denoted by $\mathscr{H}\mathscr{J}^{I}_{U}$. Let
$\mathbf{Vect}_{/\mathbb{F}_{q}(\tilde{T}_{v})}$ be the category of finite
dimensional vector spaces over $\mathbb{F}_{q}(\tilde{T}_{v})$. By [HaJu20,
Theorem 2.4.23], $\mathscr{H}\mathscr{J}^{I}_{U}$ is a neutral Tannakian
category over $\mathbb{F}_{q}(\tilde{T}_{v})$ with fiber functor
$\boldsymbol{\omega}:\mathscr{H}\mathscr{J}^{I}_{U}\rightarrow\mathbf{Vect}_{/\mathbb{F}_{q}(\tilde{T}_{v})}$
defined by:

$\mathscr{M}\longmapsto
H_{\mathscr{M}}:=\mathbb{F}_{q}(\tilde{T}_{v})\otimes_{\mathbb{F}_{q}[\tilde{T}_{v}]}H_{\mathrm{Betti}}(\mathscr{M}).$

In particular, for each object in $\mathscr{H}\mathscr{J}^{I}_{U}$, let
$\mathscr{T}_{\mathscr{M}}$ be the strictly full Tannakian subcategory of
$\mathscr{H}\mathscr{J}^{I}_{U}$ generated by $\mathscr{M}$. By Tannakian
duality, we have an affine algebraic group scheme $\Gamma_{\mathscr{M}}$
defined over $\mathbb{F}_{q}(\tilde{T}_{v})$ so that
$\mathscr{T}_{\mathscr{M}}$ is equivalent to the category
$\mathbf{Rep}(\Gamma_{\mathscr{M}})$ of the
$\mathbb{F}_{q}(\tilde{T}_{v})$-finite dimensional representations of the
algebraic group $\Gamma_{\mathscr{M}}$. We call $\Gamma_{\mathscr{M}}$ the
_$\tilde{T}_{v}$ -motivic Galois group of $\mathscr{M}$_. Note that we have
the following property of $\dim\Gamma_{\mathscr{M}}$ by Papanikolas’
difference Galois theory [P08].

Given an object $\mathscr{M}\in\mathscr{H}\mathscr{J}^{I}_{U}$, we let $r$ be
the rank of $\mathscr{M}$ over $\overline{k}[\tilde{T}_{v}]$ and fix a
$\overline{k}[\tilde{T}_{v}]$-basis $\left\\{m_{1},\ldots,m_{r}\right\\}$ of
$\mathscr{M}$. Suppose $\mathscr{M}$ is effective. Then the action of $\sigma$
on $\mathscr{M}$ is represented by a matrix
$\Phi\in\operatorname{Mat}_{r}(\overline{k}[\tilde{T}_{v}])$ in the sense that

$\sigma\begin{pmatrix}m_{1}\\\ \vdots\\\
m_{r}\end{pmatrix}=\Phi\begin{pmatrix}m_{1}\\\ \vdots\\\ m_{r}\end{pmatrix}.$

Since $\mathscr{M}$ is uniformizable, by [ABP04, P08] there exists a matrix
$\Psi\in\operatorname{GL}_{r}(\mathbb{C}_{v}\left\\{\tilde{T}_{v}\right\\})$
satisfying the system of Frobenius difference equations

(5.2.3) $\Psi^{(-1)}=\Phi\Psi.$

In [P08], Papanikolas developed a difference Galois theory for the system of
Frobenius difference equation (5.2.3) to explicitly construct an algebraic
subgroup $\Gamma_{\Psi}\subset\operatorname{GL}_{r}$ over $\mathbb{F}_{q}(t)$
for which

  * $\bullet$

$\Gamma_{\mathscr{M}}$ is isomorphic to $\Gamma_{\Psi}$ as algebraic groups
over $\mathbb{F}_{q}(t)$ (see [P08, Thm. 4.5.10]).

  * $\bullet$

$\dim\Gamma_{\mathscr{M}}={\rm{tr.deg}}_{\overline{k}(\tilde{T}_{v})}\overline{k}(\tilde{T}_{v})(\Psi)$
(see [P08, Prop. 4.3.3 and Thm. 4.3.1]),

where the notation $\overline{k}(\tilde{T}_{v})(\Psi)$ is referred to be the
field generated all entries of $\Psi$ over $\overline{k}(\tilde{T}_{v})$.
Combining the two properties above, we have the following identity:

(5.2.4)
$\dim\Gamma_{\mathscr{M}}={\rm{tr.deg}}_{\overline{k}(\tilde{T}_{v})}\overline{k}(\tilde{T}_{v})(\Psi)$

######  Remark 5.2.5.

Note that we have a faithful representation

$\Gamma_{\mathscr{M}}\hookrightarrow\operatorname{GL}(H_{\mathscr{M}}),$

which is functorial in $\mathscr{M}$ (see [P08, Thm. 4.5.3]). This implies
that

(5.2.6)
$\Gamma_{\mathscr{M}}\hookrightarrow{\operatorname{Cent}}_{\operatorname{GL}(H_{\mathscr{M}})}\left(\operatorname{End}_{\mathscr{T}_{\mathscr{M}}}(\mathscr{M})\right)\quad\textup{(cf.~{}\cite[cite]{[\@@bibref{}{CPY10}{}{},
p.~{}1448, (6)]})}$

as we have the natural embedding
$\operatorname{End}_{\mathscr{T}_{\mathscr{M}}}(\mathscr{M})\hookrightarrow\operatorname{End}(H_{\mathscr{M}})$.

5.2.7. de Rham pairing and the Grothendieck period conjecture. Let
$\mathscr{M}$ be an object of $\mathscr{H}\mathscr{J}^{I}_{U}$. Recall that
the de Rham module of $\mathscr{M}$ over $\bar{k}$ is

$H_{\text{dR}}(\mathscr{M},\bar{k}):=\operatorname{Hom}_{\bar{k}}\left(\frac{\mathscr{M}}{(\tilde{T}_{v}-\tilde{v})\cdot\mathscr{M}},\bar{k}\right).$

In particular, every differential $\omega\in
H_{\text{dR}}(\mathscr{M},\bar{k})$ can be extended to a
$\mathbb{C}_{v}$-linear functional on
$\mathbb{M}^{\dagger}=\mathbb{C}_{v}\\{\tilde{T}_{v}\\}^{\dagger}\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{M}$
vanishing on $(\tilde{T}_{v}-\tilde{v})\cdot\mathbb{M}^{\dagger}$. As the
uniformizability of $\mathscr{M}$ ensures the inclusion
$H_{\text{Betti}}(\mathscr{M})\subset\mathbb{M}^{\dagger}$, we have the
following _de Rham pairing_

(5.2.8) $\displaystyle H_{\mathrm{Betti}}(\mathscr{M})\times
H_{\mathrm{dR}}(\mathscr{M},\bar{k})$
$\displaystyle\longrightarrow\mathbb{C}_{v},$ $\displaystyle(\gamma,\omega)$
$\displaystyle\longmapsto\int_{\gamma}\omega:=\omega(\gamma).$

The primary tool in this paper of dealing with transcendence problems is the
following function field version of Grothendieck’s period conjecture, which
was achieved by Papanikolas [P08] using the so-called ABP criterion [ABP04].

######  Theorem 5.2.9.

(See [P08] and its refined version in [C09].) Let $\mathscr{M}$ be a
uniformizable Hartl–Juschka module $\mathscr{M}$ over $\bar{k}$. Suppose that
$\mathscr{M}$ is effective. Then we have that

$\dim\Gamma_{\textrm{$\mathscr{M}$}}=\bar{k}\left(\int_{\gamma}\omega\
\bigg{|}\ \gamma\in H_{\mathrm{Betti}}(\mathscr{M}),\ \omega\in
H_{\mathrm{dR}}(\mathscr{M},\bar{k})\right).$

###### Proof.

Let $\Phi\in\operatorname{Mat}_{r}(\overline{k}[\tilde{T}_{v}])$ be the matrix
represented by the $\sigma$-action on $\mathscr{M}$ with respect to a
$\overline{k}[\tilde{T}_{v}]$-basis $\left\\{m_{1},\ldots,m_{r}\right\\}$.
Since $\mathscr{M}$ is uniformizable, there exists
$\Psi\in\operatorname{GL}_{r}(\mathbb{C}_{v}\left\\{\tilde{T}_{v}\right\\})$
for which $\Psi^{(-1)}=\Phi\Psi$. Put
$\mathbf{m}=(m_{1},\ldots,m_{r})^{\rm{tr}}$, and note that the entries of
$\Psi^{-1}\mathbf{m}$ comprises an $\mathbb{F}_{q}[\tilde{T}_{v}]$-basis of
$H_{\mathrm{Betti}}(\mathscr{M})$ (see [ABP04, Lem. 4.4.12] for their
arguments without the condition that $\mathscr{M}$ is free of finite rank over
$\overline{k}[\sigma]$.) As mentioned above that
$H_{\mathrm{Betti}}(\mathscr{M})\subset\mathscr{M}^{\dagger}$, elements of
$\Psi^{-1}$ are in fact in $\mathbb{C}_{v}\\{\tilde{T}_{v}\\}^{\dagger}$.

Our hypothesis on $\det\Phi$ ensures that the conditions of the refined
version of the ABP criterion is satisfied (see [C09]), and so we have

${\rm{tr.deg}}_{\overline{k}(\tilde{T}_{v})}\overline{k}(\tilde{T}_{v})(\Psi)={\rm{tr.deg}}_{\overline{k}}\overline{k}(\Psi|_{\tilde{T}_{v}=\tilde{v}})={\rm{tr.deg}}_{\overline{k}}\overline{k}(\Psi^{-1}|_{\tilde{T}_{v}=\tilde{v}}),$

where $\overline{k}(\Psi|_{\tilde{T}_{v}=\tilde{v}})$ (resp.
$\overline{k}(\Psi^{-1}|_{\tilde{T}_{v}=\tilde{v}})$) denotes by the field
over $\overline{k}$ generated by all entries of $\Psi$ (resp. of $\Psi^{-1}$)
evaluated at $\tilde{T}_{v}=\tilde{v}$. It follows that by (5.2.4), we have

(5.2.10)
$\dim\Gamma_{\mathscr{M}}={\rm{tr.deg}}_{\overline{k}}\overline{k}(\Psi^{-1}|_{\tilde{T}_{v}=\tilde{v}}).$

As each $\omega\in H_{\mathrm{dR}}(\mathscr{M},\bar{k})$ can be viewed as a
$\overline{k}$-linear functional on $\mathscr{M}$ factoring through
$\mathscr{M}/(t-\theta)\mathscr{M}$, a natural basis of
$H_{\mathrm{dR}}(\mathscr{M},\bar{k})$ can be chosen as
$\left\\{\omega_{j}\right\\}_{j=1}^{r}\subset
H_{\mathrm{dR}}(\mathscr{M},\bar{k})$ given by
$\omega_{j}(m_{i})=\delta_{ij}$, the standard Kronecker symbol. Note that
$\left\\{\gamma_{i}:=\sum_{j=1}^{r}(\Psi^{-1})_{ij}m_{j}\right\\}_{i=1}^{r}$
is an $\mathbb{F}_{q}[\tilde{T}_{v}]$-basis of
$H_{\mathrm{Betti}}(\mathscr{M})$, by (5.2.8) we have that

$\int_{\gamma_{i}}\omega_{j}=((\Psi^{-1})_{ij})|_{\tilde{T}_{v}=\tilde{v}}=(\Psi^{-1}|_{\tilde{T}_{v}=\tilde{v}})_{ij}$

for every $1\leq i,j\leq r$. So the desired result follows from (5.2.10). ∎

###  5.3. Hodge–Pink cocharacters

Given an object $\mathscr{M}$ of $\mathscr{H}\mathscr{J}^{I}_{U}$, we take a
sufficiently large integer $n$ such that

(5.3.1)
$(\tilde{T}_{v}-\tilde{v})^{n}\mathscr{M}\subseteq\sigma\mathscr{M}\subseteq(\tilde{T}_{v}-\tilde{v})^{-n}\mathscr{M}.$

Let $r$ be the rank of $\mathscr{M}$ over $\bar{k}[\tilde{T}_{v}]$. There
exists a unique (non-ordered) $r$-tuple of integers $(w_{1},\ldots,w_{r})$ so
that

$\frac{(\tilde{T}_{v}-\tilde{v})^{-n}\mathscr{M}}{\sigma\mathscr{M}}\cong\bigoplus_{i=1}^{r}\frac{\bar{k}[\tilde{T}_{v}]}{(\tilde{T}_{v}-\tilde{v})^{n-w_{i}}\bar{k}[\tilde{T}_{v}]}.$

We call $(w_{1},\dots,w_{r})$ the _Hodge–Pink weights of $\mathscr{M}$_ and
mention that they are independent of the choices of $n$. Let
$\\{m_{1},\ldots,m_{r}\\}$ be a basis of the free
$\bar{k}[\tilde{T}_{v}]$-module $\mathscr{M}$ so that

$\sigma\mathscr{M}=\bigoplus_{i=1}^{r}(\tilde{T}_{v}-\tilde{v})^{-w_{i}}\bar{k}[\tilde{T}_{v}]\cdot
m_{i}\quad\subset\bar{k}(\tilde{T}_{v})\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{M}.$

Put

$H_{\mathscr{M},\mathbb{C}_{v}}:=\mathbb{C}_{v}\underset{\tilde{v}\mathrel{\reflectbox{$\mapsto$}}\tilde{T}_{v},\
\mathbb{F}_{q}[\tilde{T}_{v}]}{\otimes}H_{\mathscr{M}},$

and let $\lambda_{1},\ldots,\lambda_{r}\in H_{\matheur{M},\mathbb{C}_{v}}$ be
the images of $m_{1},\dots,m_{r}$ through the following identification

$\displaystyle\mathbb{C}_{v}\otimes_{\bar{k}}\frac{\mathscr{M}}{(\tilde{T}_{v}-\tilde{v})\mathscr{M}}\cong\frac{\mathbb{M}^{\dagger}}{(\tilde{T}_{v}-\tilde{v})\mathbb{M}^{\dagger}}$
$\displaystyle\cong\frac{\mathbb{C}_{v}\\{\tilde{T}_{v}\\}^{\dagger}}{(\tilde{T}_{v}-\tilde{v})\mathbb{C}_{v}\\{\tilde{T}_{v}\\}^{\dagger}}\otimes_{\mathbb{F}_{q}[\tilde{T}_{v}]}H_{\mathscr{M}}$
(5.3.2)
$\displaystyle\cong\mathbb{C}_{v}\underset{\tilde{v}\mathrel{\reflectbox{$\mapsto$}}\tilde{T}_{v},\
\mathbb{F}_{q}[\tilde{T}_{v}]}{\otimes}H_{\mathscr{M}}=H_{\mathscr{M},\mathbb{C}_{v}}.$

Then we have that:

######  Proposition 5.3.3.

(See [Pink97, Definition 7.10] and [BCPW22, Section 7.3].) Keep the notation
as above. Let
$\Gamma_{\mathscr{M},\mathbb{C}_{v}}:=\mathbb{C}_{v}\times_{\mathbb{F}_{q}(\tilde{T}_{v})}\Gamma_{\mathscr{M}}$,
the base change of $\Gamma_{\mathscr{M}}$ to $\mathbb{C}_{v}$ via the
evaluation map
$\iota:=\left(\tilde{T}_{v}\mapsto\tilde{v}\right):\mathbb{F}_{q}(\tilde{T}_{v})\hookrightarrow\mathbb{C}_{v}$.
There exists a cocharacter
$\chi_{\mathscr{M}}^{\vee}:\mathbb{G}_{m/\mathbb{C}_{v}}\rightarrow\Gamma_{\mathscr{M},\mathbb{C}_{v}}$
satisfying that after composing with the embedding
$\Gamma_{\mathscr{M},\mathbb{C}_{v}}\hookrightarrow\operatorname{GL}(H_{\mathscr{M},\mathbb{C}_{v}})$,
one has that

$\chi_{\mathscr{M}}^{\vee}(x)(\lambda_{i})=x^{w_{i}}\cdot\lambda_{i},\quad\forall
x\in\mathbb{G}_{m/\mathbb{C}_{v}}(\mathbb{C}_{v})\text{ and }1\leq i\leq r.$

Since
$\operatorname{GL}(H_{\mathscr{M},\mathbb{C}_{v}})=\mathbb{C}_{v}\underset{\tilde{v}\mathrel{\reflectbox{$\mapsto$}}\tilde{T}_{v},\mathbb{F}_{q}(\tilde{T}_{v})}{\times}\operatorname{GL}(H_{\mathscr{M}})$,
we have a natural action of
$\operatorname{Aut}_{\mathbb{F}_{q}(v)}(\mathbb{C}_{v})$ on
$\operatorname{GL}(H_{\mathscr{M},\mathbb{C}_{v}})$, which leaves
$\Gamma_{\mathscr{M},\mathbb{C}_{v}}$ invariant. Thus any
$\operatorname{Aut}_{\mathbb{F}_{q}(v)}(\mathbb{C}_{v})$-conjugate of
$\chi_{\mathscr{M}}^{\vee}$ is still a cocharacter of
$\Gamma_{\mathscr{M},\mathbb{C}_{v}}$.

######  Definition 5.3.4.

(cf. [Pink97, Definition 7.10].) Given an object $\mathscr{M}$ in
$\mathscr{H}\mathscr{J}^{I}_{U}$, a _Hodge–Pink cocharacter_ of
$\Gamma_{\mathscr{M},\mathbb{C}_{v}}$ is a cocharacter which lies in the
$\Gamma_{\mathscr{M},\mathbb{C}_{v}}\rtimes\operatorname{Aut}_{\mathbb{F}_{q}(v)}(\mathbb{C}_{v})$-conjugate
class of $\chi_{\mathscr{M}}^{\vee}$.

##  6\. Carlitz-type Hartl–Juschka modules and their periods

Throughout this section, we fix an irreducible element $v\in A_{+}$ of degree
$d$ as before and fix a positive integer $\ell$. Having sufficient tools at
hand, we are now able to interpret $\Omega_{d\ell,v,i}(v)$ for $0<i\leq d\ell$
(occuring in Theorem 4.1.2) as the periods of the restriction of scalars of
the “Carlitz-type” Hartl–Juschka modules, and establish their algebraic
independence in this section.

###  6.1. Carlitz-type Hartl–Juschka modules and restriction of scalars

######  Definition 6.1.1.

Let $\tilde{\mathscr{C}}_{d\ell}$ be the left
$\bar{k}[\tilde{T}_{v}][\sigma^{d\ell}]$-module
$\bar{k}[\tilde{T}_{v}][\tilde{t}_{v}]$ whose $\sigma^{d\ell}$-action is given
by

$\sigma^{d\ell}\cdot m=(1-\frac{\tilde{t}_{v}}{\tilde{\theta}_{v}})\cdot
m^{(-d\ell)},\quad\forall m\in\bar{k}[\tilde{T}_{v}][\tilde{t}_{v}].$

######  Remark 6.1.2.

By (5.1.3) and (5.1.4), one has that

$\tilde{T}_{v}\equiv\tilde{v}\bmod(\tilde{t}_{v}-\tilde{\theta}_{v})\bar{k}[\tilde{T}_{v}][\tilde{t}_{v}],$

for which we get

$(\tilde{T}_{v}-\tilde{v})\cdot\tilde{\mathscr{C}}_{d\ell}\subset\sigma^{d\ell}\cdot\tilde{\mathscr{C}}_{d\ell}\subset\tilde{\mathscr{C}}_{d\ell}.$

By (5.1.5), we have
$\bar{k}[\tilde{T}_{v}][\tilde{t}_{v}]=\sum_{n=0}^{\infty}\frac{1}{S^{n}}\cdot\bar{k}[\tilde{t}_{v}]$
is not finitely generated as a $\bar{k}[\tilde{t}_{v}]$-module if $d>1$. Since
$\frac{1}{S^{n}}\cdot\bar{k}[\tilde{t}_{v}]\subset\tilde{\mathscr{C}}_{d\ell}$
is invariant by $\sigma^{d\ell}$-action for every non-negative integer $n$, we
see that $\tilde{\mathscr{C}}_{d\ell}$ is not finitely generated as a
$\bar{k}[\sigma^{d\ell}]$-module when $d>1$ (i.e.
$\tilde{\mathscr{C}}_{d\ell}$ is not an ‘Anderson dual $\tilde{T}_{v}$-motive’
in the terminalogy of [ABP04]).

######  Definition 6.1.3.

The restriction of scalars of $\tilde{\mathscr{C}}_{d\ell}$ is the following
$\bar{k}[\tilde{T}_{v}][\sigma]$-module:

$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}):=\bar{k}[\tilde{T}_{v}][\sigma]\otimes_{\bar{k}[\tilde{T}_{v}][\sigma^{d\ell}]}\tilde{\mathscr{C}}_{d\ell}.$

Decomposing $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ into

$\bigoplus_{i=0}^{d\ell-1}\sigma^{i}\otimes\tilde{\mathscr{C}}_{d\ell},$

we get

$\displaystyle\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})\supset\sigma\cdot\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$
$\displaystyle=\left((1-\frac{\tilde{t}_{v}}{\tilde{\theta}_{v}})\cdot
1\otimes\tilde{\mathscr{C}}_{d\ell}\right)\oplus\bigoplus_{i=1}^{d\ell-1}\sigma^{i}\otimes\tilde{\mathscr{C}}_{d\ell}$
$\displaystyle\supset(\tilde{T}_{v}-\tilde{v})\cdot\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}).$

Thus $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ is an effective
Hartl–Juschka module $\bar{k}$.

We see from the following proposition that
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ has ‘complex
multiplication’ by $\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$.

######  Proposition 6.1.4.

Let notation be given as above. Then we have the following
$\mathbb{F}_{q}[\tilde{T}_{v}]$-algebra embedding:

$\phi:=\big{(}\alpha\mapsto\phi_{\alpha}\big{)}:\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]\hookrightarrow\operatorname{End}_{\bar{k}[\tilde{T}_{v}][\sigma]}(\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})),$

where

$\phi_{\alpha}:=\big{(}\sum_{i=0}^{d\ell-1}\sigma^{i}\otimes
m_{i}\mapsto\sum_{i=0}^{d\ell-1}\sigma^{i}\otimes\alpha m_{i}\big{)}.$

###### Proof.

From the definition of $\phi$, one sees that $\phi$ is a ring homomorphism. We
confirm that for any
$\alpha\in\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$,
$\phi_{\alpha}$ commutes with $\sigma$ as follows:

$\displaystyle\phi_{\alpha}\circ\sigma\left(\sum_{i=0}^{d\ell-1}\sigma^{i}\otimes
m_{i}\right)$ $\displaystyle\hskip
28.45274pt=\phi_{\alpha}\left(\sum_{i=0}^{d\ell-2}\sigma^{i+1}\otimes
m_{i}+1\otimes\sigma^{d\ell}m_{d\ell-1}\right)$ $\displaystyle\hskip
28.45274pt=\sum_{i=0}^{d\ell-2}\sigma^{i+1}\otimes\alpha
m_{i}+1\otimes\alpha\sigma^{d\ell}m_{d\ell-1}=\sum_{i=0}^{d\ell-2}\sigma^{i+1}\otimes\alpha
m_{i}+1\otimes\sigma^{d\ell}\alpha m_{d\ell-1}$ $\displaystyle\hskip
28.45274pt=\sigma\left(\sum_{i=0}^{d\ell-1}\sigma^{i}\otimes\alpha
m_{i}\right)=\sigma\circ\phi_{\alpha}\left(\sum_{i=0}^{d\ell-1}\sigma^{i}\otimes
m_{i}\right).$

∎

######  Remark 6.1.5.

Alternatively, we may regard $\tilde{\mathscr{C}}_{d\ell}$ as a
$(\bar{k}[\tilde{T}_{v}][\sigma^{d\ell}],\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}])$-bimodule,
and get the $\mathbb{F}_{q}[\tilde{T}_{v}]$-algebra homomorphism
$\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]\hookrightarrow\operatorname{End}_{\bar{k}[\tilde{T}_{v}][\sigma]}(\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}))$
explicitly described in Proposition 6.1.4.

6.1.6. Explicit descriptions of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$. Let
$\bar{k}[\tilde{T}_{v}][X]$ be the polynomial ring with variable $X$ over
$\bar{k}[\tilde{T}_{v}]$, and let $\tilde{g}_{\tilde{T}_{v}}(X)$ be the
minimal polynomial of $\tilde{t}_{v}$ over $\bar{k}[\tilde{T}_{v}]$, i.e., by
(5.1.4), we have

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

Then $\bar{k}[\tilde{T}_{v}][\tilde{t}_{v}]$ can be identified with

$\frac{\bar{k}[\tilde{T}_{v}][X]}{\tilde{g}_{\tilde{T}_{v}}(X)\cdot\bar{k}[\tilde{T}_{v}][X]}.$

We extend the Frobenius twists by fixing $X$ to $\bar{k}(\tilde{T}_{v})[X]$ as
follows: for each $f(X)=\sum_{m}f_{m}X^{m}$ and $n\in\mathbb{Z}$,

$f^{(n)}(X):=\sum_{m}f_{m}^{(n)}X^{m}.$

We point out that when $f(X)\in\bar{k}[X]$, i.e. $f_{m}\in\bar{k}$ for all
$m$, one has that $f^{(n)}(X)=\sum_{m}f_{m}^{q^{n}}X^{m}$. In the
decomposition

$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})=\bigoplus_{i=0}^{d\ell-1}\sigma^{i}\otimes\tilde{\mathscr{C}}_{d\ell},$

for each $i$ with $0\leq i<d\ell$ we have the $\bar{k}[\tilde{T}_{v}]$-linear
isomorphism

$\frac{\bar{k}[\tilde{T}_{v}][X]}{\tilde{g}_{\tilde{T}_{v}}^{(-i)}(X)\cdot\bar{k}[\tilde{T}_{v}][X]}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\sigma^{i}\otimes\tilde{\mathscr{C}}_{d\ell}=\sigma^{i}\otimes\left(\frac{\bar{k}[\tilde{T}_{v}][X]}{\tilde{g}_{\tilde{T}_{v}}(X)\cdot\bar{k}[\tilde{T}_{v}][X]}\right)$

defined by

$\left(\sum_{m=0}^{n}f_{m}X^{m}\right)\bmod\tilde{g}_{\tilde{T}_{v}}^{(-i)}(X)\longmapsto\sigma^{i}\otimes\left(\sum_{m=0}^{n}f_{m}^{(i)}X^{m}\bmod\tilde{g}_{\tilde{T}_{v}}(X)\right).$

Therefore:

######  Proposition 6.1.7.

We may identify $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ with the
product space

$\frac{\bar{k}[\tilde{T}_{v}][X]}{\tilde{g}_{\tilde{T}_{v}}^{(0)}(X)\cdot\bar{k}[\tilde{T}_{v}][X]}\times\frac{\bar{k}[\tilde{T}_{v}][X]}{\tilde{g}_{\tilde{T}_{v}}^{(-1)}(X)\cdot\bar{k}[\tilde{T}_{v}][X]}\times\cdots\times\frac{\bar{k}[\tilde{T}_{v}][X]}{\tilde{g}_{\tilde{T}_{v}}^{(1-d\ell)}(X)\cdot\bar{k}[\tilde{T}_{v}][X]},$

equipped with the $\sigma$-action given by:

$\displaystyle\sigma\cdot(h_{0}\bmod\tilde{g}_{\tilde{T}_{v}}^{(0)},h_{1}\bmod\tilde{g}_{\tilde{T}_{v}}^{(-1)},\cdots,h_{d\ell-1}\bmod\tilde{g}_{\tilde{T}_{v}}^{(1-d\ell)})$
$\displaystyle\hskip 42.67912pt:=\left((1-\frac{X}{\tilde{\theta}_{v}})\cdot
h_{d\ell-1}^{(-1)}\bmod\tilde{g}_{\tilde{T}_{v}}^{(0)},h_{0}^{(-1)}\bmod\tilde{g}_{\tilde{T}_{v}}^{(-1)},\cdots,h_{d\ell-2}^{(-1)}\bmod\tilde{g}_{\tilde{T}_{v}}^{(1-d)}\right).$

######  Remark 6.1.8.

As $\tilde{t}_{v,i}$ is a root of $\tilde{g}_{\tilde{T}_{v}}^{(-i)}(X)$ for
each integer $i$ with $0\leq i<d\ell$, one has that

$\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}]\cong\frac{\bar{k}[\tilde{T}_{v}][X]}{\tilde{g}_{\tilde{T}_{v}}^{(-i)}(X)\cdot\bar{k}[\tilde{T}_{v}][X]}.$

The above proposition allows us to identify
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ with

(6.1.9)
$\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,0}]\times\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,1}]\times\cdots\times\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,d\ell-1}],$

and the $\sigma$-action is given by: for every polynomials
$h_{0}(X),...,h_{d\ell-1}(X)\in\bar{k}[\tilde{T}_{v}][X]$,

$\displaystyle\sigma\cdot\Big{(}h_{0}(\tilde{t}_{v,0}),h_{1}(\tilde{t}_{v,1}),\cdots,h_{d\ell-1}(\tilde{t}_{v,d\ell-1})\Big{)}$
$\displaystyle\hskip
42.67912pt=\Big{(}(1-\frac{\tilde{t}_{v,0}}{\tilde{\theta}_{v}})\cdot
h_{d\ell-1}^{(-1)}(\tilde{t}_{v,0}),h_{0}^{(-1)}(\tilde{t}_{v,1}),\cdots,h_{d\ell-2}^{(-1)}(\tilde{t}_{v,d\ell-1})\Big{)}.$

###  6.2. Uniformizability

Recall that $\mathbb{C}_{v}\\{\tilde{T}_{v}\\}$ is the Tate algebra with
variable $\tilde{T}_{v}$ over $\mathbb{C}_{v}$. Denote by
$\mathbb{C}_{v}\\{\tilde{t}_{v}\\}$ the Tate algebra with variable
$\tilde{t}_{v}$ over $\mathbb{C}_{v}$, i.e.

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

Put

$\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v}]=\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\otimes_{\bar{k}[\tilde{T}_{v}]}\bar{k}[\tilde{T}_{v}][\tilde{t}_{v}].$

######  Lemma 6.2.1.

We have the following inclusion

$\mathbb{C}_{v}\\{\tilde{t}_{v}\\}\subset\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v}].$

###### Proof.

For each non-negative integer $i$, by (5.1.4) there exist
$f_{i,j}\in\mathbb{F}_{q^{d}}[\tilde{T}_{v}]$, $0\leq j<d$, with
$\deg_{\tilde{T}_{i}}f_{i,j}\leq i$ such that

$\tilde{t}_{v}^{i}=f_{i,0}\tilde{t}_{v}^{d-1}+\cdots+f_{i,d-1}.$

Write $f_{i,j}(\tilde{T}_{v})=\sum_{s=0}^{i}u_{i,j,s}\tilde{T}_{v}^{s}$ where
$u_{i,j,s}\in\mathbb{F}_{q^{d}}$. Then for each $f=\sum_{i\geq
0}a_{i}\tilde{t}_{v}^{i}\in\mathbb{C}_{v}\\{\tilde{t}_{v}\\}$, the property
$\lim_{i\rightarrow\infty}|a_{i}|_{v}=0$ implies that for $0\leq j<d$ and
$s\geq 0$,

$b_{j,s}:=\lim_{n\rightarrow\infty}\sum_{i=s}^{n}a_{i}u_{i,j,s}\quad\text{
converges absolutely in $\mathbb{C}_{v}$ with
}\quad\lim_{s\rightarrow\infty}b_{j,s}=0.$

Put

$\displaystyle f_{j}$ $\displaystyle:=\sum_{s\geq
0}b_{j,s}\tilde{T}_{v}^{s}=\sum_{s=0}^{\infty}\left(\sum_{i=s}^{\infty}a_{i}u_{i,j,s}\right)\tilde{T}_{v}^{s}$
$\displaystyle\,=\sum_{i=0}^{\infty}a_{i}\left(\sum_{s=0}^{i}u_{i,j,s}\tilde{T}_{v}^{s}\right)=\sum_{i=0}^{\infty}a_{i}f_{i,j}(\tilde{T}_{v})\quad\in\mathbb{C}_{v}\\{\tilde{T}_{v}\\}.$

We then get

$f=\sum_{i=0}^{\infty}a_{i}\tilde{t}_{v}^{i}=\sum_{j=0}^{d-1}\sum_{i=0}^{\infty}a_{i}f_{i,j}\tilde{t}_{v}^{j}=\sum_{j=0}^{d-1}f_{j}\cdot\tilde{t}_{v}^{j}\quad\in\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v}]$

as desired. ∎

######  Remark 6.2.2.

The inclusion $\bar{k}(\tilde{T}_{v})\subset\bar{k}(\tilde{t}_{v})$ induces a
finite morphism $\phi$ from the projective $\tilde{t}_{v}$-line
$\mathbb{P}^{1}_{\tilde{t}_{v}}$ onto the projective $\tilde{T}_{v}$-line
$\mathbb{P}^{1}_{\tilde{T}_{v}}$ (over $\bar{k}$). In particular,
$\bar{k}[\tilde{t}_{v}]$ and $\bar{k}[\tilde{T}_{v}][\tilde{t}_{v}]$ are the
affine coordinate rings of $\mathbb{A}^{1}_{\tilde{t}_{v}}$ and
$U:=\mathbb{A}^{1}_{\tilde{t}_{v}}\setminus\phi^{-1}(\tilde{v})$,
respectively. Since $\mathbb{C}_{v}\\{\tilde{t}_{v}\\}$ and
$\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v}]$ correspond to the the
affinoid domains $\mathbb{A}^{1}_{\tilde{t}_{v},\leq
1}:=\\{x\in\mathbb{A}^{1}_{\tilde{t}_{v}}(\mathbb{C}_{v})\mid|x|_{v}\leq 1\\}$
and $U_{\leq 1}:=\\{x\in U(\mathbb{C}_{v})\mid|x|_{v}\leq 1\\}$, respectively,
the natural inclusion $U_{\leq 1}\subset\mathbb{A}^{1}_{\tilde{t}_{v},\leq 1}$
alternatively implies
$\mathbb{C}_{v}\\{\tilde{t}_{v}\\}\subset\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v}]$.

To show the uniformizability of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$, we first note that the
identification of $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ extends
to

(6.2.3)
$\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})=\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,0}]\times\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,1}]\times\cdots\times\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,d\ell-1}].$

In particular, Lemma 6.2.1 implies that

(6.2.4)
$\mathbb{C}_{v}\\{\tilde{t}_{v,0}\\}\times\mathbb{C}_{v}\\{\tilde{t}_{v,1}\\}\times\cdots\times\mathbb{C}_{v}\\{\tilde{t}_{v,d\ell-1}\\}\subset\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}).$

In particular, for every
$\big{(}h_{0}(\tilde{t}_{v,0}),...,h_{d\ell-1}(\tilde{t}_{v,d\ell-1})\big{)}\in\mathbb{C}_{v}\\{\tilde{t}_{v,0}\\}\times\mathbb{C}_{v}\\{\tilde{t}_{v,1}\\}\times\cdots\times\mathbb{C}_{v}\\{\tilde{t}_{v,d\ell-1}\\}$,
one has that

$\displaystyle\sigma\cdot\Big{(}h_{0}(\tilde{t}_{v,0}),h_{1}(\tilde{t}_{v,1}),\cdots,h_{d\ell-1}(\tilde{t}_{v,d\ell-1})\Big{)}$
(6.2.5) $\displaystyle\hskip
42.67912pt=\Big{(}(1-\frac{\tilde{t}_{v,0}}{\tilde{\theta}_{v}})\cdot
h_{d\ell-1}^{(-1)}(\tilde{t}_{v,0}),h_{0}^{(-1)}(\tilde{t}_{v,1}),\cdots,h_{d\ell-2}^{(-1)}(\tilde{t}_{v,d\ell-1})\Big{)},$

and for $0\leq i<d\ell$,

$h_{i}^{(-1)}(\tilde{t}_{v,\overline{i+1}})=\sum_{m}a_{m}^{q^{-1}}\tilde{t}_{v,\overline{i+1}}^{m}\quad\text{
when }\quad
h_{i}(\tilde{t}_{v,i})=\sum_{m}a_{m}\tilde{t}_{v,i}^{m}\in\mathbb{C}_{v}\\{\tilde{t}_{v,i}\\}.$

Here $\overline{i+1}=i+1$ if $i+1<d\ell$ or $0$ if $i+1=d\ell$.

Note that since $|\tilde{\theta}_{v}|_{v}>1$, given an integer $i$ with $0\leq
i<d\ell$, we can set $\psi(\tilde{t}_{v})$ to be the following infinite
product

$\psi(\tilde{t}_{v}):=\prod_{i=1}^{\infty}\left(1-\frac{\tilde{t}_{v}}{\tilde{\theta}_{v}^{q^{d\ell
i}}}\right)^{-1}\quad\in\mathbb{C}_{v}\\{\tilde{t}_{v}\\}^{\times}\subset\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v}]^{\times}.$

Take $\xi\in\mathbb{F}_{q^{d\ell}}$ so that
$\mathbb{F}_{q^{d\ell}}=\mathbb{F}_{q}(\xi)$. For $0\leq i<d\ell$ and $0\leq
j<d$, Put

$\displaystyle\psi_{i,j}$
$\displaystyle:=\bigg{(}\xi^{i}\tilde{t}_{v,0}^{j}\psi(\tilde{t}_{v,0}),(\xi^{q^{-1}})^{i}\tilde{t}_{v,1}^{j}\psi^{(-1)}(\tilde{t}_{v,1}),\cdots,(\xi^{q^{1-d\ell}})^{i}\tilde{t}_{v,d\ell-1}^{j}\psi^{(1-d\ell)}(\tilde{t}_{v,d\ell-1})\bigg{)}$
$\displaystyle\hskip
19.91684pt\in\mathbb{C}_{v}\\{\tilde{t}_{v,0}\\}\times\mathbb{C}_{v}\\{\tilde{t}_{v,1}\\}\times\cdots\times\mathbb{C}_{v}\\{\tilde{t}_{v,d\ell-1}\\}.$

Then

######  Lemma 6.2.6.

Via the decomposition (6.2.3), we have that

$H_{\text{\rm
Betti}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})\big{)}=\bigoplus_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq j<d}}\mathbb{F}_{q}[\tilde{T}_{v}]\psi_{i,j}.$

Consequently, $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ is
uniformizable.

###### Proof.

We first note that

$\psi^{(-d\ell)}(\tilde{t}_{v})=(1-\frac{\tilde{t}_{v,0}}{\tilde{\theta}_{v}})^{-1}\cdot\psi(\tilde{t}_{v}).$

From (6.2) we see that $\psi_{i,j}\in H_{\text{\rm
Betti}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})\big{)}$ for
$0\leq i<d\ell$ and $0\leq j<d$. Since
$\operatorname{rank}_{\bar{k}[\tilde{T}_{v}]}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})=d^{2}\ell$,
it suffices to show that the set $\left\\{\psi_{i,j}|\hbox{ }0\leq
i<d\ell,0\leq j<d\right\\}$ generates
$\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\otimes_{\bar{k}[\tilde{T}_{v}]}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$
over $\mathbb{C}_{v}\\{\tilde{T}_{v}\\}$. First, from the isomorphism

$\bar{k}\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q^{d\ell}}$ | $\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}$ | $\bar{k}\times\bar{k}\times\cdots\times\bar{k}$
---|---|---
$\alpha\otimes 1$ | $\longmapsto$ | $(\alpha\ ,\alpha\ ,\cdots,\alpha)$
$1\otimes\xi$ | $\longmapsto$ |  $(\xi,\xi^{q^{-1}},\cdots,\xi^{q^{d\ell-1}})$,

one has that for each integer $i$ with $0\leq i<d\ell$, there exist
$\alpha_{i,0},...,\alpha_{i,d\ell-1}\in\bar{k}$ such that

$\sum_{m=0}^{d\ell-1}\alpha_{i,m}\xi^{mq^{-i^{\prime}}}=\begin{cases}1,&\text{
if $i^{\prime}=i$,}\\\ 0,&\text{ otherwise.}\end{cases}$

Put

$\psi_{i,j}^{\prime}:=\sum_{m=0}^{d\ell-1}\alpha_{i,m}\psi_{m,j}=\Big{(}0,...,\tilde{t}_{v,i}^{j}\psi^{(-i)}(\tilde{t}_{v,i}),0,...,0\Big{)}.$

We then obtain that

$\displaystyle\ \sum_{\genfrac{}{}{0.0pt}{}{0\leq i<d\ell}{0\leq
j<d}}\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\cdot\psi_{i,j}\ =\
\sum_{\genfrac{}{}{0.0pt}{}{0\leq i<d\ell}{0\leq
j<d}}\mathbb{C}_{v}\\{\tilde{T}_{v}\\}\cdot\psi^{\prime}_{i,j}$
$\displaystyle=$ $\displaystyle\
\Big{(}\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,0}]\cdot\psi(\tilde{t}_{v,0})\Big{)}\times\Big{(}\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,1}]\cdot\psi^{(-1)}(\tilde{t}_{v,1})\Big{)}\times\cdots\times\Big{(}\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,d\ell-1}]\cdot\psi^{(1-d\ell)}(\tilde{t}_{v,d\ell-1})\Big{)}$
$\displaystyle=$ $\displaystyle\
\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,0}]\times\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,1}]\times\cdots\times\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,d\ell-1}],$

where the last equality comes from the observation:

$\psi^{(-i)}(\tilde{t}_{v,i})\in\mathbb{C}_{v}\\{\tilde{t}_{v,i}\\}^{\times}\subset\mathbb{C}_{v}\\{\tilde{T}_{v}\\}[\tilde{t}_{v,i}]^{\times},\quad
0\leq i<d\ell.$

This completes the proof. ∎

######  Remark 6.2.7.

Since $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ has Complex
Multiplication by $\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$ (see
Remark 6.1.5), we may regard
$H_{\text{Betti}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})\big{)}$
as an $\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$-module. Then the
proof of the above lemma indicates that

$H_{\text{Betti}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})\big{)}=\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]\cdot\psi_{0,0}.$

###  6.3. Periods of $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$

Recall that the de Rham module of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ over $\bar{k}$ is

$H_{\text{dR}}(\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}):=\operatorname{Hom}_{\bar{k}}\left(\frac{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}{(\tilde{T}_{v}-\tilde{v})\cdot\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})},\bar{k}\right),$

and the de Rham pairing is given by (see (5.2.8))

$H_{\text{Betti}}\big{(}\mathscr{R}_{\sigma}(\text{$\tilde{\mathscr{C}}$}_{d\ell})\big{)}\times
H_{\text{dR}}\big{(}\mathscr{R}_{\sigma}(\text{$\tilde{\mathscr{C}}$}_{d\ell}),\bar{k}\big{)}\longrightarrow\mathbb{C}_{v},\quad(\gamma,\omega)\mapsto\int_{\gamma}\omega:=\omega(\gamma).$

In particular, for each integer $i$ with $0\leq i<d\ell$, put

(6.3.1)
$\tilde{g}_{v,i}(X):=\tilde{g}_{\tilde{T}_{v}}^{(-i)}(X)\big{|}_{\tilde{T}_{v}=\tilde{v}}=X^{d}-(\tilde{\epsilon}_{d-1}^{q^{(-i)}}\tilde{v})X^{d-1}-\cdots-(\tilde{\epsilon}_{1}^{q^{(-i)}}\tilde{v})X-\tilde{v}\in\bar{k}[X].$

The identification in Proposition 6.1.7 leads to the following isomorphism:

(6.3.2)
$\frac{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}{(\tilde{T}_{v}-\tilde{v})\cdot\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}\cong\prod_{i=0}^{d\ell-1}\frac{\bar{k}[X]}{\tilde{g}_{v,i}(X)\cdot\bar{k}[X]}.$

Moreover, for each integer $s$ with $0\leq s<d\ell$, from (5.1.4) the element

$\tilde{\theta}_{v,s}:=(\theta-\varepsilon^{q^{-s}})^{-1}=\iota(\tilde{t}_{v,s})\quad\in\bar{k}$

is a root of $\tilde{g}_{v,s}(X)$, which gives a differential
$\tilde{\omega}_{s}\in
H_{\text{dR}}(\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k})$
induced from

(6.3.3) $\displaystyle\prod_{i=0}^{d\ell-1}\frac{\bar{k}[X]}{\tilde{g}_{v,i}(X)\cdot\bar{k}[X]}$ | $\longrightarrow$ | $\displaystyle\frac{\bar{k}[X]}{\tilde{g}_{v,s}(X)\cdot\bar{k}[X]}$ | $\longrightarrow$ | $\bar{k}$
---|---|---|---|---
$\Big{(}\bar{h}_{i}(X)\bmod\tilde{g}_{v,i}\Big{)}_{0\leq i<d\ell}$ | $\longmapsto$ | $\bar{h}_{s}(X)\bmod\tilde{g}_{v,s}$ | $\longmapsto$ |  $\bar{h}_{s}(\tilde{\theta}_{v,s})$.

Therefore we obtain the following period interpretation:

######  Proposition 6.3.4.

Recall the notation (4.1.1). Given an integer $s$ with $0\leq s<d\ell$, one
has

$\int_{\psi_{0,0}}\tilde{\omega}_{s}=\frac{1}{\Omega_{d\ell,v,d\ell-s}(v)}.$

###### Proof.

From the definition of $\omega_{s}$ and the de Rham pairing, one gets that

$\displaystyle\int_{\psi_{0,0}}\tilde{\omega}_{s}$
$\displaystyle=\psi^{(-s)}(\tilde{t}_{v,s})\Big{|}_{\tilde{t}_{v,s}=\tilde{\theta}_{v,s}}$
$\displaystyle=\prod_{i=1}^{\infty}\left(1-\frac{\tilde{\theta}_{v,s}}{\tilde{\theta}_{v}^{q^{d\ell
i-s}}}\right)^{-1}=\prod_{i=1}^{\infty}\left(1-\frac{(\theta-\varepsilon^{q^{-s}})^{-1}}{((\theta-\varepsilon)^{q^{d\ell
i-s}})^{-1}}\right)^{-1}$
$\displaystyle=\prod_{i=1}^{\infty}\left(1-\frac{(\theta-\varepsilon)^{q^{d\ell
i-s}}}{\theta-\varepsilon^{q^{-s}}}\right)^{-1}=\frac{1}{\Omega_{d\ell,v,d\ell-s}(v)}.$

∎

######  Remark 6.3.5.

As $\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ has CM by
$\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$ (see Remark 6.2.7), we
have that

$\displaystyle\bar{k}\bigg{(}\int_{\gamma}\tilde{\omega}\ \bigg{|}\ \gamma\in
H_{\text{Betti}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})\big{)},\
\tilde{\omega}\in
H_{\text{dR}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}\big{)}\bigg{)}$
$\displaystyle\hskip 56.9055pt=\
\bar{k}\bigg{(}\int_{\psi_{0,0}}\tilde{\omega}\ \bigg{|}\ \tilde{\omega}\in
H_{\text{dR}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}\big{)}\bigg{)}.$

The Gronthendieck period conjecture in Theorem 5.2.9 leads to

(6.3.6)
$\operatorname{tr.deg}_{\bar{k}}\bar{k}\bigg{(}\int_{\psi_{0,0}}\omega\
\bigg{|}\ \omega\in
H_{\text{dR}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}\big{)}\bigg{)}=\dim\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})},$

where $\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}$ is the
$\tilde{T}_{v}$-motivic Galois group of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$. Next, we will determine
the dimension of $\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}$,
and complete the proof of Theorem 4.1.2.

###  6.4. $\tilde{T}_{v}$-Motivic Galois group of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ and the Hodge–Pink
cocharacters

We let $\matheur{K}_{d\ell}:=\mathbb{F}_{q^{d\ell}}(\tilde{t}_{v})$ and
$\matheur{F}:=\mathbb{F}_{q}(\tilde{T}_{v})$. Since
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ has complex multiplication
by $\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]$ by Proposition 6.1.4
and

$\operatorname{rank}_{\mathbb{F}_{q}[\tilde{T}_{v}]}\mathbb{F}_{q^{d\ell}}[\tilde{T}_{v}][\tilde{t}_{v}]=[\matheur{K}_{d\ell}:\matheur{F}]=d^{2}\ell=\dim_{\mathbb{F}_{q}(\tilde{T})}H_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})},$

by (5.2.6) we obtain that

$\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}\hookrightarrow\mathscr{R}_{K_{d\ell}/F}(\mathbb{G}_{m/\matheur{K}_{d\ell}})\subset\operatorname{GL}(H_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}),$

where $\mathscr{R}_{K_{d\ell}/F}(\mathbb{G}_{m/\matheur{K}_{d\ell}})$ is the
_restriction of scalars of $\mathbb{G}_{m/\matheur{K}_{d\ell}}$ to
$\matheur{F}$_. Our goal is to show
$\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}=\mathscr{R}_{K_{d\ell}/F}(\mathbb{G}_{m/\matheur{K}_{d\ell}})$
by analysing the images of its Hodge–Pink cocharacters.

6.4.1. Hodge–Pink cocharacters of
$\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}$. For $0\leq
i<d\ell$, let
$\tilde{\theta}_{v,i,0}(=\tilde{\theta}_{v,i}),...,\tilde{\theta}_{v,i,d-1}$
be the distinct roots of $\tilde{g}_{v,i}(X)\in\bar{k}[X]$ defined in (6.3.1).
Comparing the minimal polynomials of $\tilde{t}_{v,i}$ in (5.1.6) and
$\tilde{g}_{v,i}(X)$, we get that

$\tilde{T}_{v}-\tilde{v}\in(\tilde{t}_{v,i}-\tilde{\theta}_{v,i,j})\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}],$

whence

(6.4.2)
$(\tilde{T}_{v}-\tilde{v})\cdot\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}]=\prod_{j=0}^{d-1}(\tilde{t}_{v,i}-\tilde{\theta}_{v,i,j})\cdot\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}].$

On the other hand, under the identification (6.1.9), we get

(6.4.3)
$\displaystyle\sigma\cdot\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$
$\displaystyle=\Big{(}(\tilde{t}_{v,0}-\tilde{\theta}_{v,0,0})\cdot\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,0}]\Big{)}\times\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,1}]\times\cdots\times\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,d\ell-1}]$
$\displaystyle\supset(\tilde{T}_{v}-\tilde{v})\cdot\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),$

and the Hodge–Pink weight of
$\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ is $(w_{i,j})_{0\leq
i<d\ell,0\leq j<d}$ with

$w_{i,j}=\begin{cases}-1,&\text{ if $i=j=0$,}\\\ 0,&\text{
otherwise.}\end{cases}$

Moreover precisely, we may take a $\bar{k}[\tilde{T}_{v}]$-base
$\\{m_{i,j}\mid 0\leq i<d\ell,\ 0\leq j<d\\}$ satisfying that under the
identification

$\frac{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}{(\tilde{T}_{v}-\tilde{v})\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}\cong\prod_{i=0}^{d\ell-1}\frac{\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}]}{(\tilde{T}_{v}-\tilde{v})\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}]}\cong\prod_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq
j<d}}\frac{\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}]}{(\tilde{t}_{v,i}-\tilde{\theta}_{v,i,j})\bar{k}[\tilde{T}_{v}][\tilde{t}_{v,i}]}\cong\prod_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq j<d}}\bar{k},$

the image of $m_{i,j}$ is

$(\delta_{(i,j),(i^{\prime},j^{\prime})})_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq j<d}}\in\prod_{\genfrac{}{}{0.0pt}{}{0\leq i<d\ell}{0\leq
j<d}}\bar{k},\quad\text{ where
}\quad\delta_{(i,j),(i^{\prime},j^{\prime})}=\begin{cases}1,&\text{if
$i=i^{\prime}$ and $j=j^{\prime}$,}\\\ 0,&\text{ otherwise.}\end{cases}$

From the isomorphism (5.3), we may decompose

$\displaystyle
H_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\mathbb{C}_{v}}=\mathbb{C}_{v}\underset{\tilde{v}\mathrel{\reflectbox{$\mapsto$}}\tilde{T}_{v},\matheur{F}}{\otimes}H_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\bigoplus_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq j<d}}H_{i,j}.$

Notice that for each $0\leq i<d\ell$ and $0\leq j<d$, let
$\iota_{i,j}:\matheur{K}_{d\ell}=\mathbb{F}_{q^{d\ell}}(\tilde{t}_{v})\hookrightarrow\mathbb{C}_{v}$
be the embedding satisfying that
$\iota_{i,j}(\varepsilon)=\varepsilon^{q^{-j}}$ and
$\iota_{i,j}(\tilde{t}_{v})=\tilde{\theta}_{v,i,j}$. Regarding
$H_{\sigma}(\tilde{\mathscr{C}}_{d\ell})$ as a one dimensional vector space
over $\matheur{K}_{d\ell}$ (by Remark 6.2.7), we have that

$\alpha\cdot z=\iota_{i,j}(\alpha)\cdot z,\quad\forall z\in
H_{i,j}\quad\text{and}\quad\alpha\in\mathbb{F}_{q^{d\ell}}(\tilde{t}_{v}).$

Therefore under the identification

$\mathbb{C}_{v}\underset{\tilde{v}\mathrel{\reflectbox{$\mapsto$}}\tilde{T}_{v},\matheur{F}}{\times}\mathscr{R}_{\matheur{K}_{d\ell}/\matheur{F}}(\mathbb{G}_{m/\matheur{K}_{d\ell}})\cong\prod_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq
j<d}}\mathbb{C}_{v}\underset{\iota_{i,j},\matheur{K}_{d\ell}}{\times}\mathbb{G}_{m/\matheur{K}_{d\ell}}=\prod_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq j<d}}\operatorname{GL}(H_{i,j}),$

the cocharacter

$\chi_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}^{\vee}:\mathbb{G}_{m/\mathbb{C}_{v}}\rightarrow\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\mathbb{C}_{v}}\subset\mathbb{C}_{v}\underset{\tilde{v}\mathrel{\reflectbox{$\mapsto$}}\tilde{T}_{v},\matheur{F}}{\times}\mathscr{R}_{\matheur{K}_{d\ell}/\matheur{F}}(\mathbb{G}_{m/\matheur{K}_{d\ell}})$

is then realized as follows:

$\chi_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}^{\vee}(x)=\begin{pmatrix}x^{-1}&&&\\\
&1&&\\\ &&\ddots&\\\
&&&1\end{pmatrix}\quad\in\prod_{\genfrac{}{}{0.0pt}{}{0\leq i<d\ell}{0\leq
j<d}}\mathbb{C}_{v}\underset{\iota_{i,j},\matheur{K}_{d\ell}}{\times}\mathbb{G}_{m/\matheur{K}_{d\ell}}.$

Since $\operatorname{Aut}_{\mathbb{F}_{q}(\tilde{v})}(\mathbb{C}_{v})$ acts
transitively on the coordinates of $\prod_{\genfrac{}{}{0.0pt}{}{0\leq
i<d\ell}{0\leq
j<d}}\mathbb{C}_{v}\underset{\iota_{i,j},\matheur{K}_{d\ell}}{\times}\mathbb{G}_{m/\matheur{K}_{d\ell}}$,
the Hodge–Pink cocharacters of
$\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}$ generate the
whole group
$\mathbb{C}_{v}\underset{\tilde{v}\mathrel{\reflectbox{$\mapsto$}}\tilde{T}_{v},\matheur{F}}{\times}\mathscr{R}_{\matheur{K}_{d\ell}/\matheur{F}}(\mathbb{G}_{m/\matheur{K}_{d\ell}})$.
Consequently, we finally arrive at:

######  Theorem 6.4.4.

$\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}=\mathscr{R}_{\matheur{K}_{d\ell}/\matheur{F}}(\mathbb{G}_{m/\matheur{K}_{d\ell}}),$

whence
$\dim\Gamma_{\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell})}=[\matheur{K}_{d\ell}:\matheur{F}]=d^{2}\ell$.

Combining with (6.3.6), we finally arrive at:

######  Corollary 6.4.5.

$\displaystyle\operatorname{tr.deg}_{\bar{k}}\bar{k}\bigg{(}\int_{\psi_{0,0}}\tilde{\omega}\
\bigg{|}\ \tilde{\omega}\in H_{\text{\rm
dR}}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}\big{)}\bigg{)}$
$\displaystyle=d^{2}\ell.$

6.4.6. Proof of Theorem 4.1.2. By Proposition 6.3.4 we have that

$\Omega_{d\ell,v,i}(v)=\left(\int_{\psi_{0,0}}\tilde{\omega}_{d\ell-i}\right)^{-1},\quad
0<i\leq d\ell.$

On the other hand, for $0\leq i<d\ell$ and $0\leq j<d$, define
$\tilde{\omega}_{i,j}\in
H_{dR}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}\big{)}$
as follows:

$\displaystyle\prod_{i^{\prime}=0}^{d\ell-1}\frac{\bar{k}[X]}{\bar{g}_{v,i^{\prime}}(X)\cdot\bar{k}[X]}$ | $\longrightarrow$ | $\displaystyle\frac{\bar{k}[X]}{\bar{g}_{v,i}(X)\cdot\bar{k}[X]}$ | $\longrightarrow$ | $\bar{k}$
---|---|---|---|---
$\Big{(}\bar{h}_{i^{\prime}}(X)\bmod\bar{g}_{v,i^{\prime}}\Big{)}_{0\leq i<d\ell}$ | $\longmapsto$ | $\bar{h}_{i}(X)\bmod\bar{g}_{v,i}$ | $\longmapsto$ |  $\bar{h}_{i}(\tilde{\theta}_{v,i,j})$,

where $\tilde{\theta}_{v,i,j}$ are given in the beginning of Section 6.4. Then

$\big{\\{}\tilde{\omega}_{i,j}\ \big{|}\ 0\leq i<d\ell,\ 0\leq
j<d\big{\\}}\quad\text{ is a basis of
$H_{dR}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}\big{)}$},$

and $\tilde{\omega}_{i}=\tilde{\omega}_{i,0}$ for $0\leq i<d\ell$ by comparing
with (6.3.3). Since

$\displaystyle\dim_{\bar{k}}H_{dR}\big{(}\mathscr{R}_{\sigma}(\tilde{\mathscr{C}}_{d\ell}),\bar{k}\big{)}=d^{2}\ell=\operatorname{tr.deg}_{\bar{k}}\bar{k}\bigg{(}\int_{\psi_{0,0}}\tilde{\omega}_{i,j}\
\bigg{|}\ 0\leq i<d\ell,\ 0\leq j<d\bigg{)},$

we obtain that

$\int_{\psi_{0,0}}\tilde{\omega}_{i,j},\quad 0\leq i<d\ell,\quad 0\leq
j<d,\quad\text{ are algebraically independent over $\bar{k}$.}$

Finally, as for $0<i\leq\ell$, one has that

(6.4.7)
$\prod_{j=0}^{d-1}\Omega_{d\ell,v,j\ell+i}(v)=\prod_{j=0}^{d-1}\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{d\ell
n+j\ell+i}}}{\theta-\varepsilon^{q^{j\ell+i}}})=\prod_{n=0}^{\infty}(1-\frac{\theta_{v}^{q^{\ell
n+i}}}{\theta-\varepsilon^{q^{\ell n+i}}})=\Omega_{\ell,v,i}(v).$

Therefore the result follows.

$\Box$

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