#  $\infty$-Categorical Perverse $p$-adic Differential Equations over Stacks

Xin Tong

#### abstract

We will discuss $\infty$-categorical perverse $p$-adic differential equations
over stacks. On one hand, we are going to study some $p$-adic analogous
results of the Drinfeld’s original lemma about the étale fundamental groups in
the étale setting, in the context of $F$-isocrystals closely after Kedlaya and
Kedlaya-Xu. We expect similar things could also be considered for diamonds
after Scholze, in the context of Kedlaya-Liu’s work namely the derived
category of pseudocoherent Frobenius sheaves, which will induce some
categorical form of Drinfeld’s lemma for diamonds motivated by work of Carter-
Kedlaya-Zábrádi and Pal-Zábrádi. On the other hand, we are going to establish
the $\infty$-categorical theory of arithmetic $D$-modules after Abe and
Gaitsgory-Lurie, which will allow one to construct the rigid Gross
$G$-motives. And we are expecting to apply the whole machinery to revisit
Weil’s conjecture parallel to and after Gaitsgory-Lurie.

11footnotetext: Keywords and Phrases: $p$-adic differential equation,
$\infty$-categorical arithmetic $D$-modules, $\infty$-categorical arithmetic
$D^{\dagger}$-modules, $L$-functions.

###### Contents

  1. 1 Perverse $p$-adic Differential Equations over Quotient Stacks
    1. 1.1 The Drinfeld’s Lemma
    2. 1.2 Convergent Isocrystals
    3. 1.3 Overconvergent Isocrystals
    4. 1.4 Rephrasization by Arithmetic $D^{\dagger}$-Modules over Stacky Disks
      1. 1.4.1 Double Points
      2. 1.4.2 Derived $(\varphi,\nabla)$-Modules over Stacks
    5. 1.5 Polydisks and the Quotients
    6. 1.6 $L$-Functions and Product Formula for Perverse $p$-adic Differential Equations over Stacks
  2. 2 $\infty$-Categorical Perverse $p$-adic Differential Equations over $\mathrm{Bun}_{G}$
    1. 2.1 Introduction
      1. 2.1.1 Motivation
      2. 2.1.2 Approaches
    2. 2.2 Arithmetic $\mathcal{D}$-modules over Schemes
      1. 2.2.1 The $\infty$-category $\mathbb{D}^{\mathrm{b}}_{\mathrm{hol}}(X)$ of constructible objects
      2. 2.2.2 Inverse and Direct Images in Derived $\infty$-Category
      3. 2.2.3 $\infty$-Ind-Categories
    3. 2.3 Arithmetic $\mathcal{D}$-modules over Stacks
      1. 2.3.1 Inverse and Direct Images in Derived $\infty$-Category
    4. 2.4 Arithmetic of Rigid Gross $G$-Motives: $\mathrm{Bun}_{G}$
    5. 2.5 Arithmetic of Rigid Gross $G$-Motives: $\mathrm{Bun}^{\mathrm{Parabolic,P}}_{G}$
    6. 2.6 Arithmetic of Rigid Gross $G$-Motives: $\mathrm{Bun}^{\mathrm{Local}}_{G(K_{X,x})}$

##  Chapter 1 Perverse $p$-adic Differential Equations over Quotient Stacks

Drinfeld’s celebrated lemma relates the corresponding product of étale
fundamental groups of different schemes with the corresponding fundamental
group of some single stack coming the quotient of the product space of the
involved schemes. This is very important since in [La1] V.Lafforgue relied
heavily on the corresponding such lemma to give the corresponding
parametrisation after Langlands in the context of function field arithmetic.
Following this, Kedlaya conjectured that we should also have the chance to do
this in the context of $p$-adic cohomology theory (also see [Ked9] and [KX]),
which will generalize the work of [Abe] for linear algebraic groups.

###  1.1 The Drinfeld’s Lemma

In this section we recall some basics around the corresponding original
Drinfeld’s Lemma in the context of the étale local systems over schemes in
characteristic $p>0$.

######  Assumption 1.1.1.

We will mainly work with smooth schemes which are also assumed to be
connected, reduced and separated over $\mathbb{F}_{p}$.

Then in the current situation, suppose we have different schemes
$X_{1},...,X_{I}$ over $\mathbb{F}_{p}$ then we could consider the
corresponding product of these schemes in the absolute sense:

$\displaystyle
X:=X_{1}\times_{\mathrm{Spec}\mathbb{F}_{p}}...\times_{\mathrm{Spec}\mathbb{F}_{p}}X_{I}.$
(1.1.1)

What happens is that one might want to ask if we could have the chance to
relate the étale local systems over $X$ and the ones over each scheme. However
this is not very direct in the sense that we have to consider more
construction in order to make such relationship transparent. This is how
Drinfeld’s Lemma kicks in. Here is how the idea works. First we have the
corresponding partial Frobenius (which was observed by Drinfeld) $\varphi_{i}$
for $i=1,...,I$. We need to consider some sort of multi Frobenius structures
in order to guarantee that we could have the desired relationship. The first
construction is the following stack for any $i\in I$:

$X/\Phi:=X/\left<\varphi_{1},...,\widehat{\varphi}_{i},...\varphi_{1}\right>,$

see [CBCKSW, Sheaves, Stacks and Shtukas, Definition 4.2.10] as well for more
details. For such stack the corresponding construction is bit complicated.

######  Remark 1.1.2.

Recall from [CBCKSW, Sheaves, Stacks and Shtukas, Definition 4.2.10] that the
corresponding étale coverings of the stack $X/\Phi$ could be related to the
corresponding étale coverings of the scheme $X$, but one has to consider the
further equivariant Frobenius action coming from $\varphi_{1},...,\varphi_{I}$
(with the corresponding composite specified as those relative Frobenius
$\varphi_{Y/X}$ over $X$ for any covering $Y$), namely for such $Y$ for $X$ we
need to assume that $Y$ admits isomorphisms up to all the pullbacks by the
partial Frobenius actions.

So we can have the reasonable category $\text{F\'{E}T}(X/\Phi)$, and we will
use the corresponding notation $\text{F\'{E}T}(X)_{\Phi}$ to denote the
corresponding equivalent equivariant category:

$\displaystyle\text{F\'{E}T}(X)_{\Phi}\overset{\sim}{\longrightarrow}\text{F\'{E}T}(X/\Phi).$
(1.1.2)

So we can have the reasonable profinite fundamental group
$\pi_{1}^{\mathrm{prof}}(X/\Phi,\overline{x})$, and under the corresponding
framework of Tannakian cateogories, we will use the notation
$\pi_{1}^{\mathrm{prof}}(X,\overline{x})_{\Phi}$ to denote the Tannakian
fundamental group of the category $\text{F\'{E}T}(X)_{\Phi}$ with

$\displaystyle\text{F\'{E}T}(X)_{\Phi}\overset{\sim}{\longrightarrow}\text{F\'{E}T}(X/\Phi).$
(1.1.3)

######  Proposition 1.1.3.

(The Drinfeld’s Lemma) We have the isomorphism between the fundamental group
of the stack $X/\Phi$ and the product of the ones from each space $X_{i}$,
$i=1,...,I$. Therefore we have the corresponding equivalence between the
category of all the $\ell$-adic representations of
$\pi_{1}^{\mathrm{prof}}(X/\Phi,\overline{x})$ and the category of all the
$\ell$-adic representations of the product of fundamental groups of the each
separate schemes involved, for any prime number $\ell$ (which could be the
same as $p$).

###  1.2 Convergent Isocrystals

Now we switch to just the $p$-adic setting in the context of convergent
isocrystals.

######  Setting 1.2.1.

Recall from [Ked1, Definition 2.1], we have the corresponding category of all
the convergent $F$-isocrystals over some scheme $S$ over $\mathbb{F}_{p}$. We
use the notation $\textbf{FIsoc}(S)$ to denote the category of all the
convergent $F$-isocrystals over $S$ (the exact definition could be found in
[Ked1, Definition 2.1]). And at the same time we also have the corresponding
category of overconvergent $F$-isocrystals over the same space $S$, which we
denote it by $\textbf{FIsoc}^{\dagger}(S)$.

A corresponding convergent $F$-isocrystal over $S$ is vector bundle $M$ over
the corresponding rigid analytic generic fiber $S^{\mathrm{an}}$ over
$\mathbb{Q}_{p}$ of the corresponding lift $\mathcal{S}$ of $S$ to
$W(\mathbb{F}_{p})$ in the glueing fashion carrying an integrable connection
$\nabla$, with the corresponding action coming from the Frobenius lift
$\sigma$. In our situation we also have to consider the corresponding vector
bundles over some quotient stack $X/\Phi$ ($X$ is the product space in the
previous section). Certainly this will mean that we have to consider the
corresponding quotient from the corresponding $X$ to this quotient.

######  Definition 1.2.2.

Over the stack $X/\Phi$ we define a corresponding convergent $F$-isocrystal
$M$ to be a convergent $F$-isocrystal $(M,\nabla,\sigma)$ over $X$ with the
corresponding action from each partial Frobenius $\varphi_{i}$ ($i\in I$)
compatible with the corresponding partial Frobenius actions on the space $X$.
We can consider the corresponding category of all such objects, which will be
denoted by $\textbf{FIsoc}(X/\Phi)$.

######  Proposition 1.2.3.

With the corresponding notations established above. Consider the following
three categories. The first is the category of all the $p$-adic
representations of $\prod_{I}\pi_{1}^{\mathrm{prof}}(X_{i},\overline{x}_{i})$.
The second is the category of all the $p$-adic representations of
$\pi_{1}^{\mathrm{prof}}(X/\Phi,\overline{x})$. The third is the category of
all the convergent $F$-isocrystals over $X/\Phi$ which are unit-root for
$\sigma$ when regarded as the corresponding objects over $X$ carrying partial
Frobenius actions. Then we have that all the three categories are equivalence.
The corresponding functors are the one induced from the usual Drinfeld’s lemma
on the isomorphism of the groups on the both sides, and the one coming from
Katz-Crew’s equivalence.

###### Proof.

This would be just a combination of the usual Drinfeld’s lemma as in the
previous section, and the corresponding Katz-Crew’s equivalence [Ka], [Cr2] on
the unit-root convergent isocrystals and the corresponding $p$-adic étale
local systems as in [Cr2, Theorem 2.1]. ∎

######  Corollary 1.2.4.

Let $I$ be a set of cardinality 2. And we put $X_{2}$ to be
$\mathrm{Spec}k=\mathrm{Spec}\overline{\mathbb{F}}_{p}$. Then we have the
category of the corresponding unit root convergent $F$-isocrystals over
$X_{1,k}/\varphi_{k}$ and the category of the corresponding unit root
convergent $F$-isocrystals over $X_{1}$ are equivalent.

Beyond the corresponding unit-root situation, we should consider the
corresponding non-étale context where we need to replace the corresponding
profinite étale fundamental groups by the corresponding isocrytal Tannakian
fundamental groups $\pi^{\mathrm{Isoc}}_{1}$ (see [DK, Appendix B]). Therefore
the fundamental conjecture will be the following:

######  Conjecture 1.2.5.

The category of the corresponding $\overline{\mathbb{Q}}_{p}$-representations
of $\prod_{i=1}^{I}\pi^{\mathrm{Isoc}}_{1}(X_{i})$ is equivalent to the
corresponding category of all the $\overline{\mathbb{Q}}_{p}$-representations
of $\pi^{\mathrm{Isoc}}_{1}(X/\Phi)$, which is further equivalent to the
corresponding category of all the $\overline{\mathbb{Q}}_{p}$-convergent
isocrystals over $X/\Phi$.

###  1.3 Overconvergent Isocrystals

Now we switch to just the $p$-adic setting in the context of overconvergent
isocrystals.

######  Setting 1.3.1.

Recall from [Ked1, Definition 2.7], we have the corresponding category of all
the overconvergent $F$-isocrystals over some scheme $S$ over $\mathbb{F}_{p}$.
We use the notation $\textbf{FIsoc}^{\dagger}(S)$ to denote the category of
all the overconvergent $F$-isocrystals over $S$ (the exact definition could be
found in [Ked1, Definition 2.7]).

######  Definition 1.3.2.

Over the stack $X/\Phi$ we define a corresponding overconvergent
$F$-isocrystal $M$ to be an overconvergent $F$-isocrystal $(M,\nabla,\sigma)$
over $X$ with the corresponding action from each partial Frobenius
$\varphi_{i}$ ($i\in I$) compatible with the corresponding partial Frobenius
actions on the space $X$. We can consider the corresponding category of all
such objects, which will be denoted by $\textbf{FIsoc}^{\dagger}(X/\Phi)$.

######  Proposition 1.3.3.

With the corresponding notations established above. Consider the following two
categories. The first is the category of all the $p$-adic étale local systems
over $X/\Phi$ which are potentially unramified with respect to $X$. The second
is the category of all the overconvergent $F$-isocrystals over $X/\Phi$ which
are unit-root for $\sigma$ when regarded as the corresponding objects over $X$
carrying partial Frobenius actions. Then we have that all the two categories
are equivalence. The corresponding functors are the one induced from the usual
Drinfeld’s lemma on the isomorphism of the groups on the both sides, and the
one coming from Crew-Tsuzuki-Kedlaya’s equivalence.

###### Proof.

This would be just a combination of the usual Drinfeld’s lemma as in the
previous section, and the corresponding Crew-Tsuzuki-Kedlaya’s equivalence
[Cr1], [Cr2], [Tsu], [Ked4], [Ked5], [Ked6], [Ked7] on the unit-root
convergent isocrystals and the corresponding $p$-adic étale local systems (for
instance see [Tsu, Theorem 1.3]). ∎

######  Corollary 1.3.4.

Let $I$ be a set of cardinality 2. And we put $X_{2}$ to be
$\mathrm{Spec}k=\mathrm{Spec}\overline{\mathbb{F}}_{p}$. Then we have the
category of the corresponding unit root overconvergent $F$-isocrystals over
$X_{1,k}/\varphi_{k}$ and the category of the corresponding unit root
overconvergent $F$-isocrystals over $X_{1}$ are equivalent.

Beyond the corresponding unit-root situation, we should consider the
corresponding non-étale context where we need to replace the corresponding
profinite étale fundamental groups by the corresponding isocrytal Tannakian
fundamental groups $\pi^{\mathrm{Isoc},\dagger}_{1}$ (see [DK, Appendix B]).
Therefore the fundamental conjecture will be the following:

######  Conjecture 1.3.5.

The category of the corresponding $\overline{\mathbb{Q}}_{p}$-representations
of $\prod_{i=1}^{I}\pi^{\mathrm{Isoc},\dagger}_{1}(X_{i})$ is equivalent to
the corresponding category of all the
$\overline{\mathbb{Q}}_{p}$-representations of
$\pi^{\mathrm{Isoc},\dagger}_{1}(X/\Phi)$, which is further equivalent to the
corresponding category of all the $\overline{\mathbb{Q}}_{p}$-overconvergent
isocrystals over $X/\Phi$.

######  Remark 1.3.6.

The corresponding functoriality realizing such equivalence should be not that
transparent. The reason here why one has to be sufficiently careful is that
the corresponding six functors do not serve as a well-defined way to
functorializing the corresponding construction and operations of the
corresponding isocrystals. Instead at least for some better consideration one
should look at the corresponding relative $p$-adic differential equation
theory established in the scope of the arithmetic $D^{\dagger}$-modules and so
on.

###  1.4 Rephrasization by Arithmetic $D^{\dagger}$-Modules over Stacky Disks

####  1.4.1 Double Points

We now rephrase some of the corresponding local pictures considered above in
the framework of arithmetic $D^{\dagger}$-modules. The corresponding picture
is obviously more transparent than the global situation. Actually we will
start from the curve situation and consider then further the localization.

The geometric framework in the local setting comes from the two pointed formal
space $\mathrm{Spf}(\mathbb{Z}_{p}[[t]])$ and
$\mathrm{Spf}W(\overline{\mathbb{F}}_{p})[[t]]$. We use $s$ to denote the
single closed point and we use the corresponding notation $\eta$ to denote the
single open point.

######  Setting 1.4.1.

We now consider the corresponding setting up around the local picture in [AM].
We consider the corresponding space $X_{1}=\mathrm{Spec}\mathbb{F}_{p}((t))$.
Now we recall the corresponding picture of the relative $p$-adic differential
equations in [AM]. Recall from [AM, Section 2.1.2,2.1.3] in this case, we have
the following sheaves of the corresponding differential operators with the
corresponding along $t=0$ overconvergence. We use the corresponding notation

$\displaystyle D_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$ (1.4.1)

to denote the corresponding sheaf of analytic differential operaters along
$t=0$ having overconvergence. And we use the corresponding notation:

$\displaystyle D_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$ (1.4.2)

to denote the corresponding sheaf of overconvergent differential operaters
along $t=0$ having overconvergence. Also we have the following truncated
constructions which are also very important applications. We use the
corresponding notation

$\displaystyle D^{(m)}_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$
(1.4.3)

to denote the corresponding sheaf of analytic differential operaters along
$t=0$ having overconvergence, of order $m\geq 0$. And we use the corresponding
notation:

$\displaystyle D^{(m)}_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$ (1.4.4)

to denote the corresponding sheaf of overconvergent differential operaters
along $t=0$ having overconvergence, of order $m\geq 0$. And what we will do as
well is the corresponding sheaves over two points, where we will drop the
corresponding notation of $t=0$.

And we have the following:

######  Setting 1.4.2.

We now consider the corresponding setting up around the local picture in [AM].
We consider the corresponding space
$X_{1}=\mathrm{Spec}\overline{\mathbb{F}}_{p}((t))$. Now we recall the
corresponding picture of the relative $p$-adic differential equations in [AM].
Recall from [AM, Section 2.1.2,2.1.3] in this case, we have the following
sheaves of the corresponding differential operators with the corresponding
along $t=0$ overconvergence. We use the corresponding notation

$\displaystyle
D_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$ (1.4.5)

to denote the corresponding sheaf of analytic differential operaters along
$t=0$ having overconvergence. And we use the corresponding notation:

$\displaystyle D_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$ (1.4.6)

to denote the corresponding sheaf of overconvergent differential operaters
along $t=0$ having overconvergence. Also we have the following truncated
constructions which are also very important applications. We use the
corresponding notation

$\displaystyle
D^{(m)}_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
(1.4.7)

to denote the corresponding sheaf of analytic differential operaters along
$t=0$ having overconvergence, of order $m\geq 0$. And we use the corresponding
notation:

$\displaystyle D^{(m)}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
(1.4.8)

to denote the corresponding sheaf of overconvergent differential operaters
along $t=0$ having overconvergence, of order $m\geq 0$. And what we will do as
well is the corresponding sheaves over two points, where we will drop the
corresponding notation of $t=0$.

Over the corresponding sheaves of rings as above we will consider those
arithmetic coherent sheaves in [AM, Section 2.1.2,2.1.3,2.1.4]. And we
consider the corresponding holonomic ones. Also we consider the corresponding
Frobenius structures as well.

######  Setting 1.4.3.

We now consider the corresponding setting up around the local picture in [AM].
We consider the corresponding space $X_{1}=\mathrm{Spec}\mathbb{F}_{p}((t))$.
Now we recall the corresponding picture of the relative $p$-adic differential
equations in [AM]. Recall from [AM, Section 2.1.2,2.1.3,2.1.4] in this case,
we have the following rings of the corresponding differential operators with
the corresponding along $t=0$ overconvergence. We use the corresponding
notation

$\displaystyle d_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$ (1.4.9)

to denote the corresponding ring of analytic differential operaters along
$t=0$ having overconvergence. And we use the corresponding notation:

$\displaystyle d_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$ (1.4.10)

to denote the corresponding ring of overconvergent differential operaters
along $t=0$ having overconvergence. Also we have the following truncated
constructions which are also very important applications. We use the
corresponding notation

$\displaystyle d^{(m)}_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$
(1.4.11)

to denote the corresponding ring of analytic differential operaters along
$t=0$ having overconvergence, of order $m\geq 0$. And we use the corresponding
notation:

$\displaystyle d^{(m)}_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$ (1.4.12)

to denote the corresponding ring of overconvergent differential operaters
along $t=0$ having overconvergence, of order $m\geq 0$. And what we will do as
well is the corresponding rings over two points, where we will drop the
corresponding notation of $t=0$.

And we have the following:

######  Setting 1.4.4.

We now consider the corresponding setting up around the local picture in [AM].
We consider the corresponding space
$X_{1}=\mathrm{Spec}\overline{\mathbb{F}}_{p}((t))$. Now we recall the
corresponding picture of the relative $p$-adic differential equations in [AM].
Recall from [AM, Section 2.1.2,2.1.3,2.1.4] in this case, we have the
following rings of the corresponding differential operators with the
corresponding along $t=0$ overconvergence. We use the corresponding notation

$\displaystyle
d_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$ (1.4.13)

to denote the corresponding ring of analytic differential operaters along
$t=0$ having overconvergence. And we use the corresponding notation:

$\displaystyle d_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
(1.4.14)

to denote the corresponding ring of overconvergent differential operaters
along $t=0$ having overconvergence. Also we have the following truncated
constructions which are also very important applications. We use the
corresponding notation

$\displaystyle
d^{(m)}_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
(1.4.15)

to denote the corresponding ring of analytic differential operaters along
$t=0$ having overconvergence, of order $m\geq 0$. And we use the corresponding
notation:

$\displaystyle d^{(m)}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
(1.4.16)

to denote the corresponding ring of overconvergent differential operaters
along $t=0$ having overconvergence, of order $m\geq 0$. And what we will do as
well is the corresponding rings over two points, where we will drop the
corresponding notation of $t=0$.

Over the corresponding rings as above we will consider those arithmetic
coherent modules in [AM, Section 2.1.2,2.1.3,2.1.4]. And we consider the
corresponding holonomic ones. Also we consider the corresponding Frobenius
structures as well.

######  Setting 1.4.5.

We now consider the following sheaves:

$\displaystyle\Pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}:=D^{(0)}_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$
(1.4.17)

and

$\displaystyle\Pi_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}:=D^{(0)}_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}.$
(1.4.18)

Taking the global section we have:

$\displaystyle\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}:=d^{(0)}_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}$
(1.4.19)

and

$\displaystyle\pi_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}:=d^{(0)}_{\mathrm{con},\mathbb{Z}_{p}[[t]],t=0}.$
(1.4.20)

These are the corresponding Robba rings $\mathcal{R}$ and the corresponding
bounded ones $\mathcal{E}^{\dagger}$. And what we will do as well is the
corresponding rings over two points, where we will drop the corresponding
notation of $t=0$.

######  Setting 1.4.6.

We now consider the following sheaves:

$\displaystyle\Pi_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}:=D^{(0)}_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
(1.4.21)

and

$\displaystyle\Pi_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}:=D^{(0)}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}.$
(1.4.22)

Taking the global section we have:

$\displaystyle\pi_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}:=d^{(0)}_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
(1.4.23)

and

$\displaystyle\pi_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}:=d^{(0)}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}.$
(1.4.24)

These are the corresponding Robba rings $\mathcal{R}$ and the corresponding
bounded ones $\mathcal{E}^{\dagger}$. And what we will do as well is the
corresponding rings over two points, where we will drop the corresponding
notation of $t=0$.

Over these sheaves and rings we have the corresponding notion of finite free
$(\varphi,\nabla)$-modules which are for instance studied very extensively in
[Ked8].

######  Proposition 1.4.7.

The corresponding category of all the finite free $(\varphi,\nabla)$-modules
over $\pi_{\mathrm{an},\mathrm{con},{\mathbb{Z}}_{p}[[t]],t=0}$ is equivalent
to the category of all the arithmetic Frobenius modules over
$d_{\mathrm{an},\mathrm{con},{\mathbb{Z}}_{p}[[t]],t=0}$, which are assumed to
be holonomic.

###### Proof.

This is well-known, for instance see [AM, Proposition in Section 2.1.4]. ∎

######  Proposition 1.4.8.

The corresponding category of all the finite free $(\varphi,\nabla)$-modules
over $\pi_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$ is
equivalent to the category of all the arithmetic Frobenius modules over
$d_{\mathrm{an},\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$, which
are assumed to be holonomic.

###### Proof.

This is well-known, for instance see [AM, Proposition in Section 2.1.4]. ∎

####  1.4.2 Derived $(\varphi,\nabla)$-Modules over Stacks

In our current situation, we also consider the corresponding arithmetic
differential modules over the stack
$(\mathrm{Spec}\mathbb{F}_{p}((t)))_{k}/\varphi_{k}$ where
$k=\overline{\mathbb{F}}_{p}$. And we also consider the corresponding
$(\sigma,\nabla)$-modules over the stack
$(\mathrm{Spec}\mathbb{F}_{p}((t)))_{k}/\varphi_{k}$ as well. Here we follow
[Abe] to use the corresponding framework of all the arithmetic $D$-modules
over stacks.

######  Notation 1.4.9.

We consider the lift $\mathbb{Z}_{p}[[t]]$ of the original lifting space, and
we denote this $\mathrm{Spf}(\mathbb{Z}_{p}[[t]])$ by $\\{x,\eta\\}$ with
$\mathrm{Spf}(\mathbb{Z}_{p}[[t]])_{k}$ by
$\\{\widetilde{x},\widetilde{\eta}\\}$. We then have the corresponding stack
$\\{\widetilde{x},\widetilde{\eta}\\}/\varphi_{k}$ with two substacks
$\\{\widetilde{x}\\}/\varphi_{k}$ and $\\{\widetilde{\eta}\\}/\varphi_{k}$.

[Abe] defined the corresponding derived categories of arithmetic $D$-modules
with the desired hearts over stacks. This is not trivial at all since we have
to use the corresponding cohomological descent to realize the seemingly
virtual objects through representations from schemes. In our situation we have
three stacks: $\\{\widetilde{x},\widetilde{\eta}\\}/\varphi_{k}$ with two
substacks $\\{\widetilde{x}\\}/\varphi_{k}$ and
$\\{\widetilde{\eta}\\}/\varphi_{k}$, admitting the corresponding covering
from $\\{\widetilde{x},\widetilde{\eta}\\}$ with two substacks
$\\{\widetilde{x}\\}$ and $\\{\widetilde{\eta}\\}$:

$\displaystyle\\{\widetilde{x},\widetilde{\eta}\\}\rightarrow\\{\widetilde{x},\widetilde{\eta}\\}/\varphi_{k},$
(1.4.25)
$\displaystyle\\{\widetilde{x}\\}\rightarrow\\{\widetilde{x}\\}/\varphi_{k},$
(1.4.26)
$\displaystyle\\{\widetilde{\eta}\\}\rightarrow\\{\widetilde{\eta}\\}/\varphi_{k}.$
(1.4.27)

######  Definition 1.4.10.

We then consider the following bounded derived categories of holonomic
arithmetic $D$-modules over the sheaves of overconvergent rings of
differential operators over the stacks defined by Abe in [Abe, Chapter 2.1]:

$\displaystyle
D^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\mathbb{F}_{p})[[t]]_{k}/\varphi_{k},t=0}$
(1.4.28)

with overconvergence along $t=0$ and

$\displaystyle
D^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\mathbb{F}_{p})[[t]]_{k}/\varphi_{k}}$
(1.4.29)

without overconvergence along $t=0$. We use the following notations to denote
the corresponding hearts:

$\displaystyle
h^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\mathbb{F}_{p})[[t]]_{k}/\varphi_{k},t=0}$
(1.4.30)

with overconvergence along $t=0$ and

$\displaystyle
h^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\mathbb{F}_{p})[[t]]_{k}/\varphi_{k}}$
(1.4.31)

without overconvergence along $t=0$.

We also have the corresponding derived categories over the corresponding
spaces:

######  Definition 1.4.11.

We then consider the following bounded derived categories of holonomic
arithmetic $D$-modules over the sheaves of overconvergent rings of
differential operators over the stacks defined by Abe in [Abe, Chapter 2.1]:

$\displaystyle D^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(?)[[t]],t=0}$
(1.4.32)

with overconvergence along $t=0$ and

$\displaystyle D^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(?)[[t]]}$
(1.4.33)

without overconvergence along $t=0$. We use the following notations to denote
the corresponding hearts:

$\displaystyle h^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(?)[[t]],t=0}$
(1.4.34)

with overconvergence along $t=0$ and

$\displaystyle h^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(?)[[t]]}$
(1.4.35)

without overconvergence along $t=0$. Here
$?=\mathbb{F}_{p},\overline{\mathbb{F}}_{p}$.

By using the corresponding equivalence in proposition 1.4.7 and proposition
1.4.8 we have the following bounded derived categories with the associated
hearts of the $(\varphi,\nabla)$-modules over the Robba rings:

######  Definition 1.4.12.

We then consider the following bounded derived categories of
$(\varphi,\nabla)$-modules over the sheaves of overconvergent rings of
differential operators of order zero over the spaces:

$\displaystyle D^{\flat,\varphi,\nabla}_{\mathrm{con},W(?)[[t]],t=0}$ (1.4.36)

with overconvergence along $t=0$ and

$\displaystyle D^{\flat,\varphi,\nabla}_{\mathrm{con},W(?)[[t]]}$ (1.4.37)

without overconvergence along $t=0$. We use the following notations to denote
the corresponding hearts:

$\displaystyle h^{\flat,\varphi,\nabla}_{\mathrm{con},W(?)[[t]],t=0}$ (1.4.38)

with overconvergence along $t=0$ and

$\displaystyle h^{\flat,\varphi,\nabla}_{\mathrm{con},W(?)[[t]]}$ (1.4.39)

without overconvergence along $t=0$. Here
$?=\mathbb{F}_{p},\overline{\mathbb{F}}_{p}$.

Then we define things over the key stack involved.

######  Definition 1.4.13.

We now define the derived category of all the $(\varphi,\nabla)$-modules over
the bounded Robba rings associated with the stack
$\\{\widetilde{x},\widetilde{\eta}\\}$ with two substacks
$\\{\widetilde{x}\\}$ and $\\{\widetilde{\eta}\\}$:

$\displaystyle\overline{D}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}:=\overline{D}^{\flat}(\overline{h}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0})$
(1.4.40)

and

$\displaystyle\overline{D}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}:=\overline{D}^{\flat}(\overline{h}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}).$
(1.4.41)

Here the categories with $\overline{h}$ symbol on the right are define in the
following way. First
$\overline{h}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}$
is defined to be a stack fibered over the stack
$h^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
parametrizing the corresponding objects
$(M,\varphi,\nabla,\varphi_{1},\varphi_{k})$ where $(M,\varphi,\nabla)$ is
parametrized by the stack below. Then we consider the derived category
$D^{\flat}(\overline{h}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0})$.
And we assume the corresponding cohomology groups live in the same abelian
category. Each object in the corresponding derived category defined in this
way will be some $(M^{\bullet},\varphi,\varphi_{1},\varphi_{k})$ where
$M^{\bullet}$ is bounded complex of $(\varphi,\nabla)$-modules over the Robba
ring, and $\varphi_{1},\varphi_{k}$ are two Frobenius morphisms such that we
have:

$\displaystyle\varphi_{1}^{*}(...\rightarrow M^{m}\rightarrow
M^{m+1}...)\overset{\sim}{\longrightarrow}(...\rightarrow M^{k}\rightarrow
M^{k+1}...),$ (1.4.42) $\displaystyle\varphi_{k}^{*}(...\rightarrow
M^{m}\rightarrow M^{m+1}...)\overset{\sim}{\longrightarrow}(...\rightarrow
M^{k}\rightarrow M^{k+1}...).$ (1.4.43)

Then we do the same construction to the rest one situation. These derived
categories could be further endowed with the structure of derived stacks.

The corresponding arithmetic $D$-modules could be described in the same way:

######  Definition 1.4.14.

We now define the derived category of all the arithmetic $D$-modules over
rings of overconvergent differential operators associated with the stack
$\\{\widetilde{x},\widetilde{\eta}\\}$ with two substacks
$\\{\widetilde{x}\\}$ and $\\{\widetilde{\eta}\\}$:

$\displaystyle\overline{D}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}:=\overline{D}^{\flat}(\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0})$
(1.4.44)

and

$\displaystyle\overline{D}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}:=\overline{D}^{\flat}(\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}).$
(1.4.45)

Here the categories with $\overline{h}$ symbol on the right are define in the
following way. First
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}$
is defined to be a stack fibered over the stack
$h^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$
parametrizing the corresponding objects $(M,\varphi,\varphi_{1},\varphi_{k})$
where $(M,\varphi)$ is parametrized by the stack below. Then we consider the
derived category
$D^{\flat}(\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0})$.
And we assume the corresponding cohomology groups are coherent holonomic
arithmetic $D$-modules over the same space. Each object in the corresponding
derived category defined in this way will be some
$(M^{\bullet},\varphi,\varphi_{1},\varphi_{k})$ where $M^{\bullet}$ is bounded
complex of arithmetic $D$-modules in
$D^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]],t=0}$,
and $\varphi_{1},\varphi_{k}$ are two Frobenius morphisms such that we have:

$\displaystyle\varphi_{1}^{*}(...\rightarrow M^{m}\rightarrow
M^{m+1}...)\overset{\sim}{\longrightarrow}(...\rightarrow M^{k}\rightarrow
M^{k+1}...),$ (1.4.46) $\displaystyle\varphi_{k}^{*}(...\rightarrow
M^{m}\rightarrow M^{m+1}...)\overset{\sim}{\longrightarrow}(...\rightarrow
M^{k}\rightarrow M^{k+1}...).$ (1.4.47)

Then we do the same construction to the rest one situation. These derived
categories could be further endowed with the structure of derived stacks.

######  Theorem 1.4.15.

(Kedlaya, [Ked9, Corollary 4.7]) The category
$\overline{h}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}$
is well-defined, and the category
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}$
is well defined. The category
$\overline{h}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}$
is well-defined, and the category
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}$
is well defined. To be more precise these are abelian categories.

######  Proposition 1.4.16.

With the corresponding notations we have defined so far we have the following
equivalence on the corresponding derived categories:

$\displaystyle\overline{D}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}\overset{\sim}{\longrightarrow}\overline{D}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k},t=0}$
(1.4.48)

and

$\displaystyle\overline{D}^{\flat,\varphi,\nabla}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}\overset{\sim}{\longrightarrow}\overline{D}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},W(\overline{\mathbb{F}}_{p})[[t]]/\varphi_{k}}.$
(1.4.49)

###  1.5 Polydisks and the Quotients

Now we consider the corresponding situation where we have some product of
polydisks. We now choose to look at the corresponding setting up as in the
following:

######  Setting 1.5.1.

We now take the product of $\mathrm{Spf}\mathbb{Z}_{p}[[t_{1}]]$ and
$\mathrm{Spf}\mathbb{Z}_{p}[[t_{2}]]$, as well as the product of the rigid
analytic generic fibers namely the polydisks. We will use the corresponding
notation $I$ to represent the set of two factors. We then have the
corresponding Robba rings $\Pi_{I},\Pi_{I}^{\mathrm{bd}}$ in this multivariate
setting as those in [PZ] and [CKZ], which carry the corresponding multivariate
Frobenius structures by $\varphi_{1}$ and $\varphi_{2}$.

We have the following adic spaces:

$\mathrm{Spa}(\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0},\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0}^{+})\times_{\mathbb{Q}_{p}}\mathrm{Spa}(\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0},\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}^{+}),$

and

$\mathrm{Spa}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0},\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0}^{+})\times_{\mathbb{Q}_{p}}\mathrm{Spa}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0},\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}^{+}),$

and the corresponding $v$-sheaves:

$\mathrm{Spa}(\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0},\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0}^{+})/\varphi_{1}\times_{\mathbb{Q}_{p}}\mathrm{Spa}(\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0},\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}^{+}),$

and

$\mathrm{Spa}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0},\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0}^{+})/\varphi_{1}\times_{\mathbb{Q}_{p}}\mathrm{Spa}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0},\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}^{+}).$

We consider the corresponding picture in the convergent unit-root situation
first.

######  Proposition 1.5.2.

The following categories are equivalent:
1\. The category of all the $\varphi_{I}$-étale $\varphi_{I}$-modules over
perfect bounded Robba ring $\widetilde{\Pi}_{I}^{\mathrm{bd}}$ (as in [CKZ,
Section 2.3]);
2\. The category of all the $\mathbb{Q}_{p}$-representations of:

$\displaystyle\mathrm{Gal}(\widehat{\mathbb{F}_{p}((t_{1}^{1/p^{\infty}}))})\times\mathrm{Gal}(\widehat{\mathbb{F}_{p}((t_{2}^{1/p^{\infty}}))});$
(1.5.1)

3\. The category of all the $\mathbb{Q}_{p}$-representations of:

$\displaystyle\mathrm{Gal}({\mathbb{F}_{p}((t_{1}))})\times\mathrm{Gal}({\mathbb{F}_{p}((t_{2}))});$
(1.5.2)

4\. The category of all the $\mathbb{Q}_{p}$-representations of:

$\displaystyle\mathrm{Gal}(\mathrm{Spec}\mathbb{F}_{p}((t_{1}))/\varphi_{1}\times\mathrm{Spec}\mathbb{F}_{p}((t_{2})));$
(1.5.3)

5\. The category of all the $\varphi$-unit root convergent $F$-Isocrystals
carrying the corresponding actions of $\varphi_{1}$ and $\varphi_{2}$.

###### Proof.

From 1 to 2 this is the corresponding [CKZ, Theorem 6.16]. From 2 to 3, this
is trivial. From 3 to 4 this is the corresponding Drinfeld’s lemma in its
original form. From 4 to 5 this is following Crew-Katz [Cr2, Theorem 2.1]. ∎

We now consider the corresponding arithmetic $D$-modules over polydisks and
the corresponding quotient stacks. We first consider the corresponding
arithmetic $D$-modules over the product, namely we consider the following:

######  Setting 1.5.3.

We consider the corresponding ring $\Pi_{I}^{\mathrm{bd}}$ with the
corresponding ring of differential operators
$D_{\mathrm{con},I,t_{1}=t_{2}=0}^{\mathrm{bd}}$. We then have the
corresponding derived category of all the holonomic arithmetic $D$-modules
over $D_{\mathrm{con},I,t_{1}=t_{2}=0}^{\flat,\mathrm{holo},\varphi}$ carrying
the corresponding Frobenius $\varphi$.

######  Remark 1.5.4.

Please note that the corresponding notation $t_{1}=t_{2}=0$ does not indicate
the corresponding point $(0,0)$, which essentially means the larger union
coming from points where $t_{1}=0$ or $t_{2}=0$.

As in the corresponding situation we encountered before we consider the
following derived stacks fibered over the categories considered in the
previous setting.

######  Setting 1.5.5.

We have the corresponding derived category of all the holonomic arithmetic
$D$-modules over
$D_{\mathrm{con},I,t_{1}=t_{2}=0}^{\flat,\mathrm{holo},\varphi}$ carrying the
corresponding Frobenius $\varphi$. We now look at the corresponding quotient
regarded as adic stack:

$\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}\times_{\mathbb{Q}_{p}}\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}).$

Then we consider the corresponding bounded derived category of all the
holonomic arithmetic $D$-modules over the stack as above, which we denote it
by
$\overline{D}_{\mathrm{con},I/\varphi_{1},t_{1}=t_{2}=0}^{\flat,\mathrm{holo},\varphi}$
which is defined by using bounded derived category of the abelian category
$\overline{h}_{\mathrm{con},I/\varphi_{1},t_{1}=t_{2}=0}^{\flat,\mathrm{holo},\varphi}$
consisting of all the object taking the form of
$(M,\varphi,\varphi_{1},\varphi_{2})$ where we have that $(M,\varphi)$ lives
in the heart $h_{\mathrm{con},I,t_{1}=t_{2}=0}^{\flat,\mathrm{holo},\varphi}$.

######  Definition 1.5.6.

For any perfectoid affinoid $\mathrm{Spa}(A,A^{+})$ covering the diamond
$\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$,
we will consider those objects over the ring
$A\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$
and
$A\widehat{\otimes}_{\mathbb{Q}_{p}}D_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$.
We first define a $(\varphi_{2},\nabla_{2})$ modules over
$A\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$
to be finite projective module over
$A\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$
carrying the corresponding Frobenius and the corresponding connection $\nabla$
which is $A$ linear. Then we define over the diamond
$\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$
the corresponding relative $(\varphi_{2},\nabla_{2})$-modules which are by the
natural glueing of the families of such objects defined as in the above. Here
the corresponding Frobenius and $\nabla$ are required to be compatible as in
the absolute situation.

######  Definition 1.5.7.

For any perfectoid affinoid $\mathrm{Spa}(A,A^{+})$ covering the diamond
$\mathrm{Sp}(\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$,
we will consider those objects over the ring
$A\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{\mathrm{an,con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$
and
$A\widehat{\otimes}_{\mathbb{Q}_{p}}D_{\mathrm{an,con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$.
We first define a $(\varphi_{2},\nabla_{2})$ modules over
$A\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{\mathrm{an,con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$
to be finite projective module over
$A\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{\mathrm{an,con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$
carrying the corresponding Frobenius and the corresponding connection $\nabla$
which is $A$ linear. Then we define over the diamond
$\mathrm{Sp}(\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$
the corresponding relative $(\varphi_{2},\nabla_{2})$-modules which are by the
natural glueing of the families of such objects defined as in the above. Here
the corresponding Frobenius and $\nabla$ are required to be compatible as in
the absolute situation.

######  Definition 1.5.8.

Consider $\nabla_{2}$ and the context in the previous two definitions, we can
define the corresponding generic radius of convergence with respect to each
radius for the second factor relative to some perfectoid Banach affinoid $A$.
After realizing the corresponding bundle as some section over some interval
$[s_{2},r_{2}]$, we define the corresponding radius
$R^{\mathrm{pre}}(M,\rho_{2})$ for each $\rho_{2}\in[s_{2},r_{2}]$ to be:

$\displaystyle\min\\{p^{-1/(p-1)}\rho_{2},\varlimsup_{r\rightarrow\infty}\|{\partial_{t_{2}}^{r}}\|_{A}^{1/r}\\}.$
(1.5.4)

Here $\partial_{t_{2}}=\frac{\partial}{\partial{t_{2}}}$. We call the module
is basically solvable at $1$ relative to $A$ if we have that the following
equality:

$\lim_{\rho_{2}\rightarrow
1}p^{-1/(p-1)}R^{\mathrm{pre}}(M,\rho_{2})^{-1}\rho_{2}^{-1}=1.$

The corresponding consideration is as in the following. This sort of
observation is inspired by those due to Kedlaya [Ked10] (more precise [Ked10,
Chapter 6]). First by taking the quotient by $\varphi_{1}$ of the
corresponding product adic space, starting from any corresponding
overconvergent $F$-isocrystal $M$ over this adic stack, we can regard $M$ as a
corresponding $(\varphi_{2},\nabla_{2})$-modules over the stack relative to
the stack forming by the quotient of the first factor.

######  Conjecture 1.5.9.

For any $\Phi$-equivariant $F$-isocrystal
$(M,\varphi,\varphi_{1},\varphi_{2})$ in the category
$\overline{h}_{\mathrm{con},I/\varphi_{1},t_{1}=t_{2}=0}^{\flat,\mathrm{holo},\varphi}$.
We regard this as a corresponding object taking the structure of relative (to
the stack
$\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$)
$(\varphi_{2},\nabla_{2})$-module over the corresponding ring

$\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}\widehat{\otimes}_{\mathbb{Q}_{p}}\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{2}]],t_{2}=0}$.

Then we have that with respect to this $(\varphi_{2},\nabla_{2})$-module
structure over relative Robba ring with coefficients in the diamond [Sch], $M$
is solvable at 1 uniformly with respect each fiber $M_{x_{1}}$ for each
$x_{1}$ over the stack
$\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$)
namely we have $\lim_{\rho_{2}\rightarrow
1}R(M_{x_{1}},\rho_{2})\rho_{2}^{-1}=1$ uniformly for all
$x_{1}\in\mathrm{Sp}(\pi_{\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$.

######  Remark 1.5.10.

Over the corresponding point $x_{1}$ we have that the resulting underlying
$(\varphi_{2},\nabla_{2})$-module structure should be definitely
overconvergent namely solvable at 1. However the issue is that the
corresponding family version of this condition is not automatically
guaranteed.

######  Remark 1.5.11.

This will indicate some possible form of Drinfeld’s lemma in this context,
namely any $F$-isocrystal in the category
$\overline{h}_{\mathrm{an,con},I/\varphi_{1},t_{1}=t_{2}=0}^{\flat,\mathrm{holo},\varphi}$
is conjectured to behave as if it is overconvergent with respect the
$(\varphi_{2},\nabla_{2})$-module structure relative to the diamond
$\mathrm{Sp}(\pi_{\mathrm{an},\mathrm{con},\mathbb{Z}_{p}[[t_{1}]],t_{1}=0})/\varphi_{1}$.
Moreover this should also have the analog in the perverse $p$-adic
differential equation setting, and even in the setting after [CKZ], [KL1] and
[KL2] by considering diamond coefficients [Sch], [SW].

Let us mention a little bit about the corresponding some motivation from the
work [Ked9] around the corresponding relative Frobenius modules and relative
differential equations. Consider now the scheme $X$ we specified at the very
beginning of this paper, and take $k=\overline{\mathbb{F}}_{p}$.

######  Definition 1.5.12.

We define the category
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}/\varphi_{k}}$
to be the corresponding categories fibered over
$h^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}}$ respectively endowed
with further pullbacks of $\varphi_{1}$ and $\varphi_{k}$ realizing the
isomorphisms as in the previous section.

######  Theorem 1.5.13.

(Kedlaya [Ked9, Theorem 7.3, Corollary 7.4]) For any F-isocrystal object $M$
in
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}/\varphi_{k}}$
we have that there is a decomposition of $M$ with respect to the slopes for
$\varphi_{k}$:

$M=\bigoplus_{a/b}M_{a/b}$

such that for each $M_{a/b}$ we have that $M_{a/b}^{\varphi_{k}^{b}-p^{a}}$ is
an $F$-isocrystal over $X$.

This means that when we have that $X$ is just the formal (punctural) disc over
$\mathbb{F}_{p}$ as we considered above, and when we have the corresponding
overconvergence along $t_{1}=0$ we then have that the corresponding
decomposition as above of a general $F$-isocrystal object in
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k},t_{1}=0}$
into:

$\bigoplus_{a/b}M_{a/b}$

such that for each $M_{a/b}$ we have that $M_{a/b}^{\varphi_{k}^{b}-p^{a}}$ is
a $(\varphi_{1},\nabla_{1})$-module over the bounded Robba ring of
$\mathbb{Q}_{p}$.

In this current simplified and local situation we have the following
proposition:

######  Proposition 1.5.14.

(After Kedlaya [Ked9, Theorem 7.3, Corollary 7.4]) Consider the following two
categories. The first one is the category of all the
$(\varphi_{1},\nabla_{1})$-modules over the punctural local unit disc
$\mathbb{Z}_{p}[[t]][1/t]$. The second one is the corresponding category of
all the objects in
$\overline{h}^{\flat,\varphi,\nabla}_{\mathrm{con},X_{k}/\varphi_{k}}$ taking
the general form $M$ such that we have $M$ decomposes as:

$M=M_{0}$

in the sense discussed above, namely in the decomposition:

$\bigoplus_{a/b}M_{a/b}$

we will have just one component with $a/b=0$. Then we have that the two
categories are equivalent.

###### Proof.

The functor realize this equivalent is just taking the corresponding pullback
along the corresponding projection $X_{k}\rightarrow X$. Then to show the
corresponding essential surjectivity, we start from a corresponding
$(\varphi_{1},\varphi_{k})$-object over the corresponding product space,
denoted by $M_{k}$ satisfying the corresponding condition in the statement of
the proposition. Then result follows from Kedlaya’s theorem mentioned above. ∎

######  Corollary 1.5.15.

Let
$\overline{h}^{\flat,\mathrm{holo},\varphi,\varphi_{k}-\mathrm{ur}}_{\mathrm{con},X_{k}/\varphi_{k}}$
denote the corresponding category of objects taking the general form $m$ in
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}/\varphi_{k}}$
which as above decomposed as above (equivalently regarded as the corresponding
$(\varphi_{1},\nabla_{1})$-object):

$M=M_{0}.$

Then we have the following equivalence:

$\overline{h}^{\flat,\mathrm{holo},\varphi,\varphi_{k}-\mathrm{ur}}_{\mathrm{con},X_{k}/\varphi_{k}}\overset{\sim}{\rightarrow}\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X}$

and

$D^{\flat}(\overline{h}^{\flat,\mathrm{holo},\varphi,\varphi_{k}-\mathrm{ur}}_{\mathrm{con},X_{k}/\varphi_{k}})\overset{\sim}{\rightarrow}D^{\flat}(\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X}).$

###  1.6 $L$-Functions and Product Formula for Perverse $p$-adic Differential
Equations over Stacks

Now we define the corresponding $L$-functions. For basic materials around the
corresponding definitions around a curve, we refer closely to the work [AM].

######  Setting 1.6.1.

We work over finite level over $k=\mathbb{F}_{p^{r}}$, let $X$ be the
corresponding curve as in the previous section.

######  Definition 1.6.2.

We define the corresponding $\varphi_{k}$-cohomology of any object
$(M_{k},\varphi,\varphi_{1},\varphi_{k})$ in
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}/\varphi_{k}}$
by considering the corresponding hypercohomology of the following complex of
holonomic $D$-modules as the definition of the $\varphi_{k}$-cohomology:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{k}-1}$$\textstyle{M_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

######  Definition 1.6.3.

Let $X$ be just the disc $\mathrm{Spec}(\overline{\mathbb{F}}_{p}[[t]][1/t])$.
We define the corresponding $\varphi_{k}$-cohomology of any object
$(M_{k},\varphi,\varphi_{1},\varphi_{k})$ in
$\overline{h}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}/\varphi_{k}}$
by considering the corresponding hypercohomology of the following complex of
holonomic $D$-modules as the definition of the $\varphi_{k}$-cohomology:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{k}-1}$$\textstyle{M_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

We denote the corresponding complex in the above $C_{\varphi_{k}}^{\bullet}$.
Now we consider the corresponding projection map $f_{k}:X_{k}\rightarrow X$ in
the following development, and we will consider any geometric point
$i_{x}:\overline{x}\rightarrow X$, and we will consider the corresponding
structure map $h_{k}:X_{k}\rightarrow\mathrm{Spec}k$ with
$h:X\rightarrow\mathrm{Spec}\mathbb{F}_{p}$. We use the corresponding notation
$i^{k}_{x}$ to denote the corresponding base change of the morphism
$i_{x}:\overline{x}\rightarrow X$ for any $x\in|X|$.

######  Definition 1.6.4.

We define the corresponding $L$-function of any object
$(M^{\bullet},\varphi,\varphi_{1},\varphi_{k})$ in
$\overline{D}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}/\varphi_{k}}$
by using the corresponding object $(M^{\bullet},\varphi)$ in
$D^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X_{k}}$ which is denoted by
$L_{X_{k}/\varphi_{k}}(M^{\bullet},s)$ defined by the following formula:

$\displaystyle
L_{X_{k}/\varphi_{k}}(M^{\bullet},s):=\prod_{i}\prod_{x\in|X|}\mathrm{det}\left(1-s^{\delta_{x}}\varphi_{1}^{\delta_{x}}|\mathcal{H}^{i}(i_{x}^{+}f_{k,+}\mathrm{Tot}(C^{\bullet}_{\varphi_{k}}M^{\bullet}))\right)^{(-1)^{i+1}}.$
(1.6.1)

Now we assume that the object in the previous definition
$(M^{\bullet},\varphi,\varphi_{1},\varphi_{k})$ satisfy the corresponding
assumption:

######  Assumption 1.6.5.

This assumption requires that $M^{\bullet}$ satisfy the corresponding
condition which says that $M^{\bullet,\varphi_{k}=1}$ lives in
$\overline{D}^{\flat,\mathrm{holo},\varphi}_{\mathrm{con},X}$. Certainly the
base change of any such object in the latter category over $X$ satisfy the
corresponding condition.

Under this assumption we can now proceed to relate the corresponding
$L$-function we defined to the corresponding $L$-function of the objects over
$X$ after considering the corresponding local correspondence we considered in
the above after Kedlaya’s theorem [Ked9, Theorem 7.3, Corollary 7.4, Lemma
7.2].

######  Proposition 1.6.6.

We have the following equality:

$L_{X_{k}/\varphi_{k}}(M^{\bullet},s)=\prod_{i}\mathrm{det}\left(1-s\varphi_{1}|\mathcal{H}^{i}((M^{\bullet})^{\varphi_{k}=1})\right)^{(-1)^{i+1}}.$

###### Proof.

This is through the following computation. We do have locally the
corresponding correspondence on the corresponding local $L$-factors under the
1.6.5 by proposition 1.5.14 and corollary 1.5.15. Then we compute:

$\displaystyle L_{X_{k}/\varphi_{k}}(M^{\bullet},s)$
$\displaystyle:=\prod_{i}\prod_{x\in|X|}\mathrm{det}\left(1-s^{\delta_{x}}\varphi_{1}^{\delta_{x}}|\mathcal{H}^{i}(i_{x}^{+}f_{k,+}\mathrm{Tot}(C^{\bullet}_{\varphi_{k}}M^{\bullet}))\right)^{(-1)^{i+1}}$
(1.6.2)
$\displaystyle=\prod_{i}\prod_{x\in|X|}\mathrm{det}\left(1-s^{\delta_{x}}\varphi_{1}^{\delta_{x}}|\mathcal{H}^{i}((i_{x}^{+}(M^{\bullet}))^{\varphi_{k}=1})\right)^{(-1)^{i+1}}$
(1.6.3)
$\displaystyle=\prod_{i}\mathrm{det}\left(1-s\varphi_{1}|\mathcal{H}^{i}((M^{\bullet})^{\varphi_{k}=1})\right)^{(-1)^{i+1}}.$
(1.6.4)

∎

##  Chapter 2 $\infty$-Categorical Perverse $p$-adic Differential Equations
over $\mathrm{Bun}_{G}$

###  2.1 Introduction

####  2.1.1 Motivation

The celebrated Weil’s conjecture on the Tamagawa number has been tackled in
the number field case by Langlands [Lan], Kottwitz [Ko] and so on. In the
function field situation, Lurie and Gaitsgory [GL1] invent a robust tool for
studying the product formula in the level of $\infty$-category. We use the
following notation to establish the corresponding introduction. First we
consider $X$ a smooth proper curve defined over a finite field
$\mathbb{F}_{p}$, and we consider $G$ a smooth group scheme affine over $X$:

$G\rightarrow X.$

We are going to assume that the the generic fiber of the group scheme $G$ is
semisimple and simply-connected, and we assume that the fibers of the group
scheme $G$ are connected. Then we use $\mathrm{Bundle}_{G}(X)$ to denote the
smooth Artin moduli stacks of $G$-bundles over the the curve $X$, and we will
for any $x\in X$ use the notation $\mathrm{Bundle}_{G}(\\{x\\})$ to denote the
classifying stack of $G$-bundles at the point $x$. Then we use the notation
$K_{X}$ to denote the function field of the curve, and for each point $x\in X$
we use the notation $K_{X,x}$ to denote the corresponding completion of the
field $K_{X}$ at the point $x\in X$, with the corresponding ring of integer
$\mathcal{O}_{X,x}$, and the corresponding residue field $\kappa(x)$. In this
situation we have that the following Weil’s conjecture for the global function
field $K_{X}$ defined above:

######  Conjecture 2.1.1.

In the notations above, we have the following well-defined equality:

$\frac{|\mathrm{Bundle}_{G}(X)(\mathbb{F}_{p})|}{q^{\mathrm{dim}\mathrm{Bundle}_{G}(X)}}=\prod_{x\in
X}\frac{|\mathrm{Bundle}_{G}(\\{x\\})(\kappa(x))|}{q^{\mathrm{dim}\mathrm{Bundle}_{G}(\kappa(x))}}.$

This could be regarded as sort of local and global compatibility or a product
formula. This is now a theorem due to Lurie and Gaitsgory [GL1] by using
$\ell$-adic Grothendieck-Lefschetz trace formulas where $\ell$ is different
from the prime $p$. Actually to be more precise they proved the following
formula:

######  Theorem 2.1.2.

(Lurie-Gaitsgory [GL1, Theorem 1.4.4.1])

$\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathbb{Q}_{\ell}))=\prod_{x\in
X}\frac{|\mathrm{Bundle}_{G}(\\{x\\})(\kappa(x))|}{q^{\mathrm{dim}\mathrm{Bundle}_{G}(\kappa(x))}}.$

Actually by simple observation on the ’$\ell$-independence’, for instance see
[Abe1, Section 3.2], one has that actually the following corollary, where the
$p$-adic cohomology theory is based on [Abe1]:

######  Corollary 2.1.3.

$\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathbf{Dual}_{p}))=\prod_{x\in
X}\frac{|\mathrm{Bundle}_{G}(\\{x\\})(\kappa(x))|}{q^{\mathrm{dim}\mathrm{Bundle}_{G}(\kappa(x))}}.$

Note that by directly using the cohomological language we
have111$\mathbf{Dual}_{p}$ is the $p$-adic dualizing object as in [Abe1,
Section 1.1.4, and above 2.2.23]. :

######  Corollary 2.1.4.

$\displaystyle\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}($
$\displaystyle
X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathbf{Dual}_{p}))$
(2.1.1) $\displaystyle=\prod_{x\in
X}\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}(\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathbf{Dual}_{p})).$
(2.1.2)

From this we have many interesting results, first the left hand is of finite
type, while the right hand side is also well-defined. These are not obvious a
priori at all. So one has very good understanding on the $p$-adic cohomology
of $\mathrm{Bundle}_{G}(X)$ which is smooth but not quasi-compact. Our idea is
to study the cohomology with non-trivial coefficients in the category of
holonomic arithmetic $\mathcal{D}$-modules, for instance we use the notation
$\mathcal{E}\in D^{\mathrm{b}}_{\mathrm{hol}}(X)$ to denote such an module in
the $p$-adic setting. Moreover we have to assume $\mathcal{E}$ is augmented
$\mathbb{E}_{\infty}$-ring222In fact, the sheaves of rings of differential
operators whatever the forms are actually unfortunately noncommutative,
therefore maybe one should really focus on the corresponding larger categories
of the corresponding $A_{\infty}$ noncommutative algebras with the
augmentation. In some sense, this is also inspired by [BS], namely the
prismatic $E_{\infty}$-rings.. Our main conjecture is that we have the
following well defined statement:

######  Conjecture 2.1.5.

$\displaystyle\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}($
$\displaystyle
X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E}))$
(2.1.3) $\displaystyle=\prod_{x\in
X}\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}(\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{x}^{*}{\mathcal{E}}_{x})).$
(2.1.4)

Note that the cohomology groups involved for the the stack
$\mathrm{Bundle}_{G}(\overline{X})$ is essentially nonzero in infinite degrees
as in the $\ell$-adic setting in [GL1] but once we consider the cotangent
complex we could have more detailed description on that:

######  Conjecture 2.1.6.

$\mathrm{H}^{*}(\mathrm{cot}\mathrm{C}^{*}(\mathrm{Bundle}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E}))$

is finite dimensional over $\mathbb{Q}_{p}$.

This is essentially true since we will establish the following quasi-
isomorphism:

$\mathrm{cot}\mathrm{C}^{*}(\mathrm{Bundle}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E})\overset{\mathrm{quasi}}{\longrightarrow}\mathrm{C}^{*}((X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathcal{M}(G)^{\mathrm{Gross}}_{\mathcal{E}})$

where the latter is the refined rigid Gross $G$-motive with more general
coefficients, which is a $p$-adic analog of Gross $G$-motive in the
$\ell$-adic setting defined in [Gro1]. However, this quasi-isomorphism is not
obvious at all, since actually this is essentially the main thing to prove. A
little bit weaker result will be the following Euler product formula for the
L-functions attached to the rigid Gross $G$-motive (which we will define in
our setting which is an $\infty$-$p$-adic arithmetic $\mathcal{D}$-module)
which we define in the following:

######  Definition 2.1.7.

We define the rigid $L$-function attached to rigid Gross $G$-motive (in
general $p$-adic coefficient) as the following:

$L_{\mathcal{M}(G),\mathrm{Frob}^{-}1,\mathcal{E}}(t):=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G)_{\mathcal{E}})^{-1}$

Then we will show that we have the following Euler product formula for this
$L$-function as in the following:

$\displaystyle L_{\mathcal{M}(G),\mathrm{Frob}^{-}1,\mathcal{E}}(t)$
$\displaystyle:=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G)_{\mathcal{E}})^{-1}$
(2.1.5) $\displaystyle=\prod_{x\in
X}\mathrm{det}(1-t\mathrm{Frob}_{x}^{-1}|\mathrm{H}^{*}(\overline{X},M(G_{x})_{\mathcal{E}_{x}})^{-1}.$
(2.1.6)

Actually if $\mathcal{E}$ is just the trivial coefficient, then the special
value $L_{\mathcal{M}(G),\mathrm{Frob}^{-}1,\mathcal{E}}(1)$ is actually just
the inverse of $\prod_{x\in
X}\frac{|\mathrm{Bundle}_{G}(\\{x\\})(\kappa(x))|}{q^{\mathrm{dim}\mathrm{Bundle}_{G}(\kappa(x))}}$.

####  2.1.2 Approaches

The main goal of us is to establish a parallel story by using $p$-adic
cohomology studied by many people in the past 20 years. A well-defined at
least constructible coefficient systems for at least quasi-projective schemes
over $\mathbb{F}_{p}$ has been the central problem in the literature for a
quite long time in the past, for instance people are trying to determine down
the correct derived category to consider and the correct six operation to
consider. The work of [Ca], [Abe1] and [AC1] has given some relatively good
answer, which led some essential breakthrough from Kedlaya [Ked11], [Ked12] on
Deligne’s conjecture on petits camarades [De1]. The answer basically gives a
well-defined t-structure in the $p$-adic setting, as well as a derived
category of holonomic arithmetic $\mathcal{D}$-modules where we do have well-
defined six operations. More importantly the construction works for algebraic
stacks. For instance, the six operation are also defined for admissible stacks
over finite fields. This allows Abe to give a proof on a $p$-adic geometric
Langlands correspondence, which links the set of isocrystals and cuspidal
automorphic representation of the function fields.

In our situation, a $p$-adic story is expected. Our idea is parallel to Lurie
and Gaitsgory [GL1], namely we first consider the higher categorical
enrichment of the suitable derived category of arithmetic
$\mathcal{D}$-modules, then define the corresponding Gross motive [Gro1]
attached to the group scheme $G$ (which we will call it to be rigid Gross
$G$-motive). Then we will check the product formula for the Grothendieck-
Lefschetz trace formula, which will finish the proof as in the $\ell$-adic
proof.

###  2.2 Arithmetic $\mathcal{D}$-modules over Schemes

####  2.2.1 The $\infty$-category $\mathbb{D}^{\mathrm{b}}_{\mathrm{hol}}(X)$
of constructible objects

In this section we are going to define the $\infty$-category
$\mathbb{D}^{\mathrm{b}}_{\mathrm{hol}}(X)$ as the $\infty$-categorical
enrichment of the usual derived category of holonomic complexes defined for
quasi-projective schemes by [Abe1]. In order to be more precise we consider
the following notations. First we fix a finite extension $k/\mathbb{F}_{p}$.
And we let $X$ be a quasi-projective scheme over $k$. Then we choose some lift
of $X$ over $R$ which is a complete discretely valued ring whose residue field
is $k$, and we are going to use the notation $K$ to denote its fraction field.
We keep the notation $X$ when we talk about its lift, if this does not cause
any confusion.

######  Definition 2.2.1.

We are going to systematically use the language of $\infty$-category from
[Lurie1]. To be more precise we first consider the derived category of the
constructible holonomic $\mathcal{D}$-modules denoted by
$D^{\mathrm{b}}_{\mathrm{hol}}(X)$, which play the role of the
$D^{\mathrm{b}}_{\ell}(X/k)$ which is the $\ell$-adic complexes constructible
over $X$ as in [GL1]. Then we consider the corresponding $\infty$-categorical
enrichment, which is to say we are going to construct a simplicial set from
$D^{\mathrm{b}}_{\mathrm{hol}}(X)$, which we will denote it by
$\mathbb{D}^{\mathrm{b}}_{\mathrm{hol}}(X)$, whose homotopy category gives
rise to the derived category recalled above, which is to say
$D^{\mathrm{b}}_{\mathrm{hol}}(X)$.

We remard that here the category $\mathrm{Hol}(X)$ defined in [Abe1,
Definition 1.1.1] does not natural have enough injectives, which means one
could not directly apply the construction in [Lurie2, Section 1.3.2, Section
1.3.5]. Instead we consider the following construction:

######  Definition 2.2.2.

We will use the notation $\mathrm{Hol}(X)$ and $\mathrm{Con}(X)$ to denote the
corresponding categories of holonomic $\mathcal{D}$-modules and the
corresponding contructible ones defined in [Abe1, Definition after Proposition
1.3.3]. We use the notation $\mathrm{Ind}(\mathrm{Con}(X))$ to denote the
corresponding ind-category associated to the category $\mathrm{Con}(X)$. Note
that this is now a Grothendieck category which admits enough injectives, then
we apply the construction in [Lurie2, Section 1.3.2, Section 1.3.5] to get the
corresponding derive $\infty$-catogory, which we will denote it by
$\mathbb{D}(\mathrm{Ind}(\mathrm{Con}(X)))$. Then we define:

$\mathbb{D}_{\mathrm{hol}}(X):=\mathbb{D}(\mathrm{Con}(X)):=\mathbb{D}_{\mathrm{Con}(X)}(\mathrm{Ind}(\mathrm{Con}(X))),$

where the latter is the full subcategory of
$\mathbb{D}(\mathrm{Ind}(\mathrm{Con}(X)))$ consists of all the complex whose
homologies live in $\mathrm{Con}(X)$. Passing to the homotopy categories we
have:

$D_{\mathrm{hol}}(X):=D(\mathrm{Con}(X)):=D_{\mathrm{Con}(X)}(\mathrm{Ind}(\mathrm{Con}(X))).$

######  Remark 2.2.3.

By [Lurie2, Section 1.3] we have that the $\infty$-category
$\mathbb{D}_{\mathrm{hol}}(X)$ is a stable $\infty$-category.

###### Proof.

It suffices to prove that the cofibers are preserved under the embedding of
the abelian categories
$\mathrm{Con}(X)\rightarrow\mathrm{Ind}\mathrm{Con}(X)$. Then to finish, one
check the long exact sequence for any mapping cone $C^{\bullet}(f)$, but then
one finishes since on the level of derived category we have the corresponding
result:

$D(\mathrm{Con}(X))\overset{\sim}{\rightarrow}D_{\mathrm{Con}(X)}(\mathrm{Ind}\mathrm{Con}(X)).$

∎

####  2.2.2 Inverse and Direct Images in Derived $\infty$-Category

Recall that [Abe1] finishes the searching for the six operations at least for
quasi-projective schemes over $\mathbb{F}_{p}$, implicitly relying on
Kedlaya’s work on Shiho’s conjecture [Ked4], [Ked5], [Ked6], [Ked7].
Everything could be promoted to the derived $\infty$-categorical constructions
as above. We recall everything here. To be more precise we consider any
morphism of quasi-projective schemes over $k$, which we denote it as
$f:X\rightarrow Y$. Then we have four derived functor:

$\displaystyle f_{!}:D^{\mathrm{b}}_{\mathrm{hol}}(X/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(Y/K),$ (2.2.1) $\displaystyle
f_{+}:D^{\mathrm{b}}_{\mathrm{hol}}(X/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(Y/K),$ (2.2.2) $\displaystyle
f^{!}:D^{\mathrm{b}}_{\mathrm{hol}}(Y/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(X/K),$ (2.2.3) $\displaystyle
f^{+}:D^{\mathrm{b}}_{\mathrm{hol}}(Y/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(X/K).$ (2.2.4)

We would like to promote the operations onto the derived $\infty$-level, which
is to say that we have the following derived $\infty$-functors having the form
as in the following:

$\displaystyle
f_{!}:\mathbb{D}_{\mathrm{hol}}(X/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(Y/K),$
(2.2.5) $\displaystyle
f_{+}:\mathbb{D}_{\mathrm{hol}}(X/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(Y/K),$
(2.2.6) $\displaystyle
f^{!}:\mathbb{D}_{\mathrm{hol}}(Y/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(X/K),$
(2.2.7) $\displaystyle
f^{+}:\mathbb{D}_{\mathrm{hol}}(Y/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(X/K).$
(2.2.8)

Not that as in [Abe1], our construction of derived $\infty$-category used
essentially the techniques of inductive objects. Therefore the six operator
should be deduced from the following ones:

$\displaystyle\underline{f_{!}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(X))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(Y)),$
(2.2.9)
$\displaystyle\underline{f_{+}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(X))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(Y)),$
(2.2.10)
$\displaystyle\underline{f^{!}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(Y))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(X)),$
(2.2.11)
$\displaystyle\underline{f^{+}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(Y))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(X)).$
(2.2.12)

which are derived from the level of ind-category
$\mathrm{Ind}\mathrm{Con}(X)$. Here the definition of the derived
$\infty$-category $\mathbb{D}$ is essentially [Lurie2, Section 1.3.5]. First
we have the following lemma based on the discussion in [Abe1]. And we
following the notations (for instance the constructible t-structure). Our
discussion is essentially based on the observation from [Abe1, Section 1.2].
One could actually regard the discussion presented here as sort of
$\infty$-enrichment on the $t$-structures introduced in [Abe1], under the
philosophy of Riemann-Hilbert correspondence.

######  Lemma 2.2.4.

Let $f$ as a morphism as above, we have that the pullback
$\underline{\underline{f^{+}}}:\mathrm{Ind}\mathrm{Con}(Y)\rightarrow\mathrm{Ind}\mathrm{Con}(X)$
is well defined which sends objects in $\mathrm{Ind}\mathrm{Con}(Y)$ to those
in $\mathrm{Ind}\mathrm{Con}(X)$, and the functor is c-t-exact.

###### Proof.

See [Abe1, Section 1.3]. ∎

######  Lemma 2.2.5.

Over the level of derived $\infty$-category, we have the right adjoint
$\underline{\underline{f^{\wedge}_{+}}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(X))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(Y))$
of the functor in the previous lemma which is well defined, and which is left
c-t-exact.

Then we consider the étale morphism:

######  Lemma 2.2.6.

Let $f$ as a morphism as above that is étale, then we have that first the
pullback functor
$f^{+}:\mathrm{Ind}\mathrm{Con}(Y)\rightarrow\mathrm{Ind}\mathrm{Con}(X)$
which admits a left adjoint functor over the level of derived
$\infty$-categories
$\underline{\underline{f^{\wedge}_{!}}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(X))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(Y))$
.

###### Proof.

Everything extends from the level of abelian category to this setting. Also
see [Abe1, Section 1.2.2]. ∎

Finally for the proper morphisms we have:

######  Lemma 2.2.7.

Let $f$ as a morphism above that is proper, then the $!$-pullback functor
$\underline{\underline{f^{!}}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(Y))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(X))$
is well defined.

####  2.2.3 $\infty$-Ind-Categories

In this section, we are going to define the corresponding $\infty$-Ind-
Categorie which will contain essentially the constructible objects in our
previous discussion. This will be a $p$-adic analog of the corresponding
$\infty$-category of $\ell$-adic sheaves in [GL1]. First we give the following
definition:

######  Definition 2.2.8.

We are going to use the notation $\overline{\mathbb{D}}_{\mathrm{hol}}(X/K)$
to denote the $\infty$-category $\mathrm{Ind}\mathbb{D}_{\mathrm{hol}}(X/K)$
constructed from the $\infty$-category $\mathbb{D}_{\mathrm{hol}}(X/K)$.

Then we discuss the extension of the operations to the category
$\overline{\mathbb{D}}_{\mathrm{hol}}(X/K)$ as in the $\ell$-adic situation as
considered in [GL1].

Now base on the discussion in the previous subsection on the derived functors
over the derived $\infty$-category, one could make the discussion more general
to the derived categories defined above. So first the functor associated to
any morphism $f:X\rightarrow Y$ between any quasi-projective schemes over $k$:

$\underline{\underline{f^{+}}}:\mathbb{D}_{\mathrm{hol}}(Y/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(X/K),\underline{\underline{f^{\wedge}_{+}}}:\mathbb{D}_{\mathrm{hol}}(X/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(Y/K),$

extend uniquely to the derived $\infty$-categories:

$f^{+}:\overline{\mathbb{D}}_{\mathrm{hol}}(Y/K)\rightarrow\overline{\mathbb{D}}_{\mathrm{hol}}(X/K),f_{+}:\overline{\mathbb{D}}_{\mathrm{hol}}(X/K)\rightarrow\overline{\mathbb{D}}_{\mathrm{hol}}(Y/K).$

As above, when $f$ is étale we have that from the level of the constructible
objects we have the following morphisms:

$\underline{\underline{f^{+}}}:\mathbb{D}_{\mathrm{hol}}(Y/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(X/K),\underline{\underline{f^{\wedge}_{!}}}:\mathbb{D}_{\mathrm{hol}}(X/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(Y/K),$

extend uniquely to the derived $\infty$-categories:

$f^{+}:\overline{\mathbb{D}}_{\mathrm{hol}}(Y/K)\rightarrow\overline{\mathbb{D}}_{\mathrm{hol}}(X/K),f_{!}:\overline{\mathbb{D}}_{\mathrm{hol}}(X/K)\rightarrow\overline{\mathbb{D}}_{\mathrm{hol}}(Y/K).$

Then in the situation when $f$ is proper we have the extension of the
$!$-pullback functor.

###  2.3 Arithmetic $\mathcal{D}$-modules over Stacks

We use a general notation $\mathcal{X}$ to denote a general algebraic stack
over the base finite field essentially in [GL1, Construction 3.2.5.1]. Then we
use the functor of points techniques as in [GL1, Construction 3.2.5.1] define
all the categories as the corresponding homotopy limit through all the
functors of points $X_{i},i=1,2,...$333Taking the limit through all the
functors of points by considering opposite category.444Certainly one can
consider the categories for stacks in [Abe] directly and consider the
corresponding inductive categories to achieve $\infty$-enhancement.:

$\displaystyle\underset{X_{i}}{\mathrm{homotopylimit}}(*).$ (2.3.1)

######  Definition 2.3.1.

We are going to systematically use the language of $\infty$-category from
[Lurie1]. To be more precise we first consider the derived category of the
constructible holonomic $\mathcal{D}$-modules denoted by
$D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X})$, which play the role of the
$D^{\mathrm{b}}_{\ell}(\mathcal{X}/k)$ which is the $\ell$-adic complexes
constructible over $\mathcal{Y}$ as in [GL1]. Then we consider the
corresponding $\infty$-categorical enrichment, which is to say we are going to
construct a simplicial set from $D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X})$,
which we will denote it by
$\mathbb{D}^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X})$, whose homotopy category
gives rise to the derived category recalled above, which is to say
$D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X})$.

We remard that here the category $\mathrm{Hol}(X)$ defined in [Abe1,
Definition 1.1.1] does not natural have enough injectives, which means one
could not directly apply the construction in [Lurie2, Section 1.3.2, Section
1.3.5]. Instead we consider the following construction:

######  Definition 2.3.2.

We will use the notation $\mathrm{Hol}(X)$ and $\mathrm{Con}(X)$ to denote the
corresponding categories of holonomic $\mathcal{D}$-modules and the
corresponding contructible ones defined in [Abe1, Definition after Proposition
1.3.3]. We use the notation $\mathrm{Ind}(\mathrm{Con}(\mathcal{X}))$ to
denote the corresponding ind-category associated to the category
$\mathrm{Con}(\mathcal{X})$. Note that this is now a Grothendieck category
which admits enough injectives, then we apply the construction in [Lurie2,
Section 1.3.2, Section 1.3.5] to get the corresponding derive
$\infty$-catogory, which we will denote it by
$\mathbb{D}(\mathrm{Ind}(\mathrm{Con}(\mathcal{X})))$. Then we define:

$\mathbb{D}_{\mathrm{hol}}(\mathcal{X}):=\mathbb{D}(\mathrm{Con}(\mathcal{X})):=\mathbb{D}_{\mathrm{Con}(\mathcal{X})}(\underset{X_{i}}{\mathrm{homotopylimit}}(*)\mathrm{Ind}(\mathrm{Con}(X_{i}))),$

where the latter is the full subcategory of
$\mathbb{D}(\mathrm{Ind}(\mathrm{Con}(\mathcal{X})))$ consists of all the
complex whose homologies live in $\mathrm{Con}(\mathcal{X})$. Passing to the
homotopy categories we have:

$D_{\mathrm{hol}}(\mathcal{X}):=D(\mathrm{Con}(\mathcal{X})):=D_{\mathrm{Con}(\mathcal{X})}(\underset{X_{i}}{\mathrm{homotopylimit}}(*)\mathrm{Ind}(\mathrm{Con}(X_{i}))).$

######  Remark 2.3.3.

By [Lurie2, Section 1.3] we have that the $\infty$-category
$\mathbb{D}_{\mathrm{hol}}(\mathcal{X})$ is a stable $\infty$-category.

###### Proof.

It suffices to prove that the cofibers are preserved under the embedding of
the abelian categories
$\mathrm{Con}(\mathcal{X})\rightarrow\mathrm{Ind}\mathrm{Con}(\mathcal{X})$.
Then to finish, one check the long exact sequence for any mapping cone
$C^{\bullet}(f)$, but then one finishes since on the level of derived category
we have the corresponding result:

$D(\mathrm{Con}(\mathcal{X}))\overset{\sim}{\rightarrow}D_{\mathrm{Con}(\mathcal{X})}(\mathrm{Ind}\mathrm{Con}(\mathcal{X})).$

∎

####  2.3.1 Inverse and Direct Images in Derived $\infty$-Category

Recall that [Abe1, Section 1.1.3] finishes the searching for the six
operations at least for quasi-projective schemes over $\mathbb{F}_{p}$,
implicitly relying on Kedlaya’s work on Shiho’s conjecture [Ked4], [Ked5],
[Ked6], [Ked7]. Everything could be promoted to the derived
$\infty$-categorical constructions as above. We recall everything here. To be
more precise we consider any morphism, which we denote it as
$f:\mathcal{X}\rightarrow\mathcal{Y}$. Then we have four derived functor:

$\displaystyle f_{!}:D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{Y}/K),$ (2.3.2) $\displaystyle
f_{+}:D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{Y}/K),$ (2.3.3) $\displaystyle
f^{!}:D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{Y}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X}/K),$ (2.3.4) $\displaystyle
f^{+}:D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{Y}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(\mathcal{X}/K),$ (2.3.5)

by using555All the parallel definitions up to the details by using the limit
as in the following will be given in the same way.:

$\displaystyle\underset{X_{i},Y_{i}}{\mathrm{homotopylimit}}(*)D^{\mathrm{b}}_{\mathrm{hol}}(X_{i}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(Y_{i}/K),$ (2.3.6)
$\displaystyle\underset{X_{i},Y_{i}}{\mathrm{homotopylimit}}(*)D^{\mathrm{b}}_{\mathrm{hol}}(X_{i}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(Y_{i}/K),$ (2.3.7)
$\displaystyle\underset{X_{i},Y_{i}}{\mathrm{homotopylimit}}(*)D^{\mathrm{b}}_{\mathrm{hol}}(Y_{i}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(X_{i}/K),$ (2.3.8)
$\displaystyle\underset{X_{i},Y_{i}}{\mathrm{homotopylimit}}(*)D^{\mathrm{b}}_{\mathrm{hol}}(Y_{i}/K)\rightarrow
D^{\mathrm{b}}_{\mathrm{hol}}(X_{i}/K).$ (2.3.9)

We would like to promote the operations onto the derived $\infty$-level, which
is to say that we have the following derived $\infty$-functors having the form
as in the following:

$\displaystyle
f_{!}:\mathbb{D}_{\mathrm{hol}}(\mathcal{X}/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(\mathcal{Y}/K),$
(2.3.10) $\displaystyle
f_{+}:\mathbb{D}_{\mathrm{hol}}(\mathcal{X}/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(\mathcal{Y}/K),$
(2.3.11) $\displaystyle
f^{!}:\mathbb{D}_{\mathrm{hol}}(\mathcal{Y}/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(\mathcal{X}/K),$
(2.3.12) $\displaystyle
f^{+}:\mathbb{D}_{\mathrm{hol}}(\mathcal{Y}/K)\rightarrow\mathbb{D}_{\mathrm{hol}}(\mathcal{X}/K).$
(2.3.13)

Not that as in [Abe1, Section 1.2], our construction of derived
$\infty$-category used essentially the techniques of inductive objects.
Therefore the six operator should be deduced from the following ones:

$\displaystyle\underline{f_{!}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{X}))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{Y})),$
(2.3.14)
$\displaystyle\underline{f_{+}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{X}))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{Y})),$
(2.3.15)
$\displaystyle\underline{f^{!}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{Y}))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{X})),$
(2.3.16)
$\displaystyle\underline{f^{+}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{Y}))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{X})).$
(2.3.17)

which are derived from the level of ind-category
$\mathrm{Ind}\mathrm{Con}(\mathcal{X})$. Here the definition of the derived
$\infty$-category $\mathbb{D}$ is essentially [Lurie2, Section 1.3.5]. First
we have the following lemma based on the discussion in [Abe1]. And we
following the notations (for instance the constructible t-structure). Our
discussion is essentially based on the observation from [Abe1, Section 1.2].
One could actually regard the discussion presented here as sort of
$\infty$-enrichment on the $t$-structures introduced in [Abe1], under the
philosophy of Riemann-Hilbert correspondence.

######  Lemma 2.3.4.

Let $f$ as a morphism as above, we have that the pullback
$\underline{\underline{f^{+}}}:\mathrm{Ind}\mathrm{Con}(\mathcal{Y})\rightarrow\mathrm{Ind}\mathrm{Con}(\mathcal{X})$
is well defined which sends objects in $\mathrm{Ind}\mathrm{Con}(\mathcal{Y})$
to those in $\mathrm{Ind}\mathrm{Con}(\mathcal{X})$, and the functor is c-t-
exact.

###### Proof.

See [Abe1, Section 1.3]. ∎

######  Lemma 2.3.5.

Over the level of derived $\infty$-category, we have the right adjoint
$\underline{\underline{f^{\wedge}_{+}}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{X}))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{Y}))$
of the functor in the previous lemma which is well defined, and which is left
c-t-exact.

Then we consider the étale morphism:

######  Lemma 2.3.6.

Let $f$ as a morphism as above that is étale, then we have that first the
pullback functor
$f^{+}:\mathrm{Ind}\mathrm{Con}(\mathcal{Y})\rightarrow\mathrm{Ind}\mathrm{Con}(\mathcal{X})$
which admits a left adjoint functor over the level of derived
$\infty$-categories
$\underline{\underline{f^{\wedge}_{!}}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{X}))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{Y}))$
.

###### Proof.

Everything extends from the level of abelian category to this setting. Also
see [Abe1, 1.2.2]. ∎

Finally for the proper morphisms we have:

######  Lemma 2.3.7.

Let $f$ as a morphism above that is proper, then the $!$-pullback functor
$\underline{\underline{f^{!}}}:\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{Y}))\rightarrow\mathbb{D}(\mathrm{Ind}\mathrm{Con}(\mathcal{X}))$
is well defined.

###  2.4 Arithmetic of Rigid Gross $G$-Motives: $\mathrm{Bun}_{G}$

We now define the corresponding rigid Gross $G$-motives as in [Gro1] and [GL1]
in the corresponding $\ell$-adic situation. In the analogy of the
corresponding situation in [GL1] we use the language of the corresponding
arithmetic $D$-modules. The relatively different aspect is that we have to
focus on the corresponding $E_{\infty}$-algebras with the augmentation666In
fact, the sheaves of rings of differential operators whatever the forms are
actually unfortunately noncommutative, therefore maybe one should really focus
on the corresponding larger categories of the corresponding
$A_{\infty}$-algebras with the augmentation.. Our idea is to study the
cohomology with non-trivial coefficients in the category of holonomic
arithmetic $\mathcal{D}$-modules, for instance we use the notation
$\mathcal{E}\in\mathbb{D}_{\mathrm{hol}}(X)$ to denote such an module in the
$p$-adic setting. Our main conjecture is that we have the following well
defined statement:

######  Conjecture 2.4.1.

$\displaystyle\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}($
$\displaystyle
X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathrm{pr}_{*}h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E}))$
(2.4.1) $\displaystyle=\prod_{x\in
X}\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}(\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{x}^{*}{\mathcal{E}}_{x})).$
(2.4.2)

The corresponding cohomology groups and rings are defined in the following:

######  Definition 2.4.2.

As in [GL1, Construction 3.2.5.1], we define the following cohomology groups
associated to the corresponding arithmetic $D$-module in the $\infty$-category
involved by taking throughout all the functors of the corresponding points
associated withe involving stacks:

$\displaystyle\mathrm{H}^{*}(\mathrm{Bundle}_{G}($ $\displaystyle
X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathrm{pr}_{*}h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E})$
(2.4.3)
$\displaystyle:=\underset{X_{i}}{\mathrm{homotopylimit}}(*)\mathrm{H}^{*}(X_{i,\mathrm{Bundle}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p})},\mathrm{pr}_{*}h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E})$
(2.4.4)

where we have the following notations of morphisms:

$\displaystyle h_{\mathrm{Bundle}_{G}}:\mathrm{Bundle}_{G}(X)\times
X\rightarrow\mathrm{Classifyingstack}_{G}\rightarrow X,$ (2.4.5)
$\displaystyle\mathrm{pr}:\mathrm{Bundle}_{G}(X)\times
X\rightarrow\mathrm{Bundle}_{G}(X),$ (2.4.6)
$\displaystyle\mathrm{pr}^{\prime}:\mathrm{Bundle}_{G}(X)\times X\rightarrow
X.$ (2.4.7)

######  Definition 2.4.3.

As in [GL1, Construction 3.2.5.1], we define the following cohomology groups
associated to the corresponding arithmetic $D$-module in the $\infty$-category
involved by taking throughout all the functors of the corresponding points
associated withe involving stacks:

$\displaystyle\mathrm{H}^{*}(\mathrm{Bundle}_{G}($
$\displaystyle\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{x}^{*}{\mathcal{E}}_{x})$
(2.4.8)
$\displaystyle:=\underset{X_{i}}{\mathrm{homotopylimit}}(*)\mathrm{H}^{*}(X_{i,\mathrm{Bundle}_{G}(\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p})},h_{x}^{*}{\mathcal{E}}_{x})$
(2.4.9)

where we have the following notations of morphisms777Certainly up to taking
the corresponding base change to the corresponding algebraically closed
field.:

$\displaystyle h:\mathrm{Classifyingstack}_{G}\rightarrow X,$ (2.4.10)
$\displaystyle h_{x}:\mathrm{Bundle}_{G}(\\{x\\})\rightarrow\\{x\\}.$ (2.4.11)

Note that the cohomology groups involved for the the stack
$\mathrm{Bundle}_{G}(\overline{X})$ is essentially nonzero in infinite degrees
as in the $\ell$-adic setting in [GL1] but once we consider the cotangent
complex we could have more detailed description on that:

######  Conjecture 2.4.4.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra.

$\mathrm{H}^{*}(\mathrm{cot}\mathrm{C}^{*}(\mathrm{Bundle}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathrm{pr}_{*}h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E}))$

is finite dimensional over $\mathbb{Q}_{p}$.

This is essentially true since we will establish the following quasi-
isomorphism:

$\mathrm{cot}\mathrm{C}^{*}(\mathrm{Bundle}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{\mathrm{Bundle}_{G}}^{*}\mathcal{E})\overset{\mathrm{quasi}}{\longrightarrow}\mathrm{C}^{*}((X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathcal{M}(G)^{\mathrm{Gross}}_{\mathcal{E}})$

where the latter is the refined rigid Gross $G$-motive with more general
coefficients which is a $p$-adic analog of Gross $G$-motive in the $\ell$-adic
setting defined in [Gro1]. However, this quasi-isomorphism is not obvious at
all, since actually this is essentially the main thing to prove.

######  Definition 2.4.5.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra.

$\displaystyle\mathcal{M}(G)^{\mathrm{Gross}}_{\mathcal{E}}:=\mathrm{cotangentfibre}(h_{*}h^{*}\mathcal{E}).$
(2.4.13)

where
$h:\mathrm{BC}_{\overline{k}}\mathrm{Classifyingstack}_{G}\rightarrow\mathrm{BC}_{\overline{k}}X$.

A little bit weaker result will be the following Euler product formula for the
L-functions attached to the rigid Gross $G$-motive (which we will define in
our setting which is an $\infty$-$p$-adic arithmetic $\mathcal{D}$-module)
which we define in the following:

######  Definition 2.4.6.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra. We define the rigid $L$-function attached to
rigid Gross $G$-motive (in general $p$-adic coefficient) as the following:

$L_{\mathcal{M}(G),\mathrm{Frob}^{-1},\mathcal{E}}(t):=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G)_{\mathcal{E}})^{-1}$

Then we will show that we have the following Euler product formula for this
$L$-function as in the following:

$\displaystyle L_{\mathcal{M}(G),\mathrm{Frob}^{-}1,\mathcal{E}}(t)$
$\displaystyle:=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G)_{\mathcal{E}})^{-1}$
(2.4.14) $\displaystyle=\prod_{x\in
X}\mathrm{det}(1-t\mathrm{Frob}_{x}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G_{x})_{\mathcal{E}_{x}})^{-1}.$
(2.4.15)

Actually if $\mathcal{E}$ is just the trivial coefficient, then the special
value $L_{\mathcal{M}(G),\mathrm{Frob}^{-1},\mathcal{E}}(1)$ is actually just
the inverse of $\prod_{x\in
X}\frac{|\mathrm{Bundle}_{G}(\\{x\\})(\kappa(x))|}{q^{\mathrm{dim}\mathrm{Bundle}_{G}(\kappa(x))}}$.

We now look at some results as consequences of [De1], [Ked11] and [Ked12].
Namely we now assume $\mathcal{E}$ is some isocrystal $E$:

######  Proposition 2.4.7.

$\displaystyle\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}($
$\displaystyle
X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathrm{pr}_{*}h_{\mathrm{Bundle}_{G}}^{*}E))$
(2.4.16) $\displaystyle=\prod_{x\in
X}\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}_{G}(\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{x}^{*}{E}_{x})).$
(2.4.17)

###### Proof.

Since now $E$ is some isocrystal, therefore one can apply results of
conjectures of petits camarades from [De1, Conjecture 1.2.10] as proved in
[Ked11] and [Ked12, Theorem 0.1.1, Theorem 0.1.2]. Then one reduces the
corresponding proposition to the $\ell$-adic situation as in [GL1, Theorem
1.4.4.1]. ∎

######  Proposition 2.4.8.

We now here assume that the object $E$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra. Then we have the following product formula:

$\displaystyle L_{\mathcal{M}(G),\mathrm{Frob}^{-1},\mathcal{E}}(t)$
$\displaystyle:=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G)_{\mathcal{E}})^{-1}$
(2.4.18) $\displaystyle=\prod_{x\in
X}\mathrm{det}(1-t\mathrm{Frob}_{x}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G_{x})_{\mathcal{E}_{x}})^{-1}.$
(2.4.19)

###### Proof.

Since now $E$ is some isocrystal, therefore one can apply results of
conjectures of petits camarades from [De1, Conjecture 1.2.10] as proved in
[Ked11] and [Ked12, Theorem 0.1.1, Theorem 0.1.2]. Then one reduces the
corresponding proposition to the $\ell$-adic situation as in [GL1, Theorem
4.5.3.1]. ∎

###  2.5 Arithmetic of Rigid Gross $G$-Motives:
$\mathrm{Bun}^{\mathrm{Parabolic,P}}_{G}$

We now define the corresponding rigid Gross $G$-motives as in [Gro1] and [GL1]
in the corresponding $\ell$-adic situation. And we will consider the
corresponding context of moduli of parabolic $G$-bundles as in [GL1, Section
5.5.1]. Therefore we will fix some parabolic $P$ as in [GL1, Section 5.5.1]
which is denoted by $P_{0}$888Suitable inner form $G$ namely the corresponding
adjoint semisimple one will give us the chance to reduce everything to the
previous section by considering $P$-bundles as in [GL1, Example 5.5.1.8].. In
the analogy of the corresponding situation in [GL1] we use the language of the
corresponding arithmetic $D$-modules. The relatively different aspect is that
we have to focus on the corresponding $E_{\infty}$-algebras with the
augmentation999In fact, the sheaves of rings of differential operators
whatever the forms are actually unfortunately noncommutative, therefore maybe
one should really focus on the corresponding larger categories of the
corresponding $A_{\infty}$-algebras with the augmentation.. Our idea is to
study the cohomology with non-trivial coefficients in the category of
holonomic arithmetic $\mathcal{D}$-modules, for instance we use the notation
$\mathcal{E}\in\mathbb{D}_{\mathrm{hol}}(X)$ to denote such an module in the
$p$-adic setting. Our main conjecture is that we have the following well
defined statement:

######  Conjecture 2.5.1.

$\displaystyle\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathrm{pr}_{*}h_{\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}}^{*}\mathcal{E}))$
(2.5.1) $\displaystyle=\prod_{x\in
X}\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^{*}(\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{x}^{*}{\mathcal{E}}_{x})).$
(2.5.2)

The corresponding cohomology groups and rings are defined in the following:

######  Definition 2.5.2.

As in [GL1, Construction 3.2.5.1], we define the following cohomology groups
associated to the corresponding arithmetic $D$-module in the $\infty$-category
involved by taking throughout all the functors of the corresponding points
associated withe involving stacks:

$\displaystyle\mathrm{H}^{*}(\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}($
$\displaystyle
X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathrm{pr}_{*}h_{\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}}^{*}\mathcal{E})$
(2.5.3)
$\displaystyle:=\underset{X_{i}}{\mathrm{homotopylimit}}(*)\mathrm{H}^{*}(X_{i,\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p})},\mathrm{pr}_{*}h_{\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}}^{*}\mathcal{E})$
(2.5.4)

where we have the following notations of morphisms:

$\displaystyle
h_{\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}}:\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X)\times
X\rightarrow\mathrm{Classifyingstack}^{P}_{G}\rightarrow X,$ (2.5.5)
$\displaystyle\mathrm{pr}:\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X)\times
X\rightarrow\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X),$ (2.5.6)
$\displaystyle\mathrm{pr}^{\prime}:\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X)\times
X\rightarrow X.$ (2.5.7)

######  Definition 2.5.3.

As in [GL1, Construction 3.2.5.1], we define the following cohomology groups
associated to the corresponding arithmetic $D$-module in the $\infty$-category
involved by taking throughout all the functors of the corresponding points
associated withe involving stacks:

$\displaystyle\mathrm{H}^{*}(\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}($
$\displaystyle\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{x}^{*}{\mathcal{E}}_{x})$
(2.5.8)
$\displaystyle:=\underset{X_{i}}{\mathrm{homotopylimit}}(*)\mathrm{H}^{*}(X_{i,\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(\\{x\\}\times_{\mathrm{Spec}\mathbb{F}_{p})}\mathrm{Spec}\overline{\mathbb{F}}_{p})},h_{x}^{*}{\mathcal{E}}_{x})$
(2.5.9)

where we have the following notations of morphisms101010Certainly up to taking
the corresponding base change to the corresponding algebraically closed
field.:

$\displaystyle h:\mathrm{Classifyingstack}^{P}_{G}\rightarrow X,$ (2.5.10)
$\displaystyle
h_{x}:\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(\\{x\\})\rightarrow\\{x\\}.$
(2.5.11)

Note that the cohomology groups involved for the the stack
$\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(\overline{X})$ is essentially
nonzero in infinite degrees as in the $\ell$-adic setting in [GL1] but once we
consider the cotangent complex we could have more detailed description on
that:

######  Conjecture 2.5.4.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra.

$\mathrm{H}^{*}(\mathrm{cot}\mathrm{C}^{*}(\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathrm{pr}_{*}h_{\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}}^{*}\mathcal{E}))$

is finite dimensional over $\mathbb{Q}_{p}$.

This is essentially true since we will establish the following quasi-
isomorphism:

$\displaystyle\mathrm{cot}\mathrm{C}^{*}(\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}(X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),h_{\mathrm{Bundle}^{\mathrm{Parabolic,P}}_{G}}^{*}\mathcal{E})\overset{\mathrm{quasi}}{\longrightarrow}\mathrm{C}^{*}((X\times_{\mathrm{Spec}\mathbb{F}_{p}}\mathrm{Spec}\overline{\mathbb{F}}_{p}),\mathcal{M}(G)^{\mathrm{Gross,Parabolic}}_{\mathcal{E}})$
(2.5.13)

where the latter is the refined rigid Gross $G$-motive with more general
coefficients which is a $p$-adic analog of Gross $G$-motive in the $\ell$-adic
setting defined in [Gro1]. However, this quasi-isomorphism is not obvious at
all, since actually this is essentially the main thing to prove.

######  Definition 2.5.5.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra.

$\displaystyle\mathcal{M}(G)^{\mathrm{Gross,Parabolic}}_{\mathcal{E}}:=\mathrm{cotangentfibre}(h_{*}h^{*}\mathcal{E}).$
(2.5.14)

where
$h:\mathrm{BC}_{\overline{k}}\mathrm{Classifyingstack}^{P}_{G}\rightarrow\mathrm{BC}_{\overline{k}}X$.

A little bit weaker result will be the following Euler product formula for the
L-functions attached to the rigid Gross $G$-motive (which we will define in
our setting which is an $\infty$-$p$-adic arithmetic $\mathcal{D}$-module)
which we define in the following:

######  Definition 2.5.6.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra. We define the rigid $L$-function attached to
rigid Gross $G$-motive (in general $p$-adic coefficient) as the following:

$L_{\mathcal{M}(G)^{\mathrm{Gross,Parabolic}},\mathrm{Frob}^{-1},\mathcal{E}}(t):=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G)^{\mathrm{Gross,Parabolic}}_{\mathcal{E}})^{-1}$

######  Conjecture 2.5.7.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra.

$\displaystyle L_{\mathcal{M}(G),\mathrm{Frob}^{-}1,\mathcal{E}}(t)$
$\displaystyle:=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G)^{\mathrm{Gross,Parabolic}}_{\mathcal{E}})^{-1}$
(2.5.15) $\displaystyle=\prod_{x\in
X}\mathrm{det}(1-t\mathrm{Frob}_{x}^{-1}|\mathrm{H}^{*}(\overline{X},\mathcal{M}(G_{x})^{\mathrm{Gross,Parabolic}}_{\mathcal{E}_{x}})^{-1}.$
(2.5.16)

###  2.6 Arithmetic of Rigid Gross $G$-Motives:
$\mathrm{Bun}^{\mathrm{Local}}_{G(K_{X,x})}$

We now take a look at some local version. Here the interesting thing is that
we will have $\mathrm{Bun}_{G}$ over Fargues-Fontaine curves in equal
characteristic (namely the moduli of $t$-motivic $G$-bundles) as in [FF],
[FS], [GL2], [HP], [KL1], [KL2], [SW], around some point $x\in X$. And we fix
a local field $K_{X,x}$ coming from some point over $X$. Then for any
$t_{x}$-adic Lie group $G(K_{X,x})$ we define the following rigid local Gross
$G$-motive. Therefore now the object $\mathcal{E}$ will be over the Fargues-
Fontaine curve $\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}$111111As in [GL2],
which should be related to Scholze’s definition through $v$-stacks as in [FS],
[SW] and [Sch] and should be related to the construction in [KL2]. Here we
work over categories of formal schemes. around some neighbourhood
$\mathrm{Sp}(\widehat{\mathcal{O}}_{X,x})$ of $x$. Again we follow [GL1]
closely.

######  Remark 2.6.1.

We regard the stack $\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}$ as some formal
stack by taking presentation from some formal scheme $Y$ to the quotient
through Frobenius. Then we define the classifying stack
$\mathrm{Classifyingstack}_{G(K_{X,x}),\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}}$
over $\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}$ by taking the corresponding
quotient by the group $G(K_{X,x})$.

######  Definition 2.6.2.

And we define $\mathrm{Bun}^{\mathrm{Local}}_{G(K_{X,x}),X,x}$ as the
corresponding formal stack of $G(K_{X,x})$-bundles over the disc
$\mathrm{Sp}(\widehat{\mathcal{O}}_{X,x})$. And we define
$\mathrm{Bun}^{\mathrm{Local}}_{G(K_{X,x}),\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}}$
as the corresponding formal stack of $G(K_{X,x})$-bundles over
$\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}$.

######  Definition 2.6.3.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra over
$\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}$.

$\displaystyle\mathcal{M}(G)^{\mathrm{Gross,Local}}_{\mathcal{E}}:=\mathrm{cotangentfibre}(h_{*}h^{*}\mathcal{E}).$
(2.6.1)

where
$h:\mathrm{BC}_{\overline{k}}\mathrm{Classifyingstack}_{G(K_{X,x}),\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}}\rightarrow\mathrm{BC}_{\overline{k}}\mathrm{FF}_{\widehat{\mathcal{O}}_{X,x}}$.

A little bit weaker result will be the following Euler product formula for the
L-functions attached to the rigid Gross $G$-motive (which we will define in
our setting which is an $\infty$-$p$-adic arithmetic $\mathcal{D}$-module)
which we define in the following:

######  Definition 2.6.4.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra. We define the rigid $L$-factor attached to
rigid Gross $G$-motive (in general $p$-adic coefficient) as the following:

$\mathrm{Factor}_{\mathcal{M}(G)^{\mathrm{Gross,Local}},\mathrm{Frob}^{-1},\mathcal{E}}(t):=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{\mathrm{FF}_{\mathcal{O}_{X,x}}},\mathcal{M}(G)^{\mathrm{Gross,Local}}_{\mathcal{E}})^{-1}.$

######  Remark 2.6.5.

One may want to compare this with the one defined globally as in the previous
section. Also one can define things over the disc
$\mathrm{Sp}(\widehat{\mathcal{O}}_{X,x})$:

######  Definition 2.6.6.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra over $\mathrm{Sp}(\widehat{\mathcal{O}}_{X,x})$.

$\displaystyle\mathcal{M}(G)^{\mathrm{Gross,Local}}_{\mathcal{E}}:=\mathrm{cotangentfibre}(h_{*}h^{*}\mathcal{E}).$
(2.6.2)

where
$h:\mathrm{BC}_{\overline{k}}\mathrm{Classifyingstack}_{G(K_{X,x})}=\mathrm{BC}_{\overline{k}}\mathrm{Sp}(\widehat{\mathcal{O}}_{X,x})/{G(K_{X,x})}\rightarrow\mathrm{BC}_{\overline{k}}\mathrm{Sp}(\widehat{\mathcal{O}}_{X,x})$.

A little bit weaker result will be the following Euler product formula for the
L-functions attached to the rigid Gross $G$-motive (which we will define in
our setting which is an $\infty$-$p$-adic arithmetic $\mathcal{D}$-module)
which we define in the following:

######  Definition 2.6.7.

We now here assume that the object $\mathcal{E}$ is an augmented commutative
$\mathbb{E}_{\infty}$-algebra. We define the rigid $L$-factor attached to
rigid Gross $G$-motive (in general $p$-adic coefficient) as the following:

$\mathrm{Factor}_{\mathcal{M}(G)^{\mathrm{Gross,Local}},\mathrm{Frob}^{-1},\mathcal{E}}(t):=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^{*}(\overline{\mathrm{Sp}(\widehat{\mathcal{O}}_{X,x})},\mathcal{M}(G)^{\mathrm{Gross,Local}}_{\mathcal{E}})^{-1}.$

#### Acknowledgements

This work is based on the conversation with Professor Kedlaya who conjectured
that there should be a version of Drinfeld’s lemma for $F$-isocrystals and
suggested taking a look at the corresponding generalization of Abe’s work on
Langlands correspondence for $F$-isocrystals to general reductive groups. Also
we benefit a lot from the corresponding work [Ked9] from different
perspectives. We would like to thank Professor Kedlaya for helpful discussion
and many suggestions around this problem and related.

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