#  Regulators in the Arithmetic of Function Fields
Quentin Gazda111Affiliation: Max Planck Institut für Mathematik, Vivatsgasse
7, 53111 Bonn, Germany. Mail<EMAIL_ADDRESS>
Abstract

As a natural sequel for the study of $A$-motivic cohomology, initiated in
[Gaz22], we develop a notion of regulator for rigid analytically trivial mixed
Anderson $A$-motives. In accordance with the conjectural number field picture,
we define it as the morphism at the level of extension modules induced by the
exactness of the Hodge-Pink realization functor. The purpose of this text is
twofold: we first prove a finiteness result for $A$-motivic cohomology and,
under a weight assumption, we then show that the source and the target of
regulators have the same dimension. It came as a surprise to the author that
the image of this regulator might not have full rank, preventing the analogue
of a renowned conjecture of Beilinson to hold in our setting.

###### Contents

  1. 1 Introduction
  2. 2 Hodge-Pink structures and their extensions
    1. 2.1 Hodge-Pink structures
    2. 2.2 Infinite Frobenii
  3. 3 Rigid analytically trivial mixed $A$-motives
    1. 3.1 Definitions
    2. 3.2 The Betti realization functor
    3. 3.3 Analytic continuation
    4. 3.4 The associated mixed Hodge-Pink structure
  4. 4 Regulators and finiteness theorems
    1. 4.1 Regulators of $A$-motives
    2. 4.2 The complex $G$ and the fundamental exact sequence
    3. 4.3 Strategy of proofs
  5. 5 Shtuka models à la Mornev
    1. 5.1 $C$-shtuka models
    2. 5.2 $C\times C$-shtuka models
    3. 5.3 Shtuka models and extensions of mixed Hodge-Pink structures
  6. 6 Proof of the main theorems
    1. 6.1 Cohomological computations
    2. 6.2 Proof of Theorems 4.1 and 4.4

##  1 Introduction

Very recently in [ANDTR22], Anglès-Ngo Dac-Tavares-Ribeiro made the tremendous
exploit of establishing a class formula for a large family of Anderson
$A$-modules. This achievement sits at the top of a long list of work, dating
back to Carlitz [Car35] in 1935, and culminating with the breakthroughs of
Taelman [Tae12] and V. Lafforgue [Laf09].

In the classical picture of number fields, class formulas in this level of
generality are rather stated in terms of mixed motives in the form of
Beilinson’s conjectures. Those are far reaching conjectures on special
$L$-values formulated by Beilinson in the eighties in two celebrated papers
[Bei84], [Bei86]. The whole picture is rooted over the notion of a Beilinson
regulator, conjecturally defined as follows. Let $M$ be a mixed motive over
the field of rational numbers $\mathbb{Q}$. Here, "motive" is understood in
the wording of Deligne (e.g. [Del89, §1]). Consider the $\mathbb{Q}$-vector
space $\operatorname{Ext}^{1}_{\mathbb{Q}}(\mathbbm{1},M)$ of $1$-fold
extension of the neutral motive $\mathbbm{1}$ by $M$ in the category of mixed
motives over $\mathbb{Q}$. It is expected that one can defined a natural
subspace $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbbm{1},M)$ consisting of
extensions having everywhere good reduction (e.g. [Sch91]). The Hodge
realization functor $\mathscr{H}^{+}$, from the category of mixed motives and
with values in the category of mixed Hodge structures enriched with infinite
Frobenii, is expected to be exact and, in this respect, should induce a
morphism of the extension spaces:

$\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbbm{1},M)\longrightarrow\operatorname{Ext}^{1}_{\mathbb{R}-\operatorname{Hdg}}(\mathbbm{1}^{+},\mathscr{H}^{+}(M)).$
(1.1)

Observe that the right-hand side is an $\mathbb{R}$-vector space of finite
dimension. The above map is - conjecturally - the Beilinson’s regulator of
$M$. We denote it $\textrm{{Reg}}(M)$. The following is expected, although far
from being proved.

  1. (I)

The space $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbbm{1},M)$ has finite
dimension over $\mathbb{Q}$.

  2. (II)

If $M$ is pure of weight $<-2$, then $\textrm{{Reg}}(M)$ has dense image.

  3. (III)

There is a $\mathbb{Q}$-structure $V(M)$ on the target of $\textrm{{Reg}}(M)$,
natural in $M$, such that we have
$\det(\operatorname*{im}\textrm{{Reg}}(M))=L^{*}(M,0)\cdot\det V(M)$ as
$\mathbb{Q}$-structures on
$\det\operatorname{Ext}^{1}_{\mathbb{R}-\operatorname{Hdg}}(\mathbbm{1}^{+},\mathscr{H}^{+}(M))$.

Conjectures (II) and (III) are referred to as Beilinson’s conjectures. This
text is concerned with the function field analogue of (I) and (II), so we do
not further comment on the (conjectural) definition of the special $L$-value
$L^{*}(M,0)$ nor on the $\mathbb{Q}$-structure $V(M)$. We rather refer the
reader to the survey [Nek94] for a complete account of Beilinson’s conjectures
and their history.

The present work grew out as an attempt to bridge these two pictures, and
understand how Beilinson’s conjectures could be stated in the language of
Anderson $A$-motives. The study of motivic cohomology for $A$-motives was
initiated in [Gaz22] and this text consists in a natural sequel. Our primary
interest is the definition of a Beilinson’s regulator in this context, and the
study of the counterpart of conjectures (I) and (II). We hope to make
conjecture (III) the subject of a subsequent work.

Anderson $A$-motives. Let $(C,\pazocal{O}_{C})$ be a geometrically irreducible
smooth projective curve over a finite field $\mathbb{F}$ of characteristic
$p$, and fix $\infty$ a closed point of $C$. The $\mathbb{F}$-algebra

$A=H^{0}(C\setminus\\{\infty\\},\pazocal{O}_{C})$

consists of functions of $C$ that are regular away from $\infty$. We denote by
$K$ its fraction field. The notion of $A$-motives dates back to the pioneer
work of Anderson [And86], and generalizes prior ideas of Drinfeld [Dd74]. Let
us state the definition of an Anderson $A$-motives, leaving details for
Section 3. Throughout this text, unlabeled tensor or fiber products are over
$\mathbb{F}$.

######  Definition (3.2).

An Anderson $A$-motive $\underline{M}$ over $K$ consists of a finite locally-
free $A\otimes K$-module $M$ together with an isomorphism of $A\otimes
K$-modules

$\tau_{M}:(\tau^{*}M)|_{(\operatorname{Spec}A\otimes K)\setminus
V(\mathfrak{j})}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M|_{(\operatorname{Spec}A\otimes
K)\setminus V(\mathfrak{j})},$

where $\tau:A\otimes K\to A\otimes K$ is the ring endomorphism acting as the
identity on $A$ and as the $|\mathbb{F}|$th-power map on $K$, and where
$V(\mathfrak{j})$ is the effective Cartier divisor on
$\operatorname{Spec}A\otimes K$ associated to the locally-free ideal
$\mathfrak{j}:=\ker(A\otimes K\to K,~{}a\otimes r\mapsto ar)$.

######  Remark 1.1.

In general, $A$-motives do not carry a weight filtration; those that do are
called mixed (see [Gaz22, §3]). However, as the main results of this text do
not require mixedness - aside from obvious analogy purposes with the number
field theory - we voluntarily forget the notion of mixedness. Yet, that of
weights of an $A$-motive (see [Gaz22, Def. 3.20]) plays a major role.

A crucial assumption throughout this text however is that of rigid analyitc
triviality (Subsection 3.2 for details). Given an $A$-motive $\underline{M}$
over $K$, we consider the $A$-module

$\underline{M}_{B}:=\\{f\in M\otimes_{A\otimes K}\mathbb{C}_{\infty}\langle
A\rangle~{}|~{}f=\tau_{M}(\tau^{*}f)\\}.$

It is the Betti realization of $\underline{M}$. Here,
$\mathbb{C}_{\infty}\langle A\rangle$ is an affinoid algebra over
$\mathbb{C}_{\infty}$, the completion of an algebraic closure of the local
field $K_{\infty}$ of $(C,\pazocal{O}_{C})$ at $\infty$ (see [HJ20, §2.3.3],
[GM22b, §3] or Section 3 below for details). To ensure good properties of the
functor $\underline{M}\mapsto\underline{M}_{B}$ (e.g. exactness) we focus on
rigid analytically trivial $A$-motives; i.e. those $\underline{M}$ for which
$\underline{M}_{B}$ generates $M\otimes_{A\otimes K}\mathbb{C}_{\infty}\langle
A\rangle$ over $\mathbb{C}_{\infty}\langle A\rangle$ (Definition 3.8). The
category of rigid analytically trivial $A$-motives over $K$, denoted by
$A\mathbf{Mot}_{K}^{\operatorname{rat}}$ along this text, is an exact
$A$-linear category which shall play the role of the classical category of
mixed motives over $\mathbb{Q}$ and the functor
$\underline{M}\mapsto\underline{M}_{B}$ that of the singular realization. The
reader is invited to consult [BGHP20] for surveys on this analogy.

Let $\underline{M}$ be an object of $A\mathbf{Mot}_{K}^{\operatorname{rat}}$.
The $A$-module $\underline{M}_{B}$ carries a natural action of the Galois
group $G_{\infty}:=\operatorname{Gal}(K_{\infty}^{s}|K_{\infty})$,
$K_{\infty}^{s}$ being the separable closure of $K_{\infty}$ inside
$\mathbb{C}_{\infty}$. We have (see Proposition 3.11 and Corollary 3.17):

######  Proposition.

The profinite group $G_{\infty}$ acts continuously on the discrete $A$-module
$\underline{M}_{B}$. Further, the functor
$\underline{M}\mapsto\underline{M}_{B}$, having for source the category
$A\mathbf{Mot}_{K}^{\operatorname{rat}}$ and for target the category of
continuous $A$-linear representation of $G_{\infty}$, is exact.

The action of $G_{\infty}$ on $\underline{M}_{B}$ should be interpreted as the
counter-part of the complex conjugation acting on the Betti realization of
classical mixed motives, although here $G_{\infty}$ is a way more convoluted
that its number field avatar $\operatorname{Gal}(\mathbb{C}|\mathbb{R})$.

In [Gaz22], we established the definition of
$\operatorname{Ext}^{1}_{A}(\mathbbm{1},\underline{M})$, a natural $A$-module
consisting of integral extensions of the neutral $A$-motive $\mathbbm{1}$ by
$\underline{M}$ in the category $A\mathbf{Mot}_{K}^{\operatorname{rat}}$.
However, there are at least two reasons why this module is not finitely
generated in general, preventing the naive analogue of conjecture (I) to hold
in our context:

  1. 1.

The first reason, mentioned in details in [Gaz22, §5], is related to the fact
that taking Hodge filtrations is not an exact operation on the full class of
exact sequences. This is solved by the notion of regulated extensions,
introduced in Definition 5.7 in _loc. cit._ The sub-$A$-module
$\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})$ of
regulated extensions, however, might still not be finitely generated.

  2. 2.

The second reason, more subtle, is due to the infinite nature of the absolute
Galois group $G_{\infty}$. We solve this second point by introducing the
notion of analytic reduction at $\infty$ defined as follows.

The exactness of the Betti realization functor induces a morphism at the level
of extension modules:

$r_{B}:\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})\longrightarrow\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})$
(1.2)

where the target denotes the continuous Galois cohomology module. We say that
an extension $[\underline{E}]$ in
$\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})$ has
analytic reduction at $\infty$ if it lies in the kernel of $r_{B}$;
equivalently, if $[\underline{E}]$ splits as representations of $G_{\infty}$.
We denote by
$\operatorname{Ext}^{1,\text{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$ the
kernel of $r_{B}$, and by $\operatorname{Cl}(\underline{M})$ its cokernel. Our
first main theorem is the following (repeated from Theorem 4.1).

######  Theorem (4.1).

The $A$-modules
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
and $\operatorname{Cl}(\underline{M})$ are finitely generated. If in addition
the weights of $\underline{M}$ are all negative, then
$\operatorname{Cl}(\underline{M})$ is finite.

Let us throw some comments on Theorem 4.1.

  1. $-$

The above theorem should be understood as the analogue of conjecture (I) for
rigid analytically trivial Anderson $A$-motives. Indeed, all extensions of
classical mixed motives are regulated in the obvious sense, and hence (1.2)
corresponds classically to the morphism of $\mathbb{Q}$-vector spaces

$r_{B}:\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbbm{1},M)\longrightarrow\operatorname{H}^{1}(\operatorname{Gal}(\mathbb{C}|\mathbb{R}),M_{B})$

induced by the exactness of the Betti realization (above, $M$ is a mixed
motive over $\mathbb{Q}$, $M_{B}$ its Betti realization). Yet, in the
$\mathbb{Q}$-linear category of mixed motives, the right-hand side is zero,
which amounts to say that all extensions in
$\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbbm{1},M)$ have analytic reduction
at $\infty$.

  2. $-$

As a second remark, let us mention that Theorem 4.1 also shows that
$\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})$ is almost
never finitely generated. To wit, its size is approximately the same as
$\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})$. Yet, $G_{\infty}$ is not
topologically finitely generated: by class field theory, its wild inertia
group is topologically isomorphic to the group of $1$-unit in
$\pazocal{O}_{\infty}$, itself isomorphic to a countable product of
$\mathbb{Z}_{p}$.

  3. $-$

Finally, let us mention the work of Mornev [Mor18, Thm. 1.1], supersiding
prior results of Taelman in [Tae10], where he obtained a similar version of
Theorem 4.1 for Drinfeld modules having everywhere good reduction. We strongly
suspect a relation with our work.

This discussion hints that the module
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
is the right source of a regulator. This speculation is in chords with recent
computations obtained jointly with Maurischat [GM22a] for the Carlitz tensor
powers. The side of Hodge-Pink structures, that we portray next, seems to
confirm this insight.

Hodge-Pink structures. In an innovative unpublished monograph [Pin97], Pink
defined and studied the general theory of Hodge structures in function fields
arithmetic. The right object of study, highlighted in loc. cit. and nowadays
called Hodge-Pink structures, consists in couples
$\underline{H}=(H,\mathfrak{q}_{H})$, where

  1. 1.

$H$ is a finite dimensional $K_{\infty}$-vector space,

  2. 2.

$\mathfrak{q}_{H}$ is a $K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$-lattice in the
$K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$-vector space
$H_{K_{\infty}^{s}(\\!(\mathfrak{j})\\!)}:=H\otimes_{K_{\infty},\nu}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$

(see Section 2 for details). Here $K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$
denotes the completion of $A\otimes K_{\infty}^{s}$ for the
$\mathfrak{j}$-adic topology, $K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$ its
fraction field, and $\nu:K_{\infty}\to K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$
the morphism $a\mapsto a\otimes 1$ (which is well-defined by [Gaz22, Lem.
5.1]).

######  Remark 1.2.

In [Pin97], Pink is rather concerned with mixed Hodge-Pink structures which
are triples $(H,WH,\mathfrak{q}_{H})$, where $W=WH$ is a $\mathbb{Q}$-graded
finite filtration of $H$ by subspaces, satisfying a certain semi-stability
condition.

Classically, Hodge structures arising from motives naturally carry an
involution called the infinite Frobenius which is obtained functorially from
the action of the complex conjugation. We suggest the following function field
counterpart:

######  Definition (2.9).

An infinite Frobenius for $\underline{H}$ is a $K_{\infty}$-linear continuous
representation $\phi:G_{\infty}\to\operatorname{End}_{K_{\infty}}(H)$, $H$
carrying the discrete topology, such that for all $\sigma\in G_{\infty}$,

$\phi(\sigma)\otimes_{A}\sigma:H_{K_{\infty}^{s}(\\!(\mathfrak{j})\\!)}\longrightarrow
H_{K_{\infty}^{s}(\\!(\mathfrak{j})\\!)}$

preserves the Hodge-Pink lattice $\mathfrak{q}_{H}$. We denote by
$\mathbf{HPk}^{+}$ the category of pairs $(\underline{H},\phi_{H})$ where
$\underline{H}$ is an Hodge-Pink structure and $\phi_{H}$ is an infinite
Frobenius for $\underline{H}$.

Let $\underline{H}^{+}$ be a mixed Hodge-Pink structure equipped with a
infinite Frobenius, and let $\mathbbm{1}^{+}$ denote the neutral object in
$\mathbf{HPk}^{+}$. Contrary to the number field picture, yet similar to what
we observed for mixed $A$-motives, the space of extensions
$\operatorname{Ext}^{1}_{\mathbf{HPk}^{+}}(\mathbbm{1}^{+},\underline{H}^{+})$
is generally not of finite dimension over $K_{\infty}$. Reasons are almost
identical to those for $A$-motives: taking Hodge filtrations is not an exact
operation, and this space of extensions is intertwined with the Galois
cohomology of the profinite group $G_{\infty}$. Using Pink’s notion of Hodge
additivity - which inspired that of regulated extensions in [Gaz22] \- we
considered the subspace

$\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}^{+}}(\mathbbm{1}^{+},\underline{H}^{+})$
(1.3)

of Hodge additive extensions. We denote by
$\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}^{+},\infty}(\mathbbm{1}^{+},\underline{H}^{+})$,
or simply $\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}$ when the
category $\mathbf{HPk}^{+}$ is clear from context, the subspace of (1.3) of
extensions whose infinite Frobenius splits.

From Proposition 3.26 below, there is an exact functor
$\mathscr{H}^{+}:A\mathbf{Mot}_{K}^{\operatorname{rat}}\to\mathbf{HPk}^{+}$
(see Hartl-Juschka [HJ20] for mixedness analogue of this statement), playing
the role of the Hodge realization functor (see Definition 3.25). As a
corollary of Theorem 3.28 below, we record:

######  Proposition.

The space
$\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}(\underline{M}))$
is finite dimensional over $K_{\infty}$.

The exactness of $\mathscr{H}^{+}$ induces an $A$-linear morphism

$r_{\mathscr{H}^{+}}:\operatorname{Ext}^{1}_{A\mathbf{Mot}_{K}^{\operatorname{rat}}}(\mathbbm{1},\underline{M})\longrightarrow\operatorname{Ext}^{1}_{\mathbf{HPk}^{+}}(\mathbbm{1}^{+},\mathscr{H}^{+}(\underline{M}))$

which, almost by design, maps
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
to
$\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}(\underline{M}))$
(Lemma 4.2). All our efforts are worth to justify the next definition.

######  Definition (4.3).

We call the regulator of $\underline{M}$, and denote it
$\textrm{{Reg}}(\underline{M})$, the $A$-linear morphism

$\textrm{{Reg}}(\underline{M}):\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})\longrightarrow\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}(\underline{M}))$

induced by $r_{\mathscr{H}^{+}}$.

It is the counterpart of Beilinson’s regulator in this setting. Our second
main result is the following.

######  Theorem (4.4).

Assume that the weights of $\underline{M}$ are all negative. The rank of
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
over $A$ equals the dimension of
$\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}(\underline{M}))$
over $K_{\infty}$.

In view of conjecture (II) and the above, it is natural to ask whether the
image of $\textrm{{Reg}}(\underline{M})$ has full rank in its target. It is
surprisingly false stated as it is, even in the simplest case of the $p$th
Carlitz twist (function field analogue of Tate twists); this will be the
subject of a subsequent work. The right formulation of Conjectures (II) and
(III) in this context still is an open problem to the author and source of
current investigations.

Methods. Our proof of the main theorems took strong inspirations in the work
of Lafforgue [Laf09] and Mornev [Mor18], and hinges on the concept of shtuka
models. We associate non-canonically to $\underline{M}$ \- which sits at the
level of the affine curve $\operatorname{Spec}A\otimes K$ \- a shtuka model
$\underline{\pazocal{M}}_{0}=(\pazocal{M}_{0},\pazocal{N}_{0},\tau_{0})$ over
the surface $(\operatorname{Spec}A)\times C$ (Proposition 5.2). Attached to it
is a complex of $A$-modules222$G_{\underline{M}}$ is in fact defined
differently in Definition 4.5 below, but is quasi-isomorphic to the next
definition by Proposition 6.7. sitting in degrees $0$ and $1$:

$G_{\underline{M}}=\operatorname{R}\\!\Gamma\left(\operatorname{Spec}A\times
C,\pazocal{M}_{0}\stackrel{{\scriptstyle\tau_{0}}}{{\longrightarrow}}\pazocal{N}_{0}\right),$

which only depends on $\underline{M}$ up to quasi-isomorphisms. It plays a
similar role to that of shtuka cohomology as defined and studied extensively
in [Mor18]. We show in Proposition 4.6 that there is a natural long exact
sequence of $A$-modules:

$0\to\operatorname{Hom}_{A\mathbf{Mot}_{K}}(\mathbbm{1},\underline{M})\to\underline{M}_{B}^{G_{\infty}}\to\operatorname{H}^{0}(G_{\underline{M}})\\\
\to\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})\stackrel{{\scriptstyle
r_{B}}}{{\to}}\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})\to\operatorname{H}^{1}(G_{\underline{M}})\to
0$

where $r_{B}$ is the map defined in (1.2). The first part of Theorem 4.1 is
implied by the perfectness of the complex $G_{\underline{M}}$, which itself
follows from finiteness of coherent cohomology of the proper $A$-scheme
$(\operatorname{Spec}A)\times C\to\operatorname{Spec}A$.

To some extent, the proof of Theorem 4.4 and the second part of Theorem 4.1 is
similar but more involved. When the weights of $\underline{M}$ are non-
positive, we further associate to $\underline{M}$ a shtuka model
$\underline{\pazocal{M}}$ on $C\times C$ (Theorem 5.6). An incarnation of this
gadget in the context of Drinfeld modules with everywhere good reduction
already appeared in loc. cit. under the name of global model. The miracle
ingredient of our proof is the surprising link between the behaviour of
$\underline{\pazocal{M}}$ at $\\{\infty\\}\times\\{\infty\\}\hookrightarrow
C\times C$ and the space of Hodge additive extensions of Hodge-Pink structures
having analytic reduction at $\infty$ (Corollary 5.19): this link materializes
by a distinguish triangle of complexes of $K_{\infty}$-vector spaces:

$\underline{M}_{B}^{G_{\infty}}[0]\otimes_{A}K_{\infty}\to\operatorname{R}\\!\Gamma\left(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\pazocal{M}\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\pazocal{N}\right)\otimes_{\pazocal{O}_{\infty}}K_{\infty}\to\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}(\underline{M}))[0]\to[1].$

The middle term is isomorphic to $G_{\underline{M}}\otimes_{A}K_{\infty}$ as
we show using Grothendieck’s comparison Theorem (see Section 6.2). In
particular, $G_{\underline{M}}\otimes_{A}K_{\infty}$ sits in degree zero and
hence
$\operatorname{H}^{1}(G_{\underline{M}})\cong\operatorname{Cl}(\underline{M})$
is finite as desired. The statement on the ranks follows as well, proving
Theorem 4.4.

######  Remark 1.3.

In this text, we are concern with $A$-motives over $K$. As in the number field
setting, this contains the case of finite field extensions $F$ of $K$ by
replacing an $A$-motive $\underline{M}$ of rank $r$ over $F$ by its
restriction $\operatorname{Res}_{F/K}\underline{M}$ \- an $A$-motive of rank
$r[F:K]$ over $K$ \- obtained by seeing $\underline{M}$ as being over $K$. For
instance, one shows that

$\operatorname{Ext}^{1,\text{reg}}_{\pazocal{O}_{F}}(\mathbbm{1},\underline{M})=\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\operatorname{Res}_{F/K}\underline{M})$

where notations are the one from [Gaz22]. From this observation, the
statements of Theorems 4.1 and 4.4 could be converted accordingly for
$A$-motives over $F$.

##### Acknowledgment:

I learnt a posteriori that the question of constructing natural finitely
generated extension modules out of the category of $A$-motives - as
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
is - remained an open problem suggested by Dinesh Thakur. I am much indebt to
him for his interest in my work. In early versions of the manuscript, I have
benefited much from multiple exchanges and discussions with the following
people to whom I wish to reiterate my gratitude: Gebhard Böckle, Bhargav
Bhatt, Christopher Deninger, Urs Hartl, Annette Huber-Klawitter, Maxim Mornev
and Federico Pellarin. I am also grateful to Max Planck Institute for
Mathematics in Bonn for its hospitality and financial support.

##  2 Hodge-Pink structures and their extensions

In this section we provide all the necessary material on Hodge-Pink structures
and their extension modules. Our main source of inspiration is Pink’s
monograph [Pin97]. An innovation of ours is the introduction of infinite
Frobenii in this context.

###  2.1 Hodge-Pink structures

#### Definitions

Let $(C,\pazocal{O}_{C})$ is a geometrically irreductible smooth projective
curve over $\mathbb{F}$ and $\infty$ is a closed point of $C$. The
$\mathbb{F}$-algebra

$A=H^{0}(C\setminus\\{\infty\\},\pazocal{O}_{C})$

consists of functions of $C$ that are regular away from $\infty$.

Let $v$ be a closed point of $C$ and denote by $K_{v}$ the local function
field at $v$. Let $R$ be a Noetherian subring of $K_{v}$ which contains $A$
and such that $R\otimes_{A}K$ is a field (which is then identified with a
subfield of $K_{v}$). The ring $R$ will play the role of a coefficient ring.
In practice, $R$ should be either $A$, $K$ or $K_{v}$. Let $L$ be a separable
extension of $K_{v}$: $L$ will play the role of a base field.

If $k$ is a field which is an $A$-algebra through a morphism $\kappa$, we let
$\mathfrak{j}=\mathfrak{j}_{\kappa}$ denote the maximal ideal of $A\otimes k$
generated by the set of differences $\\{a\otimes 1-1\otimes a|a\in A\\}$
(equivalently, the kernel of $A\otimes k\to k$). We denote by
$k[\\![\mathfrak{j}]\\!]$ the completion of $A\otimes k$ along the ideal
$\mathfrak{j}_{\kappa}$, and by $k(\\!(\mathfrak{j})\\!)$ its field of
fractions. $k[\\![\mathfrak{j}]\\!]$ is a discrete valuation ring with maximal
ideal $\mathfrak{j}$ and residual field $k$. By a
$k[\\![\mathfrak{j}]\\!]$-lattice $\mathfrak{h}$ in a finite dimensional
$k(\\!(\mathfrak{j})\\!)$-vector space $V$ we shall mean a finitely generated
sub-$k[\\![\mathfrak{j}]\\!]$-module of $V$ that contains a basis.

Given an $R$-module $M$, there are (at least) two ways to obtain an
$L[\\![\mathfrak{j}]\\!]$-module out of $M$: the base-wise way would be
$M\otimes_{R}L[\\![\mathfrak{j}]\\!]$ seing $R$ as a subfield of $L$; the
coefficient-wise way would be to consider $L[\\![\mathfrak{j}]\\!]$ as an
$A$-module via the morphism $\nu:A\to A\otimes L$, $a\mapsto a\otimes 1$. By
[Gaz22, Lem. 5.1], $\nu$ extends uniquely to a morphism $\nu:K_{v}\to
L[\\![\mathfrak{j}]\\!]$ which coincides modulo $\mathfrak{j}$ with the
inclusion $K_{v}\to L$, and it thus makes sense to consider the module
$M\otimes_{R,\nu}L[\\![\mathfrak{j}]\\!]$.

Following Pink’s [Pin97, Def. 3.2], we define:

######  Definition 2.1.

An Hodge-Pink structure $\underline{H}$ with base field $L$ and coefficient
ring $R$ consists of the data of $(H,\mathfrak{q})$ where

  1. 1.

$H$ is a finitely generated $R$-module,

  2. 2.

$\mathfrak{q}=\mathfrak{q}_{H}$ is an $L[\\![\mathfrak{j}]\\!]$-lattice in the
$L(\\!(\mathfrak{j})\\!)$-vector space
$H_{L(\\!(\mathfrak{j})\\!)}:=H\otimes_{R,\nu}L(\\!(\mathfrak{j})\\!)$.

We call $\mathfrak{q}$ the Hodge-Pink lattice of $\underline{H}$. We let
$\mathfrak{p}_{H}:=H\otimes_{R,v}L[\\![\mathfrak{j}]\\!]$ and name it the
tautological lattice of $\underline{H}$.

We gather Hodge-Pink structures into an $R$-linear category
${}_{R}\mathbf{HPk}_{L}$ (the leftscript $"R"$ should disappear from the
notations when clear from the context). A morphism from
$\underline{H}=(H,W,\mathfrak{q})$ to
$\underline{H}^{\prime}=(H^{\prime},W^{\prime},\mathfrak{q}^{\prime})$ in
$\mathbf{HPk}_{L}$ is a morphism $f:H\to H^{\prime}$ of $R$-modules such that
$f_{L(\\!(\mathfrak{j})\\!)}:=f\otimes_{R,\nu}\operatorname{id}_{L(\\!(\mathfrak{j})\\!)}$
preserves the Hodge-Pink lattices; i.e.
$f_{L(\\!(\mathfrak{j})\\!)}(\mathfrak{q})\subset\mathfrak{q}^{\prime}$. We
call $f$ strict whenever we have:

$f_{L(\\!(\mathfrak{j})\\!)}(\mathfrak{q})=\mathfrak{q}^{\prime}\cap\operatorname*{im}f_{L(\\!(\mathfrak{j})\\!)}.$

#### Extensions

We are interested in computing extension modules in the category
$\mathbf{HPk}_{L}$. However, $\mathbf{HPk}_{L}$ is not abelian and we have to
clarify the notion of exact sequences we would like to consider.

######  Definition 2.2.

A sequence
$S:0\to\underline{H}^{\prime}\to\underline{H}\to\underline{H}^{\prime\prime}\to
0$ in $\mathbf{HPk}_{L}$ is called exact if its underlying sequence of
$R$-module is, and strict if morphisms of $S$ are.

It is formal to check that the category $\mathbf{HPk}$ together with the
notion of strict exact sequences form an exact category. Given
$\underline{X}$, $\underline{Y}$ two Hodge-Pink structures, we denote by
$\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}}(\underline{X},\underline{Y})$ the
$R$-module of classes of strict exact sequences of the form
$0\to\underline{Y}\to\underline{H}\to\underline{X}\to 0$.

We turn to the description of those extension modules. We write
$\underline{X}=(X,\mathfrak{q}_{X})$ and $\underline{Y}=(Y,\mathfrak{q}_{Y})$.
Given an $L(\\!(\mathfrak{j})\\!)$-linear morphism
$f:X_{L(\\!(\mathfrak{j})\\!)}\to Y_{L(\\!(\mathfrak{j})\\!)}$, consider the
object

$\underline{E}_{f}=\left(Y\oplus
X,\\{(q_{y}+f(q_{x}),q_{x})~{}|~{}(q_{y},q_{x})\in\mathfrak{q}_{Y}\oplus\mathfrak{q}_{X}\\}\right)\in\operatorname{Ob}\mathbf{HPk}_{L}.$
(2.1)

Let also $S_{f}$ be the canonical short sequence
$0\to\underline{Y}\to\underline{E}_{f}\to\underline{X}\to 0$ with obvious
morphisms. A simple computation shows that $S_{f}$ is strict exact. In
particular, the assignment $f\mapsto[S_{f}]$ defines a map

$\varphi:\operatorname{Hom}_{L(\\!(\mathfrak{j})\\!)}\left(X_{L(\\!(\mathfrak{j})\\!)},Y_{L(\\!(\mathfrak{j})\\!)}\right)\longrightarrow\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}}(\underline{X},\underline{Y})$
(2.2)

The following proposition suffices to describe the extension module when the
underlying module of $\underline{X}$ is projective:

######  Proposition 2.3.

Suppose that $X$ is a projective $R$-module. The morphism $\varphi$ is a
surjective $R$-linear morphism whose kernel is

$\operatorname{Hom}_{R}(X,Y)+\operatorname{Hom}_{L[\\![\mathfrak{j}]\\!]}(\mathfrak{q}_{X},\mathfrak{q}_{Y}).$

Here, the subscripts $"\mathfrak{q}"$ refers to the submodule of morphisms
compatible with the Hodge-Pink lattice.

###### Proof.

Let $f$ and $g$ be elements of
$\operatorname{Hom}_{L(\\!(\mathfrak{j})\\!)}\left(X_{L(\\!(\mathfrak{j})\\!)},Y_{L(\\!(\mathfrak{j})\\!)}\right)$.
A formal computation shows that the Baer sum of $[S_{f}]$ and $[S_{g}]$ is
$[S_{f+g}]$, and that the pullback of $[S_{f}]$ by
$a\cdot\operatorname{id}:\underline{X}\to\underline{X}$, for $a\in R$, is
$[S_{af}]$. This proves that $\varphi$ is $R$-linear.

To show that $\varphi$ is surjective, fix an extension $[S]$ of
$\underline{X}$ by $\underline{Y}$. By our assumption that $X$ is projective,
the underlying sequence of $R$-modules splits. In other words, $S$ is
equivalent to an exact sequence of the form

$0\longrightarrow\underline{Y}\longrightarrow\left(Y\oplus
X,\mathfrak{q}\right)\longrightarrow\underline{X}\longrightarrow 0$ (2.3)

where $\mathfrak{q}\subset Y_{L(\\!(\mathfrak{j})\\!)}\oplus
X_{L(\\!(\mathfrak{j})\\!)}$ is an $L[\\![\mathfrak{j}]\\!]$-lattice. Let us
define a morphism $f:X_{L(\\!(\mathfrak{j})\\!)}\to
Y_{L(\\!(\mathfrak{j})\\!)}$ as follows. As morphisms of (2.3) are strict, the
underlying sequence of lattices is exact:

$0\longrightarrow\mathfrak{q}_{Y}\longrightarrow\mathfrak{q}\longrightarrow\mathfrak{q}_{X}\longrightarrow
0.$

For $q_{x}\in\mathfrak{q}_{X}$, choose one of its lift $\tilde{q}_{x}$ in
$\mathfrak{q}$, unique up to an element in $\mathfrak{q}_{Y}$. The assignement
$\bar{f}(q_{x}):=\tilde{q}_{x}+\mathfrak{q}_{Y}$ defines an
$L[\\![\mathfrak{j}]\\!]$-linear morphism $\bar{f}:\mathfrak{q}_{X}\to
Y_{L(\\!(\mathfrak{j})\\!)}/\mathfrak{q}_{Y}$. As $\mathfrak{q}_{X}$ is a
projective $L[\\![\mathfrak{j}]\\!]$-module, $\bar{f}$ lifts to an
$L(\\!(\mathfrak{j})\\!)$-linear morphism $f:X_{L(\\!(\mathfrak{j})\\!)}\to
Y_{L(\\!(\mathfrak{j})\\!)}$. It is clear from construction that $S$ is
isomorphic to $S_{f}$, hence $[S]=\varphi(f)$.

It remains to describe the kernel of $\varphi$. Observe that $S_{f}$ is
equivalent to $S_{g}$ if and only if there exists an $R$-linear map $u:X\to Y$
such that $f=g+u$. Besides, $S_{f}$ splits if and only if $f$ preserves the
Hodge-Pink lattices. This proves that

$\ker\varphi=\operatorname{Hom}^{W}_{R}(X,Y)+\operatorname{Hom}_{L[\\![\mathfrak{j}]\\!]}(\mathfrak{q}_{X},\mathfrak{q}_{Y})$

as desired. ∎

Let $\mathbbm{1}$ be the Hodge-Pink structure over $L$ whose underlying
$R$-module is $R$ itself and whose mixed Hodge-Pink lattice is
$\mathfrak{q}_{\mathbbm{1}}=\mathfrak{p}_{\mathbbm{1}}=L[\\![\mathfrak{j}]\\!]$.
We call $\mathbbm{1}$ the _neutral Hodge-Pink structure_. We end this
paragraph with the following corollary:

######  Corollary 2.4.

Let $\underline{H}$ be a Hodge-Pink structure over $L$. We have a natural
isomorphism of $R$-modules

$\varphi:\frac{H_{L(\\!(\mathfrak{j})\\!)}}{H+\mathfrak{q}_{H}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}}(\mathbbm{1},\underline{H}).$

#### Hodge-additive extensions

Contrary to the number fields setting, the $R$-module
$\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}}(\underline{X},\underline{Y})$ is
almost never finitely generated over $R$, even when $R=K_{v}$. This is an
issue regarding regulators, which classically are morphism of finite
dimensional vector spaces. Following Pink, we now discuss the notion of Hodge
addivity on extensions which solve this nuisance.

Let $\underline{H}$ be an Hodge-Pink structure over $L$. We first recall how
to associate a finite decreasing filtration - the _Hodge filtration_ \- on
$H_{L}$. For $p\in\mathbb{Z}$, let $\operatorname{Fil}^{p}H_{L}$ denote the
image of $\mathfrak{p}_{H}\cap\mathfrak{j}^{p}\mathfrak{q}_{H}$ through the
composition:

${\mathfrak{p}_{H}=H\otimes_{R,\nu}L[\\![\mathfrak{j}]\\!]}$${H\otimes_{R}L=H_{L}.}$$\scriptstyle{\operatorname{mod}~{}\mathfrak{j}}$

We call $\operatorname{Fil}H_{L}=(\operatorname{Fil}^{p}H_{L})_{p}$ the _Hodge
filtration_ of $\underline{H}$. The Hodge polygon of $\underline{H}$ is
defined as the polygon of the filtration $\operatorname{Fil}H_{L}$.

######  Definition 2.5 ([Pin97, §8]).

A strict exact sequence
$0\to\underline{H}^{\prime}\to\underline{H}\to\underline{H}^{\prime\prime}\to
0$ is said to be _Hodge additive_ if the Hodge polygon of $\underline{H}$
coincide with that of $\underline{H}\oplus\underline{H}^{\prime}$.

As the Hodge polygon is invariant under isomorphism, the property of being
Hodge additive respects equivalences of extensions. Hence, the following
definition makes sense:

######  Definition 2.6.

Let $\underline{X}$, $\underline{Y}$ be objects in $\mathbf{HPk}_{L}$. We
denote by
$\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}_{L}}(\underline{X},\underline{Y})$
the subset of Hodge additive extensions of $\underline{X}$ by $\underline{Y}$
in $\mathbf{HPk}_{L}$.

One easily shows that
$\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}_{L}}$ is a
sub-$R$-module. The following result due to Pink shows that
$\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}_{L}}$ is finitely
generated.

######  Proposition 2.7 ([Pin97, Prop. 8.7]).

Let $\underline{X}$, $\underline{Y}$ be objects in $\mathbf{HPk}_{L}$ where
the underlying module of $\underline{X}$ is projective. Let
$f:X_{L(\\!(\mathfrak{j})\\!)}\to Y_{L(\\!(\mathfrak{j})\\!)}$ be an
$L(\\!(\mathfrak{j})\\!)$-linear map. Then, the sequence $S_{f}$ is Hodge
additive if and only if $f$ preserves the tautological lattices. In
particular,
$\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}_{L}}(\underline{X},\underline{Y})$
is an sub-$R$-module of
$\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}}(\underline{X},\underline{Y})$, and
$\varphi$ induces an isomorphism:

$\frac{\operatorname{Hom}_{L[\\![\mathfrak{j}]\\!]}(\mathfrak{p}_{X},\mathfrak{p}_{Y})}{\operatorname{Hom}_{R}(X,Y)+\operatorname{Hom}_{L[\\![\mathfrak{j}]\\!]}(\mathfrak{q}_{X},\mathfrak{q}_{Y})\cap\operatorname{Hom}_{L[\\![\mathfrak{j}]\\!]}(\mathfrak{p}_{X},\mathfrak{p}_{Y})}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}_{L}}(\underline{X},\underline{Y}).$

In the particular case of $\underline{X}$ being $\mathbbm{1}$, we get:

######  Corollary 2.8.

Let $\underline{H}$ be a mixed Hodge-Pink structure over $L$. The morphism
$\varphi$ of Corollary 2.4 induces:

$\varphi:\frac{\mathfrak{p}_{H}}{H+(\mathfrak{q}_{H}\cap\mathfrak{p}_{H})}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}_{L}}(\mathbbm{1},\underline{H}).$

###  2.2 Infinite Frobenii

Before introducing _infinite Frobenii_ for Hodge-Pink structures, let us
shortly recall the classical story.

#### The classical picture

According to Nekovar [Nek94, (2.4)] and Deligne [Del89, §1.4 (M7)], an
infinite Frobenius $\phi_{\infty}$ for a mixed Hodge structure
$(H,WH,\operatorname{Fil}H_{\mathbb{C}})$ (with coefficients $\mathbb{R}$ and
base $\mathbb{C}$) is an involution of the $\mathbb{R}$-vector space $H$
compatible with $WH$, and such that $\phi_{\infty}\otimes_{\mathbb{R}}c$
preserves $\operatorname{Fil}H_{\mathbb{C}}$. Mixed Hodge Structures arising
from the singular cohomology groups of a variety $X$ over $\mathbb{R}$ are
naturally equipped with an infinite Frobenius, induced by functoriality of the
action of the complex conjugation on the complex points $X(\mathbb{C})$.

We let $\mathbf{MHS}_{\mathbb{R}}^{+}$ be the category whose objects are pairs
$(\underline{H},\phi_{\infty})$ where $\underline{H}$ is a mixed Hodge
structures and $\phi_{\infty}$ is an infinite Frobenius for $\underline{H}$.
Morphisms in $\mathbf{MHS}_{\mathbb{R}}^{+}$ are the morphisms of mixed Hodge
structures over $\mathbb{C}$ which commute to infinite Frobenii.

Extension modules in the category $\mathbf{MHS}_{\mathbb{C}}$ of mixed Hodge
structures over $\mathbb{C}$ are well known. Given an object $\underline{H}$
of $\mathbf{MHS}_{\mathbb{C}}$, the complex of $\mathbb{R}$-vector spaces

$\left[W_{0}H\oplus
F^{0}W_{0}H_{\mathbb{C}}\stackrel{{\scriptstyle\left(\begin{smallmatrix}1\\\
\text{-}1\end{smallmatrix}\right)}}{{\longrightarrow}}W_{0}H_{\mathbb{C}}\right]$

represents the cohomology of
$\operatorname{RHom}_{\mathbf{MHS}_{\mathbb{C}}}(\mathbbm{1},\underline{H})$
(e.g. [Bei86, §1], [Car80, Prop. 2], [PS08, Thm. 3.31]). We obtain an
$\mathbb{R}$-linear morphism

$\frac{W_{0}H_{\mathbb{C}}}{W_{0}H+F^{0}W_{0}H_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1}_{\mathbf{MHS}_{\mathbb{C}}}(\mathbbm{1},\underline{H}).$
(2.4)

If now $\underline{H}^{+}$ denotes an object in the category
$\mathbf{MHS}^{+}_{\mathbb{R}}$ with infinite Frobenius $\phi_{\infty}$, the
complex
$\operatorname{RHom}_{\mathbf{MHS}_{\mathbb{R}}^{+}}(\mathbbm{1}^{+},\underline{H}^{+})$
is rather represented by

$\left[(W_{0}H)^{+}\oplus(F^{0}W_{0}H_{\mathbb{C}})^{+}\stackrel{{\scriptstyle\left(\begin{smallmatrix}1\\\
\text{-}1\end{smallmatrix}\right)}}{{\longrightarrow}}(W_{0}H_{\mathbb{C}})^{+}\right]$

where the subscript $+$ means the corresponding $\mathbb{R}$-subspace fixed by
$\phi_{\infty}\otimes c$ (e.g. [Bei86, §1], [Nek94, (2.5)]). We obtain an
$\mathbb{R}$-linear morphism

$\frac{(W_{0}H_{\mathbb{C}})^{+}}{(W_{0}H)^{+}+(F^{0}W_{0}H_{\mathbb{C}})^{+}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1}_{\mathbf{MHS}_{\mathbb{R}}^{+}}(\mathbbm{1},\underline{H}).$
(2.5)

#### Infinite Frobenii for Hodge-Pink structures

We keep notations from the previous subsection. Assume from now on that $L$ is
a finite separable extensions of $K_{v}$ (hence complete), and fix $L^{s}$ a
separable closure of $L$. In this subsection, we enrich Hodge-Pink structures
with a compatible continuous action of the profinite Galois group
$G_{L}:=\operatorname{Gal}(L^{s}|L)$. In several cases, we compute extensions.

Let $\underline{H}=(H,\mathfrak{q}_{H})$ be an Hodge-Pink structure with
coefficient ring $R$ and base field $L^{s}$.

######  Definition 2.9.

An infinite Frobenius for $\underline{H}$ is an $R$-linear continuous
representation $\phi:G_{L}\to\operatorname{End}_{R}(H)$, $G_{L}$ carrying the
profinite topology and $H$ the discrete topology, such that, for all
$\sigma\in G_{L}$,

$\phi(\sigma)\otimes_{A}\sigma:H_{L^{s}(\\!(\mathfrak{j})\\!)}\longrightarrow
H_{L^{s}(\\!(\mathfrak{j})\\!)}$

preserves the Hodge-Pink lattice $\mathfrak{q}_{H}$.

######  Remark 2.10.

Above, we denoted $\sigma$ its extension to $L^{s}(\\!(\mathfrak{j})\\!)$,
i.e. obtained by the functoriality of the assignment $k\mapsto
k(\\!(\mathfrak{j})\\!)$, from the category of $A$-fields to rings, applied to
$\sigma:L^{s}\to L^{s}$.

We let ${}_{R}\mathbf{HPk}_{L}^{+}$ (or $\mathbf{HPk}_{L}^{+}$) be the
category whose objects are pairs $(\underline{H},\phi_{H})$ where
$\underline{H}$ is a mixed Hodge-Pink structure over $L^{s}$ and $\phi_{H}$ is
an infinite Frobenius for $\underline{H}$. Morphisms in $\mathbf{HPk}_{L}^{+}$
are the one in $\mathbf{HPk}_{L^{s}}$ commuting the infinite Frobenii.

######  Definition 2.11.

A sequence
$S:0\to\underline{H}^{\prime}\to\underline{H}\to\underline{H}^{\prime\prime}\to
0$ in $\mathbf{HPk}_{L}^{+}$ is called exact (resp. strict) if the underlying
sequence in $\mathbf{HPk}_{L^{s}}$ is exact (resp. strict) as in Definition
2.2.

As well, $\mathbf{HPk}_{L}^{+}$ with the above class of strict exact sequences
is an exact category and we now study its extension modules. While the
ingenious analogue of (2.4) holds for Hodge-Pink structures (this was
Corollary 2.8), a description similar as (2.5) does not hold in our setting as
the action of the complex conjugation is replaced by that of the (infinite)
profinite group $G_{L}$. Therefore, the extension modules are intertwined with
the Galois cohomology of $G_{L}$, preventing an isomorphism as simple as (2.5)
to exist. In order to clarify how Galois cohomology interferes with the
computation of extension spaces, we introduce next an $R$-linear morphism
$d_{\underline{H}^{+}}$.

By definition, we have a forgetful functor from $\mathbf{HPk}_{L}^{+}$ to the
category of $R$-linear continuous representation of $G_{L}$, assigning
$\underline{H}^{+}=(\underline{H},\phi_{H})$ to $\phi_{H}$. Being exact, it
induces a natural $R$-linear morphism at the level of extensions:

$\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}^{+}}(\underline{G}^{+},\underline{H}^{+})\longrightarrow\operatorname{Ext}^{1}_{G_{L}}(\phi_{G},\phi_{H}),$
(2.6)

the right-hand side denoting extensions of continuous $R$-linear
$G_{L}$-representations.

######  Definition 2.12.

An extension
$[\underline{E}]\in\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}^{+}}(\underline{G}^{+},\underline{H}^{+})$
is said to have _analytic reduction_ if it splits as a representation of
$G_{L}$; i.e. lies in the kernel of (2.6). We denote the latter by
$\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}^{+},v}(\underline{G}^{+},\underline{H}^{+})$
or simply by
$\operatorname{Ext}^{1}_{v}(\underline{G}^{+},\underline{H}^{+})$.

Let $\mathbbm{1}$ denote the neutral mixed Hodge-Pink structure over $L^{s}$,
and denote by $\mathbbm{1}^{+}$ the object of $\mathbf{HPk}^{+}_{L}$ given by
the pair $(\mathbbm{1},\phi_{\mathbbm{1}})$, where
$\phi_{\mathbbm{1}}:G_{L}\to R$, $\sigma\mapsto 1$ is the neutral
representation.

######  Definition 2.13.

Given an object $\underline{H}^{+}$ of $\mathbf{HPk}_{L}^{+}$, we denote by

$d_{\underline{H}^{+}}:\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}^{+}}(\mathbbm{1}^{+},\underline{H}^{+})\longrightarrow\operatorname{H}^{1}(G_{L},H)$

the $R$-linear morphism (2.6) with $\underline{G}^{+}=\mathbbm{1}^{+}$, where
$\operatorname{H}^{1}$ here denotes the continuous group cohomology.

Recall that
$H_{L^{s}(\\!(\mathfrak{j})\\!)}=H\otimes_{R,\nu}L^{s}(\\!(\mathfrak{j})\\!)$
is endowed with a continuous action of $\sigma\in G_{L}$ given by
$\phi_{H}(\sigma)\otimes\sigma$. For $S$ a subset of
$H_{L^{s}(\\!(\mathfrak{j})\\!)}$ we denote by $S^{+}$ the subset of elements
fixed under this action of $G_{L}$. Let $\varphi$ be the isomorphism of
Corollary 2.8. There is an $R$-linear morphism:

$\varphi^{+}:\frac{(H_{L^{s}(\\!(\mathfrak{j})\\!)})^{+}}{H^{+}+\mathfrak{q}_{H}^{+}}\longrightarrow\operatorname{Ext}^{1}_{\mathbf{HPk}^{+}_{L}}(\mathbbm{1}^{+},\underline{H}^{+}),$

mapping the class of $h\in(H_{L^{s}(\\!(\mathfrak{j})\\!)})^{+}$ to the
extension $(\varphi(h),\left(\begin{smallmatrix}\phi_{H}&0\\\
0&1\end{smallmatrix}\right))$. By definition, the image of $\varphi^{+}$ lands
in $\operatorname{Ext}^{1}_{v}(\mathbbm{1}^{+},\underline{H}^{+})$. Under some
assumption, we can say more.

######  Proposition 2.14.

Suppose that $\operatorname{H}^{1}(G_{L},\mathfrak{q}_{H})$ is trivial. Then,
the sequence of $R$-modules:

$0\longrightarrow\frac{(H_{L^{s}(\\!(\mathfrak{j})\\!)})^{+}}{H^{+}+\mathfrak{q}_{H}^{+}}\stackrel{{\scriptstyle\varphi^{+}}}{{\longrightarrow}}\operatorname{Ext}^{1}_{\mathbf{HPk}^{+}_{L}}(\mathbbm{1}^{+},\underline{H}^{+})\stackrel{{\scriptstyle
d_{\underline{H}^{+}}}}{{\longrightarrow}}\operatorname{H}^{1}(G_{L},H)\longrightarrow
0$

is exact. In particular, under the same assumption, $\varphi^{+}$ induces an
isomorphism of $R$-modules:

$\varphi^{+}:\frac{(H_{L^{s}(\\!(\mathfrak{j})\\!)})^{+}}{H^{+}+\mathfrak{q}_{H}^{+}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1}_{v}(\mathbbm{1}^{+},\underline{H}^{+}).$

######  Remark 2.15.

It will appear that the condition
$\operatorname{H}^{1}(G_{L},\mathfrak{q}_{H})=(0)$ is always satisfied for
mixed Hodge-Pink structures arising from rigid analytically trivial mixed
$A$-motives. We refer to Lemma 3.30 below.

###### Proof.

We first show that $d_{\underline{H}^{+}}$ is surjective. Let $c:G_{L}\to H$
be a cocycle. We denote by $[c]$ the $R$-linear $G_{L}$-representation of
$H\oplus R$ given by

$[c]:G_{L}\longrightarrow\operatorname{End}_{R}(H\oplus
R),\quad\sigma\longmapsto\begin{pmatrix}\phi_{H}(\sigma)&c(\sigma)\\\
0&1\end{pmatrix}.$

Because $\operatorname{H}^{1}(G_{L},\mathfrak{q}_{H})=0$, there exists
$h\in\mathfrak{q}_{H}$ such that
$c(\sigma)=h-(\phi_{H}(\sigma)\otimes\sigma)(h)$. Recall that $\varphi(h)$ is
represented by the extension of $\mathbbm{1}$ by $\underline{H}$ whose middle
term is

$\underline{E}_{h}:=\left(H\oplus R,\begin{pmatrix}1&h\\\
0&1\end{pmatrix}\mathfrak{q}_{H}\oplus L^{s}[\\![\mathfrak{j}]\\!]\right).$

It is formal to check that $[c]$ defines an infinite Frobenius for
$\underline{E}_{h}$ so that the extension given by the pair
$(\underline{E}_{h},[c])$ defines an element of
$\operatorname{Ext}^{1}_{\mathbf{HPk}_{L}^{+}}(\mathbbm{1}^{+},\underline{H}^{+})$.
Its image through $d_{\underline{H}^{+}}$ is $c$, as desired.

Before computing the kernel of $d_{\underline{H}^{+}}$, we begin with an
observation. Let $c$ be a cocycle $G_{L}\to H$ such that
$(\underline{E}_{h},[c])$ defines an extension of $\mathbbm{1}^{+}$ by
$\underline{H}^{+}$ in $\mathbf{HPk}_{L}^{+}$. For $m\in H$, the diagram

${0}$${\underline{H}^{+}}$${\left(\underline{E}_{h},[c]\right)}$${\mathbbm{1}^{+}}$${0}$${0}$${\underline{H}^{+}}$${\left(\underline{E}_{h+m},[\sigma\mapsto
c(\sigma)+m-\phi_{H}(\sigma)(m)]\right)}$${\mathbbm{1}^{+}}$${0}$$\scriptstyle{\operatorname{id}_{\underline{H}}}$$\scriptstyle{\begin{pmatrix}\operatorname{id}_{H}&m\\\
0&1\end{pmatrix}}$$\scriptstyle{1}$

defines an equivalence in $\mathbf{HPk}_{L}^{+}$ between the extensions:

$\left(\underline{E}_{h},[c]\right)\quad\text{and}\quad\left(\underline{E}_{h+m},[\sigma\mapsto
c(\sigma)+m-\phi_{H}(\sigma)(m)]\right).$ (2.7)

Let us compute the kernel of $d_{\underline{H}^{+}}$. If $[\underline{E}^{+}]$
is an element of $\ker d_{\underline{H}^{+}}$ there exists $h^{\prime}\in
H_{L^{s}(\\!(\mathfrak{j})\\!)}$ and $m\in H$ such that $[\underline{E}^{+}]$
is congruent to an extension of the form
$(\underline{E}_{h^{\prime}},[\sigma\mapsto m-\phi_{H}(\sigma)(m)])$. By our
computation (2.7), we can assume without loss of generality that
$[\underline{E}^{+}]$ is of the form $\left(\underline{E}_{h},[0]\right)$. The
condition that the infinite Frobenius of $\underline{E}^{+}$ preserves the
Hodge filtration reads

$\forall\sigma\in
G_{L},\quad(\phi_{H}(\sigma)\otimes\sigma)(h)-h\in\mathfrak{q}_{H}.$

In particular, $h+\mathfrak{q}_{H}$ is invariant under $G_{L}$ as an element
of the quotient module $H_{L^{s}(\\!(\mathfrak{j})\\!)}/\mathfrak{q}_{H}$.
Using once more that $\operatorname{H}^{1}(G_{L},\mathfrak{q}_{H})$ is
trivial, we obtain

$\left(\frac{H_{L^{s}(\\!(\mathfrak{j})\\!)}}{\mathfrak{q}_{H}}\right)^{+}=\frac{(H_{L^{s}(\\!(\mathfrak{j})\\!)})^{+}}{\mathfrak{q}_{H}^{+}}.$

Hence, $h$ is congruent to a certain
$h_{0}\in(H_{L^{s}(\\!(\mathfrak{j})\\!)})^{+}$ modulo $\mathfrak{q}_{H}$. We
have $\underline{E}_{h}=\underline{E}_{h_{0}}$ since their Hodge-Pink lattice
are the same. To conclude, it suffices to note that any equivalence in
$\mathbf{HPk}_{L}^{+}$ between the extensions
$[\underline{E}^{+}]=\left(\underline{E}_{h_{0}},[0]\right)$ and
$\left(\underline{E}_{k_{0}},[0]\right)$ for some $k_{0}\in
H_{L^{s}(\\!(\mathfrak{j})\\!)}$ implies $h_{0}-k_{0}\in H^{+}$. ∎

Given two objects $\underline{G}^{+}$ and $\underline{H}^{+}$ of
$\mathbf{HPk}_{L}^{+}$, denote by

$\operatorname{Ext}^{1,\operatorname{ha}}_{v}(\mathbbm{1}^{+},\underline{H}^{+}):=\operatorname{Ext}^{1}_{v}(\mathbbm{1}^{+},\underline{H}^{+})\cap\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}_{L}^{+}}(\mathbbm{1}^{+},\underline{H}^{+})$

the sub-$R$-module of Hodge-additive extensions having analytic reduction. The
same argument that of the proof of Proposition 2.14 applies to show:

######  Proposition 2.16.

Suppose that
$\operatorname{H}^{1}(G_{L},\mathfrak{q}_{H}\cap\mathfrak{p}_{H})$ is trivial.
Then, the sequence of $R$-modules:

$0\longrightarrow\frac{\mathfrak{p}_{H}^{+}}{H^{+}+(\mathfrak{q}_{H}\cap\mathfrak{p}_{H})^{+}}\stackrel{{\scriptstyle\varphi^{+}}}{{\longrightarrow}}\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}^{+}_{L}}(\mathbbm{1}^{+},\underline{H}^{+})\stackrel{{\scriptstyle
d_{\underline{H}^{+}}}}{{\longrightarrow}}\operatorname{H}^{1}(G_{L},H)\longrightarrow
0$

is exact. In particular, under the same assumption, $\varphi^{+}$ induces an
isomorphism of $R$-modules:

$\varphi^{+}:\frac{\mathfrak{p}_{H}^{+}}{H^{+}+(\mathfrak{q}_{H}\cap\mathfrak{p}_{H})^{+}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1,\operatorname{ha}}_{v}(\mathbbm{1}^{+},\underline{H}^{+}).$

##  3 Rigid analytically trivial mixed $A$-motives

###  3.1 Definitions

In this subsection we review the usual setup of $A$-motives. Recall that
$(C,\pazocal{O}_{C})$ is a geometrically irreductible smooth projective curve
over a finite field $\mathbb{F}$ with $q$ elements, $\infty$ is a closed point
of $C$ and $A$ is the $\mathbb{F}$-algebra
$\pazocal{O}_{C}(C\setminus\\{\infty\\})$.

Let $R$ be an $A$-algebra through a $\mathbb{F}$-algebra morphism $\kappa:A\to
R$. We denote by $A\otimes R$ the tensor product over $\mathbb{F}$, and we let
$\mathfrak{j}=\mathfrak{j}_{\kappa}$ be the kernel of the multiplication map
$A\otimes R\to R$, $a\otimes r\mapsto\kappa(a)r$. The following observation
appears in [Har19].

######  Lemma 3.1.

The ideal $\mathfrak{j}$ is a projective $A\otimes R$-module of rank $1$. In
particular, $V(\mathfrak{j})$ defines a Cartier divisor on
$\operatorname{Spec}A\otimes R$.

###### Proof.

Denote by $\mathfrak{j}_{\operatorname{id}}$ the kernel of the multiplication
map $A\otimes A\to A$. Being projective of rank $1$ is a property stable by
base-change, and as
$\mathfrak{j}_{\kappa}=\mathfrak{j}_{\operatorname{id}}\otimes_{A\otimes
A}(A\otimes R)$, it suffices to show that $\mathfrak{j}_{\operatorname{id}}$
is projective of rank $1$ over $A\otimes A$.

We first observe that $\mathfrak{j}_{\operatorname{id}}$ is finite projective.
Indeed, $A\otimes A$ is a Noetherian domain, so it suffices to show that
$\mathfrak{j}_{\operatorname{id}}$ is flat. We use Bourbaki’s flatness
criterion: $\mathfrak{j}_{\operatorname{id}}$ is torsion free, and
$\mathfrak{j}_{\operatorname{id}}/\mathfrak{j}^{2}_{\operatorname{id}}$ is
flat as it is isomorphic to $\Omega_{A/\mathbb{F}}$ itself flat over
$A\cong(A\otimes A)/\mathfrak{j}_{\operatorname{id}}$. The criterion applies
to show that $\mathfrak{j}_{\operatorname{id}}$ is flat. To conclude that it
has constant rank $1$, it suffices to observe that
$\operatorname{Spec}(A\otimes A)$ is connected and that
$\mathfrak{j}_{\operatorname{id}}/\mathfrak{j}^{2}_{\operatorname{id}}$ has
rank $1$ over $A$. ∎

Let $\tau$ be the ring endomorphism of $A\otimes R$ acting as the identity on
$A$ and as raising to the $q$th power on $R$ (i.e. $\tau(a\otimes r)=a\otimes
r^{q}$).

######  Definition 3.2.

An _$A$ -motif of rank $r$ over $R$_ is a pair $\underline{M}=(M,\tau_{M})$
where $M$ is a locally-free module over $A\otimes R$ together with a
$\tau$-linear isomorphism outside the zero locus $V(\mathfrak{j})$ of
$\mathfrak{j}$:

$\tau_{M}:(\tau^{*}M)|_{(\operatorname{Spec}A\otimes R)\setminus
V(\mathfrak{j})}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M|_{(\operatorname{Spec}A\otimes
R)\setminus V(\mathfrak{j})}.$

We call $\underline{M}$ effective whenever $\tau_{M}$ comes from the pullback
by $\operatorname{Spec}A\otimes R\to(\operatorname{Spec}A\otimes R)\setminus
V(\mathfrak{j})$ of a morphism $\tau^{*}M\to M$.

A morphism $(M,\tau_{M})\to(N,\tau_{N})$ of $A$-motives over $R$ is an
$A\otimes R$-linear morphism $f:M\to N$ such that
$f\circ\tau_{M}=\tau^{*}f\circ\tau_{N}$. We let $A\mathbf{Mot}_{R}$ denote the
$A$-linear category of $A$-motives over $R$.

######  Remark 3.3.

Along this text, we shall denote by $M[\mathfrak{j}^{-1}]$ the module
$M|_{(\operatorname{Spec}A\otimes R)\setminus V(\mathfrak{j})}$. This agrees
with the convention taken in [Gaz22], where $M[\mathfrak{j}^{-1}]$ is rather
described as the submodule of $M\otimes_{A\otimes
R}\operatorname{Quot}(A\otimes R)$ \- where $\operatorname{Quot}(A\otimes R)$
denotes the localization of $A\otimes R$ at the non zero-divisor - consisting
in elements $x$ for which there exists a positive integer $n\geq 0$ such that
$\mathfrak{j}^{n}x\in M$.

Most of the results in this text are concern with $A$-motives over $K$ (with
$\kappa:A\to K$ being the inclusion). For those, one has a notion of weights
that we recall briefly (the reader will find all details in [Gaz22, §3]).
There is an (exact) functor

$A\mathbf{Mot}_{K}\longrightarrow{}_{\infty}\mathbf{Isoc}_{K},\quad\underline{M}\longmapsto\pazocal{I}_{\infty}({\underline{M}})$

where the targeted category is that of $\infty$-isocrystals over $K$. The
category ${}_{\infty}\mathbf{Isoc}_{K}$ admits a slope function in the sense
of [And09], hence any object admits a unique slope filtration (the Harder-
Narasimhan filtration) [Gaz22, Thm. 3.9]. It is said that $\underline{M}$ has
weights $\nu_{1}<\nu_{2}<...<\nu_{s}$ if $\pazocal{I}_{\infty}(\underline{M})$
has slopes $-\nu_{1}>-\nu_{2}>...>-\nu_{s}$.

###  3.2 The Betti realization functor

We let $K$ be the fraction field of $A$ (equivalently, the function field of
$C$). Here, we introduce the Betti realization of an $A$-motive (Definition
3.7) and discuss rigid analytically triviality (Definition 3.8). One chief aim
is to define the full subcategory $A\mathbf{Mot}_{K}^{\text{rat}}$ of
$A\mathbf{Mot}_{K}$ consisting of rigid analytically trivial $A$-motives over
$K$, which shall be the source of the Hodge-Pink realization functor to be
defined in Subsection 3.4. Historically, the notion of rigid analytic
triviality dates back to Anderson [And86, §2], and most of this subsection
owes to his work. A novelty of our account is the consideration of a natural
continuous action of $G_{\infty}$ \- the absolute Galois group at $\infty$ \-
on the Betti realization $A$-modules. The existence of canonical infinite
Frobenii attached to the associated Hodge-Pink structures will follow from
this construction.

#### Tate algebras

Let $L$ be a field over $\mathbb{F}$ complete with respect to an non-
archimedean norm $|\cdot|$, and let $\pazocal{O}_{L}$ be its valuation ring
with maximal ideal $\mathfrak{m}_{L}$.

######  Definition 3.4.

We denote by $\pazocal{O}_{L}\langle A\rangle$ the $\pazocal{O}_{L}$-algebra
given by the completion of $A\otimes\pazocal{O}_{L}$ with respect to the
$A\otimes\mathfrak{m}_{L}$-adic topology. We denote by $L\langle A\rangle$ the
$L$-algebra $L\otimes_{\pazocal{O}_{L}}\pazocal{O}_{L}\langle A\rangle$. We
again denote by $\tau$ the continuous extension of $A\otimes\pazocal{O}_{L}\to
A\otimes\pazocal{O}_{L}$, $a\otimes c\mapsto a\otimes c^{q}$ to
$\pazocal{O}_{L}\langle A\rangle$ and $L\langle A\rangle$.

######  Remark 3.5.

The notation $L\langle A\rangle$ is here to stress that it generalizes the
classical Tate algebra over $L$: if $A\cong\mathbb{F}[t]$, then $L\langle
A\rangle\cong L\langle t\rangle$. For general rings $A$, one can show that
$L\langle A\rangle$ is an affinoid algebra (see [GM22b, Prop. 3.2]).

The following preliminary lemma will be used next, in the definition of the
Betti realization functor.

######  Lemma 3.6.

Let $\kappa:A\to L$ be an $\mathbb{F}$-algebra morphism with discrete image.
We have $\mathfrak{j}_{\kappa}L\langle A\rangle=L\langle A\rangle$.

###### Proof.

Because $\kappa(A)$ is discrete in $L$, it contains an element $\alpha$ of
norm $|\alpha|>1$. Let $a\in A$ be such that $\alpha=\kappa(a)$. Then,
$\kappa(a)^{-1}\in\mathfrak{m}_{L}$ and the series

$-\sum_{n\geq 0}{a^{n}\otimes\kappa(a)^{-(n+1)}}$

converges in $\pazocal{O}_{L}\langle A\rangle$ to the inverse of $(a\otimes
1-1\otimes\kappa(a))$. ∎

#### The Betti realization of an $A$-motive

Let $K_{\infty}$ be the completion of $K$ with respect to the $\infty$-adic
topology, and denote by $\pazocal{O}_{\infty}$ its ring of integers with
maximal ideal $\mathfrak{m}_{\infty}$. We fix $K_{\infty}^{s}$ a separable
closure of $K_{\infty}$, and denote $\mathbb{C}_{\infty}$ its completion
(which is now algebraically closed and complete, by Krasner’s Lemma). The
canonical norm on $K_{\infty}$ extends uniquely to a norm $|\cdot|$ on
$\mathbb{C}_{\infty}$. The action of
$G_{\infty}=\operatorname{Gal}(K_{\infty}^{s}|K_{\infty})$ extends by
continuity to $\mathbb{C}_{\infty}$.

Let $\underline{M}=(M,\tau_{M})$ be an $A$-motive over $K$. By Lemma 3.6, the
ideal $\mathfrak{j}\subset A\otimes K$ becomes invertible in
$\mathbb{C}_{\infty}\langle A\rangle$, and thus $\tau_{M}$ induces an
isomorphism of modules over $\mathbb{C}_{\infty}\langle A\rangle$:

$\tau_{M}:\tau^{*}(M\otimes_{A\otimes K}\mathbb{C}_{\infty}\langle
A\rangle)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M\otimes_{A\otimes
K}\mathbb{C}_{\infty}\langle A\rangle$ (3.1)

which commutes with the action of $G_{\infty}$ on $M\otimes_{A\otimes
K}\mathbb{C}_{\infty}\langle A\rangle$, inherited from the right-hand side of
the tensor. We still denote by $\tau_{M}$ the isomorphism (3.1).

######  Definition 3.7.

The Betti realization of $\underline{M}$ is the $A$-module

$\underline{M}_{B}:=\left\\{\omega\in M\otimes_{A\otimes
K}\mathbb{C}_{\infty}\langle
A\rangle~{}|~{}\omega=\tau_{M}(\tau^{*}\omega)\right\\}.$

endowed with the compatible action of $G_{\infty}$ it inherits as a submodule
of $M\otimes_{A\otimes K}\mathbb{C}_{\infty}\langle A\rangle$. We let
$\underline{M}_{B}^{+}$ denote the sub-$A$-module of $\underline{M}_{B}$ of
elements fixed by $G_{\infty}$.

The next definition is borrowed from [And86, §2.3].

######  Definition 3.8.

The $A$-motive $\underline{M}$ is called rigid analytically trivial if the map

$\underline{M}_{B}\otimes_{A}\mathbb{C}_{\infty}\langle A\rangle\to
M\otimes_{A\otimes K}\mathbb{C}_{\infty}\langle A\rangle,\quad\omega\otimes
c\mapsto\omega\cdot c$

given by the multiplication is bijective.

######  Remark 3.9.

Not every $A$-motive is rigid analytically trivial. An example of $A$-motive
which is not rigid analytically trivial is given in [And86, 2.2], or [Tae09,
Ex. 3.2.10].

The following proposition rephrases [BH07, Cor.4.3]:

######  Proposition 3.10.

Let $\underline{M}$ be an $A$-motive over $F$ of rank $r$. Then
$\underline{M}_{B}$ is a finite projective $A$-module of rank $r^{\prime}$
satisfying $r^{\prime}\leq r$ with equality if and only if $\underline{M}$
rigid analytically trivial.

When $\underline{M}$ is rigid analytically trivial, in Definition 3.8 the
field $\mathbb{C}_{\infty}$ can be replaced by a much smaller field. This is
the subject of the next proposition.

######  Proposition 3.11.

Let $\underline{M}$ be a rigid analytically trivial $A$-motive over $K$. There
exists a (complete) finite separable field extension $L$ of $K_{\infty}$ in
$\mathbb{C}_{\infty}$ such that $\underline{M}_{B}$ is contained in
$M\otimes_{A\otimes K}L\langle A\rangle$. In particular, the action of
$G_{\infty}$, equipped with the profinite topology, on $\underline{M}_{B}$,
equipped with the discrete topology, is continuous.

###### Proof.

For effective $t$-motives this is proven in [And86, Thm 4], so we explain how
one may reduces to this case. Let $t$ be a non constant element of $A$. The
inclusion $\mathbb{F}[t]\subset A$ makes $A$ into a finite flat
$\mathbb{F}[t]$-module, and therefore $\underline{M}$ defines an
$\mathbb{F}[t]$-motive of rank $\deg(t)\cdot\operatorname{rank}\underline{M}$
over $K$. Using the identification $\mathbb{F}[t]\otimes K=K[t]$, we rather
write $t$ for $t\otimes 1$ and $\theta$ for $1\otimes t$. Let $n>0$ be an
integer so that $(t-\theta)^{n}\tau_{M}(\tau^{*}M)\subset M$. Let
$\underline{N}$ be the $\mathbb{F}[t]$-motive over $K$ whose underlying module
is $N=K[t]$ and where $\tau_{N}$ is the multiplication by $(t-\theta)^{n}$
(this is the dual of Carlitz $n$th twist). If $\sqrt[q-1]{-\theta}$ denotes a
$q-1$-root of $-\theta$ in $\mathbb{C}_{\infty}$, we have

$\underline{N}_{B}=(\\!\sqrt[q-1]{-\theta})^{-n}\prod_{i=0}^{\infty}{\left(1-\frac{t}{\theta^{q^{i}}}\right)^{n}}\cdot\mathbb{F}[t]\subset
K_{\infty}\\!\left(\\!\sqrt[q-1]{-\theta}\right)\langle t\rangle.$

The $\mathbb{F}[t]$-motive $\underline{N}$ has been chosen so that
$\underline{M}\otimes\underline{N}$ is effective (see Definition 3.2). We are
thus in the rang of application of [And86, Thm 4] which states that there
exists a finite extension $H$ of $K_{\infty}$ in $\mathbb{C}_{\infty}$ such
that

$\underline{M}_{B}\otimes_{\mathbb{F}[t]}\underline{N}_{B}=(\underline{M}\otimes\underline{N})_{B}\subset(M\otimes_{K[t]}N)\otimes_{K[t]}H\langle
t\rangle=M\otimes_{K[t]}H\langle t\rangle.$

It follows that there exists a finite extension $L^{\prime}$ of $K_{\infty}$
such that $\underline{M}_{B}\subset M\otimes_{K[t]}L^{\prime}\langle t\rangle$
(e.g. one can take $L^{\prime}:=H(\sqrt[q-1]{-\theta})$).

We now show that one can choose $L^{\prime}$ separable over $K_{\infty}$. Note
that $M\otimes_{K[t]}K_{\infty}^{s}\langle t\rangle$ is free of finite rank
over $K_{\infty}^{s}\langle t\rangle$. Therefore,
$(M\otimes_{K[t]}K_{\infty}^{s}\langle t\rangle)/(t^{n})$ is a finite
dimensional $K_{\infty}^{s}$-vector space for all positive integers $n$. By
Lang’s isogeny Theorem (e.g. [Kat73, Prop. 1.1]), the multiplication map

$\left\\{m\in(M\otimes_{K[t]}K_{\infty}^{s}\langle
t\rangle)/(t^{n})~{}|~{}m=\tau_{M}(\tau^{*}m)\right\\}\otimes
K_{\infty}^{s}\longrightarrow(M\otimes_{K[t]}K_{\infty}^{s}\langle
t\rangle)/(t^{n})$

is an isomorphism. In particular, the inclusion

$\left\\{m\in(M\otimes_{K[t]}K_{\infty}\langle
t\rangle)/(t^{n})|m=\tau_{M}(\tau^{*}m)\right\\}\subseteq\left\\{m\in(M\otimes_{K[t]}\mathbb{C}_{\infty}\langle
t\rangle)/(t^{n})|m=\tau_{M}(\tau^{*}m)\right\\}$

is an equality. This shows that $\underline{M}_{B}$ is both a submodule of
$M\otimes_{K[t]}K_{\infty}^{s}\langle t\rangle$ and
$M\otimes_{K[t]}L^{\prime}\langle t\rangle$. Because $M$ is free over $K[t]$,
it follows that $\underline{M}_{B}\subset M\otimes_{K[t]}L\langle t\rangle$
where $L=L^{\prime}\cap K_{\infty}^{s}$ is a finite separable extension of
$K_{\infty}$ in $\mathbb{C}_{\infty}$. As $(A\otimes K)\otimes_{K[t]}L\langle
t\rangle$ is isomorphic to $L\langle A\rangle$, we deduce that
$\underline{M}_{B}\subset M\otimes_{A\otimes K}L\langle A\rangle$. ∎

As a consequence of Proposition 3.11, by the faithful flatness of the
inclusion $L\langle A\rangle\to\mathbb{C}_{\infty}\langle A\rangle$ ([Bou07,
AC I§3.5 Prop. 9]), we have:

######  Proposition 3.12.

Let $\underline{M}$ be a rigid analytically trivial $A$-motive over $K$. Let
$L$ be as in Proposition 3.11. The multiplication map

$\underline{M}_{B}\otimes_{A}L\langle A\rangle\longrightarrow
M\otimes_{A\otimes K}L\langle A\rangle$

is an isomorphism of $L\langle A\rangle$-modules.

The first part of the next result is inspired by [BH07, Prop. 6.1]. We have
adapted its proof to allow the smaller field $K_{\infty}^{s}$ instead of
$\mathbb{C}_{\infty}$. This is needed in order to compute the $A$-module
$\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})$ of continuous Galois
cohomology.

######  Theorem 3.13.

Let $\underline{M}$ be a rigid analytically trivial $A$-motive over $K$. There
is an exact sequence of $A[G_{\infty}]$-modules:

$0\longrightarrow\underline{M}_{B}\longrightarrow M\otimes_{A\otimes
K}\mathbb{C}_{\infty}\langle
A\rangle\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}M\otimes_{A\otimes
K}\mathbb{C}_{\infty}\langle A\rangle\longrightarrow 0.$ (3.2)

Furthermore, it induces a long exact sequence of $A$-modules

$0\longrightarrow\underline{M}_{B}^{+}\longrightarrow M\otimes_{A\otimes
K}K_{\infty}\langle A\rangle\longrightarrow M\otimes_{A\otimes
K}K_{\infty}\langle
A\rangle\longrightarrow\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})\longrightarrow
0.$ (3.3)

######  Remark 3.14.

The fact that (3.2) implies (3.3) has nothing immediate. We have to descend
from the completion of the perfection of $K_{\infty}$ – which by the Ax-Sen-
Tate Theorem, corresponds to the fixed subfield of $\mathbb{C}_{\infty}$ by
$G_{\infty}$ – to the much smaller field $K_{\infty}$.

###### Proof of Theorem 3.13..

Let $\mathbb{F}[t]\to A$ be a non constant morphism of rings. We have
$\mathbb{C}_{\infty}\langle
A\rangle=A\otimes_{\mathbb{F}[t]}\mathbb{C}_{\infty}\langle t\rangle$ where
$\mathbb{C}_{\infty}\langle t\rangle$ is the Tate algebra over
$\mathbb{C}_{\infty}$ in the variable $t$.

The exactness of (3.2) follows from [BH07, Prop. 6.1]. We shall use the same
argument as in loc. cit. to show that the sequence

$0\longrightarrow\underline{M}_{B}\longrightarrow
M\otimes_{K[t]}K_{\infty}^{s}\langle
t\rangle\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}M\otimes_{K[t]}K_{\infty}^{s}\langle
t\rangle\longrightarrow 0,$ (3.4)

where the first inclusion is well-defined by Proposition 3.11, is exact. It
suffices to show the surjectivity of $\operatorname{id}-\tau_{M}$ on
$M\otimes_{K[t]}K_{\infty}^{s}\langle t\rangle$. Let $f\in
M\otimes_{K[t]}K_{\infty}^{s}\langle t\rangle$. Since $\underline{M}$ is rigid
analytically trivial, without loss of generality we can assume that
$f=c\cdot\omega$ for $c=\sum_{n\geq 0}{c_{n}t^{n}}\in K_{\infty}^{s}\langle
t\rangle$ and $\omega\in\underline{M}_{B}$. For every $n\geq 0$, let $b_{n}\in
K_{\infty}^{s}$ be a solution of $x-x^{q}=c_{n}$. The condition $|c_{n}|\to 0$
as $n$ grows implies $|b_{n}|\to 0$. Hence, the element

$g:=\left(\sum_{n=0}^{\infty}{b_{n}t^{n}}\right)\cdot\omega$

belongs to $M\otimes_{K[t]}K_{\infty}^{s}\langle t\rangle$ and satisfies
$(\operatorname{id}-\tau_{M})(g)=f$. Surjectivity follows.

We turn to the second part of the statement. By Proposition 3.11, $G_{\infty}$
acts continuously on (3.4), and taking invariants yields a long exact sequence
of $A$-modules:

$0\longrightarrow\underline{M}_{B}^{+}\longrightarrow M\otimes_{A\otimes
K}K_{\infty}\langle
A\rangle\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}M\otimes_{A\otimes
K}K_{\infty}\langle
A\rangle\longrightarrow\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})\longrightarrow\cdots$

We shall prove that the $K[t]$-module

$\operatorname{H}^{1}(G_{\infty},M\otimes_{K[t]}K_{\infty}^{s}\langle
t\rangle)$

is zero. The above is isomorphic to
$M\otimes_{K[t]}\operatorname{H}^{1}(G_{\infty},K_{\infty}^{s}\langle
t\rangle)$, hence it suffices to show that
$\operatorname{H}^{1}(G_{\infty},K_{\infty}^{s}\langle t\rangle)$ vanishes. By
continuity, it is enough to show the vanishing of
$\operatorname{H}^{1}(G_{\infty},L\langle t\rangle)$ for any subfield
$L\subset K_{\infty}^{s}$ that is a finite Galois extension of $K_{\infty}$.
For such $L$, we denote by $H$ the finite Galois group
$\operatorname{Gal}(L|K_{\infty})$. By the additive version of Hilbert’s 90
Theorem [Ser68, x.§1, Prop. 1], the module
$\operatorname{H}^{1}(G_{\infty},L)=\operatorname{H}^{1}(H,L)$ vanishes and
thus $\operatorname{H}^{1}(G_{\infty},L[\\![t]\\!])$ is zero. Therefore, there
is a long exact sequence

$0\longrightarrow K_{\infty}\langle t\rangle\longrightarrow
K_{\infty}[\\![t]\\!]\longrightarrow\left(L[\\![t]\\!]/L\langle
t\rangle\right)^{G_{\infty}}\longrightarrow\operatorname{H}^{1}(G_{\infty},L\langle
t\rangle)\to 0.$

In particular, for any continuous cocycle $c:G_{\infty}\to L\langle t\rangle$,
we can find $f\in L[\\![t]\\!]$ such that

$\forall\sigma\in G_{\infty}:\quad c(\sigma)=f^{\sigma}-f.$

Because $L$ is separable over $K_{\infty}$ its trace forme is non-degenerated,
that is, there exists $\alpha\in L$ such that $\eta:=\sum_{\sigma\in
H}{\alpha^{\sigma}}\in K_{\infty}$ is nonzero. Thus, $f$ can be written as

$f=\left(\eta^{-1}\sum_{\sigma\in
H}{\alpha^{\sigma}f^{\sigma}}\right)-\left(\eta^{-1}\sum_{\sigma\in
H}{\alpha^{\sigma}c(\sigma)}\right)\in K_{\infty}[\\![t]\\!]+L\langle
t\rangle.$

It follows that $c$ is trivial, and that
$\operatorname{H}^{1}(G_{\infty},L\langle t\rangle)=0$. This concludes the
proof. ∎

We are now ready to introduce the category of rigid analytically trivial
$A$-motives over $K$, as mentioned in the introduction.

######  Definition 3.15.

We let $A\mathbf{Mot}_{K}^{\operatorname{rat}}$ be the full subcategory of
$A\mathbf{Mot}_{K}$ whose objects are rigid analytically trivial.

Recall that a sequence of objects in $A\mathbf{Mot}_{K}$ is called exact if
the underlying sequence of modules is (see [Gaz22, Def. 2.5]), and that the
category $A\mathbf{Mot}_{K}$ endowed with the class of exact sequences forms
an exact category ([Gaz22, Prop. 2.6]). The next proposition, which ensures
that extension modules in the category
$A\mathbf{Mot}_{K}^{\operatorname{rat}}$ are well-defined, is borrowed from
[HJ20, Lem. 2.3.25].

######  Proposition 3.16.

Let
$0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ be an exact sequence in $A\mathbf{Mot}_{K}$. Then $\underline{M}$ is rigid
analytically trivial if and only if $\underline{M}^{\prime}$ and
$\underline{M}^{\prime\prime}$ are. In particular, the category
$A\mathbf{Mot}_{K}^{\operatorname{rat}}$ is exact.

We finally record that the Betti realization functor having
$A\mathbf{Mot}_{K}^{\operatorname{rat}}$ as its source is exact (this is not
true on the full category $A\mathbf{Mot}_{K}$).

######  Corollary 3.17.

The functor $\underline{M}\mapsto\underline{M}_{B}$ from
$A\mathbf{Mot}_{K}^{\operatorname{rat}}$ to the category
$\operatorname{Rep}_{A}(G_{\infty})$ of continuous $A$-linear representations
of $G_{\infty}$ is exact.

###### Proof.

This follows from Theorem 3.13 together with the Snake Lemma. ∎

#### Extensions with analytic reduction

In view of Corollary 3.17, the following definition makes sense:

######  Definition 3.18.

Let
$S:0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ be an exact sequence in $A\mathbf{Mot}_{K}^{\operatorname{rat}}$. We say
that $S$ has analytic reduction at $\infty$ if the sequence $S_{B}$ splits in
$\operatorname{Rep}_{A}(G_{\infty})$.

######  Remark 3.19.

The above notion is crucial to obtain finitely generated extension modules out
of the category $A\mathbf{Mot}_{K}^{\operatorname{rat}}$ (see Section 6). It
is the counterpart of Definition 2.13 for Hodge-Pink structures.

Given two rigid analytically trivial $A$-motives
$\underline{N},\underline{M}$, the exactness of the Betti realization functor
induces a morphism of $A$-modules:

$r_{B}(\underline{N},\underline{M}):\operatorname{Ext}^{1}_{A\mathbf{Mot}_{K}^{\operatorname{rat}}}(\underline{N},\underline{M})\longrightarrow\operatorname{Ext}^{1}_{\operatorname{Rep}_{A}(G_{\infty})}(\underline{N}_{B},\underline{M}_{B})$
(3.5)

which is the counterpart of (2.6) in the context of Hodge-Pink structures. By
definition, the kernel of $r_{B}(\underline{N},\underline{M})$ consists of
extensions having analytic reduction at $\infty$. We set:

$\operatorname{Ext}^{1}_{\infty}(\underline{N},\underline{M}):=\ker
r_{B}(\underline{N},\underline{M}).$ (3.6)

###  3.3 Analytic continuation

To associate an Hodge-Pink structure to a rigid analytically trivial
$A$-motive $\underline{M}$, it is crucial to understand the behaviour of
elements in $\underline{M}_{B}$ – which can be seen as functions over the
affinoid subdomain $\operatorname{Spm}\mathbb{C}_{\infty}\langle A\rangle$
with values in $M\otimes_{K}\mathbb{C}_{\infty}$ – near the point
$\mathfrak{j}$. However, the latter does not belong to the maximal spectrum of
$\mathbb{C}_{\infty}\langle A\rangle$, as one deduces from Lemma 3.6. Hence it
is necessary to extend elements of $\underline{M}_{B}$ to a larger domain. In
this subsection, we show that elements of $\underline{M}_{B}$ can be
meromorphically continuated to the whole rigid analytification of the affine
curve $\operatorname{Spec}A\otimes\mathbb{C}_{\infty}$, with their only poles
supported at $\mathfrak{j}$ and its iterates $\tau\mathfrak{j}$,
$\tau^{2}\mathfrak{j}$, … . In the $\deg(\infty)=1$-case, this is treated in
[HJ20, §2.3.4], for special functions of Anderson $A$-modules this is the
subject of [GM22b].

We recall some material from [GM22b, §3]. let $L$ be any complete subfield of
$\mathbb{C}_{\infty}$ that contains $K_{\infty}$ and let $|\cdot|$ be the norm
on $L$ it inherits as a subfield of $\mathbb{C}_{\infty}$. We define Gauss
norms on $A\otimes L$: let $c\in L^{\times}$ and let $\rho:=|c|>0$. For $f\in
A\otimes L$, we set:

$\|f\|_{\rho}:=\inf\left(\max_{i}\\{|\ell_{i}|\rho^{\deg a_{i}}\\}\right)$

where the infinimum is taken over all representations of $f$ as finite sums
$\sum_{i}{a_{i}\otimes\ell_{i}}$ in $A\otimes L$. The application
$\|\cdot\|_{\rho}$ is indeed a Gauss norm [GM22b, Prop. 3.2], and we denote by
$L\langle A\rangle_{\rho}$ the affinoid algebra over $L$ obtained by
completing $A\otimes L$ with respect to it. Observe that $L\langle A\rangle$
as in Definition 3.4 coincides with $L\langle A\rangle_{1}$ (see [GM21, Prop.
2.2]). If $\rho<\rho^{\prime}$, there is a canonical inclusion $L\langle
A\rangle_{\rho^{\prime}}\hookrightarrow L\langle A\rangle_{\rho}$ and we set:

$L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}:=\varprojlim_{\rho}L\langle
A\rangle_{\rho}$

where $\rho$ runs over $|L^{\times}|$. If $\mathfrak{X}=\mathfrak{X}_{L}$
denote the rigid analytic variety $(\operatorname{Spec}A\otimes
L)^{\text{rig}}$ over $L$, then $\mathfrak{X}$ is isomorphic to the glueing
$\varinjlim_{\rho}\mathfrak{X}_{\rho}$ where
$\mathfrak{X}_{\rho}:=\operatorname{Sp}L\langle A\rangle_{\rho}$. In
particular,
$L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
is the ring of global sections of $\pazocal{O}_{\mathfrak{X}}$.

As $\|\tau(f)\|_{\rho}=\|f\|_{\rho^{1/q}}^{q}$ for $f\in A\otimes L$, $\tau$
induces a morphism $L\langle A\rangle_{\rho}\to L\langle
A\rangle_{\rho^{1/q}}$ which are compatible: we still denote by $\tau$ the
resulting morphism
$L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\to
L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
and

$\tau:\mathfrak{X}_{L}\longrightarrow\mathfrak{X}_{L}.$

For $m\geq 1$, we denote by $J_{m}$ the divisor corresponding to
$\mathfrak{j}+\tau\mathfrak{j}+...+\tau^{m}\mathfrak{j}$ on $\mathfrak{X}$ and
by $J$ the rigid analytic divisor
$J_{\infty}=\mathfrak{j}+\tau\mathfrak{j}+...$. For $n\geq 0$, we introduce

$\pazocal{O}_{\mathfrak{X}}(n\cdot
J):=\operatorname{colim}_{m}\pazocal{O}_{\mathfrak{X}}(n\cdot J_{m})$

as the sheaf of meromorphic functions on $\mathfrak{X}$ with a pole of order
at most $n$ at the support of $J$.

######  Example 3.20.

Let $C=\mathbb{P}^{1}_{\mathbb{F}}$ and let $\infty$ be the point $[0:1]$. We
identify $A$ with $\mathbb{F}[t]$ and the tensor product $A\otimes K$ with
$K[t]$. We let $\theta\in K$ denote the image of $t$ so that
$\tau^{i}\mathfrak{j}$ corresponds to the principal ideal $(t-\theta^{q^{i}})$
of $K[t]$. We have

$L\langle A\rangle=L\langle
t\rangle=\left\\{\sum_{n=0}^{\infty}{a_{n}t^{n}}~{}\bigg{|}~{}a_{n}\in
L;~{}\lim_{n\to\infty}a_{n}\to 0\right\\},$
$L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}t\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}=\left\\{\sum_{n=0}^{\infty}{a_{n}t^{n}}~{}\bigg{|}~{}a_{n}\in
L;~{}\forall\rho>1:~{}\lim_{n\to\infty}a_{n}\rho^{n}\to 0\right\\}.$ (3.7)

The ring $L\langle A\rangle$ corresponds to series converging in the closed
unit disc, whereas
$L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
consists of entire series. The morphism $\tau$ acts on both rings by mapping

$f=\sum_{n=0}^{\infty}{a_{n}t^{n}}\longmapsto
f^{(1)}=\sum_{n=0}^{\infty}{a_{n}^{q}t^{n}}.$

Now, $\pazocal{O}_{\mathfrak{X}}(n\cdot J)(\mathfrak{X})$ is identified with
the subring of
$\operatorname{Quot}L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}t\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
consisting of elements $f$ such that

$\prod_{j=0}^{\infty}{\left(1-\frac{t}{\theta^{q^{j}}}\right)^{n}}\cdot f\in
L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}t\mathclose{\hbox{\set<EMAIL_ADDRESS>
By Lemma 3.6, $J$ is not supported on $\operatorname{Sp}L\langle A\rangle$ and
we have a morphism

$\pazocal{O}_{\mathfrak{X}}(n\cdot
J)(\mathfrak{X}_{L})\longrightarrow\pazocal{O}_{\mathfrak{X}}(n\cdot
J)(\operatorname{Sp}L\langle A\rangle)=L\langle A\rangle$

given by sheaf restriction. We are now in position to prove the main result of
this subsection (compare with [HJ20, Prop. 2.3.30]).

######  Theorem 3.21.

Let $\underline{M}$ be a rigid analytically trivial $A$-motive over $K$. Let
$L$ be a finite separable extension of $K_{\infty}$ such that
$\underline{M}_{B}$ is contained in $M\otimes_{A\otimes K}L\langle A\rangle$
(whose existence is guaranteed by Proposition 3.11). Then, there exists a
large enough integer $n$ for which canonical morphism

$(M\otimes_{A\otimes K}\pazocal{O}_{\mathfrak{X}}(n\cdot
J)(\mathfrak{X}_{L}))^{\tau_{M}=1}\longrightarrow(M\otimes_{A\otimes
K}L\langle A\rangle)^{\tau_{M}=1}=\underline{M}_{B}$

is an isomorphism.

Before engaging in the proof, we set some definitions. Let $t\in A$ be a non-
constant element and denote by $\theta$ the element $1\otimes t$ in $A\otimes
K$. As $\mathbb{F}[t]\to A$ is finite flat, the multiplication maps

$A\otimes_{\mathbb{F}[t]}L\langle t\rangle\to L\langle A\rangle,\quad
A\otimes_{\mathbb{F}[t]}L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}t\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\to
L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$

are isomorphisms. For $i\geq 0$, the converging product (we abbreviated
$t\otimes 1$ in $t$)

$\Pi_{t}:=\prod_{j=0}^{\infty}{\left(1-\frac{t}{\theta^{q^{j}}}\right)}$

define elements in
$L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
whose only zeros in $\mathfrak{X}$ are supported at
$\displaystyle\bigcup_{j\geq 0}V(t-\theta^{q^{j}})$.

###### Proof of Theorem 3.21.

Let $n\geq 0$ be such that $\mathfrak{j}^{n}\tau_{M}(\tau^{*}M)\subset M$ and
$t\in A$ as above. Let us first show that
$\Pi_{t}^{n}\cdot\underline{M}_{B}\subset M\otimes_{A\otimes
K}L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$.
We use again notations of the proof of Proposition 3.11: $\sqrt[q-1]{-\theta}$
is a $q-1$-root of the image of $-t$ in $L$ and, introducing

$\omega_{t}:=\sqrt[q-1]{-\theta}\prod_{i=0}^{\infty}{\left(1-\frac{t}{\theta^{q^{i}}}\right)^{-1}}=\sqrt[q-1]{-\theta}\cdot\Pi_{t}^{-1}\in\operatorname{Quot}L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}},$

we have $\underline{N}_{B}=\omega_{t}^{-n}\cdot\mathbb{F}[t]$ where
$\underline{N}$ is the $\mathbb{F}[t]$-motive over $K$ whose underlying module
is $N=K[t]$ and where $\tau_{N}$ is the multiplication by $(t-\theta)^{n}$.
The $\mathbb{F}[t]$-motive $\underline{N}$ has been chosen so that
$\underline{M}\otimes\underline{N}$ is effective and from [BH07, Prop. 3.4] we
deduce that $(\underline{M}\otimes\underline{N})_{B}\subset
M\otimes_{K[t]}L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}t\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$.
Hence $\omega_{t}^{-n}\cdot\underline{M}_{B}\subset M\otimes_{A\otimes
K}L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}$
as desired.

Now, from [GM22b, Prop. 3.8], we have

$\pazocal{O}_{\mathfrak{X}}(n\cdot
J)(\mathfrak{X}_{L})=\left\\{f\in\operatorname{Quot}L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}~{}|~{}\forall
t\in A\setminus\mathbb{F}:~{}\Pi_{t}^{n}\cdot f\in
L\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}A\mathclose{\hbox{\set<EMAIL_ADDRESS>
As $M$ is finite projective, we conclude that $M_{B}\subset M\otimes_{A\otimes
K}\pazocal{O}_{\mathfrak{X}}(n\cdot J)(\mathfrak{X}_{L})$. The theorem
follows. ∎

######  Corollary 3.22.

Under the assumptions of Theorem 3.21 the multiplication map

$\underline{M}_{B}\otimes_{A}\pazocal{O}_{\mathfrak{X}}(n\cdot
J)(\mathfrak{X}_{L})\longrightarrow M\otimes_{A\otimes
K}\pazocal{O}_{\mathfrak{X}}(n\cdot J)(\mathfrak{X}_{L}),\quad\omega\otimes
c\longmapsto\omega\cdot c$

is bijective.

###### Proof.

This follows from the fully faithful flatness of
$\pazocal{O}_{\mathfrak{X}}(n\cdot J)(\mathfrak{X}_{L})\to L\langle A\rangle$.
∎

###  3.4 The associated mixed Hodge-Pink structure

Let $\underline{M}$ be a rigid analytically trivial $A$-motive over $K$
(Definition 3.8). Let $\underline{M}_{B}$ be the Betti realization of
$\underline{M}$ (Definition 3.7). By Corollary 3.22, there exists a finite
separable extension $L$ of $K_{\infty}$ such that the multiplication map

$\underline{M}_{B}\otimes_{A}\pazocal{O}_{\mathfrak{X}}(n\cdot
J)(\mathfrak{X}_{L})\longrightarrow M\otimes_{A\otimes
K}\pazocal{O}_{\mathfrak{X}}(n\cdot J)(\mathfrak{X}_{L}),\quad\omega\otimes
c\longmapsto\omega\cdot c$ (3.8)

is bijective. Localizing at $\mathfrak{j}$, we obtain an isomorphism of
$K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$-modules:

$\underline{M}_{B}\otimes_{A,\nu}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M\otimes_{A\otimes
K}K_{\infty}^{s}(\\!(\mathfrak{j})\\!),$ (3.9)

where $\nu$ denote the morphism $A\to K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$,
$a\mapsto a\otimes 1$, introduced earlier in Section 2 in the context of
Hodge-Pink structures.

######  Definition 3.23.

We denote by $\gamma_{\underline{M}}$ the isomorphism (3.9).

A trivial yet important remark is the following:

######  Lemma 3.24.

The morphism $\gamma_{\underline{M}}$ is $G_{\infty}$-equivariant, where
$\sigma\in G_{\infty}$ acts on the right-hand side of (3.9) via
$\sigma\otimes\sigma$ and on the left via
$\operatorname{id}_{M}\otimes\sigma$.

Let $R$ be a Noetherian subring of $K_{\infty}$ containing $A$. In the next
definition, attributed to Pink, we attach an Hodge-Pink structure to
$\underline{M}$ (see also [HJ20, Def. 2.3.32]).

######  Definition 3.25.

We let $\mathscr{H}_{R}(\underline{M})$ be the Hodge-Pink structure (with base
field $K_{\infty}^{s}$, coefficients ring $R$)

  * •

whose underlying $R$-module is $\underline{M}_{B}\otimes_{A}R$,

  * •

whose Hodge-Pink lattice is
$\mathfrak{q}_{\underline{M}}=\gamma_{\underline{M}}^{-1}(M\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!])$.

The tautological lattice of $\mathscr{H}_{R}(\underline{M})$ is
$\mathfrak{p}_{\underline{M}}=\underline{M}_{B}\otimes_{A,\nu}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$.
The action of $G_{\infty}$ on $\underline{M}_{B}$ is continuous (Proposition
3.11) and defines an infinite Frobenius $\phi_{\underline{M}}$ for
$\mathscr{H}_{R}(\underline{M})$. We denote by
$\mathscr{H}^{+}_{R}(\underline{M})$ the pair
$(\mathscr{H}_{R}(\underline{M}),\phi_{\underline{M}})$.

######  Proposition 3.26.

The assignement $\underline{M}\mapsto\mathscr{H}_{R}(\underline{M})$ (resp.
$\mathscr{H}^{+}_{R}(\underline{M})$) defines an exact functor
$\mathscr{H}^{+}_{R}:A\mathbf{Mot}_{K}^{\operatorname{rat}}\to\mathbf{HPk}_{K_{\infty}^{s}}$
(resp. $\mathbf{HPk}^{+}_{K_{\infty}}$) of exact categories.

###### Proof.

Let
$S:0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ be an exact sequence in $A\mathbf{Mot}_{K}^{\text{rat}}$. By Corollary
3.17, the sequence $S_{B}$ is exact hence $\mathscr{H}_{R}(S)$ and
$\mathscr{H}_{R}^{+}(S)$ are exact. To show that they are strict, observe that
the sequence

$0\longrightarrow\mathfrak{q}_{\underline{M}^{\prime}}\longrightarrow\mathfrak{q}_{\underline{M}}\longrightarrow\mathfrak{q}_{\underline{M}^{\prime\prime}}\longrightarrow
0$

is isomorphic to $0\to M^{\prime}\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]\to M\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]\to M^{\prime\prime}\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]\to 0$ through $\gamma_{S}$, and the
latter is exact by flatness of $A\otimes K\to
K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$. ∎

We recall the notion of regulated extensions in
$A\mathbf{Mot}_{K}^{\operatorname{rat}}$, as introduced in [Gaz22, Def. 5.7]:

######  Definition 3.27.

Let
$S:0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ be an exact sequence in $A\mathbf{Mot}_{K}^{\text{rat}}$. We say that $S$
is regulated if $\mathscr{H}_{A}(S)$ is Hodge additive (Definition 2.5).

We conclude this section by giving a description of the extension modules of
Hodge-Pink structures arising from $A$-motives. It consists mainly in the
reformulation of Propositions 2.14 and 2.16 in the case of
$\underline{H}=\mathscr{H}^{+}_{R}(\underline{M})$ for a rigid analytically
trivial $A$-motive $\underline{M}$ over $K$.

######  Theorem 3.28.

Let $\underline{M}$ be a rigid analytically trivial mixed $A$-motive over $K$
whose weights are all non-positive. Let
$\underline{H}^{+}=\mathscr{H}_{R}^{+}(\underline{M})$. In the notations of
Proposition 2.14, we have an exact sequence

$0\longrightarrow\frac{M\otimes_{A\otimes
K}K_{\infty}(\\!(\mathfrak{j})\\!)}{(\underline{M}_{B}^{+})_{R}+M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}\longrightarrow\operatorname{Ext}^{1}_{\mathbf{HPk}^{+}_{K_{\infty}}}(\mathbbm{1}^{+},\underline{H}^{+})\stackrel{{\scriptstyle
d_{\underline{H}^{+}}}}{{\longrightarrow}}\operatorname{H}^{1}(G_{\infty},(\underline{M}_{B})_{R})\longrightarrow
0$

where $(\underline{M}_{B})_{R}:=\underline{M}_{B}\otimes_{A}R$. The Hodge
additive version of this exact sequence holds:

$0\longrightarrow\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{(\underline{M}_{B}^{+})_{R}+M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}\longrightarrow\operatorname{Ext}^{1,\operatorname{ha}}_{\mathbf{HPk}^{+}_{K_{\infty}}}(\mathbbm{1}^{+},\underline{H}^{+})\stackrel{{\scriptstyle
d_{\underline{H}^{+}}}}{{\longrightarrow}}\operatorname{H}^{1}(G_{\infty},(\underline{M}_{B})_{R})\longrightarrow
0.$

The theorem follows from the next two lemmas, the first of which precises the
form of $\mathfrak{p}_{\underline{M}}$ seen as a submodule of
$M\otimes_{A\otimes K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$.

######  Lemma 3.29.

We have
$\gamma_{\underline{M}}(\mathfrak{p}_{\underline{M}})=\tau_{M}(\tau^{*}M)\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$.

###### Proof.

Let $L$ be a finite separable extension of $K_{\infty}$ as in Theorem 3.21 and
let $\tau J_{m}$ denote the divisor
$\tau\mathfrak{j}+\tau^{2}\mathfrak{j}+...+\tau^{m}\mathfrak{j}$ on
$\mathfrak{X}_{L}$ and $\tau J:=\tau
J_{\infty}=\tau\mathfrak{j}+\tau^{2}\mathfrak{j}+...$. For $n\geq 0$, we also
denote by $\pazocal{O}_{\mathfrak{X}}(n\cdot\tau J)$ the sheaf on
$\mathfrak{X}_{L}$ given by
$\operatorname{colim}_{m}\pazocal{O}_{\mathfrak{X}}(n\cdot\tau J_{m})$; namely
the sheaf of meromorphic functions on $\mathfrak{X}_{L}$ that are holomomrphic
away from the support of $\tau J$. The morphism $\tau$ on $\mathfrak{X}$
induces a map

$\tau:\pazocal{O}_{\mathfrak{X}}(n\cdot
J)(\mathfrak{X}_{L})\longrightarrow\pazocal{O}_{\mathfrak{X}}(n\cdot\tau
J)(\mathfrak{X}_{L}).$

Taking the pullback of (3.8) by the above morphism produces an isomorphism:

$\underline{M}_{B}\otimes_{A}\pazocal{O}_{\mathfrak{X}}(n\cdot\tau
J)(\mathfrak{X}_{L})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\tau^{*}M)\otimes_{A\otimes
K}\pazocal{O}_{\mathfrak{X}}(n\cdot\tau J)(\mathfrak{X}_{L})$

which, localized at $\mathfrak{j}$, produces:

$\delta_{\underline{M}}:\underline{M}_{B}\otimes_{A}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\tau^{*}M)\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!].$

The latter inserts in a commutative diagram

${\underline{M}_{B}\otimes_{A}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)}$${(\tau^{*}M)\otimes_{A\otimes
K}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)}$${M\otimes_{A\otimes
K}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)}$$\scriptstyle{\delta_{\underline{M}}}$$\scriptstyle{\gamma_{\underline{M}}}$$\scriptstyle{\tau_{M}\otimes\operatorname{id}_{K_{\infty}^{s}(\\!(\mathfrak{j})\\!)}}$

Note that this already appears in [HJ20, Prop.2.3.30] under different
notations. The equality
$\gamma_{\underline{M}}(\mathfrak{p}_{\underline{M}})=\tau_{M}(\tau^{*}M)\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]$ follows from the commutativity of the
above diagram together with the fact that both $\gamma_{\underline{M}}$ and
$\delta_{\underline{M}}$ are isomorphisms. ∎

To apply Proposition 2.16 and conclude the proof of Theorem 3.28, we need a
vanishing result of Galois cohomology, supplied by the next lemma.

######  Lemma 3.30.

Let $\mathfrak{l}$ be a $K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$-lattice in
$\underline{M}_{B}\otimes_{A}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$. Then,
$\mathfrak{l}$ is $G_{\infty}$-equivariant and
$\operatorname{H}^{1}(G_{\infty},\mathfrak{l})=(0)$.

###### Proof.

The $K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$-lattice $\mathfrak{l}$ is
isomorphic to a $K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$-lattice in
$M\otimes_{A\otimes K}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$ via
$\gamma_{\underline{M}}$. By the elementary divisor Theorem in the discrete
valuation ring $K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$, there exists a
$G_{\infty}$-equivariant $K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$-linear
automorphism $\psi$ of the $K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$-vector space
$M\otimes_{A\otimes K}K_{\infty}^{s}(\\!(\mathfrak{j})\\!)$ such that

$\gamma_{\underline{M}}(\mathfrak{l})=\psi(M\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]).$

This implies that $\mathfrak{l}$ is $G_{\infty}$-equivariant and further that
$\mathfrak{l}$ is isomorphic to $M\otimes_{A\otimes
K}K_{\infty}^{s}[\\![\mathfrak{j}]\\!]$ as a
$K_{\infty}^{s}[\\![\mathfrak{j}]\\!][G_{\infty}]$-module. By the additive
Hilbert’s $90$ Theorem we have
$\operatorname{H}^{1}(G_{\infty},K_{\infty}^{s}[\\![\mathfrak{j}]\\!])=0$ and
it follows that $\operatorname{H}^{1}(G_{\infty},\mathfrak{l})=0$. ∎

##  4 Regulators and finiteness theorems

###  4.1 Regulators of $A$-motives

Let $\underline{M}=(M,\tau_{M})$ be an $A$-motive over $K$. Recall from
[Gaz22, §4] that there exists a unique $A\otimes A$-submodule $M_{A}\subset M$
with the following properties:

  1. 1.

$M_{A}$ is finitely generated and generates $M$ over $A\otimes K$,

  2. 2.

$M_{A}$ is stable through $\tau_{M}$; namely, $\tau_{M}(\tau^{*}M_{A})\subset
M_{A}[\mathfrak{j}^{-1}]$,

  3. 3.

$M_{A}$ is maximal; i.e. not strictly included in another submodule satisfying
1 and 2.

The module $M_{A}$ is called the maximal integral model of $\underline{M}$
over $A$. One can show that $M_{A}$ is a projective $A\otimes A$-module
[Gaz22, Prop. 4.43] although this will not be needed below. We also give a
name to the following submodule of $M[\mathfrak{j}^{-1}]$:

$N_{A}:=(M+\tau_{M}(\tau^{*}M))\cap M_{A}[\mathfrak{j}^{-1}].$ (4.1)

In _loc. cit._ , we introduced the sub-$A$-module of integral and regulated
extensions of $\mathbbm{1}$ by $\underline{M}$ in $A\mathbf{Mot}_{K}$, denoted
$\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})$, and proved
that the map

$\iota:\frac{N_{A}}{(\operatorname{id}-\tau_{M})(M)}\longrightarrow\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M}),$
(4.2)

assigning to the class of $m\in N_{A}\subset M[\mathfrak{j}^{-1}]$ the
extension whose middle object has underlying module $M\oplus(A\otimes K)$ and
$\tau$-morphism $\left(\begin{smallmatrix}\tau_{M}&m\\\
0&1\end{smallmatrix}\right)$ (with obvious arrows), is a natural isomorphism
of $A$-modules [Gaz22, Thm. D+Cor. 5.10]. The author also formulated a
conjecture [Gaz22, Con. 5.13] relating the above to the submodule of regulated
extensions having everywhere good reduction.

Some computations suggested that
$\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})$ is
generally _not_ finitely generated (see below), contrary to what is expected
in the number fields setting. Assuming $\underline{M}$ rigid analytically
trivial, this defect is measured by the morphism $r_{B}$ introduced in (3.5):
that there is a map:

$r_{B}(\underline{M}):\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})\longrightarrow\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})$
(4.3)

assigning to an extension of rigid analytically trivial $A$-motives the class
of the continuous cocycle associated to the induced extension of $A$-linear
representations of $G_{\infty}$. We introduce the following notations:

$\operatorname{Ext}^{1,\text{reg}}_{A,\infty}(\mathbbm{1},\underline{M}):=\ker
r_{B}(\underline{M}),\quad\operatorname{Cl}(\underline{M}):=\operatorname{coker}r_{B}(\underline{M}).$

We are ready to state our first main result.

######  Theorem 4.1.

Suppose that $\underline{M}$ is rigid analytically trivial. Both
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
and $\operatorname{Cl}(\underline{M})$ are finitely generated $A$-modules. If
the weights of $\underline{M}$ are all negative, then
$\operatorname{Cl}(\underline{M})$ is finite.

The proof of Theorem 4.1 (and Theorem 4.4 below) is postponed to the last
section. Let us comment the above statement.

  1. $-$

Classically, the $\mathbb{Q}$-vector space
$\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbbm{1},M)$, consisting in extensions
having everywhere good reduction of the unit motive by a mixed motive $M$ over
$\mathbb{Q}$, is expected to be finite dimensional (e.g. [Sch91, §III]). First
observe that, in our analogy, Theorem 4.1 is the function field counterpart of
this expectation: classically, the analogue of $r_{B}(\underline{M})$, given
in (4.3), would rather have targeted the finite $2$-group
$\operatorname{H}^{1}(\operatorname{Gal}(\mathbb{C}|\mathbb{R}),M_{B})$,
$M_{B}$ denoting the Betti realization of $M$. To that respect, the finite
generation of the kernel of $r_{B}(M)$ is the counterpart of the statement
that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbbm{1},M)$ has finite
dimension.

  2. $-$

A second observation, already announced and corroborating the analogy made in
the above paragraph, is that the $A$-module
$\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})$ is
typically not finitely generated. To wit, Theorem 4.1 roughly tells that a set
of generators has - up to a finite set - the same cardinality as one for
$\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})$. Yet, $G_{\infty}$ is not
topologically finitely generated: by class field theory, its wild inertia
group is topologically isomorphic to the group of $1$-unit in
$\pazocal{O}_{\infty}$, itself isomorphic to a countable product of
$\mathbb{Z}_{p}$.

  3. $-$

The modules
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
and $\operatorname{Cl}(\underline{M})$ seem to contain function field
arithmetic informations, as showed in the case of Carlitz twists in [GM22a].

Let $\underline{M}$ be rigid analytically trivial. We move to the definition
of the regulator of $\underline{M}$. Let $\underline{H}^{+}$ denote the mixed
Hodge-Pink structure $\mathscr{H}^{+}_{K_{\infty}}(\underline{M})$ attached to
$\underline{M}$ with coefficient ring $R=K_{\infty}$. In view of Proposition
3.26, the Hodge-Pink realization functor $\mathscr{H}^{+}$ is exact and hence
induces an $A$-linear morphism of the corresponding extension groups:

$r_{\mathscr{H}^{+}}(\underline{M}):\operatorname{Ext}^{1}_{A\mathbf{Mot}_{K}^{\text{rat}}}(\mathbbm{1},\underline{M})\longrightarrow\operatorname{Ext}^{1}_{\mathbf{HPk}^{+}_{K_{\infty}}}(\mathbbm{1}^{+},\underline{H}^{+}).$
(4.4)

From Definition 3.27, $r_{\mathscr{H}^{+}}(\underline{M})$ maps the class of
regulated exact sequences in $A\mathbf{Mot}_{K}^{\text{rat}}$ to that of Hodge
additive strict exact sequences in $\mathbf{HPk}^{+}_{K_{\infty}}$. Therefore,
there is square:

${\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})}$${\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})}$${\operatorname{Ext}^{1,\text{ha}}_{\mathbf{HPk}^{+}_{K_{\infty}}}(\mathbbm{1}^{+},\underline{H}^{+})}$${\operatorname{H}^{1}(G_{\infty},(\underline{M}_{B})_{K_{\infty}})}$$\scriptstyle{r_{B}(\underline{M})}$$\scriptstyle{r_{\mathscr{H}^{+}}(\underline{M})}$$\scriptstyle{\subset}$$\scriptstyle{d_{\underline{H}^{+}}}$
(4.5)

The left vertical arrow $d_{\underline{H}^{+}}$ was introduced in Definition
2.13. The next lemma stems from the definitions:

######  Lemma 4.2.

The square (4.5) commutes in the category of $A$-modules.

As such, $r_{\mathscr{H}^{+}}(\underline{M})$ induces a morphism from the
corresponding kernels. Accordingly, we fix the following namings:

######  Definition 4.3.

We call the regulator of $\underline{M}$, and denote it by
$\textrm{{Reg}}(\underline{M})$, the restriction of
$r_{\mathscr{H}^{+}}(\underline{M})$:

$\textrm{{Reg}}(\underline{M}):\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})\longrightarrow\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\underline{H}^{+})$

which is well-defined according to Lemma 4.2.

Our second main result is the following:

######  Theorem 4.4.

Let $\underline{M}$ be a rigid analytically trivial mixed $A$-motive over $K$
whose weights are all negative. Then, the rank of
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
as an $A$-module equals the dimension of the $K_{\infty}$-vector space
$\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\underline{H}^{+})$.

Regarding the above, it is natural to ask whether the image of
$\textrm{{Reg}}(\underline{M})$ forms a lattice of full rank in the
$K_{\infty}$-vector space
$\operatorname{Ext}^{1,\text{ha}}_{\infty}(\mathbbm{1}^{+},\underline{H}^{+})$.
This would be an analogue of Beilinson’s conjecture [Bei86, 8.4.1].
Surprisingly, this fails to hold in many situations: this fails in the simple
case where $\underline{M}$ is the Carlitz $n$th twists, $n\geq 1$ being a
multiple of the characteristic $p$ (and will be studied in a subsequent work).
Fortunately enough, such a situation does not occur for number fields.

###  4.2 The complex $G$ and the fundamental exact sequence

Let $\underline{M}$ be a rigid analytically trivial $A$-motive over $K$. In
this subsection we introduce a chain complex $G_{\underline{M}}$ which wil
play a fundamental role in the proof of Theorems 4.1 and 4.4. It deserves its
own definition:

######  Definition 4.5.

Let $G_{\underline{M}}$ denote the complex of $A$-modules sitting in degrees
$0$ and $1$:

$G_{\underline{M}}=\left[\frac{M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle}{M_{A}}\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}\frac{M\otimes_{A\otimes
K}K_{\infty}\langle A\rangle}{N_{A}}\right]$

where the arrow is induced by $m\mapsto m-\tau_{M}(\tau^{*}m)$ on
$M\otimes_{A\otimes K}K_{\infty}\langle A\rangle$.

We have a commutative diagram of $A$-modules with exact rows:

${0}$${M_{A}}$${M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle}$${G_{\underline{M}}^{0}}$${0}$${0}$${N_{A}}$${M\otimes_{A\otimes
K}K_{\infty}\langle
A\rangle}$${G_{\underline{M}}^{1}}$${0}$$\scriptstyle{\operatorname{id}-\tau_{M}}$$\scriptstyle{\operatorname{id}-\tau_{M}}$$\scriptstyle{d^{1}}$
(4.6)

The next proposition introduces what we believe to be fundamental for the
sequel:

######  Proposition 4.6.

The Snake Lemma applied to the diagram (4.6) results in a long-exact sequence
of $A$-modules:

$0\to\operatorname{Hom}_{A\mathbf{Mot}_{K}}(\mathbbm{1},\underline{M})\to\underline{M}_{B}^{+}\to\operatorname{H}^{0}(G_{\underline{M}})\\\
\to\operatorname{Ext}^{1,\operatorname{reg}}_{A}(\mathbbm{1},\underline{M})\stackrel{{\scriptstyle
r_{B}(\underline{M})}}{{\longrightarrow}}\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})\to\operatorname{H}^{1}(G_{\underline{M}})\to
0.$

###### Proof.

The complex
$[M_{A}\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}N_{A}]$
has $\operatorname{Hom}_{A\mathbf{Mot}_{K}}(\mathbbm{1},\underline{M})$ and
$\operatorname{Ext}^{1,\operatorname{reg}}_{A}(\mathbbm{1},\underline{M})$ for
$0$th and $1$th-cohomology modules (see [Gaz22, Prop. 4.23] and (4.2)). The
kernel and the cokernel of the middle vertical arrow is computed by Theorem
3.13. Hence, everything is clear but, perhaps, that the map
$r_{B}(\underline{M})$ is the true one that appears at the level of the
cokernels. To prove the latter, first observe that Theorem 3.13 gives an
isomorphism of $A$-modules:

$\frac{M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle}{(\operatorname{id}-\tau_{M})(M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle)}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{H}^{1}(G_{\infty},\underline{M}_{B}).$
(4.7)

The above map assigns to the class of $f\in M\otimes_{A\otimes
K}K_{\infty}\langle A\rangle$ the class of the cocycle
$c_{f}:\sigma\mapsto\,^{\sigma}\xi_{f}-\xi_{f}$, where $\xi_{f}\in
M\otimes_{A\otimes K}K_{\infty}^{s}\langle A\rangle$ is _any_ solution $\xi$
of the equation $\xi-\tau_{M}(\tau^{*}\xi)=f$. The class of $c_{f}$ then does
not depend on the choice of $\xi_{f}$, whose existence is provided by Theorem
3.13.

We turn to an explicit description of the map $r_{B}(\underline{M})$. Choose
$m\in N_{A}$ and let
$[\underline{E}]\in\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})$
be the extension $\iota(m)$. The set underlying the Betti realization of
$\underline{E}$ consists of pairs $(\xi,a)$, $\xi\in M\otimes_{A\otimes
K}\mathbb{C}_{\infty}\langle A\rangle$ and $a\in K_{\infty}\langle A\rangle$,
solution of the system

$\begin{pmatrix}\tau_{M}&m\\\ 0&1\end{pmatrix}\begin{pmatrix}\tau^{*}\xi\\\
\tau^{*}a\end{pmatrix}=\begin{pmatrix}\xi\\\ a\end{pmatrix}.$

It follows that $a\in A$ and $\xi-\tau_{M}(\tau^{*}\xi)=am$. A splitting of
$0\to\underline{M}_{B}\to\underline{E}_{B}\to A\to 0$ in the category of
$A$-modules corresponds to the choice of a solution $\xi_{m}\in
M\otimes_{A\otimes K}\mathbb{C}_{\infty}\langle A\rangle$ of the equation
$\xi-\tau_{M}(\tau^{*}\xi)=m$. To the choice of $\xi_{m}$ corresponds the
decomposition

$\underline{M}_{B}\oplus\mathbbm{1}_{B}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\underline{E}_{B},\quad(\omega,a)\longmapsto(\omega+a\xi_{m},a).$

An element $\sigma\in G_{\infty}$ acts on the right-hand side by

$(\omega+a\xi_{m},a)\mapsto(\,^{\sigma}\omega+a\,^{\sigma}\xi_{m},a)=(\,^{\sigma}\omega+a(\,^{\sigma}\xi_{m}-\xi_{m})+a\xi_{m},a)$

where $\,{}^{\sigma}\xi_{m}-\xi_{m}\in\underline{M}_{B}$. Hence, $\sigma$ acts
as the matrix
$\left(\begin{smallmatrix}\sigma&\,\,{}^{\sigma}\xi_{m}-\xi_{m}\\\
0&1\end{smallmatrix}\right)$. We deduce from this computation that the square

${\displaystyle\frac{N_{A}}{(\operatorname{id}-\tau_{M})(M_{A})}}$${\displaystyle\frac{M\otimes_{A\otimes
K}K_{\infty}\langle A\rangle}{(\operatorname{id}-\tau_{M})(M\otimes_{A\otimes
K}K_{\infty}\langle
A\rangle)}}$${\operatorname{Ext}^{1,\text{reg}}_{A}(\mathbbm{1},\underline{M})}$${\operatorname{H}^{1}(G_{\infty},\underline{M}_{B})}$$\scriptstyle{\iota}$$\scriptstyle{\wr}$$\scriptstyle{\eqref{eq:H1-explicit}}$$\scriptstyle{\wr}$$\scriptstyle{r_{B}(\underline{M})}$

where the top arrow stems from the inclusion $N\subset M\otimes_{A\otimes
K}K_{\infty}\langle A\rangle$, commutes in the category of $A$-modules. This
ends the proof of the Proposition. ∎

###  4.3 Strategy of proofs

In the following section, we develop the theory of shtuka models for
$\underline{M}$. Roughly speaking, they are the data of
$(\pazocal{M},\pazocal{N},\tau{M})$ where
$\pazocal{M}\hookrightarrow\pazocal{N}$ are two coherent sheaves on
$\operatorname{Spec}A\times C$, and where
$\tau_{\pazocal{M}}:\tau^{*}\pazocal{M}\to\pazocal{N}$ is a morphism which
restricts to $\tau_{M}:\tau^{*}M_{A}\to N_{A}$ on the affine part
$\operatorname{Spec}A\times\operatorname{Spec}A$ together with some vanishing
condition along $\infty$ (Definition 5.1). Shtuka models for $\underline{M}$
always exist (Proposition 5.2) though they are far from being unique. If the
weights of $\underline{M}$ are negative, our construction extends to sheaves
on $C\times C$ (Definition 5.4, Theorem 5.6).

One of the main step towards the proof of the theorems is that there is a
quasi-isomorphism

$G_{\underline{M}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{R}\\!\Gamma\\!\left(\operatorname{Spec}A\times
C,\pazocal{M}\stackrel{{\scriptstyle\iota-\tau{M}}}{{\longrightarrow}}\pazocal{N}\right)\quad(\text{Proposition~{}}\ref{prop:ge-
in-terms-of-cohomology}).$ (4.8)

That $C$ is proper over $\mathbb{F}$ leads to the perfectness of
$G_{\underline{M}}$. The fundamental exact sequence of Proposition 4.6 then
implies that both
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})$
and $\operatorname{Cl}(\underline{M})$ are finitely generated $A$-modules,
thus completing the proof of the first part of Theorem 4.1.

Theorem 4.4 and the second part of Theorem 4.1 follow from subtler
observations on $C\times C$-shtuka models along
$\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}\operatorname{Spf}\pazocal{O}_{\infty}$.
Denoting by $\hat{\pazocal{M}}$ and $\hat{\pazocal{N}}$ the corresponding
formal sheaves and by $\hat{\pazocal{M}}_{\infty}$ and
$\hat{\pazocal{N}}_{\infty}$ the
$\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}$-modules of their global
sections, the coup de théâtre of Subsection 5.3 is an exact sequence

$0\longrightarrow\underline{M}_{B}^{+}\otimes_{A}K_{\infty}\longrightarrow\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\longrightarrow\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}_{K_{\infty}}(\underline{M}))\longrightarrow
0$

(see Corollary 5.19). The material used to construct this sequence resembles
much to the techniques employed by V. Lafforgue in [Laf09, §4]. Although we do
use the theory of function fields Fontaine rings as developed in [GL11] or
[Har09], the results involved might be reminiscent of such a theory at the
place $\infty$.

Now, using classical derived algebro-geometric methods, we show that there is
a quasi-isomorphism of complexes of $K_{\infty}$-vector spaces:

$\operatorname{R}\\!\Gamma\\!\left(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{M}}\stackrel{{\scriptstyle\iota-\tau{M}}}{{\longrightarrow}}\hat{\pazocal{N}}\right)\cong\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}[0].$
(4.9)

The last ingredient in our proof is Grothendieck’s comparison Theorem between
Zariski and formal coherent cohomology, which amounts to a quasi-isomorphism
among the left-hand side of (4.9) tensored along $\pazocal{O}_{\infty}\to
K_{\infty}$ and the right-hand side of (4.8) tensored along $A\to K_{\infty}$.

This reads both that $G_{\underline{M}}\otimes_{A}K_{\infty}$ sits in degree
$0$ \- hence that $\operatorname{Cl}(\underline{M})$ is torsion - and that the
dimension of
$\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}_{K_{\infty}}(\underline{M}))$
is the same as that of
$\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})\otimes_{A}K_{\infty}$;
hence achieving the proof of the main Theorems.

##  5 Shtuka models à la Mornev

Let $\underline{M}$ be an $A$-motive over $K$ (we may not need rigid
analytically triviality). In this section, we associate non-canonically to
$\underline{M}$ a shtuka model on $(\operatorname{Spec}A)\times C$ and,
whenever $\underline{M}$ only has non-positive weights, on $C\times C$. This
powerful technique was, to the knowledge of the author, first introduce in
[Mor18, §12] in the context of Drinfeld modules with everywhere good
reduction. An incarnation of this construction seems already to appear in
[Fan15, Def. 1.13] in the setting of Anderson $t$-modules. Although our
motivations owe much to Mornev’s work, our definition of $C\times C$-shutka
models differs. The one presented below in Definition 5.4 has the nice feature
to carry an existence result by simply assuming that the weights of
$\underline{M}$ are non-positive (Theorem 5.6).

Let us introduce some notations. We still denote by $\tau:C\times C\to C\times
C$ the morphism of $\mathbb{F}$-schemes which acts as the identity on the
left-hand factor $C$ and as the absolute $q$-Frobenius on the right-hand one.
Because $C$ is separated over $\mathbb{F}$, the diagonal morphism $C\to
C\times C$ is a closed immersion and its image defines a closed subscheme
$\Delta$ of $C\times C$ of codimension $1$. It defines an effective divisor
$\Delta$ on $C\times C$. The evaluation of $\pazocal{O}(\Delta)$ at the affine
open subscheme $\operatorname{Spec}(A\otimes A)$ of $C\times C$ recovers the
ideal $\mathfrak{j}$ of $A\otimes A$.

We also borrow notations from [Gaz22]. For $R$ a Noetherian
$\mathbb{F}$-algebra, $\pazocal{A}_{\infty}(R)$ stand for the ring

$\pazocal{A}_{\infty}(R)=\varprojlim_{n}(\pazocal{O}_{\infty}\otimes
R)/(\mathfrak{m}_{\infty}^{n}\otimes R).$

This ring was considered to define isocrystals and mixedness in _loc. cit_.
Let also $\pazocal{B}_{\infty}(R)$ be the ring
$K_{\infty}\otimes_{\pazocal{O}_{\infty}}\pazocal{A}_{\infty}(R)$.
Geometrically, the formal spectrum $\operatorname{Spf}\pazocal{A}_{\infty}(R)$
corresponds to the completion of the Noetherian scheme
$C\times\operatorname{Spec}R$ along the closed subscheme $\\{\infty\\}\times
R$, that is:

$\operatorname{Spf}\pazocal{A}_{\infty}(R)=\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}\operatorname{Spec}R.$

Dually, when the completion is done at the level of the base ring instead of
at the level of coefficients, we obtain the algebra introduced in the context
of Betti realizations, namely $\pazocal{O}_{\infty}\langle A\rangle$, defined
in Section 3 as

$\pazocal{O}_{\infty}\langle
A\rangle=\varprojlim_{n}(A\otimes\pazocal{O}_{\infty})/(A\otimes\mathfrak{m}_{\infty}^{n}).$

Similarly, $\operatorname{Spf}\pazocal{O}_{\infty}\langle A\rangle$ is the
completion of $\operatorname{Spec}(A\otimes\pazocal{O}_{\infty})$ along
$\operatorname{Spec}A\times\\{\infty\\}$.

The closed subscheme $C\times\\{\infty\\}$ defines an effective divisor on
$C\times C$ which we denote $\infty_{C}$. Similarly, we let $\infty_{A}$ be
the effective divisor $(\operatorname{Spec}A)\times\\{\infty\\}$ of
$(\operatorname{Spec}A)\times C$.

###  5.1 $C$-shtuka models

######  Definition 5.1.

A $C$-shtuka model $\underline{\pazocal{M}}$ for $\underline{M}$ is the datum
$(\pazocal{N},\pazocal{M},\tau_{\pazocal{M}})$ of

  1. $(a)$

A coherent sheaf $\pazocal{N}$ on $(\operatorname{Spec}A)\times C$ such that
$\pazocal{N}(\operatorname{Spec}A\otimes A)=N_{A}$,

  2. $(b)$

A coherent subsheaf $\pazocal{M}$ of $\pazocal{N}$ such that
$\pazocal{M}(\operatorname{Spec}A\otimes A)=M_{A}$ and for which the cokernel
of the inclusion $\iota:\pazocal{M}\to\pazocal{N}$ is supported at $\Delta$,

  3. $(c)$

A morphism $\tau_{\pazocal{M}}:\tau^{*}\pazocal{M}\to\pazocal{N}(-\infty_{A})$
which coincides with $\tau_{M}:\tau^{*}M_{A}\to N_{A}$ on the affine open
subscheme $\operatorname{Spec}A\otimes A$.

######  Proposition 5.2.

A $C$-shtuka model for $\underline{M}$ exists.

###### Proof.

Let $B$ be a sub-$\mathbb{F}$-algebra of $K$ such that
$(\operatorname{Spec}A)\cup(\operatorname{Spec}B)$ forms an affine open
covering of $C$ in the Zariski topology. Let $D$ be the
sub-$\mathbb{F}$-algebra of $K$ containing both $A$ and $B$ and such that
$\operatorname{Spec}D=(\operatorname{Spec}A)\cap(\operatorname{Spec}B)$333Let
$x$ be a closed point on $C$ distinct from $\infty$. Then
$B:=H^{0}(C\setminus\\{x\\},\pazocal{O}_{C})$ works. In the latter case, we
have $D:=H^{0}(C\setminus\\{\infty,x\\},\pazocal{O}_{C})$.. For
$S\in\\{A,B,D\\}$, we let $\mathfrak{j}_{S}$ be the ideal of $A\otimes S$
given by either $\mathfrak{j}_{A}:=\mathfrak{j}$,
$\mathfrak{j}_{D}:=\mathfrak{j}(A\otimes D)$ and
$\mathfrak{j}_{B}:=\mathfrak{j}_{D}\cap(A\otimes B)$. Note that
$\pazocal{O}(\Delta)(\operatorname{Spec}A\otimes S)=\mathfrak{j}_{S}$.

Let $M_{D}$ be the $A\otimes D$-module $M_{A}\otimes_{A}D$, and let
$M_{B}^{\prime}$ be an $A\otimes B$-lattice in $M_{D}$ (for instance, if
$m_{1},...,m_{s}$ are generators of $M_{D}$, consider $M_{B}^{\prime}$ to be
the $A\otimes B$-submodule spanned by $m_{1},...,m_{s}$).

Since $\tau_{M}(\tau^{*}M_{A})\subset M_{A}[\mathfrak{j}^{-1}]$, we have
$\tau_{M}(\tau^{*}M_{D})\subset M_{D}[\mathfrak{j}_{D}^{-1}]$. However, it
might not be true that $\tau_{M}(\tau^{*}M_{B}^{\prime})\subset
M_{B}^{\prime}[\mathfrak{j}_{B}^{-1}]$. Yet, there exists $d\in B$ invertible
in $D$ such that

$\tau_{M}(\tau^{*}M_{B}^{\prime})\subset
d^{-1}M_{B}^{\prime}[\mathfrak{j}_{B}^{-1}].$

Let $r\in B$ invertible in $D$ which vanishes444Such an $r$ always exists: the
divisor $D:=\deg(x)\cdot\infty-\deg(\infty)\cdot x$ has degree zero so that
$nD$ is principal for $n$ large enough ($\operatorname{C}^{0}(K)$ is finite
[Ros02, Lem. 5.6]). Chosing $r$ such that $(r)=nD$, then $r\in B$ and $r$ is
invertible in $D$. at $\infty$ and let $M_{B}:=(rd)M_{B}^{\prime}$. We now
have

$\tau_{M}(\tau^{*}M_{B})\subset rM_{B}[\mathfrak{j}_{B}^{-1}].$

Since $r$ is invertible in $D$, the multiplication maps furnish glueing
isomorphisms

$M_{A}\otimes_{A}D\stackrel{{\scriptstyle=}}{{\longrightarrow}}M_{D}\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}M_{B}\otimes_{B}D.$
(5.1)

For $S\in\\{A,B,D\\}$, we set $N_{S}:=(M+\tau_{M}(\tau^{*}M))\cap
M_{S}[\mathfrak{j}_{S}^{-1}]$. $N_{S}$ is an $A\otimes S$-module of finite
type which contains $M_{S}$. By flatness of $D$ over $A$ (resp. $B$), the
multiplication maps also are isomorphisms:

$N_{A}\otimes_{A}D\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}N_{D}\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}N_{B}\otimes_{B}D.$
(5.2)

Let $\pazocal{M}$ (resp. $\pazocal{N}$) be the coherent sheaf on
$\operatorname{Spec}A\times C$ resulting from the glueing (5.1) (resp. (5.2))
by Zariski descent. Since $M_{A}\subset N_{A}$ and $M_{B}\subset N_{B}$,
$\pazocal{M}$ is a subsheaf of $\pazocal{N}$. We further have
$M_{A}[\mathfrak{j}^{-1}]=N_{A}[\mathfrak{j}^{-1}]$ and
$M_{B}[\mathfrak{j}^{-1}_{B}]=N_{B}[\mathfrak{j}^{-1}_{B}]$ which implies that
the cokernel of $\pazocal{M}\subset\pazocal{N}$ is supported at $\Delta$.

Because $\tau_{M}(\tau^{*}M_{S})\subset N_{S}$ for all $S\in\\{A,B,D\\}$, one
obtains by glueing a morphism of $\pazocal{O}_{(\operatorname{Spec}A)\times
C}$-modules $\tau_{\pazocal{M}}:\tau^{*}\pazocal{M}\to\pazocal{N}$. Since
$\tau_{M}(\tau^{*}M_{B})\subset rN_{B}$ and $r$ vanishes at $\infty$, we also
have $\tau_{\pazocal{M}}(\tau^{*}\pazocal{M})\subset\pazocal{N}(-\infty_{A})$.
∎

Let $\underline{\pazocal{M}}=(\pazocal{N},\pazocal{M},\tau_{\pazocal{M}})$ be
a $C$-shtuka model for $\underline{M}$. The assumption that the image of
$\tau_{\pazocal{M}}$ lands in $\pazocal{N}(-\infty_{A})$ is crucial for the
incoming cohomological considerations. In much of what follows, this is
materialized by the next lemma.

######  Lemma 5.3.

Let $i:\operatorname{Spec}\pazocal{O}_{\infty}\langle
A\rangle\to\operatorname{Spec}A\otimes\pazocal{O}_{\infty}\hookrightarrow(\operatorname{Spec}A)\times
C$ be the canonical composition of $A$-schemes. The inclusion of sheaves
$i^{*}\pazocal{M}\subset i^{*}\pazocal{N}$ is an equality and the induced
morphism

$\iota-\tau_{\pazocal{M}}:i^{*}\pazocal{M}(\operatorname{Spec}\pazocal{O}_{\infty}\langle
A\rangle)\longrightarrow
i^{*}\pazocal{N}(\operatorname{Spec}\pazocal{O}_{\infty}\langle A\rangle)$

is an isomorphism of $\pazocal{O}_{\infty}\langle A\rangle$-modules.

###### Proof.

By Lemma 3.6, we have $\mathfrak{j}\pazocal{O}_{\infty}\langle
A\rangle=\pazocal{O}_{\infty}\langle A\rangle$. In particular, $i^{*}\Delta$
is the empty divisor of $\operatorname{Spec}\pazocal{O}_{\infty}\langle
A\rangle$. The equality between $i^{*}\pazocal{M}$ and $i^{*}\pazocal{N}$
follows.

Let $\pi_{\infty}$ be a uniformizer of $\pazocal{O}_{\infty}$. We denote by
$\Xi$ the $\pazocal{O}_{\infty}\langle A\rangle$-module
$i^{*}\pazocal{M}(\operatorname{Spec}\pazocal{O}_{\infty}\langle A\rangle)$.
Because
$\tau_{\pazocal{M}}(\tau^{*}\pazocal{M})\subset\pazocal{N}(-\infty_{A})$, we
have $\tau_{\pazocal{M}}(\tau^{*}\Xi)\subset\pi_{\infty}\Xi$. In particular,
for all $\xi\in\Xi$, the series

$\psi:=\sum_{n=0}^{\infty}{\tau_{M}^{n}(\tau^{n*}\xi)}$

converges in $\Xi$. The assignment $\xi\mapsto\psi$ defines an inverse of
$\operatorname{id}-\tau_{\pazocal{M}}$ on $\Xi$. ∎

###  5.2 $C\times C$-shtuka models

We want to extend the construction of Proposition 5.2 from
$(\operatorname{Spec}A)\times C$ to $C\times C$.

######  Definition 5.4.

A $C\times C$-shtuka model $\underline{\pazocal{M}}$ for $\underline{M}$ is
the datum $(\pazocal{N},\pazocal{M},\tau_{\pazocal{M}})$ of

  1. $(a)$

a coherent sheaf $\pazocal{N}$ on $C\times C$ such that
$\pazocal{N}(\operatorname{Spec}A\otimes A)=N_{A}$,

  2. $(b)$

a coherent subsheaf $\pazocal{M}$ of $\pazocal{N}$ such that
$\pazocal{M}(\operatorname{Spec}A\otimes A)=M_{A}$ and such that the cokernel
of the inclusion $\iota:\pazocal{M}\to\pazocal{N}$ is supported at $\Delta$,

  3. $(c)$

a morphism of sheaves
$\tau_{\pazocal{M}}:\tau^{*}\pazocal{M}\to\pazocal{N}(-\infty_{C})$ which
coincides with $\tau_{M}:\tau^{*}M_{A}\to N_{A}$ on
$\operatorname{Spec}A\otimes A$.

######  Remark 5.5.

Clearly, the restriction of a $C\times C$-shtuka model for $\underline{M}$ on
$(\operatorname{Spec}A)\times C$ is a $C$-shtuka model for $\underline{M}$.

The main result of this subsection is the following:

######  Theorem 5.6.

If the weights of $\underline{M}$ are non-positive, a $C\times C$-shtuka model
for $\underline{M}$ exists.

Before initiating the proof of Theorem 5.6, which will take us the remaining
of this subsection, we shall supply some ingredients on function fields
isocrystals with negative weights.

######  Lemma 5.7.

Let $\underline{M}$ be an $A$-motive over $K$ whose weights are all non-
positive. Then $M\otimes_{A\otimes K}\pazocal{B}_{\infty}(K)$ contains an
$\pazocal{A}_{\infty}(K)$-lattice stable by $\tau_{M}$.

###### Proof.

The proof for the $A=\mathbb{F}[t]$-case is essentially [Har11, Lem. 1.5.9].
The general $A$-case follows the same lines, using Dieudonné Theory as in
[Gaz22, §3.1]. ∎

If the weights of $\underline{M}$ are further negative, the same strategy
shows the following:

######  Lemma 5.8.

Let $\underline{M}$ be an $A$-motive over $K$ whose weights are all negative.
There exist an $\pazocal{A}_{\infty}(K)$-lattice $T$ in $M\otimes_{A\otimes
K}\pazocal{B}_{\infty}(K)$ and two positive integers $d$ and $h$ such that
$\tau_{M}^{h}(\tau^{h*}T)\subset\mathfrak{m}_{\infty}^{d}T$.

###### Proof of Theorem 5.6.

We use the notations and definitions of the proof of Proposition 5.2. That is,
$B$ is a sub-$\mathbb{F}$-algebra of $K$ such that
$(\operatorname{Spec}A)\cup(\operatorname{Spec}B)$ forms an open affine cover
of $C$, $D$ is the sub-$\mathbb{F}$-algebra of $K$ containing $A$ and $B$ such
that $\operatorname{Spec}D=(\operatorname{Spec}A)\cap(\operatorname{Spec}B)$.

Let $\underline{\pazocal{M}}_{0}=(\pazocal{N}_{0},\pazocal{M}_{0},\tau_{0})$
be a $C$-shtuka model for $\underline{M}$. We recover the notations of the
proof of Proposition 5.2 by setting:

$M_{A}:=\pazocal{M}_{0}(\operatorname{Spec}A\otimes
A),~{}M_{B}:=\pazocal{M}_{0}(\operatorname{Spec}A\otimes
B),~{}M_{D}:=\pazocal{M}_{0}(\operatorname{Spec}A\otimes D),$

and similarly for $N$ and $\pazocal{N}_{0}$. Because the weights of
$\underline{M}$ are non-positive, there exists by Lemma 5.8 an
$\pazocal{A}_{\infty}(K)$-lattice $T$ in $M\otimes_{A\otimes
K}\pazocal{B}_{\infty}(K)$ stable by $\tau_{M}$. We introduce six submodules
of $T$, namely:

  1. $(i)$

two sub-$\pazocal{A}_{\infty}(A)$-modules of $T$:

$T_{A}:=T\cap\left(M_{A}\otimes_{A\otimes
A}\pazocal{B}_{\infty}(A)\right),\quad
U_{A}:=T\cap\left(N_{A}\otimes_{A\otimes A}\pazocal{B}_{\infty}(A)\right).$

  2. $(ii)$

two sub-$\pazocal{A}_{\infty}(B)$-modules of $T$:

$T_{B}:=T\cap\left(M_{B}\otimes_{A\otimes
B}\pazocal{B}_{\infty}(B)\right),\quad
U_{B}:=T\cap\left(N_{B}\otimes_{A\otimes B}\pazocal{B}_{\infty}(B)\right),$

  3. $(iii)$

and two sub-$\pazocal{A}_{\infty}(D)$-modules of $T$:

$T_{D}:=T\cap\left(M_{D}\otimes_{A\otimes
D}\pazocal{B}_{\infty}(D)\right),\quad
U_{D}:=T\cap\left(N_{D}\otimes_{A\otimes D}\pazocal{B}_{\infty}(D)\right),$

The first two $\pazocal{A}_{\infty}(A)$-modules are in fact equal. Indeed, as
$\mathfrak{j}\pazocal{B}_{\infty}(A)=\pazocal{B}_{\infty}(A)$, and since the
inclusion $A\otimes A\to\pazocal{B}_{\infty}(A)$ is flat, we have

$\displaystyle N_{A}\otimes_{A\otimes A}\pazocal{B}_{\infty}(A)$
$\displaystyle=[(M+\tau_{M}(\tau^{*}M))\cap
M_{A}[\mathfrak{j}^{-1}]]\otimes_{A\otimes A}\pazocal{B}_{\infty}(A)$
$\displaystyle=[(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
A}\pazocal{B}_{\infty}(A)]\cap[M_{A}[\mathfrak{j}^{-1}]\otimes_{A\otimes
A}\pazocal{B}_{\infty}(A)]$ $\displaystyle=[M\otimes_{A\otimes
K}\pazocal{B}_{\infty}(K)]\cap[M_{A}\otimes_{A\otimes
A}\pazocal{B}_{\infty}(A)]$ $\displaystyle=M_{A}\otimes_{A\otimes
A}\pazocal{B}_{\infty}(A).$

Our aim is to glue together $M_{A}$, $M_{B}$, $T_{A}$ and $T_{B}$ (resp.
$N_{A}$, $N_{B}$, $U_{A}$ and $U_{B}$) to obtain $\pazocal{M}$ (resp.
$\pazocal{N}$) along the covering $\operatorname{Spec}A\otimes A$,
$\operatorname{Spec}A\otimes B$, $\operatorname{Spec}\pazocal{A}_{\infty}(A)$
and $\operatorname{Spec}\pazocal{A}_{\infty}(B)$ of $C\times C$.

This covering is not Zariski, so we will use the Beauville-Lazslo Theorem
[BL95] to carry out the glueing process. By functoriality, the morphism
$\tau_{\pazocal{M}}$ will result as the glueing of

${\tau^{*}M_{A}}$${\tau^{*}M_{B}}$${\tau^{*}T_{A}}$${\tau^{*}T_{B}}$${N_{A}}$${N_{B}}$${U_{A}}$${U_{B}}$
(5.3)

along the corresponding covering. Note that the first two arrows glue together
as they arise from $\underline{\pazocal{M}}_{0}$.

##### Step 1: the modules $T_{A}$, $T_{B}$, $U_{A}$ and $U_{B}$ are finitely
generated.

We prove finite generation for $T_{A}$ (the argument for $T_{B}$, $U_{A}$ and
$U_{B}$ being similar).

Let $L$ be a finite free $A\otimes A$-module containing $M_{A}$ and let
$\mathbf{n}=(n_{1},...,n_{s})$ be a basis of $L$. For any element $m$ in
$L\otimes_{A\otimes A}\pazocal{B}_{\infty}(K)$, we denote by $v_{\infty}(m)$
the minimum of the $\infty$-valuations of the coefficients of $m$ in n. Let
also $\Lambda\subset L\otimes_{A\otimes A}\pazocal{B}_{\infty}(K)$ be the
finite free $\pazocal{A}_{\infty}(A)$-module generated by n. Clearly,
$v_{\infty}(\lambda)\geq 0$ for any $\lambda\in\Lambda$. As $T\subset
M\otimes_{A\otimes K}\pazocal{B}_{\infty}(K)$ is finitely generated over
$\pazocal{A}_{\infty}(K)$, there exists a positive integer $v_{T}$ such that
$v_{\infty}(t)\geq-v_{T}$ for all $t\in T$.

Let $x\in T_{A}\setminus\\{0\\}$. Because $A\otimes
A\to\pazocal{B}_{\infty}(A)$ is flat, we have the inclusions

$T_{A}\subset M_{A}\otimes_{A\otimes
A}\pazocal{B}_{\infty}(A)\subset\Lambda\otimes_{\pazocal{A}_{\infty}(A)}\pazocal{B}_{\infty}(A)=\bigcup_{n=0}^{\infty}{\pi_{\infty}^{-n}\Lambda},$

and there exists a non-negative integer $n$ such that $x=\pi_{\infty}^{-n}m$
for some $m\in\Lambda\setminus\pi_{\infty}\Lambda$. Comparing valuations
yields

$n=v_{\infty}(m)-v_{\infty}(x)\leq v_{\infty}(m)+v_{T}.$

The number $v_{\infty}(m)$ cannot be positive, otherwise we would have
$m\in\pi_{\infty}(\Lambda\otimes_{\pazocal{A}_{\infty}(A)}\pazocal{A}_{\infty}(K))$,
which contradicts our assumption $m\notin\pi_{\infty}\Lambda$. Thus, $n\leq
v_{T}$ and it follows that

$x\in\bigcup_{n=0}^{v_{T}}{\pi_{\infty}^{-n}\Lambda}.$

Consequently,
$T_{A}\subset\displaystyle\bigcup_{n=0}^{v_{T}}{\pi_{\infty}^{-n}\Lambda}$
and, because $\pazocal{A}_{\infty}(A)$ is Noetherian, $T_{A}$ is finitely
generated.