# Equivariant line bundles with connection on the $p$-adic upper half plane

Konstantin Ardakov  Mathematical Institute
University of Oxford
Oxford OX2 6GG<EMAIL_ADDRESS>and  Simon Wadsley  Homerton College,
Cambridge, CB2 8PH<EMAIL_ADDRESS>
###### Abstract.

Let $F$ be a finite extension of $\mathbb{Q}_{p}$, let $\Omega_{F}$ be
Drinfeld’s upper half-plane over $F$ and let $G^{0}$ the subgroup of
$GL_{2}(F)$ consisting of elements whose determinant has norm $1$. By working
locally on $\Omega_{F}$, we construct and classify the torsion
$G^{0}$-equivariant line bundles with integrable connection on $\Omega$ in
terms of the smooth linear characters of the units of the maximal order of the
quaternion algebra over $F$.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 The main result
    3. 1.3 Motivation
    4. 1.4 An overview of some related works
    5. 1.5 Acknowledgements
    6. 1.6 Conventions and Notation
  2. 2 Background from algebra
    1. 2.1 Measures on profinite sets
    2. 2.2 Some stabilisers in $G^{0}$ and their linear characters
    3. 2.3 Quaternions
    4. 2.4 Equivariant sheaves and amalgamated products
  3. 3 Topics in rigid analytic geometry
    1. 3.1 Line bundles with flat connection on smooth rigid spaces
    2. 3.2 Equivariant line bundles with flat connections
    3. 3.3 Cocycles and equivariant line bundles on affinoids
  4. 4 Applications to Drinfeld’s upper half plane
    1. 4.1 Subdomains of the rigid analytic affine line
    2. 4.2 Drinfeld’s upper half-plane
    3. 4.3 Units, measures and flat connections on $\Omega$
    4. 4.4 Proof of Theorem A

##  1\. Introduction

###  1.1. Background

Let $p$ be a prime number, $\mathbb{Q}_{p}$ the field of $p$-adic numbers and
$F$ a finite extension of $\mathbb{Q}_{p}$ with ring of integers
$\mathcal{O}_{F}$ and uniformiser $\pi_{F}$. Let $\Omega_{F}$ denote the the
Drinfeld upper half-plane: it is a rigid $F$-analytic space whose underlying
set is the set of $\operatorname{Gal}(\overline{F}/F)$-orbits in
$\mathbb{P}^{1}(\overline{F})\backslash\mathbb{P}^{1}(F)$. In [11], Drinfeld
defined a tower of $G:=GL_{2}(F)$-equivariant étale rigid $\breve{F}$-analytic
coverings of $\Omega:=\Omega_{F}\times_{F}\breve{F}$, where $\breve{F}$ is the
completion of the maximal unramified extension of $F$.

We consider a slight modification of Drinfeld’s construction, due to Rapoport-
Zink [25]. They constructed a tower of coverings

$\cdots\to\mathcal{M}_{n}\to\mathcal{M}_{n-1}\to\cdots\to\mathcal{M}_{1}\to\mathcal{M}_{0}$

of rigid $\breve{F}$-analytic spaces. Their base space $\mathcal{M}_{0}$ can
be viewed as being the disjoint union of countably many copies of $\Omega$ (cf
[25, Theorem 3.72]): there is a non-canonical isomorphism
$\mathcal{M}_{0}\cong\Omega\times\mathbb{Z}$. The tower comes equipped with an
action of $G$, so that the maps $\mathcal{M}_{n}\to\mathcal{M}_{n-1}$ are all
$G$-equivariant. The $G$-action on the base space is given by

$g\cdot(z,n)=(g\cdot z,n+v_{\pi_{F}}(\det g)).$

Here the $G$-action on $\Omega$ is the usual one by Möbius transformations.
Thus each copy of $\Omega$ is stabilised by the subgroup $G^{0}$ of $G$
consisting of matrices $g$ in $G$ such that $v_{\pi_{F}}(\det g)=0$. In this
way we may view $\mathcal{M}_{0}$ as being isomorphic to
$G\times_{G^{0}}\Omega$.

Let $D$ be the quaternion division algebra over $F$ with valuation ring
$\mathcal{O}_{D}$ and let $\Pi$ denote a generator of the unique maximal ideal
in $\mathcal{O}_{D}$. The maps $\mathcal{M}_{n}\to\mathcal{M}_{n-1}$ in the
tower are all finite étale and Galois with

$\operatorname{Gal}(\mathcal{M}_{n}/\mathcal{M}_{0})\cong\mathcal{O}_{D}^{\times}/(1+\Pi^{n}\mathcal{O}_{D}).$

Moreover, the actions of $G$ and
$\operatorname{Gal}(\mathcal{M}_{n}/\mathcal{M}_{0})$ on $\mathcal{M}_{n}$
commute.

Scholze–Weinstein have proved in [29, Theorem 7.3.1] that if $\mathbf{C}$
denotes a complete and algebraically closed extension of $\breve{F}$ and
$\widetilde{\Omega}$ is a finite étale $G$-equivariant cover of the base-
change $\Omega\times_{\breve{F}}\mathbf{C}$, then there is an integer
$m\geqslant 0$ such that
$\mathcal{M}_{m}\times_{\breve{F}}\mathbf{C}\to\Omega\times_{\breve{F}}\mathbf{C}$
factors through $\widetilde{\Omega}$.

One way to better understand these finite étale equivariant covers is through
the study of equivariant vector bundles with flat connections on the base
space: the two theories are essentially equivalent. Here we briefly sketch how
to obtain equivariant vector bundles with flat connection from the Drinfeld
tower. Suppose that $\rho$ is a smooth geometrically irreducible
representation of $\mathcal{O}_{D}^{\times}$ whose character is defined over
some finite extension $K$ of $\breve{F}$. There is some $m\geqslant 0$ such
that $\rho$ factors through
$\mathcal{O}_{D}^{\times}/(1+\Pi^{m}\mathcal{O}_{D})$ and, for all $n\geqslant
m$, $\mathcal{O}_{\mathcal{M}_{n,K}}$ has a $\rho$-isotypical component
$\mathcal{V}^{\rho}$ independent of the choice of $n$. Then the pushforward of
$\mathcal{V}^{\rho}$ down to $\mathcal{M}_{0,K}$ is a $G$-equivariant vector
bundle over $\mathcal{M}_{0,K}$ of rank $(\dim\rho)^{2}$, equipped with an
integrable connection.

In this paper we only consider the smooth $K$-linear representations of
$\mathcal{O}_{D}^{\times}$ of degree $1$, that is, the torsion characters
$\theta:\mathcal{O}_{D}^{\times}\to K^{\times}$. Then the pushforward of
$\mathcal{V}^{\theta}$ is a $G$-equivariant line bundle on $\mathcal{M}_{0,K}$
with an integrable connection. For example, when $\bf{1}$ is the trivial
representation, the corresponding line bundle is just the structure sheaf
$\mathcal{O}_{\mathcal{M}_{0}}$ with its natural $G$-action and natural
integrable connection. The set
$\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})_{\operatorname{tors}}$
of torsion characters $\mathcal{O}_{D}^{\times}\to K^{\times}$ forms a group
under pointwise multiplication of characters; given a continuous action111see
$\S\ref{PicConG}$ for the precise definitions of a topological group $H$ on a
smooth rigid $K$-analytic space $X$, we denote by
$\operatorname{PicCon}^{H}(X)$ the group of isomorphism classes of
$H$-equivariant line bundles with an integrable connection on $X$, where the
group structure comes from the tensor product of these line bundles. In this
way, the Drinfeld tower gives us a group homomorphism

(1)
$\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})_{\operatorname{tors}}\quad\longrightarrow\quad\operatorname{PicCon}^{G}(\mathcal{M}_{0,K})_{\operatorname{tors}}.$

###  1.2. The main result

In this paper we will consider, for each complete valued field extension $K$
of $F$, the subgroup
$\operatorname{PicCon}^{G^{0}}(\Omega)_{\operatorname{tors}}$ of
$\operatorname{PicCon}^{G^{0}}(\Omega)$ consisting of the isomorphism classes
of the $G^{0}$-equivariant line bundles with integrable connection on the
rigid $K$-analytic space $\Omega:=\Omega_{F}\times_{F}K$ that have finite
order under $\otimes$. This group is naturally isomorphic to
$\operatorname{PicCon}^{G}(G\times_{G_{0}}\Omega)_{\operatorname{tors}}$ via
an induction map, and therefore also, non-canonically, to
$\operatorname{PicCon}^{G}(\mathcal{M}_{0,K})_{\operatorname{tors}}$ whenever
$K$ contains $\breve{F}$. Our main result is then

######  Theorem A.

Suppose that $K$ contains the quadratic unramified extension $L$ of $F$. Then
there is an isomorphism of abelian groups

$\operatorname{PicCon}^{G^{0}}(\Omega)_{\operatorname{tors}}\quad\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\quad\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})_{\operatorname{tors}}.$

We note that Theorem A cannot be true without the condition that $K$ contains
$L$ in the statement, because
$\operatorname{PicCon}^{G^{0}}(\Omega)[p^{\prime}]$ is cyclic of order
$q^{2}-1$ for any choice of field extension $K$, but $K^{\times}$ contains no
such subgroup unless $K$ contains $L$ and so
$\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})$ cannot either. We
also note here in passing that in fact all line bundles on $\Omega$ are known
to be trivial — see, e.g. [14, Théorème A].

Our isomorphism depends on a choice of a point $z\in\Omega_{F}(L)$ and an
$F$-algebra embedding $\iota:L\hookrightarrow D$. However, there is a natural
$G$-action on $\operatorname{PicCon}^{G^{0}}(\Omega)_{\operatorname{tors}}$
and a natural conjugation $D^{\times}$-action on
$\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})_{\operatorname{tors}}$:
the first action factors through $G/G^{0}F^{\times}$ and the second action
factors through $D^{\times}/\mathcal{O}_{D}^{\times}F^{\times}$, both cyclic
groups of order $2$. The isomorphisms in Theorem A are compatible with respect
to these actions, provided we identify $G/G^{0}F^{\times}$ with
$D^{\times}/\mathcal{O}_{D}^{\times}F^{\times}$ in the only possible way.
Quotienting out by these actions on both sides, we obtain a bijection

$\operatorname{PicCon}^{G^{0}}(\Omega)_{\operatorname{tors}}/G\quad\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\quad\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})_{\operatorname{tors}}/D^{\times}$

which no longer depends on any choices. We expect, but do not check it in this
paper, that when $K=\breve{F}$, the isomorphism in Theorem A is in fact
inverse to (1) for some choice of identification of $\mathcal{M}_{0}$ with
$G\times_{G^{0}}\Omega$, depending on our other choices.

###  1.3. Motivation

We do not use the result of Scholze–Weinstein [29], nor even the existence of
the Drinfeld tower, to prove Theorem A. Instead, we give an explicit
construction of the elements
$\operatorname{PicCon}^{G^{0}}(\Omega)_{\operatorname{tors}}$ as finite order
$G^{0}$-equivariant line bundles with connection on $\Omega$ and prove
directly that there are no others. More precisely: in this paper we give an
_elementary construction_ 222elementary in the sense of not depending on the
duality of Rapoport-Zink spaces, nor on the theory of perfectoid spaces of
finite order $GL_{2}(\mathcal{O}_{F})$-equivariant line bundles with
connection on the $K$-affinoid subdomain $\Omega_{0}$ of $\Omega$ that is the
inverse image of the vertex fixed by $GL_{2}(\mathcal{O}_{F})$ in the Bruhat-
Tits tree under the reduction map, and show that each of these line bundles
extends uniquely to a $G^{0}$-equivariant line bundle with connection on all
of $\Omega$. We have provided full arguments for some results that already
appear in the literature in order to stress the elementary nature of our work.

In our forthcoming work [1], we will use this elementary construction to
better understand the structure of the strong duals of the global sections
$\mathscr{L}^{\theta}(\Omega)^{\prime}_{s}$ of these equivariant line bundles
as locally $F$-analytic representations of $G$. If
$j:\Omega\to\mathbb{P}^{1}_{K}$ denotes the open inclusion, then the results
of this paper are used to show that $j_{\ast}\mathscr{L}^{\theta}$ is a
coadmissible $G$-equivariant $\mathcal{D}$-module on $\mathbb{P}^{1}_{K}$ in
the sense of [2]. It is a formal consequence that
$\mathscr{L}^{\theta}(\Omega)^{\prime}_{s}$ is an _admissible_ $F$-locally
analytic $K$-representation of $G$, which is a consequence of a result of
Patel, Schmidt and Strauch [22, Theorem 7.2.1(iv)]. However we are able to
prove stronger results about the structure of these representations: we will
show that for each
$\theta\in\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})_{\operatorname{tors}}$,
$\mathscr{L}^{\theta}(\Omega)^{\prime}$ is an admissible locally $F$-analytic
$K$-representation of $G^{0}$ that is _topologically irreducible_ when
$\theta$ is not fixed by the $D^{\times}$-action on
$\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})_{\operatorname{tors}}$
mentioned above, and that has a codimension $1$-submodule that is
topologically irreducible otherwise.

###  1.4. An overview of some related works

Because of the central position of the theory of Drinfeld coverings and
Rapoport-Zink spaces in the local Langlands program [13], it is difficult to
make a comprehensive literature review. In his work [33], Teitelbaum found
explicit local equations for the first Drinfeld covering of $\Omega$: this
yields explicit torsion line bundles with connection on $\Omega_{0}$ together
with a (non-explicit) $GL_{2}(\mathcal{O}_{F})$-equivariant structure. However
neither Teitelbaum, nor Lue Pan in his closely-related work [21], consider the
flat connections on these line bundles, nor do they make an attempt to
classify the appropriate equivariant structures in an elementary manner.

We acknowledge our intellectual debt to the Introduction of [10], where
Dospinescu and Le Bras explain the importance of the equivariant vector
bundles on $\Omega$ in the context of locally analytic representations of
$GL_{2}(F)$, and in particular in the $p$-adic local Langlands program. We
mention here in passing that the constructions in our paper enable us to
define the first Drinfeld covering of $\Omega$ directly over $F$ rather than
over $\breve{F}$ without appealing to the theory of Weil descent. We also take
the opportunity to mention here the monumental works of Colmez, Dospinescu and
Nizioł [8], [9] that use Scholze’s pro-étale methods and build upon [10] to
show, amongst other things, that the $p$-adic étale cohomology of the Drinfeld
tower realises the $p$-adic local Langlands correspondence, at least in the
case when $F=\mathbb{Q}_{p}$.

Finally, we mention that Junger in his recent preprint [15] has classified
equivariant line bundles on Deligne’s formal model $\widehat{\Omega}$ of
$\Omega_{F}$. There are some formal similarities in our methods (for example
his Proposition 2.10 plays a similar role to our Proposition 3.2.14), however
we cannot deduce his results from ours, nor vice versa. We expect that our
results here can be naturally extended to analogues of $\Omega$ in higher
dimensions.

###  1.5. Acknowledgements

We thank Aurel Page for pointing the paper of Riehm [26] on MathOverflow. We
also thank James Taylor for his comments. The second author thanks Brasenose
College, Oxford for its hospitality.

###  1.6. Conventions and Notation

$F$ will denote a finite extension of $\mathbb{Q}_{p}$ with ring of integers
$\mathcal{O}_{F}$, uniformiser $\pi_{F}$ and residue field $k_{F}$ of order
$q$. $K$ will denote a field containing $F$ that is complete with respect to a
non-archimedean norm $|\cdot|$ such that $|p|=1/p$. We will write:

  * •

$K^{\circ}:=\\{a\in K:|a|\leqslant 1\\}$ for the valuation ring of $K$,

  * •

$K^{\circ\circ}:=\\{a\in K:|a|<1\\}$ for the maximal ideal of $K^{\circ}$,

  * •

$\overline{K}$ for a fixed algebraic closure of $K$, and

  * •

$\mathbf{C}$ for the completion of $\overline{K}$.

Let $X$ be a rigid $K$-analytic variety. When $Y$ is a subset $X$, we will say
that $Y$ is an _affinoid subdomain_ of $X$ to mean that $Y$ is an admissible
open subspace of $X$, itself isomorphic to an affinoid $K$-variety. When $X$
itself happens to be a $K$-affinoid variety, this agrees by [4, Corollary
8.2.1/4] with the standard definition found at [4, Definition 7.2.2/2]. We
will write

  * •

$|\cdot|_{X}$ to denote the (power-multiplicative) supremum seminorm on $X$,

  * •

$\mathcal{O}(X)^{\circ}:=\\{f\in\mathcal{O}(X):|f|_{X}\leqslant 1\\}$,

  * •

$\mathcal{O}(X)^{\circ\circ}:=\\{f\in\mathcal{O}(X):|f|_{X}<1\\}$,

  * •

$\mathcal{O}(X)^{\times\times}:=1+\mathcal{O}(X)^{\times\times}$, the subgroup
of _small units_ in $\mathcal{O}(X)^{\times}$, and

  * •

$\mathcal{O}(X)^{\times\times}_{r}:=\\{1+f:|f|_{X}\leqslant
r\\}\leqslant\mathcal{O}(X)^{\times\times}$ for each real number $r\in(0,1)$.

$\operatorname{Pic}(X)$ will denote the _Picard group_ of $X$ consisting of
the isomorphism classes of line bundles on $X$ with the group operation given
by tensor product.

When $K^{\prime}$ is a complete field extension of $K$, we will write
$X_{K^{\prime}}:=X\times_{K}K^{\prime}$ for the base change of $X$ to
$K^{\prime}$, and

$X(K^{\prime})=\\{\phi\colon\operatorname{Sp}K^{\prime}\to X:\phi\mbox{ is a
morphism of rigid }K\mbox{-analytic varieties}\\}$

will denote the set of _$K^{\prime}$ -valued points of $X$_.

Let $A$ be an abelian group and let $d$ be a non-zero integer. We will write

  * •

$A[d]=\\{a\in A\mid da=0\\}$ to denote the $d$-torsion subgroup of $A$,

  * •

$A[p^{\prime}]:=\bigcup_{(d,p)=1}A[d]$ to denote the prime-to-$p$ torsion
subgroup of $A$,

  * •

$A[p^{\infty}]:=\bigcup_{n=1}^{\infty}A[p^{n}]$ to denote the $p$-power
torsion subgroup of $A$.

Let a group $G$ act on a set $X$. We will write

  * •

$G_{x}:=\\{g\in G:gx=x\\}$ for the stabilizer of a point $x\in X$, and

  * •

$X^{G}:=\\{x\in X:gx=x$ for all $g\in G\\}$ for the set of elements fixed by
$G$.

When we discuss cochains, cocycles and coboundaries we will work with the
continuous cochain cohomology of Tate, [32]. That is if $G$ is a topological
group and $A$ is a topological abelian group equipped with a continuous action
of $G$ then:

  * •

$C^{n}(G,A):=\\{f\colon G^{n}\to A:f$ is continuous$\\}$ is the set of
continuous $A$-valued $n$-cochains,

  * •

$Z^{n}(G,A)$ is the set of continuous $A$-valued $n$-cocycles,

  * •

$B^{n}(G,A)$ is the set of continuous $A$-valued $n$-coboundaries,

  * •

$H^{n}(G,A)$ is the $n$th continuous cohomology group $Z^{n}(G,A)/B^{n}(G,A)$,

  * •

$\delta_{G}\colon C^{0}(G,A)=A\to C^{1}(G,A)$ is the map given by
$\delta_{G}(a)(g)=g\cdot a-a$.

Note that whenever $\theta\colon A\to B$ is a $G$-equivariant map of
topological abelian groups with continuous $G$-action, we have
$\delta_{G}\theta=\theta\delta_{G}$.

In the particular case where $G$ acts trivially on $A$, we will usually write
$\operatorname{Hom}(G,A)$ instead of $Z^{1}(G,A)$ or $H^{1}(G,A)$ to denote
the group of continuous homomorphisms from $G$ to $A$. For any continuous
group homomorphism $\varphi\colon G\to H$, we denote by

$\varphi^{\ast}:\operatorname{Hom}(H,A)\to\operatorname{Hom}(G,A)$

the map given by pre-composition by $\varphi$.

##  2\. Background from algebra

###  2.1. Measures on profinite sets

We begin by adapting some definitions from [12, Definitions 2.7.10].

######  Definition 2.1.1.

Let $Z$ be a profinite set and let $\mathfrak{{a}}$ be an abelian group.

  1. (a)

An _$\mathfrak{{a}}$ -valued measure on Z_ is a function $\nu$ from the set of
clopen subsets of $Z$ to $\mathfrak{{a}}$, satisfying
$\nu(U)=\nu(V)+\nu(U\backslash V)$ whenever $V\subseteq U$ are clopen subsets
of $Z$.

  2. (b)

$M(Z,\mathfrak{{a}})$ denotes the abelian group of all $\mathfrak{{a}}$-valued
measures on $Z$ under pointwise operations.

  3. (c)

$M_{0}(Z,\mathfrak{{a}}):=\\{\nu\in M(Z,\mathfrak{{a}}):\nu(Z)=0\\}$ is the
subgroup of _$\mathfrak{{a}}$ -valued measures on $Z$ with total value zero_.

######  Examples 2.1.2.

  1. (1)

If $Z$ is any profinite set and $\mathfrak{a}$ is a unital ring then for each
$z\in Z$ we can define a measure $\delta_{z}$ by

$\delta_{z}(U)=\begin{cases}1&\mbox{ if }z\in U\\\ 0&\mbox{
otherwise.}\end{cases}$

  2. (2)

If $Z$ happens to be finite we can define a ‘counting measure’ in
$M(Z,\mathbb{Z})$ via

$\Sigma_{Z}(U)=|U|\mbox{ for all }U\subset Z.$

Indeed $\Sigma_{Z}=\sum_{z\in Z}\delta_{z}$.

Let $C(Z,\mathbb{Z})$ be the set of continuous $\mathbb{Z}$-valued functions
on $Z$, where we give $\mathbb{Z}$ the discrete topology. Of course every such
function is locally constant.

######  Proposition 2.1.3.

Let $Z$ be a profinite set and let $\mathfrak{{a}}$ be an abelian group.

  1. (a)

There is a natural additive isomorphism
$M(Z,\mathfrak{{a}})\to\operatorname{Hom}_{\mathbb{Z}}(C(Z,\mathbb{Z}),\mathfrak{{a}})$.

  2. (b)

Let $(Z_{i})_{i\in I}$ be a filtered inverse system of finite sets. Then every
isomorphism
$Z\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}Z_{i}$
of profinite sets induces an isomorphisms of abelian groups
$M(Z,\mathfrak{{a}})\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}M(Z_{i},\mathfrak{{a}})$
and
$M_{0}(Z,\mathfrak{{a}})\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}M_{0}(Z_{i},\mathfrak{{a}})$.

  3. (c)

The functor $\mathfrak{{a}}\mapsto M(Z,\mathfrak{{a}})$ is exact.

###### Proof.

(a) Let $\nu\in M(Z,\mathfrak{{a}})$ and let $f:Z\to\mathbb{Z}$ be locally
constant. Because $Z$ is profinite and hence compact, we can choose an open
partition $\\{U_{1},\cdots,U_{m}\\}$ of $Z$ such that $f_{|U_{i}}$ is constant
for each $i$. Choose $z_{i}\in U_{i}$ for each $i=1,\cdots,m$ and define
$\langle\nu,f\rangle:=\sum\limits_{i=1}^{m}f(z_{i})\nu(U_{i})\in\mathfrak{{a}}$.
Then $\langle\nu,f\rangle$ does not depend on the choice of open partition or
the choice of the $z_{i}$’s, and $\nu\mapsto(f\mapsto\langle\nu,f\rangle)$
defines an additive map
$M(Z,\mathfrak{{a}})\to\operatorname{Hom}_{\mathbb{Z}}(C(Z,\mathbb{Z}),\mathfrak{{a}})$.

Let $1_{U}$ denote the characteristic function of the clopen subset $U$ of
$Z$. Given an additive map $g:C(Z,\mathbb{Z})\to\mathfrak{{a}}$, setting
$\nu(U):=g(1_{U})\in\mathfrak{{a}}$ for each clopen $U$ defines an element
$\nu\in M(Z,\mathfrak{{a}})$ such that $\langle\nu,f\rangle=g(f)$ for all
$f\in C(Z,\mathbb{Z})$ because the characteristic functions $1_{U}$ generate
$C(Z,\mathbb{Z})$ as an abelian group. The function $g\mapsto\nu$ is then an
inverse to
$M(Z,\mathfrak{{a}})\to\operatorname{Hom}_{\mathbb{Z}}(C(Z,\mathbb{Z}),\mathfrak{{a}})$.

(b) The isomorphism
$Z\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}Z_{i}$
induces an isomorphism of abelian groups
$C(Z,\mathbb{Z})\cong\lim\limits_{\longrightarrow}C(Z_{i},\mathbb{Z})$. The
functor $\operatorname{Hom}_{\mathbb{Z}}(-,\mathfrak{{a}})$ converts colimits
into limits; now apply part (a) to get
$M(Z,\mathfrak{{a}})\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}M(Z_{i},\mathfrak{{a}})$.
Since taking limits commutes with taking kernels the other part follows
immediately.

(c) By a theorem of Nöbeling — see [28, Theorem 5.4] — $C(Z,\mathbb{Z})$ is a
free abelian group. Hence $\operatorname{Hom}_{\mathbb{Z}}(C(Z,\mathbb{Z}),-)$
is exact; now apply part (a). ∎

######  Definition 2.1.4.

For every abelian group $\mathfrak{{a}}$ the map $Z\mapsto
M(Z,\mathfrak{{a}})$ from profinite sets to abelian groups extends to a
functor from the category of profinite sets and continuous functions to the
category of abelian groups sending a continuous function $f\colon Z_{1}\to
Z_{2}$ to the group homomorphism

$f_{\ast}\colon M(Z_{1},\mathfrak{{a}})\to M(Z_{2},\mathfrak{{a}});\hskip
5.69046ptf_{\ast}(\nu)(U)=\nu(f^{-1}(U)).$

In particular an action of group $G$ on a profinite set $Z$ by homeomorphisms
induces an action on $M(Z,\mathfrak{{a}})$ by automorphisms of abelian groups
via $g\cdot\nu=g_{\ast}(\nu)$.

######  Remark 2.1.5.

If $\nu\in M_{0}(Z_{1},\mathfrak{{a}})$ in the setting of Definition 2.1.4
then $f_{\ast}(\nu)\in M_{0}(Z_{2},\mathfrak{{a}})$ since
$f_{\ast}(\nu)(Z_{2})=\nu(f^{-1}(Z_{2}))=\nu(Z_{1})=0$ so $Z\mapsto
M_{0}(Z,\mathfrak{{a}})$ also defines a functor.

The following result will be useful in what follows.

######  Lemma 2.1.6.

Let $Z$ be a profinite set and let $d$ be a non-zero integer. Then the
sequence $0\to M_{0}(Z,\mathbb{Z})\stackrel{{\scriptstyle
d}}{{\longrightarrow}}M_{0}(Z,\mathbb{Z})\to
M_{0}(Z,\mathbb{Z}/d\mathbb{Z})\to 0$ is exact.

###### Proof.

Consider the multiplication-by-$d$ map on the short exact sequence of abelian
groups $0\to M_{0}(Z,\mathbb{Z})\to M(Z,\mathbb{Z})\to\mathbb{Z}\to 0$. This
gives a $3\times 3$ diagram of abelian groups, whose rows are exact, whose
second column is exact by Proposition 2.1.3(c) and whose third column is exact
for trivial reasons. Hence the first column is also exact by the Nine Lemma. ∎

Recall our conventions concerning continuous group cohomology from
$\S\ref{ConvNotn}$.

######  Lemma 2.1.7.

Let $G$ be a profinite group with a continuous transitive action on a finite
set $X$. Then

  1. (a)

$M(X,\mathbb{Z})^{G}=\mathbb{Z}\cdot\Sigma_{X}$;

  2. (b)

$M_{0}(X,\mathbb{Z})^{G}=0$;

  3. (c)

$H^{1}(G,M(X,\mathbb{Z}))=0$.

###### Proof.

(a) It is easy to compute that $M(X,\mathbb{Z})^{G}=\mathbb{Z}\cdot\Sigma_{X}$
where $\Sigma_{X}$ is the counting measure from Example 2.1.2(2).

(b) Use
$M_{0}(X,\mathbb{Z})^{G}=\ker\left(M(X,\mathbb{Z})^{G}\to\mathbb{Z};\nu\mapsto\nu(X)\right)$
and $n\Sigma_{X}(X)=n|X|$.

(c) Choose an arbitrary point $x\in X$. Since $X$ is finite, $M(X,\mathbb{Z})$
is isomorphic to the (co)induced module $\mathrm{Ind}^{G_{x}}_{G}\mathbb{Z}$
in the sense of [20, Chapter I, §6], where $\mathbb{Z}$ denotes the trivial
$G_{x}$-module. Then by Shapiro’s Lemma, [20, Proposition 1.6.4], we have

$H^{1}(G,M(X,\mathbb{Z}))\cong H^{1}(G_{x},\mathbb{Z}).$

Since the action of $G_{x}$ on $\mathbb{Z}$ is trivial, $G_{x}$ is profinite
and $\mathbb{Z}$ is a discrete torsion-free group we deduce that
$H^{1}(G_{x},\mathbb{Z})=\operatorname{Hom}(G_{x},\mathbb{Z})=0$. ∎

######  Proposition 2.1.8.

Let $G$ be a profinite group acting continuously and transitively on two non-
empty finite sets $X,Y$, and let $\pi\colon X\to Y$ be a $G$-equivariant
function. Let $d\geqslant 1$ be an integer and let $h:=\gcd(d,|X|)$. Then

  1. (a)

$M(X,\mathbb{Z}/d\mathbb{Z})^{G}$ is cyclic of order $d$, generated by the
image of $\Sigma_{X}$,

  2. (b)

$M_{0}(X,\mathbb{Z}/d\mathbb{Z})^{G}$ is cyclic of order $h$, generated by the
image of $\frac{d}{h}\Sigma_{X}$, and

  3. (c)

$\pi_{\ast}\left(\Sigma_{X}\right)=\frac{|X|}{|Y|}\Sigma_{Y}$.

###### Proof.

(a) The short exact sequence

$0\to M(X,\mathbb{Z})\stackrel{{\scriptstyle d}}{{\to}}M(X,\mathbb{Z})\to
M(X,\mathbb{Z}/d\mathbb{Z})\to 0$

induces a long exact sequence of cohomology

$0\to M(X,\mathbb{Z})^{G}\stackrel{{\scriptstyle
d}}{{\to}}M(X,\mathbb{Z})^{G}\to
M(X,\mathbb{Z}/d\mathbb{Z})^{G}{\to}H^{1}(G,M(X,\mathbb{Z}))$

whose final term is $0$ by Lemma 2.1.7(c). Since
$M(X,\mathbb{Z})^{G}=\mathbb{Z}\cdot\Sigma_{X}$ by Lemma 2.1.7(a) the result
follows.

(b) Note that $M_{0}(X,\mathbb{Z}/d\mathbb{Z})^{G}=\\{\nu\in
M(X,\mathbb{Z}/d\mathbb{Z})^{G}:\nu(X)=d\mathbb{Z}\\}$. By part (a), every
such $\nu$ is the image of $n\Sigma_{X}$ for some $n$. But
$n\Sigma_{X}(X)=n|X|$ so

$M_{0}(X,\mathbb{Z}/d\mathbb{Z})^{G}=\\{n\Sigma_{X}\in
M(X,\mathbb{Z}/d\mathbb{Z}):d\mbox{ divides }n|X|\\}$

giving the result.

(c) Whenever $U\subseteq Y$, we have
$\pi_{\ast}(\Sigma_{X})(U)=\Sigma_{X}(\pi^{-1}(U))=\left|\pi^{-1}(U)\right|$.
Since $G$ acts transitively on $X$ and $Y$, all the fibres have order
$\frac{|X|}{|Y|}$ and so

$\left|\pi^{-1}(U)\right|=\frac{|X|}{|Y|}|U|$

as required. ∎

###  2.2. Some stabilisers in $G^{0}$ and their linear characters

######  Notation 2.2.1.

  1. (a)

$G^{0}:=\\{g\in GL_{2}(F):v_{\pi_{F}}(\det g)=1\\}$.

  2. (b)

The _Iwahori subgroup_ $I$ of $GL_{2}(\mathcal{O}_{F})$ is

$I:=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
GL_{2}(\mathcal{O}_{F}):c\equiv
0\operatorname{mod}\pi_{F}\mathcal{O}_{F}\right\\}.$

  3. (c)

We write $w:=\begin{pmatrix}0&1\\\ \pi_{F}&0\end{pmatrix}\in GL_{2}(F)$.

  4. (d)

We write $A:=GL_{2}(\mathcal{O}_{F})$ and $B:={}^{w}A=wAw^{-1}$.

We recall from [31, §§II.1.2-3] that if $\mathcal{B}\mathcal{T}$ is the
Bruhat–Tits tree associated with $PGL_{2}(F)$, then $A,B$ and $I$ arise as
stabilisers in $G^{0}$ under the natural $G^{0}$-action on
$\mathcal{B}\mathcal{T}$ as follows: there is a vertex $s$ of
$\mathcal{B}\mathcal{T}$ such that $(s$ $ws)$ is an edge of
$\mathcal{B}\mathcal{T}$, and

$A=G^{0}_{s},\quad B=G^{0}_{ws},\quad\mbox{and}\quad I=G^{0}_{(s\mbox{ }ws)}.$

In particular, we have $I=A\cap B$. The following classical result will be
crucial to our arguments later on in $\S\ref{MainProofSec}$.

######  Theorem 2.2.2.

$G^{0}$ is the amalgamated product of its open subgroups $A$ and $B$ over
their intersection $I$:

$G^{0}=A\underset{I}{\ast}{}B.$

###### Proof.

This is [31, Theorem II.3]. ∎

Recall that $F^{\times}$ is the direct product of its subgroups
$\mu_{p^{\prime}}(F)$, $\mathcal{O}_{F}^{\times\times}$ and
$\langle\pi_{F}\rangle$:

$F^{\times}\cong\mu_{p^{\prime}}(F)\times\mathcal{O}_{F}^{\times\times}\times\langle\pi_{F}\rangle.$

We write $a\mapsto\widehat{a}$ to denote the homomorphism that is the
projection $F^{\times}\to\mu_{p^{\prime}}(F)$ with kernel
$\mathcal{O}_{F}^{\times\times}\times\langle\pi_{F}\rangle$. We will use the
same notation $\hat{\cdot}$ to denote the analogous projection
$L^{\times}\to\mu_{p^{\prime}}(L)$ for other finite extensions
$L/\mathbb{Q}_{p}$.

######  Lemma 2.2.3.

  1. (a)

$\operatorname{Hom}(A,K^{\times})[p^{\prime}]=\left\\{\widehat{\det}^{k}:k\in\mathbb{Z}/(q-1)\mathbb{Z}\right\\}$.

  2. (b)

Every element of $\operatorname{Hom}(I,K^{\times})[p^{\prime}]$ is of the form

$\begin{pmatrix}a&b\\\
\pi_{F}c&d\end{pmatrix}\mapsto\widehat{a}^{n_{1}}\widehat{d}^{n_{2}}$

for some $n_{1},n_{2}\in\mathbb{Z}/(q-1)\mathbb{Z}$.

  3. (c)

$\left(\operatorname{Hom}(I,K^{\times})[p^{\prime}]\right)^{\langle
w\rangle}=\left\\{\widehat{\det}^{k}:k\in\mathbb{Z}/(q-1)\mathbb{Z}\right\\}$.

###### Proof.

(a) For any $\theta\in\operatorname{Hom}(A,K^{\times})[p^{\prime}]$, every
pro-$p$ subgroup of $A$ is contained in $\ker\theta$. Since
$SL_{2}(\mathcal{O}_{F})$ is generated by its pro-$p$ subgroups it follows
that $\theta$ factors through $\det\colon A\to\mathcal{O}_{F}^{\times}$. Since
$\mathcal{O}_{F}^{\times\times}$ is also pro-$p$ we see that any element of
$\operatorname{Hom}(\mathcal{O}_{F}^{\times},K^{\times})[p^{\prime}]$ factors
through $\widehat{-}$. It remains to observe that
$\operatorname{Hom}(\mu_{p^{\prime}}(F),K^{\times})$ consists of maps of the
form $a\mapsto a^{k}$ for some $k\in\mathbb{Z}/(q-1)\mathbb{Z}$, because
$\mu_{p^{\prime}}(F)$ is cyclic of order $q-1$.

(b) Once again, if $\theta\in\operatorname{Hom}(I,K^{\times})[p^{\prime}]$
then any pro-$p$ subgroup of $I$ is contained in $\ker\theta$. The kernel of
the map $I\to\mu_{p^{\prime}}(F)\times\mu_{p^{\prime}}(F)$ sending
$\begin{pmatrix}a&b\\\ \pi_{F}c&d\end{pmatrix}$ to $(\widehat{a},\widehat{c})$
is pro-$p$. Now the result follows as in (a).

(c) By part (b), every $\chi\in\operatorname{Hom}(I,K^{\times})[p^{\prime}]$
is of the form

$\begin{pmatrix}a&b\\\
\pi_{F}c&d\end{pmatrix}\mapsto\widehat{a}^{n_{1}}\widehat{d}^{n_{2}}$

for some $n_{1},n_{2}\in\mathbb{Z}/(q-1)\mathbb{Z}$. Now

$(w\cdot\chi)\left(\begin{pmatrix}a&b\\\
\pi_{F}c&d\end{pmatrix}\right)=\chi\left(\begin{pmatrix}d&c\\\
\pi_{F}b&a\end{pmatrix}\right)=\widehat{a}^{n_{2}}\widehat{d}^{n_{1}}.$

Thus if $\chi=w\cdot\chi$ then $n_{1}=n_{2}$ and
$\chi=\widehat{\det}^{n_{1}}$. ∎

######  Remark 2.2.4.

It follows from the proof of Lemma 2.2.3 that any $p^{\prime}$-quotient of
$A,B$ or $I$ has exponent dividing $q-1$.

We will now recall the structure of the stabilisers in $GL_{2}(F)$ of points
in $\mathbb{P}^{1}(L)$ for quadratic extensions $L$ of $F$, under the
$GL_{2}(F)$-action by Möbius transformations.

######  Lemma 2.2.5.

Suppose that $F(z)$ is a quadratic extension of $F$. Let

$N,\operatorname{tr}\colon F(z)\to F$

be the norm and trace maps respectively.

  1. (a)

The stabiliser $GL_{2}(F)_{z}$ of $z$ in $GL_{2}(F)$ is

$GL_{2}(F)_{z}=\left\\{\begin{pmatrix}a&-cN(z)\\\
c&a-c\operatorname{tr}(z)\end{pmatrix}\,\middle|(a,c)\in
F^{2}\backslash\\{(0,0)\\}\right\\}.$

  2. (b)

There is a commutative diagram of groups

$\textstyle{GL_{2}(F)_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{z}}$$\scriptstyle{\det}$$\textstyle{F(z)^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N}$$\textstyle{F^{\times}}$

whose horizontal arrow

$j_{z}\colon\begin{pmatrix}a&-cN(z)\\\
c&a-c\operatorname{tr}(z)\end{pmatrix}\mapsto a-cz$

is an isomorphism of topological abelian groups.

  3. (c)

$j_{z}(G^{0}_{z})=\mathcal{O}_{F(z)}^{\times}$ and $j_{z}(SL_{2}(F)_{z})=\ker
N\cap\mathcal{O}_{F(z)}^{\times}$.

  4. (d)

Let $\sigma\in\operatorname{Gal}(F(z)/F)$ be the unique element of order two.
Then

$\sigma\cdot g=\det(g)g^{-1}\quad\mbox{for all}\quad g\in GL_{2}(F)_{z}$

defines an continuous action of $\operatorname{Gal}(F(z)/F)$ on
$GL_{2}(F)_{z}$ such that

    1. (i)

$j_{z}$ is $\operatorname{Gal}(F(z)/F)$-equivariant,

    2. (ii)

$G^{0}_{z}$, $SL_{2}(F)_{z}$ and the Sylow pro-$p$-subgroup $P_{z}$ of
$SL_{2}(F)_{z}$ are all $\operatorname{Gal}(F(z)/F)$-stable.

###### Proof.

(a) We can compute that $\frac{az+b}{cz+d}=z$ if and only if
$cz^{2}+(d-a)z-b=0$. Moreover the minimal polynomial of $z$ is
$t^{2}-\operatorname{tr}(z)t+N(z)$. So $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in GL_{2}(F)$ fixes $z$ if and only if
$(d-a)=-c\operatorname{tr}(z)$ and $b=-cN(z)$ as claimed.

(b) Since

$\det\begin{pmatrix}a&-cN(z)\\\
c&a-c\operatorname{tr}(z)\end{pmatrix}=a^{2}-ac\operatorname{tr}(z)+c^{2}N(z)=N(a-cz)$

the given diagram commutes. For any $(a_{1},c_{1}),(a_{2},c_{2})\in
F^{2}\backslash\\{(0,0\\}$, we have

$\begin{pmatrix}a_{1}&-c_{1}N(z)\\\
c_{1}&a_{1}-c_{1}\operatorname{tr}(z)\end{pmatrix}\begin{pmatrix}a_{2}&-c_{2}N(z)\\\
c_{2}&a_{2}-c_{2}\operatorname{tr}(z)\end{pmatrix}=\begin{pmatrix}a_{1}a_{2}-c_{1}c_{2}N(z)&*\\\
\\\ c_{1}a_{2}+a_{1}c_{2}-c_{1}c_{2}\operatorname{tr}(z)&*\end{pmatrix}\in
GL_{2}(F)_{z}$

and

$\displaystyle(a_{1}-c_{1}z)(a_{2}-c_{2}z)$ $\displaystyle=$
$\displaystyle(a_{1}a_{2}-(a_{1}c_{2}+c_{1}a_{2})z+c_{1}c_{2}z^{2})$
$\displaystyle=$
$\displaystyle(a_{1}a_{2}-c_{1}c_{2}N(z))-(c_{1}a_{2}+a_{1}c_{2}-c_{1}c_{2}\operatorname{tr}(z))z\in
F(z)^{\times}.$

This implies that $j_{z}$ is a group isomorphism.

(c) Since $G^{0}_{z}=\ker\left(v_{\pi_{F}}\circ\det\colon
GL_{2}(F)_{z}\to\mathbb{Z}\right)$, part (b) implies that

$j_{z}(G^{0}_{z})=\ker\left(v_{\pi_{F}}\circ N\colon
F(z)^{\times}\to\mathbb{Z}\right)=\mathcal{O}_{F(z)}^{\times}.$

Similarly since $SL_{2}(F)_{z}=\ker\left(\det\colon GL_{2}(F)_{z}\to
F^{\times}\right)$, we see that $j_{z}(SL_{2}(F)_{z})=\ker\left(N\colon
F(z)^{\times}\to F^{\times}\right)$, which is contained in
$\mathcal{O}^{\times}_{F(z)}$.

(d) We have $N(\lambda)=\lambda\sigma(\lambda)$ for any $\lambda\in
F(z)^{\times}$. Now it follows from (b) that

$j_{z}^{-1}(\sigma(\lambda))=\det(j_{z}^{-1}(\lambda))j_{z}^{-1}(\lambda)^{-1}\quad\mbox{for
any}\quad\lambda\in F(z)^{\times}$

and we can define the claimed action by transport of structure. Since
$SL_{2}(F)_{z}$ is $\sigma$-stable and $P_{z}$ is a characteristic subgroup of
$SL_{2}(F)_{z}$, the rest follows. ∎

######  Corollary 2.2.6.

Suppose that $F(z)$ is a quadratic extension of $F$. Then

$G^{0}_{z}\cdot SL_{2}(F)=G^{0}.$

###### Proof.

Let $g\in G^{0}$ so that $\det(g)\in\mathcal{O}_{F}^{\times}$. The norm map
$N:\mathcal{O}_{F(z)}^{\times}\to\mathcal{O}_{F}^{\times}$ is surjective by
[30, Chapter V, Corollary to Proposition 3], so we can find
$x\in\mathcal{O}_{F(z)}^{\times}$ such that $N(x)=\det(g)$. Using Lemma
2.2.5(c), we can choose $h\in G^{0}_{z}$ such that $j_{z}(h)=x$. Then Lemma
2.2.5(b) implies that $\det(h)=N(j_{z}(h))=N(x)=\det(g)$, so $g=h\cdot
h^{-1}g\in G^{0}_{z}\cdot SL_{2}(F)$ as required. ∎

######  Remark 2.2.7.

Note that, with the notation of Lemma 2.2.5,
$GL_{2}(F)_{z}=GL_{2}(F)_{\sigma\cdot z}$ but $j_{z}\neq j_{\sigma\cdot z}$.
However $j_{z}^{\ast}$ does induce a canonical bijection

$\operatorname{Hom}(G^{0}_{z},K^{\times})/\operatorname{Gal}(F(z)/F)\to\operatorname{Hom}(\mathcal{O}_{F(z)}^{\times},K^{\times})/\operatorname{Gal}(F(z)/F).$

When $F(z)$ a quadratic extension of $F$, we define a continuous homomorphism
$\widehat{j_{z}}\colon G^{0}_{z}\to F(z)^{\times}$ by setting

$\widehat{j_{z}}(g):=\widehat{j_{z}(g)}\quad\mbox{for all}\quad g\in
G^{0}_{z}.$

######  Lemma 2.2.8.

Suppose that $F(z)$ is a quadratic extension of $F$. Then

$\operatorname{Hom}(G^{0}_{z},K(z)^{\times})[p^{\prime}]=\operatorname{Hom}(G^{0}_{z},\mu_{p^{\prime}}(F(z)))=\left\langle\widehat{j_{z}}\right\rangle$

is a cyclic group. Its order is $q^{2}-1$ if $F(z)/F$ is unramified, and $q-1$
otherwise.

###### Proof.

Since $j_{z}\colon G^{0}_{z}\to\mathcal{O}_{F(z)}^{\times}$ is an isomorphism
of topological groups by Lemma 2.2.5(c), it suffices to show that every
element of
$\operatorname{Hom}(\mathcal{O}_{F(z)}^{\times},K(z)^{\times})[p^{\prime}]$ is
of the form $a\mapsto\hat{a}^{k}$ for some $k\in\mathbb{Z}$. Since the kernel
of $\hat{\cdot}\colon\mathcal{O}_{F(z)}^{\times}\to\mu_{p^{\prime}}(F(z))$ is
pro-$p$, any
$\theta\in\operatorname{Hom}(\mathcal{O}_{F(z)}^{\times},K(z)^{\times})[p^{\prime}]$
factors through $\hat{\cdot}$. Since $\mu_{p^{\prime}}(F(z))$ is cyclic of
order $q^{2}-1$ if $F(z)/F$ is unramified and cyclic of order $q-1$ if $F(z)$
is ramified the result is now straightforward. ∎

###  2.3. Quaternions

We use [24, §1.4] and [23] as basic references for this material. By [23,
Proposition 12.5b, Theorem 17.10] there is a central division algebra $D$ over
$F$ of dimension $4$ which is unique up to isomorphism. The norm on $F$
extends uniquely to a norm $|\cdot|_{D}$ on the division algebra by [23,
Proposition 17.6].

######  Notation 2.3.1.

Let $\mathcal{O}_{D}:=\\{d\in\mathcal{O}_{D}:|d|_{D}\leqslant 1\\}$ be the
maximal order in $D$, $\mathcal{P}_{D}:=\\{d\in\mathcal{O}_{D}:|d|_{D}<1\\}$
its unique maximal ideal and let $k_{D}:=\mathcal{O}_{D}/\mathcal{P}_{D}$. We
will write $\omega_{D}:\mathcal{O}_{D}^{\times}\to k_{D}^{\times}$ to denote
the reduction map modulo $\mathcal{P}_{D}$ on multiplicative groups. Let $L$
denote the unramified quadratic field extension of $F$; the reduction map
$\omega_{L}:\mathcal{O}_{L}^{\times}\to k_{L}^{\times}$ on multiplicative
groups is defined similarly.

We recall the Noether–Skolem Theorem.

######  Theorem 2.3.2 (Noether–Skolem).

Suppose that $B$ is a central simple algebra over $F$ and that $A$ is a simple
subalgebra of $B$. If $\chi\colon A\to B$ is an algebra homomorphism then
there is $b\in B^{\times}$ such that $\chi(x)=bxb^{-1}$ for all $x\in A$.

###### Proof.

See [23, Noether–Skolem Theorem 12.6]. ∎

######  Corollary 2.3.3.

$D^{\times}$ acts transitively on the set of $F$-algebra homomorphisms
$\iota\colon L\to D$ via conjugation:

$(d\cdot\iota)(x)=d\hskip 2.84544pt\iota(x)\hskip
2.84544ptd^{-1}\quad\mbox{for all}\quad d\in D^{\times},x\in L.$

Recall the reduced norm map $\operatorname{Nrd}\colon D^{\times}\to
F^{\times}$ from [23, Chapter 16].

######  Proposition 2.3.4.

The derived subgroup of $\mathcal{O}_{D}^{\times}$ is the intersection of

$\ker\left(\operatorname{Nrd}|_{\mathcal{O}_{D}^{\times}}\colon\mathcal{O}_{D}^{\times}\to\mathcal{O}_{F}^{\times}\right)\mbox{
and }\ker\left(\omega_{D}:\mathcal{O}_{D}^{\times}\to k_{D}^{\times}\right).$

###### Proof.

Since $\mathcal{O}_{F}^{\times}$ and $k_{D}^{\times}$ are both abelian groups,
the derived subgroup must be contained in the intersection given in the
statement. By a result of Riehm [26], the two subgroups are in fact equal —
see the remark following [24, Theorem 1.9]. ∎

######  Definition 2.3.5.

Let $P^{1}_{L}$ denote the following subgroup of $\mathcal{O}_{L}^{\times}$:

$P^{1}_{L}:=\ker
N_{L/F}\quad\cap\quad\ker(\omega_{L}:\mathcal{O}_{L}^{\times}\to
k_{L}^{\times}).$

Let
$\varrho:\mathcal{O}_{D}^{\times}\twoheadrightarrow(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$
denote the canonical projection.

######  Proposition 2.3.6.

The $F$-algebra homomorphism $\iota:F\hookrightarrow L$ induces an isomorphism
of profinite abelian groups

$\overline{\varrho\circ\iota}:\mathcal{O}_{L}^{\times}/P^{1}_{L}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}.$

###### Proof.

The projection $\varrho$ appears in the following diagram:

|  |  |
---|---|---|---
$\textstyle{\mathcal{O}_{L}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{N_{L/F}\times\omega_{L}}$$\textstyle{\mathcal{O}_{D}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varrho}$$\scriptstyle{\operatorname{Nrd}\times\omega_{D}}$$\textstyle{(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathcal{O}_{F}^{\times}\times
k_{L}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\times\overline{\iota}}$$\textstyle{\mathcal{O}_{F}^{\times}\times
k_{D}^{\times}}$

We have $\operatorname{Nrd}\circ\iota=N_{L/F}$ by [23, Proposition 16.2(b)].
The inclusion $\iota:L\hookrightarrow D$ induces ring homomorphisms
$\iota:\mathcal{O}_{L}\hookrightarrow\mathcal{O}_{D}$ and
$\overline{\iota}:k_{L}\hookrightarrow k_{D}$ 333In fact, $\overline{\iota}$
is an isomorphism, and we have
$\omega_{D}\circ\iota=\overline{\iota}\circ\omega_{L}$ . We can now see that
the square on the left is commutative. Since $\mathcal{O}_{F}^{\times}\times
k_{D}^{\times}$ is abelian, $\operatorname{Nrd}\times\omega_{D}$ factors
through $\varrho$, giving the diagonal arrow $q$ and making the entire diagram
commutative.

Proposition 2.3.4 tells us that
$\ker\varrho=\ker(\operatorname{Nrd}\times\omega_{D})$, so the map $q$ is
injective. Since $\overline{\iota}$ is injective, chasing the diagram shows
that

$\ker(\varrho\circ\iota)=\ker(q\circ\varrho\circ\iota)=\ker((\operatorname{Nrd}\times\omega_{D})\circ\iota)=\ker(N_{L/F}\times\omega_{L})=P^{1}_{L}.$

Hence
$\varrho\circ\iota:\mathcal{O}_{L}^{\times}\to(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$
descends to give an injective group homomorphism

$\overline{\varrho\circ\iota}:\mathcal{O}_{L}^{\times}/P^{1}_{L}\hookrightarrow(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}.$

Since both $\mathcal{O}_{L}^{\times}/P^{1}_{L}$ and
$(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$ are abelian groups that are
virtually pro-$p$, to show that $\overline{\varrho\circ\iota}$ is surjective,
it suffices to check this on the Sylow pro-$p$ subgroups of both groups, and
on the subgroups of elements of order coprime to $p$.

The Sylow pro-$p$ subgroup $S$ of
$(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$ appears in the commutative
triangle

$\textstyle{1+\pi_{F}\mathcal{O}_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{L/F}\times
1}$$\scriptstyle{\varrho\circ\iota}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{(1+\pi_{F}\mathcal{O}_{F})\times
1}$

where the diagonal arrow is surjective by [30, Chapter V, Proposition 3(a)].
Since $q$ is injective, we see that
$\varrho\circ\iota:1+\pi_{F}\mathcal{O}_{L}\to S$ is surjective as well.

Finally, $\ker\omega_{D}=1+\mathcal{P}_{D}$ is a pro-$p$ subgroup of
$\mathcal{O}_{D}^{\times}$, so
$\varrho:\mathcal{O}_{D}^{\times}\twoheadrightarrow(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$
induces a surjective homomorphism
$k_{D}^{\times}\twoheadrightarrow(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}[p^{\prime}]$.
Since $\overline{\iota}:k_{L}\to k_{D}$ is an isomorphism, this implies that
$\varrho\circ
i:\mathcal{O}_{L}^{\times}[p^{\prime}]\to(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}[p^{\prime}]$
is surjective as well. ∎

######  Corollary 2.3.7.

Let $\iota:L\hookrightarrow D$ be an $F$-algebra homomorphism.

  1. (a)

There is an isomorphism of abelian groups

$\overline{\varrho\circ\iota}^{\ast}:\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}(\mathcal{O}_{L}^{\times}/P^{1}_{L},K^{\times}).$

  2. (b)

This induces a bijection

$\overline{\overline{\varrho\circ\iota}^{\ast}}:\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})/D^{\times}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Hom}(\mathcal{O}_{L}^{\times}/P^{1}_{L},K^{\times})/\operatorname{Gal}(L/F).$

  3. (c)

The bijection $\overline{\overline{\varrho\circ\iota}^{\ast}}$ does not depend
on the choice of $\iota$.

###### Proof.

(a) This follows immediately from Proposition 2.3.6.

(b) Let $\operatorname{Gal}(L/F)=\langle\sigma\rangle$; then, by [23, Theorem
17.10, Proposition 15.1a], we can find an element $\Pi\in D$ such that
$\Pi^{2}=\pi_{F}$ and

$\Pi\hskip 2.84544pt\iota(a)\hskip
2.84544pt\Pi^{-1}=\iota(\sigma(a))\quad\mbox{for all}\quad a\in L.$

Since $\mathcal{O}_{D}^{\times}$ is normal in $D^{\times}$, there is a natural
conjugation action of $D^{\times}$ on
$(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$ which evidently factors
through $D^{\times}/F^{\times}\mathcal{O}_{D}^{\times}$, a group of order $2$
generated by the image of $\Pi$. Then the above formula shows that this action
of $D^{\times}/F^{\times}\mathcal{O}_{D}^{\times}$ on
$(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$ corresponds under the
isomorphism $\overline{\varrho\circ\iota}$ to the natural
$\operatorname{Gal}(L/F)$-action on $\mathcal{O}_{L}^{\times}/P^{1}_{L}$.
Hence, the isomorphism in part (a) is equivariant with respect to the
$D^{\times}/F^{\times}\mathcal{O}_{D}^{\times}$-action on
$\operatorname{Hom}(\mathcal{O}_{D}^{\times},K^{\times})$ and the
$\operatorname{Gal}(L/F)$-action on
$\operatorname{Hom}(\mathcal{O}_{L}^{\times}/P^{1}_{L},K^{\times})$, when we
identify $\operatorname{Gal}(L/F)$ with
$D^{\times}/F^{\times}\mathcal{O}_{D}^{\times}$ via $\sigma\mapsto\Pi$.

(c) Let $\iota^{\prime}:L\hookrightarrow D$ be another $F$-algebra
homomorphism. Then by Corollary 2.3.3 we can find $d\in D^{\times}$ such that
$\iota^{\prime}(x)=d\hskip 2.84544pt\iota(x)\hskip 2.84544ptd^{-1}$ for all
$x\in L$.

Let $c_{d}:\mathcal{O}_{D}^{\times}\to\mathcal{O}_{D}^{\times}$ and
$\overline{c_{d}}:(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}\to(\mathcal{O}_{D}^{\times})^{\operatorname{ab}}$
denote the conjugations by $d$, so that $\iota^{\prime}=c_{d}\circ\iota$ and
$\overline{c_{d}}\circ\varrho=\varrho\circ c_{d}$. Then
$\overline{c_{d}}\circ\varrho\circ\iota=\varrho\circ
c_{d}\circ\iota=\varrho\circ\iota^{\prime}$, so
$\overline{c_{d}}\circ\overline{\varrho\circ\iota}=\overline{\varrho\circ\iota^{\prime}}$.
Hence
$\overline{\varrho\circ\iota^{\prime}}^{\ast}=\overline{\varrho\circ\iota}^{\ast}\circ\overline{c_{d}}^{\ast}$
and we can now see that
$\overline{\overline{\varrho\circ\iota^{\prime}}^{\ast}}=\overline{\overline{\varrho\circ\iota}^{\ast}}$.
∎

###  2.4. Equivariant sheaves and amalgamated products

We begin by recalling some material from [2, §2.3]. Let $X$ be a set equipped
with a Grothendieck topology in the sense of [4, Definition 9.1.1/1]. Note
that we do not assume at the outset that there is a final object in the
category of admissible open subsets of $X$, as $X$ is not itself required to
be admissible open in the $G$-topology.

Let $\operatorname{Homeo}(X)$ be the group of continuous bijections from $X$
to itself. We say that a group $G$ _acts on $X$_ if there is given a group
homomorphism $\rho:G\to\operatorname{Homeo}(X)$. If this action is understood,
we write $gU$ to denote the image of an admissible open subset $U$ of $X$
under the action of $g\in G$. For every $g\in G$, there is an auto-equivalence
$\rho(g)_{\ast}$ of the category of sheaves on $X$, with inverse
$\rho(g)^{\ast}=\rho(g^{-1})_{\ast}$. To simplify the notation, we will simply
denote these auto-equivalences by $g_{\ast}$ and $g^{\ast}$, respectively.
Thus

$(g_{\ast}\mathcal{F})(U)=\mathcal{F}(g^{-1}U)\quad\mbox{and}\quad(g^{\ast}\mathcal{F})(U)=\mathcal{F}(gU)$

for all admissible open subsets $U$ of $X$ and all $g\in G$.

######  Definition 2.4.1.

Let $G$ act on $X$, and let $\mathcal{F}$ be a presheaf of $R$-modules on $X$,
where $R$ is a commutative base ring.

  1. (a)

An _$R$ -linear equivariant structure_ on $\mathcal{F}$ is a set
$\\{g^{\mathcal{F}}:g\in G\\}$, where

$g^{\mathcal{F}}:\mathcal{F}\to g^{\ast}\mathcal{F}$

is a morphism of presheaves of $R$-modules for each $g\in G$, such that

(2) $(gh)^{\mathcal{F}}=h^{\ast}(g^{\mathcal{F}})\circ
h^{\mathcal{F}}\quad\mbox{for all}\quad g,h\in G,\quad\mbox{and}\quad
1^{\mathcal{F}}=1_{\mathcal{F}}.$

  2. (b)

An _$R$ -linear $G$-equivariant presheaf_ is a pair
$(\mathcal{F},\\{g^{\mathcal{F}}\\}_{g\in G})$, where $\mathcal{F}$ is a
presheaf of $R$-modules on $X$, and $\\{g^{\mathcal{F}}\\}_{g\in G}$ is an
$R$-linear equivariant structure on $\mathcal{F}$.

  3. (c)

A _morphism_ of $R$-linear $G$-equivariant presheaves

$\varphi:(\mathcal{F},\\{g^{\mathcal{F}}\\})\to(\mathcal{F}^{\prime},\\{g^{\mathcal{F}^{\prime}}\\})$

is a morphism of presheaves of $R$-modules
$\varphi:\mathcal{F}\to\mathcal{F}^{\prime}$ such that

$g^{\ast}(\varphi)\circ
g^{\mathcal{F}}=g^{\mathcal{F}^{\prime}}\circ\varphi\quad\mbox{for all}\quad
g\in G.$

We will frequently use this abuse of notation, and simply write $\varphi(x)$
to mean $\varphi(U)(x)$ if $x$ is a section of $\mathcal{F}$ over the
admissible open subset $U$ of $X$. Note that with this abuse of notation, the
cocycle condition $(\ref{Cocycl})$ becomes simply

(3) $g^{\mathcal{F}}(h^{\mathcal{F}}(x))=(gh)^{\mathcal{F}}(x)\quad\mbox{for
all}\quad x\in\mathcal{F},g,h\in G.$

When the base ring $R$ and the $R$-linear equivariant structure on a sheaf
$\mathcal{F}$ of $R$-modules is understood, we will simply say that
$\mathcal{F}$ is a _$G$ -equivariant sheaf_, and omit the equivariant
structure from the notation.

######  Definition 2.4.2.

Let $\mathcal{F}$ be a presheaf of $R$-modules on $X$.

  1. (a)

An _automorphism of $\mathcal{F}$ over $X$_ is a pair $(\alpha,\beta)$, where
$\alpha\in\operatorname{Homeo}(X)$ and
$\beta:\mathcal{F}\to\alpha^{\ast}\mathcal{F}$ is an $R$-linear isomorphism of
presheaves on $X$.

  2. (b)

Define $\operatorname{Aut}(\mathcal{F}/X)$ to be the set of all automorphisms
of $\mathcal{F}$ over $X$.

  3. (c)

Given
$(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})\in\operatorname{Aut}(\mathcal{F}/X)$,
define

$(\alpha_{1},\beta_{1})\square(\alpha_{2},\beta_{2}):=(\alpha_{1}\alpha_{2},\alpha_{2}^{\ast}(\beta_{1})\circ\beta_{2}).$

This is again an element of $\operatorname{Aut}(\mathcal{F}/X)$.

######  Lemma 2.4.3.

Let $\mathcal{F}$ be a presheaf of $R$-modules on $X$. Then the binary
operation $\square$ turns $\operatorname{Aut}(\mathcal{F}/X)$ into a group.

###### Proof.

The identity element is $(1_{X},1_{\mathcal{F}})$. Let
$(\alpha_{i},\beta_{i})$, $i=1,2,3$ be three elements of
$\operatorname{Aut}(\mathcal{F}/X)$. Checking that the operation $\square$ is
associative boils down to the formula

$\alpha_{3}^{\ast}(\alpha_{2}^{\ast}(\beta_{1})\circ\beta_{2})\circ\beta_{3}=(\alpha_{2}\alpha_{3})^{\ast}(\beta_{1})\circ\alpha_{3}^{\ast}(\beta_{2})\circ\beta_{3},$

which is readily verified. The inverse of
$(\alpha,\beta)\in\operatorname{Aut}(\mathcal{F}/X)$ is
$(\alpha^{-1},\alpha_{\ast}(\beta)^{-1})$. ∎

By Definition 2.4.2(c), the first projection map
$\operatorname{pr}_{1}:\operatorname{Aut}(\mathcal{F}/X)\to\operatorname{Homeo}(X)$
is a group homomorphism.

######  Definition 2.4.4.

Let $G$ be a group acting on $X$ via $\rho:G\to\operatorname{Homeo}(X)$, and
let $\mathcal{F}$ be a presheaf of $R$-modules on $X$. Form the fibre product

$\operatorname{Aut}(\mathcal{F}/X/G):=G\quad\underset{\operatorname{Homeo}(X)}{\times}{}\quad\operatorname{Aut}(\mathcal{F}/X)$

with respect to the group homomorphisms

$\rho:G\to\operatorname{Homeo}(X)\quad\mbox{and}\quad\operatorname{pr}_{1}:\operatorname{Aut}(\mathcal{F}/X)\to\operatorname{Homeo}(X).$

By definition, the elements of $\operatorname{Aut}(\mathcal{F}/X/G)$ have the
form $(g,(\rho(g),\beta))$ for some $g\in G$ and some $R$-linear isomorphism
$\beta:\mathcal{F}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}g^{\ast}\mathcal{F}$;
evidently, such an element is completely determined by the pair $(g,\beta)$.
In order to simplify the notation, we will abuse notation and write

$\operatorname{Aut}(\mathcal{F}/X/G)=\left\\{(g,\beta):g\in
G,\quad\beta:\mathcal{F}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}g^{\ast}\mathcal{F}\mbox{
}R-\mbox{linear}\right\\},$

where the product is given by the formula

(4)
$(g_{1},\beta_{1})\square(g_{2},\beta_{2})=(g_{1}g_{2},g_{2}^{\ast}(\beta_{1})\circ\beta_{2}).$

######  Definition 2.4.5.

Let $G$ be a group acting on $X$ via $\rho:G\to\operatorname{Homeo}(X)$, and
let $\mathcal{F}$ be a presheaf of $R$-modules on $X$. Define

$\mathcal{S}(G,\mathcal{F}):=\left\\{\sigma\in\operatorname{Hom}(G,\operatorname{Aut}(\mathcal{F}/X/G)):\operatorname{pr}_{1}\circ\sigma=1_{G}\right\\}$

to be the set of sections of the first projection
$\operatorname{pr}_{1}:\operatorname{Aut}(\mathcal{F}/X/G)\to G$.

We make these definitions in order to formulate the following

######  Lemma 2.4.6.

Let $G$ be a group acting on $X$ via $\rho:G\to\operatorname{Homeo}(X)$, and
let $\mathcal{F}$ be a presheaf of $R$-modules on $X$. Then the rule

$\\{g^{\mathcal{F}}\\}_{g\in
G}\quad\mapsto\quad\left[g\mapsto(g,g^{\mathcal{F}})\in\operatorname{Aut}(\mathcal{F}/X/G)\right]$

defines a bijection between the set all $R$-linear $G$-equivariant structures
on $\mathcal{F}$ and $\mathcal{S}(G,\mathcal{F})$.

###### Proof.

Let $\\{g^{\mathcal{F}}\\}_{g\in G}$ be an $R$-linear $G$-equivariant
structure on $X$. Define the map
$\sigma:G\to\operatorname{Aut}(\mathcal{F}/X/G)$ by setting
$\sigma(g)=(g,g^{\mathcal{F}})$ for all $g\in G$. Using the cocycle condition
(2), we compute that for all $g,h\in G$ we have

$\sigma(gh)=(gh,h^{\ast}(g^{\mathcal{F}})\circ
h^{\mathcal{F}})=(g,g^{\mathcal{F}})\square(h,h^{\mathcal{F}})=\sigma(g)\square\sigma(h).$

Since $\sigma(1)=(1,1^{\mathcal{F}})=(1,1_{\mathcal{F}})$, we see that
$\sigma$ is a group homomorphism such that
$\operatorname{pr}_{1}\circ\sigma=1_{G}$, that is,
$\sigma\in\mathcal{S}(G,\mathcal{F})$.

Conversely, for each $\sigma\in\mathcal{S}(G,\mathcal{F})$, the set
$\\{\operatorname{pr}_{2}(\sigma(g))\\}_{g\in G}$ forms an $R$-linear
$G$-equivariant structure on $\mathcal{F}$ by reversing the above argument. ∎

Next, we recall the following definitions from [2]:

######  Definition 2.4.7.

Let $G$ act on $X$, and let $\mathcal{A}$ be a sheaf of $R$-algebras on $X$.

  1. (a)

We say that $\mathcal{A}$ is a _$G$ -equivariant sheaf of $R$-algebras_ if
there is given an $R$-linear $G$-equivariant structure
$\\{g^{\mathcal{A}}:g\in G\\}$ such that each $g^{\mathcal{A}}:\mathcal{A}\to
g^{\ast}\mathcal{A}$ is a morphism of sheaves of $R$-algebras.

  2. (b)

Let $\mathcal{A}$ be a $G$-equivariant sheaf of $R$-algebras on $X$. A _$G$
-$\mathcal{A}$-module_ is an $R$-linear $G$-equivariant sheaf $\mathcal{M}$ on
$X$, such that $\mathcal{M}$ is a sheaf of left $\mathcal{A}$-modules and
$g^{\mathcal{M}}(a\cdot m)=g^{\mathcal{A}}(a)\cdot g^{\mathcal{M}}(m)$ for all
$g\in G$, $a\in\mathcal{A},m\in\mathcal{M}$.

We want to study all possible $G$-$\mathcal{A}$-module structures on a given
$\mathcal{A}$-module $\mathcal{M}$.

######  Definition 2.4.8.

Let $G$ act on $X$, let $\mathcal{A}$ be a $G$-equivariant sheaf of
$R$-algebras on $X$ and let $\mathcal{M}$ be an $\mathcal{A}$-module. We
define

$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G):=\left\\{\begin{array}[]{rcl}(g,\beta)&\in&\operatorname{Aut}(\mathcal{M}/X/G):\\\
\beta(a\cdot m)&=&(g\cdot a)\cdot\beta(m)\quad\mbox{for all}\quad
a\in\mathcal{A},m\in\mathcal{M}\end{array}\right\\}.$

######  Lemma 2.4.9.

With the hypotheses of Definition 2.4.8,
$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ is a subgroup of
$\operatorname{Aut}(\mathcal{M}/X/G)$.

###### Proof.

It is clear that $(1,1_{\mathcal{F}})$ lies in
$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$. Let $(g_{1},\beta_{1})$
and $(g_{2},\beta_{2})$ be two elements of
$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$, so that

(5) $\beta_{i}(a\cdot m)=(g_{i}\cdot a)\cdot\beta_{i}(m)\quad\mbox{for
all}\quad a\in\mathcal{A},m\in\mathcal{M},i=1,2.$

Let $(g_{3},\beta_{3}):=(g_{1},\beta)\square(g_{2},\beta_{2})$ so that
$g_{3}=g_{1}g_{2}$ and $\beta_{3}=g_{2}^{\ast}(\beta_{1})\circ\beta_{2}$. On
local sections, $\beta_{3}$ is simply the composition $\beta_{1}\beta_{2}$.
For any $a\in\mathcal{A}$ and $m\in\mathcal{M}$, we use the fact that
$\mathcal{A}$ is a $G$-equivariant sheaf together with $(\ref{BetaiAM})$ to
compute

$\begin{array}[]{lll}\beta_{3}(a\cdot m)&=&\beta_{1}(\beta_{2}(a\cdot
m))=\beta_{1}((g_{2}\cdot a)\cdot\beta_{2}(m))=(g_{1}\cdot(g_{2}\cdot
a))\cdot\beta_{1}(\beta_{2}(m))\\\ &=&((g_{1}g_{2})\cdot
a)\cdot(\beta_{1}\beta_{2}(m))=(g_{3}\cdot a)\cdot\beta_{3}(m).\end{array}$

So, $(g_{3},\beta_{3})\in\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$
and $\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ is closed under
composition.

To show that $\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ is stable
under inversion in $\operatorname{Aut}(\mathcal{M}/X/G)$, let
$(g,\beta)\in\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$. Then for
$b:=g^{-1}\cdot a\in\mathcal{A}$ and $w:=\beta^{-1}(v)\in\mathcal{M}$ we have
$\beta(b\cdot w)=(g\cdot b)\cdot\beta(w)=a\cdot v.$ Applying $\beta^{-1}$ to
this equation gives $\beta^{-1}(a\cdot v)=b\cdot w=(g^{-1}\cdot
a)\cdot\beta^{-1}(v)$, so
$(g,\beta)^{-1}=(g^{-1},g_{\ast}(\beta^{-1}))\in\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$.
∎

######  Definition 2.4.10.

With the hypotheses of Definition 2.4.8, define

$\mathcal{S}_{\mathcal{A}}(G,\mathcal{M}):=\left\\{\sigma\in\operatorname{Hom}(G,\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)):\operatorname{pr}_{1}\circ\sigma=1_{G}\right\\}$

to be the set of sections of the first projection
$\operatorname{pr}_{1}:\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)\to
G$.

We can now give the generalisation of Lemma 2.4.6 to the case of
$\mathcal{A}$-modules: the proof is completely straightforward and is
therefore omitted.

######  Proposition 2.4.11.

Assume the hypotheses of Definition 2.4.8. Then

$\\{g^{\mathcal{M}}\\}_{g\in
G}\quad\mapsto\quad\left[g\mapsto(g,g^{\mathcal{M}})\in\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G))\right]$

defines a bijection between the set all $G$-$\mathcal{A}$-module structures on
$\mathcal{M}$ extending the given $\mathcal{A}$-module structure on
$\mathcal{M}$, and $\mathcal{S}_{\mathcal{A}}(G,\mathcal{M})$.

Next, we briefly study the functorialities of
$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ and
$\mathcal{S}_{\mathcal{A}}(G,\mathcal{M})$.

######  Lemma 2.4.12.

Assume the hypotheses of Definition 2.4.8, and let $H$ be a subgroup of $G$.

  1. (a)

$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/H)$ is a subgroup of
$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$.

  2. (b)

Restriction of functions induces a map
$\operatorname{Res}^{G}_{H}:\mathcal{S}_{\mathcal{A}}(G,\mathcal{M})\to\mathcal{S}_{\mathcal{A}}(H,\mathcal{M})$.

  3. (c)

For any subgroup $J$ of $H$, we have
$\operatorname{Res}^{G}_{J}=\operatorname{Res}^{H}_{J}\circ\operatorname{Res}^{G}_{H}$.

###### Proof.

(a) An element of $\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/H)$ is a
pair $(h,\beta)$ where $h\in H$ and $\beta:\mathcal{M}\to h^{\ast}\mathcal{M}$
is an $R$-linear isomorphism of sheaves such that $\beta(a\cdot m)=(h\cdot
a)\cdot\beta(m)$ for all $a\in\mathcal{A}$ and $m\in\mathcal{M}$. Evidently
such a pair is also an element of
$\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$.

(b) Given a group homomorphism
$\sigma:G\to\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ such that
$\operatorname{pr}_{1}\circ\sigma=1_{G}$, the restriction
$\sigma|_{H}:H\to\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ takes
values in $\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/H)$. It is still a
group homomorphism, and
$\operatorname{pr}_{1}\circ\sigma|_{H}=(\operatorname{pr}_{1}\circ\sigma)|_{H}=(1_{G})_{H}=1_{H}$.
Hence $\sigma\mapsto\sigma|_{H}$ defines the required function
$\operatorname{Res}^{G}_{H}:\mathcal{S}_{\mathcal{A}}(G,\mathcal{M})\to\mathcal{S}_{\mathcal{A}}(H,\mathcal{M})$.

(c) This is clear from the definitions. ∎

We now come to the application of the above formalism. Suppose that the group
$G$ is equal to an amalgamated product

$G=A\underset{C}{\ast}B$

of its subgroups $A$ and $B$, along their common subgroup $C$. Using Lemma
2.4.12, we see that sending
$\sigma\in\mathcal{S}_{\mathcal{A}}(G,\mathcal{M})$ to the pair
$(\operatorname{Res}^{G}_{A}(\sigma),\operatorname{Res}^{G}_{B}(\sigma))$
defines a function

(6)
$\mathcal{S}_{\mathcal{A}}(G,\mathcal{M})\longrightarrow\mathcal{S}_{\mathcal{A}}(A,\mathcal{M})\underset{\mathcal{S}_{\mathcal{A}}(C,\mathcal{M})}{\times}{}\mathcal{S}_{\mathcal{A}}(B,\mathcal{M}).$

######  Theorem 2.4.13.

Let $G$ be a group acting on $X$, let $\mathcal{A}$ be a $G$-equivariant sheaf
of $R$-algebras on $X$ and let $\mathcal{M}$ be an $\mathcal{A}$-module.
Suppose further that $G$ is equal to the amalgamated product
$G=A\underset{C}{\ast}B$ of its subgroups $A$ and $B$ along their common
subgroup $C$. Then the map $(\ref{SGM})$ is a bijection.

###### Proof.

Using Lemma 2.4.12(a), we have the commutative diagram of groups and group
homomorphisms

$\textstyle{\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G).}$$\textstyle{\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Let $\sigma_{1},\sigma_{2}\in\mathcal{S}_{\mathcal{A}}(G,\mathcal{M})$ be such
that
$\operatorname{Res}^{G}_{A}(\sigma_{1})=\operatorname{Res}^{G}_{A}(\sigma_{2})$
and
$\operatorname{Res}^{G}_{B}(\sigma_{1})=\operatorname{Res}^{G}_{B}(\sigma_{2})$.
Using the above diagram, we may regard $\sigma_{1}$ and $\sigma_{2}$ having
the same codomain $\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$. Then
$(\sigma_{1})|_{A}=(\sigma_{2})|_{A}$ and
$(\sigma_{1})|_{B}=(\sigma_{2})|_{B}$. Since $A$ and $B$ generate $G$ as a
group, it follows that $\sigma_{1}=\sigma_{2}$.

Suppose now that $(\tau,\psi)$ is an element of the fibre product on the right
hand side of $(\ref{SGM})$. Then
$\tau:A\to\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ and
$\psi:B\to\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$ have the same
restriction to $C$. By the universal property of amalgamated products — see
[31, equation $(\ast),\S$ 1.1] — $\tau$ and $\psi$ extend to a unique group
homomorphism $\sigma:G\to\operatorname{Aut}_{\mathcal{A}}(\mathcal{M}/X/G)$
such that $\sigma|_{A}=\tau$ and $\sigma|_{B}=\psi$. Then
$(\operatorname{pr}_{1}\circ\sigma)|_{A}=\operatorname{pr}_{1}\circ(\sigma|_{A})=\operatorname{pr}_{1}\circ\tau=1_{A}$
and
$(\operatorname{pr}_{1}\circ\sigma)|_{B}=\operatorname{pr}_{1}\circ(\sigma|_{B})=\operatorname{pr}_{1}\circ\psi=1_{B}$
because $\tau\in\mathcal{S}_{\mathcal{A}}(\mathcal{M}/X/A)$ and
$\psi\in\mathcal{S}_{\mathcal{A}}(\mathcal{M}/X/B)$. Since $A$ and $B$
generate $G$ as a group, and the group homomorphism
$\operatorname{pr}_{1}\circ\sigma:G\to G$ fixes both $A$ and $B$ pointwise, we
conclude that $\operatorname{pr}_{1}\circ\sigma=1_{G}$. So,
$\sigma\in\mathcal{S}_{\mathcal{A}}(\mathcal{M}/X/G)$,
$\operatorname{Res}^{G}_{A}(\sigma)=\tau$ and
$\operatorname{Res}^{G}_{B}(\sigma)=\psi$. ∎

To spell out the meaning of Theorem 2.4.13 together with Proposition 2.4.11:
the data of a $G$-$\mathcal{A}$-module structure on $\mathcal{M}$ is
equivalent to the data of an $A$-$\mathcal{A}$-module structure and a
$B$-$\mathcal{A}$-module structures whose restrictions to a
$C$-$\mathcal{A}$-module structure agree.

##  3\. Topics in rigid analytic geometry

###  3.1. Line bundles with flat connection on smooth rigid spaces

Let $X$ be a smooth rigid $K$-analytic space. By a _line bundle with flat
connection_ we mean a $\mathcal{D}$-module $\mathscr{L}$ on $X$ which is
invertible as an $\mathcal{O}$-module.

If $\mathscr{L}$ and $\mathscr{M}$ are two line bundles with flat connection
on $X$, then so are $\mathscr{L}\otimes_{\mathcal{O}}\mathscr{M}$ and
$\mathscr{L}^{\otimes-1}:=\mathpzc{Hom}_{\mathcal{O}}(\mathscr{L},\mathcal{O})$:
the tangent sheaf $\mathcal{T}_{X}$ acts via the Leibniz rule on
$\mathscr{L}\otimes_{\mathcal{O}}\mathscr{M}$, and on
$\mathscr{L}^{\otimes-1}$ via the rule

$(v\cdot f)(\ell)=v\cdot f(\ell)-f(v\cdot\ell)\quad\mbox{for all}\quad
v\in\mathcal{T}_{X},f\in\mathscr{L}^{\otimes-1},v\in\mathscr{L}.$

######  Definition 3.1.1.

  1. (a)

$\operatorname{PicCon}(X)$ denotes the abelian group of isomorphism classes of
line bundles with flat connection on $X$ under the operation
$-\otimes_{\mathcal{O}}$-.

  2. (b)

$\operatorname{Con}(X):=\ker(\operatorname{PicCon}(X)\to\operatorname{Pic}(X))$
denotes the group of isomorphism classes of line bundles with flat connection
on $X$ that are trivial after forgetting the connection.

We now show that when $X$ is connected, $\mathscr{L}$ is a simple
$\mathcal{D}_{X}$-module for any $[\mathscr{L}]\in\operatorname{PicCon}(X)$.
We start with the case where $X$ is $K$-affinoid.

######  Lemma 3.1.2.

Suppose that $X$ is a connected $K$-affinoid variety such that
$\mathcal{T}_{X}$ is a free $\mathcal{O}_{X}$-module. Then for every
$[\mathscr{L}]\in\operatorname{Con}(X)$, $\mathscr{L}(X)$ is a simple
$\mathcal{D}(X)$-module.

###### Proof.

Suppose that $n=\dim X$. Following the formalism of [19, §1] we see that
$\mathcal{O}(X)$ satisfies the conditions found in §1.1.2 of loc. cit. Let
$\partial_{1},\ldots,\partial_{n}$ denote a free generating set for
$\mathcal{T}(X)$ as a $\mathcal{O}(X)$-module so we may consider

$\mathcal{D}(X)=\mathcal{O}(X)[\partial_{1},\ldots,\partial_{n}]$

as a filtered $K$-algebra with associated graded ring

$\operatorname{gr}\mathcal{D}(X)\cong\mathcal{O}(X)[T_{1},\ldots,T_{n}]$

with $\mathcal{O}(X)$ in degree $0$ and $T_{1},\ldots,T_{n}$ in degree $1$,
the principal symbols of $\partial_{1},\ldots,\partial_{n}$ respectively.

The filtration of $\mathscr{L}(X)$ whose $0$th filtered part is
$\mathscr{L}(X)$ and whose $-1$st filtered part is $0$ is a good filtration,
so that

$\operatorname{gr}\mathscr{L}(X)\cong\mathcal{O}(X)[T_{1},\ldots,T_{n}]/(T_{1},\ldots,T_{n}).$

Since $X$ is also connected, $\mathcal{O}(X)$ is an integral domain by the
proof of [3, Proposition 4.2]. Thus any non-zero proper
$\mathcal{D}(X)$-module quotient of $\mathscr{L}(X)$ must have dimension $<n$.
However, by [19, Théorème 1.1.4, Corollaire 1.2.3], no such
$\mathcal{D}(X)$-module can exist and so $\mathscr{L}(X)$ is a simple
$\mathcal{D}(X)$-module as claimed. ∎

######  Proposition 3.1.3.

Suppose $X$ is connected. Then every $\mathscr{L}\in\operatorname{PicCon}(X)$
is simple as a $\mathcal{D}$-module.

###### Proof.

Suppose that $[\mathscr{L}]\in\operatorname{PicCon}(X)$. Since $\mathscr{L}$
is a line bundle, there is an admissible cover $\mathcal{U}$ of $X$ consisting
of $K$-affinioid subdomains such that the line bundle $\mathscr{L}|_{U}$ is
trivial for all $U\in\mathcal{U}$. By passing to a refinement, we may also
assume that for each $U\in\mathcal{U}$, $U$ is connected and that
$\mathcal{T}|_{U}$ is a free $\mathcal{O}_{U}$-module.

Suppose that $\mathcal{M}$ is a subobject of $\mathscr{L}$ as a
$\mathcal{D}_{X}$-module and consider

$\mathcal{V}_{1}:=\\{U\in\mathcal{U}:\mathcal{M}(U)=\mathscr{L}(U)\\}\mbox{
and }\mathcal{V}_{2}=\\{U\in\mathcal{U}:\mathcal{M}(U)=0\\}.$

Then $\mathcal{U}$ is the disjoint union of $\mathcal{V}_{1}$ and
$\mathcal{V}_{2}$ by Lemma 3.1.2. Now if $U\in\mathcal{V}_{1}$ and
$V\in\mathcal{V}_{2}$ with $U\cap V\neq\emptyset$, then

$\mathcal{L}(U\cap V)\cong\mathcal{O}(U\cap
V)\otimes_{\mathcal{O}(U)}\mathcal{M}(U)\cong\mathcal{M}(U\cap
V)\cong\mathcal{O}(U\cap V)\otimes_{\mathcal{O}(V)}\mathcal{M}(V)=0,$

a contradiction. Since $X$ is connected if follows that
$\mathcal{U}=\mathcal{V}_{1}$ or $\mathcal{U}=\mathcal{V}_{2}$. Hence
$\mathcal{M}=\mathscr{L}$ or $\mathcal{M}=0$ as required. ∎

The following result enables us to characterise the trivial line bundle with
trivial connection in terms of its horizontal sections, at least when $X$ is
quasi-Stein and geometrically connected.

######  Proposition 3.1.4.

Suppose that $X$ is quasi-Stein and geometrically connected and
$[\mathscr{L}]\in\operatorname{PicCon}(X)$. Then
$\mathscr{L}(X)^{\mathcal{T}(X)=0}=K$ if and only if $\mathscr{L}$ is the
trivial line bundle with trivial flat connection.

###### Proof.

First we show that if $\mathcal{O}$ is equipped with the trivial connection
then

$\mathcal{O}(X)^{\mathcal{T}(X)=0}=K.$

Since $X$ is smooth over $K$, by [5, Proposition 2.7] we can find an affinoid
subdomain $U$ of $X$ which admits an étale map $g$ to a polydisc
$\operatorname{Sp}K\langle t_{1},\ldots,t_{n}\rangle$. By recentering the
disc, we can find a point $x\in U$ such that $g(x)$ is the origin
$t_{1}=\cdots=t_{n}=0$ in this polydisc. Consider the completion
$\widehat{\mathcal{O}_{X,x}}$ of the local ring $\mathcal{O}_{X,x}$ at $x$.
Since $X$ is connected, the restriction map
$\mathcal{O}(X)\to\widehat{\mathcal{O}_{X,x}}$ is injective. This completed
local ring $\widehat{\mathcal{O}_{X,x}}$ is isomorphic to a power-series ring
$K(x)[[t_{1},\ldots,t_{n}]]$ where $K(x)$ is the residue field of
$\mathcal{O}_{X,x}$ at $x$. Since $X$ is quasi-Stein, the local vector fields
$\mathcal{T}_{X,x}=\bigoplus_{i=1}^{n}\mathcal{O}_{X,x}\partial_{t_{i}}$ are
generated as an $\mathcal{O}_{X,x}$-module by $\mathcal{T}(X)$, and then

$\mathcal{O}(X)^{\mathcal{T}(X)=0}\quad\subset\quad\widehat{\mathcal{O}_{X,x}}^{\mathcal{T}_{X,x}=0}\quad\cong\quad
K(x)[[t_{1},\ldots,t_{n}]]^{\partial_{t_{1}}=\cdots=\partial_{t_{n}}=0}=K(x).$

Note that $\mathcal{O}(X)^{\mathcal{T}(X)=0}$ is a $K$-subalgebra of
$\mathcal{O}(X)$. Since $\dim_{K}K(x)<\infty$, the $K$-subalgebra
$\mathcal{O}(X)^{\mathcal{T}(X)=0}$ of $K(x)$ is a finite field extension of
$K$, $L$ say. If $L$ was a proper field extension of $K$ then base changing
$X$ to $L$ would yield non-trivial idempotents in $L\otimes
L\subset\mathcal{O}(X)\otimes L=\mathcal{O}(X_{L})$ and show that $X_{L}$ is
not connected. Since $X$ was assumed to be geometrically connected, we
conclude that $L=\mathcal{O}(X)^{\mathcal{T}(X)=0}$ is in fact equal to $K$.

For the converse we choose $0\neq v\in\mathcal{L}(X)^{\mathcal{T}(X)=0}$,
which is possible by assumption. We may use $v$ to construct a morphism of
$\mathcal{D}_{X}$-modules $\varphi\colon\mathcal{O}\to\mathscr{L}$; $f\mapsto
fv$. Since $v\neq 0$, $\varphi\neq 0$. Since $\mathcal{O}$ and $\mathscr{L}$
are both simple by Proposition 3.1.3 it follows that $\varphi$ is an
isomorphism as required. ∎

It follows that an isomorphism between two line bundles with flat connection
is unique up to scalars. More precisely we have the following result.

######  Corollary 3.1.5.

Suppose that $X$ is quasi-Stein and geometrically connected. Let
$\varphi_{1},\varphi_{2}:\mathscr{L}_{1}\to\mathscr{L}_{2}$ be two
isomorphisms between two line bundles with flat connection on $X$. Then there
is a scalar $\lambda\in K^{\times}$ such that
$\varphi_{2}=\lambda\varphi_{1}$.

###### Proof.

By tensoring $\varphi_{1}$ and $\varphi_{2}$ by $\mathscr{L}_{2}^{\otimes-1}$
we may assume that $\mathscr{L}_{2}=\mathcal{O}$. But then
$\varphi_{1}\circ\varphi_{2}^{-1}:\mathcal{O}\to\mathcal{O}$ is a
$\mathcal{D}$-linear isomorphism, so by $(\ref{HomDOL})$ is given by
multiplication by a non-zero element of $\mathcal{O}(X)^{\mathcal{T}(X)=0}$.
Now apply Proposition 3.1.4. ∎

######  Corollary 3.1.6.

Suppose that $X$ is quasi-Stein and that $(X_{n})_{n\geqslant 0}$ is an
increasing admissible cover of $X$ by geometrically connected affinoid
subdomains. Then the family of restriction maps
$\operatorname{PicCon}(X)\to\operatorname{PicCon}(X_{n})$ induce a natural
isomorphism

$\operatorname{PicCon}(X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\operatorname{PicCon}(X_{n}).$

###### Proof.

Suppose for a contradiction that $X$ is not geometrically connected. Then
$X_{\mathbf{C}}$ has two non-empty families of non-empty admissible open
subsets $\mathcal{U}$ and $\mathcal{V}$ such that $\mathcal{U}\cup\mathcal{V}$
is an admissible cover of $X_{\mathbf{C}}$ and

$\bigcup_{U\in\mathcal{U}}U\cap\bigcup_{V\in\mathcal{V}}V=\emptyset.$

Since $(X_{n,\mathbf{C}})_{n\geqslant 0}$ is an admissible cover of $X$ that
is ordered by inclusion, we can find $n\geqslant 0$, $U\in\mathcal{U}$ and
$V\in\mathcal{V}$ such that $X_{n,\mathbf{C}}\cap U$ and $X_{n,\mathbf{C}}\cap
V$ are both non-empty. But then $\\{U\cap X_{n,\mathbf{C}}:U\in\mathcal{U}\\}$
and $\\{V\cap X_{n,\mathbf{C}}:V\in\mathcal{V}\\}$ together form an admissible
cover of $X_{n,\mathbf{C}}$ that disconnects it, giving the required
contradiction.

Now suppose that $[\mathscr{L}]\in\operatorname{PicCon}(X)$ is such that
$[\mathscr{L}|_{X_{n}}]=0\in\operatorname{PicCon}(X_{n})$ for all $n\geqslant
0$. Then $\mathscr{L}^{\mathcal{T}=0}$ is a subsheaf of $\mathscr{L}$ with
$\mathscr{L}(X_{n})^{\mathcal{T}=0}=K$ for all $n\geqslant 0$ by Proposition
3.1.4. Moreover the restriction maps
$\mathscr{L}(X_{n+1})^{\mathcal{T}=0}\to\mathscr{L}(X_{n})^{\mathcal{T}=0}$
are all injective, and thus isomorphisms, since the restriction maps
$\mathcal{O}(X_{n+1})\to\mathcal{O}(X_{n})$ are all injective. Thus by the
sheaf condition on $\mathscr{L}^{\mathcal{T}=0}$,
$\mathscr{L}(X)^{\mathcal{T}=0}=K$. Thus by Proposition 3.1.4 again we can
deduce that $[\mathscr{L}]=0\in\operatorname{PicCon}(X)$ i.e. the homomorphism
in the statement is injective.

To see that the homomorphism is also surjective we consider a family of line
bundles with connection $[\mathscr{L}_{n}]_{n\geqslant 0}\in\prod_{n\geqslant
0}\operatorname{PicCon}(X_{n})$ such that

$[\mathscr{L}_{n}|_{X_{m}}]=[\mathscr{L}_{m}]\mbox{ in
}\operatorname{PicCon}(X_{m})\quad\mbox{whenever}\quad n\geqslant m.$

Choose isomorphisms of $\mathcal{D}$-modules
$\varphi_{m}\colon\mathscr{L}_{m+1}|_{X_{m}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{L}_{m}$
for all $m\geqslant 0$. Then whenever $n\geqslant m$ we can define an
isomorphism of sheaves
$\varphi_{n,m}\colon\mathscr{L}_{n}|_{X_{m}}\to\mathscr{L}_{m}$ by

$\varphi_{n,m}:=\varphi_{n-1}|_{X_{m}}\circ\cdots\circ\varphi_{m}|_{X_{m}}.$

It is then easy to verify that the construction in [12, §4.4] gives a sheaf
$\mathscr{L}$ of $\mathcal{D}$-modules on $X$ together with isomorphisms of
$\mathcal{D}$-modules
$\mathscr{L}|_{X_{n}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{L}_{n}$.
Then $\mathscr{L}$ is a line bundle on $X$, since each $\mathscr{L}(X_{n})$ is
free of rank $1$ over $\mathcal{O}(X_{n})$. The family
$[\mathscr{L}_{n}]\in\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\operatorname{PicCon}(X_{n})$
is then the required image of $[\mathscr{L}]$. ∎

The following result will also be useful.

######  Corollary 3.1.7.

Suppose that $X$ is quasi-Stein and geometrically connected, and that $d$ is a
positive integer. Then $\mathcal{O}(X)^{\times}/K^{\times}$ has no
$d$-torsion.

###### Proof.

Suppose that $f\in\mathcal{O}(X)^{\times}$ with $f^{d}\in K^{\times}$. Then
for every $\partial\in\mathcal{T}(X)$ we see that
$0=\partial(f^{d})=df^{d-1}\partial(f)$. Since
$df^{d-1}\in\mathcal{O}(X)^{\times}$ it follows that $\partial(f)=0$. Thus by
Proposition 3.1.4, $f\in K^{\times}$. ∎

######  Lemma 3.1.8.

Suppose that a group $G$ acts on $X$. Then $G$ acts naturally on
$\operatorname{PicCon}(X)$ by abelian group automorphisms via

$g\cdot[\mathscr{L}]=[g_{\ast}\mathscr{L}]$

where $g_{\ast}\mathscr{L}$ is a $\mathcal{D}$-module via the ring isomorphsim
$(g^{-1})^{\mathcal{D}}\colon\mathcal{D}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}g_{\ast}\mathcal{D}$.

Moreover $\operatorname{Con}(X)$ is a $G$-stable subgroup of
$\operatorname{PicCon}(X)$.

###### Proof.

We can check that
$g\cdot([\mathscr{L}][\mathscr{M}])=(g\cdot[\mathscr{L}])(g\cdot[\mathscr{M}])$
for any $[\mathscr{L}],[\mathscr{M}]\in\operatorname{PicCon}(X)$, because
$g_{\ast}(\mathscr{L}\otimes_{\mathcal{O}}\mathscr{M})$ is naturally
isomorphic to
$(g_{\ast}\mathscr{L})\otimes_{g_{\ast}\mathcal{O}}(g_{\ast}\mathscr{M})$ as a
$g_{\ast}\mathcal{D}$-module.

Since $g_{\ast}\mathscr{L}$ is trivial as a line bundle whenever $\mathscr{L}$
is trivial as a line bundle the last part is immediate. ∎

######  Remark 3.1.9.

We note more generally, in the context of Lemma 3.1.8, that if $U$ is an
admissible open subspace of $X$, each $g\in G$ induces a group isomorphism
$\operatorname{PicCon}(U)\to\operatorname{PicCon}(g\cdot U)$;
$[\mathscr{L}]\mapsto[g_{\ast}\mathscr{L}]$ where again $g_{\ast}\mathscr{L}$
is a $\mathcal{D}_{gU}$-module via the ring isomorphism
$(g^{-1})^{\mathcal{D}}\colon\mathcal{D}_{gU}\to g_{\ast}\mathcal{D}_{U}$.
Moreover these restrict to isomorphisms
$\operatorname{Con}(U)\to\operatorname{Con}(g\cdot U)$.

We will now construct some connections on the trivial line bundle by using
units.

######  Lemma 3.1.10.

Suppose that $d$ is a positive integer and $u\in\mathcal{O}(X)^{\times}$. Then
there is a unique element $[\mathscr{L}_{u,d}]$ of $\operatorname{Con}(X)$
such that $\mathscr{L}_{u,d}$ has a free generator $v$ as a
$\mathcal{O}$-module with $\partial(v)=\frac{1}{d}\frac{\partial(u)}{u}v$ for
all $\partial\in\mathcal{T}$.

###### Proof.

Suppose that $\mathscr{L}=\mathcal{O}v$ is a line bundle with a flat
connection satisfying

$\partial(v)=\frac{1}{d}\frac{\partial(u)}{u}v$

for all $\partial\in\mathcal{T}$. For all $f\in\mathcal{O}$ and
$\partial\in\mathcal{T}$, necessarily

$\partial(fv)=\left(\partial(f)+\frac{1}{d}\frac{\partial(u)}{u}f\right)v$

so as $\mathcal{D}$ is generated by $\mathcal{O}$ and $\mathcal{T}$ there is
at most one element of $\operatorname{Con}(X)$ with the property given in the
statement. To prove the existence of such a line bundle with flat connection
it suffices to show that for all $\partial_{1},\partial_{2}\in\mathcal{T}$

$(\partial_{1}\partial_{2}-\partial_{2}\partial_{1})(v)=[\partial_{1}\partial_{2}](v)$

where $[--]$ denotes the Lie bracket on $\mathcal{T}$.

But

$\displaystyle\partial_{1}\partial_{2}(v)$ $\displaystyle=$
$\displaystyle\partial_{1}\left(\frac{1}{d}\frac{\partial_{2}(u)}{u}v\right)$
$\displaystyle=$
$\displaystyle\frac{1}{d}\partial_{1}\left(\frac{\partial_{2}(u)}{u}\right)v+\frac{1}{d}\frac{\partial_{2}(u)}{u}\frac{1}{d}\frac{\partial_{1}(u)}{u}v$
$\displaystyle=$
$\displaystyle\frac{1}{d}\frac{u\partial_{1}\partial_{2}(u)-\partial_{2}(u)\partial_{1}(u)}{u^{2}}v+\frac{1}{d}\frac{\partial_{2}(u)}{u}\frac{1}{d}\frac{\partial_{1}(u)}{u}v$

and so

$(\partial_{1}\partial_{2}-\partial_{2}\partial_{1})(v)=\frac{1}{d}\frac{[\partial_{1}\partial_{2}](u)}{u}v$

as required. ∎

######  Proposition 3.1.11.

Let $d$ be a non-zero integer and let $X$ be a geometrically connected,
smooth, quasi-Stein rigid $K$-analytic space. There is a homomorphism of
abelian groups $\mathcal{O}(X)^{\times}\to\operatorname{Con}(X)$ given by
$u\mapsto[\mathscr{L}_{u,d}]$. The kernel of this homomorphism is
$K^{\times}\mathcal{O}(X)^{\times d}$ and its image is
$\operatorname{Con}(X)[d]$.

###### Proof.

Suppose that $u_{1},u_{2}\in\mathcal{O}(X)^{\times}$. Then if, for $i=1,2$,
$v_{i}$ generates $\mathscr{L}_{u_{i},d}$ with
$\partial(v_{i})=\frac{1}{d}\frac{\partial(u_{i})}{u_{i}}$, $v_{1}\otimes
v_{2}$ is a generator of $\mathscr{L}_{u_{1},d}\otimes\mathscr{L}_{u_{2},d}$
and

$\displaystyle\partial(v_{1}\otimes v_{2})$ $\displaystyle=$
$\displaystyle\partial(v_{1})\otimes v_{2}+v_{1}\otimes\partial(v_{2})$
$\displaystyle=$
$\displaystyle\frac{1}{d}\left(\frac{\partial(u_{1})}{u_{1}}+\frac{\partial(u_{2})}{u_{2}}\right)(v_{1}\otimes
v_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{d}\frac{\partial(u_{1}u_{2})}{u_{1}u_{2}}(v_{1}\otimes
v_{2})$

for all $\partial\in\mathcal{T}$. Thus $u\mapsto[\mathscr{L}_{u,d}]$ does
define a homomorphism.

Note that $u\in\mathcal{O}(X)^{\times}$ is in the kernel of the homomorphism
if and only if $[\mathscr{L}_{u,d}]=[\mathcal{O}.v]=[\mathcal{O}]$ in
$\operatorname{Con}(X)$. That is $u$ is in the kernel if and only if there is
$w\in\mathcal{O}(X)^{\times}$ such that $\partial(wv)=0$ for all
$\partial\in\mathcal{T}(X)$. But

$\partial(wv)=\left(\frac{\partial(w)}{w}+\frac{1}{d}\frac{\partial(u)}{u}\right)wv.$

Now

$\frac{\partial(uw^{d})}{uw^{d}}=\frac{\partial(u)}{u}+d\frac{\partial(w)}{w}$

so $\partial(wv)=0$ if and only if $\partial(uw^{d})=0$. Thus by Proposition
3.1.4 the kernel is precisely $K^{\times}\mathcal{O}(X)^{\times d}$ as
claimed. Moreover $[\mathscr{L}_{u,d}^{\otimes
d}]=[\mathscr{L}_{u^{d},d}]=[\mathcal{O}]$ so each $[\mathscr{L}_{u,d}]$ is
indeed $d$-torsion.

Given $[\mathscr{L}]\in\operatorname{Con}(X)[d]$, we use the hypothesis that
$\mathscr{L}$ is trivial as a line bundle to pick a generator
$v\in\mathscr{L}(X)$ as an $\mathcal{O}(X)$-module, and we choose a
$\mathcal{D}$-linear isomorphism $\psi\colon\mathscr{L}^{\otimes
d}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{O}$ using the fact
that $d\cdot[\mathscr{L}]=0$ in $\operatorname{Con}(X)$. We claim that if
$\psi(v^{\otimes d})=u$ then $\partial(v)=\frac{1}{d}\frac{\partial(u)}{u}$
for all $\partial\in\mathcal{T}$ and so $[\mathscr{L}]=[\mathscr{L}_{u,d}]$.
For this we compute that $\partial(v^{\otimes
d})=\frac{\partial(u)}{u}v^{\otimes d}$ for all $\partial\in\mathcal{T}$. But
if $\partial(v)=av$ then $\partial(v^{\otimes d})=dav^{\otimes d}$ so
$a=\frac{1}{d}\frac{\partial(u)}{u}$ as claimed. ∎

######  Definition 3.1.12.

If $X$ is a geometrically connected, smooth, quasi-Stein rigid $K$-analytic
space then we define

$\theta_{d}\colon\operatorname{Con}(X)[d]\to\mathcal{O}(X)^{\times}/K^{\times}\mathcal{O}(X)^{\times
d}$

to be the inverse of the isomorphism induced by the homomorphism in
Proposition 3.1.11.

The proof of surjectivity in Proposition 3.1.11 shows that
$\theta_{d}\left([\mathcal{O}v]\right)$ is determined by the image of
$v^{\otimes d}$ under a $\mathcal{D}$-linear isomorphism
$\psi\colon(\mathcal{O}v)^{\otimes d}\to\mathcal{O}$, via

(7) $\theta_{d}([\mathcal{O}v])=\psi(v^{\otimes
d})K^{\times}\mathcal{O}(X)^{\times d}.$

######  Proposition 3.1.13.

Let $d$ be a non-zero integer and let $X$ be a geometrically connected,
smooth, quasi-Stein rigid $K$-analytic space and let $G$ be a group acting on
$X$. Then $\theta_{d}$ is a $G$-equivariant isomorphism

$\theta_{d}\colon\operatorname{Con}(X)[d]\to\mathcal{O}(X)^{\times}/K^{\times}\mathcal{O}(X)^{\times
d}.$

###### Proof.

Let $g\in G$, $[\mathscr{L}]\in\operatorname{Con}(X)[d]$ and fix a
$\mathcal{D}$-linear isomorphism $\psi:\mathscr{L}^{\otimes
d}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{O}$. The map
$g^{\mathcal{O}}:\mathcal{O}\to g^{\ast}\mathcal{O}$ is a $\mathcal{D}$-linear
isomorphism, so $g^{\mathcal{O}}\circ\psi:\mathscr{L}^{\otimes d}\to
g^{\ast}\mathcal{O}$ is also a $\mathcal{D}$-linear isomorphism. After
identifying $g_{\ast}(\mathscr{L}^{\otimes d})$ with
$(g_{\ast}\mathscr{L})^{\otimes d}$ and $g_{\ast}(g^{\ast}\mathcal{O})$ with
$\mathcal{O}$, we obtain a $g_{\ast}\mathcal{D}$-linear isomorphism

$\psi^{\prime}:=g_{\ast}(g^{\mathcal{O}}\circ\psi):(g_{\ast}\mathscr{L})^{\otimes
d}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{O}.$

Recall Lemma 3.1.8 that $g\cdot[\mathscr{L}]=[g_{\ast}\mathscr{L}]$, where
$\mathcal{D}$ acts on $g_{\ast}\mathscr{L}$ via the ring isomorphism
$(g^{-1})^{\mathcal{D}}:\mathcal{D}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}g_{\ast}\mathcal{D}$.
So $\psi^{\prime}$ becomes an $\mathcal{D}$-linear isomorphism in this way,
and we can use it to compute $\theta_{d}([g_{\ast}\mathscr{L}])$ as follows:
let $v\in\mathscr{L}(X)$ be such that $\mathscr{L}(X)=\mathcal{O}(X)v$; then
by definition of $\psi^{\prime}$ we have $\psi^{\prime}(v^{\otimes
d})=g\cdot\psi(v^{\otimes d})$, so

$\theta_{d}(g\cdot[\mathscr{L}])=\theta_{d}([g_{\ast}\mathscr{L}])=\psi^{\prime}(v^{\otimes
d})K^{\times}\mathcal{O}(X)^{\times d}=g\cdot\psi(v^{\otimes
d})K^{\times}\mathcal{O}(X)^{\times d}=g\cdot\theta_{d}([\mathscr{L}])$

as required. ∎

######  Remark 3.1.14.

Proposition 3.1.13 can be viewed as saying that

$[g_{\ast}\mathscr{L}_{u,d}]=[\mathscr{L}_{g\cdot
u,d}]\in\operatorname{Con}(X)[d].$

More generally if $U$ is an admissible open subset of $X$ and
$u\in\mathcal{O}(U)^{\times}$ then

$[g_{\ast}\mathscr{L}_{u,d}]=[\mathscr{L}_{g\cdot
u,d}]\in\operatorname{Con}(g\cdot U).$

###  3.2. Equivariant line bundles with flat connections

We now turn to a discussion of _equivariant_ line bundles with flat
connection. In this section we will assume that $G$ is a topological group
acting continuously on a smooth rigid $K$-analytic space $X$ in the sense of
[2, Definition 3.1.8]. We first consider $G$-equivariant line bundles. Our
next definition, Definition 3.2.3 below, will require some preparation.

######  Lemma 3.2.1.

Let $\mathscr{M}$ be a coherent $\mathcal{O}$-module on $X$. Suppose that
$\\{g^{\mathscr{M}}\\}_{g\in G}$ is a $G$-equivariant structure on
$\mathscr{M}$. Then for every affinoid subdomain $U$ of $X$ and $g\in G$, the
structure map

$g^{\mathscr{M}}\colon\mathscr{M}(U)\to\mathscr{M}(gU)$

is continuous with respect to canonical $K$-Banach topologies on the domain
and codomain.

###### Proof.

Let $U$ be an affinoid subdomain of $X$. By Kiehl’s Theorem — see, e.g. [12,
Theorem 4.5.2] — $\mathscr{M}(U)$ is a finitely generated module over the
affinoid algebra $\mathcal{O}(U)$ because $\mathscr{M}_{|U}$ is a coherent
$\mathcal{O}_{U}$-module. Recalling from [27, Proposition 2.1] that every
finitely generated module $M$ over an affinoid algebra $A$ carries a canonical
$K$-Banach-space topology, we see that $\mathscr{M}(U)$ carries a canonical
$K$-Banach space topology. Fixing $g\in G$, we can regard $\mathscr{M}(gU)$ as
an $\mathcal{O}(U)$-module via a twisted action, by defining $a\ast v:=(g\cdot
a)v$ for all $a\in\mathcal{O}(U)$ and $v\in\mathscr{M}(gU)$. Then
$\mathscr{M}(gU)$ is still a finitely generated $\mathcal{O}(U)$-module, and
the structure map $g^{\mathscr{M}}(U):\mathscr{M}(U)\to\mathscr{M}(gU)$ is now
a $\mathcal{O}(U)$-linear homomorphism between two finitely generated
$\mathcal{O}(U)$-modules. It is therefore automatically continuous by [12,
Corollary 1.2.4]. ∎

######  Lemma 3.2.2.

Let $\mathscr{M}$ be a coherent $\mathcal{O}_{X}$-module. Suppose that
$\\{g^{\mathscr{M}}\\}_{g\in G}$ is a $G$-equivariant structure on
$\mathscr{M}$ and that $L$ is a finite field extension of $K$. Then for every
$z\in X(L)$ there is a natural group homomorphism

$\phi_{z,\mathscr{M}}\colon G_{z}\to\operatorname{Aut}_{L}(\mathscr{M}(z))$

where $\mathscr{M}(z):=L\otimes_{\mathcal{O}_{X,z}}\mathscr{M}_{z}$ denotes
the fibre of $\mathscr{M}$ at $z$.

###### Proof.

Let $g\in G_{z}$. The $G$-equivariant structure on $\mathcal{O}$ gives us a
local $K$-algebra automorphism
$g^{\mathcal{O}}_{z}:\mathcal{O}_{X,z}\to\mathcal{O}_{X,z}$, whereas the
$G$-equivariant structure on $\mathscr{M}$ gives a $K$-linear automorphism
$g^{\mathscr{M}}_{z}:\mathscr{M}_{z}\to\mathscr{M}_{z}$, satisfying
$g^{\mathscr{M}}_{z}(a.m)=g^{\mathcal{O}}_{z}(a)\cdot g^{\mathscr{M}}_{z}(a)$
for all $a\in\mathcal{O}_{X,z}$ and $m\in\mathscr{M}_{z}$. It is now
straightforward to check that setting

$g\cdot(\lambda\otimes m):=\lambda\otimes g^{\mathscr{M}}_{z}(a)\quad\mbox{for
all}\quad g\in G_{z},\lambda\in L,m\in\mathscr{M}_{z}$

gives a well-defined $L$-linear action of $G_{z}$ on
$L\otimes_{\mathcal{O}_{X,z}}\mathscr{M}_{z}$. ∎

Suppose that $M$ is any $K$-Banach space. Then the $K$-algebra of bounded
$K$-linear endomorphisms $\mathcal{B}(M)$ is also a $K$-Banach algebra through
the operator norm $||T||:=\sup\limits_{v\in
V\backslash\\{0\\}}\frac{|Tv|}{|v|}$, so its unit group
$\mathcal{B}(M)^{\times}$ becomes a topological group — using the geometric
series, one can check that the inversion map on $\mathcal{B}(M)^{\times}$ is
continuous. If $\mathcal{M}\subset M$ is the unit ball in $M$, then the
congruence subgroups of $\mathcal{B}(M)^{\times}$

$\Gamma_{n}(\mathcal{M}):=\\{\gamma\in\mathcal{B}(M)^{\times}:(\gamma-1)(\mathcal{M})\subseteq\pi_{F}^{n}\mathcal{M}\\}$

form a fundamental system of open neighbourhoods of the identity in
$\mathcal{B}(M)^{\times}$. Since any isomorphism of $K$-Banach spaces
$M\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}N$ induces an isomorphism
of topological groups
$\mathcal{B}(M)^{\times}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{B}(N)^{\times}$
via ‘conjugation’, we see that the topology on $\mathcal{B}(M)^{\times}$ only
depends on the topology on $M$ and not on any particular choice of $K$-Banach
norm on $M$.

Let $\mathscr{M}$ be a coherent $\mathcal{O}_{X}$-module and let
$\\{g^{\mathscr{M}}\\}_{g\in G}$ be a $G$-equivariant structure on
$\mathscr{M}$. For each affinoid subdomain $U$ of $X$ and each $g\in G_{U}$,
the maps $g^{\mathscr{M}}(U):\mathscr{M}(U)\to\mathscr{M}(U)$ induce, by Lemma
3.2.1 a homomorphism

$G_{U}\to\mathcal{B}(\mathscr{M}(U))^{\times}.$

######  Definition 3.2.3.

A _$G$ -equivariant line bundle on $X$_ is a $G$-equivariant
$\mathcal{O}_{X}$-module $\mathscr{L}$ on $X$ such that

  1. (a)

$\mathscr{L}$ is invertible as an $\mathcal{O}_{X}$-module, and

  2. (b)

the action map $G_{U}\to\mathcal{B}(\mathscr{L}(U))^{\times}$ is continuous
for every affinoid subdomain $U$ of $X$.

######  Lemma 3.2.4.

$\mathcal{O}_{X}$ with its usual equivariant structure is a $G$-equivariant
line bundle on $X$.

###### Proof.

Let $U$ be an affinoid subdomain of $X$. Consider the sup norm $|\cdot|_{U}$
on $\mathcal{O}(U)$ whose unit ball in $\mathcal{A}:=\mathcal{O}(U)^{\circ}$.
For each $n\geqslant 0$, the congruence subgroup
$\Gamma_{n}(\mathcal{O}(U)^{\circ})$ of $\mathcal{B}(\mathcal{O}(U))^{\times}$
contains the group $\mathcal{G}_{\pi_{F}^{n}}(\mathcal{A})$ appearing on [2,
p. 19]. Through the Raynaud generic fibre functor $\operatorname{rig}$,
$\mathcal{G}_{\pi_{F}^{n}}(\mathcal{A})$ can be identified with a subgroup of
the group of $K$-linear automorphisms
$\operatorname{Aut}_{K}(U,\mathcal{O}_{U})$ of the $G$-ringed topological
space $(U,\mathcal{O}_{U})$. These subgroups of
$\operatorname{Aut}_{K}(U,\mathcal{O}_{U})$ form a filter base for a certain
topology on $\operatorname{Aut}_{K}(U,\mathcal{O}_{U})$ — see [2, Theorem
3.1.5] — and the action map
$G_{U}\to\operatorname{Aut}_{K}(U,\mathcal{O}_{U})$ is continuous with respect
to this topology by [2, Definition 3.1.8], because $G$ is assumed to act
continuously on $X$. It follows that the action map
$G_{U}\to\mathcal{B}(\mathcal{O}(U))^{\times}$ is continuous as required. ∎

A _morphism_ between two $G$-equivariant line bundles $\mathscr{L}$ and
$\mathscr{M}$ is a morphism of $G$-equivariant $\mathcal{O}$-modules in the
sense of [2, Definition 2.3.1(c)]. Given any such morphism
$\varphi:\mathscr{L}\to\mathscr{M}$ and an affinoid subdomain $U$ of $X$, the
map $\varphi(U):\mathscr{L}(U)\to\mathscr{M}(U)$ is then an
$\mathcal{O}(U)$-linear homomorphism between two finitely generated
$\mathcal{O}(U)$-modules, and is therefore automatically continuous.

######  Definition 3.2.5.

We let $\operatorname{Pic}^{G}(X)$ denote the set of isomorphism classes of
$G$-equivariant line bundles on $X$.

######  Lemma 3.2.6.

Let $\mathscr{L}$ and $\mathscr{M}$ be $G$-equivariant line bundles on $X$.
Then so are $\mathscr{L}\otimes_{\mathcal{O}}\mathscr{M}$ and
$\mathscr{L}^{\otimes-1}=\mathpzc{Hom}_{\mathcal{O}}(\mathscr{L},\mathcal{O})$.

###### Proof.

We can easily verify that the usual formula for tensor product and
contragredient representations satisfies Definition 3.2.3. ∎

With respect to these operations $\operatorname{Pic}^{G}(X)$ is an abelian
group with unit given by the structure $\\{g^{\mathcal{O}}\\}$ on
$\mathscr{L}=\mathcal{O}_{X}$.

######  Proposition 3.2.7.

Suppose that $L$ is a finite extension of $K$ and $z\in X(L)$. There is a
natural group homomorphism
$\phi_{z}\colon\operatorname{Pic}^{G}(X)\to\operatorname{Hom}(G_{z},L^{\times})$
given by

$\phi_{z}([\mathscr{L}])=\phi_{z,\mathscr{L}}.$

###### Proof.

We note that if $\mathscr{L}$ is a line bundle on $X$ with a $G$-equivariant
structure then $\mathscr{L}(z)$ is a one-dimensional vector space over the
residue field of the local ring $\mathcal{O}_{X,z}$, and so, by Lemma 3.2.2,
$\phi_{z,\mathscr{L}}$ can be viewed as a homomorphism $G_{z}\to L^{\times}$.

Next we show that $\phi_{z,\mathscr{L}}$ is continuous. To this end we choose
an affinoid subdomain $U$ of $X$ such that $z\colon\operatorname{Sp}(L)\to X$
factors through $U$. Since $G_{z}\cap G_{U}$ is an open subgroup of $G_{z}$ it
suffices to show that $\phi_{z,\mathscr{L}}|_{G_{U}\cap G_{z}}$ is continuous.
Since the natural map $\mathcal{B}(\mathscr{L}(U))^{\times}\to L^{\times}$ is
continuous this follows from Definition 3.2.3(b).

It remains to show that $\phi_{z}$ is a group homomorphism. This is immediate,
because whenever $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$ are elements of
$\operatorname{Pic}^{G}(X)$, there is a canonical isomorphism

$(\mathscr{L}_{1}\otimes_{\mathcal{O}}\mathscr{L}_{2})(z)\to\mathscr{L}_{1}(z)\otimes_{L}\mathscr{L}_{2}(z)$

which is compatible with the $G$-actions. ∎

This discussion leads us on to the following definition.

######  Definition 3.2.8.

A _$G$ -equivariant line bundle with flat connection on $X$_ is a
$G$-equivariant $\mathcal{D}_{X}$-module $\mathscr{L}$, such that when
$\mathscr{L}$ is viewed as a $G$-equivariant $\mathcal{O}_{X}$-module by
restriction, it is a $G$-equivariant line bundle on $X$.

It follows easily from Lemma 3.2.4 that $\mathcal{O}_{X}$ equipped with the
trivial connection and its usual equivariant structure is a $G$-equivariant
line bundle with connection.

We can also typically put other $G$-equivariant structures on the trivial line
bundle with trivial connection by considering
$\operatorname{Hom}(G,K^{\times})$, the abelian group of _continuous_ group
homomorphisms from $G$ to $K^{\times}$.

######  Definition 3.2.9.

Given $\chi\in\operatorname{Hom}(G,K^{\times})$ we define a new
$G$-equivariant structure on $\mathcal{O}$ equipped with the trivial
connection: for each affinoid subdomain $U$ of $X$ and for each $g\in G$ we
define $K$-linear continuous maps

$g^{\mathcal{O}_{\chi}}(U)\colon\mathcal{O}(U)\longrightarrow\mathcal{O}(gU)$

by $f\mapsto\chi(g)g^{\mathcal{O}}(f)$.

This family $\\{g^{\mathcal{O}_{\chi}}\\}$ then defines a $G$-equivariant line
bundle with connection on $X$ in the sense of Definition 3.2.8 that we will
denote by $\mathcal{O}_{\chi}$. Note that condition (b) in Definition 3.2.8
follows from the assumption that $\chi:G\to K^{\times}$ is continuous.

######  Lemma 3.2.10.

Let $\mathscr{L}$ and $\mathscr{M}$ be $G$-equivariant line bundles with flat
connection. Then so are $\mathscr{L}\otimes_{\mathcal{O}}\mathscr{M}$ and
$\mathscr{L}^{\otimes-1}=\mathpzc{Hom}_{\mathcal{O}}(\mathscr{L},\mathcal{O})$.

###### Proof.

We saw at the start of $\S\ref{ELBsec}$ that
$\mathscr{L}\otimes_{\mathcal{O}}\mathscr{M}$ and $\mathscr{L}^{\otimes-1}$
are $\mathcal{D}$-modules on $X$. The usual formula for the tensor product and
contragredient representations allow us to see that they also carry standard
$G$-equivariant $\mathcal{D}$-module structures and Lemma 3.2.6 shows that
this makes them $G$-equivariant line bundles.∎

######  Definition 3.2.11.

We denote the set of isomorphism classes of $G$-equivariant line bundles with
flat connection on $X$ by $\operatorname{PicCon}^{G}(X)$.

In view of Lemma 3.2.10, the operations $-\otimes_{\mathcal{O}}-$ and
$(-)^{\otimes-1}$ endow $\operatorname{PicCon}^{G}(X)$ with the structure of
an abelian group. The unit element in this group is given by the isomorphism
class of $\mathcal{O}_{X}$ equipped with the trivial connection together with
its usual $G$-equivariant structure.

######  Definition 3.2.12.

We define $\operatorname{Con}^{G}(X)$ by

$\operatorname{Con}^{G}(X):=\ker(\operatorname{PicCon}^{G}(X)\to\operatorname{Pic}(X)),$

the group of isomorphism classes of $G$-equivariant line bundles with flat
connection on $X$ that are trivial after forgetting the connection and the
$G$-action.

We record that $\operatorname{Con}$ is functorial in a natural way.

######  Proposition 3.2.13.

Let $U$ be a geometrically connected admissible open subset of $X$, and
suppose that $H$ is a closed subgroup of $G_{U}$. Then for each $g\in G$,

  1. (a)

the map $[\mathscr{L}]\mapsto[g_{\ast}\mathscr{L}]$ induces a natural
isomorphism

$g\colon\operatorname{PicCon}^{H}(U)\quad\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\quad\operatorname{PicCon}^{{}^{g}H}(gU);$

  2. (b)

for $L$ a finite extension of $K$ and $z\in U(L)$, the following diagram
commutes:

$\textstyle{\operatorname{PicCon}^{H}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{z}}$$\scriptstyle{g}$$\textstyle{\operatorname{Hom}(H_{z},L^{\times})}$$\textstyle{\operatorname{PicCon}^{{}^{g}H}(gU)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{g\cdot
z}}$$\textstyle{\operatorname{Hom}({}^{g}H_{z},L^{\times}).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{g}^{\ast}}$

###### Proof.

(a) We’ve already seen in Remark 3.1.9 that $\mathscr{L}\mapsto
g_{\ast}\mathscr{L}$ induces an isomorphism
$\operatorname{PicCon}(U)\to\operatorname{PicCon}(gU)$.

It remains to see how the $H$-equivariant structure on
$\mathscr{L}\in\operatorname{PicCon}^{H}(U)$ induces an ${}^{g}H$-equivariant
structure on $g_{\ast}\mathscr{L}$: for each $h\in H$ we have an isomorphism
$h^{\mathscr{L}}\colon\mathscr{L}\to h^{\ast}\mathscr{L}$. This induces an
isomorphism

$(ghg^{-1})^{g_{\ast}\mathscr{L}}\colon
g_{\ast}\mathscr{L}\to(ghg^{-1})^{\ast}g_{\ast}\mathscr{L}=g_{\ast}h^{\ast}\mathscr{L}$

given by $(ghg^{-1})^{g_{\ast}\mathscr{L}}=g_{\ast}(h^{\mathscr{L}})$. It is
easy to verify that this induces the desired isomorphism
$\operatorname{PicCon}^{H}(U)\to\operatorname{PicCon}^{{}^{g}H}(gU)$.

(b) Fix $[\mathscr{L}]\in\operatorname{PicCon}^{H}(U)$ and consider the stalk
$(g_{\ast}\mathscr{L})_{g\cdot z}$ of $g_{\ast}\mathscr{L}$ at $g\cdot z\in
gU$. There is a natural bijection between the affinoid subdomains of $gU$
containing $g\cdot z$, and the affinoid subdomains of $U$ containing $z$,
given by $V\mapsto g^{-1}V$. This gives a $K$-linear isomorphism
$\tau_{g}:\mathscr{L}_{z}\to(g_{\ast}\mathscr{L})_{g\cdot z}$ which is
appropriately equivariant with respect to the $H_{z}$-action on
$\mathscr{L}_{z}$ and the $H_{g\cdot z}={}^{g}H_{z}$-action on
$(g_{\ast}\mathscr{L})_{g\cdot z}$:

$\tau_{g}(h\cdot m)=c_{g}(h)\cdot\tau_{g}(m)\quad\mbox{for all}\quad h\in
H_{z},m\in\mathscr{L}_{z}.$

Now let $h\in H_{z}$. Using Lemma 3.2.2, we see that the scalar
$\phi_{z}([\mathscr{L}])(h)\in L^{\times}$ is completely determined by the
following equation inside $\mathscr{L}(z)$:

$1\otimes h\cdot m=\phi_{z}([\mathscr{L}])(h)\otimes m\quad\mbox{for all}\quad
m\in\mathscr{L}_{z}.$

Since $c_{g}(h)=ghg^{-1}$ lies in $H_{g\cdot z}$, we have a similar equation
inside $(g_{\ast}\mathscr{L})(g\cdot z)$:

$1\otimes c_{g}(h)\cdot m=\phi_{g\cdot
z}([g_{\ast}\mathscr{L}])(c_{g}(h))\otimes m\quad\mbox{for all}\quad
m\in(g_{\ast}\mathscr{L})_{g\cdot z}.$

Note that the map $\tau_{g}$ satisfies $\tau_{g}(a\cdot
m)=g^{\mathcal{O}}_{z}(a)\cdot\tau_{g}(m)$ for all
$a\in\mathcal{O}_{X,z},m\in\mathscr{L}_{z}$. Therefore
$1\otimes\tau_{g}:L\otimes_{K}\mathscr{L}_{z}\to
L\otimes_{K}(g_{\ast}\mathscr{L})_{g\cdot z}$ descends to a well-defined
$L$-linear map $\mathscr{L}(z)\to(g_{\ast}\mathscr{L})(g\cdot z)$. Applying
this map to the first equation and comparing the result with the second shows
that

$\phi_{z}([\mathscr{L}])(h)=\phi_{g\cdot
z}([g_{\ast}\mathscr{L}])(c_{g}(h))=(c_{g}^{\ast}\circ\phi_{g\cdot z}\circ
g)([\mathscr{L}])(h)\quad\mbox{for all}\quad h\in H_{z}.$

This implies the commutativity of the diagram in the statement.∎

Forgetting the $G$-equivariant structure gives us a functor
$\underline{\omega}$ from the category of $G$-equivariant line bundles with
flat connection on $X$ and isomorphisms between them to the category of line
bundles with flat connection on $X$ and isomorphisms between them. Moreover
$\underline{\omega}$ induces a group homomorphism

$\omega\colon\operatorname{PicCon}^{G}(X)\longrightarrow\operatorname{PicCon}(X).$

######  Proposition 3.2.14.

Suppose that $X$ is quasi-Stein and geometrically connected. There is an exact
sequence of abelian groups

$0\to\operatorname{Hom}(G,K^{\times})\to\operatorname{PicCon}^{G}(X)\stackrel{{\scriptstyle\omega}}{{\to}}\operatorname{PicCon}(X)^{G}$

with the first non-trivial map given by $\chi\mapsto\mathcal{O}_{\chi}$.

###### Proof.

It is easy to verify that $\chi\mapsto\mathcal{O}_{\chi}$ defines a group
homomorphism from $\operatorname{Hom}(G,K^{\times})$ to
$\operatorname{PicCon}^{G}(X)$. Moreover we observe that for any
$G$-equivariant line bundle with flat connection $\mathscr{L}$, the space of
global horizontal sections444Recall that
$\operatorname{Hom}_{\mathcal{O}}(\mathcal{O},\mathcal{F})=\mathcal{F}(X)$ for
_any_ $\mathcal{O}$-module $\mathcal{F}$ and any ringed space
$(X,\mathcal{O})$ of $\mathscr{L}$

(8)
$\operatorname{Hom}_{\mathcal{D}}(\mathcal{O},\mathscr{L})=\mathscr{L}(X)^{\mathcal{T}(X)=0}$

is a $K$-linear $G$-representation: if $g\in G$ and $\mathcal{T}(X)\cdot v=0$
for some $v\in\mathscr{L}(X)$, then $\partial\cdot(g\cdot
v)=g\cdot((g^{-1}\cdot\partial)\cdot v)=0$ for all $\partial\in\mathcal{T}(X)$
so that $g\cdot v\in\mathscr{L}(X)^{\mathcal{T}(X)=0}$ again.

Suppose that $\chi\in\operatorname{Hom}(G,K^{\times})$ is such that
$\mathcal{O}_{\chi}$ is isomorphic to $\mathcal{O}$ as a $G$-equivariant line
bundle with flat connection. Considering the global horizontal sections, we
obtain an isomorphism of continuous $G$-representations

$\mathcal{O}(X)^{\mathcal{T}(X)=0}\quad\cong\quad\mathcal{O}_{\chi}(X)^{\mathcal{T}(X)=0}.$

By Lemma 3.1.4, both of these $K$-vector spaces are $1$-dimensional and
spanned by $1\in\mathcal{O}(X)$. However the $G$-action on the first is
trivial, whereas the $G$-action on the second is through the character $\chi$.
Hence $\chi$ is the trivial character, and the map
$\operatorname{Hom}(G,K^{\times})\to\operatorname{PicCon}^{G}(X)$ is
injective.

If $\mathscr{L}$ is $G$-equivariant, then
$(g^{-1})^{\mathscr{L}}:\mathscr{L}\to g_{\ast}\mathscr{L}$ is a
$\mathcal{D}$-linear isomorphism for all $g\in G$ which means that the class
$[\mathscr{L}]$ in $\operatorname{PicCon}(X)$ is fixed by this $G$-action i.e.
the image of the map $\operatorname{PicCon}^{G}(X)\to\operatorname{PicCon}(X)$
is indeed contained in $\operatorname{PicCon}(X)^{G}$.

Finally suppose that $\mathscr{L}$ is a $G$-equivariant line bundle with flat
connection on $X$ which becomes trivial after forgetting the $G$-structure.
Then we can find a $\mathcal{D}$-linear isomorphism
$\varphi:\mathcal{O}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{L}$.
Applying the functor of global horizontal sections together with Lemma 3.1.4,
we deduce that the $K$-linear $G$-representation
$\mathscr{L}(X)^{\mathcal{T}(X)=0}$ is in fact one-dimensional. Let
$v:=\varphi(X)(1)\in\mathscr{L}(X)^{\mathcal{T}(X)=0}$; since
$1\in\mathcal{O}(X)$ generates $\mathcal{O}$ as an $\mathcal{O}$-module, we
see that $v\in\mathscr{L}(X)$ generates $\mathscr{L}$ as an
$\mathcal{O}$-module: $\mathscr{L}=\mathcal{O}\cdot v$. Let
$\chi\in\operatorname{Hom}(G,K^{\times})$ describe the $G$-action on $v$, so
that $g\cdot v=\chi(g)v$ for all $g\in G$. This gives us a character
$\chi:G\to K^{\times}$ and we will show that $\chi$ is continuous. To see
this, choose a non-empty affinoid subdomain $U$ of $X$ and let $\mathcal{A}$
be an affine formal model in $\mathcal{O}(U)$, so that $\mathcal{A}\cdot
v_{|U}$ is the unit ball in $\mathscr{L}(U)=\mathcal{O}(U)\cdot v_{|U}$ with
respect to some $K$-Banach norm defining the canonical topology on
$\mathscr{L}(U)$. By Definition 3.2.8(b), for any $n\geqslant 0$ we can find
an open subgroup $H_{n}$ of $G_{U}$ such that $h\cdot v_{|U}\equiv
v_{|U}\operatorname{mod}\pi_{F}^{n}\mathcal{A}\cdot v_{|U}$ for all $h\in
H_{n}$. Hence $\chi(h)\equiv 1\operatorname{mod}\pi_{F}^{n}K^{\circ}$ for all
$h\in H_{n}$, and $\chi:G\to K^{\times}$ is therefore continuous.

With $\chi\in\operatorname{Hom}(G,K^{\times})$ in hand, consider the
$G$-equivariant line bundle with connection
$\mathscr{L}\otimes_{\mathcal{O}}\mathcal{O}_{\chi^{-1}},$ and the
$\mathcal{D}$-linear isomorphism
$\varphi_{\chi}:\mathcal{O}\to\mathscr{L}\otimes_{\mathcal{O}}\mathcal{O}_{\chi^{-1}}$
given by $\varphi_{\chi}(f):=\varphi(f)\otimes 1$. We claim that
$\varphi_{\chi}$ is $G$-equivariant; given this claim, it follows immediately
that $\mathscr{L}\cong\mathcal{O}_{\chi}$ as a $G$-equivariant
$\mathcal{D}$-module, and then $[\mathscr{L}]\in\operatorname{PicCon}^{G}(X)$
lies in the image of $\operatorname{Hom}(G,K^{\times})$ as required.

To establish the claim, we first check that
$\varphi_{\chi}(X):\mathcal{O}(X)\to(\mathscr{L}\otimes\mathcal{O}_{\chi^{-1}})(X)$
is $G$-equivariant: for $g\in G$ and $f\in\mathcal{O}(X)$ we have

$g\cdot\varphi_{\chi}(X)(f)=g\cdot(fv\otimes 1)=(g\cdot f)(g\cdot
v)\otimes\chi^{-1}(g)=(g\cdot f)v\otimes 1=\varphi_{\chi}(X)(g\cdot f).$

By replacing $\mathscr{L}$ by $\mathscr{L}\otimes\mathcal{O}_{\chi^{-1}}$, it
now remains to show that if $\varphi:\mathcal{O}\to\mathscr{L}$ is a
$\mathcal{D}$-linear isomorphism such that
$\varphi(X):\mathcal{O}(X)\to\mathscr{L}(X)$ is $G$-equivariant, then
$\varphi$ is also $G$-equivariant. To this end, fix $g\in G$, and consider the
morphisms $g^{\ast}(\varphi)\circ g^{\mathcal{O}}:\mathcal{O}\to
g^{\ast}\mathscr{L}$ and $g^{\mathscr{L}}\circ\varphi:\mathcal{O}\to
g^{\ast}\mathscr{L}$. By precomposing the $g^{\ast}\mathcal{O}$-module
structure on $g^{\ast}\mathscr{L}$ with the ring isomorphism
$g^{\mathcal{O}}:\mathcal{O}\to g^{\ast}\mathcal{O}$ we may regard
$g^{\ast}\mathscr{L}$ to be an $\mathcal{O}$-module. Then it is coherent, and
the two maps are $\mathcal{O}$-linear. Since they have the same global
sections by our computation above and since $X$ is quasi-Stein, we conclude
using [27, Corollary 3.3] that the two maps are equal. This means that
$\varphi$ is $G$-equivariant. ∎

In order to understand $\operatorname{PicCon}^{G^{0}}(\Omega)$ the following
result will prove to be useful.

######  Proposition 3.2.15.

Suppose that $X$ is quasi-Stein and geometrically connected and $G$ is equal
to an amalgamated product $A\ast_{C}B$ of its open subgroups $A$ and $B$ along
their common subgroup $C$. Then the homomorphism

$(p_{A},p_{B})\colon\operatorname{PicCon}^{G}(X)\to\operatorname{PicCon}^{A}(X)\underset{\operatorname{PicCon}^{C}(X)}{\times}{}\operatorname{PicCon}^{B}(X)$

induced by restriction of equivariant structures is an isomorphism.

###### Proof.

Suppose that $[\mathscr{L}]\in\operatorname{PicCon}^{G}(X)$ lies in the kernel
of $(p_{A},p_{B})$. Then

$\omega([\mathscr{L}])=[\mathcal{O}]\in\operatorname{PicCon}(X).$

By Proposition 3.2.14, $[\mathscr{L}]=[\mathcal{O}^{\chi}]$ for some
$\chi\in\operatorname{Hom}(G,K^{\times})$. However, also by Proposition
3.2.14, $\chi|_{A}$ and $\chi|_{B}$ are both trivial and so, as $G$ is
generated by $A$ and $B$, $\chi$ is trivial. Thus
$[\mathscr{L}]=[\mathcal{O}]\in\operatorname{PicCon}^{G}(X)$ and
$(p_{A},p_{B})$ is injective.

Suppose now that $[\mathscr{L}_{A}]\in\operatorname{PicCon}^{A}(X)$,
$[\mathscr{L}_{B}]\in\operatorname{PicCon}^{B}(X)$ and that there is an
isomorphism
$\theta\colon\mathscr{L}_{A}|_{C}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{L}_{B}|_{C}$
of $C$-equivariant line bundles with flat connection obtained by restriction
of equivariant structures from $A$ and $B$ to $C$. We transport the
$B$-equivariant structure on $\mathscr{L}_{B}$ along $\theta$ to
$\mathscr{L}_{A}$. In this way, the $\mathcal{D}$-module
$\mathscr{L}:=\underline{\omega}(\mathscr{L}_{A})$ can be equipped with
$A$-equivariant and $B$-equivariant structures whose restrictions to $C$
agree.

By Proposition 2.4.11, for every subgroup $H$ of $G$ there is a bijection
between the set of all $H-\mathcal{D}$-module structures on $\mathscr{L}$
extending the given $\mathcal{D}$-module structure on $\mathscr{L}$, and the
set $\mathcal{S}_{\mathcal{D}}(H,\mathscr{L})$. This bijection is given by

$\\{g^{\mathscr{L}}\\}_{g\in
H}\mapsto[g\mapsto(g,g^{\mathscr{L}})\in\operatorname{Aut}_{\mathcal{D}}(\mathscr{L}/X/H)].$

On the other hand, by Theorem 2.4.13 restriction induces a bijection

$\mathcal{S}_{\mathcal{D}}(G,\mathscr{L})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{S}_{\mathcal{D}}(A,\mathscr{L})\times_{\mathcal{S}_{\mathcal{D}}(C,\mathscr{L})}\mathcal{S}_{\mathcal{D}}(B,\mathscr{L}).$

It follows that there exists a $G-\mathcal{D}$-module
$\mathscr{L}_{G}=(\mathscr{L},\\{g^{\mathscr{L}}\\}_{g\in G})$ whose
restriction to $A$ is $\mathscr{L}_{A}$, and whose restriction to $B$ is the
transport of $\mathscr{L}_{B}$ to $\mathscr{L}_{A}$ along $\theta$. Since $A$
and $B$ are open in $G$, the action map
$G_{U}\to\mathcal{B}(\mathscr{L}(U))^{\times}$ is continuous for every
affinoid subdomain $U$ of $X$, because the restrictions of this map to both
$A_{U}$ and $B_{U}$ are continuous. This shows that
$[\mathscr{L}_{G}]\in\operatorname{PicCon}^{G}(X)$. By construction, we have
$p_{A}([\mathscr{L}_{G}])=[\mathscr{L}_{A}]$ and
$p_{B}([\mathscr{L}_{G}])=[\mathscr{L}_{B}]$. Hence $(p_{A},p_{B})$ is
surjective. ∎

######  Lemma 3.2.16.

If $X$ is a geometrically connected quasi-Stein space with an admissible cover
by an increasing chain $(X_{n})$ of $G$-stable affinoid subdomains then the
restriction maps
$\operatorname{PicCon}^{G}(X)\to\operatorname{PicCon}^{G}(X_{n})$ induce an
isomorphism of groups

$\operatorname{PicCon}^{G}(X)\cong\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\operatorname{PicCon}^{G}(X_{n}).$

###### Proof.

Certainly the restriction maps induce a group homomorphism

$\alpha\colon\operatorname{PicCon}^{G}(X)\to\mathop{\underset{\scriptstyle}{\underset{\raisebox{0.0pt}[0.75346pt][-1.50694pt]{$\longleftarrow$}}{\operatorname{lim}}}}\operatorname{PicCon}^{G}(X_{n}).$

By Corollary 3.1.6, if $[\mathscr{L}]\in\ker\alpha$ then
$\omega([\mathscr{L}])=[\mathcal{O}_{X}]\in\operatorname{PicCon}(X)$. Thus
$[\mathscr{L}]=[\mathcal{O}_{X}^{\chi}]$ for some continuous character
$\chi\colon G\to K^{\times}$ by Lemma 3.2.14. But then
$[\mathscr{L}|_{X_{n}}]=[\mathcal{O}_{X_{n}}^{\chi}]$ for any $n\geqslant 0$
and so $\chi$ is the trivial character since $\alpha([\mathscr{L}])$ is
trivial. Thus $\alpha$ is injective.

Let $([\mathscr{L}_{n}])\in\prod\operatorname{PicCon}^{G}(X_{n})$ be a
compatible family of isomorphism classes of equivariant line bundles with
connection under restriction so that for each $n$ we can find an isomorphism

$\theta_{n+1,n}\colon\mathscr{L}_{n+1}|_{X_{n}}\to\mathscr{L}_{n}$

of $G$-equivariant line bundles with connection on $X_{n}$.

Thus each $l\geqslant m$, we can define
$\theta_{m,l}\colon\mathscr{L}_{l}|_{X_{m}}\to\mathscr{L}_{m}$ to be the
composite of the restrictions of
$\theta_{l,l-1},\theta_{l-1,l-2},\ldots,\theta_{m+1,m}$ to $X_{m}$. Now
$(\mathscr{L}_{l},\theta_{m,l})$ forms gluing data for the cover
$(X_{m})_{m\geqslant n}$ of $X$. The resulting sheaf $\mathscr{L}$ is an
$G$-equivariant line bundle with connection on $X$ with
$\alpha([\mathscr{L}])=([\mathscr{L}_{n}])$ and so $\alpha$ is surjective. ∎

###  3.3. Cocycles and equivariant line bundles on affinoids

In this technical subsection, we will explain how isomorphism classes of
$G$-equivariant structures on the trivial line bundle over a $K$-affinoid
variety $X$ can be classified through the language of continuous $1$-cocycles
of $G$ acting on $\mathcal{O}(X)^{\times}$. This material will be crucial to
the proof of one of the main results in $\S\ref{MainProofSec}$, namely Theorem
4.4.1.

We assume throughout $\S\ref{SecCocycELBC}$ that $X$ is a smooth and
geometrically connected $K$-affinoid space, with a topological group $G$
acting continuously on it.

######  Lemma 3.3.1.

  1. (a)

The set of $G$-equivariant structures on a trivial line bundle
$\mathscr{L}=\mathcal{O}\cdot v$ is in natural bijection with
$Z^{1}(G,\mathcal{O}(X)^{\times})$ under a function that sends
$\\{g^{\mathscr{L}}:g\in G\\}$ to the function
$\alpha:G\to\mathcal{O}(X)^{\times}$ determined by the rule

$g^{\mathscr{L}}(v)=\alpha(g)v\quad\mbox{for all}\quad g\in G.$

  2. (b)

The bijection in (a) induces an isomorphism

$\theta^{G}_{X}:\ker\left(\operatorname{Pic}^{G}(X)\to\operatorname{Pic}(X)\right)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{1}(G,\mathcal{O}(X)^{\times}).$

###### Proof.

(a) Suppose that for each $g\in G$, we have a morphism of sheaves of
$K$-vector spaces $g^{\mathscr{L}}:\mathscr{L}\to g^{\ast}\mathscr{L}$ such
that

$g^{\mathscr{L}}(fv)=g^{\mathcal{O}}(f)g^{\mathscr{L}}(v)\quad\mbox{for
all}\quad g\in G,f\in\mathcal{O}.$

This data is completely determined by the function $\alpha\colon
G\to\mathcal{O}(X)$ given by

$g^{\mathscr{L}}(v)=\alpha(g)v\quad\mbox{for all}\quad g\in G.$

We first claim that $\\{g^{\mathscr{L}}:g\in G\\}$ is a $G$-equivariant
structure on $\mathscr{L}$ if and only only $\alpha$ is a $1$-cocycle with
values in the group $\mathcal{O}(X)^{\times}$.

We see that for all $g,h\in G$

(9) $\displaystyle
h^{\ast}(g^{\mathscr{L}})h^{\mathscr{L}}(v)=h^{\ast}(g^{\mathscr{L}})(\alpha(h)v)$
$\displaystyle=$ $\displaystyle(g\cdot\alpha(h))\alpha(g)v\mbox{ and}$ (10)
$\displaystyle(gh)^{\mathscr{L}}(v)$ $\displaystyle=$
$\displaystyle\alpha(gh)v.$

Now if $\\{g^{\mathscr{L}}:g\in G\\}$ defines a $G$-equivariant structure on
$\mathscr{L}$ then $\alpha(1)=1$ and
$h^{\ast}(g^{\mathscr{L}})h^{\mathscr{L}}(v)=(gh)^{\mathscr{L}}(v)$. Thus by
(9) and (10) $\alpha(gh)=(g\cdot\alpha(h))\alpha(g)$ for all $g,h\in G$ and in
particular

$1=\alpha(gg^{-1})=(g\cdot\alpha(g^{-1}))\alpha(g)$

for all $g\in G$. Thus $\alpha$ is a $1$-cocycle with values in
$\mathcal{O}(X)^{\times}$. Conversely, if $\alpha$ is a $1$-cocycle then by
(9) and (10) again, for all $g,h\in G$,

$(gh)^{\mathscr{L}}(v)=h^{\ast}(g^{\mathscr{L}})h^{\mathscr{L}}(v)\mbox{ and
so }$ $(gh)^{\mathscr{L}}=h^{\ast}(g^{\mathscr{L}})h^{\mathscr{L}}.$

Moreover $\alpha(1^{2})=\alpha(1)^{2}$ and so, since $X$ is connected,
$\alpha(1)=1$ and $1^{\mathscr{L}}=\mathrm{id}_{\mathscr{L}}$.

It remains to observe that $\alpha$ is continuous if and only if for every
affinoid subdomain $U$ of $X$, the action map
$G_{U}\to\mathcal{B}(\mathscr{L}(U))^{\times}$ is continuous. This holds
because

$g\cdot(fv)=g^{\mathcal{O}}(f)\alpha(g)v\quad\mbox{ for all }\quad g\in
G_{U},f\in\mathcal{O}(U)$

and because $G$ acts continuously on $X$.

(b) Suppose now that $\mathscr{L}_{1}=\mathcal{O}\cdot v_{1}$ and
$\mathscr{L}_{2}=\mathcal{O}\cdot v_{2}$ are two $G$-equivariant line bundles
corresponding to $1$-cocycles $\alpha_{1}$ and $\alpha_{2}$ respectively.

Let $\varphi\colon\mathscr{L}_{1}\to\mathscr{L}_{2}$ be an isomorphism of the
underlying line bundles. Then $\varphi(v_{1})=fv_{2}$ for some
$f\in\mathcal{O}(X)^{\times}$, so for all $g\in G$ we have

$\varphi(g^{\mathscr{L}_{1}}(v_{1}))=\varphi(\alpha_{1}(g)v_{1})=\alpha_{1}(g)fv_{2}$

whereas

$g^{\mathscr{L}_{2}}(\varphi(v_{1}))=g^{\mathscr{L}_{2}}(fv_{2})=g^{\mathcal{O}}(f)\alpha_{2}(g)v_{2}.$

Hence $\varphi$ defines an isomorphism of $G$-_equivariant_ line bundles if
and only if

$\alpha_{2}(g)=\frac{g^{\mathcal{O}}(f)}{f}\alpha_{1}(g)\quad\mbox{ for all
}\quad g\in G.$

Thus the map in (a) induces a bijection

$\theta^{G}_{X}\colon\ker\left(\operatorname{Pic}^{G}(X)\to\operatorname{Pic}(X)\right)\to
H^{1}(G,\mathcal{O}(X)^{\times})$

which is a group homomorphism because $g\cdot(v_{1}\otimes
v_{2})=\alpha_{1}(g)v_{1}\otimes\alpha_{2}(g)v_{2}=(\alpha_{1}\alpha_{2})(g)(v_{1}\otimes
v_{2})$ for all $g\in G$. ∎

Recall the map $\phi_{z}$ from Proposition 3.2.7; by abuse of notation, we
will also denote its pre-composition with the forgetful map
$\operatorname{Con}^{G}(X)\to\operatorname{Pic}^{G}(X)$ by $\phi_{z}$.

######  Proposition 3.3.2.

Suppose that $L$ is a finite field extension of $K$.

  1. (a)

The isomorphism from Lemma 3.3.1(b) induces a homomorphism

$\phi^{G}_{X}\colon\operatorname{Con}^{G}(X)\to
H^{1}(G,\mathcal{O}(X)^{\times})$

by pre-composition with the forgetful map

$\operatorname{Con}^{G}(X)\to\ker(\operatorname{Pic}^{G}(X)\to\operatorname{Pic}(X)).$

  2. (b)

For every $z\in X(L)$ and every $[\mathscr{L}]\in\operatorname{Con}^{G}(X)$,
we have

$z\circ(\operatorname{res}^{G}_{G_{z}}\phi^{G}_{X}([\mathscr{L}]))=\phi_{z}([\mathscr{L}]).$

  3. (c)

For every $z\in X(L)$ and every $\chi\in\operatorname{Hom}(G,K^{\times})$, we
have

$\phi_{z}([\mathcal{O}_{\chi}])=\chi|_{G_{z}}.$

  4. (d)

Let $Y\subseteq X$ be a $G$-stable affinoid subdomain, with $z\in
Y(L)\subseteq X(L)$. Then the following diagram is commutative:

$\textstyle{\operatorname{Con}^{G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{z}}$$\scriptstyle{(-)|_{Y}}$$\textstyle{\operatorname{Con}^{G}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{z}}$$\textstyle{\operatorname{Hom}(G_{z},L^{\times}).}$

###### Proof.

(a) Since the forgetful map
$\operatorname{Con}^{G}(X)\to\ker(\operatorname{Pic}^{G}(X)\to\operatorname{Pic}(X))$
is a group homomorphism, the function $\phi^{G}_{X}$ that is the composite of
this forgetful map with $\theta^{G}_{X}$ is also a homomorphism.

(b) Suppose that $\phi^{G}_{X}([\mathscr{L}])=[\alpha]\in
H^{1}(G,\mathcal{O}(X)^{\times})$ so that we can write
$\mathscr{L}=\mathcal{O}\cdot v$ with $g\cdot v=\alpha(g)v$ for all $g\in G$.
Then working inside $L\otimes_{\mathcal{O}(X)}\mathscr{L}(X)$ we have

$\phi_{z}(g)(1\otimes v)=g\cdot(1\otimes
v)=1\otimes\alpha(g)v=(z\circ\alpha)(g)\otimes v\quad\mbox{for all}\quad g\in
G_{z}.$

(c) This follows from Lemma 3.3.1(a) and Definition 3.2.9.

(d) If $[\mathscr{L}]=[\mathcal{O}_{X}\cdot v]\in\operatorname{Con}^{G}(X)$,
then $[\mathscr{L}|_{Y}]=[\mathcal{O}_{Y}\cdot v]\in\operatorname{Con}^{G}(Y)$
and $g^{\mathscr{L}|_{Y}}(v)=g^{\mathscr{L}}(v)$ for all $g\in G$. Now use
Lemma 3.3.1 together with part (a). ∎

######  Definition 3.3.3.

Recall from $\S\ref{ConvNotn}$ the map
$\delta_{G}\colon\mathcal{O}(X)^{\times}\to Z^{1}(G,\mathcal{O}(X)^{\times})$,
given by $\delta_{G}(u)(g)=g\cdot u/u$. For each $u\in\mathcal{O}(X)^{\times}$
and $d,e\geqslant 1$, we define

$\mathcal{Z}^{G,X}_{u,d,e}:=\left\\{\alpha\in
Z^{1}(G,\mathcal{O}(X)^{\times}):\alpha^{de}=\delta_{G}(u^{e})\right\\}.$

This special set of $1$-cocycles will be useful for our explicit construction
of torsion equivariant line bundles with flat connection.

Recall from Lemma 3.1.10 the line bundle with connection $\mathscr{L}_{u,d}$:
it is the free $\mathcal{O}_{X}$-module on the $1$-element set $\\{v\\}$, and
the action of $\mathcal{T}(X)$ is determined by

(11) $\partial(v)=\frac{1}{d}\frac{\partial(u)}{u}v\quad\mbox{for
all}\quad\partial\in\mathcal{T}(X).$

######  Lemma 3.3.4.

Let $u\in\mathcal{O}(X)^{\times}$ and let $d,e\geqslant 1$.

  1. (a)

For each $\alpha\in\mathcal{Z}^{G,X}_{u,d,e}$, there is a $(de)$-torsion
$G$-equivariant line bundle with connection $\mathscr{L}^{\alpha}_{u,d}$ on
$X$ such that

$\omega([\mathscr{L}^{\alpha}_{u,d}])=\mathscr{L}_{u,d}\quad\mbox{and}\quad\phi_{X}^{G}([\mathscr{L}^{\alpha}_{u,d}])=[\alpha]\in
H^{1}(G,\mathcal{O}(X)^{\times}).$

  2. (b)

If $u,w\in\mathcal{O}(X)^{\times}$, $\alpha\in\mathcal{Z}^{G,X}_{u,d,e}$ and
$\beta\in\mathcal{Z}^{G,X}_{w,d,e}$, then
$\alpha\beta\in\mathcal{Z}^{G,X}_{uw,d,e}$ and

$\mathscr{L}_{u,d}^{\alpha}\otimes\mathscr{L}_{w,d}^{\beta}\cong\mathscr{L}_{uw,d}^{\alpha\beta}.$

  3. (c)

For each $\alpha\in\mathcal{Z}^{G,X}_{u,d,e}$, the following are equivalent:

    1. (i)

$[\mathscr{L}_{u,d}^{\alpha}]=[\mathcal{O}]$ in
$\operatorname{Con}^{G}(X)[de]$,

    2. (ii)

there is $f\in\mathcal{O}(X)^{\times}$ such that $u/f^{d}\in K^{\times}$ and
$\alpha=\delta_{G}(f)$.

  4. (d)

The map $\alpha\mapsto[\mathscr{L}_{u,d}^{\alpha}]$ defines a bijection

$\mathcal{Z}^{G,X}_{u,d,e}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\left\\{[\mathscr{L}]\in\operatorname{Con}^{G}(X)[de]:\omega([\mathscr{L}])=[\mathscr{L}_{u,d}]\right\\}.$

###### Proof.

(a) We equip $\mathscr{L}_{u,d}$ with the $G-\mathcal{O}_{X}$-module structure
associated to $\alpha$ by Lemma 3.3.1. In particular the action map
$G_{U}\to\mathcal{B}(\mathscr{L}(U))^{\times}$ is continuous for every
affinoid subdomain $U$ of $X$. We check that this is in fact a
$G-\mathcal{D}_{X}$-module. Using $\alpha^{de}=\delta_{G}(u^{e})$, we compute

$\frac{\partial(\alpha(g))}{\alpha(g)}=\frac{1}{d}\left(\frac{\partial(g\cdot
u)}{g\cdot u}-\frac{\partial(u)}{u}\right)\quad\mbox{for
each}\quad\partial\in\mathcal{T}_{X},g\in G.$

Using this together with (11) we compute

$\displaystyle(g\cdot\partial)(g\cdot v)$ $\displaystyle=$
$\displaystyle(g\cdot\partial)(\alpha(g))v+\alpha(g)(g\cdot\partial)(v)$
$\displaystyle=$ $\displaystyle\frac{1}{d}\left(\frac{(g\cdot\partial)(g\cdot
u)}{g\cdot
u}-\frac{(g\cdot\partial)u}{u}\right)\alpha(g)v+\alpha(g)\frac{1}{d}\frac{(g\cdot\partial)(u)}{u}v$
$\displaystyle=$ $\displaystyle\frac{1}{d}\frac{g\cdot(\partial(u))}{g\cdot
u}g\cdot v$ $\displaystyle=$ $\displaystyle g\cdot(\partial(v)).$

Thus we obtain a $G$-equivariant line bundle with connection on $X$ that we
denote $\mathscr{L}^{\alpha}_{u,d}$. It is evident that
$\omega\left([\mathscr{L}_{u,d}^{\alpha}]\right)=[\mathscr{L}_{u,d}]$.

Using (11), we see that $v^{\otimes de}\mapsto u^{e}$ defines an isomorphism
$\psi:\mathscr{L}_{u,d}^{\otimes
de}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{O}_{X}$ of line
bundles with flat connection. To establish that $\mathscr{L}_{u,d}^{\alpha}$
is $(de)$-torsion, it suffices to show this isomorphism is $G$-linear. Since
for all $g\in G$ we have

$g\cdot(v^{\otimes de})=\alpha^{de}(g)v^{\otimes de}\quad\mbox{and}\quad
g\cdot(u^{e})=\delta_{G}(u^{e})(g)u^{e};$

we have $\alpha^{de}=\delta_{G}(u)^{e}$ and the $G$-linearity follows.

(b) Since $\alpha^{de}=\delta_{G}(u^{e})$ and $\beta^{de}=\delta_{G}(w^{e})$
and $\delta_{G}\colon\mathcal{O}(X)^{\times}\to
Z^{1}(G,\mathcal{O}(X)^{\times})$ is a group homomorphism, we see that
$(\alpha\beta)^{e}=\delta_{G}((uw)^{e})$. Hence
$\alpha\beta\in\mathcal{Z}_{uw,d,e}^{G,X}$.

Using Proposition 3.1.11, we see that
$[\mathscr{L}_{u,d}\otimes\mathscr{L}_{v,d}]=[\mathscr{L}_{uv,d}]$ in
$\operatorname{Con}(X)[d]$. Using the definition of the $G$-equivariant
structure on $\mathscr{L}_{u,d}^{\alpha}$ in part (a) and the definition of
the product in $\operatorname{Con}^{G}(X)$ given in the proof of Lemma 3.2.10,
we also see that
$[\mathscr{L}_{u,d}^{\alpha}\otimes\mathscr{L}_{w,d}^{\beta}]=[\mathscr{L}_{uw,d}^{\alpha\beta}]$
in $\operatorname{Con}^{G}(X)$.

(c) For any $f\in\mathcal{O}(X)^{\times}$, there is an isomorphism
$\mathcal{O}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathscr{L}_{f^{d},d}^{\delta_{G}(f)}$
of $G$-equivariant line bundles with connection on $X$, sending $1$ to
$f^{-1}v$. This gives the equality
$[\mathscr{L}_{f^{d},d}^{\delta_{G}(f)}]=[\mathcal{O}]$ in
$\operatorname{Con}^{G}(X)$. Using part (b), we then also have

(12)
$[\mathscr{L}^{\alpha}_{u,d}]=[\mathscr{L}^{\alpha}_{u,d}]\cdot[\mathcal{O}]=[\mathscr{L}^{\alpha}_{u,d}]\cdot[\mathscr{L}^{\delta_{G}(f)}_{f^{d},d}]^{-1}=[\mathscr{L}_{u/f^{d},d}^{\alpha/\delta_{G}(f)}].$

Suppose now that $u=\lambda f^{d}$ for some $\lambda\in K^{\times}$ and some
$f\in\mathcal{O}(X)^{\times}$ such that $\alpha=\delta_{G}(f)$. Using (12), we
then have
$[\mathscr{L}^{\alpha}_{u,d}]=[\mathscr{L}_{\lambda,d}^{1}]=[\mathcal{O}]$ in
$\operatorname{Con}^{G}(X)$.

Conversely, suppose that $[\mathscr{L}^{\alpha}_{u,d}]=[\mathcal{O}]$ in
$\operatorname{Con}^{G}(X)$. Then
$\omega([\mathscr{L}_{u,d}^{\alpha}])=[\mathscr{L}_{u,d}]=[\mathcal{O}]$ in
$\operatorname{Con}(X)^{G}$, so using Proposition 3.1.11 we can find
$f\in\mathcal{O}(X)^{\times}$ and $\lambda\in K^{\times}$ such that $u=\lambda
f^{d}$. Then $(\ref{DivideCocycle})$ implies that
$[\mathcal{O}]=[\mathscr{L}^{\alpha}_{u,d}]=[\mathscr{L}_{\lambda,d}^{\alpha/\delta_{G}(f)}]$
in $\operatorname{Con}^{G}(X)$. Hence $G$ must fix the basis vector $v$ of
$\mathscr{L}_{\lambda,d}^{\alpha/\delta_{G}(f)}(X)^{\mathcal{T}(X)=0}$, so
$\alpha=\delta_{G}(f)$.

(d) Suppose that $[\mathscr{L}]\in\operatorname{Con}^{G}(X)[de]$ is such that
$\omega([\mathscr{L}])=[\mathscr{L}_{u,d}]$. Consider the function
$\alpha\colon G\to\mathcal{O}(X)^{\times}$ defined by $g\cdot v=\alpha(g)v$.
Then $\alpha\in Z^{1}(G,\mathcal{O}(X)^{\times})$ by Lemma 3.3.1.

Since $(de)\cdot[\mathscr{L}]=0$ in $\operatorname{Con}^{G}(X)$, there is an
isomorphism of $G$-equivariant line bundles with flat connection
$\varphi:\mathscr{L}^{\otimes
de}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{O}_{X}$. On the
other hand we also have the isomorphism $\psi:\mathscr{L}_{u,d}^{\otimes
de}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathcal{O}_{X}$ of line
bundles with flat connection constructed in the proof of part (a) above. Since
$\omega([\mathscr{L}])=\mathscr{L}_{u,d}$, Corollary 3.1.5 implies that
$\varphi=\lambda\psi$ for some $\lambda\in K^{\times}$, and then
$\varphi(v^{\otimes de})=\lambda u^{e}$. Since $\varphi$ is $G$-linear, we
have

$g\cdot\lambda u^{e}=g\cdot\varphi(v^{\otimes de})=\varphi(g\cdot v^{\otimes
de})=\alpha(g)^{de}\lambda u^{e}\quad\mbox{for all}\quad g\in G,$

so $\alpha^{de}=\delta_{G}(u^{e})$. Hence $\alpha\in\mathcal{Z}^{G,X}_{u,d,e}$
and $[\mathscr{L}]=[\mathscr{L}^{\alpha}_{u,d}]$ in
$\operatorname{Con}^{G}(X)$.

Finally, suppose that $\alpha,\beta\in\mathcal{Z}^{G,X}_{u,d,e}$ are such that
$[\mathcal{L}^{\alpha}_{u,d}]=[\mathcal{L}^{\beta}_{u,d}]$. Then
$[\mathcal{L}^{\alpha/\beta}_{1,d,e}]=[\mathcal{O}]$ by part (b), so
$\alpha/\beta=\delta_{G}(f)$ for some $f\in\mathcal{O}(X)^{\times}$ such that
$1/f^{d}\in K^{\times}$. It follows, by Corollary 3.1.7, that $f\in
K^{\times}$. Then $\delta_{G}(f)=1$ so $\alpha=\beta$ as required. ∎

######  Notation 3.3.5.

We will write $T(X)$ to denote the discrete abelian group

$T(X):=\mathcal{O}(X)^{\times}/\mathcal{O}(X)^{\times\times}$

and $\pi_{T(X)}$ to denote the natural projection map
$\pi_{T(X)}\colon\mathcal{O}(X)^{\times}\to T(X)$.

Since $\mathcal{O}(X)^{\times\times}$ is open in $\mathcal{O}(X)^{\times}$,
$\pi_{T(X)}$ is continuous. We also observe that $T(-)$ defines a functor from
affinoid varieties to abelian groups. Our next result gives conditions on the
data $u,d,e,\alpha$ that determine when
$[\mathscr{L}^{\alpha}_{u,d,e}]\in\operatorname{Con}^{G}(X)$ is in fact the
trivial element, in the case where $d,e$ are both coprime to $p$.

######  Proposition 3.3.6.