\ok\bigl(p_{\eK}(\xi,\Xi)\mid \xi\in J_{\eK}\bigr)
&= \ok\left(\int_{\gamma_0} \omega_{\eM,\xi} \ \bigg|\ \xi\in J_{\eK}\right) \\
&= \ok\bigl(\omega_i(\gamma)  \mid 1\leq i\leq r,\ \gamma \in H_{\mathrm{Betti}}(\eM)\bigr). \notag
\end{align}

Note that the last equality holds as $H_{\mathrm{Betti}}(\eM)$ is a projective $O_\eK$-module of rank one.

Put ${\mathbf{m}}\assign (m_{1},\ldots,m_{r})\in \Mat_{r\times 1}(\eM)$.
Since the entries of $\Psi^{-1}\mathbf{m}$ form an $\FF_{q}[t]$-basis of the Betti module $H_{\mathrm{Betti}}(\eM)$, the value of  $\omega_j$ specialized at the $i$-th component of $\Psi^{-1}\mathbf{m}$ is given by

\[
\omega_{\bar{m}_{j}}\left( \sum_{j=1}^{r}(\Psi^{-1})_{ij} m_{j}  \right) =(\Psi^{-1})_{ij}(\theta), \]

\[
\ok\bigl(\omega_i(\gamma) \mid 1\leq i\leq r,\ \gamma\in H_{\rm{Betti}}(\eM)\bigr)=\ok(\Psi^{-1}(\theta))
\]

and the desired identity follows from (<ref>).

Based on Proposition <ref> and Theorem <ref>, showing Theorem <ref> is then translated into finding a lower bound of the dimension of the corresponding $t$-motivic Galois group in question.
We shall use Hodge-Pink theory in the next section and prove Theorem <ref> at the end of Section <ref>.

§ HODGE-PINK THEORY ON CM DUAL T-MOTIVES

In order to apply Hodge-Pink theory, we first review the properties of the categories of Hartl and Juschka that we need.
Then we describe explicitly the Hodge-Pink filtration arising from a CM dual $t$-motive with generalized CM type $(\eK,\Xi)$, which enables us to describe the Hodge-Pink cocharacters in question.
This characterization plays a crucial role in giving the precise connection between the lower bound in Theorem <ref> on the one hand and the rank of the Galois module generated by $\Xi$ in $I_{\eK}^{0}$ on the other.

§.§ Hartl-Juschka Categories

We recall the needed properties of the Hartl-Juschka Category $\Hcal\Jcal$ introduced in <cit.> as follows:
an object of $\Hcal\Jcal$ is a free $\bar{k}[t]$-module $\eM$ of finite rank together with a $\sigma$-action on $M \assign  \bar{k}(t)\otimes_{\bar{k}[t]} \eM$ satisfying that

* $M$ is a pre-$t$-motive over $\bar{k}$;

* there exists a sufficiently large integer $n$ so that

\[
(t-\theta)^n \eM \subseteq \sigma \eM \subseteq (t-\theta)^{-n} \eM.
\]

A morphism $f : \eM \rightarrow \eM'$ in $\Hcal\Jcal$ is a $\bar{k}[t]$-module homomorphism so that the induced $\bar{k}(t)$-linear homomorphism $f: M = \bar{k}(t)\otimes_{\bar{k}[t]} \eM  \rightarrow M' = \bar{k}(t)\otimes_{\bar{k}[t]} \eM'$ satisfies

\[
f (\sigma m) = \sigma f(m), \quad \forall\, m \in M.
\]

Given two objects $\eM$ and $\eM'$ of $\Hcal\Jcal$, the set $\Hom(\eM,\eM')$ of morphisms from $\eM$ to $\eM'$ has a natural $\FF_q[t]$-module structure, and elements in

\[
\Hom^{0}(\eM,\eM')\assign  \FF_q(t)\otimes_{\FF_q[t]} \Hom(\eM,\eM')
\]

are called quasi-morphisms from $\eM$ to $\eM'$.
The Hartl-Juschka Category $\Hcal\Jcal^I$ up to isogeny has the same objects in $\Hcal\Jcal$ and the morphisms in $\Hcal\Jcal^I$ are given by quasi-morphisms.
In particular, the functor $\eM \mapsto  \bar{k}(t)\otimes_{\bar{k}[t]} \eM$ from $\Hcal\Jcal^I$ to the category  of pre-$t$-motives is faithful.

An object $\eM\in \Hcal\Jcal$ is called effective if $\sigma \eM \subseteq \eM$.
We note that a dual $t$-motive is an effective object $\eM$ of $\Hcal\Jcal$, which is also  finitely generated over $\bar{k}[\sigma]$.
Thus the category of dual $t$-motives (resp. up to isogeny) is a strictly full subcategory of $\Hcal\Jcal$ (resp. $\Hcal\Jcal^I$).

We let $\TT[\sigma]$ be the non-commutative algebra generated by $\sigma$ over $\TT$ subject to the relation

\[
\sigma f=f^{(-1)} \sigma, \quad \forall\, f\in \TT.
\]

Given an object $\eM \in \Hcal\Jcal^I$, we set
$\MM \assign  \TT \otimes_{\bar{k}[t]}\eM$, on which $\sigma$ acts diagonally, and so $\MM$ becomes a left $\TT[\sigma]$-module (as $t-\theta$ is invertible in $\TT$).

\[
H_{\mathrm{Betti}}(\eM)\assign  \{ m \in \MM \mid \sigma m = m\}.
\]

We call $\eM$ uniformizable (or rigid analytically trivial) if the natural homomorphism

\[
\TT \otimes_{\FF_q[t]} H_{\mathrm{Betti}}(\eM) \longrightarrow \MM
\]

is an isomorphism (as $\TT$-modules). It is known that (see <cit.>) if $\eM$ is uniformizable, then $H_{\mathrm{Betti}}(\eM)$ is contained in $\MM^\dagger = \TT^\dagger \otimes_{\bar{k}[t]}\eM$, and the natural homomorphism

\[
\TT^\dagger \otimes_{\FF_q[t]} H_{\mathrm{Betti}}(\eM) \longrightarrow \MM^\dagger
\]

is an isomorphism (as $\TT^\dagger$-modules).

$t$-motivic Galois groups
The full subcategory of $\Hcal\Jcal^I$ consisting of all uniformizable objects is denoted by $\Hcal\Jcal^I_U$.
By <cit.>, the functor $\eM \mapsto \bar{k}(t)\otimes_{\bar{k}[t]} \eM$ from $\Hcal\Jcal^I_U$ to the category $\cR$ of uniformizable pre-$t$-motives is fully faithful.
Now we let $\mathbf{Vect}_{/\FF_q(t)}$ be the category of finite dimensional vector spaces over $\FF_q(t)$ and note that by <cit.> $\Hcal\Jcal^I_U$ is a neutral Tannakian category over $\FF_q(t)$ with fiber functor $\bomega: \Hcal\Jcal^I_U \rightarrow \mathbf{Vect}_{/\FF_q(t)}$ defined by:

\[
\eM \longmapsto H_\eM \assign  \FF_q(t)\otimes_{\FF_q[t]} H_{\mathrm{Betti}}(\eM).
\]

Given a uniformizable dual $t$-motive $\eM$, let $M \assign  \bar{k}(t)\otimes_{\bar{k}[t]} \eM$.
Viewing $\eM$ (resp. $M$) as an object in $\Hcal\Jcal^I_U$ (resp. $\cR$),
let $\Tcal_\eM$ (resp. $\cT_M$) be the strictly full Tannakian subcategory of $\Hcal\Jcal^I_U$ (resp. $\cR$) generated by $\eM$ (resp. $M$).
By Tannakian duality, we have an affine algebraic group scheme $\Gamma_\eM$ defined over $\FF_q(t)$ so that $\Tcal_\eM$ is equivalent to the category $\mathbf{Rep}(\Gamma_\eM)$ of the $\FF_q(t)$-finite dimensional representations of the algebraic group $\Gamma_\eM$.
We call $\Gamma_\eM$ the $t$-motivic Galois group of $\eM$.
The fully faithful functor from $\Hcal\Jcal^I_U$ to $\cR$ gives an equivalence between $\Tcal_\eM$ and $\cT_M$, whence $\Gamma_\eM$ is isomorphic to $\Gamma_M$  defined in Section <ref>.

We recall the functorial property of the base change $\Gamma_{\eM,\CC_{\infty}}\assign \CC_{\infty}\times_{\FF_{q}(t)} \Gamma_{\eM}$ (via the embedding $\FF_{q}(t)\hookrightarrow \CC_{\infty}$ by sending $t$ to $\theta$) as follows.
Let $\bomega_\eM$ be the restriction of the fiber functor $\bomega$ to $\Tcal_\eM$.
For any object $\Xcal \in\Tcal_\eM $ we let $A_{\Xcal}\subset \Hom(\bomega_{\eM} (\Xcal),\bomega_{\eM} (\Xcal))$ be defined as in <cit.> and $\underline{A}_{\Xcal}\in \Tcal_\eM$ be defined as in <cit.>,
from which $\underline{A}_{\Xcal}$ is a ring in $\Tcal_\eM$ such that $\bomega_\eM(\underline{A}_{\Xcal})=A_{\Xcal}$ as $\FF_{q}(t)$-algebras.

\[
\underline{B}\assign \varinjlim_{\Xcal} \underline{A}_{\Xcal}^{\vee}
\quad \text{ and } \quad
B\assign \varinjlim_\Xcal A_{\Xcal}^{\vee}.
\]

Then by <cit.> we have that

\[
\bomega_\eM(\underline{B})=B,\quad \underline{\End}^{\otimes}(\bomega_\eM)=\Spec(\bomega_\eM(\underline{B}))=\Gamma_{\eM},
\]

and for any fiber functor $\eta: \Tcal_{\eM}\rightarrow {\mathbf{Proj}_{R}}$ where $R$ is an $\FF_{q}(t)$-algebra and ${\mathbf{Proj}_{R}}$ is the category of finitely generated projective $R$-modules,
the following equality holds:

\[
\underline{\End}^{\otimes}(\eta)=\Spec(\eta(\underline{B})).
\]

In particular, let $R = \CC_\infty$ (i.e., $\mathbf{Proj}_R = \mathbf{Vect}_{/\CC_\infty}$) and take $\eta \assign \omega_{\eM,\CC_\infty}$, the base change of $\bomega_\eM$ from $\FF_q(t)$ to $\CC_\infty$, i.e.,

\[
\bomega_{\eM,\CC_\infty}(\Xcal)\assign \CC_\infty \otimes_{\FF_q(t)}\bomega_\eM(\Xcal).
\]

Then $\bomega_{\eM,\CC_\infty}$ is still a tensor functor, and we have that

\begin{align*}
\underline{\mathrm{Aut}}^{\otimes }(\bomega_{\eM,\CC_\infty})=\underline{\End}^{\otimes}(\bomega_{\eM,\CC_\infty})& =\Spec(\bomega_{\eM,\CC_\infty}(\underline{B})) \\
& =\Spec(\CC_{\infty}\otimes_{\FF_{q}(t)} \bomega_\eM(\underline{B}))= \Gamma_{\eM,\CC_{\infty}},
\end{align*}

where the first identity comes from <cit.>.

§.§ Hodge-Pink filtration

Let $\eM$ be an object in $\Hcal\Jcal^I_U$.
Recall that $H_{\mathrm{Betti}}(\eM)$ actually lies in $\MM^\dagger$, and the natural map

\[
\TT^\dagger \otimes_{\FF_q[t]}
H_{\mathrm{Betti}}(\eM) \longrightarrow \MM^\dagger
\]

is an isomorphism as $\TT^\dagger$-modules.

\[
\eM_{\CC_\infty}\assign  \CC_\infty[t] \otimes_{\bar{k}[t]}\eM
\quad \text{ and } \quad
H_{\eM,\CC_\infty} \assign \bomega_{\eM,\CC_\infty}(\eM) = \CC_\infty \otimes_{\FF_q(t)} H_\eM.
\]

The above isomorphism induces the chain of isomorphisms

\begin{multline}\label{eqn: H-iso}
H_{\eM,\CC_\infty} \cong \CC_\infty \otimes_{\FF_q[t]} H_{\mathrm{Betti}}(\eM) \cong \frac{\TT^\dagger}{(t-\theta) \TT^\dagger} \otimes_{\FF_q[t]} H_{\mathrm{Betti}}(\eM) \\
\cong \frac{\TT^\dagger}{(t-\theta) \TT^\dagger} \otimes_{\bar{k}[t]} \eM
\cong  \frac{\eM_{\CC_\infty}}{(t-\theta)\eM_{\CC_\infty}}.
\end{multline}

(cf. <cit.>) For each object $\eM$ in the category $\Hcal\Jcal^I_U$,
the Hodge-Pink filtration of $\eM$ is $F^\bullet H_{\eM,\CC_\infty} \assign (F^i H_{\eM,\CC_\infty})_{i\in\ZZ}$, where $F^i H_{\eM,\CC_\infty}$ is the image of $\eM_{\CC_\infty} \cap (t-\theta)^i (\sigma \eM_{\CC_\infty})$ in $H_{\eM,\CC_\infty}$, through the identification (<ref>) for each $i \in \ZZ$. That is,

\[
H_{\eM,\CC_\infty} \supseteq F^i H_{\eM,\CC_\infty} \cong  \frac{\bigl(\eM \cap (t-\theta)^i (\sigma \eM_{\CC_\infty})\bigr) + (t-\theta) \eM_{\CC_\infty}}{(t-\theta)\eM_{\CC_\infty}}.
\]

Given an object $\eM$ of $\Hcal\Jcal^I_U$, there exists a sufficiently large integer $n$ such that

\begin{equation}\label{eqn: Fil}
(t-\theta)^n \eM \subseteq \sigma \eM \subseteq (t-\theta)^{-n} \eM.
\end{equation}

Let $r$ be the rank of $\eM$ over $\bar{k}[t]$.
There exists a unique (non-ordered) $r$-tuple of integers $(w_1,\ldots, w_r)$ so that

\[
\frac{(t-\theta)^{-n} \eM_{\CC_\infty}}{\sigma \eM_{\CC_\infty}} \cong \bigoplus_{i=1}^r \frac{\CC_\infty[t]}{(t-\theta)^{n-w_i} \CC_\infty[t]}.
\]

We call $(w_1,\dots, w_r)$ the Hodge-Pink weights of $\eM$.
In other words, $w_1,\dots, w_r$ are the “jumps” of the Hodge-Pink filtration of $\eM$.
In particular, $\eM$ is effective if and only if $w_1,\dots, w_r$ are all non-positive.

We fix an object $\eM\in \Hcal\Jcal^I_U$ and let $\{m_1,\ldots, m_r\}$ be a basis of the free $\CC_\infty[t]$-module $\eM_{\CC_\infty}$ so that

\[
\sigma\eM_{\CC_\infty} = \bigoplus_{i=1}^r(t-\theta)^{-w_i}\CC_\infty[t] \cdot m_i \quad \subset \CC_\infty(t)\otimes_{\CC_\infty[t]} \eM_{\CC_\infty}.
\]

Identifying $H_{\eM,\CC_\infty}$ with $\eM_{\CC_\infty}/(t-\theta)\eM_{\CC_\infty}$ via (<ref>), let $v_1,\ldots, v_r \in H_{\eM,\CC_\infty}$ be the images of $m_1,\dots, m_r$ through the identification (<ref>).
It is straightforward to see that

\[
F^{i_0} H_{\eM,\CC_\infty}
= \bigoplus_{\substack{1\leq i\leq r \\ w_{i}\geq i_0}}
\CC_\infty v_{i} \quad \forall\,i_0 \in \ZZ.
\]

In particular, we have the associated graded spaces

\begin{equation}\label{eqn: Grading}
\mathrm{Gr}^{i_0} (H_{\eM,\CC_\infty})
\assign \bigoplus_{\substack{1\leq i \leq r \\ w_{i} = i_0}} \CC_\infty v_{i} \cong
\frac{F^{i_0}H_{\eM,\CC_\infty}}{F^{i_0+1}H_{\eM,\CC_\infty}}, \quad \forall\,i_0 \in \ZZ,
\end{equation}

\begin{equation}\label{E:HM-Gr}
H_{\eM,\CC_\infty} = \bigoplus_{i_0 \in \ZZ} {\rm Gr}^{i_0}(H_{\eM,\CC_\infty}).
\end{equation}

Given two objects $\eM$ and $\eM'$ in $\Hcal\Jcal^I_U$, we
let $\eM_0 \assign  \eM \otimes_{\bar{k}[t]} \eM'\in \Hcal\Jcal^I_U$, on which $\sigma$ acts diagonally. The natural isomorphism $H_{\eM_0,\CC_\infty} \cong H_{\eM,\CC_\infty} \otimes_{\CC_\infty} H_{\eM',\CC_\infty}$

\[
\Gr^{i_0}(H_{\eM_0,\CC_\infty}) \cong \bigoplus_{j_0+j_0' = i_0} \mathrm{Gr}^{j_0} (H_{\eM,\CC_\infty})\otimes_{\CC_\infty} \mathrm{Gr}^{j_0'} (H_{\eM',\CC_\infty}), \quad \forall i_0 \in \ZZ.
\]

Let $(w_1,\ldots, w_r)$ (resp. $(w_1',\ldots, w_r')$) be the Hodge-Pink weights of $\eM$ (resp. $\eM'$).
Take $\{m_1,\ldots, m_r\}$ (resp. $\{m_1',\ldots, m_r'\}$) to be a basis of the free $\CC_\infty[t]$-module $\eM_{\CC_\infty}$ (resp. $\eM_{\CC_\infty}'$) so that

\sigma\eM_{\CC_\infty} = \bigoplus_{j=1}^r(t-\theta)^{-w_j}\CC_\infty[t] \cdot m_j \quad \text{ and } \quad
\sigma\eM_{\CC_\infty}' = \bigoplus_{j'=1}^r(t-\theta)^{-w_{j'}'}\CC_\infty[t] \cdot m_{j'}'.

Then $\{m_j\otimes m'_{j'}:1\leq j,j' \leq r\}$ is a basis of $\eM_{0,\CC_\infty}$ so that

\sigma \eM_{0,\CC_\infty} =  \sigma\eM_{\CC_\infty}\otimes_{\CC_\infty[t]} \sigma\eM_{\CC_\infty}'= \bigoplus_{1\leq j,j'\leq r} (t-\theta)^{-(w_j+w_{j'}')}\CC_\infty[t] \cdot m_j\otimes m'_{j'}.

This means that $w_j+w_{j'}'$, for $1\leq j,j' \leq r$, are the Hodge-Pink weights of $\eM_0$.

For $1\leq j,j' \leq r$, let $v_j$, $v'_{j'}$, and $v_{j,j'}$ be the image of $m_j$, $m'_{j'}$, and $m_j\otimes m'_{j'}$ in $H_{\eM,\CC_\infty}$, $H_{\eM',\CC_\infty}$, and $H_{\eM_0,\CC_\infty}$, respectively (through the identification (<ref>)).
The canonical isomorphism $H_{\eM,\CC_\infty}\otimes_{\CC_\infty} H_{\eM',\CC_\infty} \cong H_{\eM_0,\CC_\infty}$
sending $v_j\otimes v'_{j'}$ to $v_{j,j'}$ and the isomorphism in (<ref>) implies that for each $i_0 \in \ZZ$,

\begin{align*}
&\cong \bigoplus_{\substack{1\leq j,j'\leq r \\ w_{j}+w_{j'}' = i_0}} \CC_\infty v_{j,j'} \\
\bigoplus_{j_0+j_0'=i_0}\Biggl(
\bigoplus_{\substack{1\leq j,j'\leq r \\ w_{j} = j_0,\ w_{j'}'=j_0'}}\CC_\infty  v_{j,j'} \Biggr) \\
\bigoplus_{j_0+j_0'=i_0}\Biggl[\biggl(
\bigoplus_{\substack{1\leq j\leq r \\ w_{j} = j_0}}\CC_\infty v_{j}
\biggr) \otimes_{\CC_\infty}
\biggl(
\bigoplus_{\substack{1\leq j'\leq r \\ w_{j'}' = j_0'}}\CC_\infty v'_{j'}
\biggr)\Biggr] \\
\bigoplus_{j_0+j_0' = i_0} {\Gr}^{j_0} (H_{\eM,\CC_\infty})\otimes_{\CC_\infty} {\Gr}^{j_0'} (H_{\eM',\CC_\infty}).
\end{align*}

In the next subsection we shall use the Hodge-Pink filtration of the given object $\eM$ in $\Hcal\Jcal^I_U$ to construct a special family of cocharacters of the $t$-motivic Galois group $\Gamma_\eM$.

§.§ Hodge-Pink cocharacters

Given an object $\eM$ in $\Hcal\Jcal^I_U$,
let $\cT_{M}$ be the neutral Tannikian subcategory of $\cT$ over $\FF_{q}(t)$ generated by the pre-$t$-motive $M = \bar{k}(t)\otimes_{\bar{k}[t]}\eM$ (see Definition <ref> (2)).
Recall in <ref> that the Tannakian subcategory $\Tcal_\eM$ of $\Hcal\Jcal^I_U$ generated by $\eM$ is equivalent to $\cT_{M}$, and so we have the algebraic group isomorphism (over $\FF_q(t)$)

\[
\Gamma_\eM \cong \Gamma_M.
\]

Note that we have a faithful representation
$$\Gamma_M \hookrightarrow \GL(H_{\eM}),$$

which is functorial in $M$ (see <cit.>). This implies that

\begin{equation}\label{eqn: CI}
\Gamma_{M}\hookrightarrow {\Cent}_{\GL(H_{\eM})}\left( \End_{\cT}(M)\right)\quad  \textup{(cf.~\cite[p.~1448, (6)]{CPY10})}
\end{equation}

as we have the natural embedding $\End_{\cT}(M)\hookrightarrow \End(H_{\eM})$.

We let $\GG_{m/\CC_\infty}$ be the multiplicative group over $\CC_{\infty}$, and we denote by $\mathbf{Rep}(\GG_{m/\CC_\infty})$ the Tannakian category of finite dimensional algebraic representations of $\GG_{m/\CC_\infty}$ over $\CC_{\infty}$. Note that by <cit.>, $\mathbf{Rep}(\GG_{m/\CC_\infty})$ is equivalent to the category ${\mathbf{Grad.Vect}}_{/\CC_{\infty}}$ of $\ZZ$-graded vector spaces of finite dimensions over $\CC_{\infty}$.
The fiber functor, which is also the forgetful functor, is denoted by

\[
\bomega^{\GG_{m/\CC_\infty}}:\mathbf{Rep}(\GG_{m/\CC_\infty})\rightarrow {\mathbf{Vect}}_{/\CC_{\infty}}.
\]

For any object $\eM\in \Hcal\Jcal^I_U$, we have the following functor (induced by the Hodge-Pink filtrations),

\[
\Fcal:\Tcal_\eM \rightarrow {\mathbf{Grad.Vect}}_{/\CC_{\infty}}, \quad \Fcal(\eM') \assign {\oplus_{i\in \ZZ}\rm{Gr}}^{i}H_{M',\CC_{\infty}}.
\]

By Lemma <ref>, one checks that $\Fcal$ is a tensor functor.
On the other hand, recall from Remark <ref> that $\bomega_{\eM,\CC_\infty}$ denotes the base change of the fiber functor $\bomega_\eM$ from $\FF_q(t)$ to $\CC_\infty$, i.e.

\[
\bomega_{\eM,\CC_\infty}: \Tcal_\eM \rightarrow \mathbf{Vect}_{/\CC_\infty}, \quad \bomega_{\eM,\CC_\infty}(\eM') = \CC_\infty \otimes_{\FF_q(t)} H_{\eM'}.
\]

Let $\bomega^{\rm GV} : \mathbf{Grad.Vect}_{/\CC_\infty} \rightarrow \mathbf{Vect}_{/\CC_\infty}$ be the forgetful functor.
By (<ref>) we then have the following commutative diagram

\[
\SelectTips{cm}{}
\xymatrix{
({\mathbf{Rep}}(\Gamma_{\eM}) & \hspace{-0.9cm})\hspace{0.7cm}  \Tcal_\eM \ar[d]_{\bomega_{\eM,\CC_\infty}} \ar[r]^<<<<<<{\Fcal} \ar@{}[l]|*=0[@]{\approx}
&{\mathbf{Grad.Vect}}_{/\CC_{\infty}} \hspace{0.3cm} ( \ar[ld]^{\bomega^{\rm GV}} \ar@{}[r]|*=0[@]{\approx}
& \hspace{-0.3cm}\mathbf{Rep}(\GG_{m/\CC_\infty})) \\
& {\mathbf{Vect}}_{/\CC_{\infty}} &
\]

By the arguments of <cit.> we obtain a cocharacter of $\Gamma_{\eM,\CC_{\infty}}$ (induced from $\Fcal$),

\[
\chi_{\eM}^\vee:\GG_{m/\CC_{\infty}}={\rm{Aut}}^{\otimes}(\bomega^{\rm GV})\rightarrow {\rm{Aut}}^{\otimes}(\bomega_{\eM,\CC_{\infty}})=\Gamma_{\eM,\CC_{\infty}}\hookrightarrow \GL(H_{\eM,\CC_{\infty}}).
\]

Note that the image of $\chi_{\eM}^{\vee}$ is uniquely determined by the action of $\GG_{m/\CC_{\infty}}$ on
\[H_{\eM,\CC_\infty} = \bigoplus_{i \in \ZZ} {\rm Gr}^{i}(H_{\eM,\CC_\infty}),\]
i.e., for each $i\in \ZZ$ and $x\in \GG_{m/\CC_{\infty}}(\CC_{\infty})$ we have

\begin{equation}\label{E:chiMvee}
\chi_{\eM}^\vee(x) v = x^i \cdot v, \quad \forall v \in \text{Gr}^i(H_{\eM,\CC_\infty}).
\end{equation}

Since $\GL(H_{\eM,\CC_\infty})$ (resp. $\Gamma_{\eM,\CC_{\infty}}$) is the base change $\CC_{\infty}\times_{\FF_{q}(t)} \GL(H_{\eM})$ (resp. $\CC_{\infty}\times_{\FF_{q}(t)}\Gamma_{\eM}$), we have a natural action of $\Aut_{k}(\CC_{\infty})$ on $\GL(H_{\eM,\CC_\infty})$, which leaves $\Gamma_{\eM,\CC_{\infty}}$ invariant. Thus any $\Aut_k(\CC_\infty)$-conjugate of $\chi_\eM^\vee$ is still a cocharacter of $\Gamma_{\eM,\CC_\infty}$.

(cf. <cit.>) Given an object $\eM$ in $\Hcal\Jcal^I_U$, a Hodge-Pink cocharacter of $\Gamma_{\eM,\CC_{\infty}}$ is a cocharacter which lies in the $\Gamma_{\eM,\CC_\infty} \rtimes \Aut_k(\CC_\infty)$-conjugate class of $\chi_\eM^\vee$.

We shall describe the images of the Hodge-Pink cocharacters of $\Gamma_{\eM,\CC_\infty}$ in $\GL(H_{\eM,\CC_\infty})$ via (<ref>) when $\eM$ is a CM dual $t$-motive, and provide a lower bound for the dimension of $\Gamma_{\eM}$ in the next subsection.

§.§ The case of CM dual t-motives

Let $\eM$ be a CM dual $t$-motive with generalized CM type $(\eK,\Xi)$, where $\eK$ is a CM field with $[\eK:\FF_q(t)] = r$.
For $\xi \in J_\eK$, let $\Pfk_\xi$ be the maximal ideal of $O_\bK$ associated to $\xi$, i.e. $\Pfk_\xi$ is generated by $\alpha - \nu_\xi(\alpha)$ for all $\alpha \in O_\eK$.
Write $\Xi = \sum_{\xi \in J_\eK} m_\xi \xi$ and put $\Ifk_\Xi \assign  \prod_{\xi \in J_\eK} \Pfk_\xi^{m_\xi}$.
As $\eM$ is projective of rank one over $O_\bK$ and $\sigma \eM = \Ifk_\Xi \eM$ (see the equality (<ref>)), we have that

\[
\frac{\eM}{\sigma \eM} \cong \frac{O_\bK}{\Ifk_\Xi} \cong \prod_{\xi \in J_\eK}\frac{O_\bK}{\Pfk_\xi^{m_\xi}}.
\]

Since $(t-\theta)O_\bK = \prod_{\xi \in J_\eK} \Pfk_\xi$, i.e., the prime ideal $(t-\theta)\bar{k}[t]$ of $\bar{k}[t]$ splits completely in $O_\bK$,
for each $\xi \in J_\eK$ one has that

\[
\bar{k}[t] \cap \Pfk_{\xi}^{m_\xi} = (t-\theta)^{m_\xi}\cdot \bar{k}[t] \quad \text{ and } \quad \dim_{\bar{k}}(\frac{O_\bK}{\Pfk_\xi^{m_\xi}}) = m_\xi.
\]

Thus the natural inclusion $\bar{k}[t]\hookrightarrow O_{\bK}$
induces the following isomorphism (as $\bar{k}[t]$-modules)

\[
\frac{O_\bK}{\Pfk_\xi^{m_\xi}} \cong \frac{\bar{k}[t]}{(t-\theta)^{m_\xi}\bar{k}[t]} \quad \textup{ for every $\xi \in J_\eK$.}
\]

To conclude, we obtain the following description of Hodge-Pink weights of $\eM$.

Let $\eK/\FF_{q}(t)$ be a CM field and $\eM$ be a CM dual $t$-motive with generalized CM type $(\eK, \Xi)$. Write $\Xi = \sum_{\xi \in J_\eK} m_\xi \xi \in I_\eK^0$.

\[
\frac{\eM_{\CC_\infty}}{\sigma \eM_{\CC_\infty}} \cong \prod_{\xi \in J_\eK} \frac{\CC_\infty[t]}{(t-\theta)^{m_\xi}\CC_\infty[t]}.
\]

Consequently, the Hodge-Pink weights of $\eM$ are
$(-m_\xi)_{\xi \in J_\eK}$.

Let $\eM$ be a CM dual $t$-motive with generalized CM type $(\eK,\Xi)$ over $\bar{k}$,
and let $M = \bar{k}(t)\otimes_{\bar{k}[t]} \eM$ be the pre-$t$-motive associated to $\eM$. Since $O_{\eK}\subset \End_{\ok[t,\sigma]}(\eM)$,
we can embed $\eK$ into the endomorphism ring $\End_{\cT}(M)$.
From (<ref>), the closed immersion $\Gamma_\eM\hookrightarrow \GL_{\FF_q(t)}(H_\eM)$ actually factors through $T_\eK \assign  {\rm Res}_{\eK/\FF_q(t)} \GG_m$, the Weil restriction of $\GG_m$ from $\eK$ to $\FF_q(t)$.
Thus the base change $\Gamma_{\eM,\CC_\infty}$ can be viewed as a subgroup of

\[
T_{\eK,\CC_\infty} \assign  \CC_\infty \underset{\FF_q(t)}{\times} {\rm Res}_{\eK/\FF_q(t)} \GG_{m}
\cong \prod_{\xi \in J_\eK} \CC_\infty \underset{\nu_\xi, \eK}{\times} \GG_{m/\eK}.
\]

Moreover, for each $\xi \in J_\eK$, let $H_\xi$ be the unique one-dimensional subspace of $H_{\eM,\CC_\infty}$ satisfying that

\[
\alpha \cdot v = \nu_\xi(\alpha) \cdot v, \quad \forall\, \alpha \in O_\eK,\  v \in H_\xi.
\]

The decomposition

\[
H_{\eM,\CC_\infty} = \bigoplus_{\xi \in J_\eK} H_\xi
\]

gives the embedding

\[
\prod_{\xi \in J_\eK} \CC_\infty \underset{\nu_\xi, \eK}{\times} \GG_{m/\eK} \hookrightarrow \GL_{\CC_\infty}(H_{\eM,\CC_\infty}), \quad (x_\xi)_{\xi \in J_\eK} \longmapsto \begin{pmatrix} \ddots & & \\ & x_\xi & \\ & & \ddots \end{pmatrix}.
\]

Write $\Xi = \sum_\xi m_\xi \xi \in I_\eK^0$. Then for $i \in \ZZ$, we have

\[
{\Gr}^i(H_{\eM,\CC_\infty}) = \bigoplus_{\substack{\xi \in J_\eK \\ -m_\xi = i}} H_\xi \quad \textup{(see~\eqref{eqn: Grading}).}
\]

Therefore by (<ref>) the cocharacter

\[
\chi_\eM^\vee: \GG_{m/\CC_\infty} \longrightarrow \Gamma_{\eM,\CC_\infty} \subset
\cong \prod_{\xi \in J_\eK} \CC_\infty \underset{\nu_\xi, \eK}{\times} \GG_{m/\eK}
\]

is realized as

\[
\chi_\eM^\vee(x) = (x^{-m_\xi})_{\xi \in J_\eK} \quad \in \prod_{\xi \in J_\eK} \CC_\infty \underset{\nu_\xi, \eK}{\times} \GG_{m/\eK}.
\]

On the other hand, for each cocharacter $\chi^\vee:  \GG_{m/\CC_\infty} \rightarrow T_{\eK,\CC_\infty}$,
there exists a unique set of integers $\{w_\xi: \xi \in J_\eK\}$ (i.e., the coweights of $\chi^\vee$) so that

\[
\chi^\vee(x) = (x^{w_\xi})_{ \xi \in J_\eK} \quad  \in \prod_{\xi \in J_\eK} \CC_\infty \underset{\nu_\xi, \eK}{\times} \GG_{m/\eK}.
\]

We remark that the map

\[
\chi^\vee \longmapsto
\Phi_{\chi^\vee} \assign  \sum_{\xi \in J_\eK}(-w_\xi) \xi \in I_\eK
\]

gives an isomorphism between the group of cocharacters of $T_{\eK,\CC_\infty}$ and $I_\eK$, which is equivariant under the action of $\Aut_k(\CC_\infty)$, i.e.
\[
\Phi_{({}^\varsigma \chi^\vee)} = \varsigma (\Phi_{\chi^\vee}),
\quad
\forall\,\chi^\vee: \GG_{m/\CC_\infty} \rightarrow T_{\eK,\CC_\infty} \ \text{ and }\ \varsigma \in \Aut_k(\CC_\infty).
\]

In particular, we have

\begin{equation}\label{eqn: HPc-CMt}
\Phi_{\chi^\vee_\eM} = \Xi \in I_\eK^0.
\end{equation}

As $\Gamma_\eM$ is commutative, we obtain the following correspondence.

Under the above identification between the group of cocharacters of $T_{\eK,\CC_\infty}$ and $I_\eK$, the set of the Hodge-Pink cocharacters of $\Gamma_\eM(\CC_{\infty})$ corresponds bijectively to

\[
\{\varsigma (\Xi) :  \varsigma \in \Aut_k(\CC_\infty)\}.
\]

(1) Since every point $\xi \in J_\eK$ given in (<ref>) is defined over $k^{\text{sep}}$ (as $\eK/\FF_q(t)$ is separable),
one has

\[
\{\varsigma (\Xi) :  \varsigma \in \Aut_k(\CC_\infty)\} =
\{\varsigma (\Xi) :  \varsigma \in \Gal(k^{\text{sep}}/k)\}.
\]

(2) Let
$I^0_\Xi$ be the subgroup of $I_\eK^0$ generated by $\varsigma (\Xi)$ for all $\varsigma \in \Gal(k^{\text{sep}}/k)$,
and let
$T_\eM$ be the subgroup of $\Gamma_\eM$ generated by
the image of all the Hodge-Pink cocharacters of $\Gamma_\eM$.

\[
\dim \Gamma_\eM \geq \dim T_\eM = \rank_\ZZ I^0_\Xi.
\]

Given a non-degenerate CM type $\Xi$ of $\eK$, let $\eM$ be a CM dual $t$-motive with generalized CM type $(\eK,\Xi)$.
From Proposition <ref> and Theorem <ref>, we obtain that

\[
\trdeg_{\bar{k}} \bar{k}\bigl(p_\eK(\xi,\Xi)\mid \xi \in J_\eK\bigr)
= \dim \Gamma_\eM \geq \rank_\ZZ I_\Xi^0
\]

as desired.

§ APPLICATION TO PERIODS OF ABELIAN T-MODULES

Let $E_\rho$ be an abelian $t$-module over $\bar{k}$ and $\eM(\rho)$ be the Hartl-Juschka dual $t$-motive associated with $E_\rho$ (see Remark <ref>).
Based on Remark <ref>, we say that $E_\rho$ is CM with generalized CM type $(\eK,\Xi)$ if $\eM(\rho)$ is a CM dual $t$-motive with generalized CM type $(\eK,\Xi)$.
The aim of this section is to apply Theorem <ref> to prove  the algebraic independence of the coordinates of a nonzero period vector of $E_{\rho}$ when $E_{\rho}$ has a non-degenerate CM type.
To do so, we first establish the needed comparisons for the Lie spaces, Betti modules and de Rham modules in terms of $t$-motives in the following subsections.

§.§ Comparisons between Lie spaces

Let $E_\rho$ be an abelian $t$-module over $\ok$, and let $\Mscr(\rho)$ be the $\ok[t,\tau]$-module defined in Definition <ref> (2).
There is a canonical $\bar{k}$-vector space isomorphism  (see <cit.>)

\begin{equation}\label{eqn: Lie1}
\Lie(E_\rho) \cong \Hom_{\bar{k}}(\Mscr(\rho)/\tau \Mscr(\rho), \bar{k}).
\end{equation}

On the other hand, as $(t-\theta)^N\eM(\rho) \subset \sigma \eM(\rho)$ for a sufficiently large integer $N$, we may identify $\eM(\rho) \assign \Hom_{\bar{k}[t]}(\tau \Mscr(\rho),\bar{k}[t]dt)$ with

\[
\biggl\{ \eum \in \Hom_{\ok[t]}\Bigl(\Mscr(\rho), \bar{k}[t][\frac{1}{t-\theta}] dt\Bigr) \biggm| \eum(\tau \Mscr(\rho)) \subset \ok[t]dt\biggr\},
\]

and $\sigma \eM(\rho)$ corresponds to  $\Hom_{\ok[t]}(\Mscr(\rho),\ok[t]dt)$.
Thus we obtain a $\ok$-bilinear map

\[
\Mscr(\rho) \times \eM(\rho) \longrightarrow \ok[t][\frac{1}{t-\theta}] dt \xrightarrow{\Res_{t=\theta}} \bar{k},
\]

where $(m,\eum)\mapsto \Res_{t=\theta}\eum(m)$ for every $m \in \Mscr(\rho)$ and $\eum \in \eM(\rho)$.
This induces a perfect pairing

\begin{equation}\label{eqn: perfect pairing}
\frac{\Mscr(\rho)}{\tau \Mscr(\rho)} \times \frac{\eM(\rho)}{\sigma \eM(\rho)} \longrightarrow \bar{k}.
\end{equation}

The above perfect pairing and (<ref>) lead to an isomorphism of $\bar{k}$-vector spaces,

\[
\Lie(E_\rho) \cong \eM(\rho)/\sigma \eM(\rho).
\]

Viewing $\Lie(E_\rho)$ as a $\bar{k}[t] \ (=\!\!\bar{k}\otimes_{\FF_q}\FF_q[t])$-module via $\text{id}_{\bar{k}} \otimes \partial \rho$, the above isomorphism is actually $\bar{k}[t]$-linear.

§.§ Comparisons between period lattices and Betti modules

Suppose $E_\rho$ is a uniformizable abelian $t$-module over $\bar{k}$, i.e.the corresponding exponential map $\exp_\rho : \Lie(E_\rho)(\CC_\infty) \rightarrow E_\rho(\CC_\infty)$ is surjective.
Let $\Lambda_\rho$ be the period lattice of $E_\rho$, i.e.

\[
\Lambda_\rho \assign  \{\lambda \in \Lie(E)(\CC_\infty)\mid \exp_\rho(\lambda) = 0\},
\]

which is a discrete $\FF_q[t]$-submodule with

\[
\rank_{\FF_q[t]}(\Lambda_\rho) = \rank_{\bar{k}[t]} \eM(\rho).
\]

For each $\lambda \in \Lambda_\rho$, let $\Gcal_\lambda: \Mscr(\rho)\rightarrow \TT $ be the following Anderson generating function

\begin{equation}\label{eqn: AGF}
\langle m \mid \Gcal_\lambda\rangle \assign  \sum_{n=0}^\infty m\Bigl(\exp_\rho(\partial \rho_t^{-n-1}\lambda)\Bigr) t^n, \quad \forall\,m \in \Mscr(\rho).
\end{equation}

Following [28], we take

\[
\gamma_\lambda\assign  -\Gcal_\lambda\big|_{\tau \Mscr(\rho)} dt\in \Hom_{\bar{k}[t]}(\tau \Mscr(\rho), \TT\, dt) \cong \TT \otimes_{\bar{k}[t]} \eM(\rho).
\]

It is straightforward to check that $\sigma \cdot \gamma_\lambda = \gamma_\lambda$, which means that
$\gamma_\lambda \in H_{\mathrm{Betti}}(\eM(\rho))$.

The map $(\lambda\longmapsto \gamma_\lambda)$ gives an $\FF_{q}[t]$-module isomorphism from $\Lambda_\rho$ onto $H_{\mathrm{Betti}}(\eM(\rho))$.

Note that $\rank_{\FF_q[t]}\Lambda_\rho = \rank_{\bar{k}[t]} \eM(\rho) = \rank_{\FF_q[t]} H_{\mathrm{Betti}}(\eM(\rho))$. Let $\{\lambda_1,\ldots, \lambda_r\}$ be an $\FF_q[t]$-base of $\Lambda_\rho$ and $\{\tau m_1, \ldots,  \tau m_r\}$ be a $\bar{k}[t]$-base of $\tau \Mscr(\rho)$.
The result follows immediately from the fact (cf. <cit.>, see also <cit.> and <cit.>) that

\[
(\langle \tau m_i \mid \Gcal_{\lambda_j}\rangle)_{1\leq i,j\leq n} \in \GL_r(\TT).
\qedhere
\]

Recall that $\TT^\dagger$ is the ring of rigid analytic functions on $\CC_\infty\setminus \{\theta^{q^i}\mid i \in \NN\}$.
As $E_\rho$ is uniformizable, so is $\eM(\rho)$.
Thus by <cit.>  we have that

\[
H_{\mathrm{Betti}}(\eM(\rho)) \subseteq \TT^\dagger \otimes_{\bar{k}[t]}\eM(\rho) \subseteq
\CC_\infty[\![t-\theta]\!] \otimes_{\bar{k}[t]}\eM(\rho).
\]

Let $\bar{\gamma}_\lambda$ be the image of $\gamma_\lambda$ in $\Hom_{\bar{k}}(\Mscr(\rho)/\tau\Mscr(\rho),\CC_\infty)$ under the following maps

\begin{align*}
\CC_\infty[\![t-\theta]\!] \otimes_{\bar{k}[t]}\eM(\rho)\twoheadrightarrow \CC_\infty[\![t-\theta]\!]\otimes_{\bar{k}[t]}\frac{\eM(\rho)}{\sigma \eM(\rho)} & \cong \CC_\infty \otimes_{\bar{k}} \frac{\eM(\rho)}{\sigma \eM(\rho)} \\
& \cong \Hom_{\bar{k}}(\frac{\Mscr(\rho)}{\tau\Mscr(\rho)},\CC_\infty),
\end{align*}

where the last isomorphism comes from (<ref>). We obtain

\[
\bar{\gamma}_\lambda(m) = -\Res_{t=\theta}\Bigl( \langle m |\Gcal_\lambda\rangle dt\Bigr),
\]

which coincides with the image of $\lambda$ in $\Hom_{\bar{k}}(\Mscr(\rho)/\tau \Mscr(\rho),\CC_\infty)$ under the isomorphism (<ref>) by <cit.>.
In conclusion we obtain the following.

The following diagram commutes:

\[
\begin{tabular}{ccccc}
$\Lie(E)(\CC_\infty)$ & $\stackrel{\sim}{\longrightarrow}$ &
$\displaystyle\Hom_{\bar{k}}(\frac{\Mscr(\rho)}{\tau \Mscr(\rho)},\CC_\infty)$ & $\stackrel[\eqref{eqn: perfect pairing}]{\sim}{\longleftarrow}$ &
$\hspace{-0.3cm}\displaystyle \CC_\infty \otimes_{\bar{k}} \frac{\eM(\rho)}{\sigma \eM(\rho)}$ \\
$\cup$ & & $\uparrow$ & & \rotatebox{90}{\scalebox{1}[1]{$\cong$}} \\
$\Lambda_\rho$ & $\stackrel{\sim}{\longrightarrow}$ &
$H_{\mathrm{Betti}}(\eM(\rho))$ & $\longrightarrow$ & $\displaystyle \CC_\infty[\![t-\theta]\!] \otimes_{\bar{k}[t]} \frac{\eM(\rho)}{\sigma \eM(\rho)}$. \\
\rotatebox{90}{\scalebox{1}[1]{$\in$}} & & \rotatebox{90}{\scalebox{1}[1]{$\in$}} & & \\
$\lambda$ & $\longmapsto$ & $\gamma_\lambda$ & &
\end{tabular}
\]

§.§ Comparisons between de Rham modules and pairings

Let $E_\rho$ be an abelian $t$-module over $\bar{k}$.
The de Rham module of $E_\rho$ was first introduced via bi-derivations in <cit.>. When $E_\rho$ is uniformizable, the de Rham pairing of $E_\rho$ was constructed using the Anderson generating function (cf. <cit.>, see also <cit.>). Here we shall identify canonically the de Rham modules of $E_\rho$ with the one of the Hartl-Juschka dual $t$-motive $\eM(\rho)$ associated with $E_\rho$, and make a natural comparison between their de Rham pairings.

Recall that the de Rham module of $E_\rho$ is isomorphic to (see <cit.> and <cit.>)

\begin{equation}\label{E:Isom HdR and QtauM}
H_{\mathrm{dR}}(E_\rho,\bar{k}) \cong \frac{\tau \Mscr(\rho)}{(t-\theta) \cdot \tau \Mscr(\rho)}.
\end{equation}

From the canonical isomorphism $\tau \Mscr(\rho) \cong \Hom_{\bar{k}[t]}(\eM(\rho), \Omega_{\bar{k}[t]/\bar{k}})$,
we may identify $(t-\theta)\cdot \tau \Mscr(\rho)$ with $\Hom_{\bar{k}[t]}(\eM(\rho), (t-\theta)\Omega_{\bar{k}[t]/\bar{k}})$.
This induces the following isomorphism.

In the setting above, we have the following identifications:

\begin{multline*}
\hspace{-0.4cm} H_{\mathrm{dR}}(E_\rho,\bar{k}) \cong \frac{ \Hom_{\bar{k}[t]}(\eM(\rho), \Omega_{\bar{k}[t]/\bar{k}})}{\Hom_{\bar{k}[t]}(\eM(\rho), (t-\theta)\Omega_{\bar{k}[t]/\bar{k}})}
\cong \Hom_{\bar{k}[t]}\left(\eM(\rho),\frac{\Omega_{\bar{k}[t]/\bar{k}}}{(t-\theta)\Omega_{\bar{k}[t]/\bar{k}}}\right) \\
\cong \Hom_{\bar{k}}\left(\frac{\eM(\rho)}{(t-\theta)\eM(\rho)},\bar{k}\right)
= H_{\mathrm{dR}}(\eM(\rho),\bar{k}).
\end{multline*}

Assume that $E_\rho$ is uniformizable.
Let $\Lambda_\rho \subset \Lie(E_\rho)(\CC_\infty)$ be the period lattice of $E_\rho$.
For each $\lambda \in \Lambda_\rho$,
let $\Gcal_\lambda: \Mscr(\rho)\rightarrow \TT $ be the Anderson generating function in (<ref>). Recall that the de Rham pairing of $E_\rho$ is defined by (cf. <cit.>)

\begin{equation}\label{eqn: dR-P t-module}
\begin{tabular}{ccl}
$H_{\mathrm{dR}}(E_\rho,\bar{k})\times\Lambda_\rho$ & $\longrightarrow$ & $\CC_\infty$, \\
\quad \quad \quad $(\delta, \lambda)$ & $\longmapsto$ & $[\delta,\lambda]\assign  \langle \tau m_\delta | \Gcal_\lambda \rangle\big|_{t=\theta}$,
\end{tabular}
\end{equation}

where $\tau m_\delta  \in \tau \Mscr(\rho)$, whose image in $H_{\mathrm{dR}}(E_\rho,\bar{k})$ is $\delta$ via (<ref>).
On the other hand, for each $\delta \in H_{\mathrm{dR}}(E_\rho,\bar{k})$ and $\lambda \in \Lambda_\rho$,
let $\omega_\delta \in H_{\mathrm{dR}}(\eM(\rho),\bar{k})$ be the differential corresponding to $\delta \in H_{\mathrm{dR}}(E_\rho,\bar{k})$ under the isomorphism in Lemma <ref>, and
let $\gamma_\lambda \in H_{\mathrm{Betti}}(\eM(\rho))$ be the cycle
corresponding to $\lambda$ under the isomorphism in Lemma <ref>.
Identifying $\tau m_\delta$ as an element of $\Hom_{\TT^\dagger}(\TT^\dagger \otimes_{\bar{k}[t]}\eM(\rho), \TT^\dagger dt)$,
we have the following commutative diagram:

\[
\SelectTips{cm}{}
\xymatrix{
\TT^\dagger \otimes_{\bar{k}[t]} \eM(\rho) \ar[r]^>>>>>{\omega_\delta} \ar[d]_{\tau m_\delta} & \CC_\infty
\\
\TT^\dagger\, dt \ar[r]^{\sim}
& \TT^\dagger \ar[u]_{(\cdot)|_{t=\theta}}
\]

This gives us the comparison,

\[
\int_{\gamma_\lambda}\omega_\delta = \omega_{\delta}(\gamma_\lambda) = \langle \tau m_\delta | -\Gcal_\lambda\rangle|_{t=\theta} = -
\langle \tau m_\delta | \Gcal_\lambda \rangle\big|_{t=\theta} = -[\delta,\lambda],
\]

from which we conclude the following.

Let $E_\rho$ be a uniformizable abelian $t$-module over $\bar{k}$.
We have the following comparison between the de Rham pairings of $E_\rho$ and $\eM(\rho)$:

\[
\SelectTips{cm}{}
\xymatrixrowsep{0.4cm}
\xymatrixcolsep{0.05cm}
\xymatrix{
[\ ,\ ]: & H_{\mathrm{dR}}(E_\rho,\bar{k})\ar@{<->}[dd]^{\rotatebox{90}{\scalebox{1}[1]{$\sim$}}}_{\text{\rm Lem.\ \ref{lem: dR-iso}}} & \times & \Lambda_\rho\ar@{<->}[dd]^{\rotatebox{90}{\scalebox{1}[1]{$\sim$}}}_{\text{\rm Lem.\ \ref{lem: L(rho)-H(rho)}}} \ar[rrr] & & & \CC_\infty \ar@{<->}[dd]^{\rotatebox{90}{\scalebox{1}[1]{$\sim$}}} & z \ar@{<->}[dd] \\
&&&& \\
\int : & H_{\mathrm{dR}}(\eM(\rho),\bar{k}) & \times & H_{\mathrm{Betti}}(\eM(\rho)) \ar[rrr] & & & \CC_\infty & -z\\
\]

§.§ Quasi-periods of CM abelian t-modules

Let $E_\rho$ be a CM abelian $t$-module with generalized CM type $(\eK,\Xi)$ over $\bar{k}$.
By Theorem <ref>, we know that $E_\rho$ is always uniformizable.
Let $\Lambda_\rho$ be the period lattice of $E_\rho$.
For each $\delta \in H_{\mathrm{dR}}(E_\rho,\bar{k})$ and $\lambda \in \Lambda_\rho$, we call $[\delta,\lambda]$ the quasi-period with respect to $\delta$ associated to $\lambda$ (see <cit.>).
From the natural comparison between the de Rham pairings of $E_\rho$ and $\eM(\rho)$ in Proposition <ref>, we obtain the following:

Let $E_\rho$ be a CM abelian $t$-module with generalized CM type $(\eK,\Xi)$ over $\ok$.
Let $\eK^+$ be the maximal totally real subfield of $\eK$.

\begin{align}\label{eqn: trdeg}
1+\frac{([\eK:\eK^+]-1)}{[\eK:\eK^+]}\cdot [\eK:\FF_q(t)] &\geq
\trdeg_{\bar{k}}\, \bar{k}\bigl([\delta,\lambda] \bigm| \delta \in H_{\mathrm{dR}}(E_\rho,\bar{k}),\ \lambda \in \Lambda_\rho \bigr) \\
&\geq \rank_\ZZ I_\Xi^0. \notag
\end{align}

In particular, if the generalized CM type $\Xi$ of $\eK$ is non-degenerate, then

\[
\trdeg_{\bar{k}} \bar{k}\bigl([\delta,\lambda] \bigm|  \delta \in H_{\mathrm{dR}}(E_\rho,\bar{k}),\ \lambda \in \Lambda_\rho \bigr)
= 1+\frac{[\eK:\eK^+]-1}{[\eK:\eK^+]}\cdot [\eK:\FF_q(t)].
\]

From Proposition <ref>,
we have that

\[
\bar{k}\bigl([\delta,\lambda] \bigm|  \delta \in H_{\mathrm{dR}}(E_\rho,\bar{k}),\ \lambda \in \Lambda_\rho \bigr)
= \bar{k}\bigl(p_\eK(\xi,\Xi) \bigm| \xi \in J_\eK\bigr).
\]

Hence (<ref>) follows immediately from Lemma <ref> and Theorem <ref>.
When $\Xi$ is non-degenerate, one obtains

\[
\rank_\ZZ I_\Xi^0 = \rank_\ZZ I_\eK^0 = 1+\frac{[\eK:\eK^+]-1}{[\eK:\eK^+]}\cdot [\eK:\FF_q(t)].
\]

This completes the proof.

(1) Let $E_\rho$ be a CM abelian $t$-module with a  generalized CM type $(\eK,\Xi)$.
Suppose that $\Xi$ is non-degenerate.
Then all of the algebraic relations among the quasi-periods of $E_\rho$ come from the following Legendre relation of the period symbols (cf. Remark <ref>):

\begin{equation}\label{eqn: Legendre relation}
\prod_{\substack{\xi \in J_\eK \\ \pi_{\bX/\bX^+(\xi) = \xi^+}}} p_\eK(\xi,\Xi) \sim \tilde{\pi}^{\text{\rm wt}(\Xi)}, \quad \forall \xi^+ \in J_{\eK^+}.
\end{equation}

(2) Let $E_\rho = (\GG_{a/\ok},\rho)$ be a Drinfeld $\FF_q[t]$-module of rank $r$ over $\bar{k}$.
Suppose $E_\rho$ has CM by $O_\eK$, where $O_\eK$ is the integral closure of $\FF_q[t]$ in a separable imaginary field $\eK$ (see Remark <ref> (2)) and $[\eK:\FF_q(t)] = r$.
Extending $\rho$ to an $\FF_q$-algebra homomorphism $\rho: O_\eK \rightarrow \End_{\FF_q}(\GG_{a/\ok})$, its derivation $\partial \rho : O_\eK \rightarrow \Lie(\GG_{a/\ok})$ induces an embedding $\nu_\rho: \eK \hookrightarrow \CC_\infty$.
Let $\eM(\rho)$ be the Hartl-Juschka dual $t$-motive associated to $E_\rho$. Then $\eM(\rho)$ is a CM dual $t$-motive over $\ok$ with CM type $(\eK,\xi_\rho)$,
where $\xi_\rho \in J_\eK$ is the point corresponding to $\nu_\rho$.
As $\xi_\rho$ is non-degenerate, Theorem <ref> implies that

\[
\trdeg_{\bar{k}} \bar{k}\bigl([\delta,\lambda] \bigm|  \delta \in H_{\mathrm{dR}}(E_\rho,\bar{k}),\ \lambda \in \Lambda_\rho \bigr) = r.
\]

This coincides with <cit.>.

§.§ Periods of CM Hilbert-Blumenthal t-modules

Now, let $\eK$ be a CM field and $\Xi$ be a CM type of $\eK$.
Suppose $E_\rho$ is a CM abelian $t$-module with CM type $(\eK,\Xi)$ over $\bar{k}$.
Let $\eK^+$ be the maximal totally real subfield of $\eK$, and write $\Xi = \sum_{i=1}^d \xi_i$, where $d = [\eK^+:\FF_q(t)]$.
For $1\leq i \leq d$, recall that $\Pfk_i$ is the maximal ideal of $O_{\bK}$ corresponding to $\xi_i$.
By Lemma <ref> and the equality (<ref>) in the appendix, we have the following identification:

\begin{equation}\label{eqn: Lie id}
\Lie(E_\rho) \cong \frac{\eM(\rho)}{\sigma \eM(\rho)} = \frac{\eM(\rho)}{\Ifk_\Xi \eM(\rho)} \cong \prod_{i=1}^d \frac{\eM(\rho)}{\Pfk_i \eM(\rho)} \cong \ok^d.
\end{equation}

Let $\xi_i^+\assign \pi_{\bX/\bX^+}(\xi_i)$ and $\nu_i^+ = \nu_{\xi_i^+} \in \Emb(\eK^+,\CC_\infty)$.
With respect to (<ref>), the multiplication of $O_\eK^+$ on $\eM(\rho)$ induces a structure homomorphism (still denoted by) $\rho: O_\eK^+\rightarrow \End_{\rm ab.\ t-mod.}(E_\rho)$ such that the induced $\CC_\infty$-linear homomorphism $\partial \rho: O_{\eK^+}\rightarrow \End_{\CC_\infty}(\Lie(E_\rho)(\CC_\infty))$ is isomorphic to $\nu_1^+\oplus\cdots \oplus \nu_d^+$.
In other words, $E_\rho$ is actually a Hilbert-Blumenthal $O_\eK^+$-module (introduced in the beginning of Section <ref>).

Conversely, let $E_\rho$ be a Hilbert-Blumenthal $O_\eK^+$-module satisfying the property that $\rank_A(\Lambda_\rho) = [\eK:\FF_q(t)]$, and suppose that the structure homomorphism $\rho$ can be extended to a homomorphism (still denoted by $\rho$) $O_\eK \rightarrow \End_{\rm ab.\ t-mod.}(E_\rho)$.
Then $\eM(\rho)$ is pure, $\rank_{\bar{k}[t]}\eM(\rho) = [\eK:\FF_q(t)]$, and $\rho$ induces an $\FF_q[t]$-algebra embedding $O_\eK \hookrightarrow \End_{\bar{k}[t,\sigma]}(\eM(\rho))$.
Moreover, one checks that

\[
\sigma \eM(\rho) = \Ifk_{\Xi}\eM(\rho),
\]

where $\Xi$ is the CM type of $\eK$ fixed above. Therefore by Proposition <ref> we have that $E_\rho$ is a CM abelian $t$-module with CM type $(\eK,\Xi)$.

In [37], Yu showed that for any Hilbert-Blumenthal $t$-module $E_\rho$ defined over $\ok$, every coordinate of a nonzero period vector of $E_\rho$ is transcendental over $k$.
When $E_\rho$ is in particular a CM abelian $t$-module with non-degenerate CM type,
we extend Yu's result as follows.

Let $E_\rho$ be a CM abelian $t$-module of dimension $d$ with a non-degenerate CM type $(\eK,\Xi)$ over $\ok$.
Let $\eK^+$ be the maximal totally real subfield of $\eK$ and suppose $\eK \neq \eK^+$.
Fix an identification $\Lie(E_\rho) \cong \bar{k}^d$, where $d = [\eK^+:\FF_q(t)]$, and extend it to $\Lie(E_\rho)(\CC_\infty)\cong \CC_\infty^d$.
Given a nonzero period vector  $\lambda \in \Lambda_\rho \subseteq \Lie(E_\rho)(\CC_\infty)$, let $(\lambda_1,\ldots,\lambda_d)$ be the corresponding vector in $\CC_\infty^d$. Then the coordinates $\lambda_{1},\ldots,\lambda_{d}$ are algebraically independent over $\ok$.

Let $\Xi\assign \xi_{1}+\cdots+\xi_{d}$ be the given  CM type of $\eK$.
Without loss of generality, we may assume that the fixed identification $\Lie(E_\rho) \cong \bar{k}^d$ coincides with (<ref>).
Then for a nonzero period vector $\lambda \in \Lambda_\rho$  with  corresponding coordinate vector  $(\lambda_1,\dots ,\lambda_d)\in \CC_\infty^d$,
we claim that

\[
\lambda_i \sim p_\eK(\xi_i,\Xi), \quad i=1,\dots ,d.
\]

For $1\leq i \leq d$, let $\xi_i^+\assign \pi_{\bX/\bX^+}(\xi_i)$.
Then $J_{\eK^+} = \{\xi_1^+,\dots ,\xi_d^+\}$, and the result follows from Remark <ref>.

To prove the claim above, let $\eM = \eM(\rho)$.
Take the differential $\omega_{\eM,\xi_i} \in H_{\mathrm{dR}}(\eM,\bar{k})$ associated with $\xi_i$ for $1\leq i \leq d$,  each of which extends  to a nonzero $\CC_\infty$-linear functional on $\CC_\infty \otimes_{\bar{k}} (\eM/(t-\theta) \eM)$ factoring through
$\CC_\infty \otimes_{\bar{k}} (\eM/\Pfk_i \eM)$.
Let $\gamma_\lambda \in H_{\mathrm{Betti}}(\eM)$ be the cycle corresponding to $\lambda$ in Lemma <ref>.
We use the commutative diagram in Proposition <ref> together with the identification (<ref>)  extended to $\Lie(E_\rho)(\CC_\infty) \cong \CC_\infty^d$  to conclude that

\[
\lambda_i \sim \int_{\gamma_\lambda} \omega_{\eM,\xi_i} \sim p_\eK(\xi_i,\Xi), \quad 1\leq i \leq d,
\]

as desired.

§ NON-GEOMETRIC CM FIELDS AND CM TYPES

§.§ Notation on curves

Let $\eK$ be a CM field over $\FF_q(t)$.
Suppose the constant field of $\eK$ is $\FF_{q^\ell}$ with $\ell>1$.
Then $\eK$ is geometric over $\FF_{q^\ell}(t)$.
Let $X$ be the smooth, projective, geometrically connected algebraic curve defined over $\FF_{q^\ell}$ with function field $\eK$.
We call $X$ the curve associated with $\eK$.
Fix an algebraically closed field $\KK$ with $k\subset \KK \subset \CC_\infty$.
Viewing $X$ as a scheme over $\FF_q$, let $\bX$ be the base change of $X$ from $\FF_q$ to $\KK$, i.e.,

\[
\bX = \KK \times_{\FF_q} X = \coprod_{i=0}^{\ell-1}\bX_{(i)},
\quad \text{ where } \quad
\bX_{(i)}\assign  \KK \underset{\Frob_q^{-i},\FF_{q^\ell}}{\times} X, \quad i \in \ZZ,
\]

i.e. the base change of $X$ from $\FF_{q^\ell}$ to $\KK$ with respect to the Frobenius twist

\[
\Frob_q^{-i}: \FF_{q^\ell}\hookrightarrow \KK,
\quad \varepsilon \longmapsto  \varepsilon^{1/q^i}, \quad \forall \varepsilon \in \FF_{q^\ell}.
\]

In particular, we have $\bX_{(i+\ell)} = \bX_{(i)}$.
Moreover, let $\bK_{(i)}\assign \KK(\bX_{(i)})$ be the function field of $\bX_{(i)}$.
The Frobenius twist  $(\alpha \mapsto \alpha^{q^{-1}})$ on the scalar field $\KK$ induces a field isomorphism $\bK_{(i)}\stackrel{\sim}{\rightarrow} \bK_{(i+1)}$ corresponding to the (geometric) Frobenius twist

\[
\bX_{(i+1)}\longrightarrow \bX_{(i)}, \quad x \longmapsto x^{(1)}, \quad \forall x \in \bX_{(i+1)},\ i \in \ZZ.
\]

Let $\eK^+$ be the maximal totally real subfield of $\eK$ (which is always geometric over $\FF_q(t)$), and let $X^+$ be the curve over $\FF_q$ with function field $\eK^+$ as before.
The embeddings
$$\FF_q(t)\hookrightarrow \eK^+\hookrightarrow \eK$$
induce the $\FF_q$-morphisms $\pi_{X/X^+}:X\rightarrow X^+$, $\pi_{X^+/\PP^1}:X^+\rightarrow \PP^1$, and $\pi_{X/\PP^1}:X\rightarrow \PP^1$
so that

\pi_{X/\PP^1} = \pi_{X^+/\PP^1}\circ \pi_{X/X^+}.

Let $\bX^+$ be the base change of $X^+$ from $\FF_q$ to $\KK$, and denote by $\pi_{\bX/\PP^1}$, $\pi_{\bX^+/\PP^1}$, and $\pi_{\bX/\bX^+}$ the base change of respective morphisms to $\KK$.

§.§ CM types

We continue in the setting above with a CM field $\eK$ over $\FF_{q}(t)$.
Denote by

\[J_\eK\assign  \pi_{\bX/\PP^{1}}^{-1}(\theta) = \{\xi \in \bX(\CC_{\infty})\mid \pi_{\bX/\PP^{1}}(\xi) = \theta \},\]

the set of points of $\bX$ lying above $\theta$.
Let $\Emb(\eK,\CC_\infty)$ be the set of all $\FF_q$-algebra embeddings from $\eK$ into $\CC_\infty$ which send $t$ to $\theta$.
As in the geometric case, we have a bijection between $J_\eK$ and
given as follows:
Take $\xi \in J_\eK$, corresponding to an $\FF_q$-morphism

\[
\Spec(\CC_\infty)\rightarrow X.
\]

As $\xi$ is not $\overline{\FF}_q$-valued (where $\overline{\FF}_q$ is the algebraic closure of $\FF_q$ in $\CC_\infty$),
this morphism must factor through the generic point
$\Spec(\eK)\hookrightarrow X$.
Thus we obtain an $\FF_q$-morphism $\Spec(\CC_\infty)\rightarrow \Spec(\eK)$, which corresponds to an $\FF_q$-algebra embedding $\nu_\xi:\eK\hookrightarrow \CC_\infty$ sending $t$ to $\theta$ (as $\pi_{\bX/\PP^1}(\xi) = \theta$).
Conversely, for $\nu \in \Emb(\eK,\CC_\infty)$, the induced $\FF_q$-morphism

\[
\Spec(\CC_\infty)\stackrel{\nu^*}{\longrightarrow}
\Spec(\eK)\hookrightarrow X
\]

gives a $\CC_\infty$-valued point $\xi_\nu$ of $\bX$ so that $\pi_{\bX/\PP^1}(\xi_\nu) = \theta$ (as $\nu(t) = \theta$).
It is clear that

\[
\xi_{\nu_\xi} = \xi, \quad \forall \xi \in J_\eK, \quad \text{ and } \quad \nu_{\xi_\nu} = \nu, \quad \forall \nu \in \Emb(\eK,\CC_\infty).
\]

In particular, for $\xi \in J_\eK$, one has that

\[
\xi \in \bX_{(i)} \quad \text{ if and only if } \quad
\nu_\xi \big|_{\FF_{q^\ell}} = \text{Frob}_q^{-i}, \quad 0\leq i < \ell.
\]

The generalized CM types of a given CM field are defined as follows.

Let $\eK$ be a CM field over $\FF_q(t)$ with the maximal totally real subfield $\eK^+$.
Let $I_\eK$ be the free abelian group generated by elements in $J_\eK$.
Let $I_\eK^0$ be the subgroup of $I_\eK$ consisting of elements

$$ \Phi= \sum_{\xi \in J_\eK} m_\xi \xi, \quad m_\xi \in \ZZ,$$

\[
\sum_{\xi\in \pi_{\bX/\bX^+}^{-1}(\xi_1^+)} m_\xi= \sum_{\xi'\in \pi_{\bX/\bX^+}^{-1}(\xi_2^+)} m_{\xi'},
\quad \forall \xi_1^+, \xi_2^+ \in J_{\eK^+}.
\]

We put
\[\wt(\Phi)\assign \sum_{\xi\in \pi_{\bX/\bX^+}^{-1}(\xi^+)} m_\xi \quad \text{ for a } \xi^+ \in J_{\eK^+}.\]

A  generalized CM type of $\eK$ is a nonzero effective divisor in $I_\eK^0$. A CM type of $\eK$ is a generalized CM type $\Xi$ of $\eK$ for which $\text{wt}(\Xi) = 1$.

The following properties and definitions extend naturally to the case of non-geometric CM fields without difficulties.

* $I_\eK^0$ is generated by all the CM types of $\eK$  (Proposition <ref>).

* $I_\eK^0$ is invariant under the right action of $\Aut_{\FF_q(t)}(\eK)$ and the left action of $\Gal(k^{\rm sep}/k)$ (Lemma <ref> and Remark <ref>).

* The restriction and inflation maps on CM types (Section <ref>).

* The reduction points at infinity (Proposition <ref> and Definition <ref>).

Let $\eK$ be a CM field over $\FF_q(t)$ with the constant field $\FF_{q^\ell}$, and let $X$ be the curve over $\FF_{q^\ell}$ associated with $\eK$.
Given a CM type $\Xi = \sum_{\xi \in J_\eK}m_\xi \xi$ of $\eK$, we may also define the $\overline{\FF}_q$-rational divisor $\eI_\Xi$ as in Definition <ref>.
Recall that the Frobenius twist $(\alpha \mapsto \alpha^{q^{-1}})$ on the scalar field $\KK$ corresponds to the geometric Frobenius morphism on $\bX = \coprod_{i=0}^{\ell-1} \bX_{(i)}$,
which maps $\bX_{(i+1)}$ to $\bX_{(i)}$ by sending $x$ to $x^{(1)}$ for every $x \in \bX_{(i+1)}$.

\[
\Xi = \sum_{i=0}^{\ell-1} \Xi_{(i)}, \quad \text{ where } \quad \Xi_{(i)} = \sum_{\xi \in J_\eK\cap \bX_{(i)}} m_\xi \xi.
\]

We may also decompose $\eI_\Xi$ into

\[
\eI_\Xi = \sum_{i=0}^{\ell-1} \eI_{\Xi_{(i)}}, \quad \text{ where } \quad \eI_{\Xi_{(i)}} \assign  \sum_{\xi \in J_\eK \cap \bX_{(i)}} m_\xi \overline{\infty}_\xi.
\]

\begin{align}\label{eqn: Sharp}
\Xi^{\#} & \assign  (\Xi_{(1)}^{(1-\ell)}+\cdots + \Xi_{(\ell-1)}^{(-1)}) + \Xi_{(0)} \\
\text{and}\quad
\eI^\#_\Xi\  & \assign  (\eI_{\Xi_{(1)}}^{(1-\ell)}+\cdots + \eI_{\Xi_{(\ell-1)}}^{(-1)}) + \eI_{\Xi_{(0)}}. \nonumber
\end{align}

Then $\Xi^\#$ and $\eI^\#_\Xi$ are divisors on $\bX_{(0)}$ and $\Xi^\# - \eI^\#_\Xi$ has degree zero.
Next, we shall apply Lang's isogeny theorem on the Jacobian of $\bX_{(0)}$ to introduce shtuka functions for a given CM type $\Xi$ and define the CM dual $t$-motives in the case of non-geometric CM fields.

§ CM DUAL T-MOTIVES FOR NON-GEOMETRIC CASE

§.§ Shtuka functions of CM types

We retain the notations of Appendix <ref> with a CM field $\eK$ over $\FF_q(t)$ whose constant field is $\FF_{q^\ell}$.

Let $\Xi$ be a generalized CM type of $\eK$.
There exists a divisor $W \in \Div(\bX_{(0)})$ and a function $h \in \bK_{(0)} = \KK(\bX_{(0)})$ so that

\begin{equation}\label{eqn: shtuka}
\divv(h) = W^{(\ell)} - W + \Xi^\# - \eI_{\Xi}^\# \quad \in \Div(\bX_{(0)}).
\end{equation}

We call $h$ a shtuka function associated to the divisor $W$ for the generalized CM type $(\eK,\Xi)$.

As the divisor $\Xi^\#-\eI_\Xi^\#$ is of degree zero on the curve $\bX_{(0)}$ defined over $\FF_{q^\ell}$,
the proof is the same as the geometric case in Lemma <ref>.

§.§ Anderson dual t-motives with generalized CM types

Let $O_\eK$ be the integral closure of $\FF_q[t]$ in $\eK$, and put

\[
O_\bK \assign  \KK\otimes_{\FF_q} O_\eK = \prod_{i=0}^{\ell-1} O_{\bK,{(i)}}, \text{ where } O_{\bK,{(i)}}\assign  \KK \underset{\Frob_q^{-i},\FF_{q^\ell}}{\otimes} O_\eK \subset \bK_{(i)},\ i \in \ZZ.
\]

For each $n \in \ZZ$, the $n$-th Frobenius twist on $O_\bK$ is given by:

\[
(c\otimes a)^{(n)}\assign  c^{q^n} \otimes a, \quad \forall c \in \KK \text{ and } a \in O_\eK.
\]

Hence for $\alpha_{(i)} \in O_{\bK,(i)}$ one has

\[
\alpha_{(i)}^{(-n)} \in O_{\bK,(i+n)}, \quad \forall n \in \ZZ.
\]

In particular, for each $\alpha \in O_\bK$,  writing $\alpha = (\alpha_{(0)},\ldots, \alpha_{(\ell-1)}) \in O_{\bK,{(0)}}\times\ldots \times O_{\bK,{(\ell-1)}}$,
we have the following expression:

\[
\alpha^{(-1)} = (\alpha_{(\ell-1)}^{(-1)},\alpha_{(0)}^{(-1)},\alpha_{(1)}^{(-1)},\ldots, \alpha_{(\ell-2)}^{(-1)}).
\]

\[
\bU \assign  \bX - \pi_{\bX/\PP^1}^{-1}(\infty) = \Spec(O_\bK) = \coprod_{i=0}^{\ell-1} \bU_{(i)},
\]

$\bU_{(i)} = \Spec(O_{\bK,(i)}) = \bX_{(i)} \cap \bU$.
We may identify $\bK_{(i)} = \KK(\bX_{(i)})$ with the field of fractions of $O_{\bK,(i)}$ for $0\leq i < \ell$.
Let $\Xi$ be a generalized  CM type of $\eK$, and take $h \in \bK_{(0)}$ and $W \in \Div(\bX_{(0)})$ satisfying (<ref>).

\begin{equation}\label{e:Appendix M(W,h)}
\eM_{(W,h)}\assign  \prod_{i=0}^{\ell-1}\eM_{(W,h),(i)},
\end{equation}

\[
\eM_{(W,h),(0)}\assign  \Gamma\bigl(\bU_{(0)}, \Ocal_{\bX_{(0)}}(-W^{(\ell)})\bigr) \quad \subset \bK_{(0)},
\]

\[
\eM_{(W,h),(i)}\assign  \Gamma\bigl(\bU_{(i)}, \Ocal_{\bX_{(i)}}(-W^{(\ell-i)} + \sum_{j=1}^i \Xi_{(j)}^{(j-i)})\bigr) \quad \subset \bK_{(i)}, \quad 1\leq i < \ell.
\]

The $\sigma$-action on $\eM_{(W,h)}$ is given by: for $(m_{(0)},\ldots, m_{(\ell-1)}) \in \eM_{(W,h)}$,

\[
\sigma \cdot (m_{(0)},\ldots, m_{(\ell-1)}) \assign  \Bigl(h \cdot m_{(\ell-1)}^{(-1)}, m_{(0)}^{(-1)},\ldots, m_{(\ell-2)}^{(-1)}\Bigr).
\]

Then we have

\[
\sigma \cdot ( \alpha \cdot m) = \alpha^{(-1)}\cdot  (\sigma \cdot m), \quad \forall \alpha \in O_\bK \quad \text{ and } m \in \eM_{(W,h)}.
\]

\begin{multline}\label{eqn: sigma-M}
\sigma \eM_{(W,h)} = \Gamma\bigl(\bU_{(0)},\Ocal_{\bX_{(0)}}(-W^{(\ell)}-\Xi_{(0)})\bigr)  \\
{}\times \prod_{i=1}^{\ell-1}
\Gamma\Bigl(\bU_{(i)},\Ocal_{\bX{(i)}}\bigl(-W^{(\ell-i)} + (\sum_{j=1}^{i}\Xi_{(j)}^{(j-i)})-\Xi_{(i)}\bigr)\Bigr).
\end{multline}

In particular, we point out that for $(m_{(0)},\ldots, m_{(\ell-1)}) \in \eM_{(W,h)}$,

\[
\sigma^\ell \cdot (m_{(0)},\ldots, m_{(\ell-1)})
= \Bigl(h \cdot m_{(0)}^{(-\ell)}, h^{(-1)}\cdot  m_{(1)}^{(-\ell)},\ldots, h^{(1-\ell)}\cdot m_{(\ell-1)}^{(-\ell)}\Bigr).
\]

Adapting the proof of Theorem <ref>, we have  the following:

The $\KK[t,\sigma]$-module $\eM = \eM_{(W,h)}$ constructed above is a dual $t$-motive over $\KK$ equipped with an $O_\eK$-module structure commuting with the $\sigma$-action.
In particular,

$$\rank_{\KK[t]}\eM = [\eK:\FF_q(t)] \quad \text{ and } \quad \rank_{\KK[\sigma]}\eM = \deg \Xi.

As $\eM$ is a projective $O_\bK$-module of rank one, we have that $\eM$ is free over $\KK[t]$ with

\rank_{\KK[t]}\eM = \rank_{\KK[t]} O_\bK = [\eK:\FF_q(t)].

From the above description of $\sigma \eM$, we get that

(t-\theta)^{\text{wt}(\Xi)} \cdot \eM \subset \sigma \eM.

It suffices to show that $\eM$ is finitely generated as a $\KK[\sigma]$-module, and so $\eM$ is free over $\KK[\sigma]$ with

\rank_{\KK[\sigma]}\eM = \dim_{\KK}(\eM/\sigma \eM) = \sum_{i=0}^{\ell-1}\deg \Xi_{(i)} = \deg \Xi \quad \quad \text{(by Lemma~\ref{L:degE})}.

To prove this, take $\mu \in \NN$ sufficiently large so that $(\eI_{\Xi}^{\#})^{(\ell \mu)} = \eI_\Xi^\#$.

\bI\assign  \eI_\Xi^\#+ (\eI_{\Xi}^{\#})^{(-\ell)}+\cdots +
(\eI_{\Xi}^{\#})^{(\ell-\ell \mu)}.

One has that $\bI^{(-\ell)} = \bI$, and the support of $\bI$ covers $\bX_{(0)} - \bU_{(0)} = \pi_{\bX/\PP^1}^{-1}(\infty)\cap \bX_{(0)}$.
For each $n \in \ZZ$, let

\begin{align*}
\eM_n \assign\ &  \Gamma(\bX_{(0)}, \Ocal_{\bX_{(0)}}(-W^{(\ell)} + n \bI) \\
&\times \prod_{i=1}^{\ell-1}
\Gamma\Bigl(\bX_{(i)}, \Ocal_{\bX_{(i)}}\bigl(-W^{(\ell-i)} + (\sum_{j=1}^i \Xi_{(j)}^{(j-i)})+n \bI^{(-i)}\bigr)\Bigr).
\end{align*}

Thus there exists $n_0 \in \ZZ$ so that
$\eM = \cup_{n\geq n_0} \eM_n$.

Put $\mathbf{\Xi}\assign  \Xi^\# + (\Xi^{\#})^{(-\ell)}+\cdots + (\Xi^{\#})^{(\ell-\ell \mu)}$.
Multiplication by $\sigma^{\ell \mu}$ on $\eM_n$ for each $n> n_0$ induces a $\KK$-vector space isomorphism

\begin{multline*}
\sigma^{\ell \mu} : \frac{\eM_n}{\eM_{n-1}} \stackrel{\sim}{\longrightarrow} {} \frac{\Gamma\bigl(\bX_{(0)}, \Ocal_{\bX_{(0)}}(-W^{(\ell)} - \mathbf{\Xi} + (n+1) \bI)\bigr)}{\Gamma\bigl(\bX_{(0)}, \Ocal_{\bX_{(0)}}(-W^{(\ell)} - \mathbf{\Xi} + n \bI)\bigr)} \\
\times \prod_{i=1}^{\ell-1}
\frac{\Gamma\Bigl(\bX_{(i)}, \Ocal_{\bX_{(i)}}\bigl(-W^{(\ell-i)} + (\sum_{j=1}^i \Xi_{(j)}^{(j-i)})-\mathbf{\Xi}^{(-i)}+ (n+1) \bI^{(-i)}\bigr)\Bigr)}{\Gamma\Bigl(\bX_{(i)}, \Ocal_{\bX_{(i)}}\bigl(-W^{(\ell-i)} + (\sum_{j=1}^i \Xi_{(j)}^{(j-i)})-\mathbf{\Xi}^{(-i)}+ n \bI^{(-i)}\bigr)\Bigr)}.
\end{multline*}

By the Riemann-Roch theorem (cf. <cit.>), the right hand side in the above is isomorphic to $\eM_{n+1}/\eM_n$ (as $\KK$-vector spaces) under the natural inclusion map when $n$ is sufficiently large.
In other words, there exists a sufficiently large $n_1 \in \ZZ$ so that the multiplication by $\sigma^{\ell \mu}$ induces a $\KK$-vector space isomorphism

\sigma^{\ell \mu} : \frac{\eM_n}{\eM_{n-1}} \stackrel{\sim}{\longrightarrow} \frac{\eM_{n+1}}{\eM_{n}} \quad \text{ for all $n\geq n_1$}.

This implies that any $\KK$-basis of $\eM_{n_1}$ generates $\eM$ over $\KK[\sigma^{\ell \mu}]$, and so over $\KK[\sigma]$.
As $\eM_{n_1}$ is finite dimensional over $\KK$, the result follows.

Let $\eK$ be a CM field over $\FF_q(t)$ and $\Xi$ be a (resp. generalized) CM type of $\eK$.
A dual $t$-motive (over $\KK$) which is isomorphic to $\eM_{(W,h)}$ for some shtuka function $h$ associated to $W$ for $(\eK,\Xi)$ is called a
CM dual $t$-motive with the $($resp. generalized$)$  CM type $(\eK,\Xi)$.

Let $\eK$ be a CM field over $\FF_q(t)$.
For each $\xi \in J_\eK$, let $\Pfk_\xi$ be the ideal of $O_\bK$ associated with $\xi$.
Given a generalized  CM type $\Xi = \sum_{\xi \in J_\eK}m_\xi \xi$ of $\eK$,
let $\Ifk_\Xi \assign  \prod_{\xi \in J_\eK}\Pfk_\xi^{m_\xi}$.
Let $\eM$ be a CM dual $t$-motive with generalized  CM type $(\eK,\Xi)$.
From the description of $\sigma \eM$ in (<ref>) (when identifying $\eM \cong \eM_{(W,h)}$ for some pair $(W,h)$), we have that

\begin{equation}\label{eqn: sigma-M iso}
\sigma \eM = \Ifk_\Xi \cdot \eM \subset \eM.
\end{equation}

Similarly we obtain the extension of Proposition <ref>.

Let $\eK$ be a CM field.
Given CM dual $t$-motives $\eM_i$ with generalized  CM types $(\eK,\Xi_i)$ for $i=1,...,n$,
the tensor product

\[
\eM_1\otimes_{O_\bK} \cdots \otimes_{O_\bK} \eM_n
\]

with $\sigma$-action defined by

\[
\sigma (m_1\otimes \cdots \otimes m_n) \assign  \sigma  m_1 \otimes \cdots \otimes \sigma  m_n, \quad \forall m_i \in \eM_i,\ i = 1,\ldots, n,
\]

is a CM dual $t$-motive with generalized  CM type $(\eK,\Xi_1+\cdots +\Xi_n)$.

Without loss of generality, we may assume that $n=2$ and
$\eM_i = \eM_{(W_i,h_i)}$, where
$h_i \in \KK(\bX_{(0)})$ is a shtuka function associated to $W_i \in \Div(\bX_{(0)})$ for $(\eK,\Xi_i)$ for $i=1,2$.

W \assign  W_1+W_2, \quad h \assign  h_1 h_2, \quad \text{ and } \quad \Xi \assign  \Xi_1+\Xi_2.

One has

\divv(h) = W^{(\ell)}-W+\Xi^\#-\eI_\Xi^\# \quad \in \Div(\bX_{(0)}).

Let $\eM = \eM_{(W,h)}$.
For $0\leq i \leq \ell-1$,
the $\Ocal_{\bX_{(i)}}$-module isomorphism

\begin{multline*}
\Ocal_{\bX_{(i)}}\bigl(-W_1^{(\ell-i)}+(\sum_{j=1}^i (\Xi_{1})_{(j)}^{(j-i)})\bigr) \otimes_{\Ocal_{\bX_{(i)}}}  \Ocal_{\bX_{(i)}}\bigl(-W_2^{(\ell-i)}+(\sum_{j=1}^i (\Xi_{2})_{(j)}^{(j-i)})\bigr) \\
\stackrel{\sim}{\longrightarrow} \Ocal_{\bX_{(i)}}\bigl(-W^{(\ell-i)}+(\sum_{j=1}^i \Xi_{(j)}^{(j-i)})\bigr)
\end{multline*}

induces the following $O_\bK$-module isomorphism

\[
\SelectTips{cm}{}
\xymatrix{
\eM_1\otimes_{O_\bK} \eM_2 \ar[d]_{\Theta}^{\rotatebox{90}{\scalebox{1}[1]{$\sim$}}} & ((m_1)_{(1)},\ldots, (m_1)_{(\ell-1)})\otimes ((m_2)_{(1)},\ldots, (m_2)_{(\ell-1)}) \ar@{|->}[d] \\
\eM &
((m_1)_{(1)} (m_2)_{(1)},\ldots, (m_1)_{(\ell-1)}(m_2)_{(\ell-1)}).
\]

Moreover, it is straightforward to see that $\Theta$ is invariant under the action of $\sigma$.
Hence $\eM_1\otimes_{O_\bK}\eM_2$ is isomorphic to the CM dual $t$-motive $\eM$ with generalized  CM type $(\eK,\Xi)$, and the proof is complete.

To end this section, we generalize Proposition <ref> to arbitrary CM fields.

Any CM dual $t$-motives with the same generalized  CM type must be isogeneous.

Let $\eK$ be a CM field over $\FF_q(t)$ with constant field $\FF_{q^\ell}$.
Let $\eM_{(W_1,h_1)}$ and $\eM_{(W_2,h_2)}$ be two CM dual $t$-motives with the same generalized  CM type $(\eK,\Xi)$.
Take an $\FF_{q^\ell}$-rational divisor $D_0$ of the curve $X$ (associated with $\eK$) over $\FF_{q^\ell}$ with degree one (see <cit.>).
Viewing $D_0$ as a divisor of $\bX_{(0)}$, let

$$W = W_2 - W_1 - (\deg W_2-\deg W_1)\cdot D_0.$$

W^{(\ell)} - W = \divv(h_2/h_1) \quad \in \Div(\bX_{(0)}).

Thus as in the geometric case, there exist an $\FF_{q^\ell}$-rational divisor $D_1$ of $\bX_{(0)}$ with degree $0$ and $f \in \KK(\bX_{(0)})^\times = \bK_{(0)}^\times$ such that
$W = D_1 + \divv(f)$, i.e.,

\begin{equation}\label{eqn: isogeny f}
\hspace{1.5cm} W_2-W_1 +\bigl(-D_1 -(\deg W_2-\deg W_1) \cdot D_0\bigr)= \divv(f) \quad \in \Div(\bX_{(0)}).
\end{equation}

Applying the Riemann-Roch Theorem on the curve $X$ over $\FF_{q^\ell}$ as in the geometric case, we may assume that $D_1$ has been chosen so that

\[
D_3 \assign -D_1-(\deg W_2-\deg W_1)\cdot D_0 \quad
\text{is effective on $\bU_{(0)}$.}
\]

Then $W+D_3 = \divv(f)$ and

\[
\divv(h_2/h_1) = W^{(\ell)} - W = \divv(f^{(\ell)}/f) \quad \in \Div(\bX_{(0)}).
\]

Up to a $\KK$-scalar multiple, we may choose $f$ so that

\[
h_2 \cdot f =  f^{(\ell)} \cdot h_1.
\]

Define $\Theta : \eM_{(W_1,h_1)} \rightarrow \eM_{(W_2,h_2)}$ as follows:
for $((m_1)_{(0)},\ldots, (m_1)_{(\ell-1)}) \in \eM_{(W_1,h_1)}$, set

\begin{align*}
\Theta & \bigl((m_1)_{(0)},(m_1)_{(1)},\ldots, (m_1)_{(\ell-1)}\bigr) \\
& \assign
\bigl(f^{(\ell)}\cdot (m_1)_{(0)}, f^{(\ell-1)} \cdot (m_1)_{(1)},\ldots,  f^{(1)} \cdot (m_1)_{(\ell-1)}\bigr).
\end{align*}

From the equality (<ref>), the map  $\Theta$ is an injective $O_\bK$-module homomorphism and the cokernel is of finite dimension over $\KK$ (by Lemma <ref>).

\begin{align*}
\Theta {}&\bigl(\sigma \cdot ((m_1)_{(0)},(m_1)_{(1)},\ldots, (m_1)_{(\ell-1)})\bigr) \\
&= \Theta\bigl( h_1\cdot (m_1)_{(\ell-1)}^{(-1)}, (m_1)_{(0)}^{(-1)},\ldots, (m_1)_{(\ell-2)}^{(-1)}\bigr) \\
&= \bigl(f^{(\ell)} \cdot h_1
\cdot (m_1)_{(\ell-1)}^{(-1)}, f^{(\ell-1)} \cdot (m_1)_{(0)}^{(-1)},\ldots, f^{(1)}\cdot  (m_1)_{(\ell-2)}^{(-1)}\bigr) \\
&= \bigl(h_2 \cdot (f^{(1)}\cdot (m_1)_{(\ell-1)})^{(-1)}, (f^{(\ell)}\cdot  (m_1)_{(0)})^{(-1)},\ldots, (
f^{(2)} \cdot (m_1)_{(\ell-2)})^{(-1)}\bigr) \\
&= \sigma \cdot \bigl(f^{(\ell)}\cdot  (m_1)_{(0)},\ldots,  f^{(1)}\cdot (m_1)_{(\ell-1)}\bigr) \\
&= \sigma \cdot \Theta\bigl((m_1)_{(0)},(m_1)_{(1)},\ldots, (m_1)_{(\ell-1)})\bigr).
\end{align*}

Hence $\Theta:\eM_{(W_1,h_1)}\rightarrow \eM_{(W_2,h_2)}$ is an isogeny, and the statement holds.

§ UNIFORMIZABILITY OF CM DUAL T-MOTIVES

§.§ Characterization of CM dual t-motives

Fix an algebraically closed field $\KK$ with $k\subset \KK \subset \CC_\infty$.
We first characterize when a given dual $t$-motive is “CM”.

Let $\eK$ be a CM field over $\FF_q(t)$, and $O_\eK$ be the integral closure of $\FF_q[t]$ in $\eK$.
Given a dual $t$-motive $\eM$, suppose that $\rank_{\KK[t]}(\eM) = [\eK:\FF_q(t)]$ and that there is an $\FF_q[t]$-algebra embedding $O_\eK \hookrightarrow \End_{\KK[t,\sigma]}(\eM)$.
Then $\eM$ is projective of rank one over $O_\bK \assign  \KK\otimes_{\FF_q} O_\eK$.

Suppose that the constant field of $\eK$ is $\FF_{q^\ell}$. Let $e_0,\ldots, e_{\ell-1}$ be the idempotents in

$$O_\bK = \prod_{i=0}^{\ell-1}O_{\bK,{(i)}}, \quad \text{ where } \quad
\assign  \CC_\infty \underset{\Frob_q^{-i}, \FF_{q^\ell}}{\otimes} O_\eK,

so that $O_\bK \cdot e_i = O_{\bK,(i)}$ for $0\leq i <\ell$.
Put $\eM_{(i)}\assign  e_i \cdot \eM$ and then we may write

$$\eM = \prod_{i=0}^{f-1}\eM_{(i)}.

The Frobenius twisting permutes $e_0,\ldots, e_{\ell-1}$ by

\[
e_{i}^{(-1)} = e_{i+1}, \quad 0\leq i \leq \ell-2, \quad \text{ and } \quad e_{\ell-1}^{(-1)} = e_0.
\]

It follows that

\[
\sigma \cdot \eM_{(i)} \subset \eM_{(i+1)} \quad \text{ for } 0\leq i<\ell-1, \quad
\sigma \cdot \eM_{(\ell-1)} \subset \eM_{(0)}.
\]

As $\dim_{\KK} (\eM/\sigma \eM)$ is finite, we obtain the equalities

\[
\rank_{\KK[t]} \eM_{(0)} = \rank_{\KK[t]} \eM_{(1)} = \cdots = \rank_{\KK[t]} \eM_{(\ell-1)},
\]

\[
\rank_{\KK[t]} M_{(i)} = \frac{1}{\ell} \cdot \rank_{\KK[t]}(\eM) = [\eK:\FF_{q^\ell}(t)], \quad 0\leq i < \ell.
\]

$\eM_{(i)}$ is a projective $O_{\bK,(i)}$-module of rank one for $0\leq i \leq \ell-1$,
showing that $\eM$ is a projective $O_\bK$-module of rank one.

Let $\eM$ be a dual $t$-motive satisfing the conditions in the above lemma.
As $\eM$ is projective of rank one over $O_\bK$ and $(t-\theta)^N \cdot \eM \subseteq \sigma \eM$ when $N$ is sufficiently large,
there exists a unique ideal $\Ifk_\eM$ of $O_\eK$ which is of the form

\Ifk_\eM = \prod_{\xi \in J_\eK}\Pfk_\xi^{m_\xi}, \quad m_\xi \geq 0,

where $\Pfk_\xi$ is the maximal ideal of $O_\bK$ corresponding to $\xi$ for each $\xi \in J_\eK$, so that $\sigma \eM = \Ifk_\eM\cdot \eM$.
In this case, we set

$$\Xi_\eM \assign  \sum_{\xi \in J_\eK}m_\xi \xi \quad \in I_\eK.$$

Let $\eK$ be a CM field over $\FF_q(t)$,
and $O_\eK$ be the integral closure of $\FF_q[t]$ in $\eK$.
Given a dual $t$-motive $\eM$, suppose the following conditions hold:

(1) $\rank_{\KK[t]}\eM = [\eK:\FF_q(t)]$;

(2) there is an $\FF_q[t]$-algebra embedding $O_\eK \hookrightarrow \End_{\KK[t,\sigma]} \eM$;

(3) $\Xi_\eM$ is a generalized CM type of $\eK$;

(4) $\eM$ is pure (see <cit.>).

Then $\eM$ is a CM dual $t$-motive with  generalized CM type $(\eK,\Xi_\eM)$.

As in Lemma <ref>, we identify

$$ \eM = \prod_{i=0}^{\ell-1} \eM_{(i)},$$

where $\eM_{(i)} = e_i \cdot \eM$ is projective of rank one over $O_{\bK,(i)}$ for $0\leq i < \ell$.
\Xi \assign  \Xi_\eM = \sum_{\xi \in J_\eK} m_\xi \xi = \sum_{i=0}^{\ell -1} \Xi_{(i)}, \quad
\text{ where } \Xi_{(i)} = \sum_{\xi \in J_\eK\cap \bX_{(i)}} m_\xi \xi, \quad 0\leq i < \ell.

Let $\Ifk_{\eM,(i)} \assign  (\prod_{\xi \in J_\eK \cap \bX_{(i)}} \Pfk_\xi^{m_\xi}) \cdot O_{\bK,(i)}$ for $0\leq i < \ell$.
As $\sigma \eM = \Ifk_\eM \cdot \eM$, we have that
\begin{equation}\label{eqn: M(i)}
\sigma \eM_{(i)} = \Ifk_{\eM,(i+1)} \cdot \eM_{(i+1)}, \quad \textup{for $0\leq i \leq \ell-2$,} \quad \textup{and $\sigma \eM_{(\ell-1)} = \Ifk_{\eM,(0)} \cdot \eM_{(0)}$.}
\end{equation}

Take a suitable divisor $W_0$ of $\bX_{(0)}$ and $\bm \in \eM_{(0)}$ so that

\begin{equation}\label{eqn: M0-1}
\eM_{(0)} = \Gamma\bigl(\bU_{(0)}, \Ocal_{\bX_{(0)}}(-W_0^{(\ell)})\bigr) \cdot \bm.
\end{equation}

It follows from (<ref>) that for $1\leq i \leq \ell-1$,

\[
\eM_{(i)} = \Gamma\bigl(\bU_{(i)},\Ocal_{\bX_{(i)}}(-W_0^{(\ell-i)}+ \sum_{j=1}^i \Xi_{(j)}^{(j-i)})\bigr) \cdot (\sigma^i \cdot \bm).
\]

In particular,

\[
\sigma \cdot  \eM_{(\ell-1)}
= \Gamma
\bigl(\bU_{(0)},\Ocal_{\bX_{(0)}}(-W_0+ \sum_{j=1}^{\ell-1} \Xi_{(j)}^{(j-\ell)})\bigr)
\cdot (\sigma^\ell \cdot \bm),
\]

and so

\begin{align}\label{eqn: M0-2}
\eM_{(0)} &= \Ifk_{\eM,(0)}^{-1} \cdot \bigl(\sigma \cdot \eM_{(\ell-1)}\bigr) \\
\Gamma
\bigl(\bU_{(0)},\Ocal_{\bX_{(0)}}(-W_0+ (\sum_{j=1}^{\ell-1} \Xi_{(j)}^{(j-\ell)})+\Xi_{(0)})\bigr)
\cdot (\sigma^\ell \cdot \bm) \notag \\
&=  \Gamma\bigl(\bU_{(0)},\Ocal_{\bX_{(0)}}(-W_0+\Xi^\#)\bigr)\cdot (\sigma^\ell \cdot \bm), \notag
\end{align}

where $\Xi^\#$ is defined in (<ref>).

From the two expressions for $\eM_{(0)}$ in (<ref>) and (<ref>), we have an $h_\eM \in \bK_{(0)}^\times$ such that

\[
\sigma^\ell \cdot \bm = h_\eM \cdot \bm,
\]

\[
\Gamma\bigl(\bU_{(0)},\Ocal_{\bX_{(0)}}(-W_0+\Xi^\#)\bigr)\cdot h_\eM = \Gamma\bigl(\bU_{(0)},\Ocal_{\bX_{(0)}}(-W_0^{(\ell)})\bigr).
\]

\[
\divv(h_\eM) = W_0^{(\ell)}-W_0 + \Xi^\# - \eI^\#_{h_\eM} \quad \in \Div(\bX_{(0)}),
\]

where $\eI_{h_\eM}^\# \in \Div(\bX_{(0)})$ has support at the points of $\bX_{(0)}$ lying above $\infty$.
Finally, to apply the conditions (3) and (4), we invoke the following technical lemma (whose proof will then be given):

There exists $W_\infty \in \Div(\bX_{(0)})$ so that
W_\infty^{(\ell)} - W_\infty = \eI_\Xi^\# - \eI_{h_\eM}^\#

Put $W\assign  W_0+W_\infty$.
We then have

\divv(h_\eM) = W^{(\ell)}-W + \Xi^\# - \eI_\Xi^\# \quad \in \Div(\bX_{(0)}).

In conclusion, we obtain a dual $t$-motive isomorphism
$\eM_{(W,h_\eM)}\stackrel{\sim}{\rightarrow} \eM$ defined by sending
(m_{(0)},\ldots, m_{(\ell-1)})
to $(m_{(0)}\cdot \bm,m_{(1)} \cdot \sigma \bm,\ldots,  m_{(\ell-1)}\cdot \sigma^{\ell-1} \bm)$
for every $(m_{(0)},\ldots, m_{(\ell-1)})$ in $\eM_{(W,h_\eM)}$.
Therefore $\eM$ is a CM dual $t$-motive with generalized CM type $(\eK,\Xi)$.

Regarding $X$ as a curve over $\FF_{q^\ell}$, we may view $\bX_{(0)}$ as the base change of $X$ from $\FF_{q^\ell}$ to $\KK$, and $\eK$ is a geometric extension over $\FF_{q^\ell}(t)$ (corresponding to an $\FF_{q^\ell}$-morphism from $X$ to $\PP^1$).
Let $\pi_{\bX_{(0)}/\PP^1}: \bX_{(0)}\rightarrow \PP^1$ be the base change from $\FF_{q^\ell}$ to $\KK$, which coincides with the composition of the inclusion $\bX_{(0)}\subset \bX$ and $\pi_{\bX/\PP^1}: \bX \rightarrow \PP^1$.
Let $\Ocal_{\eK,\infty}$ be the integral closure of $\FF_{q^\ell}[1/t]$ in $\eK$.
Since the smoothness of $X$ (over $\FF_{q^\ell}$) is preserved under the base change from $\FF_{q^\ell}$ to $\KK$ by <cit.>,
the integral closure of $\KK[1/t]$ in $\bK_{(0)}$ $(=\KK(t)\otimes_{\FF_{q^\ell}(t)} \eK = \KK(t) \otimes_{\FF_{q^\ell}[1/t]}\Ocal_{\eK,\infty})$ can be identified with $\KK\otimes_{\FF_{q^\ell}} \Ocal_{\eK,\infty} = \KK[1/t]\otimes_{\FF_{q^\ell}[1/t]} \Ocal_{\eK,\infty}$.
For each $\overline{\infty} \in \pi^{-1}_{\bX_{(0)}/\PP^1}(\infty)$,
$\hat{\bK}_{\overline{\infty}}$ be the completion of $\bK_{(0)}$ ($=\KK(t)\otimes_{\FF_{q^\ell}(t)}\eK$) with respect to $\overline{\infty}$.
We have the following identifications,

\[
\hat{\bK}_{(0),\infty}\assign  \prod_{\overline{\infty} \in \pi_{\bX_{(0)}/\PP^1}^{-1}(\infty)}\hat{\bK}_{\overline{\infty}} = \KK(\!(1/t)\!)\otimes_{\KK(t)}\bK_{(0)}
= \KK(\!(1/t)\!)\otimes_{\FF_{q^\ell}[1/t]} \Ocal_{\eK,\infty}.
\]

Moreover, put

\[
\hat{\Ocal}_{\bX,\overline{\infty}}
\assign  \{\alpha \in \hat{\bK}_{\overline{\infty}}\mid \ord_{\overline{\infty}}(\alpha) \geq 0 \} \ \text{ and }\  \Mfk_{\overline{\infty}}\assign  \{\alpha \in \hat{\bK}_{\overline{\infty}}\mid \ord_{\overline{\infty}}(\alpha) > 0 \}.
\]

By <cit.>) we obtain

\begin{align}\label{E:Ocal_{(0),infty}}
\hat{\Ocal}_{\bX_{(0)},\infty}\assign  \prod_{\overline{\infty} \in \pi_{\bX_{(0)}/\PP^1}^{-1}(\infty)} \hat{\Ocal}_{\bX,\overline{\infty}}
\KK[\![1/t]\!] \otimes_{\KK[1/t]}(\KK[1/t]\otimes_{\FF_{q^\ell}[1/t]}\Ocal_{\eK,\infty}) \\
&= \KK[\![1/t]\!] \otimes_{\FF_{q^\ell}[1/t]} \Ocal_{\eK,\infty}. \notag
\end{align}

Now, the purity of $\eM$ (from the condition (4) in Proposition <ref>) says that there exists a  $\KK[\![1/t]\!]$-lattice $L$ of full rank in
and positive integers $u,w$
satisfying that

\[
t^u L = \sigma^w L.
\]

After iterating the action of $t^u$ on the equation above $\ell$ times, we may assume that $\ell$ divides $w$.
Since the idempotents $e_0,\ldots, e_{\ell-1} \in O_\bK = \prod_{i=0}^{\ell-1} O_{\bK,(i)}$
acting on the module $\KK(\!(1/t)\!)\otimes_{\KK[t]}\eM$ actually commute with $\sigma^\ell$,
we get that $L_0\assign  e_0 \cdot L$ is a $\KK[\![1/t]\!]$-lattice of full rank in $\eM_{(0)}(\!(1/t)\!) \assign \KK(\!(1/t)\!)\otimes_{\KK[t]} \eM_{(0)}$ satisfying

\[
t^u L_0 = \sigma^w L_0.
\]

Notice that the action of $\sigma^\ell$ on $\eM_{(0)}(\!(1/t)\!)$ commutes with multiplication by elements in $\hat{\eK}_{(0),\infty}$.
After replacing $L_0$ by $\Ocal_{\eK,\infty} \cdot L_0$, we may assume that $L_0$ is an $\hat{\Ocal}_{\bX_{(0)},\infty}$-module via (<ref>).
Identifying $\eM_{(0)}(\!(1/t)\!)$ as a free $\hat{\bK}_{(0),\infty}$-module of rank one generated by $\bm$, we
can write $L_0 = \Lcal_0 \cdot \bm$, where

\[
\Lcal_0 = \prod_{\overline{\infty} \in \pi_{\bX_{(0)}/\PP^1}^{-1}(\infty)} \Mfk_{\overline{\infty}}^{\ell_{\overline{\infty}}} \subset \prod_{\overline{\infty} \in \pi_{\bX_{(0)}/\PP^1}^{-1}(\infty)} \hat{\bK}_{\overline{\infty}} \quad
\text{ for $\ell_{\overline{\infty}} \in \ZZ$.}
\]

\[
(t^u \Lcal_0) \cdot \bm = t^u L_0 = \sigma^w L_0 = \Bigl(\prod_{i=0}^{(w/\ell)-1}h_\eM^{(-\ell i)}\Bigr) \cdot \Lcal_0^{(-w)} \cdot
\bm.
\]

Since $\pi_{\bX_{(0)}/\PP^1}^{-1}(\infty)$ is invariant under the $\ell$-th Frobenius twisting (as the maximal ideals $\Mfk_{\overline{\infty}}$ with $\overline{\infty} \in \pi_{\bX_{(0)}/\PP^1}^{-1}(\infty)$ are permuted by the $\ell$-th Frobenius twisting), there exists a positive integer $\ell_\infty$ so that

\[
\overline{\infty}^{(\ell_\infty)} = \overline{\infty}, \quad \forall \overline{\infty} \in \pi_{\bX_{(0)}/\PP^1}^{-1}(\infty).
\]

we may increase $u$ and $w$ to be multiples of $\ell_\infty$, and get $\Lcal_0^{(-w)} = \Lcal_0$.

\[
t^u \cdot \Lcal_0 = \Bigl(\prod_{i=0}^{(w/\ell)-1} h_\eM^{(-\ell i)}\Bigr) \cdot \Lcal_0 \quad \quad \subset \prod_{\overline{\infty} \in \pi_{\bX_{(0)}/\PP^1}^{-1}(\infty)}  \hat{\bK}_{\overline{\infty}}.
\]

The above equality implies that

\begin{equation}\label{eqn: purity}
u \cdot \pi_{\bX_{(0)}/\PP^1}^*(\infty)=  \sum_{i=0}^{(w/\ell)-1} (\eI_{h_\eM}^\#)^{(-\ell i)},
\end{equation}

where $\pi_{\bX_{(0)}/\PP^1}^*(\infty) \in \Div(\bX_{(0)})$ is the pull-back of the divisor $\infty$.
In particular, let $X^+$ be the curve over $\FF_q$ associated with the maximal totally real subfield $\eK^+$ of $\eK$, and let $\pi_{\bX^+/\PP^1}^{-1}(\infty) = \{\infty_1^+,\ldots, \infty_d^+\}$ where $d \assign [\eK^+:\FF_q(t)]$.

\[
\eI_{h_\eM}^\# = \sum_{\overline{\infty} \in \pi_{\bX/\PP^1}^{-1}(\infty) \cap \bX_{(0)}} n_{\overline{\infty}} \overline{\infty} = \sum_{i=1}^d\left( \sum_{\substack{\overline{\infty} \in \bX_{(0)} \\ \pi_{\bX/\bX^+}(\overline{\infty}) =   \infty_i^+}}n_{\overline{\infty}} \overline{\infty}\right).
\]

As $\deg \eI_{h_\eM}^\# = \deg \Xi^\# = d \cdot {\rm wt}(\Xi)$, equality (<ref>) assures us that

\[
\sum_{\substack{\overline{\infty} \in \bX_{(0)} \\ \pi_{\bX/\bX^+}(\overline{\infty}) =   \infty_i^+}}n_{\overline{\infty}} = {\rm wt}(\Xi), \quad \forall i = 1,\ldots, d.
\]

On the other hand,
we may write

\[
\eI_\Xi^\# = \sum_{\overline{\infty} \in \pi_{\bX/\PP^1}^{-1}(\infty)\cap \bX_{(0)}} m_{\overline{\infty}} \overline{\infty}
\ \text{ with}\!
\sum_{\substack{\overline{\infty} \in \bX_{(0)} \\ \pi_{\bX/\bX^+}(\overline{\infty}) =   \infty_i^+}}m_{\overline{\infty}} = {\rm wt}(\Xi), \ \forall i = 1,\ldots, d.
\]

For $1\leq i \leq d$, we take a point $\overline{\infty}_i \in \pi_{\bX/\bX^+}^{-1}(\infty_i^+) \cap \bX_{(0)}$.
Since $\infty_i^+$, as a place of $\eK^+$, is non-split in $\eK$, every point $\overline{\infty} \in  \pi_{\bX/\bX^+}^{-1}(\infty_i^+)\cap \bX_{(0)}$ must be equal to the Frobenius twist $(\overline{\infty}_i)^{(\ell \mu_{\overline{\infty}})}$ for some non-negative integer $\mu_{\overline{\infty}}$.

\begin{align*}
\eI_\Xi^\# - \eI_{h_\eM}^\# &=  \sum_{\overline{\infty} \in  \pi_{\bX/\PP^1}^{-1}(\infty) \cap \bX_{(0)}} (m_{\overline{\infty}}-n_{\overline{\infty}}) \overline{\infty} \\
&= \sum_{i=1}^d \left( \sum_{\substack{\overline{\infty} \in \bX_{(0)} \\ \pi_{\bX/\bX^+}(\overline{\infty}) =   \infty_i^+}} (m_{\overline{\infty}}-n_{\overline{\infty}}) ( \overline{\infty}_i^{(\ell \mu_{\overline{\infty}})} - \overline{\infty}_i)\right).
\end{align*}

Consequently we can take

\[
W_\infty \assign  \sum_{i=1}^d \left( \sum_{\substack{\overline{\infty} \in \bX_{(0)} \\ \pi_{\bX/\bX^+}(\overline{\infty}) =   \infty_i^+}} (m_{\overline{\infty}}-n_{\overline{\infty}}) \sum_{j=0}^{\mu_{\overline{\infty}}-1} \overline{\infty}_i^{(\ell j)}\right)
\]

and obtain

\[
W_\infty^{(\ell)} - W_\infty = \eI_\Xi^\# - \eI_{h_\eM}^\#.
\qedhere
\]

§.§ Dual t-motives arising from Hilbert-Blumenthal t-modules

Throughout this section,
we take $\KK = \CC_\infty$.
Let $\eK/\FF_{q}(t)$ be a CM field with maximal totally real subfield $\eK^+$.
Put $d = [\eK^+:\FF_q(t)]$.
Let $\Xi = \xi_1+\cdots + \xi_d$ be a  CM type of $\eK$.
Notice that
$\nu_{\xi_1}\big|_{\eK^+},\ldots, \nu_{\xi_d}\big|_{\eK^+}$ give all the $\FF_q$-algebra embeddings from $\eK^+$ into $\CC_\infty$ sending $t$ to $\theta$.

\nu_\Xi:\eK \rightarrow \CC_\infty^d, \quad \nu_\Xi(\lambda) \assign  (\nu_{\xi_1}(\lambda),\ldots, \nu_{\xi_d}(\lambda)), \quad \forall \lambda \in \eK.

Let $O_\eK$ be the integral closure of $\FF_q[t]$ in $\eK$. Then $\nu_\Xi(O_\eK)$ becomes a discrete $A$-lattice in $\CC_\infty^d$.

There exists a Hilbert-Blumenthal $O_{\eK^+}$-module $E_{(\eK,\Xi)} = (\GG_a^d,\rho^\Xi)$ whose period lattice is $\nu_\Xi(O_\eK)$.

This is directly from <cit.>.

From <cit.> we can actually extend $\rho^\Xi$ to an $\FF_q$-algebra homomorphism from $O_\eK$ into $\End_{\FF_q}(\GG^d_{a/\CC_\infty})$ so that
$$\partial \rho^\Xi = \nu_{\xi_1}\oplus \cdots \oplus \nu_{\xi_d} \quad \text{ on $\Lie(\GG_a^d)$}.$$

Recall in Definition <ref> that we set

\[\Mscr(\rho^\Xi)\assign  \Hom_{\FF_q}({\GG_a^d}_{/\CC_\infty}, {\GG_a}_{/\CC_\infty}),
\]

together with the $O_\bK$-module structure given by

(c \otimes a) \cdot m \assign  c \cdot m \circ \rho_\eK(a), \quad \forall c \in \CC_\infty,\ a \in O_\eK,\ m \in \Mscr(\rho_\eK),

where $O_{\bK}=\CC_{\infty}\otimes_{\FF_{q}}O_{\eK}$.
As $E_{\rho^\Xi}$ is pure and uniformizable (from the definition of a Hilbert-Blumenthal $O_{\eK^+}$-module), the $t$-motive $\Mscr(\rho^\Xi)$ is pure and uniformizable

\begin{align*}
& \rank_{\CC_\infty[t]}\Mscr(\rho^\Xi) = [\eK:\FF_q(t)] \rassign  r \\
\text{and} \quad &
\rank_{\CC_\infty[\tau]}\Mscr(\rho^\Xi) = \dim_{\CC_\infty}\left(\frac{\Mscr(\rho^\Xi)}{\tau \Mscr(\rho^\Xi)}\right) = d
\end{align*}

(see <cit.>).
In particular,

\Ifk_\Xi \assign  \prod_{i=1}^d \Pfk_i \subset O_\bK,

where $\Pfk_i$ is the maximal ideal of $O_\bK$ generated by $a - \nu_{\xi_i}(a)$ for $a \in O_\eK$ (i.e. the maximal ideal corresponding to $\xi_{i}$).
We have the following:

\begin{equation}\label{Ifk Mscr}
\Ifk_\Xi  \cdot \Mscr(\rho^\Xi) \subset \tau  \cdot \Mscr(\rho^\Xi).
\end{equation}

It suffices to show that for any $a_{1},\ldots,a_{d}\in O_{\eK}$,

\[ (a_{1}-\nu_{\xi_{1}}(a_{1})\cdots (a_{d}-\nu_{\xi_{d}}(a_{d}))\cdot m \in \tau  \cdot \Mscr(\rho^\Xi), \quad \forall m \in  \Mscr(\rho^\Xi).    \]
Now, given any $m\in \Mscr(\rho^\Xi)\cong \Mat_{1\times d}(\CC_{\infty}[\tau])$ we write

\[m=\sum_{j=0}^{s}v_{j} \tau^{j}\quad {\rm{for}} \quad v_{o},\ldots,v_{s}\in \Mat_{1\times d}(\CC_{\infty}). \]

Note that for any $a\in O_{\eK}$ and $\xi\in J_{\eK}$, we have

\[(a-\nu_{\xi}(a))\cdot m\assign  m \rho_{a}^{\Xi}- \nu_{\xi}(a)\cdot m=\left(\sum_{j=0}^{s}v_{j}\tau^{j} \right) \rho_{a}^{\Xi}-\nu_{\xi}(a) \left( \sum_{j=0}^{s}v_{j}\tau^{j}\right) \]

and hence the constant vector of $(a-\nu_{\xi}(a))\cdot m$ is given by

\[ v_{0}\cdot \left(\partial(\rho_{a}^{\Xi})-\nu_{\xi}(a)I_{d} \right) .\]

It follows that for any $a_{1},\ldots,a_{d}\in O_{\eK}$, the constant vector of

\[(a_{1}-\nu_{\xi_{1}}(a_{1}))\cdots (a_{d}-\nu_{\xi_{d}}(a_{d}))\cdot m \]

is given by

\[v_{0}\cdot\left(\partial(\rho_{a_{d}}^{\Xi})-\nu_{\xi_{d}}(a_{d})I_{d} \right)\cdots \left(\partial(\rho_{a_{1}}^{\Xi})-\nu_{\xi_{1}}(a_{1})I_{d} \right),\]
which is zero because

\[ \partial(\rho_{a}^{\Xi})=\begin{pmatrix}\nu_{\xi_{1}}(a)& &\\
& & \nu_{\xi_{d}}(a)
\end{pmatrix} \quad \forall a\in O_{\eK}.
\]

That is, $(a_{1}-\nu_{\xi_{1}}(a_{1})\cdots (a_{d}-\nu_{\xi_{d}}(a_{d}))\cdot m \in \tau  \cdot \Mscr(\rho^\Xi)$ as claimed.

In (<ref>) we define the Hartl-Juschka dual $t$-motive associated with $E_{(\eK,\Xi)}$ by

\[
\eM(\rho^\Xi)\assign \Hom_{\CC_\infty[t]}(\tau \Mscr(\rho^\Xi), \Omega_{\CC_\infty[t]/\CC_\infty}),
\]

where $\Omega_{\CC_\infty[t]/\CC_\infty} = \CC_\infty[t]dt$ is the module of Kähler differentials of $\CC_\infty[t]$ over $\CC_\infty$, together with the following $\sigma$-action

\[
(\sigma \cdot \phi)(\tau m)
\assign  \phi( \tau^2 m)^{(-1)}, \quad \forall \tau m \in \tau \Mscr(\rho^\Xi).
\]

The following result is known from <cit.>:

$\eM(\rho^\Xi)$ is pure and uniformizable with

\begin{align*}
& \rank_{\CC_\infty[t]} \eM(\rho^\Xi) = \rank_{\CC_\infty[t]} \Mscr(\rho^\Xi) = r\\
\text{and}\quad  &
\dim_{\CC_\infty} (\frac{\eM(\rho^\Xi)}{\sigma \eM(\rho^\Xi)}) = \dim_{\CC_\infty}(\frac{\Mscr(\rho^\Xi)}{\tau \Mscr(\rho^\Xi)}) = d.
\end{align*}

We note that the $O_\bK$-module structure on $\Mscr(\rho^\Xi)$ induces an $O_\bK$-module structure on $\eM(\rho_\eK)$ as follows:
for $\phi \in \eM(\rho^\Xi)$ and $\alpha \in O_\bK$,

\[
(\alpha \cdot \phi)(\tau m) \assign  \phi(\alpha \tau m), \quad \forall \tau m \in \tau \Mscr(\rho^\Xi).
\]

In particular, this induces an $\FF_q(t)$-algebra embedding $O_\eK\hookrightarrow \End_{\CC_\infty[t,\sigma]}(\eM(\rho^\Xi))$, and the following lemma holds:

$\Ifk_\Xi \cdot \eM(\rho^\Xi) = \sigma \cdot \eM(\rho^\Xi)$.

By Lemma <ref>, we know that $\eM(\rho^\Xi)$ is projective of rank one over $O_\bK$.

\dim_{\CC_\infty}\left(\frac{\eM(\rho^\Xi)}{\sigma \eM(\rho^\Xi)}\right) = d = \dim_{\CC_\infty}\left(\frac{O_\bK}{\Ifk_\Xi}\right) = \dim_{\CC_\infty}\left(\frac{\eM(\rho^\Xi)}{\Ifk_\Xi \eM(\rho^\Xi)}\right),

it suffices to show that
$\Ifk_\Xi \cdot \eM(\rho^\Xi) \subseteq \sigma \cdot \eM(\rho^\Xi)$.
Given $\alpha \in \Ifk_\Xi$ and $\phi \in \eM(\rho^\Xi)$, define $\phi'_\alpha \in \eM(\rho^\Xi)$ by

\phi'_\alpha (\tau m) \assign  \phi(\alpha m)^{(1)}, \quad \forall \tau m \in \tau   \Mscr(\rho^\Xi).

The map $\phi'_{\alpha}$ is well-defined since $\alpha m \in \alpha \Mscr(\rho^\Xi) \subset \tau \Mscr(\rho^\Xi)$, and for $a \in \CC_\infty[t]$ one has

\begin{align*}
\phi'_\alpha(a \tau m) = \phi'_\alpha(\tau a^{(-1)} m ) & = \phi(\alpha a^{(-1)} m)^{(1)} \\
& = \Bigl(a^{(-1)} \phi(\alpha m)\Bigr)^{(1)} = a  \phi(\alpha m)^{(1)} = a  \phi'_\alpha( \tau m).
\end{align*}

We shall check that $\alpha \cdot \phi = \sigma \cdot \phi_\alpha' \in \sigma \cdot \eM(\rho^\Xi)$ and the result follows.
Indeed, given any $\tau m \in \tau \Mcal(\rho^\Xi)$, we have

(\sigma \cdot \phi'_\alpha)(\tau m)
= \phi'_\alpha(\tau^2 m)^{(-1)}
=\Bigl(\phi(\alpha \tau m)^{(1)}\Bigr)^{(-1)} = \phi(\alpha \tau m) = (\alpha \cdot \phi)(\tau m),

\[
\alpha \cdot \phi = \sigma \cdot \phi_\alpha' \subset \sigma \cdot \eM(\rho^\Xi).
\]

From the characterization in Proposition <ref>, we obtain

$\eM(\rho^\Xi)$ is a CM dual $t$-motive with the CM type $(\eK,\Xi)$.

By Proposition <ref> and Lemma <ref>, the dual $t$-motive $\eM(\rho^\Xi)$ satisfies the conditions (1)–(4) in Proposition <ref>.
Hence the result follows.

[And82] A82 G. W. Anderson, Logarithmic derivatives of Dirichlet L-functions and the periods of abelian varieties, Compositio Math. 45 (1982), 3, 315-332.

[And86] Anderson86 G. W.  Anderson, $t$-motives,
Duke Math. J. 53 (1986), no. 2, 457–502.

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