# Explicit Serre weights for two-dimensional Galois representations over a
ramified base

Misja F.A. Steinmetz  Mathematical Institute, Leiden University, Niels Bohrweg
1, 2333 CA, Leiden, the Netherlands<EMAIL_ADDRESS>
(Date: February 2020)

###### Abstract.

By the work of a seminal paper of Buzzard, Diamond and Jarvis [BDJ10] and
extensions thereof (culminating in [BLGG13]), we have a formulation of Serre’s
modularity conjecture for Hilbert modular forms over totally real number
fields. For a precise formulation of the conjecture the authors need to
describe a subspace of extensions using $p$-adic Hodge theory. Assuming that
$p$ is unramified in the base field Dembélé, Diamond and Roberts [DDR16] give
an explicit conjectural reformulation of this subspace using local class field
theory and the Artin-Hasse exponential instead of $p$-adic Hodge theory.
Calegari, Emerton, Gee and Mavrides [CEGM17] proved that, indeed, the
reformulated version of the conjecture was equivalent to the original
formulation of the conjecture. In this paper we extend the techniques of
[DDR16] and its proof in [CEGM17] to hold without simplifying assumptions.

##  1\. Introduction

For a totally real field $F$, a prime $p$ and a mod $p$ Galois representation
$\rho:G_{F}\to\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$, the question
whether this representation can arise as the reduction of the Galois
representation associated to a Hilbert modular form was first rigorously
studied in [BDJ10] under the hypothesis that $p$ is unramfied in $F$. This
being a generalisation of Serre’s famous conjecture [Ser87] for
$F=\mathbf{Q}$, it was long believed that a similar conjecture could be
formulated for totally real fields. However, in the conjecture of [BDJ10], the
authors were for the first time able to predict the set of weights of Hilbert
modular forms leading to the given representation precisely. The
unramifiedness hypothesis was later removed in Schein [Sch08], [Gee11] and
[BLGG13]. Building on previous work of Gee and co-authors it was proved in
[GLS15] (and in [Wan17] for $p=2$) that the predicted set of weights of
[BDJ10] and its extensions is correct in the sense that if $\rho$ comes from a
Hilbert modular form then it must come from a form of weight predicted by the
conjectures and every predicted weight occurs.

One unifying aspect of all these conjectures is that the sets of weights are
predicted using abstract $p$-adic Hodge theory making it complicated to do
explicit computations in many cases. This problem was addressed in the paper
[DDR16] under the assumption that $p$ is unramified in $F$. The authors
conjecture an alternative explicit formulation of the predicted sets of
weights in terms of local class field theory and the Artin–Hasse exponential.
Subsequently, it was proved in [CEGM17] that their reformulation was
equivalent to the original conjectures. The main result of this paper is the
generalisation of the conjecture of [DDR16] as well as a a generalisation of
the proof of the equivalence of the two conjectures of [CEGM17] to the case of
an arbitrary totally real field $F$. As an immediate consequence, we can give
an alternative, explicit and equivalent formulation of the sets of weights
appearing in the conjectures on the modularity of two-dimensional mod $p$
Galois representations over an arbitrary totally real field.

###  1.1. Notation

In this paper $p$ will denote any prime number and $K$ will be a finite
extension of $\mathbf{Q}_{p}$ with ramification index $e$, residue degree $f$
and residue field $k$. We will fix all necessary algebraic closures and
embeddings between them; $G_{K}:=\mathrm{Gal}(\overline{K}/K)$ denotes the
absolute Galois group of $K$ and similarly for other fields. We fix an
embedding $\overline{\tau}_{0}\in\mathrm{Hom}(k,\overline{\mathbf{F}}_{p})$
and label the remaining embeddings of the residue field via the rule
$\overline{\tau}_{i+1}^{p}=\overline{\tau}_{i}.$ This is the convention as in
[GLS15] and [CEGM17] and opposite to [DDR16]. We now label the embeddings

$\mathrm{Hom}(K,\overline{\mathbf{Q}}_{p})=\\{\tau_{i,j}\mid
i=0,\dotsc,f-1;\>j=0,\dotsc,e-1\\}$

in any way such that the embedding $k\to\overline{\mathbf{F}}_{p}$ induced by
$\tau_{i,j}$ is $\overline{\tau}_{i}.$ We fix a uniformiser $\pi_{K}$ of $K$
and set $\pi_{0}:=\pi_{K}.$ For each $n\geq 1$ we choose a $p^{n}$-th root of
$\pi_{K}$ such that $\pi_{n+1}^{p}=\pi_{n}.$ We define the rising union
$K_{\infty}:=\bigcup_{n\geq 0}K(\pi_{n});$ when $p=2$ we, furthermore, require
that $\pi_{K}$ was chosen so that $K_{\infty}\cap K_{p^{\infty}}=K$, where
$K_{p^{\infty}}:=\bigcup_{n\geq 1}K(\zeta_{p^{n}})$ with $\zeta_{p^{n}}$ a
primitive $p^{n}$-th root of unity in $\overline{K}$ – it follows from [Wan17,
Lem. 2.1] that this is always possible. For any unramified extension $L$ of
$K$ we let $\mathrm{Frob}_{K}\in\mathrm{Gal}(L/K)$ denote the arithmetic
Frobenius.

Let $\pi$ be a root of $x^{p^{f}-1}+\pi_{K}$ in $\overline{K}$. Recall that we
have a character $\overline{\omega}_{\pi}:G_{K}\to k^{\times}$ defined by
$\sigma\mapsto\sigma(\pi)/\pi\bmod{\pi_{K}}.$ For an embedding
$\overline{\tau}\in\mathrm{Hom}(k,\overline{\mathbf{F}}_{p})$, we get a
character
$\omega_{\overline{\tau}}:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$ via
$\omega_{\overline{\tau}}:=\overline{\tau}\circ\overline{\omega}_{\pi}.$ For
$\overline{\tau}=\overline{\tau}_{i}$ we will sometimes simply denote this
character by $\omega_{i};$ these characters are often called the fundamental
characters of level $f$. The ($p$-adic) cyclotomic character
$\epsilon:G_{K}\to\mathbf{Z}_{p}^{\times}$ of $K$ is defined by
$\sigma(\zeta)=\zeta^{\epsilon(\sigma)}$ for any $\sigma\in G_{K}$ and $\zeta$
any compatible sequence of higher $p$-th roots of unity in
$\mu_{p^{\infty}}(\overline{K})$; by abuse of notation for its reduction
modulo $p$ we will sometimes use the same terminology:
$\epsilon:G_{K}\to\mathbf{F}_{p}^{\times}.$

Let $\mathbf{C}_{K}$ denote the completion of $\overline{K}.$ For an integer
$i,$ we write $\mathbf{C}_{K}(i)$ for the $i$-th Tate twist, i.e. the same
underlying vector space with $G_{K}$ acting as the $i$-th power of the
$p$-adic cyclotomic character. Given a crystalline representation
$\rho:G_{K}\to\mathrm{GL}(V)$ on a $\overline{\mathbf{Q}}_{p}$-vector space
$V$ we define its Hodge–Tate weights as follows: for
$\tau\in\mathrm{Hom}(K,\overline{\mathbf{Q}}_{p})$ we define the
$\tau$-labelled Hodge–Tate weights $\mathrm{HT}_{\tau}(\rho)$ of $\rho$ to be
the multiset of integers containing the integer $i$ with multiplicity

$\dim_{\overline{\mathbf{Q}}_{p}}(V\otimes_{\tau,K}\mathbf{C}_{K}(-i))^{G_{K}},$

where Galois acts diagonally. It follows from the definition that the
cyclotomic character $\epsilon$ has Hodge–Tate weight 1.

###  1.2. Background and outline of the paper

Let us now explain the problem in some more detail and give an overview of how
this paper is structured. We will limit ourselves here to defining the sets of
weights appearing in Serre’s conjecture for totally real fields. For a more
detailed discussion, including a historical overview and motivation of the
problem, we happily invite the reader to have a closer look at the first three
chapters of the author’s PhD thesis [Ste20].

####  1.2.1. Serre weights attached to a representation

Given a representation
$\rho:G_{F}\to\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$, the goal of this
paper is to describe the set of _Serre weights_ $W(\rho)$ attached to $\rho$
explicitly. Let us first define this set of weights.

One of the important aspect of Serre’s conjecture for a totally real field $F$
is that the set of weights $W(\rho)$ associated to this representation only
depends on the restriction of $\rho$ to local representations
${\left.\kern-1.2pt\rho\vphantom{|}\right|_{G_{F_{\mathfrak{p}}}}}:G_{F_{\mathfrak{p}}}\to\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$,
where $\mathfrak{p}\mid p$ is a prime of $F$ dividing $p$ and
$F_{\mathfrak{p}}$ denotes the completion of $F$ at this prime. This allows us
to write the whole paper locally, because we get back to the global case via

$W(\rho):=\left\\{V=\otimes_{\mathfrak{p}\mid
p}V_{\mathfrak{p}}\;\middle|\;V_{\mathfrak{p}}\in
W({\left.\kern-1.2pt\rho\vphantom{|}\right|_{G_{F_{\mathfrak{p}}}}})\text{ for
all }\mathfrak{p}\mid p\right\\}.$

Therefore, we assume from now on
$\rho:G_{K}\to\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p}),$ where $K$ is a
finite extension of $\mathbf{Q}_{p}$ as above. Another important feature is
that the recipe for the set of weights $W(\rho)$ associated to $\rho$ is
already completely explicit in the case that $\rho$ is an irreducible
representation. Since our goal is to obtain an explicit way of describing this
set, we may therefore assume that $\rho$ is reducible.

Let us then suppose that we can write

$\rho\sim\begin{pmatrix}\chi_{1}&*\\\
0&\chi_{2}\end{pmatrix}=\chi_{2}\otimes\begin{pmatrix}\chi&c_{\rho}\\\
0&1\end{pmatrix}$

for some characters
$\chi_{1},\chi_{2}:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$ and where
$\chi:=\chi_{1}\chi_{2}^{-1}.$ It is not hard to check that $c_{\rho}$ defines
a cocycle in $H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)),$ which is well-
defined up to multiplication by a scalar in
$\overline{\mathbf{F}}_{p}^{\times}.$ We will now define a subspace
$L_{V}\subseteq H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ depending on a
so-called _Serre weight_ $V$. The set of weights $W(\rho)$ associated to
$\rho$ is then defined via

$V\in W(\rho)\iff c_{\rho}\in L_{V}.$

In other words, an explicit description of the set of weights $W(\rho)$ is
equivalent to an explicit description of the subspace $L_{V}$ for all Serre
weights $V$. In the remainder of the paper we will try to do the latter.

A Serre weight for $K$ is an irreducible
$\overline{\mathbf{F}}_{p}$-representation of $\mathrm{GL}_{2}(k)$, which is
necessarily of the form

$V_{\underline{\eta},\underline{\theta}}:=\bigotimes_{\lambda\in\mathrm{Hom}(k,\overline{\mathbf{F}}_{p})}(\det{}^{\theta_{\lambda}}\otimes_{k}\mathrm{Sym}^{\eta_{\lambda}-\theta_{\lambda}}k^{2})\otimes_{k,\lambda}\overline{\mathbf{F}}_{p}$

for some uniquely determined integers $\eta_{\lambda},\theta_{\lambda}$ with
$\theta_{\lambda},\eta_{\lambda}-\theta_{\lambda}\in[0,p-1]$ for all $\lambda$
and $\theta_{\lambda}<p-1$ for at least one $\lambda.$ Furthermore, we say
that a crystalline representation
$\widetilde{\rho}:G_{K}\to\mathrm{GL}_{2}(\overline{\mathbf{Q}}_{p})$ has
Hodge type $(\underline{\eta},\underline{\theta})$ if
$\mathrm{HT}_{\tau_{i,0}}(\widetilde{\rho})=\\{\theta_{i},\eta_{i}+1\\}$ and
$\mathrm{HT}_{\tau_{i,j}}=\\{0,1\\}$ for $j>0.$ Following [GLS15] we will say
a representation is pseudo-Barsotti–Tate (or pseudo-BT) of weight
$\\{\eta_{i}+1\\}$ if it has Hodge type $(\eta,\underline{0}).$ We will often
write $\\{r_{i}\\}$ where $r_{i}:=\eta_{i}+1$ for the weight of a pseudo-BT
representation.

Let us now define the distinguished subspaces $L_{V}\subseteq
H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ mentioned above. For any ordered
pair of characters
$\chi_{1},\chi_{2}:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$ and a Serre
weight $V_{\underline{\eta},\underline{\theta}}$, let the distinguished
subspace $L_{V_{\underline{\eta},\underline{\theta}}}(\chi_{1},\chi_{2})$ be
defined as the subset of $H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$
obtained as the reduction of all crystalline representations of Hodge type
$(\underline{\eta},\underline{\theta})$ of the form

$\begin{pmatrix}\widetilde{\chi}_{1}&*\\\
0&\widetilde{\chi}_{2}\end{pmatrix},$

where $\widetilde{\chi}_{1}$ and $\widetilde{\chi}_{2}$ are any crystalline
lifts of $\chi_{1}$ and $\chi_{2},$ respectively. (Note that any such
reduction gives a representation which is an extension of $\chi_{2}$ by
$\chi_{1}$ and, therefore, by the process explained above a class in
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$.) When
$L_{V_{\underline{\eta},\underline{\theta}}}(\chi_{1},\chi_{2})$ is non-empty,
it is a subspace – this follows, for example, from the results of [GLS15].

It follows from the definitions that

$L_{V_{\underline{\eta},\underline{\theta}}}(\chi_{1},\chi_{2})=L_{V_{\underline{\eta}-\underline{\theta},\underline{0}}}(\prod_{i}\omega_{i}^{-\theta_{i}}\otimes\chi_{1},\prod_{i}\omega_{i}^{-\theta_{i}}\otimes\chi_{2}).$

Therefore it suffices to give an explicit description of the distinguished
subspaces (for aribtrary $\chi_{1}$ and $\chi_{2}$) in the case when the
determinant in our Serre weight is trivial, or, equivalently, we only consider
reductions of pseudo-Barsotti–Tate representations in the definition of the
distinguished subspaces.

####  1.2.2. Outline of the paper

By and large, the strategy of this paper generalises the strategies of the
papers [DDR16] and [CEGM17] to the case where $K$ is allowed to be an
arbitrary extension of $\mathbf{Q}_{p}.$ For §3 also the ideas of [GLS15] were
very important.

In §2 we define an explicit basis of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ using local class field theory
and the Artin–Hasse exponential. We do this by first defining a filtration on
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ given in terms of restrictions
to higher ramification groups. We study the jumps in this filtration. Then we
use local class field theory to relate this filtration to a filtration on the
units of a local field $M$ by higher unit groups. The Artin–Hasse exponential
proves to be the perfect tool to get good control of elements in these higher
unit groups increasingly closer to 1. Therefore we can write down a basis of
each graded piece for the filtration by unit groups and the ‘duality’ with
$H^{1}$ us to ‘translate’ this back into a basis of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$

In §3 we first use the classification theorem of [GLS15] to give the
distinguished spaces $L_{V}$ (after restriction to
$H^{1}(G_{K_{\infty}},\overline{\mathbf{F}}_{p}(\chi))$) in terms of
étale-$\varphi$ modules. Next we use Artin–Schreier theory and restriction to
the absolute Galois group of a local field $M_{\infty}$ over which the
character $\chi$ becomes trivial to describe $L_{V}$ as a subspace of
$\mathrm{Hom}(G_{M_{\infty}},\overline{\mathbf{F}}_{p}).$ The basis from §2 is
easily expressed in terms of $G_{M_{\infty}}$ using local class field theory
because it was defined in terms of higher unit groups of $M$. Then we use an
explicitly known formula for the evaluation pairing
$\mathrm{Hom}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})\times
G_{M_{\infty}}\to\overline{\mathbf{F}}_{p}$ to compare the basis elements of
§2 to $L_{V}$.

Our explicit definition of the space $L_{V}$ is given in Definition 3.14. In
this definition some unfamiliar quantities appear such as an indexing set
$J_{V}^{\mathrm{AH}}$ defined in Definition 3.12. All these quantities can be
computed easily and quickly on a computer using any programming language that
supports basic integer operations. This makes our reformulation of the
conjecture really explicit. Nonetheless, the author agrees that it would be
satisfying to find a more direct formula for the indexing set
$J_{V}^{\mathrm{AH}}$ as was done in [DDR16, §7] under the assumption that $K$
is unramified. Some progress (which will hopefully appear as a follow-up
paper) under other simplifying assumptions is made towards this question in
[Ste20, Ch. 7], but in general the problem is still open.

We remark that special cases of §2 and §3 are related to Abrashkin’s papers
[Abr89] and [Abr97]. The first paper implies cases of our conjecture and in
the second paper the Artin–Hasse exponential appears much like in our work.

###  1.3. Acknowledgements

First and foremost I would like to thank my supervisor, Fred Diamond, for his
support and guidance during these past years. I would also like to thank
Victor Abrashkin, Robin Bartlett, Laurent Berger, Dougal Davis, Toby Gee, Pol
van Hoften, Ashwin Iyengar, James Newton and Esther Polak for helpful
mathematical conversations and suggestions.

This paper is the result of the research executed by the author as part of his
PhD programme. This work was supported by the Engineering and Physical
Sciences Research Council [EP/L015234/1]. During the research the author was a
registered student at The EPSRC Centre for Doctoral Training in Geometry and
Number Theory (The London School of Geometry and Number Theory), University
College London. During this time the author was also registered as a student
at the Department of Mathematics, King’s College London, and could often be
found at Imperial College London (who were kind enough to provide him with a
workspace). In addition from February until April of 2019 the author was a
visiting researcher at the Mathematical Institute of Leiden University in the
Netherlands. The author would like to thank all the institutions above for
their support and hospitality.

##  2\. Explicit bases of spaces of extensions

In this section we define an explicit basis of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ in terms of local class field
theory and the Artin–Hasse exponential. In the next section we will use this
basis to give an explicit description of the distinguished subspaces $L_{V}$
from §1.2.1. Let us emphasise that the basis elements in this section may
depend on the choice of uniformiser we make, but in the next section we will
take an appropriate subset of basis elements spanning a subspace which is
independent of this choice.

This section is a generalisation of §3 and §5 of Dembélé, Diamond and Roberts
[DDR16]; we will not continue to mention this throughout. We invite the
curious reader to have a look at [DDR16] for a comparison.

###  2.1. The filtration on $H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$

Let $\pi$ be a root of $x^{p^{f}-1}+\pi_{K}$ in $\overline{K}.$ Recall that we
have a character $\overline{\omega}_{\pi}:G_{K}\to k^{\times}$ defined by
$\sigma\mapsto\sigma(\pi)/\pi.$ For $i=0,\dotsc,f-1,$ we get the fundamental
characters $\omega_{i}:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$ of level
$f$ by defining $\omega_{i}:=\overline{\tau}_{i}\circ\overline{\omega}_{\pi}.$
If $\chi:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$ is any continuous
character, then we can write
${\left.\kern-1.2pt\chi\vphantom{|}\right|_{I_{K}}}=\prod_{i}{\left.\kern-1.2pt\omega_{i}^{a_{i}}\vphantom{|}\right|_{I_{K}}}$
for $a_{i}\in[1,p].$111Our choice of $a_{i}\in[1,p]$ instead of
$a_{i}\in[0,p-1]$ is mainly to be consistent with the conventions of [DDR16].
None of the results in this paper depend on this convention, but some
combinatorial results may change with a different convention – it is possible
that easier combinatorics was the motivation of this choice in [DDR16]
initially. The $f$-tuple $(a_{0},a_{1},\dotsc,a_{f-1})$ is uniquely determined
by ${\left.\kern-1.2pt\chi\vphantom{|}\right|_{I_{K}}}$ if we, furthermore,
require that $a_{i}<p$ for at least one $i$. We will call this $f$-tuple the
tame signature of $\chi$; this is an element of the set

$S:=\\{1,2,\dotsc,p\\}^{f}-\\{(p,p,\dotsc,p)\\}.$

We define an action of
$\mathrm{Gal}(k/\mathbf{F}_{p})=\langle\mathrm{Frob}\rangle\cong\mathbf{Z}/f\mathbf{Z}$
on the set $S$ via

$\mathrm{Frob}\cdot(a_{0},a_{1},\dotsc,a_{f-1})=(a_{1},a_{2},\dotsc,a_{0}).$

Note that this notation makes sense since if $\chi$ has tame signature
$\vec{a},$ then $\mathrm{Frob}\circ\chi$ has tame signature
$\mathrm{Frob}(\vec{a}).$ We define the period of $\vec{a}\in S$ to be the
cardinality of its orbit in $S$ under the action of
$\mathrm{Gal}(k/\mathbf{F}_{p})$. We will call the period of the tame
signature of $\chi$ the absolute niveau of the character. We will write
$f^{\prime}$ for the absolute niveau of $\chi$ and
$f^{\prime\prime}:=f/f^{\prime}$ for the size of the stabiliser of the action
of $\mathrm{Gal}(k/\mathbf{F}_{p}).$ Whenever convenient we will consider the
indices of the $a_{i}$ to be elements of $\mathbf{Z}/f\mathbf{Z},$ so
$a_{f}=a_{0}$ etcetera. We can also write
${\left.\kern-1.2pt\chi\vphantom{|}\right|_{I_{K}}}={\left.\kern-1.2pt\omega_{i}^{n_{i}}\vphantom{|}\right|_{I_{K}}}$
for all $0\leq i<f$, where we define222We note that our definition of $n_{i}$
differs from [DDR16], since we assumed our embeddings to satisfy
$\overline{\tau}_{i+1}^{p}=\overline{\tau}_{i}$ rather than
$\overline{\tau}_{i}^{p}=\overline{\tau}_{i+1}$ as is assumed there.

$n_{i}:=\sum_{j=1}^{f}a_{i+j}p^{f-j}=a_{i+1}p^{f-1}+\dots+a_{i+f-1}p+a_{i}.$

####  2.1.1. Definition of the filtration

Suppose $\chi$ is a continuous character
$G_{K}\to\overline{\mathbf{F}}_{p}^{\times}.$ Recall from [Ser79, Ch. IV,§3]
that we have a decreasing filtration on $G_{K}$ by closed subgroups
$G_{K}^{u}$ given by the upper numbering on $G_{K}$ for $u\in\mathbf{R}$.
Recall also that $G_{K}^{u}=G_{K}$ for $u\leq-1,$ that for $-1<u\leq 0$ we
have $G_{K}^{u}=I_{K},$ the inertia subgroup of $G_{K},$ and that
$\bigcup_{u>0}G_{K}^{u}=P_{K},$ the wild inertia subgroup of $G_{K}.$ For
$s\in\mathbf{R},$ we will define an increasing filtration $\mathrm{Fil}^{s}$
on $H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ by the rule

$\mathrm{Fil}^{s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)):=\bigcap_{u>s-1}\ker\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\xrightarrow{\hskip
5.69054pt\mathrm{res}\hskip
5.69054pt}H^{1}(G^{u}_{K},\overline{\mathbf{F}}_{p}(\chi))\right).$

We will, furthermore, define

$\mathrm{Fil}^{<s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)):=\bigcup_{t<s}\mathrm{Fil}^{t}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$

and define the associated grading by

$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right):=\frac{\mathrm{Fil}^{s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))}{\mathrm{Fil}^{<s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))}.$

####  2.1.2. The filtration for $s\leq 1$

Let us first study this filtration on the interval $s\in(-\infty,1].$ We see
immediately that
$\mathrm{Fil}^{s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))=0$ for any
$s<0$. For $0\leq s<1,$ we have

$\mathrm{Fil}^{s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))=\ker\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\xrightarrow{\hskip
5.69054pt\mathrm{res}\hskip
5.69054pt}H^{1}(I_{K},\overline{\mathbf{F}}_{p}(\chi))\right).$

Since $\bigcup_{u>0}G_{K}^{u}=P_{K},$ a class $c$ of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ lies in
$\mathrm{Fil}^{1}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ if and only if
$z(P_{K})=0$ for any cocycle $z$ representing $c$. Since $P_{K}$ is the
maximal pro-$p$ subgroup of $I_{K}$, we see that
$H^{1}(I_{K}/P_{K},\overline{\mathbf{F}}_{p}(\chi))=0$ and it follows from
inflation-restriction that the restriction map

$H^{1}(I_{K},\overline{\mathbf{F}}_{p}(\chi))\to
H^{1}(P_{K},\overline{\mathbf{F}}_{p}(\chi))$

is injective. Therefore, any cocycle with trivial restriction to $P_{K}$ must
also have trivial restriction to $I_{K}$. In other words,

$\mathrm{Fil}^{1}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))=\mathrm{Fil}^{0}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$

####  2.1.3. Filtered piece as kernel

Suppose that $s>1.$ If $z$ is a cocycle representing a cohomology class in
$\mathrm{Fil}^{<s}$, then clearly $z(G_{K}^{s-1})=0.$ Conversely, if $z$ is a
continuous cocycle such that $z(G_{K}^{s-1})=0$, then it follows from the
identity $G_{K}^{s-1}=\bigcap_{v<s-1}G_{K}^{v}$ that

$\ker(z)\cup\bigcup_{v<s-1}\left(G_{K}-G_{K}^{v}\right)$

is an open cover of $G_{K}$. Compactness of $G_{K}$ then implies that there is
a $v_{0}<s-1$ such that $G_{K}^{v_{0}}\subseteq\ker(z).$ In other words, we
have just proved that

$\mathrm{Fil}^{<s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))=\ker\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\xrightarrow{\hskip
5.69054pt\mathrm{res}\hskip
5.69054pt}H^{1}(G^{s-1}_{K},\overline{\mathbf{F}}_{p}(\chi))\right).$

####  2.1.4. The jumps in the filtration

Having defined the filtration and its most basic properties, we are now
interested in the jumps of the filtration and the size of these jumps. That
is, we would like to study the dimensions of the spaces
$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right)$.
Fix a continuous character $\chi:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$
and let the $n_{i}$ for $i=0,\dotsc,f-1$ be as above. The following theorem
gives the answer.

######  Theorem 2.1.

For $s\in\mathbf{R}$, write
$d_{s}:=\dim_{\overline{\mathbf{F}}_{p}}\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right).$
Then $d_{s}=0$ unless $s=0$ or $1<s\leq 1+\frac{pe}{p-1}.$ Moreover, if
$1<s<1+\frac{pe}{p-1}$ and $d_{s}\neq 0$ then $s=1+\frac{m}{p^{f}-1}$ for an
integer $m\not\equiv 0\bmod p$. More precisely, the dimensions $d_{s}$ are
given by

  1. (1)

$d_{0}=1$ if $\chi$ is trivial and $d_{0}=0$ otherwise.

  2. (2)

if $1<s<1+\frac{pe}{p-1}$ and $s=1+\frac{m}{p^{f}-1}$ for $p\nmid m$, then

$d_{s}=\\#\left\\{i\in\\{0,\dotsc,f-1\\}\mid m\equiv
n_{i}\bmod{p^{f}-1}\right\\}.$

  3. (3)

$d_{1+\frac{ep}{p-1}}=1$ if $\chi$ is cyclotomic and $d_{1+\frac{ep}{p-1}}=0$
otherwise.

######  Remark 2.2.

Note that if $t,u\in\\{0,\dotsc,f-1\\}$ are such that $n_{t}$ and $n_{u}$ are
both congruent to $m\bmod{(p^{f}-1)}$ then the uniqueness of the $n_{i}$
modulo $(p^{f}-1)$ implies that $n_{t}=n_{u}$. Hence, either $t=u$ or
$\vec{a}$ has a non-trivial stabilizer in $\mathrm{Gal}(k/\mathbf{F}_{p}).$
This observation implies that, for $1<s<1+\frac{pe}{p-1},$ we have that
$d_{s}=f/f^{\prime}$ whenever $d_{s}>0$, where $f^{\prime}$ is the absolute
niveau of $\chi$.

For the theorem to make sense, of course, the sum of the dimensions $d_{s}$
must add up to the $\overline{\mathbf{F}}_{p}$-dimension of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$ Since we will need this in the
proof of the theorem, let us prove it first. To contrast the actual dimensions
of the graded pieces with the dimensions claimed in the second part of the
theorem above, let us write $d_{s}^{\prime}$ for the values of the dimensions
claimed in the theorem.

######  Proposition 2.3.

For $j=0,\dotsc,e-1,$ we have

$\sum_{\frac{jp}{p-1}<\frac{m}{p^{f}-1}<\frac{(j+1)p}{p-1}}d_{1+\frac{m}{p^{f}-1}}^{\prime}=f.$

######  Remark 2.4.

In fact, the proposition and Remark 2.2 above imply that for any $j\geq 0$ the
set of integers $m\in\mathbf{Z}$ satisfying

  1. (1)

$\frac{jp}{p-1}<\frac{m}{p^{f}-1}<\frac{(j+1)p}{p-1}$,

  2. (2)

$p\nmid m$ and

  3. (3)

there exists an $i$ such that $m\equiv n_{i}\bmod{(p^{f}-1)}$

has cardinality $f^{\prime},$ where $f^{\prime}$ is the absolute niveau of
$\chi.$ We will need this later in §2.3.

###### Proof.

We have the inequalities
$\frac{1}{p-1}\leq\frac{n_{i}}{p^{f}-1}<\frac{p}{p-1}$ for all $i.$ It is easy
to show explicitly that there exists a unique integer $k_{j}$ such that

$\left(\left[\frac{1}{p-1},\frac{p}{p-1}\right)+k_{j}\right)\cap\left(\frac{jp}{p-1},\frac{(j+1)p}{p-1}\right)\neq\varnothing$

and

$\left(\left[\frac{1}{p-1},\frac{p}{p-1}\right)+k_{j}+1\right)\cap\left(\frac{jp}{p-1},\frac{(j+1)p}{p-1}\right)\neq\varnothing.$

Moreover, if we write $j-1\equiv b\bmod{p-1}$ for $0\leq b<p-1$ (when $p=2$ we
simply let $b=0$), then we can show $k_{j}\equiv b+1\bmod p.$ We are
interested in counting the number of $n_{i}$ that satisfy

(1) $\frac{jp}{p-1}<\frac{n_{i}}{p^{f}-1}+k_{j}\text{ and
}n_{i}+k_{j}(p^{f}-1)\not\equiv 0\bmod{p}$

or

(2) $\frac{n_{i}}{p^{f}-1}+(k_{j}+1)<\frac{(j+1)p}{p-1}\text{ and
}n_{i}+(k_{j}+1)(p^{f}-1)\not\equiv 0\bmod{p}$

or both, in which case they should be counted twice. We don’t have to worry
about upper or lower bounds in the first or second case, respectively, since
these are automatically satisfied.

In the case of Equation (1) the inequality reduces to the inequality

$\frac{n_{i}}{p^{f}-1}>\frac{b+1}{p-1}.$

This inequality is satisfied if and only if $n_{i}$ has coordinates

$(a_{i},a_{i+1},\dotsc,a_{j},\dotsc)=(a_{i},b+1,b+1,\dotsc,b+1,a_{j},\dotsc)$

with $a_{j}>b+1.$ If we, furthermore, require that
$n_{i}+k_{j}(p^{f}-1)\not\equiv 0\bmod{p},$ we see that this requirement is
equivalent to requiring that $a_{i}\neq b+1.$ Hence, the number of such
$n_{i}$ is the same as the number $\\#\\{a_{j}>b+1\\}.$

In the case of Equation (2) the inequality reduces to the inequality

$\frac{n_{i}}{p^{f}-1}<\frac{b+2}{p-1}.$

This inequality is satisfied if and only if $n_{i}$ corresponds to a tuple

$(a_{i},a_{i+1},\dotsc,a_{j},\dotsc)=(a_{i},b+2,b+2,\dotsc,b+2,a_{j},\dotsc)$

with $a_{j}\leq b+1$. If we also require $n_{i}+(k_{j}+1)(p^{f}-1)\not\equiv
0\bmod{p},$ we find that $a_{i}\neq b+2.$ Hence, in this case we see that the
number of $n_{i}$ satisfying both conditions is $\\#\\{a_{j}\leq b+1\\}.$

It follows that the total number of $n_{i}$ satisfying one of these equations,
counted with multiplicity, is

$\\#\\{a_{j}>b+1\\}+\\#\\{a_{j}\leq b+1\\}=f.$

∎

A straightforward calculation using the local Euler–Poincaré characteristic
gives

$\dim_{\overline{\mathbf{F}}_{p}}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))=\begin{cases}ef+2&\text{if
$\chi$ is both trivial and cyclotomic;}\\\ ef+1&\text{if $\chi$ is trivial,
but not cyclotomic;}\\\ ef+1&\text{if $\chi$ is cyclotomic, but not
trivial;}\\\ ef&\text{otherwise.}\end{cases}$

From the proposition above and this observation we immediately get the
following corollary.

######  Corollary 2.5.

Let the $d_{s}^{\prime}$ denote the dimensions as claimed in Theorem 2.1. Then

$\sum_{1<s<1+\frac{ep}{p-1}}d_{s}^{\prime}=ef.$

Hence, we find
$\sum_{s}d_{s}^{\prime}=\dim_{\overline{\mathbf{F}}_{p}}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$

###### Proof of Theorem 2.1.

Let us write $d_{s}^{\prime}$ again for the values of $d_{s}$ claimed in the
second part of Theorem 2.1. By Corollary 2.5 we know that
$\sum_{s}d_{s}^{\prime}=\sum_{s}d_{s}$, hence it suffices to consider only
$s\in\mathbf{R}$ such that $d_{s}^{\prime}>0$ and prove $d_{s}^{\prime}=d_{s}$
for these values of $s.$

Let us first treat the case $s=0.$ We already saw above that

$\mathrm{Fil}^{0}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))=\ker\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\xrightarrow{\hskip
5.69054pt\mathrm{res}\hskip
5.69054pt}H^{1}(I_{K},\overline{\mathbf{F}}_{p}(\chi))\right)$

and $\mathrm{Fil}^{s}=0$ for $s<0.$ We can use the inflation-restriction exact
sequence to show that the kernel above is isomorphic to
$H^{1}\left(G_{K}/I_{K},\overline{\mathbf{F}}_{p}(\chi)^{I_{K}}\right).$ This
space is clearly 1-dimensional over $\overline{\mathbf{F}}_{p}$ if $\chi$ is
trivial and $0$-dimensional if $\chi$ is ramified. However, if $\chi$ is
unramified and non-trivial, by identifying $G_{K}/I_{K}$ with
$\hat{\mathbf{Z}}$, we may consider $\chi$ as a character
$\hat{\mathbf{Z}}\to\overline{\mathbf{F}}_{p}^{\times}$. By continuity any
representation of $\hat{\mathbf{Z}}$ is completely determined by the image of
$1.$ An extension of $\overline{\mathbf{F}}_{p}$ by
$\overline{\mathbf{F}}_{p}(\chi)$ is always split since the matrix
corresponding to 1 has distinct eigenvalues over an algebraically closed field
(as $\chi$ is non-trivial). Hence,
$\dim_{\overline{\mathbf{F}}_{p}}H^{1}\left(G_{K}/I_{K},\overline{\mathbf{F}}_{p}(\chi)^{I_{K}}\right)=1$
if $\chi$ is trivial and $0$ otherwise.

Assume that $1<s\leq 1+\frac{ep}{p-1}$ and $m:=(s-1)(p^{f}-1).$ We define an
extension $M/K$ by letting $M:=L(\pi)$ for $\pi$ a root of
$x^{p^{f}-1}+\pi_{K}$ and $L/K$ an unramified extension of degree prime to $p$
such that ${\left.\kern-1.2pt\chi\vphantom{|}\right|_{G_{M}}}$ is trivial. The
extension $M$ is now a tamely ramified Galois extension of $K$ of ramification
degree $p^{f}-1$ and $\mathrm{Gal}(M/K)$ has order prime to $p$. Combined with
the inflation-restriction sequence

$0\to H^{1}(\mathrm{Gal}(M/K),\overline{\mathbf{F}}_{p}(\chi))\to
H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\to
H^{1}(G_{M},\overline{\mathbf{F}}_{p}(\chi))^{\mathrm{Gal}(M/K)}$

this gives

$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\cong
H^{1}(G_{M},\overline{\mathbf{F}}_{p}(\chi))^{\mathrm{Gal}(M/K)}=\mathrm{Hom}_{\mathrm{Gal}(M/K)}(G_{M}^{\mathrm{ab}},\overline{\mathbf{F}}_{p}(\chi)).$

The isomorphisms of local class field theory give an isomorphism

$G_{M}^{\mathrm{ab}}\otimes\mathbf{F}_{p}\cong M^{\times}/(M^{\times})^{p},$

and we obtain

$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\cong\mathrm{Hom}_{\mathrm{Gal}(M/K)}(M^{\times}/(M^{\times})^{p},\overline{\mathbf{F}}_{p}(\chi)).$

Since $M/K$ is tamely ramified of ramification degree $p^{f}-1$, we know that
Hasse-Herbrand function is given by $\psi_{M/K}(x)=(p^{f}-1)x$ for $x\geq 0$
and that $G_{K}^{u}\subset G_{M}$ for any $u>0.$ Using Proposition IV.15
[Ser79, p. 74], it follows that for any finite extension $M^{\prime}/M$ and
any $u>0$ we have

$\mathrm{Gal}(M^{\prime}/K)^{u}=\mathrm{Gal}(M^{\prime}/M)_{\psi_{M^{\prime}/M}((p^{f}-1)u)}=\mathrm{Gal}(M^{\prime}/M)^{(p^{f}-1)u}.$

So $G_{K}^{u}=G_{M}^{(p^{f}-1)u}$ for $u>0.$ Moreover, the maps of local class
field theory map $G_{M}^{(p^{f}-1)u}$ onto the unit group
$1+\pi^{\lceil(p^{f}-1)u\rceil}\mathcal{O}_{M}$ by Corollary 3 to Theorem XV.1
in [Ser79, p. 228]. If we denote the unit group $1+\pi^{i}\mathcal{O}_{M}$ by
$U_{i}$ for any integer $i>0$, then it becomes apparent that a cocycle in
$c\in H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ has trivial restriction to
$G_{K}^{u}$ for all $u>s-1$ (to $G_{K}^{s-1}$, resp.) if and only if the
corresponding homomorphism
$M^{\times}/(M^{\times})^{p}\to\overline{\mathbf{F}}_{p}(\chi)$ factors
through $M^{\times}/(M^{\times})^{p}U_{m+1}$ (through
$M^{\times}/(M^{\times})^{p}U_{m}$, resp.), where we still let
$m=(s-1)(p^{f}-1).$ Note that this is precisely the condition such that
$c\in\mathrm{Fil}^{s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$
($c\in\mathrm{Fil}^{<s}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)),$ resp.)
and it follows that

$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right)\cong\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(\frac{U_{m}}{(U_{m}\cap(M^{\times})^{p})U_{m+1}},\overline{\mathbf{F}}_{p}(\chi)\right).$

First assume $1<s<1+\frac{ep}{p-1}$ and $p\nmid m:=(s-1)(p^{f}-1).$ This is
equivalent to the requirement that $0<m<\frac{ep(p^{f}-1)}{(p-1)}.$ We claim
that $U_{m}\cap(M^{\times})^{p}\subset U_{m+1}.$ Let $x=1+u\pi^{t}$ for
positive integer $t$ and a unit $u\in\mathcal{O}_{M}^{\times}$ and suppose
$v_{M}(x^{p}-1)\geq m.$ But

$v_{M}(x^{p}-1)=v_{M}(pu\pi^{t}+\dots+u^{p}\pi^{pt})\geq\min(e(p^{f}-1)+t,pt)$

with equality unless $e(p^{f}-1)+t=pt.$ If $pt<e(p^{f}-1)+t$, then $m=pt$,
which would contradict the condition $p\nmid m.$ On the other hand, if
$e(p^{f}-1)+t\leq pt$, then $t\geq\frac{e(p^{f}-1)}{p-1}$ gives
$e(p^{f}-1)+t\geq\frac{ep(p^{f}-1)}{(p-1)}$ and, therefore, $m=e(p^{f}-1)+t$
would contradict the upper bound on $m.$ Hence, $v_{M}(x^{p}-1)\geq m+1$ as
required. It follows from the claim that

$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right)\cong\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(U_{m}/U_{m+1},\overline{\mathbf{F}}_{p}(\chi)\right).$

We let $l$ denote the residue field of $L$. The action of
$\sigma\in\mathrm{Gal}(M/K)$ on $U_{m}/U_{m+1}$ sends $1+x\pi^{m}\mapsto
1+\sigma(x)\overline{\omega}_{\pi}(\sigma)\pi^{m}.$ Therefore, we have a
$\mathrm{Gal}(M/K)$-equivariant isomorphism

${l(\overline{\omega}_{\pi}^{m})}$${U_{m}/U_{m+1}.}$$\scriptstyle{x\xmapsto{\hskip
22.76219pt}1+x\pi^{m}}$

Let $S_{i}$ be the set of embeddings $l\to\overline{\mathbf{F}}_{p}$ which
restrict to $\overline{\tau}_{i}$ over $k$. Then the map

$\displaystyle
l(\overline{\omega}_{\pi}^{m})\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$
$\displaystyle\cong\bigoplus_{i=0}^{f-1}\left(\bigoplus_{\tau\in
S_{i}}\overline{\mathbf{F}}_{p}(\omega_{i}^{m})\right)$ $\displaystyle
x\otimes 1$ $\displaystyle\mapsto(\tau(x))_{\tau}$

is a $\mathrm{Gal}(M/K)$-equivariant isomorphism, where the action of
$\mathrm{Gal}(M/K)$ on $\bigoplus_{\tau\in
S_{i}}\overline{\mathbf{F}}_{p}(\omega_{i}^{m})$ is defined by
$g\cdot((x_{\tau})_{\tau})=(\omega_{i}^{m}(g)x_{\tau\circ g})_{\tau}.$ (This
is the well-known isomorphism

$\displaystyle l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$
$\displaystyle\cong\bigoplus_{\tau:l\hookrightarrow\overline{\mathbf{F}}_{p}}\overline{\mathbf{F}}_{p};$
$\displaystyle x\otimes 1$ $\displaystyle\mapsto(\tau(x))_{\tau},$

in which we keep track of the Galois action.) We note that

$\bigoplus_{\tau\in
S_{i}}\overline{\mathbf{F}}_{p}\cong\mathrm{Ind}_{\mathrm{Gal}(M/L)}^{\mathrm{Gal}(M/K)}\overline{\mathbf{F}}_{p}.$

The latter representation is just the regular representation of the quotient
$\mathrm{Gal}(L/K)$. Since this group is finite abelian of order prime-to-$p$,
its regular representation decomposes as a direct sum of all its 1-dimensional
representations, i.e.

$\mathrm{Ind}_{\mathrm{Gal}(M/L)}^{\mathrm{Gal}(M/K)}\overline{\mathbf{F}}_{p}\cong\bigoplus_{\nu:\mathrm{Gal}(L/K)\to\overline{\mathbf{F}}_{p}^{\times}}\overline{\mathbf{F}}_{p}(\nu),$

and hence we find that

$l(\overline{\omega}_{\pi}^{m})\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}\cong\bigoplus_{i=0}^{f-1}\left(\bigoplus_{\nu}\overline{\mathbf{F}}_{p}(\nu\omega_{i}^{m})\right).$

It follows that the dimension of
$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right)$ is
equal to

$\dim_{\overline{\mathbf{F}}_{p}}\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(\bigoplus_{i=0}^{f-1}\left(\bigoplus_{\nu}\overline{\mathbf{F}}_{p}(\nu\omega_{i}^{m})\right),\overline{\mathbf{F}}_{p}(\chi)\right)$

and this dimension is equal to the number of $i$ such that $m\equiv
n_{i}\bmod{p^{f}-1},$ which proves that $d_{s}^{\prime}=d_{s}$ for all
$1<s<1+\frac{ep}{p-1}$ for which $d_{s}^{\prime}>0.$

Finally, assume $s=1+\frac{ep}{p-1},$ or, equivalently, assume
$m=\frac{ep(p^{f}-1)}{p-1}.$ We may, furthermore, assume that $\chi$ is
cyclotomic, since we are only considering cases in which $d_{s}^{\prime}>0.$
Note that the requirement that
${\left.\kern-1.2pt\chi\vphantom{|}\right|_{G_{M}}}$ is trivial implies that
$\zeta_{p}\in M.$ To show $d_{s}=d_{s}^{\prime}=1$, by the earlier
calculation, is equivalent to showing

$\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(\frac{U_{m}}{(U_{m}\cap(M^{\times})^{p})U_{m+1}},\overline{\mathbf{F}}_{p}(\chi)\right)$

is 1-dimensional. We will prove that the quotient of unit groups is isomorphic
to $\mathbf{F}_{p}(\chi)$ as a module over $\mathbf{F}_{p}[\mathrm{Gal}(M/K)]$
from which the required result is immediate. Similarly to the calculation on
valuations above, it is easy to show $U_{m}\cap(M^{\times})^{p}=(U_{n})^{p}$
where $n=m/p.$ We can obtain the quotient
$\frac{U_{m}}{(U_{m}\cap(M^{\times})^{p})U_{m+1}},$ therefore, as the cokernel
of the $p$-power map

${U_{n}/U_{n+1}}$${U_{m}/U_{m+1}.}$$\scriptstyle{x\xmapsto{\hskip
28.45274pt}x^{p}}$

A small calculation shows that this map sends $1+s\pi^{n}\mapsto
1+(s^{p}+cs)\pi^{m},$ where $c\in\mathcal{O}_{M}^{\times}$ is defined by
$p=c\pi^{e(p^{f}-1)}.$ Since $\zeta_{p}\in M$, we know that $M$ contains
$N:=\mathbf{Q}_{p}(\zeta_{p})=\mathbf{Q}_{p}(\sqrt[p-1]{-p}).$ Therefore,
there exists a unit $u\in\mathcal{O}_{M}^{\times}$ such that
$\pi^{e(p^{f}-1)}=(u\sqrt[p-1]{-p})^{p-1}=-u^{p-1}p$ giving
$-c=(u^{-1})^{p-1}$. We conclude that $-c$ always has a $(p-1)$st root in the
residue field and that the $p$-power map above has kernel and, thus, cokernel
of size $p.$ Write $\overline{s}$ for the image of any
$s\in\mathcal{O}_{M}^{\times}$ in the cokernel. The action of
$\mathrm{Gal}(M/K)$ on this cokernel is given by
$\overline{\omega}_{\pi}^{m}\otimes\mu_{-\overline{c}}$ where
$\mu_{\alpha}:\mathrm{Gal}(M/K)\to\mathbf{F}_{p}^{\times}$ is the unramified
character sending the (absolute) arithmetic Frobenius $\mathrm{Frob}_{p}$ to
$\alpha\in\mathbf{F}_{p}^{\times}.$ For any uniformiser $\pi_{N}$ of $N,$ we
can write $\zeta_{p}=v\pi_{N}+1$ for a unit $v\in\mathcal{O}_{N}^{\times}.$
Hence, one sees that for $\sigma\in\mathrm{Gal}(M/K)$ we have

$\frac{\sigma(v\pi_{N})}{v\pi_{N}}=\frac{(1+v\pi_{N})^{\chi(\sigma)}-1}{v\pi_{N}}\equiv\chi(\sigma)\bmod{\pi}.$

Since $\sigma(v)/v\equiv 1\bmod{\pi},$ it follows that
$\sigma(\pi_{N})/\pi_{N}\equiv\chi(\sigma)\bmod\pi$ for any uniformiser
$\pi_{N}$ of $N$, in particular $\pi_{N}=\sqrt[p-1]{-p}.$ Thence, if
$\sigma\in\mathrm{Gal}(M/K)$ is a lift of $\mathrm{Frob}_{p},$ then

$\overline{\omega}_{\pi}(\sigma)^{n}=\frac{\sigma(u\sqrt[p-1]{-p})}{u\sqrt[p-1]{-p}}\equiv\overline{u}^{p-1}\chi(\sigma)=(-\overline{c})^{-1}\chi(\sigma).$

In other words, we’ve just shown that $\overline{\omega}_{\pi}^{n}$ acts as
$\chi\otimes\mu_{-\overline{c}^{-1}}$ on the cokernel and, therefore,
$\overline{\omega}_{\pi}^{m}$ must act on the cokernel in the same way since
$(p^{f}-1)\mid m-n.$ Thus, $\mathrm{Gal}(M/K)$ acts on the cokernel via the
cyclotomic character and
$U_{m}/(U_{m}\cap(M^{\times})^{p})U_{m+1}\cong\mathbf{F}_{p}(\chi),$ as
required. ∎

###  2.2. Constructing the basis elements

In this subsection we will construct explicit basis elements of the space
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)),$ which we will later use to
give an explicit version of Serre’s conjecture. We will construct the basis by
making certain choices and only at the end of the next section it will follow
that our construction is independent of the choices made.

####  2.2.1. The Artin-Hasse Exponential

Before we move on to define a basis of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ explicitly, we must first
define a homomorphism $\varepsilon_{\alpha}$ in terms of which our basis will
be defined. Recall that the Artin-Hasse exponential is given by

$E_{p}(x)=\exp\left(\sum_{n\geq 0}\frac{x^{p^{n}}}{p^{n}}\right).$

It’s a well-known fact that $E_{p}(x)$ is an element of
$\mathbf{Z}_{p}\llbracket x\rrbracket$ (see, for example, [Rob00, §7.2.2]),
i.e. its coefficients are $p$-integral elements of $\mathbf{Q}.$ Since $p$ is
fixed throughout, we will write $E(x)$ instead of $E_{p}(x)$ from now on.

Suppose $l$ is a finite field of characteristic $p$ and $L$ is the field of
fractions of the ring of Witt vectors $W(l)$ of $l$. Let $M$ be a subfield of
$\mathbf{C}_{p}$ containing $L$ and let $\alpha\in M$ be such that
$|\alpha|<1.$ We define a map

(3)
$\displaystyle\begin{split}\varepsilon_{\alpha}:l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}&\to\mathcal{O}_{M}^{\times}\otimes_{\mathbf{Z}}\overline{\mathbf{F}}_{p}\\\
a\otimes b&\mapsto E([a]\alpha)\otimes b,\end{split}$

where $[a]$ is a Teichmüller lift of $a\in l.$ It follows from [DDR16, Lem.
4.1] that this map is a homomorphism which relates the additive structure of
$l$ to the multiplicative structure of
$\mathcal{O}_{M}^{\times}\otimes\mathbf{F}_{p}.$ We will use this homomorphism
to construct an explicit basis for
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$

####  2.2.2. An $\overline{\mathbf{F}}_{p}$-dual of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$

We need a slightly more general set-up than in the previous subsection.
Suppose $K/\mathbf{Q}_{p}$ is a finite extension of ramification index $e,$ of
residue degree $f$ and with residue field $k$. Moreover, we take
$\chi:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$ to be any continuous
character. Let $M=L(\pi)$ be a totally tamely ramified extension of an
unramified prime-to-$p$-degree extension $L/K$, where the ramification degree
$e_{M}$ of $M/K$ satisfies $e_{M}\mid p^{f}-1$ and the uniformiser $\pi$ of
$M$ satisfies $\pi^{e_{M}}\in K^{\times},$ and we take $M$ sufficiently large
so that ${\left.\kern-1.2pt\chi\vphantom{|}\right|_{G_{M}}}$ is trivial. Note
that $e_{M}$ denotes the ramification degree of $M$ over $K$, and the
ramification degree $e_{M/\mathbf{Q}_{p}}$ of $M$ over $\mathbf{Q}_{p}$ is
given by $e_{M/\mathbf{Q}_{p}}=e_{M}e.$ We recover the set-up of the previous
subsection by taking $e_{M}=p^{f}-1$ and $\pi^{e_{M}}=-\pi_{K}.$ Note that the
proof of Theorem 2.1 follows in the same way as before, taking into account
that now $s=1+\frac{m}{e_{M}}.$ In particular, for $1<s\leq 1+\frac{ep}{p-1}$
we still have the isomorphism

$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right)\cong\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(\frac{U_{m}}{(U_{m}\cap(M^{\times})^{p})U_{m+1}},\overline{\mathbf{F}}_{p}(\chi)\right),$

which, if we furthermore require that $s<1+\frac{ep}{p-1}$ and $p\nmid
m:=e_{M}(s-1)$, can again be shown to simplify to

$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right)\cong\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(U_{m}/U_{m+1},\overline{\mathbf{F}}_{p}(\chi)\right).$

More generally, for this choice of $M$ we certainly still have

$\displaystyle H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$
$\displaystyle\cong\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(M^{\times},\overline{\mathbf{F}}_{p}(\chi)\right)$
$\displaystyle\cong\mathrm{Hom}_{\overline{\mathbf{F}}_{p}}\left(\left(M^{\times}\otimes_{\mathbf{Z}}\overline{\mathbf{F}}_{p}(\chi^{-1})\right)^{\mathrm{Gal}(M/K)},\overline{\mathbf{F}}_{p}\right).$

We will construct our basis of $H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$
as a dual basis to a basis for the space above, therefore it will be useful to
give this space a name. Let

$U_{\chi}:=\left(M^{\times}\otimes_{\mathbf{Z}}\overline{\mathbf{F}}_{p}(\chi^{-1})\right)^{\mathrm{Gal}(M/K)}.$

####  2.2.3. The basis elements $u_{\alpha}$

Letting $f^{\prime}$ denote the absolute niveau of $\chi$ and
$f^{\prime\prime}:=f/f^{\prime},$ recall from Section 2.1.4 that for any
$0\leq j<e$ the set of integers $m\in\mathbf{Z}$ satisfying

  1. (1)

$\frac{jp}{p-1}<\frac{m}{p^{f}-1}<\frac{(j+1)p}{p-1}$,

  2. (2)

$p\nmid m$ and

  3. (3)

there exists an $i$ such that $m\equiv n_{i}\bmod{(p^{f}-1)}$

has cardinality $f^{\prime}.$ Denote this set of integers by $W_{j}^{\prime}$
and let $W^{\prime}:=\bigcup_{j=0}^{e-1}W_{j}^{\prime},$ which is now a set of
integers of cardinality $ef^{\prime}.$ For any $m\in W^{\prime}$, we get a
uniquely defined $i_{m}\in\\{0,\dotsc,f^{\prime}-1\\}$ via the association
$m\equiv n_{i_{m}}\bmod{(p^{f}-1)}.$ (Note that there may be distinct
$m_{1}\neq m_{2}\in W^{\prime}$, possibly even lying in the same component
$W_{j}^{\prime}$, that satisfy $i_{m_{1}}=i_{m_{2}}.$) We define
$W:=W^{\prime}\times\\{0,\dotsc,f^{\prime\prime}-1\\}$ which has cardinality
$ef$. To any $\alpha=(m,k)\in W$ (with $m\in W^{\prime}$ and $0\leq
k<f^{\prime\prime}$) we can attach an embedding via
$\tau_{\alpha}:=\tau_{i_{m}+kf^{\prime}}.$ Notice that we also have the
congruence $m\equiv n_{i_{m}+kf^{\prime}}\bmod{(p^{f}-1)}$ for any
$k\in\mathbf{Z}.$

Let us return to the situation where $M=L(\pi)$ for an unramified extension
$L$ of $K$ of degree prime-to-$p$ and a uniformiser $\pi$ such that
$\pi^{e_{M}}\in K^{\times}$ and
${\left.\kern-1.2pt\chi\vphantom{|}\right|_{G_{M}}}$ is trivial. This implies
that ${\left.\kern-1.2pt\chi\vphantom{|}\right|_{I_{K}}}$ factors through
$I_{M/K}$ which has cardinality $e_{M}$, so for any $i$ we see
$({\left.\kern-1.2pt\chi\vphantom{|}\right|_{I_{K}}})^{e_{M}}=\omega_{i}^{n_{i}e_{M}}=\mathbf{1}.$
Any $n_{i}$, therefore, is divisible by $(p^{f}-1)/e_{M}.$ It follows that the
set of integers $m^{\prime}\in\mathbf{Z}$ satisfying the three conditions

  1. (1)

$\frac{jp}{p-1}<\frac{m^{\prime}}{e_{M}}<\frac{(j+1)p}{p-1}$,

  2. (2)

$p\nmid m^{\prime}$ and

  3. (3)

there exists an $i$ such that
$m^{\prime}\equiv\frac{e_{M}n_{i}}{p^{f}-1}\bmod{e_{M}}$

is in bijection with the set $W_{j}^{\prime}$ above by the explicit map
$m\mapsto m^{\prime}:=\frac{e_{M}m}{p^{f}-1}.$ Unfortunately, this notation
leads to the awkward situation that, in general, $m^{\prime}\notin
W^{\prime},$ but we will stick to it because $m^{\prime}$ is analogous to the
integers $n_{i}^{\prime}$ appearing in [DDR16]. We will write $\omega_{\pi}$
for the character $G_{K}\to K^{\times}$ defined by
$\sigma\mapsto\sigma(\pi)/\pi$ and $\overline{\omega}_{\pi}$ for its reduction
mod $\pi_{K}$. We note that this character factors through $\mathrm{Gal}(M/K)$
and, since $\sigma(\pi)$ is a root of $x^{e_{M}}-\pi^{e_{M}}\in K[x],$ the
image of $\omega_{\pi}$ is contained in $\mu_{e_{M}}(K),$ the $e_{M}$th roots
of unity in $K^{\times}.$ To contrast this with our previous fundamental
character, let us from now on write $\overline{\omega}_{\mathrm{fc}}:G_{K}\to
k^{\times}$ for the mod $p$ character defined by
$\sigma\mapsto\sigma(\beta)/\beta$ for $\beta$ a root of
$x^{p^{f}-1}+\pi_{K}.$ We have
${\left.\kern-1.2pt\overline{\omega}_{\pi}\vphantom{|}\right|_{I_{K}}}=({\left.\kern-1.2pt\overline{\omega}_{\mathrm{fc}}\vphantom{|}\right|_{I_{K}}})^{(p^{f}-1)/e_{M}},$
and, therefore, for all $0\leq i<f,$ we have

${\left.\kern-1.2pt\chi\vphantom{|}\right|_{I_{K}}}=(\tau_{i}\circ\overline{\omega}_{\pi})|_{I_{K}}^{n_{i}e_{M}/(p^{f}-1)}.$

Writing $m^{\prime}:=me_{M}/(p^{f}-1),$ we see that for any $\alpha=(m,k)\in
W$ we have

$\chi=\mu(\tau_{\alpha}\circ\overline{\omega}_{\pi})^{m^{\prime}}$

for a unique unramified character
$\mu:\mathrm{Gal}(L/K)\to\overline{\mathbf{F}}_{p}^{\times}$ independent of
$\alpha.$

Recall that we have an isomorphism

$l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}\cong\bigoplus_{\tau:k\hookrightarrow\overline{\mathbf{F}}_{p}}l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}.$

By the normal basis theorem $l$ is free of rank 1 over $k[\mathrm{Gal}(l/k)]$
and, thus, $l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}$ is free of rank 1 over
$\overline{\mathbf{F}}_{p}[\mathrm{Gal}(l/k)]$. The latter statement means
that $l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}$ is isomorphic to the regular
representation of $\mathrm{Gal}(l/k)$ over $\overline{\mathbf{F}}_{p}$, which
splits up in a direct sum over all characters by Maschke’s Theorem. Hence, for
each embedding $\tau,$ the $\mu$-eigenspace $\Lambda_{\tau,\mu}$ defined as

$\left\\{a\otimes b\in
l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}\;\middle|\;g(a)\otimes
b=(1\otimes\mu(g))a\otimes b\text{ for all }g\in\mathrm{Gal}(L/K)\right\\}$

is 1-dimensional over $\overline{\mathbf{F}}_{p}.$ We let $\lambda_{\tau,\mu}$
be any non-zero element of $\Lambda_{\tau,\mu}$; note that these elements are
linearly independent over $\overline{\mathbf{F}}_{p}$ for differing embeddings
$\tau.$

Then, for any $\alpha=(m,k)\in W$, we define

$u_{\alpha}:=\varepsilon_{\pi^{m^{\prime}}}(\lambda_{\tau_{\alpha},\mu})\in\mathcal{O}_{M}^{\times}\otimes_{\mathbf{Z}}\overline{\mathbf{F}}_{p},$

where $\varepsilon_{\pi^{m^{\prime}}}$ is defined by (3), giving $ef$ elements
$u_{\alpha}.$ Later we will prove that the elements $u_{\alpha}$ give the
explicit basis of $U_{\chi}$ that we are pursuing. In order for this even to
make sense, we will now show $u_{\alpha}\in U_{\chi}.$ For any
$g\in\mathrm{Gal}(M/K),$ it follows from the observation
$\omega_{\pi}(g)=[\overline{\omega}_{\pi}(g)]$ that

$g\cdot
E([a]\pi^{m^{\prime}})=E\left([g(a)\overline{\omega}_{\pi}(g)^{m^{\prime}}]\pi^{m^{\prime}}\right)$

for any $a\in l.$ Thus, for any $\alpha=(m,k)\in W$,

$\displaystyle
g\cdot\varepsilon_{\pi^{m^{\prime}}}(\lambda_{\tau_{\alpha},\mu})$
$\displaystyle=\varepsilon_{\pi^{m^{\prime}}}((\overline{\omega}_{\pi}(g)^{m^{\prime}}\otimes
1)g(\lambda_{\tau_{\alpha},\mu}))$
$\displaystyle=\varepsilon_{\pi^{m^{\prime}}}((\overline{\omega}_{\pi}(g)^{m^{\prime}}\otimes\mu(g))\lambda_{\tau_{\alpha},\mu}).$

Since
$\overline{\omega}_{\pi}(g)^{m^{\prime}}\otimes\mu(g)=1\otimes\mu(g)(\tau_{\alpha}\circ\overline{\omega}_{\pi}(g))^{m^{\prime}}=1\otimes\chi(g)$
in $l\otimes_{k,\tau_{\alpha}}\overline{\mathbf{F}}_{p}$, we see that
$gu_{\alpha}=(1\otimes\chi(g))u_{\alpha}$ for all $g\in\mathrm{Gal}(M/K).$
Therefore, we can view $u_{\alpha}$ as an element of $U_{\chi}.$

####  2.2.4. The definitions of $u_{\mathrm{triv}}$ and $u_{\mathrm{cyc}}$

The definition of $u_{\mathrm{triv}}$ is straightforward. For
$g\in\mathrm{Gal}(M/K),$ it is clear that $g(\pi)=\omega_{\pi}(g)\pi.$ As
noted above, the image of $\omega_{\pi}$ is contained in $\mu_{e_{M}}(K).$
Since $p$ is coprime to $e_{M}$, the $p$-th power map gives an isomorphism
$\mu_{e_{M}}(K)\xrightarrow{\hskip 5.69054pt\sim\hskip
5.69054pt}\mu_{e_{M}}(K).$ Thus, in fact, $\omega_{\pi}(g)\in(M^{\times})^{p}$
and we set

$u_{\mathrm{triv}}:=\pi\otimes 1\in
M^{\times}\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p},$

which is an element of $U_{\chi}$ when $\chi$ is the trivial character.

Suppose $\chi$ is cyclotomic. Recall from the last part of the proof of
Theorem 2.1 that we have already proved explicitly that, for
$m=\frac{epe_{M}}{p-1},$ we have

$\frac{U_{m}}{(U_{m}\cap(M^{\times})^{p})U_{m+1}}\cong\mathbf{F}_{p}(\chi).$

However, recall that the $p$-adic exponential converges on values of $y\in M$
such that $v_{M}(y)>e_{M}e/(p-1).$ For $1+x\in U_{m+1},$ we have
$v_{M}(\log(1+x))>m$ and, thus, we see that $\exp(\log(1+x)/p)$ converges to a
$p$-th root of $1+x.$ Therefore, $U_{m+1}\subset(M^{\times})^{p}$ and also
$U_{m+1}\subset(U_{m}\cap(M^{\times})^{p}).$ Thus, we have an injection of the
quotient above into
$\mathcal{O}_{M}^{\times}\otimes_{\mathbf{Z}}\mathbf{F}_{p}.$

We extend the injection by scalars to $\overline{\mathbf{F}}_{p}$ and define
$u_{\mathrm{cyc}}$ to be any non-trivial element of

$u_{\mathrm{cyc}}\in\mathrm{Im}\left(U_{m}\otimes_{\mathbf{Z}}\overline{\mathbf{F}}_{p}\xrightarrow{\hskip
14.22636pt}\mathcal{O}_{M}^{\times}\otimes_{\mathbf{Z}}\overline{\mathbf{F}}_{p}\right).$

It is now obvious from the isomorphism with $\mathbf{F}_{p}(\chi)$ that for
any element $g\in\mathrm{Gal}(M/K)$ we have that

$g(u_{\mathrm{cyc}})=(\chi(g)\otimes
1)u_{\mathrm{cyc}}=(1\otimes\chi(g))u_{\mathrm{cyc}},$

hence, indeed, $u_{\mathrm{cyc}}\in U_{\chi}.$

###  2.3. An explicit basis of $H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$

In this section we will prove that the elements as constructed in the previous
section give an explicit basis of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$. We will do this by proving the
following theorem.

######  Theorem 2.6.

Let $B$ denote the elements $u_{\alpha}$ for $\alpha\in W$ with, additionally,
the elements $u_{\mathrm{triv}}$ if $\chi$ is trivial and $u_{\mathrm{cyc}}$
if $\chi$ is cyclotomic. Then $B$ is an $\overline{\mathbf{F}}_{p}$-basis of
$U_{\chi}.$

###### Proof.

To prove this theorem, we will define a decreasing filtration on $U_{\chi},$
which is dual to the increasing filtration on
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ defined earlier. Let
$\mathrm{Fil}^{0}U_{\chi}=U_{\chi}$ and let $\mathrm{Fil}^{m}U_{\chi}$ denote
the image of the map

${(U_{m}\otimes\overline{\mathbf{F}}_{p}(\chi^{-1}))^{\mathrm{Gal}(M/K)}}$${(M^{\times}\otimes\overline{\mathbf{F}}_{p}(\chi^{-1}))^{\mathrm{Gal}(M/K)}=:U_{\chi}}$

for $m\geq 1.$ Let
$\mathrm{gr}^{m}U_{\chi}:=\mathrm{Fil}^{m}U_{\chi}/\mathrm{Fil}^{m+1}U_{\chi}$
as usual. It is immediate by duality that
$\dim_{\overline{\mathbf{F}}_{p}}U_{\chi}=\dim_{\overline{\mathbf{F}}_{p}}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$
More specifically, whenever $0<m<\frac{pee_{M}}{p-1}$ and $p\nmid m$, it
follows as in the proof of Theorem 2.1 and the discussion in §2.2.2 that

$\mathrm{gr}^{s}\left(H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\right)\cong\mathrm{Hom}_{\mathrm{Gal}(M/K)}\left(U_{m}/U_{m+1},\overline{\mathbf{F}}_{p}(\chi)\right),$

where $s:=1+\frac{m}{e_{M}}.$ We claim that

$\mathrm{gr}^{m}U_{\chi}\cong\left(U_{m}/U_{m+1}\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}(\chi^{-1})\right)^{\mathrm{Gal}(M/K)}.$

In the proof of Theorem 2.1 we show that under these assumptions on $m$ we
have that $(M^{\times})^{p}\cap U_{m}\subset U_{m+1}$. Letting
$\overline{U}_{i}$ denote the image of $U_{i}$ in
$M^{\times}\otimes\mathbf{F}_{p}$, we obtain an exact sequence of
$\mathbf{F}_{p}$-modules

${1}$${\overline{U}_{m+1}}$${\overline{U}_{m}}$${U_{m}/U_{m+1}}$${1.}$

The claim for these $m$ now follows by noting that

$\left(-\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}(\chi^{-1})\right)^{\mathrm{Gal}(M/K)}$

is exact since $\mathrm{Gal}(M/K)$ has order prime-to-$p$. In particular, it
follows from this claim that
$\dim_{\overline{\mathbf{F}}_{p}}\mathrm{gr}^{m}U_{\chi}$ equals $d_{s}$ for
$1<s:=1+\frac{m}{e_{M}}<1+\frac{pe}{p-1}$ and $d_{s}$ as in Theorem 2.1.

Since the size of $B$ is equal to the $\overline{\mathbf{F}}_{p}$-dimension of
$U_{\chi}$, it will suffice to prove that each graded is spanned by a subset
of elements of $B$. By the dimensions of Theorem 2.1, we only need to consider
integers $m$ such that $d_{s}>0.$

The rest of the proof follows from our constructions. If
$m=\frac{epe_{M}}{p-1},$ then it suffices to note $u_{\mathrm{cyc}}$ is a non-
trivial element of $\mathrm{gr}^{m}U_{\chi}=\mathrm{Fil}^{m}U_{\chi},$ which
is 1-dimensional.

If $0<m<\frac{epe_{M}}{p-1}$, then we may assume that
$m=\frac{e_{M}\widetilde{m}}{p^{f}-1}=:\widetilde{m}^{\prime}$ for some
$\widetilde{m}\in W^{\prime}$ as defined in §2.2.3. The elements

$u_{\alpha}:=\varepsilon_{\pi^{\widetilde{m}^{\prime}}}(\lambda_{\tau_{\alpha},\mu})$

for $\alpha=(\widetilde{m},k)\in W$ with $0\leq k<f^{\prime\prime}$ are all
non-trivial elements of $\mathrm{gr}^{m}U_{\chi}$ – this follows, for example,
since $E(x)\in 1+x+x^{2}\mathbf{Z}_{p}\llbracket x\rrbracket.$ It suffices to
prove that these elements are linearly independent in
$\mathrm{gr}^{m}U_{\chi}.$ Since $p\nmid m$, we showed that

$\mathrm{gr}^{m}U_{\chi}\cong\left(U_{m}/U_{m+1}\otimes\overline{\mathbf{F}}_{p}(\chi^{-1})\right)^{\mathrm{Gal}(M/K)}.$

Since the map

$\displaystyle l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$
$\displaystyle\to
U_{m}/U_{m+1}\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$ $\displaystyle
a\otimes b$ $\displaystyle\mapsto(1+[a]\pi^{m})\otimes b$

induced by $\varepsilon_{\pi^{m}}$ is an isomorphism and the elements

$\\{\lambda_{\tau_{\alpha},\mu}\mid\text{for }\alpha=(\widetilde{m},k)\text{
with }0\leq k<f^{\prime\prime}\\}$

are $\overline{\mathbf{F}}_{p}$-linearly independent in
$l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$, we see that their images
are as well (recall that we defined
$\tau_{\alpha}:=\tau_{i_{\widetilde{m}}+kf^{\prime}}$, where
$\widetilde{m}\equiv n_{i_{\widetilde{m}}}\bmod{(p^{f}-1)}$).

Lastly, if $m=0$, it suffices to note that $u_{\mathrm{triv}}=\pi\otimes 1$
does not lie in $\mathrm{Fil}^{1}U_{\chi}.$ ∎

Given the basis $B$ above we obtain the dual basis $c_{\alpha}$ for $\alpha\in
W$ with, additionally, $c_{\mathrm{ur}}$ and $c_{\mathrm{tr}}$ if $\chi$ is
trivial or cyclotomic. This dual basis must, simply by duality, form a basis
of $H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ – the subscripts refer to
the unramified and très ramifiée part of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$.

######  Corollary 2.7.

The set consisting of the classes $c_{\alpha}$ for $\alpha\in W$ with,
additionally, $c_{\mathrm{ur}}$ if $\chi$ is trivial and $c_{\mathrm{tr}}$ if
$\chi$ is cyclotomic forms a basis of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$.

Once again we emphasise that the explicit basis as defined here may depend on
the choice of extension $M$ and the choice of uniformiser $\pi$ of $M$ in
§2.2.2.

##  3\. Explicit Serre weights over ramified bases

In this section we use the explicit basis obtained in the previous section to
give an explicit definition of the distinguished subspaces $L_{V}$ defined in
§1.2.1: we will define them as a subspace spanned by a subset of our explicit
basis elements. In Theorem 3.16 we will prove the equivalence of our subspaces
spanned by explicit basis elements with the distinguished subspaces $L_{V}$,
thereby providing all the necessary theoretical foundation to give a new
explicit reformulation of the conjectures on the modularity of mod $p$ Galois
representations of totally real number fields.

Sections 3.1, 3.2 and 3.3 are generalisations of [CEGM17, §3] to the case
where $K$ is an arbitrary local field (rather than $K$ being an unramified
extension of $\mathbf{Q}_{p}$); we invite the curious reader to have a look at
[CEGM17] for a comparison.

###  3.1. Reductions of pseudo-Barsotti Tate representations

One of the essential ingredients in our proof of the equivalence of the two
formulations is the classification in [GLS15] of representations that arise as
reductions of pseudo-Barsotti–Tate representations. Since we will heavily use
their results and their notation, we will quickly recall their most important
results adapted to the case at hand. We note that in [GLS15] only the case
$p>2$ is covered. We will use Wang’s extension [Wan17] of their results to
$p=2$ for the remaining case.

####  3.1.1. Field of norms and étale $\varphi$-modules

Let us very briefly recall the theory of the field of norms and étale
$\varphi$-modules. We are given a local field $K$ and a fixed uniformiser
$\pi_{K}.$ We emphasise that our choices of uniformiser in this section and in
the previous sections must be compatible. We define a compatible system of
$p^{n}$th roots of our uniformiser as follows: let $\pi_{0}=-\pi_{K}$. We
define $\pi_{n}$ inductively for any $n>0$ as a $p$th root of $\pi_{n-1}$,
that is, satisfying $\pi_{n}^{p}=\pi_{n-1}$. Let $K_{n}:=K(\pi_{n})$ and
$K_{\infty}=\bigcup_{n=0}^{\infty}K_{n}.$ If $p=2$, then we assume that our
uniformiser $\pi_{K}$ was chosen to satisfy [Wan17, Lem. 2.1]; that is, we
assume $K_{\infty}\cap K_{p^{\infty}}=K,$ where we define
$K_{p^{\infty}}:=\bigcup_{n\geq 1}K(\zeta_{p^{n}})$ with $\zeta_{p^{n}}$ a
primitive $p^{n}$-th root of unity. The theory of the field of norms lets us
identify

$\displaystyle k((u))$ $\displaystyle\xrightarrow{\hskip 8.53581pt\sim\hskip
8.53581pt}\varprojlim_{N_{K_{n+1}/K_{n}}}K_{n}$ $\displaystyle u$
$\displaystyle\xmapsto{\hskip 25.60747pt}(\pi_{n})_{n},$

where the transition maps in the inverse limit are the norm maps. This
construction extends to extensions of these fields and we get a natural
isomorphism of absolute Galois groups $G_{k((u))}=G_{K_{\infty}}.$ On the
other hand there exists an equivalence of abelian categories between the
category of finite-dimensional $\overline{\mathbf{F}}_{p}$-representations of
$G_{k((u))}$ and the category of étale $\varphi$-modules. These are finite
$k((u))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$-modules
$\mathcal{M}$ equipped with a $\varphi$-semilinear map
$\varphi\colon\mathcal{M}\to\mathcal{M}$ such that the induced
$k((u))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$-linear map
$\varphi^{*}\mathcal{M}\to\mathcal{M}$ is an isomorphism. The functor $T$ from
étale $\varphi$-modules to representations of $G_{k((u))}$ is given by

$T\colon\mathcal{M}\to\left(k((u))^{\mathrm{sep}}\otimes_{k((u))}\mathcal{M}\right)^{\varphi=1}.$

The isomorphism
$k\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}\to\prod_{i=0}^{f-1}\overline{\mathbf{F}}_{p}$
gives a decomposition $\mathcal{M}=\prod_{i}\mathcal{M}_{i},$ where $\varphi$
now induces $\overline{\mathbf{F}}_{p}$-linear morphisms
$\varphi\colon\mathcal{M}_{i-1}\to\mathcal{M}_{i}.$ We will make use of this
theory by describing finite-dimensional
$\overline{\mathbf{F}}_{p}$-representations of $G_{K_{\infty}}$ via étale
$\varphi$-modules.

####  3.1.2. The structure theorem of Gee–Liu–Savitt

From now on let us fix a Serre weight $V=V_{\underline{\eta},\underline{0}}$
and a pair of characters
$\chi_{1},\chi_{2}:G_{K}\to\overline{\mathbf{F}}_{p}^{\times}$ as in §1.2.1.
(Recall that we were free to assume that $V$ is of the form
$V_{\underline{\eta},\underline{0}}$, since we can obtain the case of a
general Serre weight from this one by twisting.) We will write $L_{V}$ for
$L_{V_{\underline{\eta},\underline{0}}}(\chi_{1},\chi_{2})$ and we will assume
this is non-empty. Recall that we defined the integers $r_{i}\in[1,p]$ via
$r_{i}:=\eta_{i}+1$ and we write $\chi:=\chi_{1}\chi_{2}^{-1}.$ For any $0\leq
i<f$, we may decompose $\chi$ as a power of $\omega_{i}$ and an unramified
character $\mu$ independent of $i.$ Then $\mu$ factors as a character
$\mu:\mathrm{Gal}(L/K)\to\overline{\mathbf{F}}_{p}^{\times}$ for some finite
unramified extension $L$ of $K$ of prime-to-$p$ order.

First we need to get rid of an exceptional pathological case. Later we will
define the subspaces $L_{V}^{\mathrm{AH}}$ using our constructed basis and we
will claim $L^{\mathrm{AH}}_{V}=L_{V}$ as defined above. Before we continue
let us define this space in one exceptional case.

######  Definition 3.1.

If $\chi$ is cyclotomic, $\chi_{2}$ is unramified and $r_{i}=p$ for all $0\leq
i<f,$ then we define

$L_{V}^{\mathrm{AH}}(\chi_{1},\chi_{2}):=H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$

Note that it follows from the last part of the proof [GLS15, Thm. 6.1.8] that
if $\chi$ is cyclotomic, $\chi_{2}$ is unramified and $r_{i}=p$ for all $i$,
then

$L_{V}(\chi_{1},\chi_{2})=H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)),$

so in this exceptional case we indeed have that
$L_{V}^{\mathrm{AH}}(\chi_{1},\chi_{2})=L_{V}(\chi_{1},\chi_{2}).$

We will from now on assume that if $\chi$ is cyclotomic and $\chi_{2}$ is
unramified, then $r_{i}<p$ for at least one $i.$ This assumption has the
consequence that the restriction map

$\mathrm{res}:H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))\to
H^{1}(G_{K_{\infty}},\overline{\mathbf{F}}_{p}(\chi))$

becomes injective when restricted to $L_{V}$ – this is [GLS15, Lem. 5.4.2] for
$\chi$ non-cyclotomic and follows from the proof of [GLS15, Thm. 6.1.8] when
$\chi$ is cyclotomic. (Note that it follows from Wang [Wan17] that these
results also work for $p=2$.) Alternatively, the injectivity for $\chi$
cyclotomic follows from the fact that under the assumption $L_{V}$ is
contained in the _peu ramifié subspace_ of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ by [Ste20, Thm. 4.1.1]. By
[GLS15, Lem. 5.4.2] the kernel of the restriction map is the très ramifiée
line spanned by the fixed uniformiser $-\pi_{K}$. Therefore, the restriction
map is injective on this subspace. Since we are excluding the case above, we
are free to restrict our representations to $G_{K_{\infty}}$ from now on and
talk about the corresponding étale $\varphi$-modules.

####  3.1.3. The integers $t_{i},s_{i}$

Before we state the main theorem of [GLS15], we need to introduce just a
little more notation: we need to introduce the integers $t_{i},s_{i}$
corresponding to the maximal and minimal Kisin modules of [GLS15, §5.3]. We
will take an ad hoc approach. Write
$\chi_{2}|_{I_{K}}=\prod_{i}\omega_{i}^{m_{i}}$ for the uniquely determined
integers $m_{i}\in[0,p-1]$ with not all $m_{i}$ equal to $p-1.$ Now let
$\mathcal{S}$ be the set of $f$-tuples of non-negative integers
$(a_{0},\dotsc,a_{f-1})$ satisfying
$\chi_{2}|_{I_{K}}=\prod_{i}\omega_{i}^{a_{i}}$ and
$a_{i}\in[0,e-1]\cup[r_{i},r_{i}+e-1]$ for all $i.$ This set is non-empty by
our assumption that $L_{V}$ is non-empty. If we write $v_{i}$ for the
$f$-tuple $(0,\dotsc,-1,p,\dotsc,0)$ with the $-1$ in the $i$th position and
$v_{f-1}$ for $(p,0,\dotsc,0,-1)$, then it is proved in [GLS15, Lem. 5.3.1]
that there exists a subset $J\subseteq\\{0,\dotsc,f-1\\},$ possibly empty,
such that

$(m_{0},\dotsc,m_{f-1})+\sum_{i\in J}v_{i}\in\mathcal{S}$

and we can choose $J=J_{\mathrm{min}}$ minimal in the sense that it is
contained in any other subset satisfying this requirement. Then we define

$(t_{0},\dotsc,t_{f-1}):=(m_{0},\dotsc,m_{f-1})+\sum_{i\in
J_{\mathrm{min}}}v_{i}\in\mathcal{S}.$

We will define $s_{i}:=(r_{i}+e-1)-t_{i}$ for all $i.$ Note that it follows
from our construction that $s_{i},t_{i}\in[0,e-1]\cup[r_{i},r_{i}+e-1]$ and
$s_{i}+t_{i}=r_{i}+e-1$ for all $i.$

####  3.1.4. The main structure theorem of Gee–Liu–Savitt

We will understand $(a)_{i}$ to mean the following: $(a)_{i}=a$ if $i\equiv
0\bmod f$ and $(a)_{i}=1$ otherwise. Now we are ready to state the main
structure theorem of [GLS15].

######  Theorem 3.2.

The image of the subspace $L_{V}$ of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ after restriction to
$H^{1}(G_{K_{\infty}},\overline{\mathbf{F}}_{p}(\chi))$ consists precisely of
those classes that can be represented by étale $\varphi$-modules $\mathcal{M}$
of the following form: we can choose bases $e_{i},f_{i}$ of $\mathcal{M}_{i}$
for which $\varphi$ has the form

$\displaystyle\varphi(e_{i-1})$ $\displaystyle=u^{t_{i}}e_{i};$
$\displaystyle\varphi(f_{i-1})$
$\displaystyle=(a)_{i}u^{s_{i}}f_{i}+y_{i}e_{i},$

where $a=\mu(\mathrm{Frob}_{K})$ and
$y_{i}\in\overline{\mathbf{F}}_{p}\llbracket u\rrbracket$ is a polynomial of
which the degrees of the non-zero terms lie in the interval

  * •

$[0,s_{i}-1]$ if $t_{i}\geq r_{i}$ except, possibly, if $\chi$ is trivial;

  * •

$\\{t_{i}\\}\cup[r_{i},s_{i}-1]$ if $t_{i}<r_{i}$ except, possibly, if $\chi$
is trivial.

  * •

If $\chi$ is trivial, we make any single choice of $i_{0}\in[0,\dotsc,f-1]$
and for $i_{0}$ in addition to the intervals defined above we also allow terms
of degree

$\left\\{s_{i_{0}}+\frac{1}{p^{f}-1}\sum_{k=1}^{f}p^{f-k}(s_{{i_{0}}+k}-t_{{i_{0}}+k})\right\\}.$

In every case the polynomials $y_{i}$ are uniquely determined by $\mathcal{M}$
and every collection of polynomials subject to the conditions above
corresponds to a class in $L_{V}.$

###### Proof.

For $p>2$, the statement of [GLS15, Thm. 5.1.5] shows that the corresponding
Kisin modules (which are just lattices in $\mathcal{M}$) have the form above.
It follows from [GLS15, §5] in the non-cyclotomic case and [GLS15, §6] in the
cyclotomic case that we get the full subspace $L_{V}$, since we chose the
$t_{i}$ and $s_{i}$ to be minimal and maximal, respectively. In the non-
cyclotomic case it follows from the proof of [GLS15, Thm. 5.4.1] that the
corresponding elements of
$H^{1}(G_{K_{\infty}},\overline{\mathbf{F}}_{p}(\chi))$ (and hence the
corresponding étale $\varphi$-modules) are uniquely determined by $y_{i}$ and
that any collection of polynomials $\\{y_{i}\\}$ satisfying the conditions
above corresponds to a class in $L_{V}.$ In the cyclotomic case (under the
assumption that we are not in the exceptional case above) the proof of [GLS15,
Thm. 6.1.8] gives the same uniqueness and existence result. It follows from
Wang [Wan17] that the results from [GLS15] quoted here and their proofs carry
over naturally to $p=2$ (assuming we have carefully chosen our uniformiser of
$K$) giving the required result in this case as well. ∎

######  Definition 3.3.

Given $\\{r_{i},t_{i},s_{i}\mid 0\leq i<f\\}$ as defined at the beginning of
§3.1.2 and in §3.1.3, we define

  * •

$\mathcal{I}_{i}:=[0,s_{i}-1]$ if $t_{i}\geq r_{i};$

  * •

$\mathcal{I}_{i}:=\\{t_{i}\\}\cup[r_{i},s_{i}-1]$ if $t_{i}<r_{i}.$

Here we have simply given the intervals appearing in the theorem a name for
later convenience. We note that (both in the definition and the theorem) we
follow standard conventions and let $\mathcal{I}_{i}:=\varnothing$ if
$t_{i}\geq r_{i}$ and $s_{i}=0,$ whereas, if $t_{i}<r_{i}$ and $s_{i}=r_{i},$
we let $\mathcal{I}_{i}:=\\{t_{i}\\}.$

###  3.2. Restricting to $G_{M_{\infty}}$

In this subsection we will use the structure theorem of [GLS15] from the
previous subsection to find the image of $L_{V}$ (after restricting to
$G_{M_{\infty}}$) in terms of Artin–Schreier theory. Later we will use an
explicit reciprocity law (see Theorem 3.8) to compare the Artin–Schreier
theoretic image of $L_{V}$ to our explicit basis elements of the previous
section.

Recall that we let $M=L(\pi)$ be a totally tamely ramified extension of an
unramified extension $L/K$ of degree prime-to-$p$, where the uniformiser $\pi$
of $M$ is a solution of $x^{e_{M}}+\pi_{K}=0$ for $e_{M}\mid p^{f}-1$ and we
take $M$ sufficiently large so that $\chi|_{G_{M}}$ is trivial. We will denote
the residue field of $M$ by $l$.

Since $e_{M}$ and $p$ are coprime, we have that for each $n\geq 1$ there is a
unique $p^{n}$th root $\pi^{1/p^{n}}$ of $\pi$ that satisfies
$(\pi^{1/p^{n}})^{e_{M}}=\pi_{n},$ i.e. the choice of $p^{n}$th root of $\pi$
is compatible with the choice of $p^{n}$th root of $-\pi_{K}$ in the
definition of $K_{\infty}.$ Let $M_{n}:=M(\pi^{1/p^{n}})$ and
$M_{\infty}=\bigcup_{n\geq 0}M_{n}.$ It follows by the theory of the field of
norms and local class field theory applied to $l((u))$ that

$\mathrm{Gal}\left(M_{\infty}^{(p)}/M_{\infty}\right)\simeq\mathrm{Gal}\left(l((u))^{(p)}/l((u))\right)\simeq
l((u))^{\times}\otimes\mathbf{F}_{p},$

where, for any field $F$, we write $F^{(p)}$ for the maximal exponent $p$
abelian extension of $F$. Furthermore, if we let
$\psi:l((u))^{\mathrm{sep}}\to l((u)^{\mathrm{sep}}$ denote the Artin–Schreier
map defined by $\psi(x)=x^{p}-x$, then Artin–Schreier theory gives an
isomorphism

(4)
$H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})=H^{1}(G_{l((u))},\overline{\mathbf{F}}_{p})\cong(l((u))/\psi
l((u)))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}.$

By [Ser79, Ch. X, §3] the element $a\in l((u))$ concretely corresponds to the
homomorphism $f_{a}:G_{l((u))}\to\mathbf{F}_{p}$ given by $f_{a}(g)=g(x)-x,$
where $x\in l((u))^{\mathrm{sep}}$ is chosen so that $\psi(x)=a.$ Any two-
dimensional $\overline{\mathbf{F}}_{p}$-representation $\rho$ of $G_{l((u))}$
that arises as the extension of a character $\chi$ by itself corresponds
naturally to a cocyle in $H^{1}(G_{l((u))},\overline{\mathbf{F}}_{p})$ via the
standard isomorphism

(5)
$\displaystyle\mathrm{Ext}_{\mathrm{Rep}_{\overline{\mathbf{F}}_{p}}(G_{l((u))})}(\chi,\chi)$
$\displaystyle\to H^{1}(G_{l((u))},\overline{\mathbf{F}}_{p}),$
$\displaystyle\rho\cong\chi\otimes\begin{pmatrix}\mathbf{1}&c_{\rho}\\\
0&\mathbf{1}\end{pmatrix}$ $\displaystyle\mapsto[\sigma\mapsto
c_{\rho}(\sigma)].$

Since $M/K$ is an extension of degree prime-to-$p$, so is
$M_{\infty}/K_{\infty}.$ Therefore, the restriction map
$H^{1}(G_{K_{\infty}},\overline{\mathbf{F}}_{p}(\chi))\to
H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})$ is injective and we may
describe $L_{V}$ completely by its image in
$H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})$ or by the classes
corresponding to this image via Artin–Schreier theory, which is what we will
do.

If $\mathcal{M}$ is an étale $\varphi$-module with corresponding
$G_{K_{\infty}}$-representation denoted by $T(\mathcal{M}),$ then it is easy
to check that the étale $\varphi$-module corresponding to
$T(\mathcal{M})|_{G_{M_{\infty}}}$ is

$\mathcal{M}_{M}:=l((u))\otimes_{k((u)),u\mapsto u^{e_{M}}}\mathcal{M}.$

Applying this to one of the étale $\varphi$-modules arising in the statement
of Theorem 3.2, it follows that (with the obvious choice of basis
$e_{i},f_{i}$ for $\mathcal{M}_{M,i}$) the matrix of
$\varphi:\mathcal{M}_{M,i-1}\to\mathcal{M}_{M,i}$ is

$\begin{pmatrix}u^{t_{i}e_{M}}&y_{i,M}\\\
0&(a)_{i}u^{s_{i}e_{M}}\end{pmatrix},$

where we let $y_{i,M}\in\overline{\mathbf{F}}_{p}[u^{e_{M}}]$ denote the image
of $y_{i}\in\overline{\mathbf{F}}_{p}[u]$ under $u\mapsto u^{e_{M}}.$ Here the
$\mathcal{M}_{M,i}$ are periodic with period $f[l:k],$ but $t_{i},s_{i}$ and
$y_{i,M}$ only depend on $i\bmod f.$

Recall that we defined $a=\mu(\mathrm{Frob}_{K})$ and that the unramified
character $\mu$ factors as
$\mu\colon\mathrm{Gal}(L/K)\to\overline{\mathbf{F}}_{p}^{\times},$ so that
$a^{[l:k]}=1.$ We will try to find a change of basis such that the diagonal
entries of $\varphi:\mathcal{M}_{M,i-1}\to\mathcal{M}_{M,i}$ are identical.
Since

${\left.\kern-1.2pt\rho\vphantom{|}\right|_{I_{K}}}\sim\begin{pmatrix}\prod_{i}\omega_{i}^{s_{i}}&*\\\
0&\prod_{i}\omega_{i}^{t_{i}}\end{pmatrix}$

and $\chi_{1}^{e_{M}}=\chi_{2}^{e_{M}},$ it follows that
$\left(\frac{p^{f}-1}{e_{M}}\right)$ divides

$\sum_{j=0}^{f-1}(s_{i+1+j}-t_{i+1+j})p^{f-1-j}$

for all $i.$ Hence, we may define integers

$\alpha_{i}:=\left(\frac{e_{M}}{p^{f}-1}\right)\sum_{j=0}^{f-1}(s_{i+1+j}-t_{i+1+j})p^{f-1-j}.$

If we set $e_{i}^{\prime}:=u^{\alpha_{i}}e_{i}$ and $f_{i}^{\prime}=a^{\lfloor
i/f\rfloor}f_{i}$ for $0\leq i\leq f[l:k]-1,$ then after this change of basis
the matrix of $\varphi:\mathcal{M}_{M,i-1}\to\mathcal{M}_{M,i}$ becomes

$\begin{pmatrix}u^{t_{i}e_{M}+p\alpha_{i-1}-\alpha_{i}}&a^{\lfloor{i-1/f}\rfloor}y_{i,M}u^{-\alpha_{i}}\\\
0&u^{s_{i}e_{M}}\end{pmatrix}.$

That is, upon noting that $t_{i}e_{M}+p\alpha_{i-1}-\alpha_{i}=s_{i}e_{M}$,
after the change of basis the matrix of
$\varphi:\mathcal{M}_{M,i-1}\to\mathcal{M}_{M,i}$ becomes

$u^{s_{i}e_{M}}\otimes\begin{pmatrix}1&a^{\lfloor
i-1/f\rfloor}y_{i,M}u^{-(s_{i}e_{M}+\alpha_{i})}\\\ 0&1\end{pmatrix}$

for $0\leq i\leq f[l:k]-1.$ The exponents of $u$ will be important later, so
let us give them a name.

######  Definition 3.4.

For all $0\leq i<f,$ we define the constants

$\displaystyle\xi_{i}:=(p^{f}-1)s_{i}+\sum_{j=0}^{f-1}(s_{i+j+1}-t_{i+j+1})p^{f-1-j}.$

Note that for the exponent in the matrix above we have that

$s_{i}e_{M}+\alpha_{i}=\frac{e_{M}\xi_{i}}{p^{f}-1}.$

In line with our earlier notation, let us therefore write $\xi_{i}^{\prime}$
for $\frac{e_{M}\xi_{i}}{p^{f}-1}.$

######  Lemma 3.5.

We have that $\xi_{i}\equiv n_{i}\bmod{p^{f}-1}$ and, therefore, that
$\xi_{i}^{\prime}\equiv\frac{e_{M}n_{i}}{p^{f}-1}\bmod e_{M}$ for all $i.$

###### Proof.

From
${\left.\kern-1.2pt\chi\vphantom{|}\right|_{I_{K}}}={\left.\kern-1.2pt\chi_{1}\chi_{2}^{-1}\vphantom{|}\right|_{I_{K}}}$
and Theorem 3.2 above, we have that
$\prod_{i}\omega_{i}^{s_{i}-t_{i}}=\omega_{i}^{n_{i}},$ that is

$\sum_{j=0}^{f-1}(s_{i+1+j}-t_{i+1+j})p^{f-1-j}\equiv n_{i}\bmod{p^{f}-1}.$

But this sum is equal to $\xi_{i}-s_{i}(p^{f}-1)$. ∎

Recall that in Section 2.2.3 we proved that for any embedding
$\tau:k\to\overline{\mathbf{F}}_{p}$ we have that
$l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}$ is free of rank one over
$\overline{\mathbf{F}}_{p}[\mathrm{Gal}(L/K)].$ It follows that the
$\mu$-eigenspace $\Lambda_{\tau,\mu}$ defined as

$\left\\{a\otimes b\in
l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}\;\middle|\;g(a)\otimes
b=(1\otimes\mu(g))a\otimes b\text{ for all }g\in\mathrm{Gal}(L/K)\right\\}$

is 1-dimensional over $\overline{\mathbf{F}}_{p}.$ For
$a=\mu(\mathrm{Frob}_{K}),$ we define the non-zero element

$\lambda_{\tau,\mu}:=(1,a^{-1},\dotsc,a^{1-[l:k]})\in
l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}$

and note that this is in fact a basis of the one-dimensional vector space
$\Lambda_{\tau,\mu}.$ Here we have used the standard identification

$\displaystyle l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}$
$\displaystyle\xrightarrow{\hskip 8.53581pt\sim\hskip
8.53581pt}\prod_{\sigma_{i}}\overline{\mathbf{F}}_{p};$ $\displaystyle
x\otimes y$ $\displaystyle\xmapsto{\hskip 22.76219pt}(\sigma_{i}(x)y)_{i},$

where the sum runs over all $k$-embeddings $\sigma_{i}\colon
l\hookrightarrow\overline{\mathbf{F}}_{p}$ such that
${\left.\kern-1.2pt\sigma_{i}\vphantom{|}\right|_{k}}=\tau.$ Similarly,
$\lambda_{\tau,\mu^{-1}}:=(1,a,\dotsc,a^{[l:k]-1})\in
l\otimes_{k,\tau}\overline{\mathbf{F}}_{p}$ spans $\Lambda_{\tau,\mu^{-1}}.$
Using this notation we may define the element

$(a^{\lfloor
i/f\rfloor})_{i=0,\dotsc,f[l:k]-1}=(\lambda_{\overline{\tau}_{i},\mu^{-1}})_{i=0,\dotsc,f-1}\in
l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}.$

We can consider $\mathcal{M}_{M}$ as an
$l((u))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$-module with the
obvious choice of basis. Then the matrix of $\varphi_{\mathcal{M}_{M}}$
becomes

$(u^{s_{i}e_{M}})_{i=0,\dotsc,f-1}\otimes\begin{pmatrix}1&(a^{-1}\lambda_{\overline{\tau}_{i},\mu^{-1}}y_{i,M}u^{-\xi_{i}^{\prime}})_{i=0,\dotsc,f-1}\\\
0&1\end{pmatrix}.$

By the equivalence of categories $T$ between étale $\varphi$-modules and
$\overline{\mathbf{F}}_{p}$-representations of $G_{l((u))}$ combined with
Equations (4) and (5), this étale $\varphi$-module then corresponds to a class
in $l((u))/\psi l((u))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}.$ To
understand which class exactly, we prove the following proposition.

######  Proposition 3.6.

Suppose $\mathcal{M}$ is an étale $\varphi$-module over $l((u))$ such that
$\varphi_{\mathcal{M}}:\mathcal{M}\to\mathcal{M}$ is given by the matrix

$\begin{pmatrix}1&z\\\ 0&1\end{pmatrix}.$

Then the image of $\mathcal{M}$ under the isomorphism

$\mathrm{Ext}^{1}_{\mathrm{Rep}_{\mathbf{F}_{p}}\left(G_{l((u))}\right)}\left(1,1\right)\cong
l((u))/\psi l((u))$

is given by (the class of) $z.$

###### Proof.

This is a standard argument; see, for example, [Ste20, Prop. 6.2.3]. ∎

After extending coefficients to $\overline{\mathbf{F}}_{p},$ it follows from
the proposition that the Artin–Schreier extension of $l((u))$ corresponding to
$T(\mathcal{M}_{M})$ above is determined by the element
$(\lambda_{\overline{\tau}_{i},\mu^{-1}}y_{i,M}u^{-\xi_{i}^{\prime}})_{i=0,\dotsc,f-1}$
– note the absence of $a^{-1}$ because scaling by an element of
$\overline{\mathbf{F}}_{p}^{\times}$ does not change the extension. Therefore,
we have proved the following result.

######  Corollary 3.7.

The image of $L_{V}$ in
$H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})=\mathrm{Hom}(G_{l((u))},\overline{\mathbf{F}}_{p})$
is spanned by the homomorphisms
$f_{y_{i,M}\lambda_{\overline{\tau}_{i},\mu^{-1}}u^{-\xi_{i}^{\prime}}}$
corresponding via Artin–Schreier theory to the elements

$y_{i,M}\lambda_{\overline{\tau}_{i},\mu^{-1}}u^{-\xi_{i}^{\prime}}\in
l((u))\otimes_{k,\overline{\tau}_{i}}\overline{\mathbf{F}}_{p}\subseteq
l((u))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}.$

More precisely, the image is spanned by homomorphisms corresponding to

$\lambda_{\overline{\tau}_{i},\mu^{-1}}u^{de_{M}-\xi_{i}^{\prime}}\in
l((u))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$

for all $0\leq i<f$ and all $d\in\mathcal{I}_{i},$ together with the class
$\lambda_{\overline{\tau}_{i_{0}},\mu^{-1}}u^{d_{i_{0}}e_{M}-\xi_{i_{0}}^{\prime}}$
for

$d_{i_{0}}:=s_{i_{0}}+\frac{1}{p^{f}-1}\sum_{k=0}^{f-1}(s_{{i_{0}}+1+k}-t_{{i_{0}}+1+k})p^{f-1-k}$

if $\chi$ is trivial.

###  3.3. An explicit basis of $L_{V}$

In this subsection we would like to compare our description of $L_{V}$ above
to the previously constructed basis elements $c_{\alpha}$ for $\alpha\in W$
and give an explicit description of $L_{V}$ in this way. To explain our
approach we need to introduce some theory first.

We have a pairing
$\langle\cdot,\cdot\rangle:H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})\times\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})\to\overline{\mathbf{F}}_{p}$
given by evaluation. Recall that the theory of the field of norms and the
local Artin map allowed us to write
$\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})\simeq
l((u))^{\times}\otimes\mathbf{F}_{p}$ and that we used Artin–Schreier theory
to write
$H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})\simeq\left(l((u))/\psi
l((u))\right)\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}.$ The real
advantage is that we can make the pairing explicit now.

######  Theorem 3.8.

Let $\sigma_{b}\in\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})$ be the Galois
element corresponding via the local Artin map to an element $b\in
l((u))^{\times}\otimes\mathbf{F}_{p},$ and let $f_{a}$ be the element of
$H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})$ corresponding to
$a\in(l((u))/\psi l((u)))\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$
via Artin–Schreier theory. Then

$\langle
f_{a},\sigma_{b}\rangle=\mathrm{Tr}_{l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}/\overline{\mathbf{F}}_{p}}\left(\mathrm{Res}(a\cdot\frac{db}{b})\right),$

where $\mathrm{Res}(fdu)$ simply means the coefficient of $u^{-1}$ in $f$.

###### Proof.

See [Ser79, XIV, Cor. to Prop. 15]. ∎

Since we already have a description of the image of $L_{V}$ in
$H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})$ and we defined
$c_{\alpha}\in H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)),$ for $\alpha\in
W,$ as the basis dual to basis elements

$u_{\alpha}\in
U_{\chi}:=\left(M^{\times}\otimes\overline{\mathbf{F}}_{p}(\chi^{-1})\right)^{\mathrm{Gal}(M/K)}\subseteq
M^{\times}\otimes\overline{\mathbf{F}}_{p},$

we may hope to get a description of the elements $u_{\alpha}$ in
$\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})$ and look to find conditions under
which the pairing of $u_{\alpha}$ with $L_{V}$ vanishes. The local Artin map
$\mathrm{Art}_{M}^{-1}$ induces an isomorphism $\mathrm{Gal}(M^{(p)}/M)\simeq
M^{\times}\otimes\mathbf{F}_{p},$ so by extension by scalars to
$\overline{\mathbf{F}}_{p}$ we get
$\mathrm{Art}_{M}(u_{\alpha})\in\mathrm{Gal}(M^{(p)}/M)\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}.$

######  Lemma 3.9.

We have a commutative diagram of pairings

${H^{1}(G_{M_{\infty}},\overline{\mathbf{F}}_{p})}$${\times}$${\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})}$${\overline{\mathbf{F}}_{p}}$${H^{1}(G_{M},\overline{\mathbf{F}}_{p})}$${\times}$${\mathrm{Gal}(M^{(p)}/M)}$${\overline{\mathbf{F}}_{p}}$$\scriptstyle{\mathrm{pr}}$$\scriptstyle{\mathrm{res}}$

in the sense that
$\langle\mathrm{pr}(\alpha),\beta\rangle=\langle\alpha,\mathrm{res}(\beta)\rangle,$
where the pairings are given by evaluation, the map $\mathrm{res}$ is given by
restriction to $G_{M_{\infty}}$ and $\mathrm{pr}$ is given by restricting an
automorphism to $M^{(p)}.$

###### Proof.

Since
$H^{1}(G_{M},\overline{\mathbf{F}}_{p})=\mathrm{Hom}(G_{M},\overline{\mathbf{F}}_{p})$
(and similarly for $M_{\infty}$) and the pairings are given by evaluation,
this follows from the definitions.∎

By the lemma above it is enough to consider the pairing of $L_{V}$ with
elements of $\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})$ that map to the
elements $\mathrm{Art}_{M}(u_{\alpha})$ under $\mathrm{pr}.$ To be able to use
the explicit reciprocity law of Theorem 3.8, we need to use the local class
field theory of $l((u))$ and the theory of the field of norms to get an
isomorphism $\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})\simeq
l((u))^{\times}\otimes\mathbf{F}_{p}.$ From the field of norms identification
$l((u))\simeq\varprojlim_{N_{M_{n+1}/M_{n}}}M_{n}$ we get a natural map
$\mathrm{pr}_{M}:l((u))\to M$ by projecting onto the $M$-component which is
compatible with local class field theory in the following natural way.

######  Lemma 3.10.

The following diagram commutes

${\mathrm{Gal}(M_{\infty}^{(p)}/M_{\infty})}$${\mathrm{Gal}\left(l((u))^{(p)}/l((u))\right)}$${l((u))^{\times}\otimes\mathbf{F}_{p}}$${\mathrm{Gal}(M^{(p)}/M)}$${M^{\times}\otimes\mathbf{F}_{p}.}$$\scriptstyle{\mathrm{pr}}$$\scriptstyle{\sim}$$\scriptstyle{f.o.n.}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{Art}^{-1}_{l((u))}}$$\scriptstyle{\mathrm{pr}_{M}}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{Art}^{-1}_{M}}$

###### Proof.

See [CEGM17, Lem. 3.1.5].∎

The missing step is to find elements that project to our previous $u_{\alpha}$
under $\mathrm{pr}_{M}.$ Recall that $E(X)\in\mathbf{Z}_{p}\llbracket
X\rrbracket$ denotes the Artin-Hasse exponential. Then, for each $r\geq 1,$ we
get a homomorphism

$\displaystyle\varepsilon_{u^{r}}:l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$
$\displaystyle\to
l((u))^{\times}\otimes_{\mathbf{Z}}\overline{\mathbf{F}}_{p}$ $\displaystyle
a\otimes b$ $\displaystyle\mapsto E(au^{r})\otimes b$

and, analogously to our earlier definitions of $u_{\alpha}$ in §2.2.3, we will
define

$\tilde{u}_{\alpha}:=\varepsilon_{u^{m^{\prime}}}(\lambda_{\tau_{\alpha},\mu})$

for $\alpha=(m,k)\in W.$

######  Proposition 3.11.

Under the map $\mathrm{pr}_{M}:l((u))^{\times}\otimes\mathbf{F}_{p}\to
M^{\times}\otimes\mathbf{F}_{p},$ we have that

$\mathrm{pr}_{M}\left(E(au^{r})\right)=E([a]\pi^{r})$

for any $r\geq 1$ which is prime to $p.$ In particular, after extending by
scalars to $\overline{\mathbf{F}}_{p},$ we have that
$\mathrm{pr}_{M}(\tilde{u}_{\alpha})=u_{\alpha}$ for $\alpha\in W.$

###### Proof.

To see why this is true, we recall that addition in $\varprojlim M_{n}$ is
defined by the formula

$(x+y)^{(n)}=\lim_{m\to\infty}N_{M_{n+m}/M_{n}}(x^{(n+m)}+y^{(n+m)}).$

If we write $E(X)=\sum_{k\geq 0}c_{k}X^{k}\in\mathbf{Z}_{p}\llbracket
X\rrbracket,$ then we see that

$\displaystyle\overline{E}(au^{r})$
$\displaystyle=\overline{E}\left(([a]\pi^{r},([a]\pi^{r})^{1/p},([a]\pi^{r})^{1/p^{2}},\dotsc)\right)$
$\displaystyle=\sum_{k\geq
0}\overline{c}_{k}\left([a]\pi^{r},([a]\pi^{r})^{1/p},([a]\pi^{r})^{1/p^{2}},\dotsc\right)^{k}$
$\displaystyle\mapsto\lim_{m\to\infty}N_{M_{m}/M}(\sum_{k\geq
0}c_{k}([a]\pi^{r})^{k/p^{m}}),$

where we have taken the projection onto the $M$-component in the last line.
Now the proposition follows immediately from [CEGM17, Lem. 3.5.1], where it is
proved that for any $n>1$, $a\in l$ and $r\geq 1$ such that $(r,p)=1$ we have

$N_{M_{n}/M}E([a^{1/p^{n}}](\pi^{1/p^{n}})^{r})=E([a]\pi^{r}).$

The second claim follows since for all $\alpha=(m,k)\in W$ we have that
$m^{\prime}$ is coprime to $p.$ ∎

Now we have covered all the theory to state and prove our explicit version of
the space $L_{V}$ in terms of the basis elements constructed previously for
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)).$ Recall that we defined the
indexing set $W$ of all basis elements in §2.2.3. In the following definition
we will give the definition of an indexing set $J_{V}^{\mathrm{AH}}$ as a
subset of $W$ such that the span of basis elements $c_{\alpha}$ over this
indexing set will give us $L_{V}.$

######  Definition 3.12.

We let $J^{\mathrm{AH}}_{V}(\chi_{1},\chi_{2})$ denote the subset of all
$\alpha=(k,m)\in W$ such that there exist integers $0\leq i<f,$
$d\in\mathcal{I}_{i}$ and $j\geq 0$ such that

  1. 1)

$p^{j}m^{\prime}=\xi_{i}^{\prime}-de_{M}$ and

  2. 2)

$i_{m}+kf^{\prime}\equiv i-j\bmod f,$

where $i_{m}\in\\{0,\dotsc,f^{\prime}-1\\}$ is defined by $m\equiv
n_{i_{m}}\bmod{(p^{f}-1)}.$

Note that, even though we need to calculate the integers $s_{i},t_{i}$
corresponding to minimal and maximal Kisin modules to be able to define
$\xi_{i}$ and $\mathcal{I}_{i}$ concretely, these integers can be calculated
completely explicitly (as we have done above) without the use of any $p$-adic
Hodge theory. The dependence on $\\{r_{i}\\}$ or, equivalently, the dependence
on the Serre weight $V_{\underline{\eta},\underline{0}}$ for
$\eta_{i}=r_{i}-1$ is implicit in the definitions of $s_{i},t_{i}$ (hence
$\xi_{i}$) and $\mathcal{I}_{i}$.

######  Proposition 3.13.

$\left|{J^{\mathrm{AH}}_{V}(\chi_{1},\chi_{2})}\right|\leq\sum_{i=0}^{f-1}\left|{\mathcal{I}_{i}}\right|.$

###### Proof.

By multiplying the left and right hand sides of the first equation in
Definition 3.12 by $(p^{f}-1)/e_{M},$ we obtain the equation

(6) $p^{j}m=\xi_{i}-d(p^{f}-1);$

this follows from the definitions of $m^{\prime}$ and $\xi_{i}^{\prime}$.
Therefore, given an $\alpha=(m,k)\in W$, we can solve the first equation of
Definition 3.12 for some $i,d$ and $j$ if and only if we can solve Equation 6
for the same $i,d$ and $j.$ It follows that we may assume without loss of
generality that $e_{M}=p^{f}-1.$

For a fixed choice of $0\leq i<f$ and $d\in\mathcal{I}_{i}$, we set
$j:=v_{p}(\xi_{i}-d(p^{f}-1))$ and define

$a:=\frac{\xi_{i}-d(p^{f}-1)}{p^{j}}.$

It follows from Lemma 3.5 that $p^{j}a\equiv n_{i}\bmod{(p^{f}-1)}$, so

$a\equiv n_{i-j}\bmod{(p^{f}-1)}.$

Therefore, by definition of $W^{\prime}$ in §2.2.3, the first equation of
Definition 3.12 will have a solution for some $m\in W^{\prime}$ if and only if
$0<a<\frac{pe}{p-1}(p^{f}-1)$. Suppose that this is the case. Then we have a
unique $m\in W^{\prime}$ solving the equation, namely $m=a.$ Since the
integers $m\in W^{\prime}$ are distinct, this means that $\alpha=(m,k)\in W$
solving the first equation of Definition 3.12 for our fixed choice of $(i,d)$
exists and is unique up to $k\in\\{0,\dotsc,f^{\prime\prime}-1\\}$.

Note that it follows from $m\equiv n_{i_{m}}\bmod{(p^{f}-1)}$ and $m\equiv
n_{i-j}\bmod{(p^{f}-1)}$ that $i_{m}\equiv i-j\bmod{f^{\prime}}.$ The second
equation of Definition 3.12 requires that $k\equiv\frac{i-j-
i_{m}}{f^{\prime}}\bmod{f^{\prime\prime}}$, thereby determining
$k\in\\{0,\dotsc,f^{\prime\prime}-1\\}$ uniquely. In other words, we have
proved that there exists a unique $\alpha\in
J^{\mathrm{AH}}_{V}(\chi_{1},\chi_{2})$ corresponding to our fixed choice of
$(i,d)$ if

$0<\frac{\xi_{i}-d(p^{f}-1)}{p^{j}}<\frac{ep}{p-1}(p^{f}-1)$

and no $\alpha$ corresponding to $(i,d)$ otherwise. ∎

We remark that it is possible to show with a direct argument, which carefully
uses the minimality (resp. maximality) of the integers $t_{i}$ (resp.
$s_{i}$), that the inequalities on $a$ in the proof above are always
satisfied. This also follows as a consequence of Theorem 3.16 which implies
that
$\left|{J^{\mathrm{AH}}_{V}(\chi_{1},\chi_{2})}\right|=\sum_{i=0}^{f-1}\left|{\mathcal{I}_{i}}\right|.$

Let $c_{\alpha}$ denote the basis elements of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi))$ defined in §2.3, where we
assume we define these making the same choices in the definition of $M=L(\pi)$
as in the current section, that is, they are defined using the same fixed
uniformiser $\pi_{K}\in K$, the same unramified extension $L$ of $K$ and the
same $\pi$ satisfying $\pi^{e_{M}}+\pi_{K}=0.$

######  Definition 3.14.

We define $L_{V}^{\mathrm{AH}}(\chi_{1},\chi_{2})$ to be the span of

$\\{c_{\alpha}\mid\alpha\in J^{\mathrm{AH}}_{V}(\chi_{1},\chi_{2})\\}$

together with $c_{\mathrm{un}}$ if $\chi$ is trivial and $c_{\mathrm{tr}}$ if
$\chi$ is cyclotomic, $\chi_{2}$ unramified and $r_{i}=p$ for all $i.$

######  Corollary 3.15.

$\dim_{\overline{\mathbf{F}}_{p}}L_{V}^{\mathrm{AH}}(\chi_{1},\chi_{2})\leq\dim_{\overline{\mathbf{F}}_{p}}L_{V}(\chi_{1},\chi_{2}).$

###### Proof.

This follows immediately from the definitions, Proposition 3.13 and Theorem
3.2 (since $y_{i}$ uniquely determines $\mathcal{M}$ and vice-versa). ∎

######  Theorem 3.16.

$L_{V}(\chi_{1},\chi_{2})=L_{V}^{\mathrm{AH}}(\chi_{1},\chi_{2}).$

###### Proof.

By Corollary 3.15 it is enough to prove $L_{V}\subseteq L_{V}^{\mathrm{AH}}.$
We defined $L_{V}^{\mathrm{AH}}$ in terms of the basis elements $c_{\alpha}$
dual to $u_{\alpha},$ so we need to prove that the image of every class of
$L_{V}$ in $H^{1}(G_{M},\overline{\mathbf{F}}_{p})$ is orthogonal to the
elements $\mathrm{Art}_{M}(u_{\alpha})$ in $\mathrm{Gal}(M^{(p)}/M)$ for
$\alpha\notin J_{V}^{\mathrm{AH}}$ under the pairing of Lemma 3.9. If $\chi$
is cyclotomic, we also need to prove orthogonality under pairing with
$u_{\mathrm{cyc}}$, which we will do first.

If $\chi$ is cyclotomic, then the classes that are orthogonal to
$u_{\mathrm{cyc}}$ are exactly the peu ramifiées classes. We see this, for
example, because the subspace orthogonal to $u_{\mathrm{cyc}}$ is spanned by
the basis elements $c_{\alpha}$ for $\alpha\in W$ and $c_{\mathrm{ur}}$ if
$\chi$ is trivial. Since $c_{\mathrm{tr}}$ spans the one-dimensional subspace
defined as
$\mathrm{Fil}^{1+\frac{ep}{p-1}}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)),$
it follows that these basis elements span the space
$\mathrm{Fil}^{<1+\frac{ep}{p-1}}H^{1}(G_{K},\overline{\mathbf{F}}_{p}(\chi)),$
which equals the peu ramifié subspace by [Ste20, Cor. 4.4.1]. However, since
we excluded the exceptional case in which $L_{V}$ equals
$H^{1}(G_{K},\overline{\mathbf{F}}_{p})$, we have that $L_{V}$ is always
contained in the peu ramifié subspace of
$H^{1}(G_{K},\overline{\mathbf{F}}_{p})$ by [Ste20, Thm. 4.1.1]. Therefore,
$L_{V}$ is always contained in the subspace orthogonal to $u_{\mathrm{cyc}}.$

As outlined above we apply the explicit reciprocity law of Theorem 3.8 to the
elements $\tilde{u}_{\alpha}$ and the elements of Corollary 3.7. We must show
that for all $0\leq i<f$, $d\in\mathcal{I}_{i}$ and $\alpha=(m,k)\notin
J_{V}^{\mathrm{AH}}$

$\mathrm{Tr}_{l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}/\overline{\mathbf{F}}_{p}}\mathrm{Res}\left(\mathrm{dlog}(\tilde{u}_{\alpha})\cdot\lambda_{\tau_{i},\mu^{-1}}u^{de_{M}-\xi_{i}^{\prime}}\right)=0$

and if $\chi=1$ we also need to show that the pairing with
$\lambda_{\overline{\tau}_{i_{0}},\mu^{-1}}u^{d_{i_{0}}e_{M}-\xi_{i_{0}}^{\prime}}$
vanishes. Since

$\mathrm{dlog}E(X)=(X+X^{p}+X^{p^{2}}+\dots)\mathrm{dlog}X$

and $\mathrm{dlog}(\lambda u^{n})=n\cdot u^{-1}$, the pairing evaluates to

$\mathrm{Tr}_{l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}/\overline{\mathbf{F}}_{p}}\mathrm{Res}\left(\sum_{j\geq
0}m^{\prime}\mathrm{Frob}^{j}(\lambda_{\tau_{\alpha},\mu})u^{m^{\prime}p^{j}-1}\cdot\lambda_{\tau_{i},\mu^{-1}}u^{de_{M}-\xi_{i}^{\prime}}\right),$

where $\mathrm{Frob}:l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}\to
l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}$ is induced by the $p$-th
power map on $l.$ The residue is given by the coefficient of $u^{-1}$, hence
this can only possibly be non-zero when
$p^{j}m^{\prime}=\xi_{i}^{\prime}-de_{M}$ for some $j\geq 0.$ If $\chi=1$ we
must also consider $p^{j}m^{\prime}=\xi_{i_{0}}^{\prime}-d_{i_{0}}e_{M},$ but
it follows easily from the observation $\xi_{i_{0}}^{\prime}=d_{i_{0}}e_{M}$
that this can never have solutions. In the former case, if such an $j$ exists,
then the pairing evaluates to

$m^{\prime}\mathrm{Tr}_{l\otimes_{\mathbf{F}_{p}}\overline{\mathbf{F}}_{p}/\overline{\mathbf{F}}_{p}}\left(\mathrm{Frob}^{j}(\lambda_{\tau_{\alpha},\mu})\cdot\lambda_{\tau_{i},\mu^{-1}}\right).$

Since

$\mathrm{Frob}^{j}(\lambda_{\tau_{\alpha},\mu})\cdot\lambda_{\tau_{i},\mu^{-1}}=\mathrm{Frob}^{j}(\lambda_{\tau_{\alpha},\mu}\cdot\lambda_{\tau_{i-j},\mu^{-1}}),$

we find that this expression is non-zero if and only if
$\tau_{\alpha}=\tau_{i-j},$ i.e. when $i_{m}+kf^{\prime}\equiv i-j\bmod f.$

In conclusion, we have proved that for $\alpha=(m,k)\in W$ the pairing of
$u_{\alpha}$ with $L_{V}$ is non-zero when there exist integers $0\leq i<f,$
$d\in\mathcal{I}_{i}$ and $j\geq 0$ such that

  1. 1)

$p^{j}m^{\prime}=\xi_{i}^{\prime}-de_{M}$ and

  2. 2)

$i_{m}+kf^{\prime}\equiv i-j\bmod{f}.$

These are precisely the conditions that imply that $\alpha\in
J_{V}^{\mathrm{AH}}$, as required. ∎

######  Corollary 3.17.

The space $L_{V}^{\mathrm{AH}}(\chi_{1},\chi_{2})$ is independent of the
choice of the extension $M$ of $K$ and the choice of the uniformiser $\pi$ of
$M$.

###### Proof.

This follows immediately since $L_{V}(\chi_{1},\chi_{2})$ is independent of
the choices and we have just proved that
$L_{V}^{\mathrm{AH}}(\chi_{1},\chi_{2})=L_{V}(\chi_{1},\chi_{2})$ for any
suitable set of choices. ∎

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