z\in\mathbb{Q}$ with $(q^{\deg v}-1)\cdot z\in\mathbb{Z}$, the special value
$\Gamma_{\mathrm{ari},v}(z)$ is related to Thakur’s analogue of Gauss sums.
Consequently, in this case one knows that

$\Gamma_{\mathrm{ari},v}(z)\in\mathbb{F}_{q^{\deg
v}}(\theta)\Big{(}\lambda_{v},\sqrt[q^{\deg v}-1]{-v}\Big{)},$

where $\lambda_{v}$ is a non-zero Carlitz $v$-torsion.

As a consequence of Thakur’s analogue of the Gross–Koblitz formula, we have
the following monomial relations:

######  Corollary 2.2.7.

Let $v\in A_{+}$ be irreducible. Given a positive integer $n$ with $p\nmid n$,
let $r$ be the order of $q^{\deg v}\bmod n$ in
$(\mathbb{Z}/n\mathbb{Z})^{\times}$. For every $z\in\mathbb{Q}$ satisfying
$n\cdot z\in\mathbb{Z}$, we have that

$\prod_{i=0}^{r-1}\Gamma_{\mathrm{ari},v}(q^{i\deg v}\cdot
z)\in\bar{k}^{\times}.$

###### Proof.

Without loss of generality, we may assume that $z=1-\frac{a}{n}$ with $0\leq
a<n$. The assumption of $n$ indicates that $n\mid q^{r\deg v}-1$. Write
$q^{r\deg v}-1=n\cdot n^{\prime}$. Then there exist $z_{0},...,z_{r\deg
v-1}\in\mathbb{Z}$ with $0\leq z_{0},...,z_{r\deg v-1}<q$ satisfying that

$\frac{a}{n}=\frac{an^{\prime}}{q^{r\deg v}-1}=\sum_{j=0}^{r\deg
v-1}\frac{z_{j}q^{j}}{q^{r\deg v}-1}.$

Therefore

$\displaystyle\prod_{i=0}^{r-1}\Gamma_{\mathrm{ari},v}(q^{i\deg v}\cdot z)$
$\displaystyle=\prod_{i=0}^{r-1}\Gamma_{\mathrm{ari},v}\Big{(}q^{i\deg
v}\cdot(1-\frac{a}{n})\Big{)}$
$\displaystyle\sim\prod_{i=0}^{r-1}\prod_{j=0}^{r\deg
v-1}\Gamma_{\mathrm{ari},v}\Big{(}1-\frac{z_{j}q^{j+i\deg v}}{q^{r\deg
v}-1}\Big{)}$ $\displaystyle\sim\prod_{j=0}^{r\deg
v-1}\Gamma_{\mathrm{ari},v}\Big{(}1-\frac{\sum_{i=0}^{r-1}q^{j+i\deg
v}}{q^{r\deg v}-1}\Big{)}^{z_{j}}$ $\displaystyle=\prod_{j=0}^{r\deg
v-1}\Gamma_{\mathrm{ari},v}\Big{(}1-\frac{q^{j}}{q^{\deg
v}-1}\Big{)}^{z_{j}}\quad\in\bar{k}^{\times}\quad\text{(by Theorem~{}\ref{thm:
GK-formula})}.$

Here, for $x,y\in\mathbb{C}_{v}^{\times}$, we denote by $x\sim y$ if
$x/y\in\bar{k}^{\times}$. ∎

######  Remark 2.2.8.

Since when $n\mid(q^{\deg v}-1)$, the order of $q^{\deg v}\bmod n$ in
$(\mathbb{Z}/n\mathbb{Z})^{\times}$ is $1$. Therefore Corollary 2.2.7 is in
fact equivalent to Theorem 2.2.5.

The above monomial relations among special values of $\Gamma_{\mathrm{ari},v}$
at rational $p$-adic integers give rise to the following upper bound of the
transcendence degree in question.

######  Theorem 2.2.9.

Let $v\in A_{+}$ be irreducible. Given $\ell\in\mathbb{N}$, we have

$\operatorname{tr.deg}_{\bar{k}}\bar{k}\Big{(}\Gamma_{\mathrm{ari},v}(z)\
\Big{|}\ z\in\mathbb{Q}\text{ with }(q^{\ell}-1)\cdot
z\in\mathbb{Z}\Big{)}\leq\ell-\gcd(\ell,\deg v).$

###### Proof.

The functional equation in Proposition 2.2.1 (2) and the monomial relation in
Proposition 2.2.2 show that

$\bar{k}\Big{(}\Gamma_{\mathrm{ari},v}(z)\ \Big{|}\ z\in\mathbb{Q}\text{ with
}(q^{\ell}-1)\cdot
z\in\mathbb{Z}\Big{)}=\bar{k}\Big{(}\Gamma_{\mathrm{ari},v}(1-\frac{q^{i}}{q^{\ell}-1})\
\Big{|}\ 0\leq i\leq\ell-1\Big{)}.$

Let $m=\gcd(\ell,\deg v)$ and $\ell^{\prime}=\ell/m$. By Proposition 2.2.2 and
Theorem 2.2.5, we get

(2.2.10)
$\prod_{j=0}^{\ell^{\prime}-1}\Gamma_{\mathrm{ari},v}(1-\frac{q^{i+mj}}{q^{\ell}-1})=\Gamma_{\mathrm{ari},v}(1-\frac{q^{i}}{q^{m}-1})\in\bar{k},\quad\forall
i\in\mathbb{Z},\ 0\leq i<m,$

since $(q^{m}-1)$ divides $(q^{\deg v}-1)$. The monomial relations (2.2.10)
for $0\leq i\leq m-1$ give rise to $m$ independent algebraic relations, from
which the result follows. ∎

We will prove in Section 4 that the inequality stated in Theorem 2.2.9 is, in
fact, an equality. Before doing so, we first provide “period interpretations”
of the $v$-adic arithmetic gamma values in question.

###  2.3. Product expansions of $v$-adic arithmetic gamma values

We begin with the following observation:

######  Lemma 2.3.1.

Let $v\in A_{+}$ be irreducible of degree $d$. Given $\ell,s\in\mathbb{N}$,
let $n_{s}$ be the smallest non-negative integer such that $s+n_{s}\ell\geq
d$. We have that

$\displaystyle\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s}}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s-1}}{q^{\ell}-1})^{q}}$
$\displaystyle=\frac{\displaystyle\prod_{n=0}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}{\displaystyle\prod_{n=n_{s}+1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell-d}}}{\theta-\varepsilon^{q^{s+n\ell-d}}}\bigg{)}}\cdot\bigg{(}\prod_{n=0}^{n_{s}}(\varepsilon^{q^{s+n\ell}}-\theta)\Big{)}$
$\displaystyle\hskip
79.6678pt\cdot\begin{cases}(\theta^{q^{s+n_{s}\ell-d}}-\theta)^{-1},&\text{ if
$s+n_{s}\ell>d$,}\\\ v^{-1},&\text{ if $s+n_{s}\ell=d$.}\end{cases}$

###### Proof.

From the definition of $\Gamma_{\mathrm{ari},v}$, we get

$\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s}}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s-1}}{q^{\ell}-1})^{q}}=\prod_{n=0}^{\infty}\frac{D_{s+n\ell,v}}{D_{s-1+n\ell,v}^{q}}.$

Note that for every $n\in\mathbb{N}$, one has that

$\frac{D_{n}}{D_{n-1}^{q}}=(\theta^{q^{n}}-\theta)\quad\text{ and }\quad
D_{n,v}=\begin{cases}\displaystyle\frac{D_{n}}{v^{q^{n-d}}D_{n-d}},&\text{ if
$n\geq d$,}\\\ D_{n},&\text{ if $n<d$.}\end{cases}$

Thus

(2.3.2)
$\frac{D_{n,v}}{D_{n-1,v}^{q}}=\begin{cases}\displaystyle\frac{\theta^{q^{n}}-\theta}{\theta^{q^{n-d}}-\theta}&\text{
if $n>d$,}\\\ \displaystyle\frac{\theta^{q^{n}}-\theta}{v},&\text{ if
$n=d$,}\\\ \theta^{q^{n}}-\theta,&\text{ if $0<n<d$.}\end{cases}$

Observe that:

(2.3.3)
$\theta^{q^{n}}-\theta=(\varepsilon^{q^{n}}-\theta)\cdot\Big{(}1-\frac{(\theta-\varepsilon)^{q^{n}}}{\theta-\varepsilon^{q^{n}}}\Big{)}$

and

(2.3.4)
$\frac{\theta^{q^{n}}-\theta}{\theta^{q^{n-d}}-\theta}=\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{n}}}{\theta-\varepsilon^{q^{n}}}\bigg{)}\bigg{/}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{n-d}}}{\theta-\varepsilon^{q^{n}}}\bigg{)}\quad\text{
when $n>d$.}$

As $n_{s}$ is the smallest non-negative integer such that $s+n_{s}\ell\geq d$,
we obtain that

$\displaystyle\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s}}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s-1}}{q^{\ell}-1})^{q}}=\prod_{n=0}^{\infty}\frac{D_{s+n\ell,v}}{D_{s-1+n\ell,v}^{q}}$
$\displaystyle=\ $
$\displaystyle\left(\prod_{n=n_{s}+1}^{\infty}\frac{D_{s+n\ell,v}}{D_{s-1+n\ell,v}^{q}}\right)\cdot\left(\prod_{n=0}^{n_{s}-1}\frac{D_{s+n\ell,v}}{D_{s-1+n\ell,v}^{q}}\right)\cdot\frac{D_{s+n_{s}\ell,v}}{D_{s-1+n_{s}\ell,v}^{q}}$
$\displaystyle=\ $
$\displaystyle\left(\prod_{n=n_{s}+1}^{\infty}\frac{\theta^{q^{s+n\ell}}-\theta}{\theta^{q^{s+n\ell-d}}-\theta}\right)\cdot\left(\prod_{n=0}^{n_{s}-1}(\theta^{q^{s+n\ell}}-\theta)\right)\cdot\begin{cases}\displaystyle\frac{\theta^{q^{s+n_{s}\ell}}-\theta}{\theta^{q^{s+n_{s}\ell-d}}-\theta},&\text{
if $s+n_{s}\ell>d$,}\\\
\displaystyle\frac{\theta^{q^{s+n_{s}\ell}}-\theta}{v},&\text{ if
$s+n_{s}\ell=d$}\end{cases}$ (by (2.3.2)) $\displaystyle=\ $
$\displaystyle\left(\prod_{n=n_{s}+1}^{\infty}\frac{1-\displaystyle\frac{(\theta-\epsilon)^{q^{s+n\ell}}}{\theta-\epsilon^{q^{s+n\ell}}}}{1-\displaystyle\frac{(\theta-\epsilon)^{q^{s+n\ell-d}}}{\theta-\epsilon^{q^{s+n\ell}}}}\right)\cdot\prod_{n=0}^{n_{s}}(\epsilon^{q^{s+n\ell}}-\theta)(1-\frac{(\theta-\epsilon)^{q^{s+n\ell}}}{\theta-\epsilon^{q^{s+n\ell}}})$
$\displaystyle\cdot\begin{cases}(\theta^{q^{s+n_{s}\ell-d}}-\theta)^{-1},&\text{
if $s+n_{s}\ell>d$},\\\ v^{-1},&\text{ if $s+n_{s}\ell=d$}\end{cases}\hskip
142.26378pt\text{(by \eqref{E:th^{q^n}-th} \&
\eqref{E:(th^{q^{n}}-th)/(th^{q^{n-d}}-th)})}$ $\displaystyle=\ $
$\displaystyle\frac{\displaystyle\prod_{n=0}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}{\displaystyle\prod_{n=n_{s}+1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell-d}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot\bigg{(}\prod_{n=0}^{n_{s}}(\varepsilon^{q^{s+n\ell}}-\theta)\bigg{)}\begin{cases}(\theta^{q^{s+n_{s}\ell-d}}-\theta)^{-1},&\text{
if $s+n_{s}\ell>d$,}\\\ v^{-1},&\text{ if $s+n_{s}\ell=d$,}\end{cases}$

where $\prod_{n=0}^{n_{s}-1}\frac{D_{s+n\ell,v}}{D_{s-1+n\ell,v}^{q}}$
occurring in the second identity and
$\prod_{n=0}^{n_{s}-1}(\theta^{q^{s+n\ell}}-\theta)$ occurring in the third
identity are referred to be $1$ in the case when $n_{s}=0$, the forth and
fifth identities come from (2.3.4) and (2.3.3) respectively. ∎

Consequently, for each $z\in\mathbb{Q}$, let $\langle
z\rangle_{\operatorname{ari}}\in\mathbb{Q}$ be the fractional part of $z$,
i.e. $0\leq\langle z\rangle_{\operatorname{ari}}<1$ and $z-\langle
z\rangle_{\operatorname{ari}}\in\mathbb{Z}$. We get:

######  Corollary 2.3.5.

Let $v\in A_{+}$ be irreducible of degree $d$. Given $\ell\in\mathbb{N}$ and
$s\in\mathbb{Z}_{\geq 0}$ with $0\leq s<\ell$, write
$s+d=s_{0}+n_{s}^{\prime}\ell$ where $s_{0}\in\mathbb{Z}$ with $0\leq
s_{0}<\ell$ and $n_{s}^{\prime}\in\mathbb{Z}_{\geq 0}$. Then

$\displaystyle\frac{\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\displaystyle\frac{q^{s+d}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}}{\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\displaystyle\frac{q^{s+d-1}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}^{q}}$
$\displaystyle=\frac{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s_{0}+n\ell}}}{\theta-\varepsilon^{q^{s_{0}+n\ell}}}\bigg{)}}{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot\left(\prod_{n=0}^{n_{s}^{\prime}-1}(\varepsilon^{q^{s_{0}+n\ell}}-\theta)\right)\cdot\frac{\alpha_{s_{0}}}{\alpha_{s}}\cdot
C_{s}^{-1},$

where

(2.3.6)
$\alpha_{s}:=\begin{cases}1-\displaystyle\frac{(\theta-\varepsilon)^{q^{s}}}{\theta-\varepsilon^{q^{s}}},&\text{
if $s>0$,}\\\ \displaystyle\frac{1}{\theta-\varepsilon},&\text{ if
$s=0$,}\end{cases}\quad\text{ and }\quad C_{s}:=\begin{cases}1,&\text{ if
$s>0$,}\\\ -v,&\text{ if $s=0$.}\end{cases}$

###### Proof.

Recall that $n_{s_{0}}$ denotes the smallest non-negative integer so that
$s_{0}+n_{s_{0}}\ell\geq d$. Then $n_{s}^{\prime}\geq n_{s_{0}}$ and

$0\leq(n_{s}^{\prime}-n_{s_{0}})\ell=(s_{0}+n_{s}^{\prime}\ell)-(s_{0}+n_{s_{0}}\ell)\leq
s+d-d=s<\ell.$

Thus $n_{s}^{\prime}=n_{s_{0}}$ and $s_{0}+n_{s_{0}}\ell-d=s$.

Suppose $s_{0}>0$. Then

$\displaystyle\
\frac{\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\displaystyle\frac{q^{s+d}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}}{\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\displaystyle\frac{q^{s+d-1}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}^{q}}=\frac{\Gamma_{\mathrm{ari},v}\Big{(}1-\displaystyle\frac{q^{s_{0}}}{q^{\ell}-1}\Big{)}}{\Gamma_{\mathrm{ari},v}\Big{(}1-\displaystyle\frac{q^{s_{0}-1}}{q^{\ell}-1}\Big{)}^{q}}$
$\displaystyle=$ $\displaystyle\
\frac{\displaystyle\prod_{n=0}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s_{0}+n\ell}}}{\theta-\varepsilon^{q^{s_{0}+n\ell}}}\bigg{)}}{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot\bigg{(}\prod_{n=0}^{n_{s}^{\prime}}(\varepsilon^{q^{s_{0}+n\ell}}-\theta)\bigg{)}\begin{cases}(\theta^{q^{s}}-\theta)^{-1},&\text{
if $s>0$,}\\\ v^{-1},&\text{ if $s=0$}\end{cases}$ $\displaystyle=$
$\displaystyle\
\frac{\displaystyle\alpha_{s_{0}}\cdot\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s_{0}+n\ell}}}{\theta-\varepsilon^{q^{s_{0}+n\ell}}}\bigg{)}}{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot(\theta-\varepsilon^{q^{s}})\cdot\bigg{(}\prod_{n=0}^{n_{s}^{\prime}-1}(\varepsilon^{q^{s_{0}+n\ell}}-\theta)\bigg{)}\begin{cases}(\theta-\theta^{q^{s}})^{-1},&\text{
if $s>0$,}\\\ (-v)^{-1},&\text{ if $s=0$}\end{cases}$ $\displaystyle=$
$\displaystyle\
\frac{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s_{0}+n\ell}}}{\theta-\varepsilon^{q^{s_{0}+n\ell}}}\bigg{)}}{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot\left(\prod_{n=0}^{n_{s}^{\prime}-1}(\varepsilon^{q^{s_{0}+n\ell}}-\theta)\right)\cdot\frac{\alpha_{s_{0}}}{\alpha_{s}}\cdot
C_{s}^{-1}.$

When $s_{0}=0$, one has that $n_{s}^{\prime}=n_{s_{0}}=n_{0}>0$ and
$n_{\ell}=n_{0}-1$ (as $s+d=0+n_{0}\ell=\ell+(n_{0}-1)\ell$). Then

$\displaystyle\
\frac{\Gamma_{\mathrm{ari},v}(1-\left\langle\displaystyle\frac{q^{s+d}}{q^{\ell}-1}\right\rangle)}{\Gamma_{\mathrm{ari},v}(1-\left\langle\displaystyle\frac{q^{s+d-1}}{q^{\ell}-1}\right\rangle)^{q}}=\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{1}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{\ell-1}}{q^{\ell}-1})^{q}}=(-1)\cdot\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{\ell}}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{\ell-1}}{q^{\ell}-1})^{q}}$
$\displaystyle=$ $\displaystyle\
\frac{\displaystyle\prod_{n=0}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{\ell+n\ell}}}{\theta-\varepsilon^{q^{\ell+n\ell}}}\bigg{)}}{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot\bigg{(}\prod_{n=0}^{n_{\ell}}(\varepsilon^{q^{\ell+n\ell}}-\theta)\bigg{)}\cdot(-1)\cdot\begin{cases}(\theta^{q^{s}}-\theta)^{-1},&\text{
if $s>0$,}\\\ v^{-1},&\text{ if $s=0$}\end{cases}$ $\displaystyle=$
$\displaystyle\
\frac{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{n\ell}}}{\theta-\varepsilon^{q^{n\ell}}}\bigg{)}}{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot\bigg{(}\prod_{n=0}^{n_{0}-1}(\varepsilon^{q^{n\ell}}-\theta)\bigg{)}\cdot\frac{\theta-\varepsilon^{q^{s+d}}}{\theta-\varepsilon}\cdot\begin{cases}(\theta-\theta^{q^{s}})^{-1},&\text{
if $s>0$,}\\\ (-v)^{-1},&\text{ if $s=0$}\end{cases}$ $\displaystyle=$
$\displaystyle\
\frac{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{n\ell}}}{\theta-\varepsilon^{q^{n\ell}}}\bigg{)}}{\displaystyle\prod_{n=1}^{\infty}\bigg{(}1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\bigg{)}}\cdot\left(\prod_{n=0}^{n_{s}^{\prime}-1}(\varepsilon^{q^{n\ell}}-\theta)\right)\cdot\frac{\alpha_{0}}{\alpha_{s}}\cdot
C_{s}^{-1}.$

Therefore the proof is complete. ∎

######  Remark 2.3.7.

Let notation be given as above. Suppose $\ell\geq d$. Since

$\Gamma_{\mathrm{ari},v}(1-\frac{q^{s}}{q^{\ell}-1})=\Gamma_{\mathrm{ari},v}(1-\frac{q^{s+\ell}}{q^{\ell}-1})\cdot(-D_{s,v}),$

we compute that for every integer $s$ with $0\leq s<\ell$,

$\displaystyle\Gamma_{\mathrm{ari},v}(1-\frac{q^{s}}{q^{\ell}-1})^{1-q^{\ell}}$
$\displaystyle=\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s}}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s}}{q^{\ell}-1})^{q^{\ell}}}=\frac{(-D_{s,v})\cdot\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s+\ell}}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s}}{q^{\ell}-1})^{q^{\ell}}}$
$\displaystyle=(-D_{s,v})\cdot\prod_{j=0}^{\ell-1}\left(\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s+\ell-j}}{q^{\ell}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{s+\ell-1-j}}{q^{\ell}-1})^{q}}\right)^{q^{j}}$
$\displaystyle=D_{s,v}^{1-q^{\ell}}\cdot(-D_{s+\ell,v})\cdot\prod_{j=0}^{\ell-1}\prod_{n=2}^{\infty}\left(\displaystyle\frac{1-\frac{\displaystyle(\theta-\varepsilon)^{q^{s+n\ell-j}}}{\displaystyle\theta-\varepsilon^{q^{s+n\ell-j}}}}{\displaystyle
1-\frac{(\theta-\varepsilon)^{q^{s+n\ell-
j-d}}}{\theta-\varepsilon^{q^{s+n\ell-j}}}}\right)^{q^{j}}.$

In particular, when $\ell=d$, these values are actually algebraic (the
Gross–Koblitz phenomenon). We get

$\displaystyle\Gamma_{\mathrm{ari},v}(1-\frac{q^{s}}{q^{d}-1})^{1-q^{d}}$
$\displaystyle=D_{s,v}^{1-q^{d}}\cdot(-D_{s+d,v})\cdot\prod_{j=0}^{d-1}(1-\frac{(\theta-\varepsilon)^{q^{s+d}}}{(\theta-\varepsilon^{q^{s-j}})^{q^{j}}})^{-1}$
$\displaystyle=(-1)\cdot D_{s}^{1-q^{d}}\cdot\frac{D_{s+d}}{v^{q^{s}}\cdot
D_{s}}\cdot\prod_{j=0}^{d-1}\frac{\theta^{q^{j}}-\varepsilon^{q^{s}}}{\theta^{q^{j}}-\theta^{q^{s+d}}}$
$\displaystyle=(-1)^{d-1}\cdot
v^{-q^{s}}\cdot\prod_{j=0}^{d-1}(\theta^{q^{j}}-\varepsilon^{q^{s}})$
$\displaystyle=(-1)^{d-1}\cdot\prod_{j=0}^{d-1}(\theta-\varepsilon^{q^{s-j}})^{-q^{s}}\cdot\prod_{j=0}^{d-1}(\theta-\varepsilon^{q^{s-j}})^{q^{j}}$
$\displaystyle=(-1)^{d-1}\cdot\prod_{j=0}^{d-1}(\theta-\varepsilon^{q^{s-j}})^{-(q^{s}-q^{j})},$

which coincides with Thakur’s formula in [T88, Theorem V], indicating the
exponents of the above decomposition corresponds to the ‘Stickelberger
element’ associated with the constant field extension of degree $d$.

In the following section, we will connect the infinite product

$\prod_{n=0}^{\infty}\left(1-\frac{(\theta-\varepsilon)^{q^{s+n\ell}}}{\theta-\varepsilon^{q^{s+n\ell}}}\right),\quad
s\in\mathbb{Z}\text{ with }0<s\leq\ell,$

with the transcendental periods arising from the $v$-adic “crystalline–de Rham
comparison isomorphism” for the Carlitz $t$-motive of degree $\ell$.

##  3\. Carlitz $t$-motive and the crystalline–de Rham comparison

###  3.1. Carlitz $t$-motive

For each (commutative) $\mathbb{F}_{q}$-algebra $\mathscr{A}$, let
$\mathscr{A}[t]$ be the polynomial ring with another variable $t$ over
$\mathscr{A}$. Let $\mathscr{A}[t,\tau]=\mathscr{A}[t][\tau]$ be the twisted
polynomial ring over $\mathscr{A}[t]$ satisfying the following multiplication
law:

$\tau\cdot f(t)=f^{(1)}(t)\cdot\tau,\quad\forall f(t)\in\mathscr{A}[t],$

where for every $f(t)=\sum_{i=0}^{r}a_{i}t^{i}\in\mathscr{A}[t]$, $f^{(n)}(t)$
is the $n$th Frobenius twist of $f$:

$f^{(n)}(t):=\sum_{i=0}^{r}a_{i}^{q^{n}}t^{i}\in\mathscr{A}[t],\quad\forall
n\in\mathbb{Z}_{\geq 0}.$

Throughout this section, we fix a positive integer $\ell$.

######  Definition 3.1.1.

The _Carlitz $t$-motive of degree $\ell$ over $\mathbb{F}_{q}[\theta]$_ is the
$\mathbb{F}_{q}[\theta][t,\tau^{\ell}]$-module
$\matheur{C}_{\ell}:=\mathbb{F}_{q}[\theta][t]$ with the following
$\tau^{\ell}$-action

$\tau^{\ell}\cdot m:=(t-\theta)\cdot m^{(\ell)},\quad\forall
m\in\matheur{C}_{\ell}.$

The _restriction of scalars of $\matheur{C}_{\ell}$_ is the following
$\mathbb{F}_{q}[\theta][t,\tau]$-module:

$\matheur{R}_{\tau}(\matheur{C}_{\ell}):=\mathbb{F}_{q}[\theta][\tau]\otimes_{\mathbb{F}_{q}[\theta][\tau^{\ell}]}\big{(}\mathbb{F}_{q}[\theta][t]\big{)}.$

Note that we may identify

(3.1.2)
$\matheur{R}_{\tau}(C_{\ell})=\bigoplus_{i=0}^{\ell-1}\tau^{i}\otimes\big{(}\mathbb{F}_{q}[\theta][t]\big{)}\cong\prod_{i=0}^{\ell-1}\mathbb{F}_{q}[\theta][t],$

and the equipped $\tau$-action is given by

$\tau\cdot(m_{0},m_{1},\cdots,m_{\ell-1})=\big{(}(t-\theta)\cdot
m_{\ell-1}^{(1)},m_{0}^{(1)},\cdots,m_{\ell-2}^{(1)}\big{)}.$

For clarity in the subsequent discussions, we set up the following notation:

######  Definition 3.1.3.

Let $d$ and $\ell$ be two positive integers. For every integer $i$ with $0\leq
i<\ell$, we put

$n_{i}=\min\\{n\in\mathbb{Z}_{\geq 0}\mid i+n\ell\geq d\\}\quad\text{ and
}\quad i_{1}:=i+n_{i}\ell-d\quad(\text{for which }0\leq i_{1}<\ell).$

Also, we take $i_{0}$ to be the unique integer with $0\leq i_{0}<\ell$ and
$i+d\equiv i_{0}\bmod\ell$, and set
$n_{i}^{\prime}:=(i+d-i_{0})/\ell\in\mathbb{Z}_{\geq 0}$.

In particular, the following explicit $\tau^{d}$-action on
$\matheur{R}_{\tau}(C_{\ell})$ will be used for period interpretation of
$v$-adic special gamma values in the crystalline-de Rham comparison
isomorphism when $d=\deg v$.

######  Lemma 3.1.4.

Let notation be given as above. Then we have

$\displaystyle\tau^{d}\cdot(m_{0},...,m_{\ell-1})$
$\displaystyle=(m_{0}^{\prime},...,m_{\ell-1}^{\prime}),$ $\displaystyle\text{
where
}m_{i}^{\prime}=\left(\prod_{n=0}^{n_{i}-1}(t-\theta^{q^{i+n\ell}})\right)\cdot
m_{i_{1}}^{(d)},\quad\text{ for }0\leq i<\ell.$

###### Proof.

For each integer $i$ with $0\leq i<\ell$, we write
$i+d=i_{0}+n_{i}^{\prime}\ell$ for unique integers $i_{0}$ and
$n_{i}^{\prime}$ with $0\leq i_{0}<\ell$. Then

$m_{i_{0}}^{\prime}=\left(\prod_{n=0}^{n_{i}^{\prime}-1}(t-\theta^{q^{i_{0}+n\ell}})\right)\cdot
m_{i}^{(d)}.$

Based on Definition 3.1.3, as $n_{i}^{\prime}\geq n_{i_{0}}$ and

$d\leq i_{0}+n_{i_{0}}\ell\leq i_{0}+n_{i}^{\prime}\ell=i+d<\ell+d,$

we get

$0\leq n_{i}^{\prime}\ell-n_{i_{0}}\ell<(\ell+d)-(d)=\ell,$

whence $n_{i}^{\prime}=n_{i_{0}}$. Hence $i=i_{0}+n_{i_{0}}\ell-d$ =
$(i_{0})_{1}$ and the result holds. ∎

In this section, our primary goal is to explicitly illustrate the
crystalline–de Rham comparison isomorphism for
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ at $v$.

###  3.2. Notation

Put $d:=\deg v$ as before, and write

$v=\theta^{d}+\epsilon_{1}\theta^{d-1}+\cdots+\epsilon_{d-1}\theta+\epsilon_{d},\quad\text{
where $\epsilon_{1},...,\epsilon_{d}\in\mathbb{F}_{q}$.}$

Let $\iota:\mathbb{F}_{q}[t]\cong\mathbb{F}_{q}[\theta]$ be the
$\mathbb{F}_{q}$-algebra isomorphism with $\iota(t)=\theta$. Set

$T_{v}:=\iota^{-1}(v)=t^{d}+\epsilon_{1}t^{d-1}+\cdots+\epsilon_{d-1}t+\epsilon_{d},$

which is monic irreducible of degree $d$ in $\mathbb{F}_{q}[t]$. We may embed
$\mathbb{F}_{q}[t]$ into the formal power series ring
$\mathbb{F}_{q^{d}}[\\![T_{v}]\\!]$ (identified with the completion of
$\mathbb{F}_{q}[t]$ at $T_{v}$) which is compatible with $\iota$, i.e.
$t\equiv\varepsilon\bmod T_{v}$, and $\iota$ is extended to an
$\mathbb{F}_{q^{d}}$-isomorphism
$\mathbb{F}_{q^{d}}[\\![T_{v}]\\!]\cong\mathbb{F}_{q^{d}}[\\![v]\\!]$ sending
$T_{v}$ to $v$. These settings can be illustrated in the following diagram:

$\textstyle{\mathbb{F}_{q^{d}}[\\![t-\varepsilon]\\!]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}[\\![T_{v}]\\!]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbb{F}_{q^{d}}[\\![v]\\!]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}[\\![\theta-\varepsilon]\\!]}$$\textstyle{\mathbb{F}_{q^{d}}[t]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathbb{F}_{q^{d}}[\theta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q}[t]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\iota}$$\textstyle{\mathbb{F}_{q}[\theta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

In particular, the embedding
$\mathbb{F}_{q}[t]\hookrightarrow\mathbb{F}_{q^{d}}[\\![T_{v}]\\!]$ sends $t$
to a power series in $T_{v}$ as

(3.2.1) $\displaystyle
t(T_{v})=\sum_{n=0}^{\infty}a_{n}T_{v}^{n}\in\mathbb{F}_{q^{d}}[\\![T_{v}]\\!],\quad\text{
where }\quad a_{0}=\varepsilon\text{ and
}a_{1}=\prod_{i=1}^{d-1}(\varepsilon-\varepsilon^{q^{i}})^{-1},$ (3.2.2)
$\displaystyle\text{and}\quad
t(T_{v})\big{|}_{T_{v}=v}=\varepsilon+\left(\prod_{i=1}^{d-1}(\varepsilon-\varepsilon^{q^{i}})^{-1}\right)v+\cdots=\theta\quad\in\mathbb{F}_{q^{d}}[\\![v]\\!].$

3.2.3. Comparison between twistings. Let
$A_{v}:=\mathbb{F}_{q^{d}}[\\![v]\\!]$ be the valuation ring of $k_{v}$.
Following Hartl–Kim we let

$A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}:=\Big{\\{}\sum_{i\in\mathbb{Z}}b_{i}T_{v}^{i}\in
A_{v}[\\![T_{v},T_{v}^{-1}]\\!]\ \Big{|}\
\lim_{i\rightarrow-\infty}b_{i}v^{-ni}=0,\quad\forall n>0\Big{\\}},$

on which we define the $n$-th Frobenius twist as follows.

######  Definition 3.2.4.

Given $f=\sum_{i\in\mathbb{Z}}f_{i}T_{v}^{i}\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}$, its $n$-th Frobenius twist is given by

$f^{(n)}:=\sum_{i\in\mathbb{Z}}f_{i}^{q^{n}}T_{v}^{i}.$

Note that the embedding
$\mathbb{F}_{q}[t]\hookrightarrow\mathbb{F}_{q^{d}}[\\![T_{v}]\\!]$ given in
(3.2.1) induces an $A_{v}$-algebra embedding $A_{v}[t]\hookrightarrow
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}$. However, the respective $n$-th Frobenius
twists on $A_{v}[t]$ and on $A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}$ are only
compatible when $d$ divides $n$. Indeed, regarding $t=t(T_{v})$ as a power
series in $\mathbb{F}_{q^{d}}[\\![T_{v}]\\!]$ given in (3.2.1), we get

$\displaystyle t^{(n)}(T_{v})$
$\displaystyle=\bigg{(}\varepsilon+\Big{(}\prod_{i=1}^{d-1}(\varepsilon-\varepsilon^{q^{i}})\Big{)}\cdot
T_{v}+\cdots\bigg{)}^{(n)}=\varepsilon^{q^{n}}+\Big{(}\prod_{i=1}^{d-1}(\varepsilon-\varepsilon^{q^{i}})\Big{)}^{q^{n}}\cdot
T_{v}+\cdots$
$\displaystyle=\varepsilon+\Big{(}\prod_{i=1}^{d-1}(\varepsilon-\varepsilon^{q^{i}})\Big{)}\cdot
T_{v}+\cdots=t(T_{v})\quad\in\mathbb{F}_{q^{d}}[\\![T_{v}]\\!]\quad\text{ if
and only if $d\mid n$.}$

We would like to remind readers to be aware of the above “non-compatibility”
between these two types of Frobenius twistings.

######  Remark 3.2.5.

The twisted polynomial ring $A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\tau]$ over
$A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}$ satisfies

$\tau\cdot f(T_{v})=f^{(1)}(T_{v})\tau,\quad\forall f(T_{v})\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}.$

Because of the non-compatibility of the Frobenius twists on $A_{v}[t]$ and on
$A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}$, we remark that $A_{v}[t][\tau]$ is not
a subring of $A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\tau]$ when $d>1$, but both
do contain the subring $A_{v}[t][\tau^{d}]$.

###  3.3. Crystalline module of $\matheur{R}_{\tau}(\matheur{C}_{\ell})$

Put $\theta_{v}:=\theta-\varepsilon$. Let

$A_{v}:=\mathbb{F}_{q^{d}}[\\![\theta_{v}]\\!]=\mathbb{F}_{q^{d}}[\\![v]\\!]\subset
k_{v}\quad\text{ and }\quad\mathfrak{m}_{v}:=v\cdot A_{v}=\theta_{v}\cdot
A_{v}.$

be the valuation ring of $k_{v}$ with maximal ideal

$\mathfrak{m}_{v}:=v\cdot A_{v}=\theta_{v}\cdot A_{v}.$

Note that as $\mathbb{F}_{q^{d}}\cong A_{v}/\mathfrak{m}_{v}$ is an
$\mathbb{F}_{q}[\theta]$-algebra, we define
$\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}$ to be the reduction of
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ modulo $m_{v}$, i.e.

(3.3.1)
$\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}=\frac{A_{v}}{\mathfrak{m}_{v}}\otimes_{\mathbb{F}_{q}[\theta]}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell})\big{)}.$

By (3.1.2), we have the following identification

(3.3.2)
$\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}=\prod_{i=0}^{\ell-1}\mathbb{F}_{q^{d}}[t],$

which becomes an $\mathbb{F}_{q^{d}}[t,\tau]$-module with the $\tau$-action
given by

$\tau\cdot\big{(}\bar{m}_{0},\bar{m}_{1},\cdots,\bar{m}_{\ell-1}\big{)}=\Big{(}(t-\varepsilon)\cdot\bar{m}_{\ell-1}^{(1)},\bar{m}_{0}^{(1)},\cdots,\bar{m}_{\ell-2}^{(1)}\Big{)},$

where we note that $\theta\equiv\epsilon$ (mod $v$).

######  Definition 3.3.3.

(See [HaKi20, Definition 3.5.14]) The $v$-adic crystalline module of the
$t$-motive $\matheur{R}_{\tau}(\matheur{C}_{\ell})$ is

$H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\big{)}:=\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\otimes_{\mathbb{F}_{q^{d}}[t]}\Big{(}\tau^{d}\cdot\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}\Big{)}.$

######  Remark 3.3.4.

We have a natural $\tau^{d}$-action on
$H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\big{)}$
as follows:

$\displaystyle\tau^{d}\cdot\left(\big{(}\sum_{i}f_{i}T_{v}^{i}\big{)}\otimes
m\right)$
$\displaystyle:=\big{(}\sum_{i}f_{i}^{q^{d}}T_{v}^{i}\big{)}\otimes\tau^{d}m$
$\displaystyle=\big{(}\sum_{i}f_{i}T_{v}^{i}\big{)}\otimes\tau^{d}m,\quad\forall\sum_{i}f_{i}T_{v}^{i}\in\mathbb{F}_{q^{d}}(\\!(T_{v})\\!),\
m\in\tau^{d}\cdot\overline{\matheur{R}_{\tau}(C_{\ell})}.$

(The pair
$\big{(}H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\big{)},\tau^{d}\big{)}$
is called the _$T_{v}$ -isocrystal of $\matheur{R}_{\tau}(C_{\ell})$_, see
[HaKi20, Example 3.5.7].)

###  3.4. de Rham module of $\matheur{R}_{\tau}(\matheur{C}_{\ell})$

Let $\mathscr{R}_{\tau}(\matheur{C}_{\ell})$ be the base change of
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ to $k$, i.e.

$\mathscr{R}_{\tau}(\matheur{C}_{\ell})=k\otimes_{A}\matheur{R}_{\tau}(\matheur{C}_{\ell}),$

and the $\tau$-action is extended by

$\tau\cdot(\alpha\otimes m):=\alpha^{q}\otimes(\tau\cdot
m),\quad\forall\alpha\in k,\ m\in\matheur{R}_{\tau}(\matheur{C}_{\ell}).$

The de Rham module of $\matheur{R}_{\tau}(\matheur{C}_{\ell})$ over $k$ is
defined as follows:

(3.4.1) $\displaystyle
H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}$
$\displaystyle:=\frac{\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}{(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}.$

Here $\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}$ is the
$k$-subspace of $\mathscr{R}_{\tau}(\matheur{C}_{\ell})$ spanned by
$\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})$.

######  Remark 3.4.2.

The definition of the de Rham module follows from Hartl–Juschka [HaJu20,
§2.3.5]. Regarding $\mathscr{R}_{\tau}(\matheur{C}_{\ell})$ as the $t$-motive
associated to the “Carlitz module of degree $\ell$” over $k$, the de Rham
module of $\mathscr{R}_{\tau}(\matheur{C}_{\ell})$ can be defined
alternatively via the theory of biderivations developed by Anderson, Deligne,
Gekeler, and Yu for Drinfeld modules (see [Ge89], and [Yu90]), and extended by
Brownawell–Papanikolas for higher dimensional $t$-modules (see [BP02]). For
the equivalence between these two definitions, we refer the reader to [NP22,
Prop. 4.1.3] for a detailed comparison.

######  Lemma 3.4.3.

The natural inclusion
$\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\subset\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})$
induces the $k$-linear isomorphism

$\frac{\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}{(t-\theta)\cdot\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{\mathrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}.$

###### Proof.

From the desciption of the $\tau^{d}$-action on
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$ in Lemma 3.1.4, we observe that

$\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}\supset\left(\prod_{i=1}^{d-1}(t-\theta^{q^{i}})\right)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}.$

By Chinese Remainder Theorem, we get

$\displaystyle(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}+\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}$
$\displaystyle\supset\ $
$\displaystyle(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}+\left(\prod_{i=1}^{d-1}(t-\theta^{q^{i}})\right)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}$
$\displaystyle=\ $
$\displaystyle\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k},$

whence the canonical map

$\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}\longrightarrow\frac{\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}{(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}=H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}$

is surjective with kernel
$(t-\theta)\cdot\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}.$
∎

Consider the following base change:

$H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}:=k_{v}\otimes_{k}H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)},$

called the de Rham module of $\matheur{R}_{\tau}(\matheur{C}_{\ell})$ over
$k_{v}$. Also, we put

(3.4.4) $\displaystyle
H_{\textrm{dR}}\big{(}\mathscr{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}$
$\displaystyle:=k_{v}[\\![T_{v}-v]\\!]\otimes_{k[t]}\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}.$

Then the above lemma leads to the following $k_{v}$-linear isomorphism

(3.4.5)
$\frac{H_{\textrm{dR}}\big{(}\mathscr{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}}{(T_{v}-v)\cdot
H_{\textrm{dR}}\big{(}\mathscr{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{\mathrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}.$

This will be used in the crystalline–de Rham comparison isomorphism in the
next subsection.

###  3.5. Crystalline–de Rham comparison isomorphism

In order to describe the crystalline–de Rham comparison isomorphism, we recall
in Section 3.2 that following Hartl–Kim, we define

$A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}:=\Big{\\{}\sum_{i\in\mathbb{Z}}b_{i}T_{v}^{i}\in
A_{v}[\\![T_{v},T_{v}^{-1}]\\!]\ \Big{|}\
\lim_{i\rightarrow-\infty}b_{i}v^{-ni}=0,\quad\forall n>0\Big{\\}}.$

We further consider the $v$-adic Anderson’s scattering (cf. [ABP04, GL11,
Har11], and [HaKi20, (3.5.3)])

$\varpi(T_{v}):=\prod_{i\in\mathbb{Z}_{\geq
0}}(1-\frac{v^{q^{di}}}{T_{v}})\quad\in\
A_{v}\\{\\!\\{T_{v}^{-1}\\}\\!\\}:=A_{v}[\\![T_{v}^{-1}]\\!]\cap
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}.$

######  Lemma 3.5.1.

There exists a unique isomorphism

$\displaystyle
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q^{d}}[t]}\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q}[\theta][t]}\matheur{R}_{\tau}(\matheur{C}_{\ell}),$

which is compatible with the $\tau^{d}$-action on both sides and becomes the
identity map modulo $\mathfrak{m}_{v}$.

The above lemma actually holds for other $t$-motives (in a more general
setting), see [HaKi20, Lemma 3.5.8], [GL11, Lemma 7.4], and [Har11, Lemma
2.3.1]. We illustrate the isomorphism in Lemma 3.5.1 in concrete terms as
follows. Recall $\theta_{v}:=\theta-\varepsilon$. We first observe that:

######  Lemma 3.5.2.

We identify $t$ with the power series $t(T_{v})$ in (3.2.1). For every non-
negative integer $i$ and positive integer $\ell$, we define

$\Omega_{\ell,v,i}(T_{v}):=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{t-\varepsilon^{q^{\ell n+i}}}).$

Then we have

$\Omega_{\ell,v,i}(T_{v})\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}\cap\left(A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\right)^{\times}.$

###### Proof.

Given a non-negative integer $i$, from the $T_{v}$-expansion of $t$ in (3.2.1)
and $t\equiv\varepsilon$ (mod $T_{v}$) we get

(3.5.3) $\displaystyle t-\varepsilon^{q^{i}}$
$\displaystyle\in\begin{cases}\bigg{(}\displaystyle\prod_{j=1}^{d-1}(\varepsilon-\varepsilon^{q^{j}})^{-1}\bigg{)}\cdot
T_{v}\cdot\big{(}1+T_{v}A_{v}[\\![T_{v}]\\!]\big{)},&\text{ if $i\equiv 0\bmod
d$,}\\\ (\varepsilon-\varepsilon^{q^{i}})+T_{v}A_{v}[\\![T_{v}]\\!],&\text{ if
$i\not\equiv 0\bmod d$.}\end{cases}$

Writing

$\Omega_{\ell,v,i}(T_{v})=\prod_{\genfrac{}{}{0.0pt}{}{n\geq 0}{\ell n+i\equiv
0\bmod d}}(1-\frac{\theta_{v}^{q^{\ell n+i}}}{t-\varepsilon^{q^{\ell
n+i}}})\cdot\prod_{\genfrac{}{}{0.0pt}{}{n\geq 0}{\ell n+i\not\equiv 0\bmod
d}}(1-\frac{\theta_{v}^{q^{\ell n+i}}}{t-\varepsilon^{q^{\ell n+i}}}),$

from (3.5.3) one has that

(3.5.4) $\displaystyle\prod_{\genfrac{}{}{0.0pt}{}{n\geq 0}{\ell n+i\not\equiv
0\bmod d}}(1-\frac{\theta_{v}^{q^{\ell n+i}}}{t-\varepsilon^{q^{\ell n+i}}})$
$\displaystyle\in
1+\theta_{v}A_{v}[\\![T_{v}]\\!]\subset\big{(}A_{v}[\\![T_{v}]\\!]\big{)}^{\times}\subset\left(A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\right)^{\times}.$

On the other hand, if $\gcd(d,\ell)\nmid i$, then $\ell n+i\not\equiv 0\bmod
d$ for every $n\in\mathbb{Z}$. Hence

$\Omega_{\ell,v,i}(T_{v})=\prod_{\genfrac{}{}{0.0pt}{}{n\geq 0}{\ell
n+i\not\equiv 0\bmod d}}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{t-\varepsilon^{q^{i}}})$

and the result holds.

Suppose $\gcd(d,\ell)\mid i$. Write $\text{lcm}(d,\ell)=d\ell_{0}$, and take
$s_{i}\in\mathbb{Z}_{\geq 0}$ to be minimal so that $s_{i}\ell+i\equiv 0\bmod
d$. Put $i^{\prime}=s_{i}\ell+i$. Then

$\prod_{\genfrac{}{}{0.0pt}{}{n\geq 0}{\ell n+i\equiv 0\bmod
d}}(1-\frac{\theta_{v}^{q^{\ell n+i}}}{t-\varepsilon^{q^{\ell
n+i}}})=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{d\ell_{0}n+i^{\prime}}}}{t-\varepsilon}).$

Let

$g(X):=X^{d}+\epsilon_{1}X^{d-1}+\cdots+\epsilon_{d-1}X+\epsilon_{d}-v\quad\in\mathbb{F}_{q}[v][X]\subset\overline{\mathbb{F}_{q}}[v,X]=\overline{\mathbb{F}_{q}}[X][v],$

which is an irreducible element as it is of degree one in $v$ over
$\overline{\mathbb{F}_{q}}[X]$. It follows that $g(X)$ is the minimal
polynomial of $\theta$ over $\mathbb{F}_{q^{d}}(v)$. Since

$\displaystyle g(X)$ $\displaystyle\equiv
X^{d}+\epsilon_{1}X^{d-1}+\cdots+\epsilon_{d-1}X+\epsilon_{d}\quad\bmod v$
$\displaystyle\hskip
14.22636pt=\prod_{i=0}^{d-1}(X-\varepsilon^{q^{i}})\quad\quad\in\mathbb{F}_{q^{d}}[X],$

by Hensel’s lemma we get that $g(X)$ splits completely in
$\mathbb{F}_{q^{d}}[\\![v]\\!][X]$. We denote by
$\theta_{j}\in\mathbb{F}_{q^{d}}[\\![v]\\!]=A_{v}$, $0\leq j<d$, the roots of
$g(X)$ satisfying $\theta_{j}\equiv\varepsilon^{q^{j}}\bmod\mathfrak{m}_{v}$,
and take $u_{j}\in A_{v}$ so that
$u_{j}\theta_{v}=\theta_{j}-\varepsilon^{q^{j}}\in\mathfrak{m}_{v}$. Note that
$\theta_{0}=\theta-\varepsilon=\theta_{v}$ and $u_{0}=1$. From the equalities

(3.5.5)
$T_{v}-v=(t^{d}+\epsilon_{1}t^{d-1}+\cdots+\epsilon_{d-1}t+\epsilon_{d})-v=g(t)=(t-\theta)\cdot\prod_{j=1}^{d-1}(t-\theta_{j})$

and

(3.5.6) $T_{v}=\prod_{j=0}^{d-1}(t-\varepsilon^{q^{j}}),$

we obtain that

(3.5.7)
$h(T_{v}):=\frac{1-\displaystyle\frac{v}{T_{v}}}{1-\displaystyle\frac{\theta_{v}}{t-\varepsilon}}=\frac{t-\varepsilon}{T_{v}}\cdot\frac{T_{v}-v}{t-\theta}=\prod_{j=1}^{d-1}\frac{t-\theta_{j}}{t-\epsilon^{q^{j}}}=\prod_{j=1}^{d-1}(1-\frac{u_{j}\theta_{v}}{t-\varepsilon^{q^{j}}})\quad\in
1+\theta_{v}A_{v}[\\![T_{v}]\\!],$

where the third identity comes from (3.5.5) and (3.5.6). Considering the
Frobenius twists of the identity

$1-\frac{\theta_{v}}{t-\varepsilon}=(1-\frac{v}{T_{v}})\cdot h(T_{v})^{-1},$

we obtain that (as $i^{\prime}\equiv 0\bmod d$)

(3.5.8)
$\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{d\ell_{0}n+i^{\prime}}}}{t-\varepsilon})=\prod_{n=0}^{\infty}(1-\frac{v^{q^{d\ell_{0}n+i^{\prime}}}}{T_{v}})\cdot\prod_{n=0}^{\infty}h^{(d\ell_{0}n+i^{\prime})}(T_{v})^{-1}\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}.$

Finally, since

$\displaystyle\prod_{n=0}^{\infty}(1-\frac{v^{q^{dn}}}{T_{v}})=\varpi(T_{v})\quad\in\left(A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\right)^{\times}$
$\displaystyle=\ $
$\displaystyle\left(\prod_{\genfrac{}{}{0.0pt}{}{n\in\mathbb{Z}_{\geq
0}}{dn\not\equiv i^{\prime}\bmod
d\ell_{0}}}(1-\frac{v^{q^{dn}}}{T_{v}})\prod_{\genfrac{}{}{0.0pt}{}{0\leq
n<i^{\prime}/d}{dn\equiv i^{\prime}\bmod
d\ell_{0}}}(1-\frac{v^{q^{dn}}}{T_{v}})\right)\cdot\left(\prod_{n=0}^{\infty}(1-\frac{v^{q^{d\ell_{0}n+i^{\prime}}}}{T_{v}})\right)$
$\displaystyle=\ $
$\displaystyle\left(\prod_{\genfrac{}{}{0.0pt}{}{n\in\mathbb{Z}_{\geq
0}}{dn\not\equiv i^{\prime}\bmod
d\ell_{0}}}(1-\frac{v^{q^{dn}}}{T_{v}})\prod_{\genfrac{}{}{0.0pt}{}{0\leq
n<i^{\prime}/d}{dn\equiv i^{\prime}\bmod
d\ell_{0}}}(1-\frac{v^{q^{dn}}}{T_{v}})\right)\cdot\left(\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{d\ell_{0}n+i^{\prime}}}}{t-\varepsilon})\cdot
h^{(d\ell_{0}n+i^{\prime})}(T_{v})\right)$ $\displaystyle=\ $
$\displaystyle\left(\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{d\ell_{0}n+i^{\prime}}}}{t-\varepsilon})\right)\cdot\left(\prod_{\genfrac{}{}{0.0pt}{}{n\in\mathbb{Z}_{\geq
0}}{dn\not\equiv i^{\prime}\bmod
d\ell_{0}}}(1-\frac{v^{q^{dn}}}{T_{v}})\prod_{\genfrac{}{}{0.0pt}{}{0\leq
n<i^{\prime}/d}{dn\equiv i^{\prime}\bmod
d\ell_{0}}}(1-\frac{v^{q^{dn}}}{T_{v}})\prod_{n=0}^{\infty}h^{(d\ell
n+i)}(T_{v})\right),$

one gets

$\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{d\ell_{0}n+i^{\prime}}}}{t-\varepsilon})\quad\in\left(A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\right)^{\times}$

because of (3.5.7). Together with (3.5.4) and (3.5.8), we obtain that

$\displaystyle\Omega_{\ell,v,i}(T_{v})=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{d\ell_{0}n+i^{\prime}}}}{t-\varepsilon})$
$\displaystyle\cdot\prod_{\genfrac{}{}{0.0pt}{}{n\geq 0}{\ell n+i\not\equiv
0\bmod d}}(1-\frac{\theta_{v}^{q^{\ell n+i}}}{t-\varepsilon^{q^{\ell n+i}}})$
$\displaystyle\hskip 28.45274pt\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}\cap\left(A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\right)^{\times}$

as desired. ∎

3.5.9. Description of the isomorphism in Lemma 3.5.1. From the identification
of $\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}$ in (3.3.2), we get

$\displaystyle
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q^{d}}[t]}\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}$
$\displaystyle\hskip
56.9055pt=\prod_{i=0}^{\ell-1}A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q^{d}}[t]}\mathbb{F}_{q^{d}}[t]$
$\displaystyle\hskip
56.9055pt\cong\prod_{i=0}^{\ell-1}A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}].$

On the other hand, by (3.1.2) we have that

$\displaystyle
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q}[\theta][t]}\matheur{R}_{\tau}(\matheur{C}_{\ell})$
$\displaystyle\hskip
56.9055pt=\prod_{i=0}^{\ell-1}A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q}[\theta][t]}\mathbb{F}_{q}[\theta][t].$
$\displaystyle\hskip
56.9055pt\cong\prod_{i=0}^{\ell-1}A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}].$

Then the isomorphism in Lemma 3.5.1 is given as follows: every
$(m_{0},...,m_{\ell-1})$ in
$\prod_{i=0}^{\ell-1}A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]$
is sent to

(3.5.10)
$\left(m_{0}\Omega_{\ell,v,0},m_{1}\Omega_{\ell,v,1},\cdots,m_{\ell-1}\Omega_{\ell,v,\ell-1}\right).$

To verify this, note that

$\Omega_{\ell,v,i}(T_{v})\equiv
1\bmod\mathfrak{m}_{v}\quad\in\frac{A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}}{\mathfrak{m}_{v}\cdot
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}}\cong\mathbb{F}_{q^{d}}(\\!(T_{v})\\!).$

Lemma 3.5.2 ensures the bijectivity of (3.5.10), which becomes the identity
map modulo $\mathfrak{m}_{v}$. Moreover, for each integer $i$ with $0\leq
i<\ell$, write $i+d=i_{0}+n_{i}^{\prime}\ell$ where $i_{0},n_{i}^{\prime}$ are
unique integers with $0\leq i_{0}<\ell$ and $n_{i}^{\prime}\geq 0$. Then

$\displaystyle\Omega_{\ell,v,i}^{(d)}(T_{v})$
$\displaystyle=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i+d}}}{t-\varepsilon^{q^{\ell
n+i+d}}})=\prod_{n=n_{i}^{\prime}}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i_{0}}}}{t-\varepsilon^{q^{\ell n+i_{0}}}})$
$\displaystyle=\left(\prod_{n=0}^{n_{i}^{\prime}-1}\frac{t-\varepsilon^{q^{\ell
n+i_{0}}}}{t-\theta^{q^{\ell
n+i_{0}}}}\right)\cdot\Omega_{\ell,v,i_{0}}(T_{v})\quad\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\},$

where the product
$\left(\displaystyle\prod_{n=0}^{n_{i}^{\prime}-1}\frac{t-\varepsilon^{q^{\ell
n+i_{0}}}}{t-\theta^{q^{\ell n+i_{0}}}}\right)$ is defined to be $1$ whenever
$n_{i}^{\prime}=0$. We recall the notation $i_{1}$ given in Definition 3.1.3.
Since $n_{i}^{\prime}=n_{i_{0}}$ and $(i_{0})_{1}=i$, one has that for each
$0\leq i<\ell$,

$\left(\prod_{n=0}^{n_{i}-1}(t-\theta^{q^{\ell
n+i}})\right)\cdot\Omega_{\ell,v,i_{1}}^{(d)}(T_{v})=\left(\prod_{n=0}^{n_{i}-1}(t-\varepsilon^{q^{\ell
n+i}})\right)\cdot\Omega_{\ell,v,i}(T_{v})\quad\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}.$

Finally, for every $(m_{0},...,m_{\ell-1})\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q^{d}}[t]}\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}$,
by Lemma 3.1.4 we obtain that

$\displaystyle\hskip
42.67912pt\tau^{d}\cdot\left(m_{0}\Omega_{\ell,v,0},\cdots,m_{\ell-1}\Omega_{\ell,v,\ell-1}\right)\quad\big{(}\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q}[\theta][t]}\matheur{R}_{\tau}(\matheur{C}_{\ell})\big{)}$
$\displaystyle\hskip
28.45274pt=\Big{(}\left(\prod_{n=0}^{n_{0}-1}(t-\theta^{q^{\ell
n}})\right)\cdot
m_{(0)_{1}}^{(d)}\Omega_{\ell,v,(0)_{1}}^{(d)},\cdots,\left(\prod_{n=0}^{n_{\ell-1}-1}(t-\theta^{q^{\ell
n+\ell-1}})\right)\cdot
m_{(\ell-1)_{1}}^{(d)}\Omega_{\ell,v,(\ell-1)_{1}}^{(d)}\Big{)}$
$\displaystyle\hskip
28.45274pt=\big{(}m_{0}^{\prime}\Omega_{\ell,v,0},\cdots,m_{\ell-1}^{\prime}\Omega_{\ell,v,\ell-1}\big{)},$

where

$\displaystyle(m_{0}^{\prime},...,m_{\ell-1}^{\prime})$
$\displaystyle=\Big{(}\left(\prod_{n=0}^{n_{0}-1}(t-\varepsilon^{q^{\ell
n}})\right)\cdot
m_{(0)_{1}}^{(d)},\cdots,\left(\prod_{n=0}^{n_{\ell-1}-1}(t-\theta^{q^{\ell
n+\ell-1}})\right)\cdot m_{(\ell-1)_{1}}^{(d)}\Big{)}$
$\displaystyle=\tau^{d}\cdot(m_{0},...,m_{\ell-1})\quad\big{(}\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi(T_{v})^{-1}]\otimes_{\mathbb{F}_{q^{d}}[t]}\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}\big{)}.$

Therefore the constructed isomorphism in (3.5.10) indeed preserves the
$\tau^{d}$-action on both sides. $\Box$

From the embedding $\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\hookrightarrow
k_{v}[\\![T_{v}-v]\\!]$ sending $T_{v}$ to $v+(T_{v}-v)$, we view
$\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)$ as a subfield of $k_{v}[\\![T_{v}-v]\\!]$,
and set

$\displaystyle
H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(C_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}$
$\displaystyle:=k_{v}[\\![T_{v}-v]\\!]\otimes_{\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)}H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\big{)}$
$\displaystyle=k_{v}[\\![T_{v}-v]\\!]\otimes_{\mathbb{F}_{q^{d}}[t]}\Big{(}\tau^{d}\cdot\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}\Big{)}.$

On the other hand, recall that in (3.4.4) we set

$\displaystyle
H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}$
$\displaystyle:=k_{v}[\\![T_{v}-v]\\!]\otimes_{k[t]}\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}.$

Regarding $A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}[\varpi^{(d)}(T_{v})^{-1}]$ as a
subring of $k_{v}[\\![T_{v}-v]\\!]$ (see [HaKi20, (3.5.7)]), Lemma 3.5.1
induces the following crystalline–de Rham comparison isomorphism (see [HaJu20,
Theorem 3.5.18]):

######  Theorem 3.5.11.

Let notation be given as above. Then we have the following isomorphism as
$k_{v}[\\![T_{v}-v]\\!]$-modules:

$\Phi_{v}:H_{\text{\rm
cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}\cong
H_{\text{\rm
dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}.$

######  Remark 3.5.12.

Put

$\displaystyle
H_{\textrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}$
$\displaystyle:=k_{v}\underset{\theta\mathrel{\reflectbox{$\mapsto$}}t,\
\mathbb{F}_{q^{d}}[t]}{\otimes}\Big{(}\tau^{d}\cdot\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}\Big{)}$
$\displaystyle=k_{v}\underset{v\mathrel{\reflectbox{$\mapsto$}}T_{v},\
\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)}{\otimes}H_{\textrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\big{)}$
$\displaystyle\cong\displaystyle\frac{H_{\textrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}}{(T_{v}-v)\cdot
H_{\textrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}}.$

As (3.4.5) gives us

$\frac{H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}}{(T_{v}-v)\cdot
H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}}\cong
H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)},$

the isomorphism $\Phi_{v}$ in Theorem 3.5.11 induces a $k_{v}$-linear
isomorphism

(3.5.13)
$\phi_{v}:H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}\cong
H_{\mathrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}.$

###  3.6. Crystalline–de Rham period matrix

We will choose suitable “algebraic bases” and determine the crystalline–de
Rham period matrix associated to the isomorphism $\phi_{v}$ in (3.5.13).

First, for each integer $i$ with $0\leq i<\ell$, recall that
$\Omega_{\ell,v,i}(T_{v})\in A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\}$ is
introduced in Lemma 3.5.2. Recall the notation given in Definition 3.1.3,
$n_{i}$ is the smallest non-negative integer so that $i+n_{i}\ell\geq d$, and
$i_{1}=i+n_{i}\ell-d$. In the case of $n_{i}>0$, we have

$\Big{(}\prod_{n=0}^{n_{i}-1}(t-\theta^{q^{\ell
n+i}})\Big{)}\cdot\Omega_{\ell,v,i_{1}}^{(d)}=\Big{(}\prod_{n=0}^{n_{i}-1}(t-\varepsilon^{\ell
n+i})\Big{)}\cdot\Omega_{\ell,v,i}.$

Moreover, by Lemma 3.1.4 we have the following rank-$\ell$ module

$\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}=\prod_{i=0}^{\ell-1}\big{(}\prod_{n=0}^{n_{i}-1}(t-\theta^{q^{i+n\ell}})\big{)}\cdot
k[t],$

and the same arguments show that

$\tau^{d}\overline{\text{$\matheur{R}$}_{\tau}(\matheur{C}_{\ell})}=\prod_{i=0}^{\ell-1}\big{(}\prod_{n=0}^{n_{i}-1}(t-\varepsilon^{q^{i+n\ell}})\big{)}\cdot\mathbb{F}_{q^{d}}[t].$

It follows that

$\displaystyle
H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}$
$\displaystyle=k_{v}[\\![T_{v}-v]\\!]\otimes_{\mathbb{F}_{q^{d}}[t]}\Big{(}\tau^{d}\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}\Big{)}$
$\displaystyle=\prod_{i=0}^{\ell-1}\big{(}\prod_{n=0}^{n_{i}-1}(t-\varepsilon^{q^{i+n\ell}})\big{)}\cdot
k_{v}[\\![T_{v}-v]\\!]$

and

$\displaystyle H_{\text{\rm
dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}[\\![T_{v}-v]\\!]\big{)}$
$\displaystyle=k_{v}[\\![T_{v}-v]\\!]\otimes_{k[t]}\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}$
$\displaystyle=\prod_{i=0}^{\ell-1}\big{(}\prod_{n=0}^{n_{i}-1}(t-\theta^{q^{i+n\ell}})\big{)}\cdot
k_{v}[\\![T_{v}-v]\\!].$

The induced crystalline–de Rham comparison isomorphism $\Phi_{v}$ in Theorem
3.5.11 is given explicitly as follows:

$\displaystyle\Phi_{v}\Big{(}\big{(}\prod_{n=0}^{n_{0}-1}(t-\varepsilon^{q^{n\ell}})\big{)}m_{0},\cdots,\big{(}\prod_{n=0}^{n_{\ell-1}-1}(t-\varepsilon^{q^{\ell-1+n\ell}})\big{)}m_{\ell-1}\Big{)}$
$\displaystyle\hskip
28.45274pt=\Big{(}\Omega_{\ell,v,0}\cdot\big{(}\prod_{n=0}^{n_{0}-1}(t-\varepsilon^{q^{n\ell}})\big{)}m_{0},\cdots,\Omega_{\ell,v,\ell-1}\cdot\big{(}\prod_{n=0}^{n_{\ell-1}-1}(t-\varepsilon^{q^{\ell-1+n\ell}})\big{)}m_{\ell-1}\Big{)}$
(3.6.1) $\displaystyle\hskip
28.45274pt=\Big{(}\Phi_{v,0}\cdot\big{(}\prod_{n=0}^{n_{0}-1}(t-\theta^{q^{n\ell}})\big{)}m_{0},\cdots,\Phi_{v,\ell-1}\cdot\big{(}\prod_{n=0}^{n_{\ell-1}-1}(t-\theta^{q^{\ell-1+n\ell}})\big{)}m_{\ell-1}\Big{)},$

where $\Phi_{v,i}(T_{v})$ are given by infinite products

$\Phi_{v,i}=\left(\prod_{n=0}^{n_{i}-1}\frac{t-\varepsilon^{q^{n\ell+i}}}{t-\theta^{q^{n\ell+i}}}\right)\cdot\Omega_{\ell,v,i}(T_{v})=\prod_{n=n_{i}}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{t-\varepsilon^{q^{\ell n+i}}}),\quad 0\leq i<\ell.$

In particular, by (3.2.2), evaluating $\Phi_{v,i}(T_{v})$ at $T_{v}=v$ equals

$\Phi_{v,i}(T_{v})\big{|}_{T_{v}=v}=\left(\prod_{n=n_{i}}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{t-\varepsilon^{q^{\ell
n+i}}})\right)\Bigg{|}_{t=\theta}=\prod_{n=n_{i}}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{\theta-\varepsilon^{q^{\ell n+i}}}),\quad 0\leq i<\ell.$

Via the identification (3.1.2), we take
$\mathfrak{B}=\\{b_{0},...,b_{\ell-1}\\}$ to be the standard
$\mathbb{F}_{q}[\theta][t]$ basis of $\matheur{R}_{\tau}(\matheur{C}_{\ell})$,
i.e. for $0\leq i<\ell$,

(3.6.2)
$b_{i}=(b_{i,0},...,b_{i,\ell-1})\quad\in\matheur{R}_{\tau}(\matheur{C}_{\ell})\quad\text{
with }\quad b_{i,j}=\begin{cases}1,&\text{ if $i=j$,}\\\ 0,&\text{
otherwise.}\end{cases}$

For every integer $s$ with $0\leq s<\ell$, recall that $n_{s}$ denotes the
smallest non-negative integer with $s+n_{s}\cdot\ell\geq d$ and
$s_{1}=s+n_{s}\cdot\ell-d$. Then by Lemma 3.1.4 we get

(3.6.3) $b_{v,s}:=\tau^{d}\cdot
b_{s_{1}}=(b_{s,0}^{\prime},...,b_{s,\ell-1}^{\prime})\quad\text{ where }\quad
b_{s,j}^{\prime}=\begin{cases}\displaystyle\prod_{n=0}^{n_{s}-1}(t-\theta^{q^{\ell
n+s}}),&\text{ if $j=s$,}\\\ 0,&\text{ otherwise.}\end{cases}$

Recall in Lemma 3.4.3 that the inclusion
$\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\subset\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})$
induces a surjective map

$\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}\twoheadrightarrow\frac{\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}{(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}=H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}.$

Set

(3.6.4)
$\omega_{v,s}:=b_{v,s}\mod(t-\theta)\quad\in\frac{\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}{(t-\theta)\cdot\langle\tau^{d}\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}\cong
H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}.$

Then $\beta_{v}:=\\{\omega_{v,0},...,\omega_{v,\ell-1}\\}$ is a $k$-basis of
$H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}$. On
the other hand, for each $0\leq s<\ell$, put

(3.6.5)
$\lambda_{v,s}:=b_{v,s}\mod\mathfrak{m}_{v}\quad\in\tau^{d}\cdot\overline{\matheur{R}_{\tau}(\matheur{C}_{\ell})}.$

Then $\Lambda_{v}:=\\{\lambda_{v,0},...,\lambda_{v,\ell-1}\\}$ gives a
$k_{v}$-basis of
$H_{\textrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}$
through the identification in Remark 3.5.12. From the description of
$\Phi_{v}$ in (3.6), we then obtain that:

######  Theorem 3.6.6.

For every integer $s$ with $0\leq s<\ell$, we have

$\phi_{v}(\lambda_{v,s})=\prod_{n=n_{s}}^{\infty}(1-\frac{(\theta-\varepsilon)^{q^{\ell
n+s}}}{\theta-\varepsilon^{q^{\ell n+s}}})\cdot\omega_{v,s}.$

Here $n_{s}$ is the smallest non-negative integer so that $s+n_{s}\ell\geq d$.
Consequently, the period matrix of isomorphism $\phi_{v}$ in (3.5.13) with
respect to $\beta_{v}$ and $\Lambda_{v}$ is

$\begin{pmatrix}\displaystyle\prod_{n=n_{0}}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n}}}{\theta-\varepsilon^{q^{\ell n}}})&&0\\\ &\ddots&\\\
0&&\displaystyle\prod_{n=n_{\ell-1}}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+\ell-1}}}{\theta-\varepsilon^{q^{\ell n+\ell-1}}})\end{pmatrix}.$

###  3.7. Chowla–Selberg-type formula

Note that the action of $\tau_{v}:=\tau^{d}$ on the $T_{v}$-isocrystal
$\big{(}H_{\mathrm{cris}}\big{(}\text{$\mathscr{R}_{\tau}$}(\matheur{C}_{\ell}),\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\big{)},\tau_{v}\big{)}$
introduced in Remark 3.3.4 is actually
$\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)$-linear. Through the isomorphism

$\displaystyle
k_{v}\underset{v\mathrel{\reflectbox{$\mapsto$}}T_{v},\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)}{\otimes}H_{\mathrm{cris}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),\mathbb{F}_{q^{d}}(\\!(T_{v})\\!)\big{)}$
$\displaystyle=H_{\mathrm{cris}}\big{(}\text{$\mathscr{R}_{\tau}$}(\matheur{C}_{\ell}),k_{v}\big{)}\stackrel{{\scriptstyle\phi_{v}}}{{\cong}}H_{\mathrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)},$

there exists a $k_{v}$-linear automorphism $\varrho_{v}$ on
$H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}$
satisfying

(3.7.1)
$\varrho_{v}\cdot\omega=\phi_{v}\Big{(}\tau_{v}\cdot\phi_{v}^{-1}(\omega)\Big{)},\quad\forall\omega\in
H_{\text{$\textrm{dR}$}}\big{(}\mathscr{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}.$

That is, $\varrho_{v}$ is the existence of $k_{v}$-linear automorphism so that
the following diagram commutes:

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35.52934pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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From Theorem 3.6.6, we are able to realize the $\varrho_{v}$-action on
$H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}$
via $v$-adic arithmetic gamma values in the following.

For each integer $s$ with $0\leq s<\ell$, recall that $n_{s}$ is the smallest
non-negative integer so that $s+n_{s}\ell\geq d$, and we write

$s+d=s_{0}+n_{s}^{\prime}\ell\ (\geq d),\quad\text{ where
}s_{0},n_{s}^{\prime}\in\mathbb{Z}\text{ and }0\leq s_{0}<\ell.$

From the inequalities

$\ell+d>s+d=s_{0}+n_{s}^{\prime}\ell\geq s_{0}+n_{s_{0}}\ell\geq d,$

we get

(3.7.2) $n_{s}^{\prime}=n_{s_{0}}.$

By using argument similar to Lemma 3.1.4, we have

$\tau^{d}\cdot\lambda_{v,s}=\left(\prod_{n=0}^{n_{s}-1}(t-\varepsilon^{q^{s+n\ell}})\right)\cdot\lambda_{v,s_{0}},$

and hence Theorem 3.6.6 implies the following identity:

(3.7.3) $\displaystyle\varrho_{v}\cdot\omega_{v,s}$
$\displaystyle=\left(\displaystyle\frac{\displaystyle\prod_{n=n_{s_{0}}}^{\infty}(1-\displaystyle\frac{\theta_{v}^{q^{\ell
n+s_{0}}}}{\theta-\varepsilon^{q^{\ell
n+s_{0}}}})}{\displaystyle\prod_{n=n_{s}}^{\infty}(1-\displaystyle\frac{\theta_{v}^{q^{\ell
n+s}}}{\theta-\varepsilon^{q^{\ell
n+s}}})}\right)\cdot\left(\prod_{n=0}^{n_{s}-1}(\theta-\varepsilon^{q^{s+n\ell}})\right)\cdot\omega_{v,s_{0}}$

Finally, recall from the definition that

$H_{\textrm{dR}}\big{(}\mathscr{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}=\frac{\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}{(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}.$

By (3.1.2) we may identify

$\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}=(t-\theta)\cdot
k[t]\times\prod_{i=1}^{\ell-1}k[t].$

For $0\leq s<\ell$, let
$m_{s}=\big{(}(t-\theta)m_{s,0},m_{s,1}...,m_{s,\ell-1}\big{)}\in\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}$,
where $m_{s,j}=1$ if $s=j$ or $0$ otherwise, and set

(3.7.4)
$\omega_{s}:=m_{s}\mod(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}\quad\in\frac{\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}{(t-\theta)\cdot\langle\tau\matheur{R}_{\tau}(\matheur{C}_{\ell})\rangle_{k}}=H_{\text{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}.$

Then the description of $b_{v,s}$ in (3.6.3) says in particular that

$b_{v,s}=\begin{cases}\left(\displaystyle\prod_{n=1}^{n_{s}-1}(t-\theta^{q^{s+n\ell}})\right)\cdot
m_{s},&\text{ if $s=0$,}\\\
\left(\displaystyle\prod_{n=0}^{n_{s}-1}(t-\theta^{q^{s+n\ell}})\right)\cdot
m_{s},&\text{ if $s>0$.}\end{cases}$

for which we get

(3.7.5)
$\omega_{v,s}=\begin{cases}\left(\displaystyle\prod_{n=1}^{n_{s}-1}(\theta-\theta^{q^{s+n\ell}})\right)\cdot\omega_{s},&\text{
if $s=0$,}\\\
\left(\displaystyle\prod_{n=0}^{n_{s}-1}(\theta-\theta^{q^{s+n\ell}})\right)\cdot\omega_{s},&\text{
if $s>0$.}\end{cases}$

By (3.7.3), we obtain that

$\displaystyle\varrho_{v}\cdot\omega_{s}$
$\displaystyle=\left(\displaystyle\frac{\displaystyle\prod_{n=n_{s_{0}}}^{\infty}(1-\displaystyle\frac{\theta_{v}^{q^{\ell
n+s_{0}}}}{\theta-\varepsilon^{q^{\ell
n+s_{0}}}})}{\displaystyle\prod_{n=1}^{\infty}(1-\displaystyle\frac{\theta_{v}^{q^{\ell
n+s}}}{\theta-\varepsilon^{q^{\ell
n+s}}})}\right)\cdot\alpha_{s}^{-1}\cdot\omega_{v,s_{0}}$ (3.7.6)
$\displaystyle=\left(\displaystyle\frac{\displaystyle\prod_{n=1}^{\infty}(1-\displaystyle\frac{\theta_{v}^{q^{\ell
n+s_{0}}}}{\theta-\varepsilon^{q^{\ell
n+s_{0}}}})}{\displaystyle\prod_{n=1}^{\infty}(1-\displaystyle\frac{\theta_{v}^{q^{\ell
n+s}}}{\theta-\varepsilon^{q^{\ell
n+s}}})}\right)\cdot\left(\prod_{n=0}^{n_{s_{0}}-1}(\theta-\varepsilon^{q^{s_{0}+n\ell}})\right)\cdot\frac{\alpha_{s_{0}}}{\alpha_{s}}\cdot\omega_{s_{0}},$

where $\alpha_{s}$ is given in (2.3.6). As $n_{s_{0}}=n_{s}^{\prime}$ in
(3.7.2), combining with Corollary 2.3.5 we arrive at the following $v$-adic
Chowla–Selberg-type formula:

######  Theorem 3.7.7.

Let $\omega_{0},...,\omega_{\ell-1}$ be the $k$-basis of $H_{\text{\rm
dR}}\big{(}\text{$\mathscr{R}$}_{\tau}(\matheur{C}_{\ell}),k\big{)}$ chosen in
(3.7.5). Given an integer $s$ with $0\leq s<\ell$, write
$s+d=s_{0}+n_{s}^{\prime}\ell$ for unique integers $s_{0}$ and
$n_{s}^{\prime}$ with $0\leq s_{0}<\ell$. Then

$\varrho_{v}\cdot\omega_{s}=(-1)^{n_{s}^{\prime}}\cdot
C_{s}\cdot\displaystyle\frac{\displaystyle\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\frac{q^{s+d}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}}{\displaystyle\Gamma_{\mathrm{ari},v}\Big{(}1-\Big{\langle}\frac{q^{s+d-1}}{q^{\ell}-1}\Big{\rangle}_{\operatorname{ari}}\Big{)}^{q}}\cdot\omega_{s_{0}},\quad\text{
where }\quad C_{s}:=\begin{cases}-v,&\text{ if $s=0$,}\\\ 1,&\text{ if
$s>0$,}\end{cases}$

and $\langle z\rangle_{\operatorname{ari}}$ is the fractional part of every
$z\in\mathbb{Q}$, i.e. $0\leq\langle z\rangle_{\operatorname{ari}}<1$ and
$z-\langle z\rangle_{\operatorname{ari}}\in\mathbb{Z}$.

######  Remark 3.7.8.

Theorem 3.7.7 indicates that the matrix presentation of the automorphism
$\varrho_{v}$ on $H_{\text{\rm
dR}}\big{(}\text{$\matheur{R}$}_{\tau}(\matheur{C}_{\ell}),k_{v}\big{)}$ with
respect to the basis $\beta:=\\{\omega_{0},...,\omega_{\ell-1}\\}$ can be
concretely illustrated in terms of $v$-adic arithmetic gamma values.

######  Remark 3.7.9.

The basis $\beta=\\{\omega_{0},...,\omega_{\ell-1}\\}$ of the de Rham module
$H_{\textrm{dR}}\big{(}\matheur{R}_{\tau}(\matheur{C}_{\ell}),k\big{)}$ is
actually independent of the chosen finite place $v$ of $k$. Hence Theorem
3.7.7 shows in particular that with respect to this global basis $\beta$, we
are able to realize the “isocrystal” action $\varrho_{v}$ simultaneously for
every finite place $v$ of $k$. In fact, this basis is also utilized implicitly
in the $\infty$-adic Chowla–Selberg formula in [CPTY10] (see also [Wei22]),
and we would have a natural consistency between these formulas (as in [O89,
Theorem 3.15] for the number field case).

##  4\. Algebraic relations among the $v$-adic arithmetic gamma values

In the remaining sections of this paper, we will determine the algebraic
relations among the $v$-adic arithmetic gamma values.

###  4.1. Transcendence result of the $v$-adic arithmetic gamma values

Given $\ell\in\mathbb{N}$, recall in Lemma 3.5.2 that we have defined

$\Omega_{\ell,v,i}(T_{v})=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{t-\varepsilon^{q^{\ell n+i}}})\quad\in
A_{v}[\\![T_{v},T_{v}^{-1}\\}\\!\\},\quad\forall i\in\mathbb{Z}_{\geq 0}.$

In particular, one observes that

(4.1.1)
$\displaystyle\Omega_{\ell,v,i}(v):=\Omega_{\ell,v,i}(T_{v})\big{|}_{T_{v}=v}$
$\displaystyle=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{\theta-\varepsilon^{q^{\ell n+i}}})\quad\in k_{v}^{\times}$
$\displaystyle=(1-\frac{\theta_{v}^{q^{i}}}{\theta-\varepsilon^{q^{i}}})\cdot\Omega_{\ell,v,i+\ell}(v),\quad\forall
i\in\mathbb{N}.$

We first assert that:

######  Theorem 4.1.2.

Given $\ell\in\mathbb{N}$, the values

$\Omega_{\ell,v,i}(v)=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{\theta-\varepsilon^{q^{\ell n+i}}}),\quad 0<i\leq\ell,$

are algebraically independent over $\bar{k}$.

######  Remark 4.1.3.

By Theorem 3.6.6, the values $\Omega_{\ell,v,i}(v)$, $0<i\leq\ell$ are (up to
$k^{\times}$-multiples) the $v$-adic periods of the crystalline–de Rham
comparison isomorphism for the $t$-motive
$\matheur{R}_{\tau}(\matheur{C}_{\ell})$. Theorem 4.1.2 implies that

$\operatorname{tr.deg}_{\bar{k}}\bar{k}\Big{(}\Omega_{\ell,v,i}(v)\ \Big{|}\
0<i\leq\ell\Big{)}=\ell=\dim\Gamma_{\matheur{R}_{\tau}(\matheur{C}_{\ell})},$

where $\Gamma_{\matheur{R}_{\tau}(\matheur{C}_{\ell})}$ is the “$t$-motivic
Galois group of the $t$-motive
$\bar{k}\otimes_{\mathbb{F}_{q}[\theta]}\matheur{R}_{\tau}(\matheur{C}_{\ell})$”
(see [CPTY10, Lemma 3.2.1]). This coincidence seems to provide an evidence for
a suitable $v$-adic analogue of Papanikolas’ theorem in [P08] (Grothendieck
period conjecture).

Assuming Theorem 4.1.2, whose proof will be given in Section 6.4, we are able
to show that:

######  Theorem 4.1.4.

Given $\ell\in\mathbb{N}$, we have the following equality

$\operatorname{tr.deg}_{\bar{k}}\bar{k}\Big{(}\Gamma_{\mathrm{ari},v}(z)\
\Big{|}\ z\in\mathbb{Q}\text{ with }(q^{\ell}-1)\cdot
z\in\mathbb{Z}\Big{)}=\ell-\gcd(\ell,\deg v).$

###### Proof.

First, assume that $d=\deg v\mid\ell$, and write $\ell=d\ell_{0}$. If
$\ell_{0}=1$, then the result follows from Theorem 2.2.5. Suppose
$\ell_{0}\geq 2$. For every integer $s$ with $0\leq s<d\ell_{0}$, by Lemma
2.3.1 we get

$\displaystyle\frac{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{d\ell_{0}-s}}{q^{d\ell_{0}}-1})}{\Gamma_{\mathrm{ari},v}(1-\displaystyle\frac{q^{d\ell_{0}-s-1}}{q^{d\ell_{0}}-1})^{q}}\sim\prod_{n=2}^{\infty}\left(\frac{\displaystyle
1-\frac{\theta_{v}^{q^{d\ell_{0}n-s}}}{\theta-\varepsilon^{q^{-s}}}}{\displaystyle
1-\frac{\theta_{v}^{q^{d\ell_{0}n-s-d}}}{\theta-\varepsilon^{q^{-s}}}}\right)=\frac{\Omega_{d\ell_{0},v,2d\ell_{0}-s}(v)}{\Omega_{d\ell_{0},v,2d\ell_{0}-s-d}(v)}=:u_{s}.$

In particular, for every integer $s$ with $0\leq s<d(\ell_{0}-1)$, let
$s_{d}\in\mathbb{Z}$ with $0\leq s_{d}<d$ and $s_{d}\equiv s\bmod d$. Write
$s=s_{d}+d\ell_{s}$ with $0\leq\ell_{s}<\ell_{0}-1$. Then one has that

$u^{\prime}_{s}:=u_{s}u_{s+d}\cdots
u_{s+(\ell_{0}-2-\ell_{s})d}=\frac{\Omega_{d\ell_{0},v,2d\ell_{0}-s}(v)}{\Omega_{d\ell_{0},v,d\ell_{0}+d-s_{d}}(v)}\sim\frac{\Omega_{d\ell_{0},v,d\ell_{0}-s}(v)}{\Omega_{d\ell_{0},v,d-s_{d}}(v)}.$

As $d<d\ell_{0}-s\leq d\ell_{0}$ and $0<d-s_{d}\leq d$, Theorem 4.1.2 ensures
the algebraic independence of $u_{s}^{\prime}$ over $\bar{k}$ for $0\leq
s<d(\ell_{0}-1)$. Hence

$\operatorname{tr.deg}_{\bar{k}}\bar{k}\Big{(}\Gamma_{\mathrm{ari},v}(z)\
\Big{|}\ z\in\mathbb{Q}\text{ with }(q^{\ell}-1)\cdot
z\in\mathbb{Z}\Big{)}\geq d(\ell_{0}-1)=\ell-\gcd(\ell,\deg v).$

By combining with the upper bound in Theorem 2.2.9, the result follows.

Consequently, the following monomial relations

(4.1.5)
$\displaystyle\prod_{m=0}^{\ell_{0}-1}\Gamma_{\mathrm{ari},v}(1-\frac{q^{j+dm}}{q^{d\ell_{0}}-1})$
$\displaystyle=\Gamma_{\mathrm{ari},v}(1-\frac{q^{j}+q^{j+d}+\cdots+q^{j+d(\ell_{0}-1)}}{q^{d\ell_{0}}-1})$
$\displaystyle=\Gamma_{\mathrm{ari},v}(1-\frac{q^{j}}{q^{d}-1}),\quad\text{
for $0\leq j<d$,}$

generate all algebraic relations among
$\Gamma_{\mathrm{ari},v}(1-q^{s}/(q^{d\ell_{0}}-1))$ for $0\leq
s<d\ell_{0}-1$. In particular, let
$\vec{\delta}_{0},...,\vec{\delta}_{d-1}\in\mathbb{Z}^{d\ell_{0}}$ be the
“exponent vectors” corresponding to the above monomial relations, i.e.

$\vec{\delta}_{j}=(\delta_{j,0},...,\delta_{j,d\ell_{0}-1})\quad\text{ with
}\quad\delta_{j,s}=\begin{cases}1,&\text{ if $j\equiv s\bmod d$,}\\\ 0,&\text{
otherwise.}\end{cases}$

Then

$\prod_{s=0}^{d\ell_{0}-1}\Gamma_{\mathrm{ari},v}(1-\frac{q^{s}}{q^{d\ell_{0}}-1})^{\delta_{j,s}}=\Gamma_{\mathrm{ari},v}(1-\frac{q^{j}}{q^{d}-1})\quad\in\bar{k},$

Moreover, as
$\vec{\delta}_{0},...,\vec{\delta}_{d-1}\in\mathbb{Z}^{d\ell_{0}}$ span a
$d$-dimensional “relation subspace” in $\mathbb{Q}^{d\ell_{0}}$, for arbitrary
given vectors $\vec{n}_{1},...,\vec{n}_{r}\in\mathbb{Z}^{d\ell_{0}}$ with
$\vec{n}_{j}=(n_{j,0},...,n_{j,d\ell-1})$, $j=1,...,r$, we have that

$\displaystyle\ \ \operatorname{tr.deg}_{\bar{k}}\bar{k}\left(\prod_{0\leq
s<d\ell}\Gamma_{\mathrm{ari},v}(1-\frac{q^{s}}{q^{d\ell_{0}}-1})^{n_{j,s}}\
\Bigg{|}\ 1\leq j\leq r\right)$ (4.1.6) $\displaystyle=$ $\displaystyle\ \
\dim_{\mathbb{Q}}\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1},\vec{n}_{1},...,\vec{n}_{r}\rangle_{\mathbb{Q}}-d.$

Here
$\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1},\vec{n}_{1},...,\vec{n}_{r}\rangle_{\mathbb{Q}}$
is the subspace of $\mathbb{Q}^{d\ell_{0}}$ spanned by
$\vec{\delta}_{0},...,\vec{\delta}_{d-1},\vec{n}_{1},...,\vec{n}_{r}$ over
$\mathbb{Q}$.

In the general case, for a given $\ell\in\mathbb{N}$, write
$\text{lcm}(\ell,d)=d\ell_{0}=d_{0}\ell$ with $\ell_{0},d_{0}\in\mathbb{N}$ so
that $\text{gcd}(\ell_{0},d_{0})=1$. Note that for an integer $j$ with $0\leq
j<\ell$, one has

$\Gamma_{\mathrm{ari},v}(1-\frac{q^{j}}{q^{\ell}-1})=\prod_{s=0}^{d_{0}-1}\Gamma_{\mathrm{ari},v}(1-\frac{q^{j+s\ell}}{q^{d_{0}\ell}-1}).$

Let $\vec{\lambda}_{0},...,\vec{\lambda}_{\ell-1}\in\mathbb{Z}^{d\ell_{0}}$
satisfying that

$\vec{\lambda}_{j}=(\lambda_{j,0},...,\lambda_{j,d\ell_{0}-1})\quad\text{ with
}\quad\lambda_{j,s}=\begin{cases}1,&\text{ if $j\equiv s\bmod\ell$,}\\\
0,&\text{ otherwise.}\end{cases}$

As for $0\leq j<\ell$,

$\prod_{s=0}^{d\ell_{0}-1}\Gamma_{\mathrm{ari},v}(1-\frac{q^{s}}{q^{d\ell_{0}}-1})^{\lambda_{j,s}}=\Gamma_{\mathrm{ari},v}(1-\frac{q^{j}}{q^{\ell}-1}),$

by the equality (4.1), we obtain that

$\displaystyle\
\operatorname{tr.deg}_{\bar{k}}\bar{k}\Big{(}\Gamma_{\mathrm{ari},v}(z)\
\Big{|}\ z\in\mathbb{Q}\text{ with }(q^{\ell}-1)\cdot z\in\mathbb{Z}\Big{)}$
(4.1.7) $\displaystyle=$ $\displaystyle\
\operatorname{tr.deg}_{\bar{k}}\bar{k}\bigg{(}\Gamma_{\mathrm{ari},v}(1-\frac{q^{j}}{q^{\ell}-1})\
\bigg{|}\ 0\leq j<\ell\bigg{)}$ $\displaystyle=$ $\displaystyle\
\dim_{\mathbb{Q}}\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1},\vec{\lambda}_{0},...,\vec{\lambda}_{\ell-1}\rangle_{\mathbb{Q}}-d.$

Since

$\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1}\rangle_{\mathbb{Q}}=\Big{\\{}(\alpha_{0},...,\alpha_{d\ell_{0}-1})\in\mathbb{Q}^{d\ell_{0}}\
\Big{|}\ \alpha_{i}=\alpha_{j}\text{ if }i\equiv j\bmod d\Big{\\}}$

and

$\langle\vec{\lambda}_{0},...,\vec{\lambda}_{\ell-1}\rangle_{\mathbb{Q}}=\Big{\\{}(\beta_{0},...,\beta_{d_{0}\ell_{0}-1})\in\mathbb{Q}^{d_{0}\ell}\
\Big{|}\ \beta_{i}=\beta_{j}\text{ if }i\equiv j\bmod\ell\Big{\\}},$

one has that

$\displaystyle\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1}\rangle_{\mathbb{Q}}\cap\langle\vec{\lambda}_{0},...,\vec{\lambda}_{\ell-1}\rangle_{\mathbb{Q}}$
$\displaystyle\hskip
56.9055pt=\Big{\\{}(\gamma_{0},...,\gamma_{d_{0}\ell_{0}-1})\in\mathbb{Q}^{d_{0}\ell}\
\Big{|}\ \gamma_{i}=\gamma_{j}\text{ if }i\equiv j\bmod\gcd(d,\ell)\Big{\\}}.$

Therefore

$\dim_{\mathbb{Q}}\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1},\vec{\lambda}_{0},...,\vec{\lambda}_{\ell-1}\rangle_{\mathbb{Q}}=\ell+d-\gcd(d,\ell),$

for which the result follows from (4.1). ∎

######  Remark 4.1.8.

Theorem 4.1.4 says that all algebraic relations among $v$-adic arithmetic
gamma values at rational $p$-adic integers are generated by the funcional
equations in Proposition 2.2.1 and Thakur’s analogue of the Gross–Koblitz
formula in Theorem 2.2.5.

One immediate consequence of Theorem 4.1.4 is the following.

######  Corollary 4.1.9.

Let $a$ and $b$ be nonzero integers so that $\gcd(a,b)=1$ and $p\nmid b$. If
$q^{d}\not\equiv 1\bmod b$, then

$\Gamma_{\mathrm{ari},v}(\frac{a}{b})\quad\text{ is transcendental over $k$.}$

###### Proof.

Without loss of generality, we may assume that
$\frac{a}{b}=1-\frac{a^{\prime}}{b}$ with $0\leq a^{\prime}<b$ and
$\gcd(a^{\prime},b)=1$. Let $\ell_{0}$ be the order of $q^{d}\bmod b$ in
$(\mathbb{Z}/b\mathbb{Z})^{\times}$. Then the assumption of $b$ implies that
$\ell_{0}>1$. Write $q^{d\ell_{0}}-1=b\cdot b^{\prime}$, and take
$a_{0},...,a_{d\ell_{0}-1}\in\mathbb{Z}$ with $0\leq
a_{0},...,a_{d\ell_{0}-1}<q$ satisfying that

$\frac{a^{\prime}}{b}=\frac{a^{\prime}b^{\prime}}{q^{d\ell_{0}}-1}=\sum_{i=0}^{d\ell_{0}-1}\frac{a_{i}q^{i}}{q^{d\ell_{0}}-1}.$

Let $\vec{a}=(a_{0},a_{1},...,a_{d\ell_{0}-1})\in\mathbb{Z}^{d\ell_{0}}$. If
$\vec{a}\in\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1}\rangle_{\mathbb{Q}}$,
then we get $a_{i}=a_{j}$ if $i\equiv j\bmod d$. Hence

$\frac{a^{\prime}}{b}=\sum_{i=0}^{d-1}\frac{a_{i}(q^{i}+q^{d+i}+\cdots+q^{(\ell_{0}-1)d+i})}{q^{d\ell_{0}}-1}=\frac{a_{0}+a_{1}q+\cdots+a_{d-1}q^{d-1}}{q^{d}-1},$

for which $b\mid q^{d}-1$, a contradiction. Therefore
$\vec{a}\notin\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1}\rangle_{\mathbb{Q}}$,
and by (4.1) we get

$\operatorname{tr.deg}_{\bar{k}}\bar{k}\Big{(}\Gamma_{\mathrm{ari},v}(1-\frac{a^{\prime}}{b})\Big{)}=\dim_{\mathbb{Q}}\langle\vec{\delta}_{0},...,\vec{\delta}_{d-1},\vec{a}\rangle_{\mathbb{Q}}=(d+1)-d=1$

as desired. ∎

######  Remark 4.1.10.

Let $a$ and $b$ be nonzero integers so that $\gcd(a,b)=1$ and $p\nmid b$.
Combining Thakur’s result in Theorem 2.2.5 and Corollary 4.1.9, we have the
following equivalence:

$b\mid q^{\deg v}-1\quad{\hbox{ if and only if
}}\quad\Gamma_{\mathrm{ari},v}(\frac{a}{b})\in\bar{k}^{\times}.$

###  4.2. $v$-adic arithmetic gamma distribution

Let $\mathbb{Z}_{(p)}:=\mathbb{Z}_{p}\cap\mathbb{Q}$, and
$\mathscr{A}^{\text{ari}}$ be the free abelian group generated by all elements
in $\mathbb{Z}_{(p)}/\mathbb{Z}$. We identify $\mathscr{A}^{\text{ari}}$ with
a subgroup of
$\mathscr{A}^{\text{ari}}_{\mathbb{Q}}:=\mathbb{Q}\otimes_{\mathbb{Z}}\mathscr{A}^{\text{ari}}$.
Every element in $\mathscr{A}^{\text{ari}}$ (resp.
$\mathscr{A}^{\text{ari}}_{\mathbb{Q}}$) can be written uniquely as a formal
sum

${\bf z}=\sum_{z\in\mathbb{Z}_{(p)}/\mathbb{Z}}n_{z}[z],\quad\text{where
$n_{z}\in\mathbb{Z}$ (resp.\ $\mathbb{Q}$) and $n_{z}=0$ for almost all $z$.}$

Given $\ell\in\mathbb{N}$, let $\mathscr{A}^{\text{ari}}_{\ell}$ (resp.
$\mathscr{A}^{\text{ari}}_{\ell,\mathbb{Q}}$) be the subgroup of
$\mathscr{A}^{\text{ari}}$ (resp. subspace of
$\mathscr{A}^{\text{ari}}_{\mathbb{Q}}$) generated by
$z\in\frac{1}{q^{\ell}-1}\mathbb{Z}/\mathbb{Z}$. Also, recall that the
_arithmetic diamond bracket_ of $z\in\mathbb{Z}_{(p)}$, introduced by Thakur
[T91], comes from the fractional part of $z$, i.e. $\langle
z\rangle_{\operatorname{ari}}\in\mathbb{Z}_{(p)}$ is the unique number with

$0\leq\langle z\rangle_{\operatorname{ari}}<1\quad\text{ and }\quad
z\equiv\langle z\rangle_{\operatorname{ari}}\bmod\mathbb{Z}.$

We may regard $\langle\cdot\rangle_{\operatorname{ari}}$ as a function on
$\mathbb{Z}_{(p)}/\mathbb{Z}$. Consider the subspace
$\mathscr{R}^{\text{ari}}_{\ell}$ of
$\mathscr{A}^{\text{ari}}_{\ell,\mathbb{Q}}$ spanned by

$[z]-\sum_{i=0}^{\ell-1}z_{i}[\frac{q^{i}}{1-q^{\ell}}],\quad\forall
z\in\frac{1}{q^{\ell}-1}\mathbb{Z}/\mathbb{Z}\quad\text{ (see
\eqref{E:Standard FE})},$

where $z_{0},...,z_{\ell-1}\in\mathbb{Z}$ with $0\leq z_{0},...,z_{\ell-1}<q$
so that

$\sum_{i=0}^{\ell-1}\frac{z_{i}q^{i}}{q^{\ell}-1}=\langle-z\rangle_{\operatorname{ari}}.$

Put
$\mathscr{U}^{\text{ari}}:=\mathscr{A}^{\text{ari}}_{\mathbb{Q}}/\mathscr{R}^{\text{ari}}$,
where $\mathscr{R}^{\text{ari}}:=\cup_{\ell}\mathscr{R}^{\text{ari}}_{\ell}$.
We call $\mathscr{U}^{\text{ari}}$ the _universal distribution associated to
the arithmetic diamond bracket relations_ (see [Wei22, §4.3].

Following the $\infty$-adic case, we define

$\tilde{\Gamma}_{\text{ari},v}:\mathbb{Z}_{(p)}/\mathbb{Z}\longrightarrow\mathbb{C}_{v}^{\times},\quad\tilde{\Gamma}_{\text{ari},v}(z\bmod\mathbb{Z}):=\Gamma_{\mathrm{ari},v}(1-\langle-z\rangle_{\operatorname{ari}}),$

which induces a $\mathbb{Q}$-linear homomorphism
$\hat{\Gamma}_{\text{ari},v}:\mathscr{U}_{\mathbb{Q}}^{\text{ari}}\rightarrow\mathbb{C}_{v}^{\times}/\bar{k}^{\times}$
(by the monomial relation in Proposition 2.2.2). We may call
$\hat{\Gamma}_{\text{ari},v}$ the _$v$ -adic arithmetic gamma distribution_.
From Corollary 2.2.7 and Theorem 4.1.4 together with (4.1.5), we obtain that:

######  Theorem 4.2.1.

The kernel of the $v$-adic arithmetic gamma distribution
$\hat{\Gamma}_{\text{ari},v}$ is spanned by the Gross–Koblitz–Thakur relations

$\sum_{i=0}^{r-1}[q^{i\deg v}z],\quad\forall z\in\frac{1}{n}\mathbb{Z}$

where $n$ runs through all positive integers with $p\nmid n$, and $r$ is the
order of $q^{\deg v}\bmod n$ in $(\mathbb{Z}/n\mathbb{Z})^{\times}$.

##  5\. Transcendence theory of Hartl–Juschka modules

In this section, we briefly review the necessary properties in the
trancendence theory of Hartl–Juschka modules for the proof of Theorem 4.1.2.
To utilize this well-developed theory in the $v$-adic setting, our initial
step is to regard $v$ as the infinite place of the “projective
$\frac{1}{v}$-line” over $\mathbb{F}_{q}$.

###  5.1. Regarding $v$ as the infinite place

Fix $d=\deg v$ and a positive integer $\ell$ as before. Set
$k_{d\ell}:=\mathbb{F}_{q^{d\ell}}(\theta)=\mathbb{F}_{q^{d\ell}}(\theta_{v})\subset\bar{k}$,
where $\theta_{v}=\theta-\varepsilon$, and
$k_{d\ell,v}:=\mathbb{F}_{q^{d\ell}}(\\!(\theta_{v})\\!)$. Let

$\tilde{\theta}_{v}:=(\theta-\varepsilon)^{-1}\quad\text{ and
}\tilde{v}:=v^{-1}.$

Then we have

$\mathbb{F}_{q}(\\!(v)\\!)=\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)\subset\mathbb{F}_{q^{d\ell}}(\\!(\tilde{\theta}_{v}^{-1})\\!)=\mathbb{F}_{q^{d\ell}}(\\!(\theta_{v})\\!)=k_{d\ell,v}\subset\mathbb{C}_{v}.$

The trick of viewing $\mathbb{C}_{v}$ as an ‘$\infty_{\tilde{v}}$-adic’ field
comes from the following translation.

  * $\bullet$

$\mathbb{F}_{q}(v)=\mathbb{F}_{q}(\tilde{v})$ is the fraction field of
$\mathbb{F}_{q}[\tilde{v}]$, the polynomial ring in $\tilde{v}$ over
$\mathbb{F}_{q}$.

  * $\bullet$

The $\infty_{\tilde{v}}$-adic valuation on
$\mathbb{F}_{q}(v)=\mathbb{F}_{q}(\tilde{v})$ is given by
${\rm{ord}}_{\infty_{\tilde{v}}}:={\rm{ord}}_{\tilde{v}^{-1}}={\rm{ord}}_{v}$.

  * $\bullet$

$\mathbb{F}_{q}(v)=\mathbb{F}_{q}(\tilde{v})$ is the function field of the
projective $\tilde{v}$-line over $\mathbb{F}_{q}$ with $v=\tilde{v}^{-1}$ as a
uniformizer of the infinite place.

  * $\bullet$

$\mathbb{F}_{q}(\\!(v)\\!)=\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)$ is the
completion of $\mathbb{F}_{q}(v)=\mathbb{F}_{q}(\tilde{v})$ with respect to
the $\infty_{\tilde{v}}$-adic valuation
${\rm{ord}}_{\infty_{\tilde{v}}}={\rm{ord}}_{v}$.

  * $\bullet$

$k_{v}\cong\mathbb{F}_{q^{d}}(\\!(v)\\!)=\mathbb{F}_{q^{d}}(\\!(\tilde{v}^{-1})\\!)$
is a finite extension of the $\infty_{\tilde{v}}$-adic field
$\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)$.

  * $\bullet$

A fixed algebraic closure $\overline{\mathbb{F}_{q^{d}}(\\!(v)\\!)}$ of
$\mathbb{F}_{q^{d}}(\\!(v)\\!)$ is the same as a particularly chosen algebraic
closure $\overline{\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)}$ of
$\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)$.

  * $\bullet$

$\mathbb{C}_{v}$ is the $v$-adic completion of
$\overline{\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)}$, which is the same as the
$\infty_{\tilde{v}}$-adic completion of
$\overline{\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)}$.

Since we deal with values in $\mathbb{C}_{v}$, which is viewed as an
$\infty_{\tilde{v}}$-adic field, one can apply Papanikolas’ “$\infty$-adic”
transcendence theory [P08]. The following comparison/replacement illustrates
why we consider Frobenius modules over $\overline{k}[\tilde{T_{v}}]$ in the
following subsection when applying Papanikolas’ theory fitting into our
situation.

$\begin{array}[]{rrcc|ccll}{\hbox{Over }}k_{\infty}&&&&&&&{\hbox{Over
}}k_{v}\\\ \hline\cr\hline\cr{\hbox{uniformizer at
}}\infty&\rightarrow&1/\theta&&&\tilde{v}^{-1}&\leftarrow&{\hbox{uniformizer
at }}{\tilde{\infty}_{v}}\\\
t&\rightarrow&\theta&&&\tilde{v}&\leftarrow&\tilde{T}_{v}\\\ {\hbox{polynomial
ring}}&\rightarrow&\overline{k}[t]&&&\overline{k}[\tilde{T}_{v}]&\leftarrow&{\hbox{polynomial
ring}}\\\ {\hbox{Tate
algebra}}&\rightarrow&\mathbb{C}_{\infty}\\{t\\}&&&\mathbb{C}_{v}\\{\tilde{T}_{v}\\}&\leftarrow&{\hbox{Tate
algebra}}\end{array}$

The relations of the field extensions to be used is illustrated in the
following diagram:

---
$\textstyle{\mathbb{F}_{q^{d\ell}}(\\!(\theta_{v})\\!)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d\ell}}(\\!(\tilde{\theta}_{v}^{-1})\\!)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k_{d\ell,v}}$$\textstyle{k_{d\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d\ell}}(\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d\ell}}(\tilde{\theta}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d\ell}}(\theta_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(\\!(\theta_{v})\\!)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(\\!(\tilde{\theta}_{v}^{-1})\\!)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k_{v}}$$\textstyle{\mathbb{F}_{q^{d}}(\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(\tilde{\theta}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(\theta_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(\\!(v)\\!)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(\\!(\tilde{v}^{-1})\\!)}$$\textstyle{\mathbb{F}_{q}(\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(\tilde{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q^{d}}(v)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q}(\\!(v)\\!)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q}(\\!(\tilde{v}^{-1})\\!)}$$\textstyle{\mathbb{F}_{q}(\tilde{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q}(v)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Recall that in Section 3.2 we wrote

$v=\theta^{d}+\epsilon_{1}\theta^{d-1}+\cdots+\epsilon_{d-1}\theta+\epsilon_{d},\quad\text{
where }\epsilon_{1},...,\epsilon_{d}\in\mathbb{F}_{q}.$

As

(5.1.1)
$v=\prod_{i=0}^{d-1}(\theta-\varepsilon^{q^{i}})=\prod_{i=0}^{d-1}\big{(}\theta_{v}-(\varepsilon^{q^{i}}-\varepsilon)\big{)},$

There exists
$\tilde{\epsilon}_{1},...,\tilde{\epsilon}_{d-1}\in\mathbb{F}_{q^{d}}$ such
that

(5.1.2)
$v=\theta_{v}^{d}+\tilde{\epsilon}_{1}\theta_{v}^{d-1}+\cdots+\tilde{\epsilon}_{d-1}\theta_{v}\quad\in\mathbb{F}_{q^{d}}[\theta_{v}]=\mathbb{F}_{q^{d}}[\theta].$

Multiplying $\tilde{v}\tilde{\theta}_{v}^{d}$ on both sides of the equation
(5.1.2), one gets

(5.1.3)
$\tilde{\theta}_{v}^{d}-\Big{(}(\tilde{\epsilon}_{d-1}\tilde{v})\cdot\tilde{\theta}_{v}^{d-1}+\cdots+(\tilde{c}_{1}\tilde{v})\tilde{\theta}_{v}+\tilde{v}\Big{)}=0.$

On the other hand, recall in Section 3.2 that
$\iota:\mathbb{F}_{q}(t)\cong\mathbb{F}_{q}(\theta)$ is the
$\mathbb{F}_{q}$-algebra isomorphism sending $t$ to $\theta$, and so
$\iota(T_{v})=v$. Extending $\iota$ to an $\mathbb{F}_{q^{d\ell}}$-algebra
isomorphism
$\iota:\mathbb{F}_{q^{d\ell}}(t)\cong\mathbb{F}_{q^{d\ell}}(\theta)=k_{d\ell}$,
we set

$\tilde{t}_{v}:=\frac{1}{t-\varepsilon}\quad\text{ and
}\quad\tilde{T}_{v}:=\frac{1}{T_{v}}.$

Then $\iota(\tilde{t}_{v})=\tilde{\theta}_{v}$ and
$\iota(\tilde{T}_{v})=\tilde{v}$. Moreover, the following equality holds:

(5.1.4)
$\tilde{t}_{v}^{d}-(\tilde{\epsilon}_{d-1}\tilde{T}_{v})\tilde{t}_{v}^{d-1}-\cdots-(\tilde{\epsilon}_{1}\tilde{T}_{v})\tilde{t}_{v}-\tilde{T}_{v}=0.$

In other words, the minimal polynomial of $\tilde{t}_{v}$ over
$\mathbb{F}_{q^{d\ell}}(\tilde{T}_{v})$ is

$\tilde{g}_{\tilde{T}_{v}}(X)=X^{d}-(\tilde{\epsilon}_{d-1}\tilde{T}_{v})X^{d-1}-\cdots-(\tilde{\epsilon}_{1}\tilde{T}_{v})X-\tilde{T}_{v}\quad\in\mathbb{F}_{q^{d}}[\tilde{T}_{v}][X],$

In particular, the field $\mathbb{F}_{q^{d\ell}}(\tilde{t}_{v})$ is a finite
separable extension of $\mathbb{F}_{q}(\tilde{T}_{v})$ of degree $d^{2}\ell$,
and the integral closure of $\mathbb{F}_{q}[\tilde{T}_{v}]$ in
$\mathbb{F}_{q^{d\ell}}(\tilde{t}_{v})$ is
$\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$. Indeed, as

$\tilde{T}_{v}=\frac{1}{T_{v}}=\prod_{i=0}^{d-1}\frac{1}{t-\varepsilon^{q^{i}}}=\tilde{t}_{v}^{d}\cdot\frac{1}{S},$

where $S=\prod_{i=1}^{d-1}(1-(\varepsilon^{q^{i}}-\varepsilon)\tilde{t}_{v})$
is coprime to $\tilde{t}_{v}^{d}$ in $\mathbb{F}_{q^{d}}[\tilde{t}_{v}]$.
Hence

(5.1.5)
$\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]=\mathbb{F}_{q^{d\ell}}[\tilde{t}_{v}][\frac{1}{S}],$

which is a localization of the PID $\mathbb{F}_{q^{d\ell}}[\tilde{t}_{v}]$. It
follows that $\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$ is a PID,
which implies integrally closed property, and hence it equals the integral
closure of $\mathbb{F}_{q}[\tilde{T}_{v}]$ in
$\mathbb{F}_{q^{d\ell}}(\tilde{t}_{v})$. Moreover, since
$\tilde{g}_{\tilde{T}_{v}}(X)$ is Eisenstein with respect to the prime ideal
$(\tilde{T}_{v})$ of $\mathbb{F}_{q^{d}}[\tilde{T}_{v}]$, the prime ideal
$(\tilde{T}_{v})$ of $\mathbb{F}_{q}[\tilde{T}_{v}]$ is totally ramified in
$\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$, and the unique prime
ideal of $\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$ lying above
$(\tilde{T}_{v})$ is the principal ideal generated by $\tilde{t}_{v}$. Set

$\tilde{t}_{v,i}:=\frac{1}{t-\varepsilon^{q^{-i}}}=\frac{\tilde{t}_{v}}{1-(\varepsilon^{q^{-i}}-\varepsilon)\tilde{t}_{v}},\quad
0\leq i<d\ell.$

One checks that

  * (1)

$\tilde{\theta}_{v,i}:=\iota(\tilde{t}_{v,i})=(\theta-\varepsilon^{q^{-i}})^{-1}$.

  * (2)

$\tilde{t}_{v,i}=\tilde{t}_{v,i^{\prime}}$ if and only if $i\equiv
i^{\prime}\bmod d$;

  * (3)

$\tilde{T}_{v}=\tilde{t}_{v,0}\cdots\tilde{t}_{v,d-1}$;

  * (4)

The minimal polynomial of $\tilde{t}_{v,i}$ over
$\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}]$ is

(5.1.6)
$\tilde{g}_{\tilde{T}_{v}}^{(-i)}(X)=X^{d}-\Big{(}\tilde{\epsilon}_{d-1}^{q^{-i}}\tilde{T}_{v}X^{d-1}+\cdots+\tilde{\epsilon}_{1}^{q^{-i}}\tilde{T}_{v}\cdot
X+\tilde{T}_{v}\Big{)}\quad\in\mathbb{F}_{q^{d}}[\tilde{T}_{v}][X];$

  * (5)

$(\tilde{t}_{v})=(\tilde{t}_{v,0})=(\tilde{t}_{v,1})=\cdots=(\tilde{t}_{v,d\ell-1})\subset\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$.

The above setting (on the “$t$-variable side”) can be illustrated by the
following diagram:

---
$\textstyle{\mathbb{F}_{q^{d\ell}}(t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{=}$$\textstyle{\mathbb{F}_{q^{d\ell}}(\tilde{t}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supset}$$\textstyle{\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supset}$$\textstyle{(\tilde{t}_{v,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{=}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{=}$$\textstyle{(\tilde{t}_{v,d-1})}$$\textstyle{\mathbb{F}_{q^{d\ell}}(T_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{=}$$\textstyle{\mathbb{F}_{q^{d\ell}}(\tilde{T}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supset}$$\textstyle{\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supset}$$\textstyle{(\tilde{T}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{q}(T_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{=}$$\textstyle{\mathbb{F}_{q}(\tilde{T}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supset}$$\textstyle{\mathbb{F}_{q}[\tilde{T}_{v}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supset}$$\textstyle{(\tilde{T}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Now, let $\bar{k}(\tilde{T}_{v})[\sigma]$ (resp.
$\bar{k}(\tilde{t}_{v})[\sigma]$) be the twisted polynomial ring over
$\bar{k}(\tilde{T}_{v})$ (resp. $\bar{k}(\tilde{t}_{v})$) satisfying the
following multiplication law

$\sigma\cdot f=f^{(-1)}\cdot\sigma,\quad\forall f\in\bar{k}(\tilde{T}_{v})\
\text{ (resp.\ $\bar{k}(\tilde{t}_{v})$)},$

where $f^{(n)}$ is the $n$-th Frobenius twist of $f$, i.e. raising the
coefficients of powers of $\tilde{T}_{v}$) of $f$ to the $q^{n}$-power for
each $n\in\mathbb{Z}$.

######  Lemma 5.1.7.

$\bar{k}(\tilde{T}_{v})[\sigma]$ is a subring of
$\bar{k}(\tilde{t}_{v})[\sigma]$ if and only if
$\tilde{\epsilon}_{1},...,\tilde{\epsilon}_{d-1}\in\mathbb{F}_{q}$.

###### Proof.

By definition one observes that $\bar{k}(\tilde{T}_{v})[\sigma]$ is a subring
of $\bar{k}(\tilde{t}_{v})[\sigma]$ if and only if
$\tilde{T}_{v}\cdot\sigma=\sigma\cdot\tilde{T}_{v}$ in
$\bar{k}(\tilde{t}_{v})[\sigma]$, which is equivalent to

$\displaystyle\
\Big{(}\tilde{t}_{v}^{d}-(\tilde{\epsilon}_{d-1}\tilde{T}_{v})\tilde{t}_{v}^{d-1}-\cdots-(\tilde{\epsilon}_{1}\tilde{T}_{v})\tilde{t}_{v}\Big{)}\cdot\sigma$
$\displaystyle=$ $\displaystyle\
\tilde{T}_{v}\cdot\sigma=\sigma\cdot\tilde{T}_{v}$ $\displaystyle=$
$\displaystyle\
\sigma\cdot\Big{(}\tilde{t}_{v}^{d}-(\tilde{\epsilon}_{d-1}\tilde{T}_{v})\tilde{t}_{v}^{d-1}-\cdots-(\tilde{\epsilon}_{1}\tilde{T}_{v})\tilde{t}_{v}\Big{)}$
$\displaystyle=$ $\displaystyle\
\Big{(}\tilde{t}_{v}^{d}-(\tilde{\epsilon}_{d-1}^{q^{-1}}\tilde{T}_{v})\tilde{t}_{v}^{d-1}-\cdots-(\tilde{\epsilon}_{1}^{q^{-1}}\tilde{T}_{v})\tilde{t}_{v}\Big{)}\cdot\sigma.$

This equality holds if and only if
$\tilde{\epsilon}_{1},...,\tilde{\epsilon}_{d-1}\in\mathbb{F}_{q}$. ∎

######  Remark 5.1.8.

The above lemma indicates that $\bar{k}(\tilde{T}_{v})[\sigma]$ might not be a
subring of $\bar{k}(\tilde{t}_{v})[\sigma]$ when $d>1$. However, both do
contain a subring $\bar{k}(\tilde{T}_{v})[\sigma^{d}]$.

In the remaining part of this section, we will review the necessary properties
of the Hartl–Juschka category $\mathscr{H}\mathscr{J}$ of over $\bar{k}$
introduced in [HaJu20, Definition 2.4.1] (with respect to the embedding
$\iota:\mathbb{F}_{q}(\tilde{T}_{v})\cong\mathbb{F}_{q}(\tilde{v})\subset\bar{k}\subset\mathbb{C}_{v}$),
as well as their corresponding “Hodge–Pink cocharacters”.

###  5.2. Hartl–Juschka Category

The Hartl–Juschka category over $\bar{k}$ is defined as follows: an object of
$\mathscr{H}\mathscr{J}$ is a free left $\bar{k}[\tilde{T}_{v}]$-module
$\mathscr{M}$ of finite rank together with an invertible $\sigma$-action on
$\mathit{M}:=\bar{k}(\tilde{T}_{v})\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{M}$
so that $\mathit{M}$ becomes a left $\bar{k}(\tilde{T}_{v})[\sigma]$-module
satisfying that there exists a sufficiently large integer $n$ so that

$(\tilde{T}_{v}-\iota(\tilde{T}_{v}))^{n}\mathscr{M}\subseteq\sigma\mathscr{M}\subseteq(\tilde{T}_{v}-\iota(\tilde{T}_{v}))^{-n}\mathscr{M}\quad(\subset\mathit{M}).$

We call such an object $\mathscr{M}$ a _Hartl–Juschka module over $\bar{k}$_.
In particular, we say that $\mathscr{M}$ is effective if
$\sigma\mathscr{M}\subset\mathscr{M}$. A morphism
$f:\mathscr{M}\rightarrow\mathscr{M}^{\prime}$ in $\mathscr{H}\mathscr{J}$ is
a left $\bar{k}[\tilde{T}_{v}]$-module homomorphism so that the induced
$\bar{k}(\tilde{T}_{v})$-linear homomorphism
$f:\mathit{M}=\bar{k}(\tilde{T}_{v})\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{M}\rightarrow\mathit{M}^{\prime}=\bar{k}(\tilde{T}_{v})\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{M}^{\prime}$
satisfies

$f(\sigma m)=\sigma f(m),\quad\forall\,m\in\mathit{M}.$

Given two objects $\mathscr{M}$ and $\mathscr{M}^{\prime}$ in
$\mathscr{H}\mathscr{J}$, the set
$\operatorname{Hom}(\mathscr{M},\mathscr{M}^{\prime})$ of morphisms from
$\mathscr{M}$ to $\mathscr{M}^{\prime}$ has a natural
$\mathbb{F}_{q}[\tilde{T}_{v}]$-module structure, and elements in

$\operatorname{Hom}^{0}(\mathscr{M},\mathscr{M}^{\prime}):=\mathbb{F}_{q}(\tilde{T}_{v})\otimes_{\mathbb{F}_{q}[\tilde{T}_{v}]}\operatorname{Hom}(\mathscr{M},\mathscr{M}^{\prime})$

are called quasi-morphisms from $\mathscr{M}$ to $\mathscr{M}^{\prime}$. The
Hartl-Juschka Category $\mathscr{H}\mathscr{J}^{I}$ up to isogeny has the same
objects in $\mathscr{H}\mathscr{J}$ but the morphisms in
$\mathscr{H}\mathscr{J}^{I}$ are given by quasi-morphisms.

5.2.1. Uniformizability. Let $\mathbb{C}_{v}\\{\tilde{T}_{v}\\}$ be the Tate
algebra in the variable $\tilde{T}_{v}$ over $\mathbb{C}_{v}$, i.e.

$\mathbb{C}_{v}\\{\tilde{T}_{v}\\}:=\Big{\\{}\sum_{i\geq
0}a_{i}\tilde{T}_{v}^{i}\in\mathbb{C}_{v}[\\![\tilde{T}_{v}]\\!]\ \Big{|}\
\lim_{i\rightarrow\infty}|a_{i}|_{v}=0\Big{\\}}.$

We let $\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\sigma]$ be the non-commutative
algebra generated by $\sigma$ over $\mathbb{C}_{v}\\{\tilde{T}_{v}\\}$ subject
to the relation

$\sigma f=f^{(-1)}\sigma,\quad\forall\,f\in\mathbb{C}_{v}\\{\tilde{T}_{v}\\},$

where for every $f=\sum_{i\geq
0}a_{i}\tilde{T}_{v}^{i}\in\mathbb{C}_{v}\\{\tilde{T}_{v}\\}$, we define

$f^{(n)}:=\sum_{i\geq 0}a_{i}^{q^{n}}\tilde{T}_{v}^{i},\quad\forall
n\in\mathbb{Z}.$

Given an object $\mathscr{M}$ of $\mathscr{H}\mathscr{J}$, we set
$\mathbb{M}:=\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{M}$.
As $\tilde{T}_{v}-\iota(\tilde{T}_{v})=\tilde{T}_{v}-\tilde{v}$ is invertible
in $\mathbb{C}_{v}\\{\tilde{T}_{v}\\}$, one has that
$\mathit{M}\subset\mathbb{M}$, and $\mathbb{M}$ becomes a left
$\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\sigma]$-module with the $\sigma$-action
given by

$\sigma\cdot(f\otimes m):=f^{(-1)}\otimes\sigma m,\quad\forall
f\in\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\text{ and }m\in\mathscr{M}.$

Put

$H_{\mathrm{Betti}}(\mathscr{M}):=\\{m\in\mathbb{M}\mid\sigma m=m\\}.$

We call $\mathscr{M}$ _uniformizable_ (or _rigid analytically trivial_) if the
natural homomorphism

$\displaystyle\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\otimes_{\mathbb{F}_{q}[\tilde{T}_{v}]}H_{\mathrm{Betti}}(\mathscr{M})$
$\displaystyle\longrightarrow\mathbb{M}$

is an isomorphism. In particular, let
$\mathbb{C}_{v}\\{\tilde{T}_{v}\\}^{\dagger}$ be the algebra of rigid-analytic
functions on $\mathbb{C}_{v}\setminus\\{\tilde{v}^{q^{n}}\mid
n\in\mathbb{N}\\}$. It is known that (see [HaJu20, Proposition 2.4.27]) if
$\mathscr{M}$ is uniformizable, then $H_{\mathrm{Betti}}(\mathscr{M})$ is
contained in
$\mathbb{M}^{\dagger}=\mathbb{C}_{v}\\{\tilde{T}_{v}\\}^{\dagger}\otimes_{\bar{k}[t]}\mathscr{M}$,
and the natural homomorphism

$\displaystyle\mathbb{C}_{v}\\{\tilde{T}_{v}\\}^{\dagger}\otimes_{\mathbb{F}_{q}[\tilde{T}_{v}]}H_{\text{Betti}}(\mathscr{M})$
$\displaystyle\longrightarrow\mathbb{M}^{\dagger}$

is an isomorphism.

5.2.2. $\tilde{T}_{v}$-Motivic Galois group. The full subcategory of
$\mathscr{H}\mathscr{J}^{I}$ consisting of all uniformizable objects is