# On congruent isomorphisms for tori

Anne-Marie Aubert  Sorbonne Université and Université Paris Cité, CNRS, IMJ-
PRG, F-75005 Paris, France<EMAIL_ADDRESS>and  Sandeep Varma
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha
Road, Colaba, Mumbai, India<EMAIL_ADDRESS>
###### Abstract.

Let $F$ and $F^{\prime}$ be two $l$-close nonarchimedean local fields, where
$l$ is a positive integer, and let $\mathrm{T}$ and $\mathrm{T}^{\prime}$ be
two tori over $F$ and $F^{\prime}$, respectively, such that their cocharacter
lattices can be identified as modules over the “at most $l$-ramified” absolute
Galois group
$\Gamma_{F}/I_{F}^{l}\cong\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$. In the
spirit of the work of Kazhdan and Ganapathy, for every positive integer $m$
relative to which $l$ is large, we construct a congruent isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\cong\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$,
where $\mathrm{T}(F)_{m}$ and $\mathrm{T}(F^{\prime})_{m}$ are the minimal
congruent filtration subgroups of $\mathrm{T}(F)$ and
$\mathrm{T}(F^{\prime})$, respectively, defined by J.-K. Yu. We prove that
this isomorphism is functorial and compatible with both the isomorphism
constructed by Chai and Yu and the Kottwitz homomorphism for tori. We show
that, when $l$ is even larger relative to $m$, it moreover respects the local
Langlands correspondence for tori.

##  1\. Introduction

###  1.1. A crude version of the main result

Two nonarchimedean local fields $F$ and $F^{\prime}$ are said to be $l$-close,
where $l$ is a positive integer, if
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l}\cong\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{l}$,
where $\mathfrak{O}_{?}$ stands for the ring of integers of $?$, and and
$\mathfrak{p}_{?}$ for the maximal ideal of $\mathfrak{O}_{?}$.

If $F$ and $F^{\prime}$ are $l$-close, then P. Deligne ([Del84]) constructs an
isomorphism
$\Gamma_{F}/I_{F}^{l}\rightarrow\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$ now
known as a Deligne isomorphism, where $\Gamma_{?}$ denotes the Galois group of
a chosen separable closure over $?$, and $I_{?}^{l}$ stands for the $l$-th
upper ramification filtration subgroup of the inertia subgroup
$I_{?}\subset\Gamma_{?}$. If further $F$ and $F^{\prime}$ have finite residue
fields, Kazhdan isomorphisms (see [Kaz86]), pioneered by D. Kazhdan and
studied by various others, notably by R. Ganapathy (see, e.g., [Gan15] and
[Gan22]), allow us to relate harmonic analysis on reductive groups over $F$ to
that on reductive groups over $F^{\prime}$. Thus, for instance, one could hope
to study local Langlands correspondence for a group over $F^{\prime}$ by using
local Langlands correspondence for a group over $F$, if the latter is known.

One has a good understanding of Kazhdan isomorphisms for split groups, by
[Kaz86] and [Gan15]. For reductive groups that may not be split, Kazhdan
isomorphisms have been constructed by Ganapathy in [Gan22]. However, a lot of
the properties of these isomorphisms remain to be studied, and such a study is
being pursued by Ganapathy and her collaborators.

In the present paper, we will stick to tori, and investigate questions related
to Kazhdan isomorphisms
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\cong\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$
for tori, when $F$ and $F^{\prime}$ are $l$-close and
$\mathrm{T}^{\prime}/F^{\prime}$ is a transfer of $\mathrm{T}/F$, that is, we
have an identification $X^{*}(\mathrm{T})=X^{*}(\mathrm{T}^{\prime})$ of
character lattices, or equivalently an identification
$X_{*}(\mathrm{T})=X_{*}(\mathrm{T}^{\prime})$ of cocharacter lattices, as
modules over $\Gamma_{F}/I_{F}^{l}$, identified via [Del84] with
$\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$, under the implicitly imposed
assumption that $I_{F}^{l}$ acts trivially on $X^{*}(\mathrm{T})$ and
$I_{F^{\prime}}^{l}$ on $X^{*}(\mathrm{T}^{\prime})$. Here the filtrations
$\\{\mathrm{T}(F)_{m}\\}_{m\geq 0}$ and
$\\{\mathrm{T}(F^{\prime})_{m}\\}_{m\geq 0}$ are the minimal congruent
filtrations of $\mathrm{T}(F)$ and $\mathrm{T}(F^{\prime})$, respectively, as
defined in [Yu15].

A crude version of our main result, which we will state in greater detail in
Theorem 1.2.1 below, is as follows:

######  Theorem 1.1.1.

Suppose a local field $F$ is $l$-close to a local field $F^{\prime}$, and a
torus $\mathrm{T}^{\prime}/F^{\prime}$ is a transfer of a torus
$\mathrm{T}/F$. Then:

  1. (i)

If $l$ is large relative to $m$, then there exists a (necessarily unique)
“congruent” isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$.

  2. (ii)

These isomorphisms are suitably functorial.

  3. (iii)

They are compatible with the isomorphisms constructed by Chai and Yu (see
[CY01]) and with Kottwitz homomorphisms for tori (see [Kot97, Section 7] and
[KP23, Section 11.1]).

  4. (iv)

If $l$ is even larger relative to $m$, these isomorphisms respect the local
Langlands correspondence for tori.

###  1.2. Statement of the main result — more precise version

Now we state our main result, Theorem 1.2.1 below, in terms of objects and
notation defined in later sections, especially Section 2; however, let us give
an introduction to the main objects involved:

  1. (i)

If we say $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$
(see Notation 2.3.1(viii)), we roughly mean that $F$ and $F^{\prime}$ are
discretely valued Henselian fields with perfect residue field that are
$l$-close to each other ($F\leftrightarrow_{l}F^{\prime}$), and that the torus
$\mathrm{T}^{\prime}$ over $F^{\prime}$ is a transfer of the torus
$\mathrm{T}$ over $F$.

  2. (ii)

For $(F,\mathrm{T})$ as above, $h(F,\mathrm{T})$ is a positive integer from
[CY01, Section 8.1], sort of upper-bounding the nontriviality of the
smoothening process required to arrive at the Néron model of $\mathrm{T}$.

  3. (iii)

We will be interested in “congruent isomorphisms”
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$
(Definition 3.1.3(ii)). These are isomorphisms of abelian groups.

  4. (iv)

Interpolating the $\mathrm{T}(F)_{m}$, with $m$ varying over nonnegative
integers, are the “minimal congruent filtration subgroups” $\mathrm{T}(F)_{r}$
of J.-K. Yu, with $r$ varying over nonnegative real numbers.

  5. (v)

Each torus $\mathrm{T}$ determines a relation
“$m\mathrel{\underline{\ll}}_{\mathrm{T}}l$” meaning that $l$ is sufficiently
large relative to $m$ and the Herbrand function of a minimal splitting field
for $\mathrm{T}$ (see Notation 2.3.1(vii)).

Our more precise version of Theorem 1.1.1 is as follows; note that it has
individual assertions that are more precise versions of the corresponding
assertions of Theorem 1.1.1.

######  Theorem 1.2.1.

  1. (i)

Suppose $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$,
set $h=h(F,\mathrm{T})$, and suppose $m$ is a positive integer, with
$0<m+3h(F,\mathrm{T})\mathrel{\underline{\ll}}_{\mathrm{T}}l$. Then there
exists a (unique) congruent isomorphism

$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}.$

Moreover, if $m+3h(F,\mathrm{T})+1\mathrel{\underline{\ll}}_{\mathrm{T}}l$,
then this isomorphism respects the minimal congruent filtration, i.e., takes
the image of $\mathrm{T}(F)_{r}$ to that of
$\mathrm{T}^{\prime}(F^{\prime})_{r}$, for $0\leq r\leq m$.

  2. (ii)

The isomorphisms of (i) satisfy the following functoriality. Whenever
$(F,\mathrm{T}_{i})\leftrightarrow_{l}(F^{\prime},\mathrm{T}_{i}^{\prime})$
for $i=1,2$, with the same underlying $F\leftrightarrow_{l}F^{\prime}$, and
$0<m+3h(F,\mathrm{T}_{i})\mathrel{\underline{\ll}}_{\mathrm{T}_{j}}l$ for
$i,j\in\\{1,2\\}$, and we are given homomorphisms
$\mathrm{T}_{1}\rightarrow\mathrm{T}_{2}$ and
$\mathrm{T}_{1}^{\prime}\rightarrow\mathrm{T}_{2}^{\prime}$ inducing the same
homomorphism $X^{*}(\mathrm{T}_{2})=X^{*}(\mathrm{T}_{2}^{\prime})\rightarrow
X^{*}(\mathrm{T}_{1}^{\prime})=X^{*}(\mathrm{T}_{1})$, the following diagram
is commutative:

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60.229pt\raise-40.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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where the vertical arrows are as in (i), and the horizontal arrows are induced
by the homomorphisms $\mathrm{T}_{1}\rightarrow\mathrm{T}_{2}$ and
$\mathrm{T}_{1}^{\prime}\rightarrow\mathrm{T}_{2}^{\prime}$.

  3. (iii)

In the setting of (i), we have the following compatibility with the Chai-Yu
isomorphisms and Kottwitz homomorphisms, in the sense that the following
diagram is commutative:

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where $\mathrm{T}(F)_{b}$ (resp., $\mathrm{T}(F^{\prime})_{b}$) denotes the
maximal bounded subgroup of $\mathrm{T}(F)$ (resp., $\mathrm{T}(F^{\prime})$),
the left vertical arrow is induced by the Chai-Yu isomorphism of [CY01,
Theorem 8.5], the middle vertical arrow is as in (i), the right vertical arrow
is induced by the
$\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$-equivariant
identification $X_{*}(\mathrm{T})=X_{*}(\mathrm{T}^{\prime})$, and the second
horizontal arrow of either row is the Kottwitz homomorphism.

  4. (iv)

In the setting of (i), if $F$ and $F^{\prime}$ are complete and their residue
field $\kappa_{F}=\kappa_{F^{\prime}}$ is finite, and we assume the stronger
inequality $0<m+4h(F,\mathrm{T})\mathrel{\underline{\ll}}_{\mathrm{T}}l$, we
have the following compatibility with the local Langlands correspondence for
tori. We have a commutative diagram

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where the horizontal arrows are given by the local Langlands correspondence
for tori, the left vertical arrow is induced by the isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$
of (i), and the right vertical arrow is obtained by combining the Deligne
isomorphism $W_{F}/I_{F}^{l}\cong W_{F^{\prime}}/I_{F^{\prime}}^{l}$ together
with the
$\Gamma_{F}/I_{F}^{l}\cong\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$-equivariant
identification
$\hat{\mathrm{T}}=\operatorname{Hom}(X_{*}(\mathrm{T}),\mathbb{C}^{\times})=\operatorname{Hom}(X_{*}(\mathrm{T}^{\prime}),\mathbb{C}^{\times})=\hat{\mathrm{T}}^{\prime}$.
Here, to make sense of the top row (to which the bottom row is analogous),
part of the assertion, implicitly, is that the image of the subset

$\operatorname{Hom}(\mathrm{T}(F)/\mathrm{T}(F)_{m},\mathbb{C}^{\times})\subset\operatorname{Hom}_{\mathrm{cts}}(\mathrm{T}(F),\mathbb{C}^{\times})$

under the local Langlands correspondence is contained in the subset of
$H^{1}(W_{F},\hat{\mathrm{T}})$ obtained by inflation from
$H^{1}(W_{F}/I_{F}^{l},(\hat{\mathrm{T}})^{I_{F}^{l}})=H^{1}(W_{F}/I_{F}^{l},\hat{\mathrm{T}})$.

If further $\mathrm{T}$ is weakly induced in the sense of [KP23], i.e.,
satisfies the condition (T) of [Yu15], i.e., becomes an induced torus after
base-change to some tamely ramified extension, one can replace
$h(F,\mathrm{T}),h(F,\mathrm{T}_{1})$ and $h(F,\mathrm{T}_{2})$ by $0$ in the
above statements.

###  1.3. The case of split tori

The case of split tori is an obvious extension of the $\mathrm{GL}_{1}$-case
covered by [Del84], and is a very special case of [Kaz86].

####  1.3.1. Deligne’s triples.

We first digress to remark on some fine print. For all these considerations,
choosing isomorphisms is important to ensure that various constructions are
well-defined. Thus, Deligne works not with isomorphisms
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l}\cong\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{l}$
of truncated discrete valuation rings, but rather with slightly more
rigidified data in the form of isomorphisms
$(\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l},\mathfrak{p}_{F}/\mathfrak{p}_{F}^{l+1},\epsilon)\rightarrow(\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{l},\mathfrak{p}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{l+1},\epsilon^{\prime})$
of triples, where $\epsilon$ is the obvious map
$\mathfrak{p}_{F}/\mathfrak{p}_{F}^{l+1}\rightarrow\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l}$,
and $\epsilon^{\prime}$ is analogous. Fixing such an isomorphism is what lets
one construct the Deligne isomorphism
$\Gamma_{F}/I_{F}^{l}\cong\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$ of [Del84]
and show that it is well-defined up to an inner conjugation.

####  1.3.2. The case of $\mathrm{GL}_{1}$, from [Del84]

First, if $\mathrm{T}=\mathrm{GL}_{1}/F$ and
$\mathrm{T}^{\prime}=\mathrm{GL}_{1}/F^{\prime}$ compatibly, the required
isomorphism, say when $m=l$, is the isomorphism

(3)
$F^{\times}/(1+\mathfrak{p}_{F}^{l})\rightarrow{F^{\prime}}^{\times}/(1+\mathfrak{p}_{F^{\prime}}^{l})$

constructed by Deligne from the realization $F\leftrightarrow_{l}F^{\prime}$,
in a canonical manner starting from the truncated data, in [Del84, Section
1.2]. A more concrete but slightly less obviously canonical description for
(3) requires it to

  * •

restrict to
$(\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l})^{\times}\cong(\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{l})^{\times}$
on
$\mathfrak{O}_{F}^{\times}/(1+\mathfrak{p}_{F}^{l})=(\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l})^{\times}$,
and

  * •

send the image of a uniformizer $\varpi_{F}$ to that of a uniformizer
$\varpi_{F^{\prime}}$ whenever $\varpi_{F}$ and $\varpi_{F^{\prime}}$ are
compatible under
$\mathfrak{p}_{F}/\mathfrak{p}_{F}^{l+1}\rightarrow\mathfrak{p}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{l+1}$.

####  1.3.3. General split tori

When $\mathrm{T}$ and $\mathrm{T}^{\prime}$ are split but otherwise general,
the datum relating $\mathrm{T}^{\prime}$ to $\mathrm{T}$ amounts to just an
isomorphism $X^{*}(\mathrm{T})\rightarrow X^{*}(\mathrm{T}^{\prime})$ of
abelian groups, say $\chi\mapsto\chi^{\prime}$. Then our isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$
is defined so as to match the images of $t\in\mathrm{T}(F)$ and
$t^{\prime}\in\mathrm{T}^{\prime}(F^{\prime})$ precisely when for each
$\chi\in X^{*}(\mathrm{T})$ identifying with $\chi^{\prime}\in
X^{*}(\mathrm{T}^{\prime})$, the images of $\chi(t)$ and
$\chi^{\prime}(t^{\prime})$ correspond under (3).

For general tori, in the spirit of the above discussion, we find it convenient
to specify the isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$
by forcing compatibilities that characterize it.

###  1.4. Standard and congruent isomorphisms

In addition to the congruent filtration subgroups $\mathrm{T}(F)_{m}$, we will
need the “naive” filtration subgroups
$\mathrm{T}(F)^{\operatorname{naive}}_{r}$ ($r\geq 0$):

(4)
$\mathrm{T}(F)^{\operatorname{naive}}_{r}=\\{t\in\mathrm{T}(F)_{b}\mid\operatorname{val}_{F}(\chi(t)-1)\geq
r,\;\forall\,\chi\in X^{*}(\mathrm{T})\\},$

for a suitable extension $\operatorname{val}_{F}$ of the normalized discrete
valuation on $F$.

When $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$, one
defines:

  1. (i)

for $0<r\mathrel{\underline{\ll}}_{\mathrm{T}}l$, a standard isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$
to be one that matches the images of $t\in\mathrm{T}(F)$ and
$t^{\prime}\in\mathrm{T}^{\prime}(F^{\prime})$ whenever $\chi(t)$ and
$\chi^{\prime}(t^{\prime})$ have images that match under a suitable extension
of (3) (see Definition 3.1.3(i) for more details).

  2. (ii)

a congruent isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$
to be an isomorphism induced by a standard isomorphism after passage to
maximal unramified extensions (see Definition 3.1.3(ii) for more details).

Standard isomorphisms are unique when they exist, and have good functoriality
properties, compatibility with the Kottwitz homomorphism, and (in the case of
complete fields with finite residue field) compatibility with the local
Langlands correspondence. Congruent isomorphisms inherit the first three of
these four properties, and under stronger assumptions the fourth too.

Like with [Gan22], we too make use of an argument following the construction
of the Kottwitz homomorphism for tori (see [KP23, Proposition 11.1.1]): the
following simple yet not entirely obvious fact gets us started (Proposition
3.3.1).

######  Proposition 1.4.1.

If $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$, $F$ is
strictly Henselian, and $0<r\mathrel{\underline{\ll}}_{\mathrm{T}}l$, then a
standard isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$
exists.

The difficulty is that we cannot see an obvious way to descend this to the
non-strictly-Henselian case, without going through congruent isomorphisms.
This difficulty is what motivates congruent isomorphisms for us,
notwithstanding their unpleasantness: indeed, if $\tilde{F}/F$ is a maximal
unramified extension and $\tilde{F}\leftrightarrow_{l}\tilde{F}^{\prime}$
“lies over” $F\leftrightarrow_{l}F^{\prime}$, then a “Galois invariant”
isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})_{m}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$,
on taking Galois invariants, gives us an isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$:
$\mathrm{T}(\tilde{F})_{m}$ has trivial
$\operatorname{Gal}(\tilde{F}/F)$-cohomology, being sort of pro-unipotent over
the (perfect) residue field of $\mathfrak{O}_{F}$ ([KP23, Proposition
13.8.1]).

###  1.5. Using the work of Chai and Yu

Thus, the main question now becomes: when can a standard isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}$
induce an isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})_{m}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$?
This seems to require a much deeper ingredient: the spectacular work of Chai
and Yu ([CY01]).

Under the assumption
$m+3h(F,\mathrm{T})\mathrel{\underline{\ll}}_{\mathrm{T}}l$ of Theorem 1.2.1,
Chai and Yu construct a canonical isomorphism
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{T}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$,
where $\mathcal{T}^{\mathrm{ft}}$ is the finite type Néron model of
$\mathrm{T}$, and ${\mathcal{T}^{\prime}}^{\mathrm{ft}}$ that of
$\mathrm{T}^{\prime}$. The properties that characterize their isomorphism are
implicit in their construction. A careful examination of their construction,
together with a few arguments that are tedious but not difficult (see
Proposition 3.5.1), tells us that their isomorphism
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{T}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
is characterized by the fact that, upon evaluating on
$\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}=\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m}$,
it is induced by a restriction of a standard isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h(F,\mathrm{T})}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{m+h(F,\mathrm{T})}$.
This is the key observation that lets us construct congruent (and hence also
standard) isomorphisms outside the strictly Henselian and “weakly induced”
cases.

Thus, more generally, we define a Chai-Yu isomorphism to be an isomorphism
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{T}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
that, when evaluated at
$\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}=\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m}$,
is induced by a standard isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}$
for some $0<r\mathrel{\underline{\ll}}_{\mathrm{T}}l$ such that
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\subset\mathrm{T}(\tilde{F})_{m}$
and
$\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}\subset\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$.
The existence of a Chai-Yu isomorphism easily gives the existence of a
congruent isomorphism that is tautologically compatible with the Chai-Yu
isomorphism.

###  1.6. The organization of the paper

In Section 2, we define notation and recall some material, as well as provide
some simple arguments such as an explanation as to why various results of
[Del84], though stated for “local fields” (quotient fields of complete
discrete valuation rings with perfect residue field) automatically extend to
the case of Henselian discretely valued fields with perfect residue fields.
This extension is convenient because our arguments involve passage to maximal
unramified extensions, which preserves Henselian-ness but not completeness. We
also review the result of Chai and Yu of interest to us, stating their
isomorphism and articulating the characterization implicit in their work, in
Theorem 2.5.3.

In Section 3, we study standard isomorphisms, congruent isomorphisms and Chai-
Yu isomorphisms. Much of the content of this section has been summarized
above.

In Section 4, we restrict to the case of tori that are weakly induced in the
sense of [KP23]. These tori are much simpler than general tori. In this case,
one can construct a Chai-Yu isomorphism under the milder, natural, assumption
$0<m\mathrel{\underline{\ll}}_{\mathrm{T}}l$ (Proposition 4.2.2), which is
what lets us replace $h(F,\mathrm{T})$ by $0$ in the statement of Theorem
1.2.1 for weakly induced tori.

Finally, in Section 5, we put things together and prove Theorem 1.2.1.

Acknowledgements. We are grateful to Radhika Ganapathy, whose work, especially
[Gan15] and [Gan22], is the primary inspiration for this paper. We also thank
Tasho Kaletha, Arnaud Mayeux, Gopal Prasad and Dipendra Prasad for helpful and
educative discussions as well as encouragement. The recent appearance in
literature of the book [KP23] proved propitious to the writing of this paper,
by allowing us to make several simplifications and improvements.

##  2\. Notation and review.

###  2.1. Discretely valued Henselian fields

We will abbreviate “discretely valued Henselian field” to “DVHF”. We will be
interested in DVHFs with perfect (but not necessarily finite) residue fields,
unlike [Del84], which additionally imposes completeness. This is because we
will need to pass to maximal unramified extensions $\tilde{F}$ of fields $F$
of interest (see [Ber93, just before Corollary 2.4.6]; valued Henselian fields
are the quasi-complete fields of [Ber93, Definition 2.3.1 and Proposition
2.4.3]), and doing so preserves only Henselian-ness, not completeness.

####  2.1.1. Objects associated to a DVHF

For a DVHF $F$, we will denote by $\mathfrak{O}_{F}$ its ring of integers,
$\mathfrak{p}_{F}\subset\mathfrak{O}_{F}$ the maximal ideal of
$\mathfrak{O}_{F}$, $\kappa_{F}=\mathfrak{O}_{F}/\mathfrak{p}_{F}$ its residue
field, and $\operatorname{val}_{F}$ the normalized discrete valuation of $F$
as well as its own extension to any algebraic extension of $F$. Given a field
$F$, it will often be implicitly understood that a separable closure
$F^{\mathrm{sep}}$ has been chosen. For a DVHF $F$, we will write $\Gamma_{F}$
and $I_{F}$ respectively for the absolute Galois group
$\operatorname{Gal}(F^{\mathrm{sep}}/F)$ and the inertia group of $F$, and
$\Gamma_{E/F}$ for the Galois group of any Galois extension $E/F$. Let
$\Gamma_{\kappa_{F}}=\Gamma_{F}/I_{F}$; it is isomorphic to the absolute
Galois group of $\kappa_{F}$. In case $\kappa_{F}$ is finite,
$\Gamma_{\kappa_{F}}$ is topologically generated by a Frobenius element, and
we will write $W_{F}\subset\Gamma_{F}$ for the Weil group, namely the inverse
image in $\Gamma_{F}$ of the subgroup of $\Gamma_{\kappa_{F}}$ abstractly
generated the Frobenius element. Given a DVHF $F$ with perfect residue field,
we will often be interested in separable finitely ramified extensions $E/F$,
which may not be finite or Galois. In such a situation, we will use without
further comment that $E$ is also a DVHF with perfect residue field (thus,
$\kappa_{E}$ is algebraically closed if $E$ contains a maximal unramified
extension of $F$), and denote by $e(E/F)$ the associated ramification degree.

####  2.1.2. Passage to completion

Write $\hat{F}$ for the completion of any DVHF $F$; it is a complete DVHF. Let
$F$ be a DVHF. For the following, we refer to [KP23, Proposition 2.1.6] and
[Ber93, Proposition 2.4.1]. If $E/F$ is a finite separable extension, noting
that $E$ is also a DVHF, the obvious map
$E\otimes_{F}\hat{F}\rightarrow\hat{E}$ of rings is an isomorphism. We have an
equivalence of categories between the category $\operatorname{ext}F$ of finite
separable extensions of $F$ and the analogous category
$\operatorname{ext}\hat{F}$, given by $E\mapsto\hat{E}$, so that choosing an
embedding of $F^{\mathrm{sep}}$ into
${\hat{F}}^{\mathrm{sep}}:=(\hat{F})^{\mathrm{sep}}$ gives a canonical
identification $\Gamma_{F}\rightarrow\Gamma_{\hat{F}}$.

####  2.1.3. Ramification theory

We assume the setting and notation from Subsubsection 2.1.2 above, but assume
also that $\kappa_{F}$ is perfect. Let $E/F$ be a finitely ramified separable
extension. For each $r\in[-1,\infty)$, we have the “lower ramification
(equivalence) relation” $\Xi_{r}$ on
$\operatorname{Hom}_{F-\mathrm{alg}}(E,F^{\mathrm{sep}})$ as in [Del84,
(A.3.3)] (whose $R_{u}$ is our $\Xi_{r}$), under which $\sigma$ and $\tau$ are
equivalent if and only if for some (or equivalently, any) finitely ramified
Galois extension $M/F$ contained in $F^{\mathrm{sep}}$ and containing a normal
closure of $E$, we have $\operatorname{val}_{M}(\sigma(x)-\tau(x))\geq
e(M/E)(r+1)$ for all $x\in\mathfrak{O}_{E}$. When $E=M$ is Galois, the lower
ramification subgroup
$\operatorname{Gal}(E/F)_{r}\subset\operatorname{Gal}(E/F)$ is the equivalence
class of the identity element under $\Xi_{r}$, which is then just “lies in the
same $\operatorname{Gal}(E/F)_{r}$-coset”.

Choosing any $M$ as above, $\operatorname{Gal}(M/F)$ has a well-defined
transitive action on the set of $\Xi_{r}$-equivalence classes, so that they
all have the same cardinality. This cardinality is easily seen to be bounded
above by $e(E/F)$ for $r>-1$: if
$\sigma,\tau\in\operatorname{Hom}_{F-\mathrm{alg}}(E,F^{\mathrm{sep}})$ belong
to the same class under $\Xi_{r}$ with $r>-1$, it is an easy exercise to see
that they agree on the maximal unramified subextension of $E/F$. For $r>-1$,
let $1\leq g_{r}\leq e(E/F)$ be the cardinality of each $\Xi_{r}$-equivalence
class (our $g_{r}$ is the $r_{u}$ of [Del84]). This also lets us define the
Herbrand function associated to each finitely ramified separable extension
$E/F$ by the familiar integral as in [Del84, (A.4.3)], for $r\in[0,\infty)$:

(5) $e(E/F)^{-1}r\leq\varphi_{E/F}(r)=\int_{0}^{r}dt/(g_{0}/g_{t})\leq r.$

$\varphi_{E/F}$ is a piecewise linear self-homeomorphism of $[0,\infty)$. In
fact, one can show that this also defines a self-homeomorphism of
$[-1,\infty)$, but we will only be interested in its values on $[0,\infty)$.
Let its inverse be $\psi_{E/F}$, another self-homeomorphism of $[0,\infty)$.

The following allow us to reduce the study of $\varphi_{E/F}$ and $\psi_{E/F}$
to the case where $E/F$ is finite and $F$ is complete:

  * •

For any subextension $E_{\circ}/F$ of $E/F$ with $e(E_{\circ}/F)=e(E/F)$ —
note that there exist finite such $E_{\circ}/F$ — we have
$\varphi_{E/F}=\varphi_{E_{\circ}/F}$ and $\psi_{E/F}=\psi_{E_{\circ}/F}$. To
see this, make use of the same argument that was used above to prove that
$g_{r}\leq e(E/F)$ for $r>-1$, to see that the value of $g_{r}$ associated to
$E/F$ equals that associated to $E_{\circ}/F$.

  * •

If $E/F$ is finite, it is easy to see that the identification

$\operatorname{Hom}_{F-\mathrm{alg}}(E,F^{\mathrm{sep}})\rightarrow\operatorname{Hom}_{\hat{F}-\mathrm{alg}}(\hat{E},(\hat{F})^{\mathrm{sep}})$

respects each of the equivalence relations $\Xi_{r}$, so that
$\varphi_{E/F}=\varphi_{\hat{E}/\hat{F}}$ and
$\psi_{E/F}=\psi_{\hat{E}/\hat{F}}$ as functions
$[0,\infty)\rightarrow[0,\infty)$.

We claim that for finitely ramified separable extensions $E_{2}/E_{1}/F$, we
have $\varphi_{E_{2}/F}=\varphi_{E_{1}/F}\circ\varphi_{E_{2}/E_{1}}$, or
equivalently $\psi_{E_{2}/F}=\psi_{E_{2}/E_{1}}\circ\psi_{E_{1}/F}$. To see
this, reduce using the arguments above to the case where $E_{2}/F$ is finite,
and then to the case where $F,E_{1}$ and $E_{2}$ are complete, then to the
case where $E_{2}/F$ is Galois, and use [Del84, (A.4.1) and Proposition
A.4.2].

For $E/F$ finitely ramified separable, define the “upper ramification
relations” $\Xi^{r}:=\Xi_{\psi_{E/F}(r)}$. If $E/F$ is Galois, this is the
“belongs to the same coset” relation for the upper ramification subgroup
$\operatorname{Gal}(E/F)^{r}:=\operatorname{Gal}(E/F)_{\psi_{E/F}(r)}\subset\operatorname{Gal}(E/F)$.

When $E/F$ is finite, it is easy to check that
$\operatorname{Hom}_{F-\mathrm{alg}}(E,F^{\mathrm{sep}})\rightarrow\operatorname{Hom}_{\hat{F}-\mathrm{alg}}(\hat{E},(\hat{F})^{\mathrm{sep}})$
preserves the “lower ramification relations” $\Xi_{r}$, and hence (using
$\psi_{E/F}=\psi_{\hat{E}/\hat{F}}$) also the “upper ramification relations”
$\Xi^{r}$. The $\Xi^{r}$ have the following advantage: if $E/F$ is a finite
extension and $M/F$ is a finite Galois extension in $F^{\mathrm{sep}}$
containing a normal closure of $E$, then $\Xi^{r}$ is the same as “lies in the
same $\operatorname{Gal}(M/F)^{r}$-orbit”: to see this, pass to completion and
use [Del84, (A.3.2) and the last sentence of A.4]. This nice behavior under
quotients lets us give $\Gamma_{F}$ an upper ramification filtration
$\\{I_{F}^{r}\\}_{r\geq 0}$: by definition, the upper ramification filtration
subgroup $I_{F}^{r}\subset\Gamma_{F}$ is the subgroup of elements that map to
$\operatorname{Gal}(M/F)^{r}$ for each finite Galois subextension $M/F$ of
$F^{\mathrm{sep}}/F$. Each such map
$I_{F}^{r}\rightarrow\operatorname{Gal}(M/F)^{r}$ is then seen to be
surjective. The isomorphism $\Gamma_{F}\rightarrow\Gamma_{\hat{F}}$ maps
$I_{F}^{r}$ to $I_{\hat{F}}^{r}$ and quotients to an isomorphism
$\Gamma_{F}/I_{F}^{r}\rightarrow\Gamma_{\hat{F}}/I_{\hat{F}}^{r}$, for each
$r\geq 0$. The objects associated to $F$ that we have defined above
($\psi_{E/F},I_{F}^{r}$ etc.) are intrinsic, and their definitions did not
make use of the embedding
$F^{\mathrm{sep}}\hookrightarrow{\hat{F}}^{\mathrm{sep}}$.

####  2.1.4. At most $l$-ramified extensions

Let $F$ be a DVHF with perfect residue field, and $l$ a nonnegative integer. A
separable (algebraic) extension $E/F$ will be called at most $l$-ramified if
for every finite subextension $E_{\circ}/F$ of it, the relation $\Xi^{l}$
associated to $E_{\circ}/F$ is trivial. Using [Del84, Proposition A.6.1], this
can be shown to be equivalent to requiring that $I_{F}^{l}$ fixes any
$F$-algebra embedding $E\hookrightarrow F^{\mathrm{sep}}$. Let
$(\operatorname{ext}F)^{l}$ denote the category of finite at most $l$-ramified
(and hence separable by definition) extensions of $F$. The discussion in the
previous subsubsection implies that the functor $E\mapsto\hat{E}$ from
Subsubsection 2.1.2 induces an equivalence of categories
$(\operatorname{ext}F)^{l}\rightarrow(\operatorname{ext}\hat{F})^{l}$. The
category of the ind-objects of $(\operatorname{ext}F)^{l}$ is equivalent to
the category of the algebraic (not necessarily finite or finitely ramified) at
most $l$-ramified extensions of $F$.

######  Remark 2.1.1.

Here are some properties of a finitely ramified at most $l$-ramified extension
$E/F$, that will be used without further comment in what follows:

  1. (i)

$l(1):=\psi_{E/F}(l)$ is an integer, equal to
$e(E/F)(l+1)-\operatorname{val}_{F}(\text{the different of $E/F$})-1$ if $E/F$
is finite (reduce to the case of $E/F$ finite, and see around [Del84,
Proposition A.6.1]; this uses that $E/F$ is at most $l$-ramified). Note also
that $l\leq l(1)\leq le(E/F)$, with the latter equality holding if and only if
$E/F$ is tamely ramified (use, e.g., (5) and the previous sentence).

  2. (ii)

$I_{F}^{l}=I_{E}^{l(1)}$: reduce to the case where $E/F$ is finite, and note
that for any finite Galois extension $M/E$ we have
$\operatorname{Gal}(M/F)^{l}\subset\operatorname{Gal}(M/E)$, and then:

$\operatorname{Gal}(M/F)^{l}=\operatorname{Gal}(M/E)\cap\operatorname{Gal}(M/F)_{\psi_{M/F}(l)}=\operatorname{Gal}(M/E)_{\psi_{M/E}\circ\psi_{E/F}(l)}=\operatorname{Gal}(M/E)^{\psi_{E/F}(l)}.$

###  2.2. The Krasner-Deligne theory ([Del84])

####  2.2.1. Deligne’s triples

In Subsubsections 2.2.2 and 2.2.3 below, we will write $\mathfrak{T}$ for the
category of triples $(A,\mathfrak{m},\epsilon)$ as in [Del84, Sections 1.1 and
1.4]: $A$ is a truncated discrete valuation ring with perfect residue field,
$\mathfrak{m}$ is a free $A$-module of rank $1$, and
$\epsilon\colon\mathfrak{m}\rightarrow A$ is an $A$-module epimorphism from
$\mathfrak{m}$ to the maximal ideal of $A$. For an object
$S=(A,\mathfrak{m},\epsilon)$ in this category, $l(S)$ will denote the length
of $A$ as an $A$-module.

For each object $S=(A,\mathfrak{m},\epsilon)$ of $\mathfrak{T}$ of length say
$l$, Deligne has defined a category $(\operatorname{ext}S)^{l}$ in [Del84,
Definition 2.7], which we will refer to in this subsection. Its objects are
the “finite flat” objects over $S$ in $\mathfrak{T}$ satisfying an “at most
$l$-ramified” condition, but its morphisms are only certain equivalence
classes of morphisms in $\mathfrak{T}$.

####  2.2.2. Deligne’s triple $\operatorname{Tr}_{l}F$ associated to $F$

If $F$ is a DVHF with perfect residue field, and $l$ a positive integer, we
will write $\operatorname{Tr}_{l}F$ for the object
$(\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l},\mathfrak{p}_{F}/\mathfrak{p}_{F}^{l+1},\epsilon)$
in $\mathfrak{T}$, where
$\epsilon\colon\mathfrak{p}_{F}/\mathfrak{p}_{F}^{l+1}\rightarrow\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l}$
is induced by the inclusion $\mathfrak{p}_{F}\subset\mathfrak{O}_{F}$.

[Del84, Theorem 2.8] says that if $F$ is complete, then $\operatorname{Tr}$
extends to a well-defined functor from $(\operatorname{ext}F)^{l}$ to the
category $(\operatorname{ext}\operatorname{Tr}_{l}F)^{l}$ (see Subsubsection
2.2.1), which is in fact an equivalence of categories. However, even when $F$
is not complete, using the equivalence of categories
$(\operatorname{ext}F)^{l}\rightarrow(\operatorname{ext}\hat{F})^{l}$
(Subsubsection 2.1.4) and the tautological isomorphism
$\operatorname{Tr}_{l}F\rightarrow\operatorname{Tr}_{l}\hat{F}$, we formally
get an equivalence of categories
$(\operatorname{ext}F)^{l}\rightarrow(\operatorname{ext}\operatorname{Tr}_{l}F)^{l}$.
Moreover, it is immediate that this equivalence of categories has an intrinsic
description independent of $\hat{F}$, exactly as the one used for $\hat{F}$ in
[Del84, Theorem 2.8]. In particular, any isomorphism
$\operatorname{Tr}_{l}F\rightarrow\operatorname{Tr}_{l}F^{\prime}$ determines
an equivalence of categories
$(\operatorname{ext}F)^{l}\rightarrow(\operatorname{ext}F^{\prime})^{l}$; this
takes each $E$ in $(\operatorname{ext}F)^{l}$ to some $E^{\prime}$ in
$(\operatorname{ext}F^{\prime})^{l}$ such that we have an isomorphism
$\operatorname{Tr}_{el}E=\operatorname{Tr}_{el}E^{\prime}$ in the category
$(\operatorname{ext}\operatorname{Tr}_{l}F)^{l}=(\operatorname{ext}\operatorname{Tr}_{l}F^{\prime})^{l}$,
where $e=e(E/F)=e(E^{\prime}/F^{\prime})$.

####  2.2.3. Close local fields

The notation $F\leftrightarrow_{l}F^{\prime}$ will mean not only that $F$ and
$F^{\prime}$ are DVHFs with perfect residue fields that are $l$-close in the
sense that we have an isomorphism
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{l}\cong\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{l}$
of rings, but also that the following additional data have been chosen:

  * •

An identification $\operatorname{Tr}_{l}F=\operatorname{Tr}_{l}F^{\prime}$ has
been chosen in $\mathfrak{T}$, as also an equivalence of categories
$U\colon(\operatorname{ext}F)^{l}\rightarrow(\operatorname{ext}F^{\prime})^{l}$
as in Subsubsection 2.2.2 above, which then as in [Del84, Section 3.5],
determines the inner class of an isomorphism
$\Gamma_{F}/I_{F}^{l}\cong\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$.

  * •

An identification
$\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$ from the inner
class mentioned above has been chosen by means of a choice of a fixed
isomorphism
$U((F^{\mathrm{sep}})^{I_{F}^{l}})\rightarrow({F^{\prime}}^{\mathrm{sep}})^{I_{F^{\prime}}^{l}}$
over $F^{\prime}$, as follows:

$\Gamma_{F}/I_{F}^{l}=\operatorname{Aut}_{F-\mathrm{alg}}(({F}^{\mathrm{sep}})^{I_{F}^{l}})\overset{U}{\rightarrow}\operatorname{Aut}_{F^{\prime}-\mathrm{alg}}(U((F^{\mathrm{sep}})^{I_{F}^{l}}))=\operatorname{Aut}_{F^{\prime}-\mathrm{alg}}(({F^{\prime}}^{\mathrm{sep}})^{I_{F^{\prime}}^{l}})=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l},$

where
$U\colon(\operatorname{ext}F)^{l}\cong(\operatorname{ext}F^{\prime})^{l}$ is
now extended to the level of the ind-objects.

We refer to [Del84, Section 3.5, especially Section 3.5(c)] for some of the
details, which do not need the assumption that $F$ is complete.

This involves choices of $F^{\mathrm{sep}}$ and ${F^{\prime}}^{\mathrm{sep}}$
among other things, changing which will change associated objects in an
appropriate sense, e.g., up to an inner automorphism for Galois groups.

####  2.2.4. A variation

Given $F\leftrightarrow_{l}F^{\prime}$, it will be helpful to consider the
following variant of $U$. Let $(\operatorname{ext}F)^{l,+}$ be the category of
embeddings $E\hookrightarrow F^{\mathrm{sep}}$, where $E/F$ is finitely
ramified and at most $l$-ramified. Similarly, we have
$(\operatorname{ext}F^{\prime})^{l,+}$. These categories have the following
advantage which is important for us: between any two objects in them is either
a unique morphism, or none at all, depending on whether the stabilizer of the
former in $\Gamma_{F}$ or $\Gamma_{F^{\prime}}$ contains that of the latter.
Clearly, $U$, considered at the level of ind-objects, defines an equivalence
of categories
$U^{+}\colon(\operatorname{ext}F)^{l,+}\rightarrow(\operatorname{ext}F^{\prime})^{l,+}$;
it sends $E\hookrightarrow F^{\mathrm{sep}}$ to $U(E)\hookrightarrow
U((F^{\mathrm{sep}})^{I_{F}^{l}})\rightarrow({F^{\prime}}^{\mathrm{sep}})^{I_{F^{\prime}}^{l}}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$,
where
$U((F^{\mathrm{sep}})^{I_{F}^{l}})\rightarrow({F^{\prime}}^{\mathrm{sep}})^{I_{F^{\prime}}^{l}}$
is part of the datum $F\leftrightarrow_{l}F^{\prime}$. Moreover, it is an easy
exercise to describe when
$E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ is isomorphic in
$(\operatorname{ext}F^{\prime})^{l,+}$ to $U^{+}(E\hookrightarrow
F^{\mathrm{sep}})$: this is so if and only if $E\hookrightarrow
F^{\mathrm{sep}}$ and $E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$
have the same stabilizer in
$\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$.
Upshot: the datum of an extension $E^{\prime}/F$ and an isomorphism
$U(E)\rightarrow E^{\prime}$, is the same as that of an embedding
$E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ with the same
stabilizer as $E\hookrightarrow F^{\mathrm{sep}}$ in
$\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$.

####  2.2.5. Close extensions of close local fields

Given $F\leftrightarrow_{l}F^{\prime}$, we will typically need to work with
realizations $E\leftrightarrow_{l(1)}E^{\prime}$ involving finitely ramified
at most $l$-ramified extensions $E$ and $E^{\prime}$ of $F$ and $F^{\prime}$
with $U(E)\cong E^{\prime}$, where $l(1)=\psi_{E/F}(l)$ is an integer. We will
use without further mention that, for any such $E^{\prime}$, we have
$\psi_{E/F}(l)=\psi_{E^{\prime}/F^{\prime}}(l)$: use the discussion in [Del84,
Section 1.5.3] (expressing $\psi_{E/F}(l)$ in terms of truncated data).

Let us describe how a choice of an isomorphism $U(E)\rightarrow E^{\prime}$
gives a realization $E\leftrightarrow_{l(1)}E^{\prime}$. As in [Del84,
Construction 3.4.1] (and using a direct limit argument to reduce to the case
of finite extensions), this choice determines an isomorphism
$\operatorname{Tr}_{l(1)}E\rightarrow\operatorname{Tr}_{l(1)}E^{\prime}$; by
Subsubsection 2.2.2 above this does not need the assumption that $F$ is
complete. Moreover, we have obvious choices of the additional data needed to
upgrade this to a realization $E\rightarrow_{l(1)}E^{\prime}$, as follows. An
analogue $U_{E}$ of $U$ for this isomorphism can be obtained by restricting
$U$ to extensions of $E$ and thinking of $U(E)\rightarrow E^{\prime}$ as an
identification, which we can also use to choose $F^{\mathrm{sep}}$ and
${F^{\prime}}^{\mathrm{sep}}$ as algebraic closures of $E$ and $E^{\prime}$,
and get an identification
$U_{E}((F^{\mathrm{sep}})^{I_{E}^{l(1)}})=U((F^{\mathrm{sep}})^{I_{F}^{l}})\rightarrow({F^{\prime}}^{\mathrm{sep}})^{I_{F^{\prime}}^{l}}=({F^{\prime}}^{\mathrm{sep}})^{I_{E^{\prime}}^{l(1)}}$.

######  Remark 2.2.1.

The above construction of $E\leftrightarrow_{l(1)}E^{\prime}$ has the
following properties, which will be used without further mention in what
follows:

  1. (i)

$\Gamma_{E}$ and $\Gamma_{E^{\prime}}$, as realized in
$E\leftrightarrow_{l(1)}E^{\prime}$, identify with the stabilizers of
$E\hookrightarrow F^{\mathrm{sep}}$ in $\Gamma_{F}$ and
$E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ in
$\Gamma_{F^{\prime}}$, and the resulting isomorphism
$\Gamma_{E}/I_{E}^{l(1)}\rightarrow\Gamma_{E^{\prime}}/I_{E^{\prime}}^{l(1)}$
is simply the restriction of
$\Gamma_{F}/I_{F}^{l}\rightarrow\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$.

  2. (ii)

We get a bijection

(6)
$\Gamma_{E/F}\rightarrow(\Gamma_{F}/I_{F}^{l})/(\Gamma_{E}/I_{E}^{l(1)})\rightarrow(\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l})/(\Gamma_{E^{\prime}}/I_{E^{\prime}}^{l(1)})\rightarrow\Gamma_{E^{\prime}/F^{\prime}},$

which is an isomorphism of groups if $E/F$ is Galois, in which case these
groups act compatibly on
$\operatorname{Tr}_{l(1)}E=\operatorname{Tr}_{l(1)}E^{\prime}$ over
$\operatorname{Tr}_{l}F=\operatorname{Tr}_{l}F^{\prime}$.

  3. (iii)

Suppose $L/F$ is a finitely ramified at most $l$-ramified extension
‘containing’ $E/F$. Then any extension $L^{\prime}/F^{\prime}$ together with
an isomorphism $U(L)\rightarrow L^{\prime}$ determines an extension
$L^{\prime}/E^{\prime}$ (via $E^{\prime}\rightarrow U(E)\rightarrow
U(L)\rightarrow L^{\prime}$), together with an isomorphism
$U_{E}(L)=U(L)\rightarrow L^{\prime}$, and vice versa. Given any such
$L^{\prime}$ and $U(L)\rightarrow L^{\prime}$, it is easily verified that the
realization $L\leftrightarrow_{\psi_{L/F}(l)}L^{\prime}$, obtained using the
above construction starting from $F\leftrightarrow_{l}F^{\prime}$ and
$U(L)\rightarrow L^{\prime}$, is the same as the realization
$L\leftrightarrow_{\psi_{L/E}(\psi_{E/F}(l))}L^{\prime}$, obtained using the
above construction starting from $E\leftrightarrow_{\psi_{E/F}(l)}E^{\prime}$
and $U_{E}(L)=U(L)\rightarrow L^{\prime}$.

However, it seems inconvenient to keep track of fixed isomorphisms
$U(E)\rightarrow E^{\prime}$, or even to keep referring to $U$. This is why we
had Subsubsection 2.2.4: the discussion there shows that, given
$E\hookrightarrow F^{\mathrm{sep}}$ in $(\operatorname{ext}F)^{l,+}$, any
$E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ with the same
stabilizer as it in
$\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$ determines a
unique isomorphism $U(E)\rightarrow E^{\prime}$, and hence a realization
$E\leftrightarrow_{l(1)}E^{\prime}$.

######  Notation 2.2.2.

If a realization $F\leftrightarrow_{l}F^{\prime}$ is understood, and we talk
of compatible embeddings $E\hookrightarrow F^{\mathrm{sep}}$ and
$E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$, we will mean that
$E/F$ and $E^{\prime}/F^{\prime}$ are finitely ramified at most $l$-ramified
extensions, and that $E\hookrightarrow F^{\mathrm{sep}}$ and
$E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ have the same
stabilizer in $\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$.
Thus, the compatible embeddings $E\hookrightarrow F^{\mathrm{sep}}$ and
$E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ give a realization
$E\leftrightarrow_{l(1)}E^{\prime}$, where $l(1)=\psi_{E/F}(l)$, “lying over”
$F\leftrightarrow_{l}F^{\prime}$.

######  Remark 2.2.3.

By the above discussion above, Remark 2.2.1 can be stated in terms of
compatible embeddings, with, in particular, Remark 2.2.1(iii) taking the
following shape (with a slight change of notation): if $L_{1}/L_{2}/F$ are
finitely ramified at most $l$-ramified extensions, and $L_{i}\hookrightarrow
F^{\mathrm{sep}}$ and
$L_{i}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ are compatible
embeddings for $i=1,2$, then
$L_{1}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ is the composite of
$L_{2}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ with an embedding
$L_{1}^{\prime}\hookrightarrow L_{2}^{\prime}$. Moreover, with
$l_{1}=\psi_{L_{1}/F}(l)$ and
$l_{2}=\psi_{L_{2}/F}(l)=\psi_{L_{2}/L_{1}}(l_{1})$, the realization
$L_{2}\leftrightarrow_{l_{2}}L_{2}^{\prime}$ produced from
$F\leftrightarrow_{l}F^{\prime}$ is the same as the
$L_{2}\leftrightarrow_{\psi_{L_{2}/L_{1}}(l_{1})}L_{2}^{\prime}$ produced from
the $L_{1}\leftrightarrow_{l_{1}}L_{1}^{\prime}$ in turn produced from
$F\leftrightarrow_{l}F^{\prime}$.

####  2.2.6. Relating the multiplicative groups of close local fields

Suppose that $F\leftrightarrow_{l}F^{\prime}$, and that we have embeddings
$L_{1}\hookrightarrow L_{2}\hookrightarrow F^{\mathrm{sep}}$, with $L_{1}/F$
and $L_{2}/F$ finitely ramified and at most $l$-ramified. Assume that for
$i=1,2$, $L_{i}\hookrightarrow F^{\mathrm{sep}}$ and
$L_{i}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ are compatible
embeddings, so that $L_{1}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$
factors through $L_{2}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$
(Remark 2.2.3). Set $l_{i}=\psi_{L_{i}/F}(l)$ for $i=1,2$. The inclusion
$L_{1}^{\times}\hookrightarrow L_{2}^{\times}$ induces
$L_{1}^{\times}/(1+\mathfrak{p}_{L_{1}}^{l_{1}})\rightarrow
L_{2}^{\times}/(1+\mathfrak{p}_{L_{2}}^{l_{2}})$, because
$l_{2}=\psi_{L_{2}/L_{1}}(l_{1})\leq l_{1}e(L_{2}/L_{1})$, by Remark 2.1.1(i).
Part of the datum defining the map
$\operatorname{Tr}_{l_{1}}L_{1}\rightarrow\operatorname{Tr}_{l_{2}}L_{2}$ (as
in [Del84, Section 1.4]) is an
$\mathfrak{O}_{L_{1}}/\mathfrak{p}_{L_{1}}^{l_{1}}$-linear map
$\mathfrak{p}_{L_{1}}/\mathfrak{p}_{L_{1}}^{l_{1}+1}\rightarrow(\mathfrak{p}_{L_{2}}/\mathfrak{p}_{L_{2}}^{l_{2}+1})^{\otimes
e(L_{2}/L_{1})}$ that sends a generator of the source to a generator of the
target as an $\mathfrak{O}_{L_{2}}/\mathfrak{p}_{L_{2}}^{l_{2}}$-module. The
description of $L_{i}^{\times}/(1+\mathfrak{p}_{L_{i}}^{l_{i}})$ ($i=1,2$) in
terms of $\operatorname{Tr}_{l_{i}}L_{i}$, given in [Del84, Section 1.2] as
the group of homogeneous units of the graded
$\mathfrak{O}_{L_{i}}/\mathfrak{p}_{L_{i}}^{l_{i}}$-algebra
$\bigoplus_{n\in\mathbb{Z}}(\mathfrak{p}_{L_{i}}/\mathfrak{p}_{L_{i}}^{l_{i}+1})^{\otimes
n}$, implies that the map
$L_{1}^{\times}/(1+\mathfrak{p}_{L_{1}}^{l_{1}})\rightarrow
L_{2}^{\times}/(1+\mathfrak{p}_{L_{2}}^{l_{2}})$ can be described in terms of
the extension
$\operatorname{Tr}_{l_{1}}L_{1}\rightarrow\operatorname{Tr}_{l_{2}}L_{2}$, as
obtained by putting together the various
$(\mathfrak{p}_{L_{1}}/\mathfrak{p}_{L_{1}}^{l_{1}+1})^{\otimes
n}\rightarrow(\mathfrak{p}_{L_{2}}/\mathfrak{p}_{L_{2}}^{l_{2}+1})^{\otimes
e(L_{2}/L_{1})n}$ (this does not use the completeness of $F$ or of the
$L_{i}$). Using the isomorphism
$\operatorname{Tr}_{l_{i}}L_{i}\rightarrow\operatorname{Tr}_{l_{i}}L_{i}^{\prime}$,
we get an isomorphism
$L_{i}^{\times}/(1+\mathfrak{p}_{L_{i}}^{l_{i}})\rightarrow{L_{i}^{\prime}}^{\times}/(1+\mathfrak{p}_{L_{i}^{\prime}}^{l_{i}})$.
Now it is clear that, whenever $F\leftrightarrow_{l}F^{\prime}$ and we have
$L_{1}\leftrightarrow_{l_{1}}L_{1}^{\prime}$ and
$L_{2}\leftrightarrow_{l_{2}}L_{2}^{\prime}$ as above, we have an obvious
commutative diagram:

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By the discussion at the end of Subsubsection 2.2.5, if some $L_{i}/F$ is
Galois, then the vertical arrow in the above diagram involving $L_{i}$ is
invariant under $\Gamma_{L_{i}/F}=\Gamma_{L_{i}^{\prime}/F^{\prime}}$.

######  Notation 2.2.4.

Let $F$ be a DVHF with perfect residue field, and $L/F$ be an algebraic
extension. For $r,l>0$, we say that $r\mathrel{\underline{\ll}}_{L}l$ (or
$l\mathrel{\underline{\gg}}_{L}r$), if $L/F$ is finitely ramified, at most
$l$-ramified, and satisfies that $r\leq\psi_{L/F}(l)/e(L/F)$ (usually $l$ will
be an integer for our purposes).

Note that, if $F\leftrightarrow_{l}F^{\prime}$,
$r\mathrel{\underline{\ll}}_{L}l$, and $L\hookrightarrow F^{\mathrm{sep}}$ and
$L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ are compatible
extensions, then with $l(1)=\psi_{E/F}(l)$, the isomorphisms
$\mathfrak{O}_{L}/\mathfrak{p}_{L}^{l(1)}\rightarrow\mathfrak{O}_{L^{\prime}}/\mathfrak{p}_{L^{\prime}}^{l(1)}$
and
$L^{\times}/(1+\mathfrak{p}_{L}^{l(1)})\rightarrow{L^{\prime}}^{\times}/(1+\mathfrak{p}_{L^{\prime}}^{l(1)})$
induce isomorphisms $\mathfrak{O}_{L}/\mathfrak{p}_{L}^{\lceil
e(L/F)r\rceil}\mathfrak{O}_{L}\rightarrow\mathfrak{O}_{L^{\prime}}/\mathfrak{p}_{L^{\prime}}^{\lceil
e(L/F)r\rceil}\mathfrak{O}_{L^{\prime}}$ and
$L^{\times}/(1+\mathfrak{p}_{L}^{\lceil
e(L/F)r\rceil})\rightarrow{L^{\prime}}^{\times}/(1+\mathfrak{p}_{L^{\prime}}^{\lceil
e(L/F)r\rceil})$.

###  2.3. Notation related to tori over DVHFs

######  Notation 2.3.1.

  1. (i)

Henceforth we will subscripting to indicate base-change: the base-change of a
scheme $\mathrm{X}/S$ to $S^{\prime}$ will be denoted by
$\mathrm{X}_{S^{\prime}}$.

  2. (ii)

For any torus $\mathrm{T}$ over a field $F$ with a chosen separable closure
$F^{\mathrm{sep}}$, we will denote by $X^{*}(\mathrm{T})$ and
$X_{*}(\mathrm{T})$ respectively the character lattice and the cocharacter
lattice of the base-change $\mathrm{T}_{F^{\mathrm{sep}}}$ of $\mathrm{T}$ to
$F^{\mathrm{sep}}$, viewed with the obvious action of
$\operatorname{Gal}(F^{\mathrm{sep}}/F)$ on these. If an embedding
$E\hookrightarrow F^{\mathrm{sep}}$ is understood from the context, where
$E/F$ a separable extension splitting $\mathrm{T}$, we may use it to view each
$\chi\in X^{*}(\mathrm{T})$ as a homomorphism
$\mathrm{T}_{E}\rightarrow\mathbb{G}_{m}/E$, $\chi(t)$ as an element of
$E^{\times}$ for $\chi\in X^{*}(\mathrm{T})$ and $t\in\mathrm{T}(E)$, etc.

  3. (iii)

If $\mathrm{T}$ is a torus over a DVHF $F$ with perfect residue field, its ft-
Néron model and connected Néron model (see [KP23, Definition B.8.9]) will be
denoted by $\mathcal{T}^{\mathrm{ft}}$ and $\mathcal{T}$, respectively (thus,
we follow [Gan22] in writing $\mathcal{T}$ for the $\mathcal{T}^{0}$ of
[KP23]).

  4. (iv)

If $\mathrm{T}$ is a torus over a DVHF $F$ with perfect residue field, then
$\mathrm{T}(F)_{b}\subset\mathrm{T}(F)$ will denote its maximal bounded
subgroup; thus, $\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{F})$ identifies with
$\mathrm{T}(F)_{b}$.

  5. (v)

Let $\mathrm{T}$ be a torus over a DVHF $F$ with perfect residue field. We
will consider three filtrations of $\mathrm{T}(F)$:

$\\{\mathrm{T}(F)^{\operatorname{naive}}_{r}\\}_{r\geq
0},\quad\\{\mathrm{T}(F)^{\operatorname{std}}_{r}\\}_{r\geq
0}\quad\text{and}\quad\\{\mathrm{T}(F)_{r}\\}_{r\geq 0}.$

For $\\{\mathrm{T}(F)^{\operatorname{naive}}_{r}\\}_{r\geq 0}$ (defined in
(4)), we have for each $r\geq 0$:

(7)
$\mathrm{T}(F)^{\operatorname{naive}}_{r}=\\{t\in\mathrm{T}(F)_{b}\mid\operatorname{val}_{F}(\chi(t)-1)\geq
r,\;\forall\,\chi\in X^{*}(\mathrm{T})\\}$

(here, each $\chi(t)$ is valued in $F^{\mathrm{sep}}$, and we recall that the
normalized valuation $\operatorname{val}_{F}$ is canonically extended to
algebraic extensions of $F$). The filtration
$\\{\mathrm{T}(F)^{\operatorname{std}}_{r}\\}_{r\geq 0}$, is the standard or
Moy-Prasad filtration of $\mathrm{T}(F)$ (see [KP23, Definition B.5.1]):

(8)
$\mathrm{T}(F)^{\operatorname{std}}_{r}=\\{t\in\mathrm{T}(F)^{0}\mid\operatorname{val}_{F}(\chi(t)-1)\geq
r,\;\forall\,\chi\in X^{*}(\mathrm{T})\\},$

where $\mathrm{T}(F)^{0}$ is the Iwahori subgroup of $\mathrm{T}(F)$ as
defined in [KP23, Definition 2.5.13], a subgroup of finite index of
$\mathrm{T}(F)_{b}$ (which can be strictly contained in $\mathrm{T}(F)_{b}$).
The filtration $\\{\mathrm{T}(F)_{r}\\}_{r\geq 0}$ is the minimal congruent
filtration of $\mathrm{T}(F)$, originally introduced by Yu in [Yu15], but
interpreted as in [KP23, Section B.10]. We have
$\mathrm{T}(F)^{\operatorname{std}}_{r}=\mathrm{T}(F)_{0}\cap\mathrm{T}(F)^{\operatorname{naive}}_{r}$
for each $r\geq 0$, and one can check that
$\mathrm{T}(F)^{\operatorname{naive}}_{0}=\mathrm{T}(F)_{b}$. If $r=m$ is an
integer, then $\mathrm{T}(F)_{m}$ is the group of $\mathfrak{O}_{F}$-points of
what is defined in [KP23, Definition A.5.12] as the $m$-th congruence subgroup
scheme of $\mathcal{T}$ (this is what the “congruent” of “minimal congruent
filtration” refers to). By [KP23, Remark A.5.14], it also equals
$\ker(\mathcal{T}(\mathfrak{O}_{F})\rightarrow\mathcal{T}(\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}))$,
and hence is what [Gan22] denotes as $T_{m}$.

  6. (vi)

In the setting of (v) above, if $E/F$ is a finitely ramified separable
algebraic extension, we set
$\mathrm{T}(E)^{\operatorname{naive}}_{r}:=\mathrm{T}_{E}(E)^{\operatorname{naive}}_{r}$.
In slight contrast, if $E/F$ is unramified (algebraic), we let
$\mathrm{T}(E)_{r}$ be $\mathcal{T}_{r}(\mathfrak{O}_{E})$, where
$\mathcal{T}_{r}$ is the $r$-th minimal congruence filtration group scheme of
$\mathrm{T}$ from [KP23, Definition B.10.8(3)] (thus,
$\mathrm{T}(E)_{r}\cap\mathrm{T}(F)=\mathrm{T}(F)_{r}$, but we do not know if
$\mathrm{T}(E)_{r}$ equals $\mathrm{T}_{E}(E)_{r}$).

  7. (vii)

Let $\mathrm{T}$ be a torus over a DVHF $F$ with perfect residue field, and
let $r,l>0$. We say that $r\mathrel{\underline{\ll}}_{\mathrm{T}}l$ (or
$l\mathrel{\underline{\gg}}_{\mathrm{T}}r$), if there exists a (finitely
ramified at most $l$-ramified) extension $L/F$ that splits $\mathrm{T}$, such
that $r\mathrel{\underline{\ll}}_{L}l$ (see Notation 2.2.4; usually $l$ will
be an integer for our purposes).

  8. (viii)

If we write
$(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$, we will
mean that $F\leftrightarrow_{l}F^{\prime}$, and that
$\mathrm{T},\mathrm{T}^{\prime}$ are tori over $F,F^{\prime}$ such that the
actions of $I_{F}^{l}$ on $X^{*}(\mathrm{T})$ and $I_{F^{\prime}}^{l}$ on
$X^{*}(\mathrm{T}^{\prime})$ are trivial (in other words, $\mathrm{T}$ and
$\mathrm{T}^{\prime}$ are “at most $l$-ramified”), and that an isomorphism
$X^{*}(\mathrm{T})\rightarrow X^{*}(\mathrm{T}^{\prime})$ has been chosen that
is equivariant for the actions of
$\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$ (recall that the
identification $\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$
is part of the datum defining $F\leftrightarrow_{l}F^{\prime}$).

  9. (ix)

Suppose $F\leftrightarrow_{l}F^{\prime}$, and that $E\hookrightarrow
F^{\mathrm{sep}}$ and $E^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$
are compatible embeddings, so (see Subsubsection 2.2.5) we have
$E\leftrightarrow_{l(1)}E^{\prime}$, where $l(1)=\psi_{E/F}(l)$. Suppose
further that
$(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$ and
$(E,\mathrm{S})\leftrightarrow_{l(1)}(E^{\prime},\mathrm{S}^{\prime})$ extend
our $F\leftrightarrow_{l}F^{\prime}$ and $E\leftrightarrow_{l(1)}E^{\prime}$,
respectively. Then we will implicitly work with realizations
$(E,\mathrm{T}_{E})\leftrightarrow_{l(1)}(E^{\prime},\mathrm{T}^{\prime}_{E^{\prime}})$
and
$(F,\mathrm{R}:=\operatorname{Res}_{E/F}\mathrm{S})\leftrightarrow_{l}(F^{\prime},\mathrm{R}^{\prime}:=\operatorname{Res}_{E^{\prime}/F^{\prime}}\mathrm{S}^{\prime})$,
extending $E\leftrightarrow_{l(1)}E^{\prime}$ and
$F\leftrightarrow_{l}F^{\prime}$, obtained from the following identifications:

    1. (a)

$X^{*}(\mathrm{T}_{E})=X^{*}(\mathrm{T})\rightarrow
X^{*}(\mathrm{T}^{\prime})=X^{*}(\mathrm{T}^{\prime}_{E^{\prime}})$, which is
equivariant for $\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$
and hence for
$\Gamma_{E}/I_{E}^{l(1)}=\Gamma_{E^{\prime}}/I_{E^{\prime}}^{l(1)}$; and

    2. (b)

$X^{*}(\mathrm{R})=\operatorname{Ind}_{\Gamma_{E}}^{\Gamma_{F}}X^{*}(\mathrm{S})=\operatorname{Ind}_{\Gamma_{E}/I_{E}^{l(1)}}^{\Gamma_{F}/I_{F}^{l}}X^{*}(\mathrm{S})=\operatorname{Ind}_{\Gamma_{E^{\prime}}/I_{E^{\prime}}^{l(1)}}^{\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}}X^{*}(\mathrm{S}^{\prime})=\operatorname{Ind}_{\Gamma_{E^{\prime}}}^{\Gamma_{F^{\prime}}}X^{*}(\mathrm{S}^{\prime})=X^{*}(\mathrm{R}^{\prime})$,
where we recall that $\mathrm{S}$ and $\mathrm{S}^{\prime}$ are at most
$l(1)$-ramified, use the discussion of (i) and (ii) of Remark 2.2.1, and use
the canonical identifications
$X^{*}(\mathrm{R})=\operatorname{Ind}_{\Gamma_{E}}^{\Gamma_{F}}X^{*}(\mathrm{S})=\mathbb{Z}[\Gamma_{F}]\otimes_{\mathbb{Z}[\Gamma_{E}]}X^{*}(\mathrm{S})$
and
$X^{*}(\mathrm{R}^{\prime})=\operatorname{Ind}_{\Gamma_{E^{\prime}}}^{\Gamma_{F^{\prime}}}X^{*}(\mathrm{S})$
reviewed in Remark 2.3.2 below.

As usual, the above notation will be adapted in obvious ways: for any torus
$\mathrm{T}_{1}$ over any DVHF $F^{\flat}$, we will make sense of
$\mathcal{T}^{\mathrm{ft}}_{1}$ and $\mathrm{T}_{1}(F^{\flat})_{m}$, etc.

######  Remark 2.3.2.

Let $E/F$ be a finite separable field extension, $\mathrm{S}/E$ a torus, and
$\mathrm{R}:=\operatorname{Res}_{E/F}\mathrm{S}$. Let us recall the canonical
realization
$X^{*}(\mathrm{R})=\operatorname{Ind}_{\Gamma_{E}}^{\Gamma_{F}}X^{*}(\mathrm{S})$,
where $\Gamma_{F}=\operatorname{Gal}(F^{\mathrm{sep}}/F)$ and
$\Gamma_{E}=\operatorname{Gal}(F^{\mathrm{sep}}/E)$. We have a “universal”,
surjective, homomorphism $\mathrm{R}_{E}\rightarrow\mathrm{S}_{E}$, which, at
the level of $A$-valued points for an $E$-algebra $A$, is the map
$\mathrm{R}(A)=\mathrm{S}(E\otimes_{F}A)\rightarrow\mathrm{S}(A)$, obtained by
applying the functor $\mathrm{S}$ to the multiplication map
$E\otimes_{F}A\rightarrow A$ of $E$-algebras (where $E\otimes_{F}A$ is an
$E$-algebra via the first factor). This map has a well-known universal
property: for any multiplicative type group scheme $\mathrm{T}$ over $F$,
base-changing to $E$ followed by composition with
$\mathrm{R}_{E}\rightarrow\mathrm{S}_{E}$ gives us a functorial bijection
between homomorphisms $\mathrm{T}\rightarrow\mathrm{R}$ and homomorphisms
$\mathrm{T}_{E}\rightarrow\mathrm{S}_{E}$. Hence, composition with the
injection $X^{*}(\mathrm{S})\rightarrow X^{*}(\mathrm{R})$ dual to
$\mathrm{R}_{E}\rightarrow\mathrm{S}_{E}$ gives a functorial identification
$\operatorname{Hom}_{\Gamma_{E}}(X^{*}(\mathrm{S}),X^{*}(\mathrm{T}))\rightarrow\operatorname{Hom}_{\Gamma_{F}}(X^{*}(\mathrm{R}),X^{*}(\mathrm{T}))$
(the notation $X^{*}(\mathrm{T})$ extends to the case where $\mathrm{T}$ is a
multiplicative type group scheme). Hence Frobenius reciprocity gives an
identification

$X^{*}(\mathrm{R})=\operatorname{Ind}_{\Gamma_{E}}^{\Gamma_{F}}X^{*}(\mathrm{S})=\mathbb{Z}[\Gamma_{F}]\otimes_{\mathbb{Z}[\Gamma_{E}]}X^{*}(\mathrm{S}).$

######  Remark 2.3.3.

We will often use without further comment the following nice property of the
naive filtration (Notation 2.3.1(v)): for any injective homomorphism
$\mathrm{T}_{1}\hookrightarrow\mathrm{T}_{2}$ of tori over a DVHF $F$ with
perfect residue field, $r\geq 0$, and a finitely ramified separable extension
$E/F$,
$\mathrm{T}_{1}(F)^{\operatorname{naive}}_{r}=\mathrm{T}_{1}(F)\cap\mathrm{T}_{2}(F)^{\operatorname{naive}}_{r}=\mathrm{T}_{1}(F)\cap\mathrm{T}_{2}(E)^{\operatorname{naive}}_{e(E/F)r}$.
For the first equality, use that $X^{*}(\mathrm{T}_{2})\rightarrow
X^{*}(\mathrm{T}_{1})$ is surjective. For the second, recall that by the
convention in Notation 2.3.1(vi), each $\mathrm{T}_{i}(E)_{s}$ is defined
using the normalized discrete valuation on $E$.

###  2.4. Weil restriction for tori across close local fields

The following lemma is implicit in the last sentence of [CY01, Section 3.6].

######  Lemma 2.4.1.

Suppose $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$.
Let $L\hookrightarrow F^{\mathrm{sep}}$ and
$L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ be compatible
embeddings, with $L/F$ finite, so that by Notation 2.3.1(ix), we have a
realization
$(F,\mathrm{R})\leftrightarrow_{l}(F^{\prime},\mathrm{R}^{\prime})$, where
$\mathrm{R}=\operatorname{Res}_{L/F}\mathrm{T}_{L}$ and
$\mathrm{R}^{\prime}=\operatorname{Res}_{L^{\prime}/F^{\prime}}\mathrm{T}_{L^{\prime}}^{\prime}$.
Then the “diagonal” inclusions $\mathrm{T}\hookrightarrow\mathrm{R}$ and
$\mathrm{T}^{\prime}\hookrightarrow\mathrm{R}^{\prime}$ induce the same
(necessarily surjective) homomorphisms
$X^{*}(\mathrm{R})=X^{*}(\mathrm{R}^{\prime})\rightarrow
X^{*}(\mathrm{T}^{\prime})=X^{*}(\mathrm{T})$.

###### Proof.

$L^{\prime}/F^{\prime}$ is also finite, so the statement makes sense. By the
universal property of $\operatorname{Res}_{L/F}$, the “diagonal” map
$\mathrm{T}\hookrightarrow\mathrm{R}$ is the unique homomorphism that, when
base-changed to $L$ and composed with
$\mathrm{R}_{L}\rightarrow\mathrm{T}_{L}$, yields the identity map
$\mathrm{T}_{L}\rightarrow\mathrm{T}_{L}$. Dually,
$X^{*}(\mathrm{R})\rightarrow X^{*}(\mathrm{T})$ is the unique homomorphism of
$\Gamma_{F}$-modules that, when viewed as a homomorphism of
$\Gamma_{L}$-modules and pre-composed with the “universal” $\Gamma_{L}$-module
homomorphism $X^{*}(\mathrm{T})\hookrightarrow X^{*}(\mathrm{R})$, yields the
identity. In the previous sentence, we can replace $\Gamma_{F}$ by
$\Gamma_{F}/I_{F}^{l}=\Gamma_{F^{\prime}}/I_{F^{\prime}}^{l}$ and $\Gamma_{E}$
by $\Gamma_{E}/I_{E}^{l(1)}=\Gamma_{E^{\prime}}/I_{E^{\prime}}^{l(1)}$, where
$l(1)=\psi_{L/F}(l)$. Since an analogous assertion applies for
$(F^{\prime},\mathrm{T}^{\prime})$, we are done. ∎

###  2.5. A review of some results of Chai and Yu

Unfortunately, we will need to quote from the proofs, and not just the lemmas,
of [CY01]. Therefore, we summarize what we will need from that paper in this
subsection.

######  Notation 2.5.1.

  1. (i)

If $\mathrm{T}$ is a torus over a DVHF $F$ with perfect residue field, and
$L/F$ is a finite Galois extension splitting $\mathrm{T}$, we will denote by
$h(F,\mathrm{T},L)$ the nonnegative integer
$h(\mathfrak{O}_{F},\mathfrak{O}_{L},\Gamma_{F},X_{*}(\mathrm{T}))$ defined as
in [CY01, Section 8.1, just before the lemma]. If $L/F$ is a minimal splitting
extension for $\mathrm{T}$, i.e., isomorphic to the fixed field of the kernel
of $\Gamma_{F}\rightarrow\operatorname{Aut}(X^{*}(\mathrm{T}))$, we write
$h(F,\mathrm{T})=h(F,\mathrm{T},L)$. By [CY01, Lemma 8.1], whenever
$(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$,
$L\hookrightarrow F^{\mathrm{sep}}$ and
$L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ are compatible
extensions, and $h(F,\mathrm{T},L)<\lfloor\psi_{L/F}(l)/e(L/F)\rfloor$, we
have $h(F,\mathrm{T},L)=h(F^{\prime},\mathrm{T}^{\prime},L^{\prime})$ and
$h(F,\mathrm{T})=h(F,\mathrm{T}^{\prime})$: indeed, if we set
$h=h(F,\mathrm{T},L)$ and $e=e(L/F)$, then $L^{\prime}/F^{\prime}$ splits
$\mathrm{T}^{\prime}$ (minimally if $L$ does), and we have an identification
$\operatorname{Tr}_{e(h+1)}L\cong\operatorname{Tr}_{e(h+1)}L^{\prime}$ over
$\operatorname{Tr}_{h+1}F\cong\operatorname{Tr}_{h+1}F^{\prime}$, feeding into
the hypothesis of [CY01, Lemma 8.1].

  2. (ii)

Suppose $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$.
Let $L/F$ be an at most $l$-ramified finite Galois extension splitting
$\mathrm{T}$, and assume that $L\hookrightarrow F^{\mathrm{sep}}$ and
$L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ are compatible
embeddings. Recall that we have
$(F,\mathrm{R}:=\operatorname{Res}_{L/F}\mathrm{T}_{L})\leftrightarrow_{l}(F^{\prime},\mathrm{R}^{\prime}:=\operatorname{Res}_{L^{\prime}/F^{\prime}}\mathrm{T}_{L^{\prime}})$
(see Notation 2.3.1(ix)). Whenever $0<m\mathrel{\underline{\ll}}_{L}l$, [CY01,
the proof of Proposition 8.4(ii)] gives us an isomorphism

(9)
$\mathcal{R}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{R}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m},$

(we recall that $\mathcal{R}^{\mathrm{ft}}$ (resp.,
${\mathcal{R}^{\prime}}^{\mathrm{ft}}$) is the finite type Néron model of
$\mathrm{R}$ (resp., $\mathrm{R}^{\prime}$)). Explicitly, the identification
$\mathfrak{O}_{L}/\mathfrak{p}_{F}^{m}\mathfrak{O}_{L}=\mathfrak{O}_{L^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}\mathfrak{O}_{L^{\prime}}$
(from $L\leftrightarrow_{\psi_{L/F}(l)}L^{\prime}$) allows us to identify both
sides, at the level of $A$-valued points for an
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}=\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$-algebra
$A$, with

$A\mapsto\operatorname{Hom}(X^{*}(\mathrm{T}),((\mathfrak{O}_{L}/\mathfrak{p}_{F}^{m}\mathfrak{O}_{L})\otimes_{\mathfrak{O}_{F}}A)^{\times})=\operatorname{Hom}(X^{*}(\mathrm{T}^{\prime}),((\mathfrak{O}_{L^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}\mathfrak{O}_{L^{\prime}})\otimes_{\mathfrak{O}_{F^{\prime}}}A)^{\times}).$

######  Remark 2.5.2.

Assume the setting of Notation 2.5.1(ii). The following is from [CY01, Section
8.1].

  1. (i)

Note that
$\operatorname{Res}_{\mathfrak{O}_{L}/\mathfrak{O}_{F}}\mathbb{G}_{m}$ has an
obvious realization as a closed subscheme of the affine space
$\mathbb{A}^{[L:F]+1}/\mathfrak{O}_{F}$ associated to the free
$\mathfrak{O}_{F}$-module $\mathfrak{O}_{L}\oplus\mathfrak{O}_{F}$ of rank
$[L:F]+1$ (with some chosen basis). Using a compatible basis of
$\mathfrak{O}_{L^{\prime}}\oplus\mathfrak{O}_{F^{\prime}}$, we have
$\operatorname{Res}_{\mathfrak{O}_{L^{\prime}}/\mathfrak{O}_{F^{\prime}}}\mathbb{G}_{m}\hookrightarrow\mathbb{A}^{[L:F]+1}/\mathfrak{O}_{F^{\prime}}$.
Note that the obvious isomorphism
$(\mathbb{A}^{[L:F]+1}/\mathfrak{O}_{F})\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow(\mathbb{A}^{[L:F]+1}/\mathfrak{O}_{F^{\prime}})\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
restricts to the isomorphism
$\operatorname{Res}_{\mathfrak{O}_{L}/\mathfrak{O}_{F}}\mathbb{G}_{m}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}\rightarrow\operatorname{Res}_{\mathfrak{O}_{L^{\prime}}/\mathfrak{O}_{F^{\prime}}}\mathbb{G}_{m}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$
defined as in (in fact as a special case of) Notation 2.5.1(ii).

  2. (ii)

Choosing bases $\\{\chi_{i}=\chi_{i}^{\prime}\\}$ of
$X^{*}(\mathrm{T})=X^{*}(\mathrm{T}^{\prime})$, we can realize
$\mathcal{R}^{\mathrm{ft}}$, which sends an $\mathfrak{O}_{F}$-algebra $A$ to
$\operatorname{Hom}(X^{*}(\mathrm{T}),(\mathfrak{O}_{L}\otimes_{\mathfrak{O}_{F}}A)^{\times})$,
as a product of copies of
$\operatorname{Res}_{\mathfrak{O}_{L}/\mathfrak{O}_{F}}\mathbb{G}_{m}$ indexed
by $\\{\chi_{i}=\chi_{i}^{\prime}\\}$, giving using (i) an embedding
$\mathcal{R}^{\mathrm{ft}}\hookrightarrow\mathbb{A}^{\dim\mathrm{T}([L:F]+1)}/\mathfrak{O}_{F}$,
as a closed subscheme. Similarly, we get
${\mathcal{R}^{\prime}}^{\mathrm{ft}}\hookrightarrow\mathbb{A}^{\dim\mathrm{T}([L:F]+1})/\mathfrak{O}_{F^{\prime}}$.
It is immediate that the obvious isomorphism
$(\mathbb{A}^{[L:F]+1}/\mathfrak{O}_{F})\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow(\mathbb{A}^{[L:F]+1}/\mathfrak{O}_{F^{\prime}})\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
restricts to the isomorphism
$\mathcal{R}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{R}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
of Notation 2.5.1(ii).

The following is one of the main results of [CY01]:

######  Theorem 2.5.3 (Chai and Yu).

Let $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$.
Suppose $m$ is a positive integer such that
$m+3h(F,\mathrm{T},L)\mathrel{\underline{\ll}}_{L}l$, for a fixed at most
$l$-ramified finite Galois extension $L/F$ splitting $\mathrm{T}$ (which
exists since
$(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$). Let
$L\hookrightarrow F^{\mathrm{sep}}$ and
$L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ be compatible
embeddings, so that
$(F,\mathrm{R}:=\operatorname{Res}_{L/F}\mathrm{T}_{L})\leftrightarrow_{l}(F^{\prime},\mathrm{R}^{\prime}:=\operatorname{Res}_{L^{\prime}/F^{\prime}}\mathrm{T}_{L^{\prime}})$.
Set $h=h(F,\mathrm{T},L)$. Then there exists a unique isomorphism
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{T}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
satisfying the following property: for some (or equivalently, any) compatible
embeddings $\tilde{F}\hookrightarrow F^{\mathrm{sep}}$ and
$\tilde{F}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$, where
$\tilde{F}/F$ is a maximal unramified extension, evaluating this isomorphism
(resp., (9)) at
$\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}=\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m}$
(resp.,
$\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m+h}=\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m+h}$)
gives us the right-most (resp., the left-most) vertical arrow of an obvious
commutative diagram

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Here, we have identified $\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})$
with $\mathrm{T}(\tilde{F})_{b}$, and hence
$\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m})$
with $\mathrm{T}(\tilde{F})_{b}/\mathrm{T}(\tilde{F})_{m}$, etc. That the left
vertical arrow induces the middle vertical arrow, and the inclusions
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}\subset\mathrm{T}(\tilde{F})_{m}$
and
$\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{m+h}\subset\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$
needed to make sense of the right horizontal arrows in the two rows, are part
of the assertion.

The above description is not present right at the point of statement of [CY01,
Theorem 8.5], but can be assembled from various parts of [CY01]. To help the
reader do so, we will give more references later in this subsection. To this
end, we now make some preparation.

The following assertion is a special case of [CY01, Proposition 4.2],
understood using [CY01, the proof of Proposition 8.4(iii)]:

######  Proposition 2.5.4 (Chai and Yu).

Let $F\leftrightarrow_{l}F^{\prime}$. Let $\mathrm{X},\mathrm{X}^{\prime}$ be
smooth algebraic schemes over $\mathfrak{O}_{F},\mathfrak{O}_{F^{\prime}}$,
$\mathrm{W}\subset\mathrm{X}_{\kappa_{F}}$ and
$\mathrm{W}^{\prime}\subset\mathrm{X}_{\kappa_{F^{\prime}}}^{\prime}$ closed
smooth subschemes, and $\mathrm{Y},\mathrm{Y}^{\prime}$ the dilatations of
$\mathrm{W},\mathrm{W}^{\prime}$ on $\mathrm{X},\mathrm{X}^{\prime}$. Assume
that, for some $0\leq m<l$, we are given an isomorphism
$\mathrm{X}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}\rightarrow\mathrm{X}^{\prime}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$
over
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}=\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$,
that on tensoring with $\kappa_{F}=\kappa_{F^{\prime}}$ identifies
$\mathrm{W}$ with $\mathrm{W}^{\prime}$. Then there is a unique isomorphism
$\mathrm{Y}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow\mathrm{Y}^{\prime}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
over
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}=\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
with the following property: for some (or equivalently, any) compatible
embeddings $\tilde{F}\hookrightarrow F^{\mathrm{sep}}$ and
$\tilde{F}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$, where
$\tilde{F}/F$ is a maximal unramified extension, it maps the image of
$y\in\mathrm{Y}(\mathfrak{O}_{\tilde{F}})\subset\mathrm{X}(\mathfrak{O}_{\tilde{F}})$
in $\mathrm{Y}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$
(by which we mean its image in
$\mathrm{Y}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m})$) to that
of
$y^{\prime}\in\mathrm{Y}^{\prime}(\mathfrak{O}_{\tilde{F}^{\prime}})\subset\mathrm{X}(\mathfrak{O}_{\tilde{F}^{\prime}})$
in
$\mathrm{Y}^{\prime}\times_{\mathfrak{O}_{\tilde{F}^{\prime}}}\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m}$
whenever
$\mathrm{X}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}\rightarrow\mathrm{X}^{\prime}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$
does so.

###### References for the proof.

First we address the uniqueness, assuming the existence. It is immediately
seen that $\tilde{F}^{\prime}/F^{\prime}$ is automatically a maximal
unramified extension. Given
$y\in\mathrm{Y}(\mathfrak{O}_{\tilde{F}})\subset\mathrm{X}(\mathfrak{O}_{\tilde{F}})$,
its image in
$\mathrm{X}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m+1})=\mathrm{X}^{\prime}(\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m+1})$
has image in
$\mathrm{X}(\kappa_{\tilde{F}})=\mathrm{X}^{\prime}(\kappa_{\tilde{F}^{\prime}})$
that belongs to $\mathrm{W}=\mathrm{W}^{\prime}$. Thus, any
$y^{\prime}\in\mathrm{X}^{\prime}(\mathfrak{O}_{\tilde{F}^{\prime}})$ that
lifts this image (such $y^{\prime}$ exist as $\mathrm{X}^{\prime}$ is smooth)
belongs to
$\mathrm{Y}^{\prime}(\mathfrak{O}_{\tilde{F}^{\prime}})\subset\mathrm{X}^{\prime}(\mathfrak{O}_{\tilde{F}^{\prime}})$.
Thus,
$\mathrm{Y}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow\mathrm{Y}^{\prime}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
is pinned down on the image of $\mathrm{Y}(\mathfrak{O}_{\tilde{F}})$ in
$\mathrm{Y}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m})$. Now the
uniqueness follows by the schematic density of this set of points, as asserted
in [CY01, Lemma 8.5.1], which applies since $\mathrm{Y}$ is smooth (see
[BLR90, Section 3.2, Proposition 3], or [KP23, Lemma A.5.10] or [MRR,
Proposition 2.16]). Note that this argument does not use the completeness
assumption of [CY01, Section 8].

Now we discuss the existence. The non-dependence on $\tilde{F}\hookrightarrow
F^{\mathrm{sep}}$ and
$\tilde{F}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ is easy. Write
$\mathrm{X}=\operatorname{Spec}C$ and
$\mathrm{X}^{\prime}=\operatorname{Spec}C^{\prime}$, for an
$\mathfrak{O}_{F}$-algebra $C$ and an $\mathfrak{O}_{F^{\prime}}$-algebra
$C^{\prime}$. Then
$\mathrm{X}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}\rightarrow\mathrm{X}^{\prime}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$
is dual to the inverse of an isomorphism $C/\mathfrak{p}_{F}^{m+1}C\rightarrow
C^{\prime}/\mathfrak{p}_{F^{\prime}}^{m+1}C^{\prime}$. Let the subscheme
$\mathrm{W}\subset\mathrm{X}_{\kappa_{F}}\subset\mathrm{X}$ be defined by an
ideal $I_{W}$ generated by $\varpi_{F},g_{1},\dots,g_{s}$, where
$\varpi_{F}\in\mathfrak{O}_{F}$ is a uniformizer, and $g_{1},\dots,g_{s}\in
C$. Then the subscheme $\mathrm{W}^{\prime}\subset\mathrm{X}^{\prime}$ is
defined by the ideal $I_{W^{\prime}}$ generated by
$\varpi_{F^{\prime}},g_{1}^{\prime},\dots,g_{s}^{\prime}$, where
$\varpi_{F^{\prime}}\in\mathfrak{O}_{F^{\prime}}$ is a uniformizer matching
$\varpi_{F}$ under
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}=\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$,
and where $g_{i}^{\prime}$ matches $g_{i}$ under
$C/\mathfrak{p}_{F}^{m+1}C\rightarrow
C^{\prime}/\mathfrak{p}_{F^{\prime}}^{m+1}C^{\prime}$ for $1\leq i\leq s$.

Over the isomorphism
$C/\mathfrak{p}_{F}^{m+1}C=C^{\prime}/\mathfrak{p}_{F^{\prime}}^{m+1}C^{\prime}$
of
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}=\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$-algebras
lies an isomorphism

(11)
$\frac{C[x_{1},\dots,x_{s}]}{(\varpi_{F}x_{1}-g_{1},\dots,\varpi_{F}x_{s}-g_{s})}\otimes_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}\rightarrow\frac{C^{\prime}[x_{1},\dots,x_{s}]}{(\varpi_{F^{\prime}}x_{1}-g_{1}^{\prime},\dots,\varpi_{F^{\prime}}x_{s}-g_{s})}\otimes_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1},$

induced by sending $x_{i}$ to $x_{i}$ for each $i$. The ring
$C[x_{1},\dots,x_{s}]/(\varpi_{F}x_{1}-g_{1},\dots,\varpi_{F}x_{s}-g_{s})$,
modulo its $\varpi_{F}^{\infty}$-torsion, is the coordinate ring
$\mathfrak{O}_{F}[\mathrm{Y}]$ of $\mathrm{Y}$ (see, e.g., [KP23, the
discussion of Remark A.5.9]). A similar assertion applies with $F^{\prime}$ in
place of $F$. While
$\mathfrak{O}_{F}[\mathrm{Y}]\otimes_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$
and
$\mathfrak{O}_{F^{\prime}}[\mathrm{Y}^{\prime}]\otimes_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
are quotients of the left-hand side and the right-hand side of the isomorphism
(11), it is not obvious that (11) induces an isomorphism between these
quotients. Nevertheless, it does so, by [CY01, Proposition 4.2], as we now
explain.

We will superscript with $[m]$ to denote base-change to
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$ or
$\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$. In particular,
$I_{W}^{[m]}=I_{W}\otimes_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$.
Since $I_{W}\supset\varpi_{F}\mathfrak{O}_{F}[X]$, we get
$\varpi_{F}^{m}I_{W}\supset\varpi_{F}^{m+1}\mathfrak{O}_{F}[X]$. This allows
us to identify $I_{W}^{[m]}=I_{W}/\varpi_{F}^{m}I_{W}$ with
$I_{W^{\prime}}^{[m]}=I_{W^{\prime}}/\varpi_{F^{\prime}}^{m}I_{W^{\prime}}$.
One considers the following diagram:

$\textstyle{(C[x_{1},\dots,x_{s}]/(\varpi_{F}x_{1}-g_{1},\dots,\varpi_{F}x_{s}-g_{s}))^{[m]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\textstyle{\\!\left(\bigoplus_{t\geq
0}\text{Sym}^{t}_{C^{[m]}}I_{W}^{[m]}\right)_{(\varpi_{F})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\textstyle{\\!(\mathfrak{O}_{F}[\mathrm{Y}])^{[m]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C^{\prime}[x_{1},\dots,x_{s}]/(\varpi_{F^{\prime}}x_{1}-g_{1}^{\prime},\dots,\varpi_{F^{\prime}}x_{s}-g^{\prime}_{s}))^{[m]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\textstyle{\\!\left(\bigoplus_{t\geq
0}\text{Sym}^{t}_{{C^{\prime}}^{[m]}}I_{W^{\prime}}^{[m]}\right)_{(\varpi_{F^{\prime}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\\!}$$\textstyle{\\!(\mathfrak{O}_{F^{\prime}}[\mathrm{Y}^{\prime}])^{[m]}}$

where the top middle term refers to the homogeneous localization of
$\bigoplus_{t\geq 0}\text{Sym}^{t}_{C^{[m]}}I_{W}^{[m]}$ at the homogeneous
element of degree $1$ given by the image of $\varpi_{F}\in I_{W}$, so that its
spectrum is an open subset of the scheme
$\mathrm{Bl}^{\prime}(\mathrm{X},\mathrm{W})\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$
defined as in [CY01, Section 4.2.1]. Thus, the right square and its
commutativity, and the fact that the right vertical arrow is an isomorphism,
follow from the canonicity description of [CY01, Section 4.2.1]. The top left
arrow maps the image of $x_{i}$ to $g_{i}/\varpi_{F}$ for each $i$, and the
bottom left arrow is similar, so that the commutativity of the left square is
clear.

Thus, (11) indeed quotients to an isomorphism
$\mathfrak{O}_{F}[\mathrm{Y}]\otimes_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow\mathfrak{O}_{F^{\prime}}[\mathrm{Y}^{\prime}]\otimes_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$,
i.e., an isomorphism
$\mathrm{Y}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow\mathrm{Y}^{\prime}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$.
It remains to show that this morphism is as described in the proposition, so
let $y,y^{\prime}$ be as in it. It suffices to show that
$f(y)=f^{\prime}(y^{\prime})$ in
$\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}=\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m}$
whenever $f$ and $f^{\prime}$ match under (11). Without loss of generality,
$f$ and $f^{\prime}$ are both represented by $x_{i}$ for some $1\leq i\leq s$.
But since we have $\varpi_{F}f=g_{i}$ and
$\varpi_{F^{\prime}}f^{\prime}=g_{i}^{\prime}$, this follows from the fact
that $g_{i}$ and $g_{i}^{\prime}$ match each other under
$C/\mathfrak{p}_{F}^{m+1}C\rightarrow
C^{\prime}/\mathfrak{p}_{F^{\prime}}^{m+1}C^{\prime}$, as do $\varpi_{F}$ and
$\varpi_{F^{\prime}}$ under
$\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+1}\rightarrow\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+1}$,
so that $g_{i}(y)$ and $g_{i}^{\prime}(y^{\prime})$ match under
$\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m+1}\rightarrow\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m+1}$.
∎

###### References for the proof of Theorem 2.5.3.

The uniqueness follows as in the proof of Proposition 2.5.4. Using the latter
argument of [CY01, Remark 8.6] ($\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$,
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$
and
$\mathcal{R}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}$
remain unchanged when $F$ is replaced by its completion), together with [KP23,
Proposition 2.3.4(2)], from which it follows that
$\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})=\mathrm{T}(\tilde{F})_{b}$
is dense in the analogous group associated to the completion of $F$ (so that
the source and target of middle vertical arrow are unchanged when we replace
$F$ by its completion), we may and shall assume that $F$ and $F^{\prime}$ are
complete. This is being done so that we may use results from [CY01, Section
8].

We will denote by $\underline{\mathrm{T}}^{i}$ and
$\underline{\mathrm{R}}^{i}$ the integral models $\underline{T}^{i}$ for
$\mathrm{T}$ and $\underline{R}^{i}$ for $\mathrm{R}$ from [CY01, Sections
3.2, 3.4, 3.6]. By definition,
$\underline{\mathrm{R}}^{0}:=\mathcal{R}^{\mathrm{ft}}$, where
$\mathrm{R}:=\operatorname{Res}_{L/F}\mathrm{T}_{L}$, and
$\underline{\mathrm{T}}^{0}$ (see [CY01, Section 3.6]) is the schematic
closure of $\mathrm{T}$ in $\underline{\mathrm{R}}^{0}$. Thus,
$\underline{\mathrm{T}}^{0}$ is the standard model of $\mathrm{T}$ in the
sense of [KP23, Section B.4]. If $\underline{\mathrm{T}}^{i}$ and
$\underline{\mathrm{R}}^{i}$ are defined, then $\underline{\mathrm{T}}^{i+1}$
(resp., $\underline{\mathrm{R}}^{i+1}$) is the dilatation of a smooth
subscheme
$\mathrm{Z}^{i}\subset\underline{\mathrm{T}}^{i}\times_{\mathfrak{O}_{F}}\kappa_{F}$
on $\underline{\mathrm{T}}^{i}$ (resp.,
$\mathrm{W}^{i}\subset\underline{\mathrm{R}}^{i}\times_{\mathfrak{O}_{F}}\kappa_{F}$
on $\underline{\mathrm{R}}^{i}$). Further, $\underline{\mathrm{T}}^{i}$ can be
realized as the schematic closure of $\mathrm{T}$ in
$\underline{\mathrm{R}}^{i}$ ([CY01, Lemma 3.5]). We have similar objects
$(\underline{\mathrm{T}}^{\prime})^{i}$ and
$(\underline{\mathrm{R}}^{\prime})^{i}$ associated to
$\mathrm{T}^{\prime}/F^{\prime}$.

By [CY01, Corollary 8.2.4], or rather its proof, thanks to the inequalities
$\psi_{L/F}(l)/e(L/F)>2h$ and $\psi_{L/F}(l)/e(L/F)-h\geq m+h$, the
isomorphism
$\underline{\mathrm{R}}^{0}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h}\rightarrow(\underline{\mathrm{R}}^{\prime})^{0}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h}$
given by (ii) of Notation 2.5.1 restricts to an isomorphism
$\underline{\mathrm{T}}^{0}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h}\rightarrow(\underline{\mathrm{T}}^{\prime})^{0}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h}$,
as the following two sentences explain. Recall the chains of inclusions of
closed subschemes
$\underline{\mathrm{T}}^{0}\subset\underline{\mathrm{R}}^{0}\subset\mathbb{A}^{\dim\mathrm{T}\cdot([L:F]+1)}/\mathfrak{O}_{F}$
and
$(\underline{\mathrm{T}^{\prime}})^{0}\subset(\underline{\mathrm{R}^{\prime}})^{0}\subset\mathbb{A}^{\dim\mathrm{T}\cdot({[L:F]+1)}}/\mathfrak{O}_{F^{\prime}}$
from [CY01, Section 8.1 and the beginning of Section 8.3], reviewed in Remark
2.5.2. The isomorphism
$\underline{\mathrm{R}}^{0}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h}\rightarrow(\underline{\mathrm{R}}^{\prime})^{0}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h}$
from (ii) of Notation 2.5.1 has been observed to be a restriction of the
obvious identification of affine spaces
$\mathbb{A}^{\dim\mathrm{T}\cdot([L:F]+1)}/(\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h})=\mathbb{A}^{\dim\mathrm{T}\cdot([L:F]+1)}/(\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h})$,
and the isomorphism
$\underline{\mathrm{T}}^{0}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h}\rightarrow(\underline{\mathrm{T}}^{\prime})^{0}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h}$
is constructed in [CY01, Corollary 8.2.4] to also satisfy this property.

Since
$\underline{\mathrm{T}}^{0}(\mathfrak{O}_{\tilde{F}})\subset\underline{\mathrm{R}}^{0}(\mathfrak{O}_{\tilde{F}})$
identifies with the inclusion
$\mathrm{T}(\tilde{F})_{b}\subset\mathrm{R}(\tilde{F})_{b}$, and
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}=\mathrm{T}(\tilde{F})\cap\mathrm{R}(\tilde{F})_{m+h}$,
and similar assertions hold for $F^{\prime}$, the above paragraph implies that
we indeed have a well-defined middle vertical arrow of (10) making the left
square of that diagram commute.

A positive integer $\delta$ is introduced in [CY01, Section 8.5], and it is
observed that $\delta\leq h$. As observed there (the invocation of [CY01,
Lemma 5.5]) and using our assumption that $m+3h\leq l$, working with
$F^{\prime}$ instead of $F$ does not change $\delta$. Thus, as observed at
[CY01, the beginning of Section 8.5.2], $\mathcal{T}^{\mathrm{ft}}$ equals
$\underline{\mathrm{T}}^{h}$ and ${\mathcal{T}^{\prime}}^{\mathrm{ft}}$ equals
$(\underline{\mathrm{T}}^{\prime})^{h}$.

$\underline{\mathrm{R}}^{h}$ and $(\underline{\mathrm{R}}^{\prime})^{h}$ are
obtained from $\underline{\mathrm{R}}^{0}$ and
$(\underline{\mathrm{R}}^{\prime})^{0}$ by a series of $h$ dilatations. At the
$i$-th step, one inductively assumes given an identification
$\underline{\mathrm{R}}^{i-1}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h+1-i}\rightarrow(\underline{\mathrm{R}}^{\prime})^{i-1}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h+1-i}$,
and dilatates $\underline{\mathrm{R}}^{i-1}$ and
$(\underline{\mathrm{R}}^{\prime})^{i-1}$ with respect to the same subscheme
of
$\underline{\mathrm{R}}^{i-1}\times_{\mathfrak{O}_{F}}\kappa_{F}=\underline{(}\mathrm{R}^{\prime})^{i-1}\times_{\mathfrak{O}_{F^{\prime}}}\kappa_{F^{\prime}}$
to get $\underline{\mathrm{R}}^{i}$ and
$(\underline{\mathrm{R}}^{\prime})^{i}$, yielding by Proposition 2.5.4 an
isomorphism
$\underline{\mathrm{R}}^{i}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h-i}\rightarrow(\underline{\mathrm{R}}^{\prime})^{i}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h-i}$
(see [CY01, Proposition 8.4]). It therefore follows from $h$-many applications
of Proposition 2.5.4 that there is a unique isomorphism
$\underline{\mathrm{R}}^{h}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow(\underline{\mathrm{R}}^{\prime})^{h}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
that maps the image of
$y\in\underline{\mathrm{R}}^{h}(\mathfrak{O}_{\tilde{F}})\subset\mathcal{R}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})$
to that of
$y^{\prime}\in(\underline{\mathrm{R}}^{\prime})^{h}(\mathfrak{O}_{\tilde{F}^{\prime}})\subset{\mathcal{R}^{\prime}}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}^{\prime}})$
whenever the image of $y$ maps to that of $y^{\prime}$ under
$\underline{\mathrm{R}}^{0}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m+h}\rightarrow(\underline{\mathrm{R}}^{\prime})^{0}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m+h}$
(these applications are justified by the fact that the
$\underline{\mathrm{R}}^{i}$ and the $(\underline{\mathrm{R}}^{\prime})^{i}$
are smooth, unlike the $\underline{\mathrm{T}}^{i}$ and the
$(\underline{\mathrm{T}}^{\prime})^{i}$).

It is argued in [CY01, Section 8.5.2] that
$\underline{\mathrm{R}}^{h}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow(\underline{\mathrm{R}}^{\prime})^{h}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
restricts to an isomorphism
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}=\underline{\mathrm{T}}^{h}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow(\underline{\mathrm{T}}^{\prime})^{h}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}={\mathcal{T}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$.
Hence, under this isomorphism, the image of
$y\in\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})\subset\mathcal{R}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})$
maps to that of
$y^{\prime}\in{\mathcal{T}^{\prime}}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}^{\prime}})\subset{\mathcal{R}^{\prime}}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}^{\prime}})$
whenever, under the middle vertical arrow of (10), the image of $y$ maps to
that of $y^{\prime}$. This makes sense of the right square of (10) and gives
its commutativity. ∎

Note that in the above proof, the implicit assertions such as that
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}\subset\mathrm{T}(\tilde{F})_{m}$
have been implicitly taken care of. This could be compared with the easier
containment
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\supset\mathrm{T}(\tilde{F})_{r}$
(use [KP23, Propositions B.10.4 and B.10.13]), which can be proper for tori
that are not “weakly induced”. In any case, let us record the containment
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}\subset\mathrm{T}(\tilde{F})_{m}$,
since it applies in greater generality (without assuming
$m+3h(F,\mathrm{T})\mathrel{\underline{\ll}}_{\mathrm{T}}l$), and does not
need the strength of [CY01, Proposition 4.2] (i.e., of Proposition 2.5.4):

######  Lemma 2.5.5.

Let $\mathrm{T}$ be a torus over a DVHF $F$ with perfect residue field. Assume
that $h:=h(F,\mathrm{T})<\lfloor\psi_{L/F}(l)/e(L/F)\rfloor$ for some at most
$l$-ramified finite Galois extension $L/F$ splitting $\mathrm{T}$. Then for
any positive integer $m$, we have
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}=\mathrm{T}(\tilde{F})^{\operatorname{std}}_{m+h}\subset\mathrm{T}(\tilde{F})_{m}$.

######  Remark 2.5.6.

In the situation of the above lemma, we claim that whenever
$F\leftrightarrow_{m+h}F^{\prime}$, and $\tilde{F}\hookrightarrow
F^{\mathrm{sep}}$ and
$\tilde{F}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ are compatible
embeddings, where $\tilde{F}/F$ is a maximal unramified extension, we also
have an analogous containment
$\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{m+h}=\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{std}}_{m+h}\subset\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$:
this is because $h(F,\mathrm{T})=h(F^{\prime},\mathrm{T}^{\prime})$ by the
discussion of Notation 2.5.1(i).

###### Proof of Lemma 2.5.5.

It suffices to show that
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}\subset\mathrm{T}(\tilde{F})_{m}$,
since it will then follow that

$\mathrm{T}(\tilde{F})^{\operatorname{std}}_{m+h}\subset\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}=\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}\cap\mathrm{T}(\tilde{F})_{m}\subset\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{m+h}\cap\mathrm{T}(\tilde{F})_{0}=\mathrm{T}(\tilde{F})^{\operatorname{std}}_{m+h}.$

Let $\mathrm{R}=\operatorname{Res}_{L/F}\mathrm{T}$. Since the connected Néron
model $\mathcal{T}$ is obtained from ${\mathcal{T}}^{\mathrm{ft}}$ by
dilatating with respect to the identity component of the special fiber, it
follows that the group
$\mathrm{T}(\tilde{F})_{m}=\ker(\mathcal{T}(\mathfrak{O}_{\tilde{F}})\rightarrow\mathcal{T}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}))$
also equals
$\ker(\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})\rightarrow\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}))$.
The kernel of
$\mathcal{R}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})\rightarrow\mathcal{R}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m+h})$
equals $\mathrm{R}(\tilde{F})_{m+h}$, which equals
$\mathrm{R}(\tilde{F})^{\operatorname{naive}}_{m+h}$ by [KP23, Corollary
B.10.13] (and the equality $\mathcal{R}=\mathcal{R}^{\mathrm{ft}}$).

We will use notation from the above (outline of) proof. It suffices to show
that, if $t\in\underline{\mathrm{T}}^{0}(\mathfrak{O}_{\tilde{F}})$ has
trivial image in
$\mathcal{R}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m+h})=\underline{\mathrm{R}}^{0}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m+h})$,
then $t$ has trivial image in
$\underline{\mathrm{R}}^{h}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m})$
(and hence belongs to
$\ker(\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}})\rightarrow\mathcal{T}^{\mathrm{ft}}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}))$).
Indeed, one shows by induction on $0\leq i\leq h$ that $t$ has trivial image
in
$\underline{\mathrm{R}}^{i}(\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m+h-i})$.
The induction step is as in the arguments around and below (11) in the proof
of Proposition 2.5.4: if the coordinate ring of $\mathrm{R}^{i}$ is $C$, write
the coordinate ring of $\mathrm{R}^{i+1}$ as a quotient of
$C[x_{1},\dots,x_{s}]/(\varpi_{F}x_{1}-g_{1},\dots,\varpi_{F}x_{s}-g_{s})$,
and note that the induction hypothesis that
$f(t)\in\varpi_{F}^{m+h-i}\mathfrak{O}_{\tilde{F}}$ for all $f\in C$ implies
$f(t)\in\varpi_{F}^{m+h-i-1}\mathfrak{O}_{\tilde{F}}$ for all $f\in
C[x_{1},\dots,x_{s}]/(\varpi_{F}x_{1}-g_{1},\dots,\varpi_{F}x_{s}-g_{s})$. ∎

Note that the proof of the above lemma shows that we can replace
$h=h(F,\mathrm{T})$ by any $i$ such that
$\underline{\mathrm{T}}^{i}=\mathcal{T}^{\mathrm{ft}}$, with
$\underline{\mathrm{T}}^{i}$ as in the proof of Theorem 2.5.3 (e.g., $i$ could
be the $\delta$ of that proof).

##  3\. Standard, congruent and Chai-Yu isomorphisms

###  3.1. The definition of standard and congruent isomorphisms

######  Notation 3.1.1.

Let $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$, and
let $r>0$. Elements
$t\mathrm{T}(F)_{r}^{\operatorname{naive}}\in\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}$
and
$t^{\prime}\mathrm{T}^{\prime}(F^{\prime})_{r}^{\operatorname{naive}}\in\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$
(or by abuse of notation, $t\in\mathrm{T}(F)$ and
$t^{\prime}\in\mathrm{T}^{\prime}(F^{\prime})$) are said to be standard
correspondents of each other if for some (and hence by Lemma 3.2.1 below, any)
compatible embeddings $L\hookrightarrow F^{\mathrm{sep}}$ and
$L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$, such that $L$ splits
$\mathrm{T}$ and $r\mathrel{\underline{\ll}}_{L}l$ (see Notation 2.2.4), the
following holds: for every $\chi=\chi^{\prime}\in
X^{*}(\mathrm{T}^{\prime})=X^{*}(\mathrm{T})$, $\chi(t)$ and
$\chi^{\prime}(t^{\prime})$ have images that match under the isomorphism
$L^{\times}/(1+\mathfrak{p}_{L}^{\lceil
e(L/F)r\rceil})\rightarrow{L^{\prime}}^{\times}/(1+\mathfrak{p}_{L^{\prime}}^{\lceil
e(L/F)r\rceil})$ (described below Notation 2.2.4).

######  Remark 3.1.2.

  1. (i)

Note that the condition in Notation 3.1.1 does not change if $t$ or
$t^{\prime}$ is replaced by another element of
$t\mathrm{T}(F)^{\operatorname{naive}}_{r}$ or
$t^{\prime}\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$ (this
follows from the definition of the naive filtration subgroups
$\mathrm{T}(F)^{\operatorname{naive}}_{r}$ and
$\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$).

  2. (ii)

We will see in Lemma 3.2.1 below that, in the setting of Notation 3.1.1, every
element of $\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}$ has either
a unique standard correspondent in
$\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$,
or none at all (and vice versa). This will sometimes be used without further
comment in what follows.

######  Definition 3.1.3.

Let $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$.

  1. (i)

Let $r>0$ be a positive real number. An isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$
of abelian groups is said to be a standard isomorphism if it maps every
element of its source to a standard correspondent of it.

  2. (ii)

Let $m$ be a positive integer. An isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)_{m}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})_{m}$
is said to be a congruent isomorphism if for some (or equivalently by Lemma
3.4.1(i) below, any) compatible embeddings $\tilde{F}\hookrightarrow
F^{\mathrm{sep}}$ and
$\tilde{F}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$, where
$\tilde{F}/F$ is a maximal unramified extension, and some $r>0$ such that
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\subset\mathrm{T}(\tilde{F})_{m}$
and
$\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}\subset\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$,
the given isomorphism is a restriction of an isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})_{m}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$,
which in turn is induced by a standard isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}$
associated to
$(\tilde{F},\mathrm{T}_{\tilde{F}})\leftrightarrow_{l}(\tilde{F}^{\prime},\mathrm{T}^{\prime}_{\tilde{F}^{\prime}})$.

  3. (iii)

Let $m$ be a positive integer. An isomorphism
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{T}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
is said to be a Chai-Yu isomorphism if for some (or equivalently by Lemma
3.4.1(i) below, any) compatible embeddings $\tilde{F}\hookrightarrow
F^{\mathrm{sep}}$ and
$\tilde{F}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$, where
$\tilde{F}/F$ is a maximal unramified extension, and some $r>0$ such that
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\subset\mathrm{T}(\tilde{F})_{m}$
and
$\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}\subset\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$,
the isomorphism
$\mathrm{T}(\tilde{F})_{b}/\mathrm{T}(\tilde{F})_{m}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{b}/\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$
obtained by evaluating the given isomorphism at
$\mathfrak{O}_{\tilde{F}}/\mathfrak{p}_{\tilde{F}}^{m}=\mathfrak{O}_{\tilde{F}^{\prime}}/\mathfrak{p}_{\tilde{F}^{\prime}}^{m}$
is induced by a “restricted standard isomorphism”
$\mathrm{T}(\tilde{F})_{b}/\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{b}/\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}$,
namely, one that maps each element of its source to a standard correspondent
of it, for the realization
$(\tilde{F},\mathrm{T}_{\tilde{F}})\leftrightarrow_{l}(\tilde{F}^{\prime},\mathrm{T}^{\prime}_{\tilde{F}^{\prime}})$.

The following remark helps make sense of the above definition.

######  Remark 3.1.4.

  1. (i)

Note that, by definition, a standard isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$
does not exist unless $0\leq r\mathrel{\underline{\ll}}_{\mathrm{T}}l$, i.e.,
$r\leq\psi_{L/F}(l)/e(L/F)$ for some finitely ramified at most $l$-ramified
separable extension $L/F$ splitting $\mathrm{T}$. Similarly, a congruent
isomorphism
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})_{m}\rightarrow\mathrm{T}^{\prime}(\tilde{F}^{\prime})/\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$
or a Chai-Yu isomorphism
$\mathcal{T}^{\mathrm{ft}}\times_{\mathfrak{O}_{F}}\mathfrak{O}_{F}/\mathfrak{p}_{F}^{m}\rightarrow{\mathcal{T}^{\prime}}^{\mathrm{ft}}\times_{\mathfrak{O}_{F^{\prime}}}\mathfrak{O}_{F^{\prime}}/\mathfrak{p}_{F^{\prime}}^{m}$
does not exist unless there exist $r\geq 0$ such that
$l\mathrel{\underline{\gg}}_{\mathrm{T}}r$,
$\mathrm{T}(\tilde{F})^{\operatorname{naive}}_{r}\subset\mathrm{T}(\tilde{F})_{m}$
and
$\mathrm{T}^{\prime}(\tilde{F}^{\prime})^{\operatorname{naive}}_{r}\subset\mathrm{T}^{\prime}(\tilde{F}^{\prime})_{m}$
(these latter conditions force $r\geq m$, since
$\mathrm{T}(F)_{r}\subset\mathrm{T}(F)^{\operatorname{naive}}_{r}$).

  2. (ii)

In Definition 3.1.3(ii), the notion of restriction from
$\mathrm{T}(\tilde{F})/\mathrm{T}(\tilde{F})_{m}$ to
$\mathrm{T}(F)/\mathrm{T}(F)_{m}$ makes sense, because
$\mathrm{T}(\tilde{F})_{m}\cap\mathrm{T}(F)=\mathrm{T}(F)_{m}$ (see Notation
2.3.1(vi)). For $L/F$ finitely ramified separable with $e=e(L/F)$, since
$\mathrm{T}(L)^{\operatorname{naive}}_{er}\cap\mathrm{T}(F)=\mathrm{T}(F)^{\operatorname{naive}}_{r}$
(Remark 2.3.3), we may restrict from
$\mathrm{T}(L)/\mathrm{T}(L)^{\operatorname{naive}}_{er}$ to
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}$.

  3. (iii)

To relate the definition of a Chai-Yu isomorphism to isomorphisms constructed
in [CY01], see Proposition 3.5.1 below.

###  3.2. Some first properties of standard isomorphisms

######  Lemma 3.2.1.

Let $(F,\mathrm{T})\leftrightarrow_{l}(F^{\prime},\mathrm{T}^{\prime})$, and
let $r>0$. If $t\in\mathrm{T}(F)$ and
$t^{\prime}\in\mathrm{T}^{\prime}(F^{\prime})$ satisfy the conditions in the
definition of a standard correspondent (Notation 3.1.1) with respect to some
choice of $L\hookrightarrow F^{\mathrm{sep}}$ and
$L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$ (with $L/F$ finitely
ramified, such that $L$ splits $\mathrm{T}$ and
$r\mathrel{\underline{\ll}}_{L}l$), then it satisfies those conditions with
respect to any other such choice, say $L_{1}\hookrightarrow F^{\mathrm{sep}}$
and $L_{1}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$. Moreover,
every element of $\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}$ has
either a unique standard corresondent in
$\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$,
or none at all.

###### Proof.

We first prove the former assertion. Assume first that $L/F$ is minimal, i.e.,
isomorphic to the fixed field of
$\ker(\Gamma_{F}\rightarrow\operatorname{Aut}(X^{*}(\mathrm{T})))$. In this
case, $L\hookrightarrow F^{\mathrm{sep}}$ factors as the composite of
$L_{1}\hookrightarrow F^{\mathrm{sep}}$ and some $F$-algebra embedding
$L\hookrightarrow L_{1}$. Moreover, $L^{\prime}$ is automatically a minimal
splitting extension for $\mathrm{T}^{\prime}$, and we similarly get
$L^{\prime}\hookrightarrow
L_{1}^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$. In this case, the
lemma follows from the commutative diagram of Subsubsection 2.2.6. A similar
argument, with the roles of $L$ and $L_{1}$ swapped, reduces the case of
general $L$ to that of minimal $L$: note that replacing $L$ with a smaller
splitting extension increases $\psi_{L/F}(l)/e(L/F)$, and hence preserves the
relation $r\mathrel{\underline{\ll}}_{L}l$.

To see the uniqueness assertion, setting $e=e(L/F)=e(L^{\prime}/F^{\prime})$
and recalling that
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\subset\mathrm{T}(L)/\mathrm{T}(L)^{\operatorname{naive}}_{er}$,
use the identifications

$\mathrm{T}(L)/\mathrm{T}(L)^{\operatorname{naive}}_{er}\overset{\cong}{\rightarrow}\operatorname{Hom}(X^{*}(\mathrm{T}),L^{\times}/(1+\mathfrak{p}_{L}^{\lceil
er\rceil}))\overset{\cong}{\rightarrow}\operatorname{Hom}(X^{*}(\mathrm{T}^{\prime}),{L^{\prime}}^{\times}/(1+\mathfrak{p}_{L^{\prime}}^{\lceil
er\rceil}))\\\
\overset{\cong}{\rightarrow}\mathrm{T}^{\prime}(L^{\prime})/\mathrm{T}^{\prime}(L^{\prime})^{\operatorname{naive}}_{er},$

where the first map is $t\mapsto(\chi\mapsto\chi(t))$, and note that $t$ and
$t^{\prime}$ are standard correspondents if and only if they define the same
element of this identified object. ∎

######  Remark 3.2.2.

Assume the setting of the above lemma.

  1. (i)

If $\mathrm{T}/F$ is split, a unique standard isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$
exists:

$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\operatorname{Hom}(X^{*}(\mathrm{T}),F^{\times}/1+\mathfrak{p}_{F}^{\lceil
r\rceil})=\operatorname{Hom}(X^{*}(\mathrm{T}^{\prime}),{F^{\prime}}^{\times}/1+\mathfrak{p}_{F^{\prime}}^{\lceil
r\rceil})\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}.$

  2. (ii)

For general $\mathrm{T}$, there is either a unique standard isomorphism

$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r},$

or none at all. Indeed, choose compatible embeddings $L\hookrightarrow
F^{\mathrm{sep}}$ and $L^{\prime}\hookrightarrow{F^{\prime}}^{\mathrm{sep}}$,
with $L/F$ a splitting extension for $\mathrm{T}$ such that
$r\mathrel{\underline{\ll}}_{L}l$. Set $e=e(L/F)=e(L^{\prime}/F^{\prime})$. If
the standard isomorphism
$\mathrm{T}(L)/\mathrm{T}(L)^{\operatorname{naive}}_{er}\rightarrow\mathrm{T}^{\prime}(L^{\prime})/\mathrm{T}^{\prime}(L^{\prime})^{\operatorname{naive}}_{er}$
associated to
$(L,\mathrm{T}_{L})\leftrightarrow_{\psi_{L/F}(l)}(L^{\prime},\mathrm{T}^{\prime}_{L^{\prime}})$
(made sense of using Notation 2.3.1(ix)) restricts to an isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$,
then this restriction defines a standard isomorphism. If not, there is no
standard isomorphism
$\mathrm{T}(F)/\mathrm{T}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}^{\prime}(F^{\prime})/\mathrm{T}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$.

######  Lemma 3.2.3.

Standard isomorphisms have the following functoriality. Let
$(F,\mathrm{T}_{i})\leftrightarrow_{l}(F^{\prime},\mathrm{T}_{i}^{\prime})$
for $i=1,2$, with the same underlying $F\leftrightarrow_{l}F^{\prime}$, and
let $r>0$. Let $f:\mathrm{T}_{1}\rightarrow\mathrm{T}_{2}$ and
$f^{\prime}:\mathrm{T}_{1}^{\prime}\rightarrow\mathrm{T}_{2}^{\prime}$ be
homomorphisms inducing the same homomorphism
$X^{*}(\mathrm{T}_{2}^{\prime})=X^{*}(\mathrm{T}_{2})\rightarrow
X^{*}(\mathrm{T}_{1})=X^{*}(\mathrm{T}_{1}^{\prime})$ at the level of
character lattices. Then, if standard isomorphisms
$\mathrm{T}_{i}(F)/\mathrm{T}_{i}(F)^{\operatorname{naive}}_{r}\rightarrow\mathrm{T}_{i}^{\prime}(F^{\prime})/\mathrm{T}_{i}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}$
exist for $i=1,2$, they are the vertical arrows of the following commutative
diagram:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
40.14836pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{T}_{1}(F)/\mathrm{T}_{1}(F)^{\operatorname{naive}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.05807pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 65.68835pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
65.68835pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{T}_{2}(F)/\mathrm{T}_{2}(F)^{\operatorname{naive}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
104.29672pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{T}_{1}^{\prime}(F^{\prime})/\mathrm{T}_{1}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
46.50809pt\raise-35.20334pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30556pt\hbox{$\scriptstyle{f^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.14836pt\raise-41.87pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.14836pt\raise-41.87pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{T}_{2}^{\prime}(F^{\prime})/\mathrm{T}_{2}^{\prime}(F^{\prime})^{\operatorname{naive}}_{r}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

######  Remark 3.2.4.

The lemma would be immediate if we had $r\mathrel{\underline{\ll}}_{L}l$ for
some $L/F$ splitting both $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$. We are not
making this assumption, hence the longer proof.

###### Proof of Lemma 3.2.3.

The assertion of the lemma is equivalent to the following statement: if
$t_{1}\in\mathrm{T}_{1}(F)$ is a standard correspondent of
$t_{1}^{\prime}\in\mathrm{T}_{1}^{\prime}(F^{\prime})$, then
$t_{2}:=f(t_{1})\in\mathrm{T}_{2}(F)$ is a standard correspondent of
$t_{2}^{\prime}:=f^{\prime}(t_{1}^{\prime})\in\mathrm{T}_{2}^{\prime}(F^{\prime})$.

First, we consider a slightly different situation. We make the stronger