\text{and }\quad \divv(h_{(x,y)}) &= - (q^\ell-1)\cdot \ell \text{\rm wt}_0(x,y) \infty_{X_\nfk} + \Xi_{(x,y)}^{\#} +W_{(x,y)}^{(\ell)} - W_{(x,y)}.
\end{align*}

§.§ Generalized soliton dual t-motives

We now apply the geometric method from <cit.> to construct the desired CM dual $t$-motives.

\[
O_{\nfk,\ell} \assign \FF_{q^\ell} \otimes_{\FF_q} O_{\nfk}
\quad \text{ and } \quad
O_{\bK_{\nfk,\ell}} \assign \bar{k}\otimes_{\FF_q} O_{\nfk,\ell} \cong \prod_{i=0}^{\ell-1} O_{\bK_{\nfk,\ell},(i)},
\]

where for $0\leq i<\ell$,

\[
O_{\bK_{\nfk,\ell},(i)} \assign \bar{k}\underset{\text{Frob}_q^{-i},\FF_{q^\ell}}{\otimes} O_{\nfk,\ell} \quad
(\cong \bar{k} \otimes_{\FF_q} O_\nfk = O_{\bK_{\nfk}})
\]

is the affine coordinate ring of $\bU_{\nfk,\ell,(i)}$.
The field of fractions of $O_{\bK_{\nfk,\ell},(i)}$ is denoted by $\bK_{\nfk,\ell,(i)}$, which is the function field of $\bX_{\nfk,\ell,(i)}$.

Given $x \in \frac{1}{\nfk(\theta)}A/A$ and $y \in \frac{1}{q^\ell-1}\ZZ/\ZZ$,
let $*$ be either $x$, $y$, or $(x,y)$, and

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

\eM_{(W_*,h_*),(0)}\assign  \Gamma\big(\bU_{\nfk,\ell,(0)}, \Ocal_{\bX_{\nfk,\ell,(0)}}(-W_*^{(\ell)})\big) \quad \subset O_{\bK_{\nfk,\ell},(0)}, $$

\eM_{(W_*,h_*),(i)}\assign  \Gamma\big(\bU_{\nfk,\ell,(i)}, \Ocal_{\bX_{\nfk,\ell, (i)}}(-W_*^{(\ell-i)} + \sum_{j=1}^i \Xi_{*,(j)}^{(j-i)})\big) \quad \subset \bK_{\nfk,\ell,(i)}, \quad 1\leq i < \ell.

The $\sigma$-action on $\eM_{(W_*,h_*)}$ is given by

\sigma \cdot (m_{(0)},\ldots, m_{(\ell-1)}) \assign  \Big(h_* \cdot m_{(\ell-1)}^{(-1)}, m_{(0)}^{(-1)},\ldots, m_{(\ell-2)}^{(-1)}\Big), \quad
\forall (m_{(0)},\ldots, m_{(\ell-1)}) \in \eM_{(W_*,h_*)}.

By <cit.>, we have the following:

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
Given $x \in \frac{1}{\nfk(\theta)}A/A$
and $y \in \frac{1}{q^\ell-1}\ZZ/\ZZ$,
take $*$ to be either $x$, $y$, or $(x,y)$.
$\eM(*)$ is a CM dual $t$-motive with generalized CM type $(\eK_{\nfk,\ell},\Xi_{*})$ over $\ok$.

In particular, Theorem <ref> shows that:

Keep the notation from Lemma <ref>.
Let $\OO_\nfk \assign  \TT \otimes_{\FF_{q}[t]} O_\nfk $.
The infinite product

\[
\psi_{*} \assign  \prod_{j=1}^\infty \left(h_*^{-1}\right)^{(\ell j)} \quad \text{ lies in }\quad  \OO_{\nfk}^\times,
\]

\[
\tilde{\psi}_* \assign (\psi_*,\psi_*^{(-1)},...,\psi_*^{(1-\ell)})
\in H_{\rm Betti}(\eM(*)) \subset \MM(*) \assign \TT \otimes_{\bar{k}[t]} \eM(*).
\]

The proof is adapted from <cit.>.
We include the details for completeness.

The support of $W^{(\ell)}_*$ in $\bX_\nfk$ is contained in the pre-image of $\{t=\theta^{q^i}\mid i \in \NN\}$.
Thus when identifying $O_{\bK_{\nfk,\ell},(0)}$ with $O_{\bK_\nfk}$,
there exist $f_{1},...,f_{\ell} \in \bK_\nfk$ (the function field of $\bX_\nfk$) with possible poles lying over $\{t=\theta^{q^i}\mid i \in \NN \}$ and $m_1,...,m_\ell \in \eM_{(W_*,h_*),(0)}$ such that

f_{1} m_1 + \cdots + f_{\ell} m_\ell = 1 \in O_{\bK_\nfk}.

Then the restriction on the possible poles of $f_{1},...,f_{\ell}$ forces that $f_1,...,f_\ell \in \OO_{\nfk}$, which means that $1 \in \MM_{(W_*,h_*),(0)} \assign \TT\otimes_{\bar{k}[t]} \eM_{(W_*,h_*),(0)}$ and

\MM_{(W_*,h_*),(0)} = \OO_\nfk.

Moreover, by Remark <ref> we are able to show that $h_*^{(\ell j)}$ lies in $\OO_\nfk^\times$ for all $j \in \NN$,
the condition on the coefficients of $g_x$ in Theorem <ref> ensures that the infinite product $\psi_*$ lies in $\OO_\nfk^\times$.

To prove the second statement,
observe that when identifying $\bK_{\nfk,\ell,(i)}$ with $\bK_{\nfk}$ for $1\leq i < \ell$,
one has that

\[
\eM_{(W_*,h_*),(i)} \supset
\Gamma\big(\bU_{\nfk}, \Ocal_{\bX_{\nfk}}(-W_*^{(\ell-i)})\big),
\]

\[
\MM_{(W_*,h_*),(i)} \assign \TT\otimes_{\bar{k}[t]} \eM_{(W_*,h_*),(i)}  \ \supset \TT\otimes_{\bar{k}[t]} \Gamma\big(\bU_{\nfk}, \Ocal_{\bX_{\nfk}}(-W_*^{(\ell-i)})\big) = \OO_\nfk \ \ni \psi^{(-i)}.
\]

In particular,
$(\psi_*,\psi_{*}^{(-1)},...,\psi_*^{(1-\ell)}) \in \MM(*)$.
Finally, it is straightforward that

\sigma\cdot
= (\psi_{*}^{(-\ell)} h_*, \psi_{*}^{(-1)},...,\psi_*^{(1-\ell)})
= (\psi_{*},\psi_{*}^{(-1)},...,\psi_*^{(1-\ell)}).

Therefore $(\psi_{*},\psi_{*}^{(-1)},...,\psi_*^{(1-\ell)}) \in H_{\text{Betti}}(\eM(*))$.

Identifying $O_{\bK_{\nfk,\ell},(0)}$ with $O_{\bK_\nfk}$,
the above proof actually shows that

\[
\psi_* \in
\TT^\dagger \otimes_{\bar{k}[t]} O_{\bK_{\nfk,\ell},(0)} \rassign \OO_{\nfk,\ell,(0)}^\dagger.
\]

This enables us to evaluate $\psi_*$ at $\xi_{1,0} = (\theta, \lambda_\nfk) \in \bU_{\eK_{\nfk,\ell},(0)}$:

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
Given nonzero $x \in \frac{1}{\nfk(\theta)}A$
with $|x|_\infty <1$ and $y = q^c/(q^\ell-1)$
for an integer $c$ with $0\leq c <\ell$, we have that

\[
\psi_{(x,y)}(\xi_{1,0})
= \prod_{j=0}^\infty \left(\prod_{\subfrac{a \in A}{\deg a = c+\ell j}}\Big(1+\frac{x}{a}\Big)^{1-q^\ell}\right)
\]

\[
\psi_{y}(\xi_{1,0})
\prod_{j=0}^\infty\left(
\prod_{i=0}^{c-1}(1-\frac{\theta^{q^i}}{\theta^{q^{c+\ell j}}}) \cdot \prod_{i=c}^{\ell-1}(1-\frac{\theta^{q^i}}{\theta^{q^{c+\ell+\ell j}}})\right)^{-1}.
\]

Given nonzero $x \in \frac{1}{\nfk(\theta)}A$ with $|x|_\infty <1$ and a non-negative integer $N$,
from <cit.> one has that

\[
g_x^{(N+1)}(\theta,\lambda_\nfk) = \prod_{\subfrac{a \in A_{\scaleto{+}{4pt}}}{\deg a = N}}\left(1+\frac{x}{a}\right).
\]

Thus for $0\leq c < \ell$ and $j \in \NN$,
we get

\[
(h_{x,c}^{-1})^{(\ell j)}(\xi_{1,0}) = g_x^{(c+1-\ell+\ell j)}(\theta,\lambda_\nfk)^{1-q^\ell} = \prod_{\subfrac{a \in A_{\scaleto{+}{4pt}}}{\deg a = c+\ell (j-1)}}\left(1+\frac{x}{a}\right)^{1-q^\ell}.
\]

Moreover, from the definition of $h_{\ari,c}$ it is straightforward that

\[
(h_{\ari,c}^{-1})^{(\ell j)}(\xi_{1,0})
= \left(\prod_{i=0}^{c-1}(1-\frac{\theta^{q^i}}{\theta^{c-\ell+\ell j}}) \prod_{i=c}^{\ell-1}(1-\frac{\theta^{q^i}}{\theta^{q^{c+\ell j}}})\right)^{-1}.
\]

Therefore the result holds.

Using the notation in Proposition <ref>,
we get in particular that

\[
\psi_x(\xi_{1,0}) = \prod_{c=0}^{\ell-1}\psi_{(x,\frac{q^c}{q^\ell-1})}(\xi_{1,0})
= \prod_{i=0}^\infty \prod_{\subfrac{a \in A_{\scaleto{+}{4pt}}}{\deg a = i}}\left(1+\frac{x}{a}\right)^{1-q^\ell} = \prod_{a \in A_{\scaleto{+}{4pt}}}\left(1+\frac{x}{a}\right)^{1-q^\ell} .
\]

§.§ Algebraic relations among special gamma values

Let $\ZZ_p$ be the ring of $p$-adic integers (i.e. the completion of $\ZZ_{(p)}$).
Put $-A_+\assign \{-a\mid a \in A_+\}$ and

\[
A_{+,i}\assign \{a \in A_+\mid \deg a = i\} \quad \text{for each non-negative integer $i$.}
\]

Recall the following factorial functions (see [19] and [39]): for $x \in \CC_\infty \setminus -A_+$ and $y = \sum_{i=0}^\infty y_i q^i \in \ZZ_p$ with $0\leq y_i<q$ for each $i \in \ZZ_{\geq 0}$,

\begin{align*}
\Pi_{\text{ari}}(y) & \assign
\prod_{i=1}^\infty\left(\prod_{j=0}^{i-1}(1-\frac{\theta^{q^j}}{\theta^{q^{i}}})
\right)^{y_i}, &
\Pi_{\text{geo}}(x)
& \assign
\prod_{a \in A_{\scaleto{+}{4pt}}}\left(1+\frac{x}{a}\right)^{-1}, \\
\Pi_{\geo}(x,y)
\prod_{i=0}^{\infty}
\left(
\prod_{a \in A_{{\scaleto{+}{4pt}},i}} \Big(1+\frac{x}{a}\Big)^{-y_i}\right),  &
\Pi(x,y)
&\assign \frac{\Pi_{\geo}(x,y)}{\Pi_{\text{ari}}(y)}.
\end{align*}

In particular, for $x \in \CC_\infty \setminus -A_+$ one has that

\[
\Pi_{\geo}(x,\frac{1}{1-q}) = \Pi_{\text{geo}}(x)
\quad \text{and}\quad
\Pi(x,\frac{1}{1-q}) = \frac{\Pi_{\text{geo}}(x)}{\Pi_{\text{ari}}(1/(1-q))}.
\]

The corresponding gamma functions are defined as follows:
$x \in \CC_\infty \setminus (-A_+ \cup \{0\})$ and $y = \sum_{i=0}^\infty y_i q^i \in \ZZ_p$ with $0\leq y_i<q$ for $i \in \ZZ_{\geq 0}$,

\begin{align*}
\Gamma_{\text{ari}}(y) &\assign
\Pi_{\text{ari}}(y-1),
\Gamma_{\text{geo}}(x)
&\assign \frac{1}{x} \cdot \Pi_{\text{geo}}(x), \\
\Gamma_{\geo}(x,y)
\frac{1}{x}\cdot \Pi_{\geo}(x,y-1), &
\Gamma(x,y)
& \assign
\frac{1}{x} \cdot \Pi(x,y-1) = \frac{\Gamma_{\geo}(x,y)}{\Gamma_{\text{ari}}(y)}.
\end{align*}

In particular, for $x \in \CC_\infty \setminus (-A_+ \cup \{0\})$ one has that

\[
\Gamma_{\geo}(x,1-\frac{1}{q-1}) = \Gamma_{\text{geo}}(x)
\quad \text{and}\quad
\Gamma(x,1-\frac{1}{q-1}) = \Gamma_{\text{geo}}(x)\Big/ \Gamma_{\text{ari}}(1-\frac{1}{q-1}).
\]

The original Goss gamma function is equal to (see <cit.>):

\[
\Gamma_{\text{Goss}}(x,y)\assign \Pi_{\geo}(-x,-y)\cdot \Gamma_{\ari}(y).
\]

From the reflection formula (cf. <cit.>):

\[
\Gamma_{\ari}(y) \Gamma_{\ari}(1-y) =\prod_{i=1}^{\infty} \biggl(
1-\frac{\theta}{\theta^{q^{i}}} \biggr)^{-1}
= (-\theta)^{\frac{-q}{q-1}}\cdot \tilde{\pi},
\]

we get that

\[
\Gamma_{\text{Goss}}(x,y) = (-\theta)^{\frac{-q}{q-1}}\cdot \tilde{\pi} \cdot  \Pi(-x,-y)
= (-\theta)^{\frac{-q}{q-1}}\cdot \tilde{\pi} \cdot (-x)\cdot \Gamma(-x,1-y).
\]

We list the monomial relations among gamma values in the following (see [39] for arithmetic gamma values and [19] for two-variable gamma values):

For $x \in \CC_\infty\setminus (-A_+\cup\{0\})$ and $y \in \ZZ_p$,

\[
\Gamma_{\ari}(y)\Gamma_{\ari}(1-y) \sim \tilde{\pi}
\quad \text{and}\quad
\Gamma(x,y)\cdot\Gamma(x,1-y) \sim \frac{\Gamma_{\geo}(x)^{q-1}}{\tilde{\pi}}.
\]

For $x \in k\setminus (-A_+\cup\{0\})$, $y \in \ZZ_p$, $a \in A \setminus (-A_+\cup \{0\})$ and $N \in \NN$,

\[
\Gamma_{\geo}(a,y) = \Gamma(a,y)\cdot \Gamma_{\ari}(y) \sim
\Gamma_{\ari}(N) \sim
\Gamma(x,N) \sim 1.
\]

(3) Given $x \in k \setminus A$, $y \in \ZZ_{(p)}\setminus \ZZ$, $a \in A \setminus (-A_+\cup \{0\})$ and $N \in \NN$,

\[
\Gamma(x+a,y+N)\sim \Gamma(x,y)
\quad\text{ and }\quad
\Gamma_{\ari}(y+N) \sim \Gamma_{\ari}(y).
\]

Given $x \in k\setminus (-A_+\cup\{0\})$ and $y \in \frac{1}{q^\ell-1}\ZZ \setminus \ZZ$,
write $\langle -y\rangle_{\ari} = \sum_{i=0}^{\ell-1} y_i q^i/(q^\ell-1)$ where $y_0,...,y_{\ell-1}$ are integers with $0\leq y_0,...,y_{\ell-1}<q$.

\[
\Gamma(x,y) \sim \prod_{i=0}^{\ell-1} \Gamma(x,1-\frac{q^i}{q^\ell-1})^{y_i}
\quad \text{ and } \quad
\Gamma_{\ari}(y) \sim \prod_{i=0}^{\ell-1} \Gamma_{\ari}(1-\frac{q^i}{q^\ell-1})^{y_i}.
\]

Given $x \in k\setminus A$ and $y \in \frac{1}{q^\ell-1}\ZZ \setminus \ZZ$,

\[
\prod_{c=0}^{\ell-1}\prod_{\epsilon \in \FF_q^\times}\Gamma(\epsilon x,q^c y) \sim 1
\quad \text{ and } \quad
\prod_{c=0}^{\ell-1}\Gamma_{\ari}(q^c y) \sim \tilde{\pi}^{\ell \text{\rm wt}(y)/(q-1)}.
\]

For $x \in k\setminus A$, $y\in \ZZ_p$, and
$\nfk \in A_+$,

\[
\prod_{\subfrac{a \in A}{\deg a < \deg \nfk}}\Gamma(\frac{x+a}{\nfk},y)
\sim \Gamma(x,|\nfk|_\infty y).
\]

\[\Gamma_{\geo}(x) = \Gamma(x,1-\frac{1}{q-1})\cdot \Gamma_{\ari}(1-\frac{1}{q-1}), \quad \forall x \in \CC_\infty \setminus (-A_+\cup \{0\}),
\]

we have the following relations for geometric gamma values (see also [3]):

\[
\Gamma_{\geo}(a) \sim 1, \quad \forall a \in A\setminus (-A_+\cup \{0\})
\quad \text{ and } \quad
\Gamma_{\geo}(x+a) \sim \Gamma_{\geo}(x);
\]

Given $x \in k\setminus A$ and $\nfk \in A_+$,

\[
\prod_{\epsilon \in \FF_q^\times} \Gamma_{\geo}(\epsilon x) \sim \tilde{\pi}
\quad \text{ and } \quad
\prod_{\subfrac{a \in A}{\deg a < \deg \nfk}}
\Gamma_{\geo}(\frac{x+a}{\nfk}) \sim \Gamma_{\geo}(x)\cdot \tilde{\pi}^{\frac{|\nfk|_\infty-1}{q-1}}.
\]

For each $x \in k_\infty$, put $\{x\} \in k_\infty$ to be the fractional part of $x$, i.e. $|\{x\}|_\infty <1$ and $x \equiv \{x\} \bmod A$.
We may view $\{\cdot \}$ as a function on $k_\infty/A$.
Let $\nfk \in \eA_+$ and $\ell \in \NN$.
The formulas for the evaluations of $\psi_{x},\psi_{\ari,c}, \psi_{x,c} \in \OO_{\nfk,\ell,(0)}^\dagger$ at $\xi_{1,0}$
in Proposition <ref>
for every $x \in \frac{1}{\nfk(\theta)}A/A$ and $0\leq c < \ell$
give us the following key identities:

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
Given $x \in \frac{1}{\nfk(\theta)}A/A$ and $y \in \frac{1}{q^\ell-1}\ZZ/\ZZ$,
we have that

\[
\psi_{x}(\xi_{1,0}) = \Pi_{\geo}(\{x\})^{q^\ell-1},
\quad
\psi_{y}(\xi_{1,0})
=\Pi_{\text{\rm ari}}(-\langle y\rangle_{\ari})^{q^\ell-1},
\]

\[
\psi_{(x,y)}(\xi_{1,0}) = \Pi_{\geo}(\{x\},-\langle y\rangle_{\ari})^{q^\ell-1},
\quad
\left(\frac{\psi_{(x,y)}(\xi_{1,0})}{ \psi_{y}(\xi_{1,0})}\right)
= \Pi(\{x\},-\langle y\rangle_{\ari})^{q^\ell-1}.
\]

Finally, define $\tilde{\Gamma}_{\ari}:\ZZ_{(p)}/\ZZ \rightarrow \CC_\infty^\times$,
$\tilde{\Gamma}_{\geo}:k/A\rightarrow \CC_\infty^\times$,
$\tilde{\Gamma}: (k/A)\times (\ZZ_{(p)}/\ZZ)\rightarrow \CC_\infty^\times$ as follows:
for $x \in k/A$ and $y \in \ZZ_{(p)}/\ZZ$,

\[
\tilde{\Gamma}_{\ari}(y) \assign \Gamma_{\ari}(1-\langle -y \rangle_{\ari}) ,
\quad
\tilde{\Gamma}(x,y) \assign
\begin{cases}
\Gamma(\{x\},1-\langle -y\rangle_{\ari})  , & \text{ if $x \neq 0 \in k/A$;} \\
\tilde{\Gamma}_{\ari}(y)^{-1}, & \text{ otherwise,}
\end{cases}
\]

\[
\tilde{\Gamma}_{\geo}(x) \assign
\tilde{\Gamma}(\{x\},\frac{1}{1-q}) = \Gamma_{\ari}(1- \frac{1}{q-1})^{-1}\cdot
\begin{cases}
\Gamma_{\geo}(\{x\}), &\text{ if $x \neq 0 \in k/A$;}\\
1,&\text{ otherwise.}
\end{cases}
\]

We may extend $\tilde{\Gamma}_{\ari}$, $\tilde{\Gamma}_{\geo}$, and $\tilde{\Gamma}$
to $\QQ$-linear homomorphisms from $\Acal^{\ari}_\QQ$, $\Acal^{\geo}_\QQ$, and $\Acal^{\cyc}_\QQ$ to $\CC_\infty^\times$, respectively, and their compositions with the quotient map $\CC_\infty^\times \rightarrow \CC_\infty^\times/\bar{k}^\times$ are denoted by $\hat{\Gamma}_{\ari}$, $\hat{\Gamma}_{\geo}$, and $\hat{\Gamma}$, respectively.
The above proposition
leads us to the following analogue of Deligne's theorem on gamma distributions (stated in <cit.>):

\[
\hat{\Gamma}_{\text{\rm ari}}
= \Pscr^{\cyc}_{\nu_1} \circ \St^{\ari},
\quad
\hat{\Gamma}_{\text{\rm geo}}
= \Pscr^{\cyc}_{\nu_1} \circ \St^{\geo},
\quad
\text{ and } \quad
\hat{\Gamma}
= \Pscr^{\cyc}_{\nu_1} \circ \text{\rm St}.
\]

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
By Lemma <ref>, we have that for $x \in \frac{1}{\nfk(\theta)}A/A$ and $y \in \frac{1}{q^\ell-1}\ZZ/\ZZ$,

\begin{align*}
\Pscr_{\nu_1}^{\cyc}\circ \St^{\ari}(y)^{q^\ell-1}
& = \Pcal_{\eK_{\nfk,\ell}}(\xi_{1,0},\Xi_{\ari,-y}) = \psi_{\ari,-y}(\xi_{1,0}) \cdot \bar{k}^\times \quad \in \CC_\infty^\times/\bar{k}^\times, \\
\Pscr_{\nu_1}^{\cyc}\circ \St^{\geo}(x)^{q^\ell-1}
& = \Pcal_{\eK_{\nfk,\ell}}(\xi_{1,0},\Phi_x) = \left(\frac{\psi_{\geo,x}(\xi_{1,0})}{\psi_{\ari,1/(q-1)}(\xi_{1,0})}\right) \cdot \bar{k}^\times \quad \in \CC_\infty^\times/\bar{k}^\times, \\
\Pscr_{\nu_1}^{\cyc}\circ \St(x,y)^{q^\ell-1}
& = \Pcal_{\eK_{\nfk,\ell}}(\xi_{1,0},\Phi_{(x,y)}) = \left(\frac{\psi_{x,-y}(\xi_{1,0})}{\psi_{\ari,-y}(\xi_{1,0})}\right) \cdot \bar{k}^\times \quad \in \CC_\infty^\times/\bar{k}^\times.
\end{align*}

Comparing with the definition of $\tilde{\Gamma}_{\ari}$, $\tilde{\Gamma}_{\geo}$, and $\tilde{\Gamma}$,
the result thereby follows from Proposition <ref>.

For each $\varphi \in \Sscr(\eG^{\cyc})$, we let $\wp_{\nu_1}^{\cyc}(\varphi) \in \CC_\infty^\times$ be an arbitrary representative of $\Pscr_{\nu_1}^{\cyc}(\varphi)$.
Then Theorem <ref> says that for $\boy \in \Acal_\QQ^{\ari}$, $\boxx \in \Acal_\QQ^{\geo}$, and $\ba \in \Acal_\QQ^{\cyc}$,

\[
\tilde{\Gamma}_{\ari}(\boy) \sim \wp_{\nu_1}^{\cyc}\big(\St^{\ari}(\boy)\big),\quad
\tilde{\Gamma}_{\geo}(\boxx) \sim \wp_{\nu_1}^{\cyc}\big(\St^{\geo}(\boxx)\big),
\quad\text{and}\quad
\tilde{\Gamma}(\ba)  \sim \wp_{\nu_1}^{\cyc}\big(\St(\ba)\big).
\]

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
We have:

\begin{align*}
& \trdeg_{\bar{k}} \bar{k}\Big(\Gamma_{\geo}(x),\Gamma_{\ari}(y),\ \Gamma(x,y)\ \Big|\ x \in \frac{1}{\nfk(\theta)}A \setminus (-A_+ \cup \{0\}),\ y \in \frac{1}{q^\ell-1}\ZZ\Big) \\
1+ (\ell-\frac{1}{(q-1)^{\epsilon_\nfk}}) \cdot \#(A/\nfk)^\times.
\end{align*}

where $\epsilon_\nfk = 1$ if $\deg \nfk >0$ and $0$ otherwise.

Notice that for $y \in \ZZ_{(p)}$ and $0\neq x \in k/A$,

\[
\tilde{\Gamma}(0,y) = \Gamma_{\ari}(1-\langle -y\rangle_{\ari})^{-1}
\quad \text{ and } \quad
\tilde{\Gamma}(x,\frac{1}{1-q})
=\Gamma_{\geo}(\{x\}) \cdot \Gamma_{\ari}(1-\frac{1}{q-1})^{-1}.
\]

Thus by Proposition <ref> (1)–(3) we get

\begin{align*}
& \ \bar{k}\Big(\Gamma_{\geo}(x),\Gamma_{\ari}(y), \Gamma(x,y)\ \Big|\ x \in \frac{1}{\nfk(\theta)}A \setminus (-A_+ \cup \{0\}),\ y \in \frac{1}{q^\ell-1}\ZZ\Big) \\
\bar{k}\Big(\tilde{\Gamma}(x,y)\ \Big|\ x \in \frac{1}{\nfk(\theta)}A/A,\ y \in \frac{1}{q^\ell-1}\ZZ/\ZZ\Big).
\end{align*}

From Proposition <ref> (3), Theorem <ref>, and Remark <ref>, we know that the field
$\bar{k}\big(\wp_{\nu_1}^{\cyc}(\varphi)\mid \varphi \in \Sscr(\eG_{\nfk,\ell})\big)$
is algebraic over $\bar{k}\Big(\tilde{\Gamma}(x,y)\ \Big|\ x \in \frac{1}{\nfk(\theta)}A/A,\ y \in \frac{1}{q^\ell-1}\ZZ/\ZZ\Big)$.

\begin{align*}
& \trdeg_{\bar{k}}\bar{k}\Big(\tilde{\Gamma}(x,y)\ \Big|\ x \in \frac{1}{\nfk(\theta)}A/A,\ y \in \frac{1}{q^\ell-1}\ZZ/\ZZ\Big) \\
= & \trdeg_{\bar{k}} \bar{k}\big(\wp_{\nu_1}^{\cyc}(\varphi)\mid \varphi \in \Sscr(\eG_{\nfk,\ell})\big) \\
=&1+ (\ell-\frac{1}{(q-1)^{\epsilon_\nfk}}) \cdot \#(A/\nfk)^\times \quad \text{(by Theorem~\ref{thm: trdeg})}.
\end{align*}

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
By (<ref>) we have

\[
1+(\ell-\frac{1}{(q-1)^{\epsilon_\nfk}}) \cdot \#(A/\nfk)^\times
\dim_\QQ(\Ucal^{\cyc}_{\nfk,\ell})
\]

where $\Ucal^{\cyc}_{\nfk,\ell}$ is defined in Section <ref> III.
As $\nfk$ and $\ell$ are chosen arbitrarily, we arrive at:

All algebraic relations among gamma values $\Gamma_{\geo}(x)$, $\Gamma_{\ari}(y)$, and
$\Gamma(x,y)$ for $x \in \ek\setminus (-A_+\cup \{0\})$ and $y \in \ZZ_{(p)}$ are explained by the monomial relations listed in Proposition <ref>.

(1) Taking $\ell = 1$ in the above equalities, we re-prove the Lang-Rohrlich conjecture for geometric gamma values in <cit.>:

\[
\trdeg_{\bar{k}} \bar{k}\Big(\tilde{\pi},\Gamma_{\geo}(x)\ \Big|\ x \in \frac{1}{\nfk(\theta)}A \setminus (-A_+ \cup \{0\})\Big)
= 1+(1-\frac{1}{(q-1)^{\epsilon_\nfk}}) \cdot (\eA/\nfk)^\times,
\]

where $\epsilon_\nfk = 1$ if $\deg \nfk >0$ and $0$ otherwise.

(2) Taking $\nfk =1$ in the above equalities, we re-prove <cit.>:

\[
\trdeg_{\bar{k}} \bar{k}\Big(\Gamma_{\ari}(y)\ \Big|\ y \in \frac{1}{q^\ell-1}\ZZ\Big) = \ell.
\]

\[
\tilde{\Gamma}_{\geo}(x) \sim \tilde{\Gamma}(x,\frac{1}{1-q})
= \prod_{i=0}^{\ell-1}\tilde{\Gamma}(x,\frac{q^i}{1-q^\ell}),
\]

We get that

\begin{align*}
&\ \trdeg_{\bar{k}} \bar{k}\Big(\Gamma_{\geo}(x),\Gamma_{\ari}(y)\ \Big|\ x \in \frac{1}{\nfk(\theta)}A \setminus (-A_+ \cup \{0\}),\ \ y \in \frac{1}{q^\ell-1}\ZZ\Big) \\
\ell + (1-\frac{1}{(q-1)^{\epsilon_\nfk}})\cdot (\eA/\nfk)^\times,
\end{align*}

and so all algebraic relations among arithmetic and geometric gamma values come from their relations with the Carlitz period $\tilde{\pi}$.

§ THE CHOWLA–SELBERG PHENOMENON

Theorem <ref> enables us to connect the “CM periods”, when the CM field is contained in a cyclotomic function field, with special gamma values.
In this section, we shall first recall the relationship between the periods of dual $t$-motives and the periods of “abelian $t$-modules”, and then apply Theorem <ref> to derive an analogue of the Chowla–Selberg formula.

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

An abelian $t$-module of dimension $d>0$ over $\bar{k}$ is a pair $(E,\rho)\rassign E_\rho$ with the following properties.

* $E$ is a $d$-dimensional additive group ${\GG_{a}^{d}}_{/\bar{k}}$ defined over $\ok$.

* $\rho:\eA\rightarrow \End_{\FF_{q}} ({\GG_{a}^{d}}_{/\ok})$ is an $\FF_{q}$-algebra homomorphism so that $ \partial\rho_{t}-\theta I_{d}$ acts nilpotently on $\Lie (\GG_{a}^{d})$, where $\partial \rho_{t}$ denotes the  differential of the homomorphism $\rho_{t}:{\GG_{a}^{d}}_{/\ok}\rightarrow {\GG_{a}^{d}}_{/\ok}$.

* $\Mscr(\rho) \assign \Hom_{\FF_q}(E,\GG_{a})$ is finitely generated over $\ok[t]$ with respect to the $\eA$-module structure via $\rho$.

It is known that $\Mscr(\rho)$ must be free of finite rank over $\ok[t]$, and $\rank_{\ok[t]}\Mscr(\rho)$ is called the rank of $E_\rho$.

\begin{equation}\label{eqn: M(rho)}
\eM(\rho)\assign \Hom_{\bar{k}[t]}(\tau\Mscr(\rho),\bar{k}[t]dt)
\end{equation}

which is constructed by Hartl-Juschka [24],
where $\tau$ is identified with the  $q$-power Frobenius endomorphism on $\GG_a$.
It is known that $\eM(\rho)$ is a dual $t$-motive (and $E_\rho$ is always “$A$-finite”, see <cit.>).
We call $\eM(\rho)$ the Hartl-Juschka dual $t$-motive associated with $E_\rho$.

Suppose that $E_\rho$ is uniformizable,
in which case so is $\eM(\rho)$, see <cit.>.
Let $\Lambda_\rho$ be the period lattice of $E_\rho$ and let $H_{\mathrm{dR}}(E_\rho,\bar{k})$ be the de Rham module of $E_\rho$ (see <cit.> and <cit.>).
Let $[\cdot,\cdot]:H_{\mathrm{dR}}(E_\rho,\bar{k}) \times \Lambda_\rho \rightarrow \CC_\infty$ be the de Rham pairing introduced in <cit.>.
We call $[\delta,\lambda]$ for $\delta \in H_{\mathrm{dR}}(E_\rho,\bar{k})$
and $\lambda \in \Lambda_\rho$ a quasi-period of $E_\rho$.
Recall that we have the following natural comparison between the associated de Rham pairings
(cf. <cit.>):

\begin{equation}\label{eqn: dR-P comp}
\SelectTips{cm}{}
\xymatrixrowsep{0.4cm}
\xymatrixcolsep{0.05cm}
\xymatrix{
[\ ,\ ]: & H_{\mathrm{dR}}(E_\rho,\bar{k})\ar@{<->}[dd]^{\rotatebox{90}{\scalebox{1}[1]{$\sim$}}} & \times & \Lambda_\rho\ar@{<->}[dd]^{\rotatebox{90}{\scalebox{1}[1]{$\sim$}}}  \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\\
\end{equation}

We say that $E_\rho$ is a CM abelian $t$-module with generalized CM type $(\eK,\Xi)$ if $\eM(\rho)$ is a CM dual $t$-motive with generalized CM type $(\eK,\Xi)$.
In this case, the above comparison of de Rham pairings implies that the space of quasi-periods of $E_\rho$ coincides with (cf. <cit.>)

\[
\sum_{\xi \in J_\eK}\bar{k}\cdot p_\eK(\xi,\Xi).
\]

In particular, suppose $\eK \subset \eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.
Let $\varphi_{\eK,\Xi} \in \Sscr(\eG^{\cyc})$ be the Stickelberger function associated to the CM type $(\eK,\Xi)$ given in (<ref>).
Then Lemma <ref> guarantees that there exists $\ba \in \Acal^{\cyc}_{\nfk,\ell,\QQ}$ such that

\[
\varphi_{\eK,\Xi} = \St(\ba) \quad \in \Sscr(\eG^{\cyc}).
\]

In particular, consider $\xi_0 \in J_\eK$ whose corresponding embedding $\nu_{\xi_0}$ is $\nu_1\big|_{\eK}$ (where $\nu_1:\eK^{\cyc}\hookrightarrow \CC$ is chosen in (<ref>)).
As $\eK$ is Galois over $\ek$, one has
$J_\eK = \{\xi_0^{\varrho}\mid \varrho \in \eG^{\cyc}/\eH_\eK\}$.
By Theorem <ref> (2) the space of quasi-periods of $E_\rho$ becomes

\begin{equation}\label{eqn: CM-qp}
\sum_{\varrho \in \eG_\eK} \ok \cdot p_\eK(\xi_0^{\varrho},\Xi) =
\sum_{\varrho \in \eG_\eK} \ok \cdot p_\eK(\xi_0,\Xi^{\varrho^{-1}}) = \sum_{\varrho \in \eG_\eK} \ok \cdot \wp_{\nu_1}^{\cyc}(\varphi_{\eK,\Xi^{\varrho^{-1}}}).
\end{equation}

Since $\varphi_{\eK,\Xi^{\varrho^{-1}}} = \varrho \cdot \varphi_{\eK,\Xi} = \St(\varrho \star \ba)$ for every $\varrho \in \eG^{\cyc}$, by Theorem <ref> (and Remark <ref>)
we obtain

Let $E_\rho$ be a CM abelian $t$-module with generalized CM type $(\eK,\Xi)$ over $\bar{k}$, where $\eK$ is contained in $\eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.
Let $\eH_\eK = \gal(\eG^{\cyc}/\eK)$.
There exists $\ba \in \Acal^{\cyc}_{\nfk,\ell,\QQ}$ such that the space of quasi-periods of $E_\rho$ is spanned over $\bar{k}$ by

\[
\tilde{\Gamma}(\varrho \star \ba) \ (\sim \wp_{\nu_1}^{\cyc}(\varrho \cdot \varphi_{\eK,\Xi})), \quad  \varrho \eH_\eK \in \eG^{\cyc}/\eH_\eK.
\]

§.§ Period vectors of CM Hilbert–Blumenthal t-modules

Let $\eK$ be a CM field with maximal totally real subfield denoted by $\eK^+$, and let
$\Xi$ be a CM type of $\eK$.
Write $\Xi = \xi_1+\cdots+ \xi_d$, where $d = [\eK^+:\ek]$.
Let $E_\rho$ be a CM abelian $t$-module with CM type $(\eK,\Xi)$ over $\bar{k}$.
By <cit.>, $E_\rho$ is actually a Hilbert-Blumenthal $O_{\eK^+}$-module with CM by $O_{\eK}$.
Put $\eM = \eM(\rho)$.
Let $\omega_{\eM,\xi_1},...,\omega_{\eM,\xi_d}$ be the differentials of $\eM$ associated with $\xi_1,...,\xi_d$, respectively.
From the natural $\bar{k}$-linear isomorphism $\Lie(E_\rho) \cong \eM/\sigma \eM$ (see <cit.>),
we get a $\bar{k}$-linear isomorphism $\Lie(E_\rho) \cong \bar{k}^d$ with respect to $\omega_{\eM,\xi_1},...,\omega_{\eM,\xi_d}$.
Moreover, the image of every period vector $\lambda \in \Lambda_\rho \subset \Lie(E_\rho)(\CC_\infty)$ in $\CC_\infty^d$ under this isomorphism is

\[
(\int_{\gamma_\lambda} \omega_{\eM,\xi_1},...,\int_{\gamma_\lambda}\omega_{\eM,\xi_1}) \quad \in \CC_\infty^d,
\]

where $\gamma_\lambda \in H_{\rm Betti}(\eM)$ is the cycle corresponding to $\lambda$ via the natural isomorphism in the comparison diagram (<ref>).

Suppose $\eK$ is contained in $\eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.
By Lemma <ref>, there exists $\ba \in \Acal^{\cyc}_{\nfk,\ell,\QQ}$ so that

\varphi_{\eK,\Xi} = \St(\ba) \quad \in \Sscr(\eG^{\rm cyc}).

Let $\xi_0 \in J_\eK$ with $\nu_{\xi_0} = \nu_1\big|_{\eK}$.
For $1\leq i \leq d$, take $\varrho_i \in \eG^{\cyc}$ so that $\nu_1 \circ \varrho_i \big|_\eK = \nu_{\xi_i}$.
Write $\Lambda_\rho = \Ifk_{\rho}\cdot \lambda_0$ where $\Ifk_{\rho}$ is an ideal of $O_\eK$.
By Theorem <ref> and <ref> we have that

\begin{equation}\label{eqn: HB-period}
\int_{\gamma_{\lambda_0}} \omega_{\eM,\xi_i}
\sim p_\eK(\xi_i, \Xi)
\sim p_{\eK}(\xi_0,\Xi^{\varrho_i^{-1}})
\sim \wp^{\cyc}_{\nu_1}\big(\varphi_{\eK,\Xi^{\varrho_i^{-1}}}\big)
\sim \wp^{\cyc}_{\nu_1}\big(\varrho_i \cdot \varphi_{\eK,\Xi}\big)
\sim \tilde{\Gamma}(\varrho_i \star \ba).
\end{equation}

$\eK$ be a CM field whose maximal totally real subfield is denoted by $\eK^+$.
Suppose $\eK \subset \eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.
Put $d = [\eK^+:\ek]$.
Let $\Xi = \xi_1+\cdots+\xi_d$ be a CM type of $\eK$.
For $1\leq i \leq d$, take $\varrho_i \in \eG^{\cyc}$ so that $\nu_{1}\circ \varrho_i\big|_\eK = \nu_{\xi_i}$.
Let $E_\rho$ be a CM abelian $t$-module with CM type $(\eK,\Xi)$ over $\bar{k}$.
There exist $\mathbf{a} \in \Acal^{\cyc}_{\nfk,\ell,\QQ}$, an ideal $\Ifk_\rho$ of $O_\eK$, and a suitable $\bar{k}$-linear isomorphism $\Lie(E_\rho) \cong \bar{k}^d$ so that
image of the period lattice $\Lambda_\rho \subset \Lie(E_\rho)(\CC_\infty)$ in $\CC_\infty^d$ is

\left\{
\Big(\nu_{\xi_1}(a)\tilde{\Gamma}(\varrho_1\star \ba),...,\nu_{\xi_d}(a)\tilde{\Gamma}(\varrho_d \star \ba)\Big)\ \bigg|\ a \in \Ifk_\rho
\right\}.

Now, let $\eK$ be an imaginary field over $\ek$ (i.e. the infinite place $\infty$ of $\ek$ does not split in $\eK$).
Let $E_\rho$ be a Drinfeld $\eA$-module of rank $[\eK:\ek]$ over $\ok$ which has CM by $O_\eK$ (the integral closure of $\eA$ in $\eK$).
Extending $\rho$ to an $\FF_q$-algebra homomorphism (still denoted by)
$\rho: O_\eK \rightarrow \End_{\FF_q}(E_\rho)$, we have that the differential $\partial \rho : O_\eK \rightarrow \End_{\ok}(\Lie(E_\rho))$ induces an embedding $\nu_\rho: \eK \hookrightarrow \ok\subset \CC_\infty$.
Let $\xi_\rho \in J_\eK$ be the point corresponding to the embedding $\nu_\rho$.
We may regard $E_\rho$ as a CM abelian $t$-module with CM type $(\eK,\xi_\rho)$.

Suppose that $\eK\subset \eK^{\cyc}$ (which is equivalent to saying that $\eK/\ek$ is an abelian extension and the ramification index of $\infty$ in $\eK$ is divided by $q-1$ from class field theory).
By Theorem <ref>,
We are able to establish the following result which in particular verifies Thakur's recipe/conjecture on the Chowla–Selberg phenomenon in

Let $E_\rho$ be a Drinfeld $\eA$-module over $\ok$ with CM by $O_\eK$, where $O_\eK$ is the integral closure of $\eA$ in an imaginary function field $\eK$ with $\eK\subset \eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.
Let $\eH_\eK = \gal(\eK^{\cyc}/\eK)$.
Take any $\ba \in \Acal^{\cyc}_{\nfk,\ell,\QQ}$ so that $\mathbf{1}_{\eH_\eK} = \St(\ba) \in \Sscr(\eG^{\cyc})$.

\[
\lambda \sim \wp^{\cyc}_{\nu_1}(\mathbf{1}_{\eH_\eK}) \sim \tilde{\Gamma}(\ba), \quad \text{ for every nonzero period } \lambda \in \Lambda_\rho \subset \CC_\infty.
\]

\[\tilde{\Gamma}(\varrho\star \ba) \ (\sim \wp^{\cyc}_{\nu_1}(\varrho \cdot \mathbf{1}_{\eH_\eK})), \quad \varrho \eH_\eK \in \eG^{\cyc}/\eH_\eK
\]

are algebraically independent over $\bar{k}$ and
form a $\bar{k}$-basis of the space of quasi-periods of the CM Drinfeld module $E_\rho$.

As $\eK$ is imaginary,
we have that $\eK^+=\ek$ and the characteristic function $\mathbf{1}_{\eH_\eK}$ of $\eH_\eK$ is contained in $\Sscr(\eG_{\nfk,\ell})$. Hence
Lemma <ref> assures us that there exist $\ba \in \Acal^{\cyc}_{\nfk,\ell,\QQ}$ so that

\[
\mathbf{1}_{\eH_\eK} = \St(\ba).
\]

Let $\xi_0 \in J_\eK$ such that $\nu_{\xi_0} = \nu_1\big|_{\eK}$.
Then Theorem <ref> implies that

\[
\lambda \sim  p_\eK(\xi_0,\xi_0)\sim \tilde{\Gamma}(\ba), \quad \text{ for every nonzero period } \lambda \in \Lambda_\rho \subset \CC_\infty.
\]

Moreover, Theorem <ref> shows that the space of quasi-periods of $E_\rho$ is spanned over $\ok$ by

\[
\tilde{\Gamma}(\varrho\star \ba), \quad \varrho \eH_\eK \in \eG^{\cyc}/\eH_\eK,
\]

\begin{align*}
\trdeg_{\ok}\ok(\tilde{\Gamma}(\varrho\star \ba)\mid \varrho \eH_\eK \in \eG^{\cyc}/\eH_\eK)
&= \trdeg_{\ok}\ok\Big(\wp_{\nu_1}^{\cyc}\big(\Sscr(\eG_{\eK})\big)\Big) \\
&= [\eK:\ek] \quad \text{ (by Proposition~\ref{thm: trdeg}).}
\end{align*}

Next, we shall provide explicit constructions for the $\ba$'s in Theorem <ref>, Theorem <ref>, and Theorem <ref>,
and present a function field analogue of the Chowla–Selberg formula.

§.§ The Chowla–Selberg formula

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
Identifying $\eG_{\nfk,\ell}$ with $(\eA/\nfk)^\times \times (\ZZ/\ell \ZZ)$ via the Artin map as in (<ref>),
every $\bochi \in \widehat{\eG}_{\nfk,\ell}$ can also be regarded as a character on $(\eA/\nfk)^\times \times (\ZZ/\ell \ZZ)$.
We put $\cfk_{\bochi} = \cfk_{\bochi_f}$, where $\bochi_f = \bochi\big|_{(\eA/\nfk)^\times}$.
Recall that for $c \in \ZZ/\ell \ZZ$ and $\alpha \in (\eA/\cfk_{\bochi})^\times$,
by Lemma <ref>
we have

\begin{align}\label{eqn: generating-chi}
\left(\St(\frac{\alpha(\theta)}{c_{\bochi}(\theta)},\frac{q^c}{1-q^\ell})\ \bigg|\ \bochi\right)_{\cfk_{\bochi},\ell}
&= \sum_{(a,i) \in (\eA/\cfk_{\bochi})^\times \times (\ZZ/\ell\ZZ)}\St(\frac{\alpha(\theta)}{c_{\bochi}(\theta)},\frac{q^c}{1-q^\ell})(a,i) \cdot \overline{\bochi(a,i)} \nonumber \\
& = \bochi(\alpha,c+\deg \cfk_{\bochi}) \cdot L_\eA(0,\overline{\bochi}).
\end{align}

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
For each character $\bochi \in \widehat{\eG}_{\nfk,\ell}$, the following equality holds:

\[
\sum_{(a,i) \in (\eA/\cfk_{\bochi})^\times \times (\ZZ/\ell \ZZ)}
\overline{\bochi(a,i+\deg \cfk_{\bochi})} \cdot \St(\frac{a(\theta)}{\cfk_{\bochi}(\theta)},\frac{q^i}{1-q^\ell})
= L_\eA(0,\overline{\bochi}) \cdot \bochi.
\]

For every $(\alpha,c) \in (\eA/\nfk)^\times \times (\ZZ/\ell \ZZ)$, we get

\begin{align*}
& \sum_{(a,i) \in (\eA/\cfk_{\bochi})^\times \times (\ZZ/\ell \ZZ)}
\overline{\bochi(a,i+\deg \cfk_{\bochi})} \cdot \St(\frac{a(\theta)}{\cfk_{\bochi}(\theta)},\frac{q^i}{1-q^\ell})(\alpha,c)\\
= & \sum_{(a,i) \in (\eA/\cfk_{\bochi})^\times \times (\ZZ/\ell \ZZ)} \overline{\bochi(a,i)}  \cdot \left(\langle \frac{a (\theta)\alpha(\theta)}{\cfk_{\bochi}(\theta)},\frac{q^{i+c}}{q^\ell-1}\rangle -
\langle \frac{q^{i+c}}{q^\ell-1}\rangle_{\ari} \right) \cdot \overline{\bochi(1,\deg \cfk_{\bochi})} \\
= &
\left(\St(\frac{\alpha(\theta)}{\cfk_{\bochi}(\theta)},\frac{q^{c}}{1-q^\ell})\ \bigg|\ \bochi\right)_{\cfk_{\bochi},\ell} \cdot \overline{\bochi(1,\deg \cfk_{\bochi})}
= L_\eA(0,\overline{\bochi}) \cdot \bochi(\alpha,c) & \text{(by \eqref{eqn: generating-chi})}
\end{align*}

as desired.

Given a CM field $\eK$ over $\ek$ which is contained in $\eK_{\nfk,\ell}$, let $\eG_\eK$ be the Galois group $\gal(\eK/\ek)$.
For a generalized CM type $\Xi$ of $\eK$,
from the bijection (<ref>)
we may write

\begin{equation}\label{eqn: Xi-varrho}
\Xi = \sum_{\varrho \in \eG_\eK}m_{\varrho} \cdot \xi_\varrho,
\end{equation}

where for each $\varrho \in \eG_\eK$, $\xi_\varrho \in J_\eK$ is the point corresponding to the embedding $\nu_1\circ \varrho$.
Let $\eK^+$ be the maximal totally real subfield of $\eK$ and put $\eG_{\eK^+} = \gal(\eK^+/\ek)$.
By Remark <ref>, we may express

\[
\varphi_{\eK,\Xi} =\frac{1}{\#(\eG_{\eK})}\cdot \left(\sum_{\varrho \in \eG_\eK}m_\varrho\right) +  \frac{1}{\#(\eG_{\eK})}\cdot \sum_{\bochi \in \widehat{\eG}_{\eK}\setminus \widehat{\eG}_{\eK^+}}\left(\sum_{\varrho \in \eG_\eK}m_\varrho \overline{\bochi(\varrho)}\right) \cdot \bochi.
\]

Notice that $\sum\limits_{\varrho \in \eG_\eK} m_\varrho = [\eK^+:\ek] \cdot \wt(\Xi)$, and for $\bochi \in \widehat{\eG}_\eK \setminus \widehat{\eG}_{\eK^+}$ we have (by Remark <ref> and Lemma <ref>):

\[
\bochi = \sum_{\substack{a \in (\eA/\cfk_{\bochi})^\times \\ i \in \ZZ/\ell \ZZ}}\frac{\overline{\bochi(a,i+\deg \cfk_{\bochi})}}{L_\eA(0,\overline{\bochi})} \cdot \St(\frac{a(\theta)}{\cfk_{\bochi}(\theta)},\frac{q^i}{1-q^\ell}).
\]

\begin{align*}
\varphi_{\eK,\Xi} &=
\frac{\wt(\Xi)}{[\eK:\eK^+]}
+ \frac{1}{[\eK:\ek]}
\sum_{\bochi \in \widehat{\eG}_{\eK}\setminus \widehat{\eG}_{\eK^+}}\!\!
\left(\sum_{\varrho \in \eG_\eK}m_\varrho \overline{\bochi}(\varrho)\right)\!\! \cdot \!\!
\sum_{\substack{a \in (\eA/\cfk_{\bochi})^\times \\ i \in \ZZ/\ell \ZZ}}\frac{\overline{\bochi}(a,i+\deg \cfk_{\bochi})}{L_\eA(0,\overline{\bochi})}\cdot \St(\frac{a(\theta)}{\cfk_{\bochi}(\theta)},\frac{q^i}{1-q^\ell}) \\
\frac{\wt(\Xi)}{[\eK:\eK^+]}
+ \frac{1}{[\eK:\ek]}  \sum_{\cfk\mid \nfk}\sum_{\substack{a \in (\eA/\cfk)^\times \\ i \in \ZZ/\ell \ZZ}} \left(\sum_{\varrho \in \eG_\eK} m_\rho \!\!\sum_{\subfrac{\bochi \in \widehat{\eG}_\eK\setminus \widehat{\eG}_{K^+}}{\cfk_{\bochi} = \cfk}}\!\!
\frac{\bochi(\varrho)\bochi(a,i+\deg \cfk)}{L_\eA(0,\bochi)} \right) \cdot \St(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell}).
\end{align*}

For each $\varrho \in \eG_\eK$, put

\begin{equation}\label{eqn: defn-nc}
n_\cfk(\varrho,a,i) \assign \sum_{\subfrac{\bochi \in \widehat{\eG}_\eK\setminus \widehat{\eG}_{K^+}}{\cfk_{\bochi} = \cfk}}
\frac{\bochi(\varrho)\bochi(a,i+\deg \cfk)}{L_\eA(0,\bochi)}, \quad \text{ which lies in $\QQ$.}
\end{equation}

\[
\varphi_{\eK,\Xi} =
\frac{\wt(\Xi)}{[\eK:\eK^+]}
+ \frac{1}{[\eK:\ek]}
\sum_{\cfk \mid \nfk}\sum_{\substack{a \in (\eA/\cfk)^{\times} \\ i\in \ZZ/\ell\ZZ}}
\left(\sum_{\varrho \in \eG_\eK}m_\varrho \cdot  n_\cfk(\varrho,a,i)\right)
\cdot \St(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell}).
\]

In general, we obtain that:

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
Given a CM field $\eK$ contained in $\eK_{\nfk,\ell}$ and a generalized CM type $\Xi$ of $\eK$, write $\Xi = \sum_{\varrho \in \eG_\eK}m_\varrho \xi_\varrho$ as in (<ref>).
For each $\varrho_0 \in \eG_\eK$, we have the following expression for the Stickelberger function $\varphi_{\eK,\Xi^{\varrho_0}}$:

\[
\varphi_{\eK,\Xi^{\varrho_0}} =
\frac{\wt(\Xi)}{[\eK:\eK^+]}
+ \frac{1}{[\eK:\ek]}
\sum_{\cfk \mid \nfk}\sum_{\substack{a \in (\eA/\cfk)^\times \\ i\in \ZZ/\ell\ZZ}}
\left(\sum_{\varrho \in \eG_\eK}m_\varrho \cdot  n_\cfk(\varrho\varrho_0,a,i)\right)
\cdot \St(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell}).
\]

Suppose $\varrho_0 \in \eG_\eK$ is given.

\[
\Xi^{\varrho_0} = \sum_{\varrho \in \eG_\eK}m_\varrho \xi_{\varrho}^{\varrho_0}
=\sum_{\varrho \in \eG_\eK}m_\varrho \xi_{\varrho\varrho_0}
\sum_{\varrho \in \eG_\eK}m_{\varrho\varrho_0^{-1}}\xi_\varrho,
\]

we obtain that

\begin{align*}
\varphi_{\eK,\Xi^{\varrho_0}}
+ \frac{1}{[\eK:\ek]}
\sum_{\cfk \mid \nfk}\sum_{\substack{a \in (\eA/\cfk)^\times \\ i\in \ZZ/\ell\ZZ}}
\left(\sum_{\varrho \in \eG_\eK}m_{\varrho\varrho_0^{-1}} \cdot  n_\cfk(\varrho,a,i)\right)
\cdot \St(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})\\
\frac{\wt(\Xi)}{[\eK:\eK^+]}
+ \frac{1}{[\eK:\ek]}
\sum_{\cfk \mid \nfk}\sum_{\substack{a \in (\eA/\cfk)^\times \\ i\in \ZZ/\ell\ZZ}}
\left(\sum_{\varrho \in \eG_\eK}m_\varrho \cdot  n_\cfk(\varrho\varrho_0,a,i)\right)
\cdot \St(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell}).
\end{align*}

Let $\eK$ be a CM field and $\eK^+$ its maximal totally real subfield.
Suppose that $\eK$ is contained in $\eK_{\nfk,\ell}$ for some $\nfk \in \eA_+$ and $\ell \in \NN$.
It is known that (cf. <cit.>)

\[
\prod_{\bochi \in \widehat{\eG}_\eK \setminus \widehat{\eG}_{\eK^+}}L_{\eA}(0,\bochi) = \frac{\#\Pic(O_\eK)\cdot \Rcal(O_\eK)}{w_\eK \cdot \#\Pic(O_{\eK^+})\cdot \Rcal(O_{\eK^+})},
\]

* $\Pic(O_\eK)$ (resp. $\Pic(O_{\eK^+})$) is the ideal class group of $O_\eK$ (resp. $O_{\eK^+}$);

* $w_\eK \assign \#(\FF_\eK^\times)/\#(\FF_q^\times)$, and $\FF_\eK$ is the constant field of $\eK$;

* $\Rcal(O_\eK)$ (resp. $\Rcal(O_{\eK^+})$) is the regulator of $O_\eK$ (resp. $O_{\eK^+}$), see <cit.>.

Let $h(\eK/\eK^+)$ be the following “relative class number”:

\[
\frac{\#\Pic(O_\eK)\cdot \Rcal(O_\eK)}{ \#\Pic(O_{\eK^+})\cdot \Rcal(O_{\eK^+})} \quad \in \NN.
\]

Then for $\cfk \in \eA_+$ with $\cfk \mid \nfk$, $a \in (\eA/\cfk)^\times$, $i \in \ZZ/\ell\ZZ$, and $\varrho \in \eG_\eK$, the expression of $\nfk_{\cfk}(\varrho,a,i)$ in (<ref>) tells us in particular that

\[
h(\eK/\eK^+)\cdot \nfk_{\cfk}(\varrho,a,i) \ \in \ZZ.
\]

Proposition <ref> leads us to the following Chowla–Selberg formula for quasi-periods of CM abelian $t$-modules:

Let $E_\rho$ be a CM abelian $t$-module with generalized CM type $(\eK,\Xi)$ over $\bar{k}$, where $\eK$ is contained in $\eK_{\nfk,\ell}$ for some $\nfk \in \eA_+$ and $\ell \in \NN$.
Let $\eG_\eK = \gal(\eK/\ek)$ and write

\[
\Xi = \sum_{\varrho \in \eG_\eK}m_\varrho \xi_\varrho \ \in I_\eK^0 \quad \text{ as in \eqref{eqn: Xi-varrho}}.
\]

(1) The space of quasi-periods of $E_\rho$ is spanned over $\bar{k}$ by

\[
\tilde{\pi}^{\frac{\wt(\Xi)}{[\eK:\eK^+]}}
\cdot  \prod_{\varrho \in \eG_\eK} \left(\prod_{\cfk\mid \nfk}\prod_{\substack{a \in (\eA/\cfk)^\times \\ i \in \ZZ/\ell \ZZ}} \tilde{\Gamma}(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})^{n_{\cfk}(\varrho\varrho_0,a,i)}\right)^{\frac{m_{\varrho}}{[\eK:\ek]}}, \quad \text{ for } \varrho_0 \in \eG_\eK,
\]

where $n_\cfk(\varrho,a,i)$ is defined in (<ref>).

(2) Suppose $\Xi$ is a CM type of $\eK$.
Let $\eK^+$ be the maximal totally real subfield of $\eK$, and put $d = [\eK^+:\ek]$.
Write $\Xi = \xi_{\varrho_1}+\cdots + \xi_{\varrho_d}$, and let $\nu_{\varrho_j}\assign (\nu_1\big|_\eK) \circ \varrho_j$ for $1\leq j \leq d$.
There exist an ideal $\Ifk_\rho$ of $O_\eK$ and a suitable $\bar{k}$-linear isomorphism  $\Lie(E_\rho) \cong \bar{k}^d$ so that the
image of every period $\lambda \in \Lambda_\rho \subset \Lie(E_\rho)(\CC_\infty)$ in $\CC_\infty^d$ has the form $(\nu_{\varrho_1}(\alpha)\lambda_1,...,\nu_{\varrho_d}(\alpha)\lambda_d) \in \CC_\infty^d$,
where $\alpha \in \Ifk_\rho$ and for $1\leq j \leq d$,

\[
\lambda_j \assign \tilde{\pi}^{\frac{1}{[\eK:\eK^+]}}
\cdot  \prod_{j'=1}^d \prod_{\cfk\mid \nfk}\prod_{\substack{a \in (\eA/\cfk)^\times \\ i \in \ZZ/\ell\ZZ}} \tilde{\Gamma}(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})^{\frac{n_{\cfk}(\varrho_{j'}\varrho_{j}^{-1},a,i)}{[\eK:\ek]}}.
\]

(3) Suppose further that $E_\rho$ is a Drinfeld
$\eA$-module over $\ok$ with CM by $O_\eK$, where $O_\eK$ is the integral closure of $\eA$ in an imaginary field $\eK$.

\[
\lambda^{[\eK:\ek]} \sim
\tilde{\pi}
\cdot \prod_{\cfk\mid \nfk}\prod_{\substack{a \in (\eA/\cfk)^\times \\ i\in \ZZ/\ell\ZZ}} \tilde{\Gamma}(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})^{n_{\cfk}(a,i)}
\quad \text{ for every nonzero period } \lambda \in \Lambda_\rho \subset \CC_\infty,
\]

\[n_{\cfk}(a,i) \assign n_{\cfk}(\text{\rm id}_{\eK},a,i)
= \sum_{\subfrac{\bochi \in \widehat{\eG}_\eK\setminus \widehat{\eG}_{K^+}}{\cfk_{\bochi} = \cfk}}
\frac{\bochi(a,i+\deg \cfk)}{L_\eA(0,\bochi)}.
\]

\[
\varpi_\eK^{\varrho}\assign \tilde{\pi}^{\frac{1}{[\eK:\ek]}}
\cdot \prod_{\cfk\mid \nfk}\prod_{\substack{a \in (\eA/\cfk)^\times \\ i\in \ZZ/\ell\ZZ}} \tilde{\Gamma}(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})^{\frac{n_{\cfk}(\varrho,a,i)}{[\eK:\ek]}},
\quad \text{ for } \varrho \in \eG_\eK,
\]

are algebraically independent over $\bar{k}$ and
form a $\bar{k}$-basis of the space of quasi-periods of the CM Drinfeld module $E_\rho$.

For $\varrho_0 \in \eG_\eK$, take

\[
\ba_{\eK,\Xi^{\varrho_0}} \assign
\sum_{\cfk \mid \nfk}\sum_{\substack{a \in (\eA/\cfk)^{\times} \\ i\in \ZZ/\ell\ZZ}}
\left(\sum_{\varrho \in \eG_\eK}m_\varrho \cdot  n_\cfk(\varrho\varrho_0,a,i)\right)
\cdot [\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell}] \quad \in \Acal_{\nfk,\ell,\QQ}^{\cyc}.
\]

Proposition <ref> implies that

\begin{equation}\label{eqn: phi-ST}
\varphi_{\eK,\Xi^{\varrho_0}} = \frac{\wt(\Xi)}{[\eK:\eK^+]}\cdot \mathbf{1}_{\eG_\eK} + \frac{1}{[\eK:\ek]}\St(\ba_{\eK,\Xi^{\varrho_0}}).
\end{equation}

Theorem <ref> implies that the space of quasi-period of $E_\rho$ is spanned over $\ok$ by

\begin{align*}
\wp_{\nu_1}^{\cyc}(\varphi_{\eK,\Xi^{\varrho_0}}) & \sim \wp_{\nu_1}^{\cyc}\left(\frac{\wt(\Xi)}{[\eK:\eK^+]}\cdot \mathbf{1}_{\eG_\eK} + \frac{1}{[\eK:\ek]}\St(\ba_{\eK,\Xi^{\varrho_0}})\right) \\
& \sim \tilde{\pi}^{\frac{\wt(\Xi)}{[\eK:\eK^+]}}
\cdot  \prod_{\varrho \in \eG_\eK} \left(\prod_{\cfk\mid \nfk}\prod_{\substack{a \in (\eA/\cfk)^\times \\ i \in \ZZ/\ell \ZZ}} \tilde{\Gamma}(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})^{n_{\cfk}(\varrho\varrho_0,a,i)}\right)^{\frac{m_{\varrho}}{[\eK:\ek]}}, \quad \text{ for } \varrho_0 \in \eG_\eK.
\end{align*}

This completes the proof of (1).

Moreover, from the expression (<ref>),
(2) and (3) follow directly from Theorem <ref> and Theorem <ref>, respectively.

(Chowla–Selberg formula for constant field extensions.)
Given $\ell \in \NN$, put $\eK = \eK_{1,\ell} = \FF_{q^\ell}(t)$.

\[
\text{$\eG_\eK =\gal(\eK/\ek)\cong \gal(\FF_{q^\ell}/\FF_q) \cong \ZZ/\ell \ZZ$ \quad as in \eqref{eqn: ari-Artin},}
\]

and $\cfk_{\bochi} = 1$ for every $\bochi \in \widehat{\eG}_\eK$.
Put $\zeta_\ell \assign \exp(2\pi\sqrt{-1}/\ell) \in \CC^\times$.
Recall that we write $\bochi(1,1) = q^{-s_{\bochi}} = \zeta_\ell^{d_{\bochi}}$ for each $\bochi \in \widehat{\eG}_\eK$.
For $i \in \ZZ/\ell \ZZ$, we calculate

\[
n_1(1,i) = \sum_{1\neq \bochi \in \widehat{\eG}_\eK}\frac{\bochi(1,i)}{L_{\eA}(0,\bochi)}
= \sum_{d=1}^{\ell-1} \zeta_\ell^{id}(1-q\zeta_\ell^d)
= q-1+ \begin{cases}
\ell, &\text{ if $i = 0$,}\\
-\ell q, &\text{ if $i=\ell-1$,}\\
0, &\text{ otherwise.}
\end{cases}
\]

Let $E_\rho$ be a Drinfeld $\eA$-module of rank $\ell$ over $\ok$ with CM by $\FF_{q^\ell}[t]$.
Theorem <ref> (3) implies that
for every nonzero period $\lambda \in \Lambda_\rho$, we have that

\begin{align*}
\lambda & \sim \tilde{\pi}^{\frac{1}{\ell}} \cdot \left(\prod_{i=0}^{\ell-1}\tilde{\Gamma}(0,\frac{q^i}{1-q^\ell})^{\frac{q-1}{\ell}}\right) \cdot \left(\tilde{\Gamma}(0,\frac{1}{1-q^\ell})\Big/\tilde{\Gamma}(0,\frac{q^{\ell-1}}{1-q^\ell})^q\right) \\
& \sim \Gamma_{\ari}(1-\frac{q^{\ell-1}}{q^\ell-1})^q\Big/\Gamma_{\ari}(1-\frac{1}{q^\ell-1}),
\end{align*}

where the last equivalence comes from the monomial relations among gamma values in Proposition <ref> (2) and (5).
In other words, the first statement of Theorem <ref> (3) implies Thakur's analogue of Chowla–Selberg formula in <cit.>.
the second statement of Theorem <ref> (3) shows the result in <cit.> that

\[
\Gamma_{\ari}\Big(1-\frac{q^{c}}{q^\ell-1}\Big)^q\Big/\Gamma_{\ari}\Big(1-\frac{q^{c+1}}{q^\ell-1}\Big), \quad  c = 0,...,\ell-1,
\]

are algebraically independent over $\bar{k}$ and
form a $\bar{k}$-basis of the space of quasi-periods of $E_\rho$.

(Chowla–Selberg formula for the $t$-th Carlitz cyclotomic function field.)
Let $\eK = \eK_{t,1} = \ek(\sqrt[q-1]{-t})$, which is a geometric extension over $\ek$.

\[
\eG_\eK \cong \left(\frac{\eA}{(t)}\right)^{\times} \cong \FF_q^\times,
\quad
\cfk_{\bochi} = t,
\text{ and } \quad
L_\eA(0,\bochi) = 1,
\quad \forall \  1 \neq \bochi \in \widehat{\eG}_\eK.
\]

Thus for $\epsilon \in \FF_q^\times$, we get

\[
n_t(\epsilon,0) = \sum_{1\neq \bochi \in \widehat{\eG}_\eK}\bochi(\epsilon,0)
= -1+
\begin{cases}
q-1, & \text{ if $\epsilon = 1$,}\\
0, & \text{ otherwise.}
\end{cases}
\]

Let $E_\rho$ be a Drinfeld $\eA$-module of rank $q-1$ over $\ok$ with CM by $\FF_q[\sqrt[q-1]{-t}]$.
Theorem <ref> (3) implies that for every nonzero period $\lambda \in \Lambda_\rho$, we have that

\[
\lambda \sim \tilde{\pi}^{\frac{1}{q-1}} \cdot
\left(\prod_{\epsilon \in \FF_q^\times}\Gamma(\frac{\epsilon}{\theta},1-\frac{1}{q-1})^{\frac{-1}{q-1}}\right)
\cdot \Gamma(\frac{1}{\theta},1-\frac{1}{q-1})
\sim \Gamma_{\geo}(\frac{1}{\theta}),
\]

where the last equivalence comes from Proposition <ref> (5).
This matches Thakur's formula in <cit.>.
Moreover, the second statement of Theorem <ref> (3) shows that

\[
\Gamma_{\geo}(\frac{\epsilon}{\theta}),\quad \epsilon \in \FF_q^\times,
\]

form a $\ok$-basis of the space of quasi-periods of $E_\rho$ (coinciding with <cit.> when $q=3$), and that they are algebraically independent over $\ok$.

(Chowla–Selberg formula for imaginary quadratic extensions.)
Here we apply Theorem <ref> (3) to present an analogue of the Chowla–Selberg formula for CM Drinfeld $\eA$-modules of rank $2$ over $\bar{k}$.
Note that by the class field theory, an imaginary quadratic extension $\eK$ over $\ek$ is contained in $\eK^{\cyc}$ if and only if the ramification index of the infinite place $\infty$ of $\ek$ in $\eK$ divides $q-1$.
In this case, let $\dfk \in \eA_+$ be the generator of the discriminant ideal of $O_\eK/\eA$ (where $O_\eK$ is the integral closure of $\eA$ in $\eK$).
Then $\eK$ is actually contained in $\eK_{\dfk,2}$, and $\eG_{\eK}$ is a group of order $2$.

Let $\bochi_\eK:\eG_\eK \rightarrow \{\pm 1\}$ be the unique non-trivial character in $\widehat{\eG}_\eK$.
Then $\cfk_{\bochi_\eK} = \dfk$, and for $a \in (\eA/\dfk)^\times$ and $i \in \ZZ/2\ZZ$, we get

\[
n_{\dfk}(\varrho,a,i) = \frac{\bochi_\eK(a,i+\deg \dfk)}{L_{\eA}(0,\bochi)}\cdot
\begin{cases}
1, & \text{ if $\varrho =\mathrm{id}_{\eK}$,}\\
-1,& \text{ otherwise.}
\end{cases}
\]

Since $L_\eA(0,\bochi) = \#\Pic(O_\eK)/w_\eK$ (see Remark <ref>), the first statement of Theorem <ref> (3) implies that for every nonzero period $\lambda$ of a Drinfeld $\eA$-module of rank $2$ over $\ok$ with CM by $O_\eK$, we have

\[
\lambda^{2\#\Pic(O_\eK)} \sim \tilde{\pi}^{\#\Pic(O_\eK)} \cdot \prod_{\substack{a \in (\eA/\dfk)^\times \\ i \in \ZZ/2\ZZ}}\Gamma(\frac{a(\theta)}{\dfk(\theta)},1-\frac{q^i}{q^2-1})^{w_\eK \bochi(a,i+\deg \dfk)}.
\]

Moreover, put

\begin{equation}\label{eqn: pi-K-pm}
\varpi_{\eK}^{\pm} \assign
\sqrt{\tilde{\pi}} \cdot \prod_{(a,i) \in (\eA/\dfk)^\times\times \ZZ/2\ZZ} \Gamma\Big(\frac{a(\theta)}{\dfk(\theta)},1-\frac{q^i}{q^2-1}\Big)^{\pm w_\eK \frac{\bochi_\eK(a,i+\deg \dfk)}{2\#\Pic(O_\eK)}}.
\end{equation}

The second statement of Theorem <ref> (3) says that the space of quasi-periods of $E_\rho$ is spanned over $\ok$ by $\{\omega_\eK^+,\omega_\eK^-\}$, and $\omega_\eK^+,\omega_\eK^-$ are algebraically independent over $\ok$.

Here we use the notation in Example <ref>.
Assume further that $\bochi_\eK(1,i) = 1$ for any $i \in \ZZ/2\ZZ$.
Put $\chi_\eK \assign \bochi_\eK\big|_{(\eA/\dfk)^\times}$.
Then $\eK \subset \eK_{\dfk,1}$, which is a Carlitz cyclotomic function field.
In particular, the infinite place $\infty$ of $\ek$ is totally ramified in $\eK$.
These ensure that $q$ is odd, $\dfk$ is square-free with $\deg \dfk$ odd, $\eK = \ek(\sqrt{-\dfk})$, and $w_\eK = 1$.
In this case, we may write

\[
\varpi_{\eK}^{\pm}
\sqrt{\tilde{\pi}} \cdot
\prod_{a \in (\eA/\dfk)^\times}
\Gamma_{\geo}\left(\frac{a(\theta)}{\dfk(\theta)}\right)^{\pm \frac{\chi_\eK(a)}{2\#\Pic(O_\eK)}},
\]

and get a precise analogue of the classical Chowla–Selberg formula described by Gross in <cit.>.

§.§ The normalization of gamma values and a Lerch-type formula

Given a $p$-adic integer $y = \sum_{i=0}^\infty y_i q^i \in \ZZ_p$, set (see <cit.>)

\[
\partial(y) \assign \sum_{i=0}^\infty i y_i q^i \quad \in p\ZZ_p.
\]

In particular, let $\ell \in \NN$.
For each integer $c$ with $0\leq c<\ell$, one checks that

\[
\partial(\frac{q^c}{1-q^\ell}) = \frac{cq^c}{1-q^\ell} + \frac{\ell q^{\ell+c}}{(1-q^\ell)^2} \quad \in \QQ.
\]

Moreover, given a character $\bochi: \eG_{1,\ell} \cong \ZZ/\ell \ZZ \rightarrow \CC^\times$, we have the following “$L'$-evaluator” property:
writing $\bochi(1) =q^{-s_{\bochi}}$ where $s_{\bochi} = -\frac{2\pi \sqrt{-1}}{\ln q} \cdot \frac{d_{\bochi}}{\ell}$ for a unique integer $d_{\bochi}$ with $0\leq d_{\bochi}<\ell$,

\begin{align*}
&\ \ln q \cdot \sum_{c=0}^{\ell-1}\bochi(c) \cdot \partial(\frac{q^c}{1-q^\ell}) \\
= & \ \ln q \cdot
\left[
\frac{1}{1-q^\ell} \sum_{c=0}^{\ell-1}cq^{c(1-s_{\bochi})} + \frac{\ell q^\ell}{(1-q^\ell)^2} \sum_{c=0}^{\ell-1}q^{c(1-s_{\bochi})}
\right] \\
= & \
\ln q \cdot
\left[
\frac{1}{1-q^\ell} \cdot\left( \frac{(1-q^{\ell(1-s_{\bochi})} \cdot q^{1-s_{\bochi}}}{(1-q^{1-s_{\bochi}})^2} - \frac{\ell q^{\ell(1-s_{\bochi})}}{1-q^{1-s_{\bochi}}}\right)
+ \frac{\ell q^\ell}{(1-q^\ell)^2} \cdot \frac{1-q^{\ell(1-s_{\bochi})}}{1-q^{1-s_{\bochi}}}
\right] \\
= &\ \ln q \cdot \frac{q^{1-s_{\bochi}}}{(1-q^{1-s_{\bochi}})^2} \ = \ L_\eA'(0,\bochi).
\end{align*}

For each $y \in \ZZ_{(p)}$, we follow <cit.> to normalize $\Gamma_{\ari}(y)$ by setting

\begin{equation}
\Gamma^*_{\ari}(y) \assign \theta^{-\partial(y-1)}\cdot \Gamma_{\ari}(y) = \theta^{-\partial(y-1)} \cdot \Pi_{\ari}(y-1)
\end{equation}

to obtain

For $\ell \in \NN$ and $\bochi \in \widehat{\eG}_{1,\ell}$ we have that

\[
L_\eA'(0,\bochi) = -\sum_{c=0}^{\ell-1}\bochi(c)\cdot \ln\left|\Gamma_{\ari}^*(1-\frac{q^c}{q^\ell-1})\right|_\infty.
\]

One consequence of the above lemma is that

\begin{align}
\ln \left|\tilde{\pi} \cdot \prod_{i=0}^{\ell-1}\Gamma_{\ari}^*(1-\frac{q^i}{q^\ell-1})^{-n_1(1,i)}\right|_\infty
&= \frac{q \ln q}{q-1} + \ln q \cdot \sum_{i=0}^{\ell-1} n_1(1,i)\cdot \partial(\frac{q^i}{1-q^\ell})  \nonumber \\
&= \frac{\zeta_\eA'(0)}{\zeta_\eA(0)} + \sum_{1\neq \bochi \in \widehat{\eG}_{1,\ell}}\frac{L_\eA'(0,\bochi)}{L_\eA(0,\bochi)} \nonumber \\
&= \frac{\zeta_{O_{1,\ell}}'(0)}{\zeta_{O_{1,\ell}}(0)}. \nonumber
\end{align}

In order to extend the above result to arbitrary cyclotomic function fields, we define

\[
\partial(x,y) \assign \langle x,-y\rangle, \quad \forall x \in k,\ y \in \ZZ_{(p)}.
\]

Then for each character $\bochi \in \widehat{\eG}_{\nfk,\ell}$ with $\cfk_{\bochi} \neq 1$, from the expression of $L_A^{\nfk}(s,\bochi)$ in (<ref>) the following “$L'$-evaluator” property holds:

\[
-\!\!\!\! \sum_{\substack{a \in \eA,\ \deg a<\deg \nfk \\ \text{gcd}(a,\nfk) = 1} }\  \sum_{i=0}^{\ell-1} \bochi(a,i+\deg \nfk)\cdot \partial(\frac{a(\theta)}{\nfk(\theta)},\frac{q^i}{1-q^\ell})\cdot \ln\left|\frac{a(\theta)}{\nfk(\theta)}\right|_\infty = (L_{\eA}^{\nfk})'(0,\bochi) + L_{\eA}^{\nfk}(0,\bochi)\cdot \ln|\nfk(\theta)|_\infty.
\]

We normalize the two-variable gamma values $\Gamma(x,y)$ as follows:

\begin{equation}\label{eqn: gamma-normalization}
\Gamma^*(x,y) \assign \frac{x^{-\partial(x,y-1)}\cdot \Pi_{\geo}(x,y-1)}{\Gamma_{\ari}^*(x,y)} = x^{1-\partial(x,y-1)} \cdot \theta^{\partial(y-1)} \cdot  \Gamma(x,y).
\end{equation}

In particular, $\Gamma^*(0,y) = \Gamma^*_{\ari}(y)^{-1}$ for all $y \in \ZZ_{(p)}$.
Then we have the following Lerch-type formula:

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
Given a character $\bochi \in \eG_{\nfk,\ell}$, the following equality holds:

\[
L_\eA'(0,\bochi) = -L_\eA(0,\bochi)\ln|\cfk_{\bochi}|_\infty + \sum_{\substack{a \in \eA,\ \deg a<\deg \cfk_{\bochi} \\ \text{\rm gcd}(a,\cfk_{\bochi}) = 1}} \ \sum_{i=0}^{\ell-1}\bochi(a,i+\deg \cfk_{\bochi})\cdot \ln\left|\Gamma^*(\frac{a(\theta)}{\cfk_{\bochi}(\theta)},1-\frac{q^i}{q^\ell-1})\right|_\infty.
\]

Consequently, Theorem <ref> leads us to:

Let $\nfk \in \eA_+$ and $\ell \in \NN$.
Given an imaginary field $\eK \subset \eK_{\nfk,\ell}$, we have that

\[
\ln\left|\tilde{\pi}\cdot \prod_{\cfk \mid \nfk} \prod_{\substack{a \in \eA,\ \deg a < \deg \cfk \\ \text{\rm gcd}(a,\cfk) = 1}} \
\prod_{i=0}^{\ell-1}
\Gamma^*(\frac{a(\theta)}{\cfk(\theta)},1-\frac{q^i}{q^\ell-1})^{n_{\cfk}(a,i)}\right|_\infty
= \frac{\zeta_{O_\eK}'(0)}{\zeta_{O_\eK}(0)} + \ln|\dfk(O_\eK/\eA)|_\infty,
\]

where $\dfk(O_\eK/\eA) \in \eA_+$ is the monic generator of the discriminant ideal of $O_\eK$ over $\eA$.

From the formula of $\tilde{\pi}$ in Remark <ref> one gets

\[
\ln|\tilde{\pi}|_\infty=\frac{q\ln q}{q-1} = \frac{\zeta_{\eA}'(0)}{\zeta_{\eA}(0)}.
\]

As $\eK$ is imaginary, one has that $\eK^+=\ek$.
Thus for $\cfk \in \eA_+$ with $\cfk\mid \nfk$, $a \in \eA$ with $\deg a < \deg c$ and $\text{gcd}(a,c) = 1$, and $0\leq i < \ell$,

\[
n_\cfk(a,i) = \sum_{\substack{1\neq \bochi \in \widehat{\eG}_\eK\\ \cfk_{\bochi}=\cfk}} \frac{\bochi(a,i+\deg \cfk)}{L_{\eA}(0,\bochi)}.
\]

Therefore we obtain that

\begin{align*}
& \ln\left|\tilde{\pi}\cdot \prod_{\cfk \mid \nfk} \prod_{\substack{a \in \eA,\ \deg a < \deg \cfk \\ \text{gcd}(a,\cfk) = 1}} \
\prod_{i=0}^{\ell-1}
\Gamma^*(\frac{a(\theta)}{\cfk(\theta)},1-\frac{q^i}{q^\ell-1})^{n_{\cfk}(a,i)}\right|_\infty \\
=\ &
\frac{\zeta_{\eA}'(0)}{\zeta_{\eA}(0)}+ \sum_{\cfk \mid \nfk} \sum_{\substack{a \in \eA,\ \deg a<\deg \cfk \\ \text{gcd}(a,\cfk)=1}} \sum_{i=0}^{\ell-1}n_\cfk(a,i) \cdot \ln \left|\Gamma^*(\frac{a(\theta)}{\cfk(\theta)},1-\frac{q^i}{q^\ell-1})\right|_\infty \\
=\ &
\frac{\zeta_{\eA}'(0)}{\zeta_{\eA}(0)} + \sum_{1\neq \bochi \in \widehat{\eG}_\eK}
\sum_{\substack{a \in \eA,\ \deg a<\deg \cfk_{\bochi} \\ \text{gcd}(a,\cfk_{\bochi}) = 1}} \ \sum_{i=0}^{\ell-1}\frac{\bochi(a,i+\deg \cfk_{\bochi})}{L_\eA(0,\bochi)}\cdot \ln\left|\Gamma^*(\frac{a(\theta)}{\cfk_{\bochi}(\theta)},1-\frac{q^i}{q^\ell-1})\right|_\infty \\
=\ &
\frac{\zeta_{\eA}'(0)}{\zeta_{\eA}(0)} + \sum_{1\neq \bochi \in \widehat{\eG}_\eK} \left(\frac{L_\eA'(0,\bochi)}{L_\eA(0,\bochi)} + \ln|\cfk_{\bochi}|_\infty\right) \hspace{2cm}  \text{(by Theorem~\ref{thm: Lerch-type formula})} \\
=\ &
\frac{\zeta_{O_\eK}'(0)}{\zeta_{O_\eK}(0)} + \ln|\dfk(O_\eK/\eA)|_\infty,
\end{align*}

where the last equality comes from the fact that $\zeta_{O_\eK}(s) = \prod_{\bochi \in \widehat{\eG}_\eK}L_\eA(s,\bochi)$ and the conductor-discriminant formula.

Let $\eK$ be an imaginary field contained in $\eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.
Take $\nu\assign \nu_1\big|_{\eK} :\eK \hookrightarrow \ok\subset \CC_\infty$, and let $K\assign \nu(\eK)$ (resp. $O_K\assign \nu(O_\eK)$).
Put $r = [\eK:\ek]$.
Every fractional ideal of $O_K$ becomes a discrete $A$-lattice of rank $r$ in $\CC_\infty$.
The Kronecker limit formula established in [42] implies in particular that (see <cit.>)

\[
\frac{\zeta_{O_\eK}'(0)}{\zeta_{O_\eK}(0)} = \frac{1}{\#\Pic(O_K)}\sum_{[\Afk] \in \Pic(O_K)} \ln\Big(N(\Afk)|\Delta(\Afk)|_\infty^{\frac{r}{q^r-1}}\Big),
\]

* $N(\Afk) \assign \#(O_K/\Afk)$ for each nonzero ideal $\Afk$ of $O_K$; and

* $\Delta(\cdot)$ is the Drinfeld discriminant function (on $A$-lattices of rank $r$ in $\CC_\infty$, see <cit.>).

Therefore Corollary <ref> leads us to the following “analytic” Chowla–Selberg formula:

\begin{align}\label{eqn: A-CSF}
& \frac{1}{\#\Pic(O_K)}\sum_{[\Afk] \in \Pic(O_K)} \ln\Big(N(\Afk)|\Delta(\Afk)|_\infty^{\frac{r}{q^r-1}}\Big) \nonumber \\
&= \ln \left(\frac{|\tilde{\pi}|_\infty}{|\dfk(O_\eK/\eA)|_\infty}\right) +
\sum_{\cfk \mid \nfk} \sum_{\substack{a \in \eA,\ \deg a < \deg \cfk \\ \text{gcd}(a,\cfk) = 1}} \
\sum_{i=0}^{\ell-1}
n_{\cfk}(a,i)\ln \Big|\Gamma^*(\frac{a(\theta)}{\cfk(\theta)},1-\frac{q^i}{q^\ell-1})\Big|_\infty.
\end{align}

Let $E_\rho$ be a Drinfeld $\eA$-module over $\ok$ of rank $r$ which has CM by $O_\eK$, where $\eK$ is an imaginary field of degree $r$ over $\ek$.
Suppose that $\eK$ is separable over $\ek$ and tamely ramified at $\infty$.
Then the Colmez-type formula derived in <cit.> (together with <cit.>) expresses the “stable Taguchi height of $E_\rho$” as follows:

\[
h_{\mathrm{Tag}}^{\mathrm{st}}(E_\rho) = -\frac{1}{2r}\cdot \ln |\dfk(O_\eK/\eA)|_\infty - \frac{1}{r}\cdot \frac{\zeta_{O_\eK}'(0)}{\zeta_{O_\eK}(0)}.
\]

When $\eK \subset \eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$,
(<ref>) implies that

\begin{align}\label{eqn: height}
h_{\mathrm{Tag}}^{\mathrm{st}}(E_\rho) &= \frac{1}{2r}\cdot \ln |\dfk(O_\eK/\eA)|_\infty-\frac{1}{r}\ln|\tilde{\pi}|_\infty \nonumber \\
&-\sum_{\cfk \mid \nfk} \sum_{\substack{a \in \eA,\ \deg a < \deg \cfk \\ \text{gcd}(a,\cfk) = 1}} \
\sum_{i=0}^{\ell-1}\frac{n_{\cfk}(a,i)}{r}\cdot \ln \Big|\Gamma^*(\frac{a(\theta)}{\cfk(\theta)},1-\frac{q^i}{q^\ell-1})\Big|_\infty.
\end{align}

Suppose further that $E_\rho$ is “sgn-normalized” (see <cit.>) and the period lattice of $E_\rho$ is of the form:

\[
\Lambda_\rho = O_K \cdot \varpi_\rho, \quad \text{ where \quad $\varpi_\rho \in \CC_\infty$.}
\]

Then $\varpi_\rho$ is unique up to $O_K^\times$-multiples, and we can show that

\[
= -\frac{1}{2r}\cdot \ln |\dfk(O_\eK/\eA)|_\infty
- \ln |\varpi_\rho|_\infty.
\]

Combining this with (<ref>), we arrive at the following “absolute” Chowla–Selberg formula:

\begin{equation}\label{eqn: A-CSF-2}
|\varpi_\rho|_\infty^r =  \frac{|\tilde{\pi}|_\infty}{|\dfk(O_\eK/\eA)|_\infty} \cdot \prod_{\cfk \mid \nfk}\prod_{\substack{a \in \eA,\ \deg a < \deg \cfk \\ \text{gcd}(a,\cfk) = 1}} \
\prod_{i=0}^{\ell-1}\left|\Gamma^*(\frac{a(\theta)}{\cfk(\theta)},1-\frac{q^i}{q^\ell-1})\right|_\infty^{n_{\cfk}(a,i)}.
\end{equation}

§ THE DELIGNE–GROSS PERIOD CONJECTURE

In this section, we shall apply the work of Pink and Hartl–Juschka (in [35], [25], [28], and [24]) on the Hodge conjecture over function fields, to derive an analogue of the Deligne–Gross period conjecture.

§.§ Hodge–Pink structures

Let $H$ be a finite dimensional vector space over $\ek$. An exhaustive and separated increasing $\QQ$-filtration $W_\bullet H$ is a collection of $\ek$-subspaces $W_\mu H \subset H$ for $\mu \in \QQ$ with $W_{\mu'} H \subset W_{\mu} H$ whenever $\mu'<\mu$, such that the associated graded space

\[
\Gr^W H \assign \bigoplus_{\mu \in \QQ} \Gr^W_\mu H,
\quad \text{ where } \quad \Gr^W_\mu H \assign W_\mu H/ (\cup_{\mu'<\mu} W_{\mu'}H),
\]

has the same dimension over $\ek$ as $H$.

(See <cit.>.) A (mixed) $\ek$-pre-Hodge–Pink structure is a triple $\eH = (H,W_\bullet H,\qfk)$, where

* $H$ is a finite dimensional vector space over $\ek$;

* $W_\bullet H$ is an exhaustive and separated increasing $\QQ$-filtration;

* $\qfk \subset \CC_\infty(\!(t-\theta)\!)\otimes_\ek H$ is a $\CC_\infty[\![t-\theta]\!]$-lattice of full rank.

Here the tensor product is with respect to the following inclusion

\[
\ek \subset \CC_\infty(t) = \CC_\infty(t-\theta) \subset \CC_\infty(\!(t-\theta)\!).
\]

In particular, $\eH$ is called pure (of weight $\mu$) if $H \cong \Gr^W_{\mu} H$ for some $\mu \in \QQ$.

Given two (mixed) $\ek$-pre-Hodge–Pink structures $\eH = (H,W_\bullet H, \qfk)$ and $\eH' = (H', W_\bullet H', \qfk')$, a morphism $f:\eH \rightarrow \eH'$ is a $\ek$-linear homomorphism $f: H \rightarrow H'$ such that $f(W_\mu H) \subset W_\mu H'$ for every $\mu \in \QQ$ and the induced homomorphism $\mathrm{id}\otimes f : \CC_\infty(\!(t-\theta)\!)\otimes_\ek H \rightarrow \CC_\infty(\!(t-\theta)\!)\otimes_\ek H'$ satisfies $(\mathrm{id}\otimes f)(\qfk) \subseteq \qfk'$.

In fact, $\ek$ is contained in  $\CC_\infty[\![t-\theta]\!]$, as every nonzero $a \in \eA$ is contained in $\CC_\infty[\![t-\theta]\!]^\times$.
In particular,

\[
\frac{1}{t} =
\big(\theta + (t-\theta)\big)^{-1} = \sum_{n=0}^\infty (-1)^n\cdot (\frac{1}{\theta})^{n+1} \cdot (t-\theta)^n.
\]

It is known that (see <cit.>) the inclusion $\ek\subset \CC_\infty[\![t-\theta]\!]$ can be extended to an $\ek$-algebra embedding $\ek_\infty \hookrightarrow \CC_\infty[\![t-\theta]\!]$.
Thus it is natural to extend the above definition to (mixed) $\ek_\infty$-pre-Hodge–Pink structures and the morphisms between them without difficulty.
Every $\ek$-pre-Hodge–Pink structure $\eH = (H,W_\bullet H,\qfk)$ can be extended to a $\ek_\infty$-pre-Hodge–Pink structure
$\eH_\infty = (H_\infty, W_\bullet H_\infty,\qfk)$ where $H_\infty \assign \ek_\infty \otimes_\ek H$ and $W_\mu H_\infty \assign \ek_\infty \otimes_\ek W_\mu H$ for every $\mu \in \QQ$.

A (mixed) $\ek$-Hodge–Pink structure is a (mixed) $\ek$-pre-Hodge–Pink structure $\eH = (H,W_\bullet H,\qfk)$
satisfying the local semistable condition (at $\infty$): for every $\ek_\infty$-subspace $H_\infty'$ of $H_\infty$, put $W_\mu H_\infty' \assign H_\infty' \cap W_\mu H_\infty$ for every $\mu \in \QQ$, $\qfk' \assign \qfk \cap \CC_\infty(\!(t-\theta)\!)\otimes_{\ek_\infty} H'_\infty$, and $\pfk' \assign \CC_\infty[\![t-\theta]\!]\otimes_{\ek_\infty} H_\infty'$.

\[
\dim_{\CC_\infty}\left(\frac{\qfk'}{\qfk'\cap \pfk'}\right) - \dim_{\CC_\infty}\left(\frac{\pfk'}{\qfk'\cap \pfk'}\right)\leq \sum_{\mu \in \QQ} \mu \cdot \dim_{\ek_\infty} \Gr^W_\mu H'_\infty,
\]

and equality holds when $H'_\infty = \ek_\infty \otimes_\ek W_\mu H$ for every $\mu \in \QQ$.

The local semistable condition can be also defined on $\ek_\infty$-pre-Hodge–Pink structures, and we may say as well that a (mixed) $\ek_\infty$-pre-Hodge–Pink structure is a (mixed) $\ek_\infty$-Hodge–Pink structure if it satisfies the local semistable condition.
In particular, if $\eH$ is a $\ek$-Hodge–Pink structure, then $\eH_\infty$ is a $\ek_\infty$-Hodge–Pink structure.

Given two (mixed) $\ek$-Hodge–Pink structures $\eH = (H,W_\bullet H, \qfk)$ and $\eH' = (H', W_\bullet H', \qfk')$,
let $f: \eH\rightarrow \eH'$ be a morphism.
The local semistable condition on $\eH$ and $\eH'$ ensures that $f$ must be strict (see <cit.>), i.e.

\[
f(W_\mu H) = f(H) \cap W_\mu H', \quad \forall \mu \in \QQ,
\quad \text{ and } \quad
(\mathrm{id}\otimes f)(\qfk) = \CC_\infty(\!(t-\theta)\!)\otimes_\ek f(H) \cap \qfk'.
\]

We denote by $\Hom_{\mathrm{HP}}(\eH,\eH')$ the space of the morphisms from $\eH$ to $\eH'$, and $\End_{\mathrm{HP}}(\eH)$ the endomorphism ring of $\eH$.

Let $\eH = (H,W_\bullet H,\qfk)$ be a (mixed) $\ek$-Hodge–Pink structure.
A $\ek$-Hodge–Pink sub-structure of $\eH$ is a triple $\eH' =(H',W_\bullet H', \qfk')$, where
$H'$ is a $\ek$-vector suspace of $H$,
$W_\mu H' = H' \cap W_\mu H$ for every $\mu \in \QQ$, and
$\qfk' = \qfk \cap \CC_\infty(\!(t-\theta)\!) \otimes_\ek H'$.
In particular, if $\eH$ is pure of weight $\mu$, then so is $\eH'$.

One natural example of a pure $\ek$-Hodge–Pink structure comes from pure uniformizable dual $t$-motives.
Let $\eM$ be a pure uniformizable dual $t$-motive of weight $\wt(\eM)$ over $\CC_\infty$. Recall in Section <ref> that
$H^\eM = \Hom_{\eA}(H_{\mathrm{Betti}}(\eM),\ek)$.

\[
\qfk^\eM \assign \Hom_{\CC_\infty[t]}\big(\sigma \eM,\CC_\infty[\![t-\theta]\!]\big)
\quad \text{ and }\quad
W_\mu H^\eM \assign
\begin{cases}
0, & \text{ if $\mu<-\wt(\eM)$,}\\
H^\eM, &\text{ if $\nu\geq -\wt(\eM)$.}
\end{cases}
\]

By <cit.>,
the triple $\eH(\eM)\assign (H^\eM,W_\bullet H^\eM,\qfk^\eM)$ is a pure Hodge–Pink structure of weight $\mu_\eM$, where

\[
\mu_\eM =-w(\eM)= \frac{\rank_{\CC_\infty[\sigma]}\eM}{\rank_{\CC_\infty[t]}\eM}.
\]

Moreover, given two pure uniformizable dual $t$-motives $\eM$ and $\eM'$ over $\CC_\infty$, every morphism $f:\eM\rightarrow \eM'$ (over $\CC_\infty$) induces a morphism

\[
f_{\mathrm{HP}}: \eH(\eM')\rightarrow \eH(\eM),
\]

and $(f\mapsto f_{\mathrm{HP}})$ gives a $\ek$-linear (resp. $\ek$-algebra) isomorphism (resp. when $\eM = \eM'$) (see <cit.>)

\[
\ek \otimes_\eA \Hom_{\CC_\infty[t,\sigma]}(\eM,\eM') \cong \Hom_{\mathrm{HP}}(\eH(\eM'),\eH(\eM)).
\]

The following fact comes from the function field analogue of the Hodge conjecture for dual $t$-motives (see <cit.>):

Let $\eM$ be a pure uniformizable dual $t$-motive over $\CC_\infty$.
Given a $\ek$-Hodge–Pink sub-structure $\eH'$ of $\eH(\eM)$, there exists a pure uniformizable dual $t$-motive $\eM'$ over $\CC_\infty$ together with an essentially surjective morphism $f: \eM \rightarrow \eM'$ such that the corresponding morphism $f_{\mathrm{HP}}: \eH(\eM')\hookrightarrow \eH(\eM)$ gives an isomorphism $\eH(\eM')\cong \eH'$.

It is worth pointing out that
the “effectiveness” of $\eM'$ (i.e. $\eM'$ is indeed a $\CC_\infty[\sigma]$-module) comes from the “duality” in <cit.>, and the essential surjectivity of the morphism $f: \eM\rightarrow \eM'$ implies that $\eM'$ is finitely generated over $\CC_\infty[\sigma]$.
Thus $\eM'$ is indeed a dual $t$-motive in our setting.

§.§ CM Hodge–Pink structures and CM dual t-motives

Let $\eH = (H,W_\bullet H, \qfk)$ be a pure $\ek$-Hodge–Pink structure of weight $w(\eH)$.
Put $\pfk \assign \CC_\infty[\![t-\theta]\!]\otimes_{\ek} H$, and identify

\[
\frac{\pfk}{(t-\theta) \pfk} \cong \CC_\infty \underset{v_\theta,\ek}{\otimes} H \rassign H_{\CC_\infty}.
\]

The Hodge–Pink filtration of $H_{\CC_\infty}$ is $F^\bullet H_{\CC_\infty} \assign (F^i H_{\CC_\infty})_{i\in\ZZ}$, where $F^i H_{\CC_\infty}$ is the subspace of $H_{\CC_\infty}$ corresponding to

\[
\frac{\pfk\cap (t-\theta)^i \qfk}{(t-\theta)\pfk\cap (t-\theta)^i \qfk} \quad \cong \quad \frac{\pfk \cap (t-\theta)^i \qfk+(t-\theta) \pfk}{(t-\theta)\pfk} \quad \subset \quad \frac{\pfk}{(t-\theta)\pfk}.
\]

Let $r = \dim_{\ek} H$.
Take an integer $n$ sufficiently large so that $(t-\theta)^n\pfk \subset \qfk$.
There exist $r$ integers $w_1,...,w_r$ satisfying that

\[
\frac{\qfk}{(t-\theta)^n \pfk} \cong \bigoplus_{i=1}^r \frac{\CC_\infty[\![t-\theta]\!]}{(t-\theta)^{n+w_i}},
\]

which is independent of the choice of $n$.
We call $w_1,...,w_r$ the Hodge–Pink weights of $\eH$.
In particular, the local semistability of $\eH$ implies that

\[
r \cdot w(\eH) = w_1+\cdots + w_r.
\]

Suppose further that $\eH$ has full-CM by a CM field $\eK$, i.e. $\dim_\ek H = [\eK:\ek]$ and there exists a $\ek$-algebra homomorphism

\begin{equation}\label{eqn: HCM}
\eK \hookrightarrow \End_{\mathrm{HP}}(\eH).
\end{equation}

Then $\pfk$ becomes a free module of rank one over $\CC_\infty[\![t-\theta]\!]\otimes_{\ek} \eK$.
Recall that $O_\eK$ denotes the integral closure of $\eA$ in $\eK$, and here we put $O_{\bK} = \CC_\infty \otimes_{\FF_q}O_\eK$.
Observe that

\[
\CC_\infty[\![t-\theta]\!] \otimes_{\ek} \eK \cong \CC_\infty[\![t-\theta]\!] \otimes_{\eA} O_\eK \cong \prod_{\xi \in J_\eK} \hat{O}_{\bK,\xi}
\quad \text{and}\quad
\CC_\infty(\!(t-\theta)\!) \otimes_{\ek} \eK \cong
\prod_{\xi \in J_\eK} \hat{\bK}_{\xi},
\]

where for each $\xi \in J_\eK$, $\hat{O}_{\bK,\xi}$ is the completion of $O_{\bK}$ with respect to the maximal ideal $\Pfk_\xi$ and $\hat{\bK}_{\xi}$ is the field of fraction of $\hat{O}_{\bK,\xi}$.
As $\qfk$ is also a $\CC_\infty[\![t-\theta]\!] \otimes_{\ek} \eK$-module and there exists a sufficiently large integer $n$ so that

\[(t-\theta)^n\cdot \pfk \subset \qfk \subset (t-\theta)^{-n} \cdot \pfk \quad \subset \CC_\infty(\!(t-\theta)\!) \otimes_{\ek} H,
\]

we obtain that $\qfk$ is actually a free $\CC_\infty[\![t-\theta]\!] \otimes_{\ek} \eK$-module of rank one, and there exist unique integers $w_\xi$ for $\xi \in J_\eK$ satisfying that

\[
\frac{\qfk}{(t-\theta)^n \pfk} \cong \prod_{\xi \in J_\eK}\frac{\hat{O}_{\bK,\xi}}{\Pfk_\xi^{n+w_\xi}\hat{O}_{\bK,\xi}} \cong \prod_{\xi \in J_\eK}\frac{\CC_\infty[\![t-\theta]\!]}{(t-\theta)^{n+w_\xi}},
\]

where the last isomorphism is due to that
$(t-\theta)O_\bK = \prod_{\xi \in J_\eK}\Pfk_\xi$ (i.e. $(t-\theta)$ splits completely in $O_\bK$).
Thus the numbers $w_\xi$ for $\xi \in J_\eK$ coincide with the Hodge–Pink weights of $\eH$.

Let $\eH$ be a $\ek$-Hodge–Pink structure with full-CM by a CM field $\eK$ over $\ek$.
The Hodge–Pink type of $\eH$ is

\[
\Phi_{\eH} \assign \sum_{\xi \in J_\eK} w_\xi \xi \quad \in I_\eK.
\]

In fact, we have the following:

Let $\eH$ be a pure $\ek$-Hodge–Pink structure of weight $w(\eH)$ which has full-CM by a CM field $\eK$ over $\ek$.
Then $\Phi_\eH$ lies in $I_\eK^0$ with $\wt(\Phi_\eH) = [\eK:\eK^+]\cdot w(\eH)$,
where $\eK^+$ is the maximal totally real subfield of $\eK$.

Write $\eH = (H,W_\bullet H,\qfk)$, and denote by $w(\eH)$ be the weight of $\eH$.
Recall that we let $\eH_\infty\assign (H_\infty, W_\bullet H_\infty, \qfk)$ where $H_\infty = \ek_\infty \otimes_\ek H$ and $W_\mu H_\infty = \ek_\infty \otimes_\ek W_\mu H$, which is a $\ek_\infty$-Hodge–Pink structure.
Then we have an embedding

\[
\ek_\infty  \otimes_\ek \eK \hookrightarrow \End_{\mathrm{HP}}(\eH_\infty),
\]

which makes $H_\infty$, $W_\mu H_\infty$ for every $\mu \in \QQ$, and $\qfk$ become $\ek_\infty \otimes_\ek \eK$-modules.
On the other hand, as $\eK$ is a CM field over $\ek$, we have that

\[
\ek_\infty \otimes_{\ek}\eK \cong \prod_{\xi^+ \in J_{\eK^+}} \eK_{\infty,\xi^+},
\]

where for each $\xi^+ \in J_{\eK^+}$, $\eK_{\infty,\xi^+} \assign \ek_\infty \underset{\tilde{\nu}_{\xi^+},\eK^+}{\otimes} \eK$ is the compositum of $\eK$ and $\ek_\infty$ over $\eK^+$ with respect to the embedding

\[
\tilde{\nu}_{\xi^+}: \eK^+\hookrightarrow \ek_\infty \quad \text{ so that } \quad
v_\theta \circ \tilde{\nu}_{\xi^+} = \nu_{\xi^+} \in
\Emb(\eK^+,\CC_\infty).
\]

In particular, one has that
$[\eK_{\infty,\xi^+}:\ek_\infty] = [\eK:\eK^+]$ for every $\xi^+ \in J_{\eK^+}$.

For each $\xi^+ \in J_{\eK^+}$, we take $e_{\xi^+}$ to be the  idempotent of $\ek_\infty \otimes_\ek \eK$ corresponding to $\eK_{\infty,\xi^+}$.
Then $e_{\xi^+} \eH_\infty \assign (e_{\xi^+} H_\infty, W_\bullet (e_{\xi^+} H_\infty), e_{\xi^+} \qfk)$ is a pure $\ek_\infty$-Hodge–Pink sub-structure of $\eH_\infty$, and

\[
\eH_\infty = \bigoplus_{\xi \in J_{\eK^+}} e_{\xi^+}\eH_{\infty}.
\]

Moreover, comparing with the decomposition

\begin{align*}
\prod_{\xi \in J_\eK}\hat{O}_{\bK,\xi}
& \cong \CC_\infty[\![t-\theta]\!] \otimes_{\ek} \eK
\cong
\CC_\infty[\![t-\theta]\!] \otimes_{\ek_\infty} (\ek_\infty \otimes_{\ek} \eK) \\
& \cong
\prod_{\xi^+ \in J_{\eK^+}} \CC_\infty[\![t-\theta]\!] \otimes_{\ek_\infty} \eK_{\infty,\xi^+}
\cong
\prod_{\xi^+ \in J_{\eK^+}} \CC_\infty[\![t-\theta]\!] \underset{\tilde{\nu}_{\xi^+},\eK^+}{\otimes} \eK \\
& \cong
\prod_{\xi^+ \in J_{\eK^+}} \CC_\infty[\![t-\theta]\!] \underset{\tilde{\nu}_{\xi^+},O_{\eK^+}}{\otimes} O_{\eK}
\cong
\prod_{\xi^+ \in J_{\eK^+}}
\left(\prod_{\substack{\xi \in J_{\eK} \\ \pi_{\bX/\bX^+}(\xi) = \xi^+}} \hat{O}_{\bK,\xi}\right),
\end{align*}

we get

\[
e_{\xi^+} = \sum_{\substack{\xi \in J_{\eK} \\ \pi_{\bX/\bX^+}(\xi) = \xi^+}} \hat{e}_{\xi},
\quad \text{ where $\hat{e}_{\xi}$ is the
idempotent of $\prod_{\xi \in J_\eK}\hat{O}_{\bK,\xi}$ corresponding to $\hat{O}_{\bK,\xi}$.}
\]

Consequently, taking sufficiently large integer $n$ so that $(t-\theta)^n \pfk \subset \qfk$, we obtain that

\[
\frac{e_{\xi^+}\qfk}{(t-\theta)^n e_{\xi^+}\pfk}\cong
\prod_{\substack{\xi \in J_\eK \\ \pi_{\bX/\bX^+}(\xi) = \xi^+}} \frac{ \hat{O}_{\eK,\xi}}{\Pfk_{\xi}^{n+w_\xi} \hat{O}_{\eK,\xi}}.
\]

The local semistable condition of $\eH$ (at $\infty$) ensures that

\begin{align}\label{eqn: wt-equality}
\sum_{\substack{\xi \in J_\eK \\ \pi_{\bX/\bX^+}(\xi) = \xi^+}} w_\xi
& =  \dim_{\CC_\infty}\left(\frac{e_{\xi^+}\qfk}{e_{\xi^+}\qfk \cap e_{\xi^+}\pfk}\right) - \dim_{\CC_\infty}\left(\frac{e_{\xi^+}\pfk}{e_{\xi^+}\qfk \cap e_{\xi^+}\pfk}\right) \nonumber \\
& \leq \dim_{\ek_\infty} (e_{\xi^+} H_\infty) \cdot w(\eH)  = [\eK:\eK^+] \cdot w(\eH) .
\end{align}

\[
\sum_{\xi^{+} \in J_{\eK}^{+}} \left(\sum_{\substack{\xi \in J_\eK \\ \pi_{\bX/\bX^+}(\xi) = \xi^+}} w_\xi\right)= \sum_{\xi \in J_\eK} w_\xi = \dim_{\ek} (H) \cdot w(\eH)  = \sum_{\xi^{+} \in J_{\eK}^{+}} \bigg([\eK:\eK^+] \cdot w(\eH) \bigg),
\]

the inequality (<ref>) is actually an equality, whence $\Phi_{\eH} \in I_\eK^0$ with $\wt(\Phi_{\eH}) = [\eK:\eK^+] \cdot w(\eH) $ as desired.

Let $\eH=(H,W_\bullet H,\qfk)$ be a pure $\ek$-Hodge–Pink structure of weight $w(\eH)$ which has full-CM by a CM field $\eK$ over $\ek$.
Suppose $\eH$ is effective, i.e.

\[
\pfk = \CC_\infty[\![t-\theta]\!] \otimes_{\ek} H \subset \qfk.
\]

Then $w_\xi \geq 0$ for every $\xi \in J_\eK$, for which $\Phi_\eH$ becomes a generalized CM type of $\eK$ with

\[
\wt(\Phi_\eH) = [\eK:\eK^+]\cdot w(\eH).
\]

Let $\eM$ be a pure uniformizable dual $t$-motive of weight $w(\eM)$ over $\CC_\infty$.
Given a $\ek$-Hodge–Pink sub-structure $\eH'$ of $\eH(\eM)$, suppose that $\eH'$ has full-CM by a CM field $\eK$ over $\ek$.
Then $\eH'$ is effective and we may take $\eM'$ in Theorem <ref> to be a CM dual $t$-motive with generalized CM type $(\eK,\Phi_{\eH'})$ over $\CC_\infty$, where $\wt(\Phi_{\eH'}) = -[\eK:\eK^+]\cdot w(\eM)$.

Recall that $\eH(\eM) = (H^{\eM},W_\bullet H^{\eM},\qfk^\eM)$ is a pure $\ek$-Hodge–Pink structure of weight $-w(\eM)$, and

\begin{align*}
\qfk^\eM = \Hom_{\CC_\infty[t]}(\sigma \eM, \CC_\infty[\![t-\theta]\!]) &\supset \Hom_{\CC_\infty[t]}(\eM,\CC_\infty[\![t-\theta]\!])
= \Hom_{\TT^\dagger}(\MM^\dagger,\CC_\infty[\![t-\theta]\!])\\
= \Hom_{\eA}(H_{\mathrm{Betti}}(\eM), \CC_\infty[\![t-\theta]\!])
= \CC_\infty[\![t-\theta]\!]\otimes_{\ek} H^{\eM} \rassign \pfk^{\eM}.
\end{align*}

Here $\TT^\dagger$ and $\MM^\dagger$ are introduced in Section <ref>.
Then $\eH' = (H',W_\bullet H', \qfk')$, where
$H'$ is a $\ek$-subspace of $H^\eM$, $W_\mu H' = W_\mu H^{\eM}\cap  H'$, and

\[\qfk' = \qfk \cap \CC_\infty(\!(t-\theta)\!) \otimes_\ek H' \supset \pfk^\eM \cap \CC_\infty(\!(t-\theta)\!) \otimes_\ek H' = \CC_\infty[\![t-\theta]\!] \otimes_\ek H' = \pfk'.
\]

Thus $\eH'$ is pure of weight $-w(\eM)$ and effective.
By Theorem <ref>, there exists a pure uniformizable dual $t$-motive $\eM'$ over $\CC_\infty$ together with an essentially surjective morphism $f: \eM \rightarrow \eM'$ over $\CC_\infty$ such that $f_{\mathrm{HP}}: \eH(\eM') \cong \eH'$.
As $\eH'$ has full-CM by $\eK$, one has that $\rank_{\CC_\infty[t]}(\eM') = \dim_{\ek}(\eH') = [\eK:\ek]$, and

\[
\eK \hookrightarrow \End_{\mathrm{HP}}(\eH') \cong
\ek \otimes_{\eA}\End_{\CC_\infty[t,\sigma]}(\eM').
\]

In particular, the intersection of $\eK$ and $\End_{\CC_\infty[t,\sigma]}(\eM')$ via the above embedding is an $\eA$-order $\Ocal$ in $\eK$.
Let $O_\eK$ be the integral closure of $\eA$ in $\eK$.
Replacing $\eM'$ by $O_\eK \otimes_{\Ocal} \eM'$ if necessary, we may assume that $\Ocal = O_\eK$.

From the isomorphism $f_{\mathrm{HP}}: \eH(\eM') \cong \eH'$, we have that

\[
\CC_\infty[\![t-\theta]\!]\otimes_{\CC_\infty[t]} \left(\frac{\eM'}{\sigma \eM'}\right) \cong \frac{\qfk^{\eM'}}{\pfk^{\eM'}} \cong \frac{\qfk'}{\pfk'} \cong \prod_{\xi \in J_\eK}\frac{\hat{O}_{\eK,\xi}}{\Pfk_\xi^{w_\xi'}\hat{O}_{\eK,\xi}},
\]

where $\Phi_{\eH'} = \sum_{\xi \in J_\eK} w_{\xi}' \xi$ is the Hodge–Pink type of $\eH'$.
Let $\Ifk_{\Phi_{\eH'}}\assign \prod_{\xi \in J_\eK} \Pfk^{w_\xi'} \subset O_{\bK}$ to get $\sigma \eM' = \Ifk_{\Phi_{\eH'}} \eM'$.
By Lemma <ref> and Remark <ref>, we know that $\Phi_{\eH'}$ is a generalized CM type of $\eK$.
Hence by Definition <ref>, $\eM'$ is a CM dual $t$-motive with CM type $(\eK,\Phi_{\eH'})$ over $\CC_\infty$,

\[
\wt(\Phi_{\eH'}) = [\eK:\eK^+]\cdot w(\eH') = -[\eK:\eK^+]\cdot w(\eM).
\]

Keep the notations in Proposition <ref>.
By Theorem <ref> (3) we may choose $\eM'$ to be defined over $\ok$.
Suppose further that $\eM$ is also defined over $\ok$, then by Lemma <ref> the essentially surjective morphism $\eM \rightarrow \eM'$ is over $\ok$ as well.

Let $\eM$ be a pure uniformizable dual $t$-motive over $\ok$, and $\eH'$ a $\ek$-Hodge–Pink substructure of $\eH(\eM)$ with full-CM by a CM field $\eK$ over $\ek$.
A dual $t$-motive $\eM'$ over $\ok$ given in Proposition <ref> is called a CM dual $t$-motive associated to $\eH'$ over $\ok$.
By Theorem <ref> (3), such $\eM'$ is unique up to isogeny.

§.§ The Deligne–Gross conjecture

Let $\eM$ be a pure uniformizable dual $t$-motive defined over $\ok$.
Given a $\ek$-Hodge–Pink substructure $\eH' = (H', W_\bullet H', \qfk')$ of $\eH(\eM)$, via the period isomorphism $H^{\eM}_{\CC_\infty} \cong H_{\mathrm{dR}}(\eM,\CC_\infty)$ (see (<ref>)), one may identify $H'_{\CC_\infty} \assign \CC_\infty \otimes_{\ek} H' \subset H^{\eM}_{\CC_\infty}$ as a subspace of $H_{\mathrm{dR}}(\eM,\CC_\infty)$.
Suppose $\eH'$ has full-CM by a CM field $\eK \subset \ek^{\sep}$, i.e. $\dim_\ek H'= [\eK:\ek]$ and $\eK$ embeds into the endomorphism algebra $\End_{\mathrm{HP}}(\eH')$ over $\ek$.
Note that $H'_{\CC_\infty}$ becomes a $\CC_\infty\underset{v_\theta,\ek}{\otimes} \eK$-module, and

\[
\CC_\infty\underset{v_\theta,\ek}{\otimes} \eK \cong \prod_{\nu \in \Emb(\eK,\CC_\infty)} \CC_\infty.
\]

For each embedding $\nu \in \Emb(\eK,\CC_\infty)$,
there exists a nonzero $\omega_{\nu}^{\eH'} \in H_{\mathrm{dR}}(\eM,\CC_\infty) \cap H'_{\CC_\infty}$, which is unique up to $\CC_\infty^\times$-multiples, satisfying that

\[
\alpha (\omega_{\nu}^{\eH'}) = \nu(\alpha) \cdot \omega_{\nu}^{\eH'}, \quad \forall \alpha \in \eK,
\]

The main result in this section is to derive an explicit description of the periods

\[
\int_{\gamma} \omega_{\nu}^{\eH'}, \quad \forall 0\neq \gamma \in H_{\mathrm{Betti}}(\eM),
\]

when $\omega_{\nu}^{\eH'}$ is algebraic (i.e.  $\omega_{\nu}^{\eH'} \in H_{\mathrm{dR}}(\eM,\ok)\cap H'_{\CC_\infty}$), in terms of two-variable gamma values when the CM field $\eK$ is contained in a cyclotomic function field.

Suppose $\eK \subset \eK_{\nfk,\ell}$ where $\nfk \in \eA_+$ and $\ell \in \NN$.
Let $\eM'$ be a CM dual $t$-motive associated to $\eH'$ over $\ok$.
Then $\eM'$ has generalized CM type $(\eK, \Phi_{\eH'})$, where $\Phi_{\eH'} \in I_{\eK}^0$ is the Hodge–Pink type of $\eH'$.
By Proposition <ref> (3), there exists a function $\varepsilon^{\eH'}:(\frac{1}{\nfk(\theta)}A/A) \times (\frac{1}{q^{\ell}-1} \ZZ/\ZZ) \rightarrow \QQ$ such that

\[
\varphi_{\eK,\Phi_{\eH'}} = \sum_{\substack{x \in \frac{1}{\nfk(\theta)}A/A \\ y \in \frac{1}{q^{\ell}-1} \ZZ/\ZZ}}\varepsilon^{\eH'}(x,y)\cdot \St(x,y).
\]

By Theorem <ref>, we prove the following analogue of Deligne–Gross period conjecture:

Let $\eM$ be a pure uniformizable dual $t$-motive defined over $\ok$.
Given a $\ek$-Hodge–Pink sub-structure $\eH' = (H',W_\bullet H', \qfk')$ of $\eH(\eM)$, suppose $\eH'$ has full-CM by a CM field $\eK$, where $\eK \subset \eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.
For each $\varrho \in \eG_{\nfk,\ell}$, put $\nu_{\varrho}\assign \nu_1 \circ \varrho\big|_\eK : \eK \hookrightarrow \CC_\infty$.
Identifying $H'_{\CC_\infty} \assign \CC_\infty \otimes_{\ek} H' \subset H^{\eM}_{\CC_\infty}$ with a subspace of $H_{\mathrm{dR}}(\eM,\CC_\infty)$ via the period isomorphism in (<ref>), let  $\omega_{\varrho}^{\eH'} \in H_{\mathrm{dR}}(\eM,\CC_\infty) \cap \eH'_{\CC_\infty}$ be a nonzero differential satisfying

\[
\alpha (\omega_{\varrho}^{\eH'}) = \nu_{\varrho}(\alpha) \cdot \omega_{\varrho}^{\eH'}, \quad \forall \alpha \in \eK.
\]

$\varepsilon^{\eH'}:(\frac{1}{\nfk(\theta)}A/A) \times (\frac{1}{q^{\ell}-1} \ZZ/\ZZ) \rightarrow \QQ$
such that

\[
\varphi_{\eK,\Phi_{\eH'}}
= \sum_{\substack{x \in \frac{1}{\nfk(\theta)}A/A \\ y \in \frac{1}{q^{\ell}-1} \ZZ/\ZZ}}\varepsilon^{\eH'}(x,y)\cdot \St(x,y) \quad \in \Sscr(\eG_{\nfk,\ell}),
\]

where $\Phi_{\eH'}$ is the Hodge–Pink type of $\eH'$, see Definition <ref>.
Suppose that $\omega_\varrho^{\eH'}$ is algebraic, i.e. $\omega_\varrho^{\eH'} \in H_{\mathrm{dR}}(\eM,\ok)$.
For $\gamma \in \eH_{\mathrm{Betti}}(\eM)$ with $\int_{\gamma} \omega_{\varrho}^{\eH'} \neq 0$,
we then have that

\[
\int_{\gamma} \omega_{\varrho}^{\eH'}
\sim \prod_{\substack{x \in \frac{1}{\nfk(\theta)}A/A \\ y \in \frac{1}{q^{\ell}-1}\ZZ/\ZZ}}\tilde{\Gamma}(x, y)^{\varepsilon^{\eH'}(\varrho^{-1}\star x,\varrho^{-1}\star y))}.
\]

Let $\eM'$ be a CM dual $t$-motive associated to $\eH'$ over $\ok$, and $f: \eM \rightarrow \eM'$ the essentially surjective homomorphism over $\ok$ such that $f_{\mathrm{HP}}: \eH(\eM') \cong \eH' \subset \eH(\eM)$ as in Proposition <ref>.
Let $\xi_{\varrho} \in J_\eK$ be the point so that $\nu_{\xi_{\varrho}} = \nu_\varrho$.
Let $\omega_{\eM',\xi_{\varrho}} \in H_{\mathrm{dR}}(\eM',\ok)$ be the differential associated with $\xi_\varrho$.
Up to a $\ok^\times$-multiple, we then have $\omega_{\varrho}^{\eH'} = f^*\omega_{\eM',\xi_{\varrho}}$.
Note that $\eM'$ has generalized CM type $(\eK,\Phi_{\eH'})$.
Hence for each $\gamma \in H_{\mathrm{Betti}}(\eM)$ such that $\int_{\gamma} \omega_{\varrho}^{\eH'}$ is nonzero, we obtain that

\begin{align*}
\int_{\gamma} \omega_{\varrho}^{\eH'} &\sim \int_{\gamma} f^* \omega_{\eM',\xi_{\varrho}} \sim \int_{f_* \gamma } \omega_{\eM',\xi_{\varrho}}\\
&\sim p_{\eK}(\xi_{\varrho}, \Phi_{\eH'}) \sim p_{\eK}(\xi_1, \Phi_{\eH'}^{\varrho^{-1}}) \sim \wp^{\cyc}_{\nu_1}\big(\varrho \cdot  \varphi_{\eK,\Phi_{\eH'}}\big)
\sim \prod_{\substack{x \in \frac{1}{\nfk(\theta)}A/A \\ y \in \frac{1}{q^{\ell}-1}\ZZ/\ZZ}}\wp_{\nu_1}^{\cyc}\big(\St(\varrho\star x,\varrho\star y)\big)^{\varepsilon^{\eH'}(x,y)} \\
& \sim \prod_{\substack{x \in \frac{1}{\nfk(\theta)}A/A \\ y \in \frac{1}{q^{\ell}-1}\ZZ/\ZZ}}\wp_{\nu_1}^{\cyc}\big(\St(x,y)\big)^{\varepsilon^{\eH'}(\varrho^{-1}\star x,\varrho^{-1} \star y)}
\sim
\prod_{\substack{x \in \frac{1}{\nfk(\theta)}A/A \\ y \in \frac{1}{q^{\ell}-1}\ZZ/\ZZ}}\tilde{\Gamma}(x,y)^{\varepsilon^{\eH'}(\varrho^{-1}\star x,\varrho^{-1} \star y)},
\end{align*}

where the last equivalence is from Theorem <ref> (and Remark <ref>).

As $\eM'$ is defined over $\ok$ and $\eH'$ is isomorphic to $\eH(\eM')$ via the essentially surjective morphism $f:\eM\rightarrow \eM'$ over $\ok$, we are always able to choose the differential $\omega_{\varrho}^{\eH'}$ to be algebraic for every $\varrho \in \eG_{\nfk,\ell}$ by taking $\omega_{\varrho}^{\eH'} = f^* \omega_{\eM',\xi_\varrho}$ as in the proof above.

Keeping the notation in Theorem <ref>,
write the Hodge–Pink type of $\eH'$ as $\Phi_{\eH'} = \sum_{\varrho \in \eG_\eK}m_{\varrho} \cdot \xi_{\varrho} \in I_\ek^0$.
Although the choice of the function $\varepsilon^{\eH'}$ might not be unique, we can follow Theorem <ref> and take $\varepsilon^{\eH'}$ to be the following explicit one: for $x \in \frac{1}{\nfk(\theta)}A/A$ and $y \in \frac{1}{q^\ell-1}\ZZ/\ZZ$, write $x = \frac{a(\theta)}{\cfk(\theta)} \bmod A$ where $a \in \eA$ and $\cfk \in \eA_+$ with $\cfk \mid \nfk$ and $\text{gcd}(a,\cfk) = 1$, put

\[
\varepsilon^{\eH'}(x,y) \assign \begin{cases}
\displaystyle \frac{(1-q)\wt(\Phi_{\eH'})}{[\eK:\eK^+]} + \frac{m_\varrho n_{\cfk}(\varrho,a,i)}{[\eK:\ek]}, & \text{ if $\cfk = 1$ and $y = \displaystyle \frac{q^i}{1-q^\ell} \bmod \ZZ$ for $0\leq i <\ell$,}\\
\displaystyle\frac{m_\varrho n_{\cfk}(\varrho,a,i)}{[\eK:\ek]}, & \text{ if $\cfk \neq 1$ and $y = \displaystyle \frac{q^i}{1-q^\ell} \bmod \ZZ$ for $0\leq i <\ell$,}\\
0, & \text{ otherwise.}
\end{cases}
\]

Here $n_\cfk(\varrho,a,i)$ is defined in (<ref>).
As in (<ref>), we get

\begin{align*}
& \sum_{\substack{x \in \frac{1}{\nfk(\theta)}A/A \\ y \in \frac{1}{q^\ell-1}\ZZ/\ZZ}}\varepsilon^{\eH'}(x,y)\St(x,y) \\
&= \frac{\wt(\Phi_{\eH'})}{[\eK:\eK^+]} \cdot \mathbf{1}_{\eG_\eK}+\sum_{\cfk \mid \nfk}\sum_{\substack{a \in (\eA/\cfk)^\times \\ i \in \ZZ/\ell \ZZ}}\left(\sum_{\varrho \in \eG_\eK} m_\varrho \cdot n_\cfk(\varrho,a,i)\right)\cdot \St(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell}) \\
&= \varphi_{\eK,\Phi_{\eH'}}.
\end{align*}

Hence given $\varrho_0 \in \eG_\eK$,  we obtain that

\[
\int_{\gamma}\omega_{\varrho_0}^{\eH'} \sim
\tilde{\pi}^{\frac{\wt(\Phi_{\eH'})}{[\eK:\eK^+]}}
\cdot  \prod_{\varrho \in \eG_\eK} \left(\prod_{\cfk\mid \nfk}\prod_{\substack{a \in (\eA/\cfk)^\times \\ i \in \ZZ/\ell \ZZ}} \tilde{\Gamma}(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})^{n_{\cfk}(\varrho\varrho_0^{-1},a,i)}\right)^{\frac{m_{\varrho}}{[\eK:\ek]}}
\]

whenever $\gamma \in H_{\mathrm{Betti}}(\eM)$ such that $\int_{\gamma}\omega_{\varrho_0}^{\eH'}$ is nonzero.

Let $E_\rho$ be an abelian $t$-module over $\ok$ which is pure and uniformizable (i.e. the Hartl–Juschka dual $t$-motive $\eM(\rho)$ is pure and uniformizable).
Put $\eH(E_\rho) \assign \eH(\eM(\rho))$, and for each $N \in \NN$ we define the $N$-th Hodge–Pink structure of $E_\rho$ by

\[
\eH^N(E_\rho) \assign \bigwedge^N_{\ek} \eH(E_\rho) = \eH\Big(\bigwedge^N_{\ok[t]} \eM(\rho)\Big).
\]

Let $d$ be the dimension of $E_\rho$ and $r$ be the rank of $E_\rho$.
Suppose further that $E_\rho$ has “complex multiplication” by $O_\eK$, where $\eK$ is a CM field over $\ek$ with $[\eK:\ek] = n$ and $O_\eK$ is the integral closure of $\eA$ in $\ek$.
This means that there exists an $\eA$-algebra homomorphism from $O_\eK \hookrightarrow \End_{\text{ab.~$t$-mod.}}(E_\rho)$,
which corresponds to a $\ek$-algebra embedding $\eK \hookrightarrow \End_{\mathrm{HP}}(\eH(E_\rho))$.
In particular, one has that $n\mid r$.
Put $d_0\assign r/n$.
We may identify

\[
\eH^{d_0}_{\eK}(\rho)\assign \bigwedge^{d_0}_{\eK} \eH(E_\rho) = \eH\Big(\bigwedge^{d_0}_{O_\bK}\eM(\rho)\Big)
\hookrightarrow
\eH\Big(\bigwedge^{d_0}_{\ok[t]} \eM(\rho)\Big) = \eH^{d_0}(E_\rho).
\]

In particular, the Hodge–Pink structure $\eH^{d_0}_{\eK}(\rho)$ is pure of weight $d_0 \cdot \frac{d}{r} = \frac{d}{n}$ and has full-CM by $\eK$.
This is equivalent to saying that $\bigwedge^{d_0}_{O_\bK}\eM(\rho)$ is a CM dual $t$-motive with generalized CM type $(\eK,\Phi_{\eH^{d_0}_{\eK}(\rho)})$, where $\Phi_{\eH^{d_0}_{\eK}(\rho)}$ is the Hodge–Pink type of $\eH^{d_0}_{\eK}(\rho)$ with

\[
\wt(\Phi_{\eH^{d_0}_{\eK}(\rho)}) = \frac{d}{n} \cdot [\eK:\eK^+] = \frac{d}{[\eK^+:\ek]}.
\]

Now, suppose $\eK \subset \eK_{\nfk,\ell}$ for $\nfk \in \eA_+$ and $\ell \in \NN$.

\[
\Phi_{\eH^{d_0}_{\eK}(\rho)} = \sum_{\varrho \in \eG_{\eK}}m_\varrho \xi_\varrho \quad \in I_\eK^0.
\]

Then by Remark <ref> we have that for $\varrho_0 \in \eG_\eK$,

\begin{align*}
\int_{\gamma_1\wedge\cdots \wedge\gamma_{d_0}}\omega_{\varrho_0}^{\eH^{d_0}_{\eK}(\rho)} & \sim
\tilde{\pi}^{\frac{d}{[\eK:\ek]}}
\cdot  \prod_{\varrho \in \eG_\eK} \left(\prod_{\cfk\mid \nfk}\prod_{\substack{a \in (\eA/\cfk)^\times \\ i \in \ZZ/\ell \ZZ}} \tilde{\Gamma}(\frac{a(\theta)}{\cfk(\theta)},\frac{q^i}{1-q^\ell})^{n_{\cfk}(\varrho\varrho_0^{-1},a,i)}\right)^{\frac{m_{\varrho}}{[\eK:\ek]}}
\end{align*}

$\gamma_1\wedge \cdots \wedge \gamma_{d_0} \in H_{\mathrm{Betti}}\big(\bigwedge^{d_0}_{\ok[t]}\eM(\rho)\big)$ so that the period integral on the left hand side is nonzero.
In particular, when $\eK$ is imaginary and quadratic, we may write $\eG_\eK = \{\varrho_1 = \mathrm{id}_\eK,\varrho_2\}$, and

\[
\Phi_{\eH^{d_0}_{\eK}(\rho)} = m_1 \xi_{\varrho_1} + m_2 \xi_{\varrho_2} \quad \text{ with } \quad
m_1+m_2 = d \quad (\text{as $\eK^+ =\ek$}).
\]

Then we get the following analogue of the Gross formula in <cit.>:

\[
\int_{\gamma_1\wedge\cdots \wedge\gamma_{d_0}}\omega_{\varrho_1}^{\eH^{d_0}_{\eK}(\rho)} \sim (\varpi_\eK^+)^{m_1}\cdot (\varpi_{\eK}^-)^{m_2} \sim (\varpi_\eK^+)^{m_1} \cdot \left(\frac{\tilde{\pi}}{\varpi_\eK^+}\right)^{m_2},
\]

\[
\int_{\gamma_1\wedge\cdots \wedge\gamma_{d_0}}\omega_{\varrho_2}^{\eH^{d_0}_{\eK}(\rho)}  \sim (\varpi_\eK^+)^{m_2}\cdot (\varpi_{\eK}^-)^{m_1} \sim (\varpi_\eK^+)^{m_2} \cdot \left(\frac{\tilde{\pi}}{\varpi_\eK^+}\right)^{m_1},
\]

where $\varpi_\eK^{\pm}$ are given in (<ref>).

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