# The $p$-arithmetic homology of mod $p$ representations of
$\text{GL}_{2}(\mathbb{Q}_{p})$

Guillem Tarrach

###### Abstract.

We compute the non-Eisenstein systems of Hecke eigenvalues contributing to the
$p$-arithmetic homology of irreducible smooth mod $p$ representations $\pi$ of
$\text{GL}_{2}(\mathbb{Q}_{p})$ and to the cohomology of their duals. We show
that in most cases they are associated to odd irreducible 2-dimensional Galois
representations whose local component at $p$ corresponds under the mod $p$
local Langlands correspondence to a smooth representation that contains $\pi$
as a subrepresentation.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $p$-arithmetic homology of $\pi(r,\lambda,\chi)$
  4. 4 Preparation for the non-generic cases
  5. 5 Proof of Theorem 1.1

##  1\. Introduction

Let $p\geq 5$ be a prime number and $N\geq 3$ an integer coprime to $p$. Let
$\Gamma_{1}^{p}(N)$ be the subgroup of matrices in
$\text{GL}_{2}(\mathbb{Z}[1/p])$ that have positive determinant and are
congruent modulo $N\mathbb{Z}[1/p]$ to a matrix of the form
$\begin{pmatrix}*&*\\\ 0&1\end{pmatrix}$. The goal of this article is to
compute the systems of Hecke eigenvalues contributing to the homology of
$\Gamma_{1}^{p}(N)$ with coefficients in the irreducible mod $p$
representations of $\text{GL}_{2}(\mathbb{Q}_{p})$ and the cohomology of their
duals. More specifically, we prove the following local-global compatibility
result.

######  Theorem 1.1.

Let $\pi$ be an irreducible smooth mod $p$ representation of
$\text{GL}_{2}(\mathbb{Q}_{p})$ over $\overline{\mathbb{F}}_{p}$, $\pi^{\vee}$
its abstract $\overline{\mathbb{F}}_{p}$-dual. Then:

  1. (i)

The $p$-arithmetic homology $H_{*}(\Gamma_{1}^{p}(N),\pi)$ and cohomology
$H^{*}(\Gamma_{1}^{p}(N),\pi^{\vee})$ are finite-dimensional and vanish in
degrees outside the range $[0,3]$.

  2. (ii)

To any system of Hecke eigenvalues in these spaces, one can associate a
2-dimensional odd semisimple mod $p$ Galois representation of
$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ satisfying the usual relations
at primes not dividing $pN$.

  3. (iii)

Let
$\rho\colon\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\text{GL}_{2}(\overline{\mathbb{F}}_{p})$
be a 2-dimensional odd irreducible Galois representation. Then, $\rho$
contributes to $H_{*}(\Gamma_{1}^{p}(N),\pi)$ and
$H^{*}(\Gamma_{1}^{p}(N),\pi^{\vee})$ if and only if $N$ is a multiple of the
minimal level $N(\rho)$ attached to $\rho$ by Serre [Ser87], and one of the
following is satisfied:

    1. (a)

$\pi$ is a subrepresentation of the representation associated to the
restriction of $\rho$ at a decomposition group $\mathcal{G}_{p}$ at $p$ by the
mod $p$ local Langlands correspondence for $\text{GL}_{2}(\mathbb{Q}_{p})$. In
this case, $\rho$ contributes to (co)homology in degrees 1, 2 and 3, unless
$\pi$ is a twist of the Steinberg representation, in which case $\rho$
contributes to cohomology in degrees 1 and 2.

    2. (b)

$\pi$ is a character, say $\pi=\chi\circ\det$, and $\rho|_{\mathcal{G}_{p}}$
is an extension of $\chi\omega^{-1}$ by $\chi$, where we have identified
$\chi$ with a character of $\mathcal{G}_{p}$ via local class field theory and
$\omega$ denotes the mod $p$ cyclotomic character. In this case, $\rho$
contributes to (co)homology in degrees 2 and 3.

The proof of the theorem is obtained by combining the explicit construction of
the irreducible mod $p$ representations of $\text{GL}_{2}(\mathbb{Q}_{p})$ due
to Barthel–Livné [BL94] and Breuil [Bre03], a result relating $p$-arithmetic
homology to arithmetic homology in the spirit of [KS12] and [Tar22], and
classical results on the weight part of Serre’s conjecture. These are already
enough to prove the generic case where $\pi$ is supersingular or principal
series. The cases of (twists of) the trivial and Steinberg representations
require more work, and involve the group cohomological analogue of
multiplication of mod $p$ modular forms by the Hasse invariant studied by
Edixhoven–Khare [EK03] and an interpretation of this map in terms of the
representation theory of the local group $\text{GL}_{2}(\mathbb{Q}_{p})$.

###  1.1. Notation and conventions

Write $G=\text{GL}_{2}(\mathbb{Q}_{p})$, $K=\text{GL}_{2}(\mathbb{Z}_{p})$ and
$Z$ for the center of $G$, so that $Z\simeq\mathbb{Q}_{p}^{\times}$. Let
$\alpha=\begin{pmatrix}1&0\\\ 0&p\end{pmatrix}\in G$ and
$\beta=\begin{pmatrix}p&0\\\ 0&p\end{pmatrix}\in Z$. Write also $B$ for the
subgroup of upper-triangular matrices in $G$ and $I=K\cap\alpha K\alpha^{-1}$
for the Iwahori subgroup of matrices in $K$ that are upper-triangular modulo
$p$. Let $G^{+}\subseteq G$ be the submonoid of matrices whose entries lie in
$\mathbb{Z}_{p}$, and $G^{-}=(G^{+})^{-1}$. We will write $\omega$ for the
character $\mathbb{Q}_{p}^{\times}\longrightarrow\mathbb{F}_{p}^{\times}$
defined by $x\longmapsto x|x|\mod p$. Write
$k=\overline{\mathbb{F}}_{p}^{\times}$.

We normalise local class field theory so that uniformisers correspond to
geometric Frobenii, and for each prime $\ell$ we let $\text{Frob}_{\ell}$ be
the geometric Frobenius corresponding to $\ell$. Choose embeddings
$\overline{\mathbb{Q}}\xhookrightarrow{}\overline{\mathbb{Q}}_{\ell}$ for all
$\ell$, and let $\mathcal{G}_{\ell}$ denote the corresponding decomposition
group at $\ell$ in $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. We will use
our normalisation of local class field theory to identify characters of
$\mathcal{G}_{\ell}$ and $\mathbb{Q}_{\ell}^{\times}$ without coming. Write
$\varepsilon\colon\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow
k^{\times}$ for the mod $p$ cyclotomic character, it satisfies
$\varepsilon(\text{Frob}_{\ell})=\ell^{-1}\mod p$ and
$\varepsilon(\text{Frob}_{p})=1$. Its restriction to $\mathcal{G}_{p}$ at $p$
corresponds to $\omega$ under local class field theory. Write
$\mathcal{I}_{p}\subseteq\mathcal{G}_{p}$ for the inertia subgroup. Let
$\omega_{2}\colon\mathcal{I}_{p}\longrightarrow\mu_{p^{2}-1}(\overline{\mathbb{Q}}_{p}^{\times})\subseteq\overline{\mathbb{F}}_{p}^{\times}$
be Serre’s fundamental character of level 2, defined by
$\omega_{2}(g)=(gp^{1/(p^{2}-1)})/p^{1/(p^{2}-1)}$, and for $0\leq s\leq p$
let $\operatorname{Ind}(\omega_{2}^{s})$ be the irreducible representation of
$\mathcal{G}_{p}$ over $\chi$ with determinant $\omega^{s}$ and
$\operatorname{Ind}(\omega_{2}^{s})|_{\mathcal{I}_{p}}=\omega_{2}^{s}\oplus\omega_{2}^{ps}$.
All irreducible 2-dimensional representations of $\mathcal{G}_{p}$ over $k$
are of the form $\operatorname{Ind}(\omega_{2}^{s})\otimes\chi$ for some $s$
as above and character $\chi$. Given a two-dimensional odd and irreducible
representation $\rho$ of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over
$k$, we will write $N(\rho)$ for the minimal level attached to $\rho$ by Serre
in [Ser87].

Given $b\in k$, we will write $\mathrm{unr}_{{b}}$ for the $k$-valued
unramified characters of $\mathbb{Q}_{p}^{\times}$ and of
$\text{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ sending $p$ and
$\text{Frob}_{p}$ respectively to $b$. Thus, all continuous characters
$\mathbb{Q}_{p}^{\times}\longrightarrow k^{\times}$ are of the form
$\omega^{a}\mathrm{unr}_{{b}}$ for some $b\in k$ and $0\leq a\leq p-2$. If $V$
is any representation of $G$ (resp. $K$) and $\chi$ is a $k$-valued continuous
character of $\mathbb{Q}_{p}^{\times}$ (resp. $\mathbb{Z}_{p}^{\times}$), we
will write $V\otimes\chi$ instead of $V\otimes(\chi\circ\det)$.

### Acknowledgements

The author is grateful to Jack Thorne for suggesting the question of studying
the $p$-arithmetic cohomology of smooth mod $p$ representations of
$\text{GL}_{2}(\mathbb{Q}_{p})$ and for his comments on earlier drafts of this
article.

##  2\. Preliminaries

###  2.1. Arithmetic and $p$-arithmetic (co)homology

Let $U^{p}\subseteq\text{GL}_{2}(\mathbb{A}^{p\infty})$ be a compact open
subgroup of the form $\prod_{\ell\nmid
N}\text{GL}_{2}(\mathbb{Z}_{\ell})\times U_{N}$ for some $N\geq 1$ coprime to
$p$ and open compact subgroup $U_{N}\subseteq\prod_{\ell\mid
N}\text{GL}_{2}(\mathbb{Q}_{\ell})$. Assume $U^{p}$ is neat. Let
$\text{GL}_{2}(\mathbb{R})^{\circ}$ be the subgroup of
$\text{GL}_{2}(\mathbb{R})$ consisting of matrices with positive determinant.
The (arithmetic) homology of level $U^{p}K$ of a left $k[K]$-module $M$, is
defined as

$H_{i}(U^{p}K,M):=\text{Tor}_{i}^{k[\text{GL}_{2}(\mathbb{Q})\times
U^{p}\times
K\times\text{GL}_{2}(\mathbb{R})^{\circ}]}(k[\text{GL}_{2}(\mathbb{A})],M).$

Here, we view $M$ as a $\text{GL}_{2}(\mathbb{Q})\times U^{p}\times
K\times\text{GL}_{2}(\mathbb{R})^{\circ}$ module by letting
$\text{GL}_{2}(\mathbb{Q})\times U^{p}\times\text{GL}_{2}(\mathbb{R})^{\circ}$
act trivially, and $k[\text{GL}_{2}(\mathbb{A})]$ is acted on by
$\text{GL}_{2}(\mathbb{Q})$ by multiplication on the left and by $U^{p}\times
K\times\text{GL}_{2}(\mathbb{R})^{\circ}$ by multiplication on the right. The
(arithmetic) cohomology of $M$ in level $U^{p}K$ is defined analogously as

$H_{i}(U^{p}K,M):=\text{Ext}^{i}_{k[\text{GL}_{2}(\mathbb{Q})\times
U^{p}\times
K\times\text{GL}_{2}(\mathbb{R})^{\circ}]}(k[\text{GL}_{2}(\mathbb{A})],M).$

Similarly, if $M$ is a $k[G]$-module, the $p$-arithmetic homology and
cohomology of $M$ in level $U^{p}$ are defined as

$\displaystyle H_{i}(U^{p},M)$
$\displaystyle:=\text{Tor}_{i}^{k[\text{GL}_{2}(\mathbb{Q})\times
U^{p}\times\text{GL}_{2}(\mathbb{Q}_{p})\times\text{GL}_{2}(\mathbb{R})^{\circ}]}(k[\text{GL}_{2}(\mathbb{A})],M),$
$\displaystyle H^{i}(U^{p},M)$
$\displaystyle:=\text{Ext}^{i}_{k[\text{GL}_{2}(\mathbb{Q})\times
U^{p}\times\text{GL}_{2}(\mathbb{Q}_{p})\times\text{GL}_{2}(\mathbb{R})^{\circ}]}(k[\text{GL}_{2}(\mathbb{A})],M).$

For both arithmetic and $p$-arithmetic homology (and similarly for
cohomology), one can canonically define complexes computing them as in [Tar22,
Section 5.1], where they were denoted $C^{\text{ad}}_{\bullet}(U^{p}K,M)$ and
$C^{\text{ad}}_{\bullet}(U^{p},M)$; here we will denote them by
$C_{\bullet}(U^{p}K,M)$ and $C_{\bullet}(U^{p},M)$ respectively. One can also
speak of arithmetic and $p$-arithmetic hyperhomology and hypercohomology of
complexes of $k[K]$ or $k[G]$-modules; these are just the derived tensor
products and derived Hom corresponding to the Tor and Ext modules above in
their corresponding derived category. In this article we will only be
interested in the case where

$U^{p}=U^{p}_{1}(N):=\prod_{\ell\nmid
N}\text{GL}_{2}(\mathbb{Z}_{\ell})\times\prod_{\ell\mid
N}\left\\{g\in\text{GL}_{2}(\mathbb{Z}_{\ell}):g\equiv\begin{pmatrix}*&*\\\
0&1\end{pmatrix}\mod\ell^{v_{\ell}(N)}\right\\}$

and $N\geq 3$. In this case, there are canonical isomorphisms

$\displaystyle H_{*}(U^{p}_{1}(N)K,{-})$ $\displaystyle\simeq
H_{*}(\Gamma_{1}(N),{-}),$ $\displaystyle H^{*}(U^{p}_{1}(N)K,{-})$
$\displaystyle\simeq H^{*}(\Gamma_{1}(N),{-}),$ $\displaystyle
H_{*}(U^{p}_{1}(N),{-})$ $\displaystyle\simeq H_{*}(\Gamma^{p}_{1}(N),{-}),$
$\displaystyle H^{*}(U^{p}_{1}(N),{-})$ $\displaystyle\simeq
H^{*}(\Gamma^{p}_{1}(N),{-}),$

where the right-hand sides denote group homology or cohomology,
$\Gamma_{1}(N)\subseteq\text{SL}_{2}(\mathbb{Z})$ is the usual congruence
subgroup and $\Gamma^{p}_{1}(N)$ is defined as in the introduction. The
arithmetic (resp. $p$-arithmetic) (co)homology groups are non-zero in degrees
outside the range $[0,1]$ (resp. $[0,3]$).

###  2.2. Hecke operators

Let $H$ is any locally profinite group, $H_{0}$ a compact open subgroup and
$H_{+}\subseteq H$ a submonoid containing $H_{0}$, and write
$H_{-}=H_{+}^{-1}$. If $M$ is a (left) $k[H_{-}]$-module (resp.
$k[H_{+}]$-module), then the $H_{0}$-coinvariants $M_{H_{0}}$ (resp.
$H_{0}$-invariants $M^{H_{0}}$) are naturally a right (resp. left) module for
the Hecke algebra $\mathcal{H}(H_{+},H_{0})_{k}$, the algebra of smooth
compactly supported $H_{0}$-biinvariant functions $H_{+}\longrightarrow H$
under convolution.

This discussion applies in particular to arithmetic and $p$-arithmetic
(co)homology. Let $U^{p}$ and $N$ be as in the previous section. Let
$\mathbb{T}(pN)$ denote the abstract unramified Hecke algebra for
$\text{GL}_{2}$ away from $pN$ with coefficients in $\mathbb{Z}$, that is, the
restricted tensor product of the local Hecke algebras
$\mathcal{H}(\text{GL}_{2}(\mathbb{Q}_{\ell}),\text{GL}_{2}(\mathbb{Z}_{\ell}))$
with $\ell\nmid pN$. It is a commutative algebra freely generated by Hecke
operators $T_{\ell}$, corresponding to the double coset of
$\begin{pmatrix}1&0\\\ 0&\ell\end{pmatrix}$, and invertible operators
$S_{\ell}$, corresponding to the double coset of $\begin{pmatrix}\ell&0\\\
0&\ell\end{pmatrix}$, where $\ell\nmid pN$. Fix also a submonoid
$G_{+}\subseteq G$ containing $K$. Let $V$ be a representation of $G_{-}$
(resp. $G_{+}$) on a $k$-vector space. Then, the arithmetic homology
$H_{*}(U^{p}K,V)$ (resp. the arithmetic cohomology $H^{*}(U^{p}K,V)$) is the
$U^{p}K$-coinvariants (resp. invariants) of a representation of
$\text{GL}_{2}(\mathbb{A}^{p\infty})\times G_{+}$, and is thus endowed with
commuting actions of $\mathbb{T}(pN)$ and $\mathcal{H}(G_{+},K)_{k}$, the
latter being a right (resp. left) action. For us, $\mathcal{H}(G_{+},K)_{k}$
will always be a commutative algebra so we will not distinguish between left
and right actions. Similarly, if $V$ is a representation of $G$ on a
$k$-vector space, then the $p$-arithmetic homology $H_{*}(U^{p},V)$ (resp. the
$p$-arithmetic cohomology $H^{*}(U^{p},V)$) is endowed with an action of
$\mathbb{T}(pN)$. If $V$ is a representation of $G_{-}$ (resp. $G$) and
$V^{\vee}$ denotes its (abstract) contragredient, then there are
$\mathbb{T}(pN)\otimes\mathcal{H}(G_{+},K)$-equivariant (resp.
$\mathbb{T}(pN)$-equivariant) isomorphisms $H^{*}(U^{p}K,V^{\vee})\simeq
H_{*}(U^{p}K,V)^{\vee}$ (resp. $H^{*}(U^{p},V^{\vee})\simeq
H_{*}(U^{p},V)^{\vee}$). In particular, the systems of eigenvalues in both
spaces are the same, so all the statements for cohomology in Theorem 1.1
follow from the corresponding statements for homology. For this reason, we
will now work almost exclusively with homology.

Let $\rho$ be a $k$-valued continuous representation of
$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ unramified outside $N$, and
consider the maximal ideal $\mathfrak{m}_{\rho}$ of $\mathbb{T}(pN)$ defined
by the kernel of the homomorphism $\mathbb{T}(pN)\longrightarrow k$ defined by
$T_{\ell}\longmapsto\operatorname{tr}\rho(\text{Frob}_{\ell})$ and $\ell
S_{\ell}\longmapsto\det\rho(\text{Frob}_{\ell})$ for $\ell\nmid p$, where
$\text{Frob}_{\ell}$ is a geometric Frobenius at $\ell$. For example,
$\mathfrak{m}_{1\oplus\varepsilon^{-1}}$ is generated by
$T_{\ell}-(1+\ell),\ell S_{\ell}-\ell$. Given a $\mathbb{T}(pN)$-module $M$,
we will say that $\rho$ contributes to, or appears in, $M$ is the localisation
of $M_{\mathfrak{m}_{\rho}}$ is non-zero. We will sometimes write $M_{\rho}$
instead of $M_{\mathfrak{m}_{\rho}}$. If $V$ is an irreducible representation
of $K$, then any system of Hecke eigenvalues in $H^{*}(\Gamma_{1}(N),V)$
corresponds to a semisimple 2-dimensional Galois representation as above. For
such a $V$, the cohomology $H^{0}(\Gamma_{1}(N),V)$ vanishes unless $V$ is the
trivial representation, in which case it is one-dimensional and the Hecke
operators act via $T_{\ell}=1+\ell,S_{\ell}=1$. In particular, the only
semisimple Galois representation $\rho$ that can contribute to
$H^{0}(\Gamma_{1}(N),V)$ is $1\oplus\varepsilon^{-1}$. In particular,
irreducible Galois representations can only contribute to arithmetic
cohomology in level $\Gamma_{1}(N)$ only in degree 1.

Given a character $\chi\colon\mathbb{Q}_{p}^{\times}\longrightarrow
k^{\times}$, write $k(\chi)$ for the $\mathbb{T}(pN)$-module whose underlying
module is $k$ and where $T_{\ell}$ (resp. $S_{\ell}$) act via $\chi(\ell)$
(resp. $\chi(\ell)^{2}$). For any $\mathbb{T}(pN)$-module $M$, write
$M(\chi):=M\otimes_{k}k(\chi)$ for the twist of $M$ by $k(\chi)$.

###  2.3. Irreducible mod $p$ representations of
$\text{GL}_{2}(\mathbb{Q}_{p})$

In this section we will recall the construction of the smooth irreducible mod
$p$ representations of $G$ and some facts about them. Given $0\leq r\leq p-1$,
consider the representation $\operatorname{Sym}^{r}(k^{2})^{\vee}$ of $K$ over
$k$. Note that $\operatorname{Sym}^{r}(k^{2})$ naturally extends to a
representation of the monoid $G^{+}$ of matrices with entries in
$\mathbb{Z}_{p}$, and also (perhaps less naturally) to a representation of
$KZ$ where $\beta$ acts trivially. In particular,
$\operatorname{Sym}^{r}(k^{2})^{\vee}$ extends to representations of $G^{-}$
and of $KZ$. The first action defines an action of $\mathcal{H}(G^{+},K)$ on
the compact induction
$\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee})$ and the
second defines an action of $\mathcal{H}(KZ,K)$. We let $T$ denote the
operator on
$\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee})$
corresponding to the double coset of $\alpha$ under the first action and $S$
the operator corresponding to the double coset of $\beta$ under the second,
which is an invertible operator. Explicitly, these operators are described as
follows. Given $g\in G$ and $v\in\operatorname{Sym}^{r}(k^{2})^{\vee})$, let
$[g,v]$ denote the element of
$\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee})$ that is
supported on $gK$ and maps $g$ to $v$. We will use the same notation for
compact inductions from other groups and of other representations. The Hecke
operators above are then defined by the formulas

$\displaystyle T[g,v]$ $\displaystyle=\sum_{x\in
K/I}[gx\alpha,\alpha^{-1}x^{-1}v],$ $\displaystyle S[g,v]$
$\displaystyle=[\beta g,v],$

Given a continuous character $\chi\colon\mathbb{Q}_{p}^{\times}\longrightarrow
k^{\times}$ and $\lambda\in k$, define

$\pi(r,\lambda,\chi):=\frac{\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee})}{(T-\lambda,S-1)}\otimes_{k}\chi\omega^{r}$

(we have included the twist by $\omega^{r}$ in order for this notation to
match that of the existing literature).

######  Remark 2.1.

We will later need to consider a variation of this definition that is defined
in families. Let $R$ be a $k$-algebra. One can define Hecke operators $T$ and
$S$ on
$\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\chi)$
for any character $\chi\colon G\longrightarrow R^{\times}$ in the same way as
for $R=k$ and $\chi=1$, and the natural isomorphism
$\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\chi)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee})\otimes_{k}\chi$
intertwines $T$ and $S$ on the source with $T\otimes 1$ and $S\otimes 1$
respectively on the target. Moreover, if $b\in R^{\times}$ and $\lambda\in R$,
then there are isomorphisms of representations of $G$

$\displaystyle\frac{\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}R)}{(T-\lambda,S-1)}\otimes_{R}(\chi\mathrm{unr}_{{b}})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\frac{\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}(\chi\mathrm{unr}_{{b}}))}{(T-\lambda,S-1)}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\frac{\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\chi)}{(T-\lambda
b,S-b^{2})}.$

Let us define for $\tau\in R$, $\sigma\in S$ and $s\in\mathbb{Z}$,

$\widetilde{\pi}(r,\tau,\sigma,s)_{R}:=\frac{\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R)}{(T-\tau,S-\sigma)}.$

In particular, when $R=k$, then
$\pi(r,\lambda,\omega^{a}\mathrm{unr}_{{b}})\simeq\widetilde{\pi}(r,\tau,\sigma,s)_{k}$
for $\tau=\lambda b,\sigma=b^{2}$ and $s=a+r$.

The following results are due to Barthel–Livné [BL94] and Breuil [Bre03].

######  Theorem 2.2.

  1. (i)If $(r,\lambda)\neq(0,\pm 1),(p-1,\pm 1)$, then $\pi(r,\lambda,\chi)$ is irreducible.

  2. (ii)

If $\lambda\neq 0$ and $(r,\lambda)\neq(0,\pm 1)$ then there is a canonical
isomorphism

$\pi(r,\lambda,\chi)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ind}_{B}^{G}(\chi\mathrm{unr}_{{\lambda^{-1}}}\otimes\chi\mathrm{unr}_{{\lambda}}\omega^{r}).$

Here, $\operatorname{Ind}_{B}^{G}({-})$ denotes smooth parabolic induction.
The representation $\pi(r,\lambda,\chi)$ is said to be a principal series
representation.

  3. (iii)

If $r=0$ and $\lambda=\pm 1$ there is a canonical homomorphism

$\pi(r,\lambda,\chi)\longrightarrow\operatorname{Ind}_{B}^{G}(\chi\mathrm{unr}_{{\lambda^{-1}}}\otimes\chi\mathrm{unr}_{{\lambda}})$

with kernel and cokernel isomorphic to
$\operatorname{St}\otimes\chi\mathrm{unr}_{{\lambda}}$ and image isomorphic to
$\chi\mathrm{unr}_{{\lambda}}\circ\det$. Here, $\operatorname{St}$ is the
Steinberg representation of $G$ over $k$, i.e. the quotient of
$\operatorname{Ind}_{B}^{G}(k)$ by the trivial representation $k$.

  4. (iv)

If $\lambda=0$, then $\pi(r,0,\chi)$ is not isomorphic to a principal series
representation; it is said to be supersingular.

  5. (v)

Any irreducible representation of $G$ over $k$ is isomorphic to a principal
series, a supersingular, a twist of the Steinberg, or a twist of the trivial
representation.

  6. (vi)

If $\lambda\neq 0$, the only isomorphisms between various of these
representations are

$\pi(r,\lambda,\chi)\simeq\pi(r,-\lambda,\chi\mathrm{unr}_{{-1}})$

and for $\lambda\neq\pm 1$

$\pi(0,\lambda,\chi)\simeq\pi(p-1,\lambda,\chi).$

  7. (vii)

If $\lambda=0$, the isomorphisms between various of these representations are

$\displaystyle\pi(r,0,\chi)$
$\displaystyle\simeq\pi(r,0,\chi\mathrm{unr}_{{-1}})$
$\displaystyle\simeq\pi(p-1-r,0,\chi\omega^{r})$
$\displaystyle\simeq\pi(p-1-r,0,\chi\omega^{r}\mathrm{unr}_{{-1}}).$

In particular, there is a non-zero map, unique up to scalar,
$\pi(p-1,1,\chi)\longrightarrow\pi(0,1,\chi)$ (resp.
$\pi(0,1,\chi)\longrightarrow\pi(p-1,1,\chi)$), which factors through a twist
of the Steinberg (resp. trivial) representation. In Section 4, we will
describe these maps explicitly.

###  2.4. The mod $p$ Langlands correspondence for
$\text{GL}_{2}(\mathbb{Q}_{p})$

Throughout this section, we assume $p\geq 5$. We will use the same conventions
for the mod $p$ local Langlands correspondence as in [Eme], however we also
include twists of extensions of the cyclotomic character by the trivial
character in our discussion. Let $\operatorname{MF}$ be Colmez’s magical
functor, defined by $\operatorname{MF}(\pi):=\mathbf{V}(\pi)\otimes\omega$ for
$\mathbf{V}$ as in [Col10].

######  Theorem 2.3.

Let
$\rho\colon\text{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\longrightarrow\text{GL}_{2}(k)$
be a continuous representation. Then, there exists a finite length smooth
representation of $G$ over $k$, unique up to isomorphism, satisfying the
following properties:

  1. (i)

$\operatorname{MF}(\pi)\simeq\rho$,

  2. (ii)

$\pi$ has central character corresponding to $(\det\rho)\omega$ under local
class field theory,

  3. (iii)

$\pi$ has no finite-dimensional $G$-invariant subrepresentations or quotients.

More specifically, $\pi$ can be described as follows:

  1. (i)

If $\rho$ is irreducible, say
$\rho=\operatorname{Ind}(\omega_{2}^{r+1})\otimes\chi$ with $0\leq r\leq p-1$,
then $\pi=\pi(r,0,\chi\omega)$ is supersingular.

  2. (ii)

If $\rho\simeq\begin{pmatrix}\chi_{1}&*\\\ 0&\chi_{2}\end{pmatrix}$ with
$\chi_{1}\neq\chi_{2}\omega^{\pm 1}$, then $\pi$ is an extension

$0\longrightarrow\pi(r,\lambda,\chi)\longrightarrow\pi\longrightarrow\pi([p-3-r],\lambda^{-1},\chi\omega^{r+1})\longrightarrow
0,$

where

$\displaystyle\chi_{1}$
$\displaystyle=\omega^{r}\chi\mathrm{unr}_{{\lambda}},$
$\displaystyle\chi_{2}$
$\displaystyle=\omega^{-1}\chi\mathrm{unr}_{{\lambda^{-1}}}$

with $0\leq r\leq p-1$, and $[p-3-r]$ is the unique integer between $0$ and
$p-2$ that is congruent to $p-3-r$ modulo $p-1$. This extension is split if
and only if $\rho$ is semisimple.

  3. (iii)

If $\rho$ is a non-split extension $\begin{pmatrix}\chi&*\\\
0&\chi\omega^{-1}\end{pmatrix}$, then $\pi$ has a unique Jordan–Hölder series,
which is of the form

$0\subseteq\pi_{1}\subseteq\pi_{2}\subseteq\pi$

where
$\pi_{1}\simeq\operatorname{St}\otimes\chi,\pi_{2}/\pi_{1}\simeq\chi\circ\det$
and $\pi/\pi_{2}\simeq\pi(p-3,1,\chi\omega)$.

  4. (iv)

If $\rho$ is an extension $\begin{pmatrix}\chi\omega^{-1}&*\\\
0&\chi\end{pmatrix}$, then $\pi$ is an extension

$0\longrightarrow\pi(p-3,1,\chi\omega)\longrightarrow\pi\longrightarrow\operatorname{St}\otimes\chi\longrightarrow
0.$

This extension is split if and only if $\rho$ is semisimple. On both the
Galois side and the $\text{GL}_{2}(\mathbb{Q}_{p})$ side, there is a unique
class of non-trivial extensions, so this property determines $\pi$.

Proof. All the statements follow from the work of Colmez [Col10, Section
VII.4] except for case (iv), which follows from the end of the proof [Paš13,
Lemma 10.35] by taking into account that there is only one isomorphism class
of Galois representations that are non-split extensions of 1 by $\omega^{-1}$.
∎

For $\rho$ as in the theorem, we will say that $\pi$ is the representation
corresponding to $\rho$ under the mod $p$ local Langlands correspondence. One
could argue (for example, following [CEG+18, Remark 7.7]) that the “true” mod
$p$ local Langlands correspondence in case (iv) should be (up to isomorphism)
a non-trivial extension of $\pi$ as above by two copies of the trivial
character. However, we are only interested in the socle of $\pi$, which
remains the same and can be easily described in general by the following
result.

######  Proposition 2.4.

Let $\rho$ be a representation
$\text{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\longrightarrow\text{GL}_{2}(k)$
and let $\pi$ be the corresponding representation of
$\text{GL}_{2}(\mathbb{Q}_{p})$. Then, the following statements hold.

  1. (i)

The representation $\pi(r,0,\chi)$ is a subrepresentation of $\pi$ if and only
if $\rho\simeq\operatorname{Ind}(\omega_{2}^{r+1})\otimes\chi\omega^{-1}$.

  2. (ii)

For $\lambda\neq 0$ and $(r,\lambda)\neq(0,\pm 1),(p-1,\pm 1)$,
$\pi(r,\lambda,\chi)$ is a subrepresentation of $\pi$ if and only if

$\rho\simeq\begin{pmatrix}\omega^{r}\mathrm{unr}_{{\lambda}}&*\\\
0&\omega^{-1}\mathrm{unr}_{{\lambda^{-1}}}\end{pmatrix}\otimes\chi.$

  3. (iii)

$\operatorname{St}\otimes\chi$ is a subrepresentation of $\pi$ if and only if

$\rho\simeq\begin{pmatrix}1&*\\\ 0&\omega^{-1}\end{pmatrix}\otimes\chi.$

  4. (iv)

$\chi\circ\det$ is never a subrepresentation of $\pi$.

Proof. This follows from Theorem 2.3 and Theorem 2.2 (vi). ∎

###  2.5. The weight part of Serre’s conjecture

In this section we will recall the answer to the following question: given a
continuous representation
$\rho\colon\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\text{GL}_{2}(k)$,
for what $0\leq r\leq p-1$ and $s$ does $\rho$ contribute to
$H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{r}(k^{2})\otimes\omega^{-s})$ and,
when it does, what are its eigenvalues for the Hecke operators at $p$? What we
mean by Hecke operators at $p$ is the following. One can define the action of
Hecke operators $T$ and $S$ on the arithmetic homology complex
$C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})$
for any tame level $U^{p}\subseteq\text{GL}_{2}(\mathbb{A}^{p\infty})$ as we
explained in Section 2.2 by using the actions of $G^{-}$ and $KZ$ on
$\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})$, and similarly
for the cohomology of the dual. When $s=0$, these operators correspond under
the Eichler–Shimura isomorphism followed by reduction mod $p$ to the usual
Hecke operators $T_{p}$ and $\langle p\rangle$ respectively for modular forms
of level $\Gamma_{1}(N)$ and weight $r+2$.

The answer to the question is contained in the proof of [BDJ10, Theorem 3.17].
We warn the reader that our conventions are different to those in [BDJ10]:
$\rho$ contributes to the cohomology of a Serre weight $V$ in the sense of
this article if and only if $\rho^{\vee}$ is modular of weight $V$ in the
sense of [BDJ10] (equivalently, if and only if $\rho$ is modular of weight
$V^{\vee}\otimes\omega^{-1}$ in the sense of [BDJ10]). In particular, it
follows from [BDJ10, Corollary 2.11] that $\rho$ contributes to the cohomology
of $V\otimes\omega^{a}$ if and only if $\rho\varepsilon^{a}$ contributes to
the cohomology of $V$ for any $a$.

######  Theorem 2.5.

Let
$\rho\colon\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\text{GL}_{2}(k)$
be an odd irreducible Galois representation. Let $0\leq r\leq p-1$ and
$a\in\mathbb{Z}$. Let $\lambda\in k,b\in k^{\times}$ and set $\tau=\lambda b$,
$\sigma=b^{2}$ and $s=a+r$. Then, $\rho$ contributes to the
$(T=\tau,S=\sigma)$-eigenspace in
$H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{r}(k^{2})\otimes\omega^{-s})$ if and
only if $N(\rho)$ divides $N$ and one of the following holds:

  1. (i)

$\lambda=0$ and
$\rho|_{\mathcal{G}_{p}}\simeq\operatorname{Ind}(\omega_{2}^{r+1})\otimes\omega^{a-1}\mathrm{unr}_{{b}}$,

  2. (ii)

$\lambda\neq 0$, $(r,\lambda)\neq(0,\pm 1)$ and

$\rho|_{\mathcal{G}_{p}}\simeq\begin{pmatrix}\omega^{r}\mathrm{unr}_{{\lambda}}&*\\\
0&\omega^{-1}\mathrm{unr}_{{\lambda^{-1}}}\end{pmatrix}\otimes\omega^{a}\mathrm{unr}_{{b}}.$

  3. (iii)

$r=0$, $\lambda=\pm 1$ and

$\rho|_{\mathcal{G}_{p}}\simeq\begin{pmatrix}1&*\\\
0&\omega^{-1}\end{pmatrix}\otimes\omega^{a}\mathrm{unr}_{{\lambda b}}.$

where $*$ denotes a peu ramifiée extension.

Proof. According to [Edi92, Theorem 2.5 and Theorem 2.6], only $\rho$ whose
restriction to $\mathcal{G}_{p}$ is irreducible (resp. reducible) can
contribute to the eigenspaces where $T=0$ (resp. $T\neq 0$). By [Edi92,
Theorem 2.6], the representations $\rho$ appearing in the
$(T=0,S=b^{2})$-eigenspace of
$H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{r}(k^{2})\otimes\omega^{-s})$ are
those satisfying
$\rho|_{\mathcal{I}_{p}}\omega^{-s}\simeq(\omega_{2}^{r+1}\oplus\omega_{2}^{p(r+1)})\otimes\omega^{-r-1}$
and $(\det\rho)(\text{Frob}_{p})=b^{2}$ (recall that $\text{Frob}_{p}$ is a
Frobenius mapping to $p$ under class field theory, so it is a well-defined
element of $\mathcal{G}_{p}^{\mathrm{ab}}$). In particular, the restriction to
$\mathcal{G}_{p}$ of such a representation has determinant
$\omega^{2s-r-1}\mathrm{unr}_{{b^{2}}}$, so it must be isomorphic to

$\operatorname{Ind}(\omega_{2}^{r+1})\otimes\omega^{s-r-1}\mathrm{unr}_{{b}}\simeq\operatorname{Ind}(\omega_{2}^{r+1})\otimes\omega^{a-1}\mathrm{unr}_{{b}}.$

For the case where $\lambda\neq 0$, [Edi92, Theorem 2.5] states that, given a
system of Hecke eigenvalues in the $(T=\tau,S=\sigma)$-eigenspace of
$H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{r}(k^{2})\otimes\omega^{-s})$, then

$\rho|_{\mathcal{G}_{p}}\omega^{-s}\simeq\begin{pmatrix}\mathrm{unr}_{{\tau}}&*\\\
0&\omega^{-r-1}\mathrm{unr}_{{\tau^{-1}\sigma}}\end{pmatrix}.$

If $\tau=\lambda b$ and $\sigma=b^{2}$, this is equivalent to

$\rho|_{\mathcal{G}_{p}}\simeq\begin{pmatrix}\omega^{r}\mathrm{unr}_{{\lambda}}&*\\\
0&\omega^{-1}\mathrm{unr}_{{\lambda^{-1}}}\end{pmatrix}\otimes\omega^{a}\mathrm{unr}_{{b}}.$

Conversely, [BDJ10, Theorem 3.17] and its proof show that a representation of
this form does contribute to
$H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{r}(k^{2})\otimes\omega^{-s})$
provided that the extension is peu ramifiée whenever $r=0$ and $\lambda=\pm 1$
(and, in this case, it never contributes if the extension is très ramifiée).
It remains to show that it contributes to the $(T=\lambda
b,S=b^{2})$-eigenspace. When $r\neq p-2$ or $\rho|_{\mathcal{G}_{p}}$ is non-
split, [Edi92, Theorems 2.5 and 2.6] again show that this the only eigenspace
for Hecke operators at $p$ to which $\rho$ can contribute, so it must do so.
In the case when $r=p-2$ and
$\rho|_{\mathcal{G}_{p}}\simeq\omega^{a-1}\mathrm{unr}_{{\lambda
b}}\oplus\omega^{a-1}\mathrm{unr}_{{\lambda^{-1}b}}$, the same results show
that $\rho$ can only appear in the eigenspaces for $(T=\lambda b,S=b^{2})$ and
$(T=\lambda^{-1}b,S=b^{2})$. We know that $\rho$ contributes to at least one
of these, and we must show that $\rho$ contributes to both. When $\lambda=\pm
1$ this is clear, since both systems of eigenvalues are actually the same.
When $\lambda\neq\pm 1$, this follows from [Gro90, Theorem 13.10]. ∎

##  3\. $p$-arithmetic homology of $\pi(r,\lambda,\chi)$

###  3.1. $p$-arithmetic and arithmetic homology

In this section we will relate the $p$-arithmetic homology of the
representations $\pi(r,\lambda,\chi)$ to the arithmetic homology of Serre
weights $\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes\omega^{s}$. The argument
is the same as that of [Tar22]. As in Remark 2.1, $R$ is a $k$-algebra.

######  Lemma 3.1.

For any $0\leq r\leq p-1$ and $s\in\mathbb{Z}$, $\tau\in R$ and $\sigma\in
R^{\times}$, we have

$\text{Tor}^{i}_{R[T,S]}(R[T,S]/(T-\tau,S-\sigma),\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R))=0$

for $i>0$.

Proof. This follows from the fact that $(S-\sigma,T-\tau)$ is a regular
sequence for the $k[T,S]$-module
$\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R)$,
which can be seen by studying how $T$ and $S$ modify the support of a function
using the Cartan decomposition. See [CEG+18, Lemma 4.10] for the details. ∎

This shows that the $G$-module $\widetilde{\pi}(r,\tau,\sigma,s)_{R}$ from
Remark 2.1 can be written not just as the eigenquotient

$\displaystyle\operatorname{c-Ind}_{K}^{G}$
$\displaystyle(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R)/(T-\tau,S-\sigma)$
$\displaystyle\simeq\frac{R[T,S]}{(T-\tau,S-\sigma)}\otimes_{R[T,S]}\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R)$

but also as a _derived_ eigenquotient: there is an isomorphism in the derived
category of (abstract111In our arguments involving homological algebra, we
will always work with categories of abstract representations (of $G$ or other
groups) and never with categories of smooth representations)
$R[G][T,S]$-modules

$\widetilde{\pi}(r,\tau,\sigma,s)_{R}\simeq\frac{R[G][T,S]}{(T-\tau,S-\sigma)}\otimes^{\mathbb{L}}_{R[G][T,S]}\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R).$

Fix $U^{p}$ and $N$ as in Section 2.1. We can define an action of Hecke
operators $T$ and $S$ in arithmetic homology over $R$ in the same way as
described in the beginning of Section 2.5, and the arguments in [Tar22,
Section 5.8] show that we have an isomorphism in the derived category of
$\mathbb{T}(pN)\otimes_{\mathbb{Z}}R[T,S]$-modules for the $p$-arithmetic
homology complex

$\displaystyle C_{\bullet}(U^{p},\widetilde{\pi}(r,\tau,\sigma,s)_{R})$
$\displaystyle\simeq
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R)\otimes^{\mathbb{L}}_{R[T,S]}\frac{R[T,S]}{(T-\tau,S-\sigma)}.$

Moreover,

$\displaystyle
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}\otimes_{k}R)\simeq
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})\otimes^{\mathbb{L}}_{k}R.$

Thus, in fact

$\displaystyle C_{\bullet}(U^{p},\widetilde{\pi}(r,\tau,\sigma,s)_{R})$
$\displaystyle\simeq
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})\otimes^{\mathbb{L}}_{k[T,S]}\frac{R[T,S]}{(T-\tau,S-\sigma)}.$

######  Remark 3.2.

The reason why we have considered representations in families is the
following. Assume that $R=k[\tau,\sigma,\sigma^{-1}]$ for two indeterminate
variables $\tau$ and $\sigma$. Then,

$\displaystyle C_{\bullet}(U^{p},\widetilde{\pi}(r,\tau,\sigma,s)_{R})$
$\displaystyle\simeq
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})\otimes^{\mathbb{L}}_{k[T,S]}\frac{R[T,S]}{(T-\tau,S-\sigma)}$
$\displaystyle\simeq
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})\otimes^{\mathbb{L}}_{k[T,S,S^{-1}]}\frac{R[T,S,S^{-1}]}{(T-\tau,S-\sigma)}$
$\displaystyle\simeq
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})\otimes^{\mathbb{L}}_{k[T,S,S^{-1}]}\frac{k[T,S,S^{-1},\tau,\sigma,\sigma^{-1}]}{(T-\tau,S-\sigma)}$
$\displaystyle\simeq
C_{\bullet}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s}),$

where the last term is viewed as an $R$-module by letting $\tau$ act as $T$
and $\sigma$ as $S$. In other words, the $p$-arithmetic homology over $R$
coincides with the corresponding arithmetic homology, and not just a (derived)
eigenquotient of it.

######  Proposition 3.3.

There is a spectral sequence converging to
$H_{*}(U^{p},\widetilde{\pi}(r,\tau,\sigma,s)_{R})$ whose $E^{2}$ page is

$E^{2}_{i,j}=\text{Tor}_{i}^{k[T,S]}\left(\frac{R[T,S]}{(T-\tau,S-\sigma)},H_{j}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{s})\right).$

In particular, there is a spectral sequence converging to
$H_{*}(U^{p},\pi(r,\lambda,\omega^{a}\mathrm{unr}_{{b}}))$ whose $E^{2}$ page
is

$E^{2}_{i,j}=\text{Tor}_{i}^{k[T,S]}\left(\frac{k[T,S]}{(T-\lambda
b,S-b^{2})},H_{j}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{a+r})\right).$

######  Remark 3.4.

When $R=k$, the $k[T,S]$-module resolution

$k[T,S]\xrightarrow{(S-\sigma)\oplus(\tau-T)}k[T,S]^{2}\xrightarrow{(T-\tau,S-\sigma)}k[T,S]$

of $k[T,S]/(T-\tau,S-\sigma)$ shows that for any $k[T,S]$-module $V$ that is
finite-dimensional as a $k$-vector space,
$\text{Tor}^{k[T,S]}_{i}(k[T,S]/(T-\tau,S-\sigma),V)$ vanishes in degrees
outside the range $[0,2]$, and is isomorphic in degree 2 (resp. degree 0) to
the $(T=\tau,S=\sigma)$-eigenspace (resp. eigenquotient) of $V$. Moreover, the
Tor modules in degree 1 lie in a short exact sequence

$\displaystyle 0$
$\displaystyle\longrightarrow\frac{k[T]}{(T-\tau)}\otimes_{k[T]}\text{Hom}_{k[S]}\left(\frac{k[S]}{(S-\sigma)},V\right)$
$\displaystyle\longrightarrow\text{Tor}^{k[T,S]}_{1}\left(\frac{k[T,S]}{(T-\tau,S-\sigma)},V\right)$
$\displaystyle\longrightarrow\text{Hom}_{k[T]}\left(\frac{k[T]}{(T-\tau)},\frac{k[S]}{(S-\sigma)}\otimes_{k[S]}V\right)\longrightarrow
0.$

In particular, the Tor groups vanish if and only if they vanish in at least
one of the degrees 0, 1 or 2. The dimensions $d_{0}$ and $d_{2}$ of the Tor
spaces in degrees 0 and 2 are equal, and the dimension in degree 1 is
$2d_{0}=2d_{2}$.

###  3.2. Proof of Theorem 1.1 in the generic case

Parts (i) and (ii) of Theorem 1.1 follow immediately from Proposition 3.3 for
supersingular and principal series representations, as well as their analogue
for the reducible representations $\pi(0,\pm 1,\chi)$ and $\pi(p-1,\pm
1,\chi)$ (by the corresponding results for arithmetic homology).

Moreover, if $\rho$ is an odd irreducible 2-dimensional Galois
representations, then the localisation at $\rho$ of the spectral sequence from
Proposition 3.3 satisfies $(E^{2}_{i,j})_{\rho}=0$ for $j\neq 1$, so we may
conclude that

$H_{i+1}(\Gamma^{p}_{1}(N),\pi(r,\lambda,\omega^{a}\mathrm{unr}_{{b}}))_{\rho}\simeq\text{Tor}_{i}^{k[T,S]}\left(\frac{k[T,S]}{(T-\lambda
b,S-b^{2})},H_{1}(U^{p}K,\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes_{k}\omega^{a+r})_{\rho}\right).$

In particular, taking into account Remark 3.4, the following are equivalent:

  1. (i)

$\rho$ contributes to the $p$-arithmetic homology
$H_{*}(\Gamma^{p}_{1}(N),\pi(r,\lambda,\omega^{a}\mathrm{unr}_{{b}}))$, and it
does exactly in degrees 1, 2 and 3,

  2. (ii)

$\rho$ contributes to the degree 1 $p$-arithmetic homology
$H_{1}(\Gamma^{p}_{1}(N),\pi(r,\lambda,\omega^{a}\mathrm{unr}_{{b}}))$,

  3. (iii)

$\rho$ contributes to the $p$-arithmetic homology
$H_{*}(\Gamma^{p}_{1}(N),\pi(r,\lambda,\omega^{a}\mathrm{unr}_{{b}}))$,

  4. (iv)

$\rho$ contributes to the $(T=\lambda b,S=b^{2})$-eigenspace of
$H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{r}(k^{2})^{\vee}\otimes\omega^{a+r})$,

  5. (v)

$\rho$ contributes to the $(T=\lambda b,S=b^{2})$-eigenspace of
$H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{r}(k^{2})\otimes\omega^{-a-r})$.

By Theorem 2.5 and Proposition 2.4, when
$\pi(r,\lambda,\omega^{a}\mathrm{unr}_{{b}})$ is irreducible, these are
equivalent to $N(\rho)$ dividing $N$ and this representation appearing in the
socle of the smooth representation of $\text{GL}_{2}(\mathbb{Q}_{p})$
associated to $\rho|_{\mathcal{G}_{p}}$ by the mod $p$ local Langlands
correspondence of Theorem 2.3, which proves part (iii) of Theorem 1.1 in this
case. For later reference, we also record the following proposition, which
follows in the same way from Proposition 3.3 and Theorem 2.5.

######  Proposition 3.5.

Let
$\rho\colon\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\text{GL}_{2}(k)$
be an odd irreducible representation. Then, the space
$H_{1}(\Gamma^{p}_{1}(N),\pi(p-1,1,\chi))$ (resp. to
$H_{1}(\Gamma^{p}_{1}(N),\pi(0,1,\chi))$) is finite-dimensional, and any
system of Hecke eigenvalues in it has an attached Galois representation.
Moreover, $\rho$ contributes to this space if and only if $N(\rho)$ divides
$N$, and $\rho|_{\mathcal{G}_{p}}$ is isomorphic to an extension (resp. a peu
ramifiée extension)

$\begin{pmatrix}1&*\\\ 0&\omega^{-1}\end{pmatrix}\otimes\chi.$

##  4\. Preparation for the non-generic cases

In order to deal with the Steinberg and trivial cases, we will need a few
preliminaries on the non-zero maps
$\pi(p-1,1,\chi)\longrightarrow\pi(0,1,\chi)$ and
$\pi(0,1,\chi)\longrightarrow\pi(p-1,1,\chi)$ and the corresponding maps on
$p$-arithmetic homology. They turn out to be related to the degeneracy maps
from modular forms of level $\Gamma_{1}(N)$ to level
$\Gamma_{1}(N)\cap\Gamma_{0}(p)$ induced by $\tau\longmapsto\tau$ and
$\tau\longmapsto p\tau$ and to the group cohomological avatar of
multiplication by the Hasse invariant studied by Edixhoven–Khare in [EK03].
Our next goal is to study these maps.

###  4.1. The map $\pi(p-1,1,1)\longrightarrow\pi(0,1,1)$

Recall from Section 2.3 that there is a unique-up-to-scalaras non-zero map
$\pi(p-1,1,\chi)\longrightarrow\pi(0,1,\chi)$. The goal of this section is to
give an explicit description of this map.

We may assume that $\chi=1$. First, let us observe that there is a $K$-module
isomorphism
$k\oplus\operatorname{Sym}^{p-1}(k^{2})^{\vee}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Map}(\mathbb{P}^{1}(\mathbb{F}_{p}),k)$,
which identifies the trivial representation with the subrepresentation of
constant functions $\mathbb{P}^{1}(\mathbb{F}_{p})\longrightarrow k$ and
$\operatorname{Sym}^{p-1}(k^{2})^{\vee}$ with the subrepresentation of
functions whose total sum equals 0. As usual, one can also identify
$\operatorname{Sym}^{p-1}(k^{2})^{\vee}$ with the space of homogeneous
polynomial functions of degree $p-1$ in two variables. The former
identification is then given by sending a homogeneous polynomial function $Q$
of degree $p-1$ in two variables to the function $(x:y)\longmapsto Q(x,y)$.

In fact, this can be upgraded to a $G^{-}$-equivariant isomorphism in a
natural way. The action of the monoid $G^{+}$ on $\mathbb{F}_{p}^{2}$ descends
to an action on
$\mathbb{F}_{p}^{2}/\mathbb{F}_{p}^{\times}=\mathbb{P}^{1}(\mathbb{F}_{p})\cup\\{0\\}$.
It is easy to check (for example, using the Cartan decomposition of $G$) that
an element $g\in G^{+}$ can act in three ways: invertibly (if $g\in K$), by
sending everything to 0 (if all the entries of $g$ are multiples of $p$), or
by mapping 0 and one point of $\mathbb{P}^{1}(\mathbb{F}_{p})$ to 0 and all
other points of $\mathbb{P}^{1}(\mathbb{F}_{p})$ to another (fixed) point of
$\mathbb{P}^{1}(\mathbb{F}_{p})$. Thus, we get an action of $G^{-}$ on
$\operatorname{Map}(\mathbb{P}^{1}(\mathbb{F}_{p})\cup\\{0\\},k)$, and it
follows from the previous sentence that the subspace of functions such that
$f(0)=\sum_{P\in\mathbb{P}^{1}(\mathbb{F}_{p})}f(P)$ is stable under this
action. This space can be naturally identified with
$\operatorname{Map}(\mathbb{P}^{1}(\mathbb{F}_{p}),k)$ by restriction, and the
resulting action of $G^{-}$ on this space makes the isomorphisms in the
previous paragraph $G^{-}$-equivariant. Naturally, these isomorphisms are also
$KZ$-equivariant when we instead extend the action of $K$ to one of $KZ$ by
letting $\beta$ act trivially. There is a $K$-equivariant isomorphism
$\operatorname{Map}(\mathbb{P}^{1}(\mathbb{F}_{p}),k)\longrightarrow\operatorname{c-Ind}_{I}^{K}(k)$
given by sending a function $f$ to $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\longmapsto f(a:c)$, and we will view the target as a
$G^{-}$-module and a $KZ$-module (whose underlying $K$-module structures
agree) by transport of structure. In particular, compactly inducing to $G$ we
obtain actions of Hecke operators $T$ and $S$ as usual, and the maps above
induce $k[T,S]$-module isomorphisms

$\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{p-1}(k^{2}))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{c-Ind}_{K}^{G}(\operatorname{c-Ind}_{I}^{K}(k)).$

Consider the following two maps
$\phi_{1},\phi_{2}\colon\operatorname{c-Ind}_{I}^{G}(k)\longrightarrow\operatorname{c-Ind}_{K}^{G}(k)$.
The first is given simply by $\phi_{1}([g,a])=[g,a]$. The second map
$\phi_{2}$ is defined by $\phi_{2}([g,a])=[g\alpha,a]$. It will be useful to
view this map as the composition of the map
$[g,a]\longmapsto[g,a]\colon\operatorname{c-Ind}_{I}^{G}(k)\longrightarrow\operatorname{c-Ind}_{\alpha
K\alpha^{-1}}^{G}(k)$ and the intertwining isomorphism

(4.1) $\displaystyle\begin{split}\operatorname{c-Ind}_{\alpha
K\alpha^{-1}}^{G}(k)&\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{c-Ind}_{K}^{G}(k)\\\
[g,a]&\longmapsto[g\alpha,a].\end{split}$

It is tedious, but straightforward, to check that the resulting maps

$\operatorname{c-Ind}_{K}^{G}(\operatorname{c-Ind}_{I}^{K}(k))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{c-Ind}_{I}^{G}(k)\rightrightarrows\operatorname{c-Ind}_{K}^{G}(k)$

are $k[T,S]$-equivariant. One can also check that the composition

$\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{p-1}(k^{2})^{\vee})\longrightarrow\operatorname{c-Ind}_{K}^{G}(\operatorname{c-Ind}_{I}^{K}(k))\xrightarrow{\phi_{1}\oplus\phi_{2}}\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{K}^{G}(k)$

is of the form $\begin{pmatrix}1&0\\\ T&\phi\end{pmatrix},$ where $\phi$ is
also $k[T,S]$-equivariant.

######  Lemma 4.2.

The reduction mod $(T-1,S-1)$ of the homomorphism $\phi$ is a non-zero map
$\pi(p-1,1,1)\longrightarrow\pi(0,1,1)$.

Proof. The following argument is a representation theoretic analogue of the
proof of [EK03, Lemma 2] (in fact, this lemma can be deduced literally from
loc. cit., for example as a consequence of Proposition 4.6 below). Write
${}^{\circ}G=\\{g\in G:v_{p}(\det(g))=0\\}$. Then, by [Ser80, II.1.4 Theorem
3], ${}^{\circ}G$ is the amalgamated product of $K$ and $\alpha K\alpha^{-1}$
along $I$. Thus, there is a Mayer-Vietoris exact sequence in the group
homology of $k[G]$,

$0\longrightarrow\operatorname{c-Ind}_{I}^{G}(k)\longrightarrow\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{\alpha
K\alpha^{-1}}^{G}(k)\longrightarrow\operatorname{c-Ind}_{{}^{\circ}G}^{G}(k)\longrightarrow
0.$

Composing with the intertwining isomorphism 4.1, we obtain an exact sequence

(4.3) $\displaystyle
0\longrightarrow\operatorname{c-Ind}_{K}^{G}(\operatorname{c-Ind}_{I}^{K}(k))\stackrel{{\scriptstyle\phi_{1}\oplus\phi_{2}}}{{\longrightarrow}}\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{K}^{G}(k)\longrightarrow\operatorname{c-Ind}_{{}^{\circ}G}^{G}(k)\longrightarrow
0$

where the last map is given by
$([g_{1},a_{1}],[g_{2},a_{2}])\longmapsto[g_{1},a_{1}]-[g_{2}\alpha^{-1},a_{2}]$.
This exact sequence is $k[T,S]$-equivariant if we endow
$\operatorname{c-Ind}_{{}^{\circ}G}^{G}(k)$ with the action of $T$ (resp. $S$)
given by acting by $\alpha$ (resp. $\beta$). Taking the quotient of the exact
sequence above by the ideal $(T-1,S-1)$, we obtain an exact sequence

$\pi(0,1,1)\oplus\pi(p-1,1,1)\longrightarrow\pi(0,1,1)\oplus\pi(0,1,1)\longrightarrow
k\longrightarrow 0,$

where the first map is given by $\begin{pmatrix}1&0\\\
1&\overline{\phi}\end{pmatrix}$, where $\overline{\phi}$ is the map induced by
$\phi$. Looking at the Jordan-Hölder constituents of the terms in the exact
sequence, it is clear that $\overline{\phi}$ cannot be zero. ∎

Let us also remark that if $R$ is a $k$-algebra, $\tau\in R,\sigma\in
R^{\times}$ and $s\in\mathbb{Z}$, then tensoring $\phi$ with
$\omega^{s}\otimes_{k}R$ and quotienting by $(T-\tau,S-\sigma)$ we get a map
$\widetilde{\pi}(p-1,\tau,\sigma,s)_{R}\longrightarrow\widetilde{\pi}(0,\tau,\sigma,s)_{R}$.

###  4.2. The map $\pi(0,1,1)\longrightarrow\pi(p-1,1,1)$

There is also a unique-up-to-scalar non-zero map
$\pi(0,1,\chi)\longrightarrow\pi(p-1,1,\chi)$, which factors through
$\chi\circ\det$. The goal of this section is to show that this map comes from
specialising a map
$\widetilde{\pi}(0,\tau,\sigma,s)_{R}\longrightarrow\widetilde{\pi}(p-1,\tau,\sigma,s)_{R}$
as in the setting of the end of the previous section. As in the previous
section, this is essentially equivalent to the existence of a lift of the map
$\pi(0,1,1)\longrightarrow\pi(p-1,1,1)$ to a $k[T,S]$-equivariant map

$\operatorname{c-Ind}_{K}^{G}(k)\longrightarrow\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{p-1}(k^{2})^{\vee}).$

We will construct such a map by dualising the procedure of the previous
section. Consider the composition

(4.4)
$\displaystyle\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{K}^{G}(k)\xrightarrow{\text{id}\oplus\lx@cref{creftypecap~refnum}{eqn:
intertwining
isomorphism}^{-1}}\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{\alpha
K\alpha^{-1}}^{G}(k)\longrightarrow\operatorname{c-Ind}_{I}^{G}(k),$

where the last map is the sum of inclusions. It is $k[T,S]$-equivariant and
the resulting map

$\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{K}^{G}(k)\longrightarrow\operatorname{c-Ind}_{K}^{G}(k)\oplus\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{p-1}(k^{2})^{\vee})$

is of the form $\begin{pmatrix}1&S^{-1}T\\\ 0&-\psi\end{pmatrix}$. Explicitly,
$\psi([g,1])=\sum_{x\in K/I}[\beta^{-1}gx\alpha,e^{*}]$, where $e^{*}$ is the
element of $\operatorname{Sym}^{p-1}(k^{2})^{\vee}$ corresponding to the
polynomial function $Q(x,y)=x^{p-1}$.

######  Lemma 4.5.

The reduction mod $(T-1,S-1)$ of the homomorphism $\psi$ is a non-zero map
$\pi(0,1,1)\longrightarrow\pi(p-1,1,1)$.

Proof. As for Lemma 4.2, this follows from another result for group cohomology
(namely, Proposition 4.10 below), but we give another proof in a more
representation-theoretic spirit. Note that $\alpha^{-1}x^{-1}e^{*}$ is equal
to $e^{*}$ for any $x\in K$ which is not in the same left $I$-coset as
$w:=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$, and vanishes if $x\in wI$. In
particular, $\psi([g,1])=T[\beta^{-1},e^{*}]+[\beta^{-1}w\alpha,e^{*}]$.
Hence, it’s enough to show that $[1,e^{*}]+[w\alpha,e^{*}]$ defines a non-zero
element of $\pi(p-1,1,1)$. To do this, we will check that its image under the
map of Theorem 2.2 (ii) is non-zero. This map is defined in [BL94, Section
6.2] and sends $[g,Q]$ to $h\longmapsto Q(x(1:0))$ where we have written
$g^{-1}h=xb$ with $x\in K$ and $b\in B$. The image of
$[1,e^{*}]+[w\alpha,e^{*}]$ maps $w$ to $1$, so in particular it is non-zero
(in fact, as we would expect, it is the constant function with value 1). ∎

###  4.3. The resulting maps on arithmetic cohomology

######  Proposition 4.6.

Let $R=k[\tau,\sigma,\sigma^{-1}]$ be as in Remark 3.2 and $s\in\mathbb{Z}$.
Then, the map $\widetilde{\phi}\colon
H_{*}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee}\otimes\omega^{s})\longrightarrow
H_{*}(\Gamma_{1}(N),\omega^{s}\circ\det)$ induced from the map

$\widetilde{\pi}(p-1,\tau,\sigma,s)_{R}\longrightarrow\widetilde{\pi}(0,\tau,\sigma,s)_{R}$

defined at the end of Section 4.1 is a twist of the dual of the map in [EK03,
Lemma 2]. In particular, if $p\geq 5$, this map is surjective in degree 1.

Proof. By [BDJ10, Corollary 2.11], we may assume $s=0$. The first sentence
follows from the constructions of both maps, and the second follows from
[EK03, Lemma 2]. ∎

As mentioned above, the proof of Lemma 4.2 is a representation-theoretic
analogue the proof of [EK03, Lemma 2]. The latter can then be recovered by
taking $p$-arithmetic homology of the exact sequence 4.3. To see this, we need
the following result.

######  Lemma 4.7.

If $p\geq 5$, the $p$-arithmetic homology
$H_{1}(\Gamma^{p}_{1}(N),\operatorname{c-Ind}_{{}^{\circ}G}^{G}(k))$ vanishes.

Proof. As $G=\Gamma_{1}^{p}(N){}^{\circ}G$ and
$\Gamma_{1}^{p}(N)\cap{}^{\circ}G=\Gamma_{1}^{p}(N)\cap\text{SL}_{2}(\mathbb{Q})$,
the natural restriction map
$\operatorname{c-Ind}_{{}^{\circ}G}^{G}(k)\longrightarrow\operatorname{c-Ind}_{\Gamma_{1}^{p}(N)\cap\text{SL}_{2}(\mathbb{Q})}^{\Gamma_{1}^{p}(N)}(k)$
is an isomorphism of representations of $\Gamma^{p}_{1}(N)$. The lemma follows
by Shapiro’s lemma and [EK03, Proof of Lemma 1]. ∎

Thus, when $p\geq 5$, we have a commutative diagram

${{H_{1}(\Gamma_{1}(N)\cap\Gamma_{0}(p),k)}}$${{H_{1}(\Gamma_{1}(N),k)\oplus
H_{1}(\Gamma_{1}(N),k)}}$${{H_{1}(\Gamma_{1}(N),k)\oplus
H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee}).}}$$\scriptstyle{\sim}$$\scriptstyle{{\begin{pmatrix}\text{id}&0\\\
T&\widetilde{\phi}\end{pmatrix}}}$

In particular, $\widetilde{\phi}$ is surjective, so we have indeed recovered
[EK03, Lemma 2]. Moreover, we see that the kernel of $\widetilde{\phi}$ is
isomorphic to the kernel of the map

(4.8) $\displaystyle H_{1}(\Gamma_{1}(N)\cap\Gamma_{0}(p),k)\twoheadrightarrow
H_{1}(\Gamma_{1}(N),k)\oplus H_{1}(\Gamma_{1}(N),k).$

This is the homomorphism induced by the maps between open modular curves
determined by $\tau\longmapsto\tau$ and $\tau\longmapsto p\tau$ on the upper-
half plane. A generalisation by Wiles of a lemma of Ribet determines the
Galois representations that contribute to this kernel.

######  Proposition 4.9.

Let $\rho$ be a 2-dimensional odd irreducible representation of
$\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over $k$ such that $N(\rho)$
divides $N$ and $\rho|_{\mathcal{G}_{p}}\simeq\begin{pmatrix}1&*\\\
0&\omega^{-1}\end{pmatrix}\otimes\mathrm{unr}_{{b}}$. Then, the localisation
at the maximal ideal $\mathfrak{m}_{\rho}$ of $\mathbb{T}(pN)$ corresponding
to $\rho$ of the kernel of 4.8 is non-zero.

Proof. If the extension $*$ in the statement is très ramifiée, then it is
clear that $\rho$ contributes to the kernel as it contributes to the source
but not the target. Therefore, we may assume that the extension is peu
ramifiée. Let $f$ be a normalised newform of weight 2, level $\Gamma_{1}(N)$
and character $\chi$ whose associated Galois representation over $k$ is
isomorphic to $\rho$. If $a_{p}$ is its $p$-th Fourier coefficient, let
$\alpha$ be the root of $x^{2}-a_{p}x-\chi(p)p$ that is a $p$-adic unit (there
should be no confusion with our previous use of the letter $\alpha$). Then
$\alpha^{2}\equiv a_{p}^{2}\equiv\chi(p)\mod p$, the second congruence
following from [Edi92, Theorem 2.5].

There is an eigenclass in $H_{1}(\Gamma_{1}(N),\overline{\mathbb{Z}}_{p})$ for
the Hecke operators away from $N$ with the same system of eigenvalues as $f$
and whose reduction to $H_{1}(\Gamma_{1}(N),k)$ is non-zero. As $\rho$ is
irreducible, the localisation of $H_{1}(\Gamma_{1}(N),k)$ at
$\mathfrak{m}_{\rho}$ is isomorphic to
$(k\otimes_{\mathbb{F}_{p}}J_{1}(N)[p])_{\mathfrak{m}_{\rho}}$, where
$J_{1}(N)$ is the Jacobian of the compactified modular curve of level
$\Gamma_{1}(N)$. Write $P$ for the element of
$(k\otimes_{\mathbb{F}_{p}}J_{1}(N)[p])_{\mathfrak{m}_{\rho}}$ corresponding
to the reduction of the eigenclass above. Similarly,
$H_{1}(\Gamma_{1}(N)\cap\Gamma_{0}(N),k)_{\mathfrak{m}_{\rho}}\simeq(k\otimes_{\mathbb{F}_{p}}J_{1}(N,p)[p])_{\mathfrak{m}_{\rho}}$,
where $J_{1}(N,p)$ is the Jacobian of the compactified modular curve of level
$\Gamma_{1}(N)\cap\Gamma_{0}(p)$. Consider now the image of
$(P,0)=(P,-(a_{p}-\alpha)P)$ under the map

$(k\otimes_{\mathbb{F}_{p}}J_{1}(N)[p])_{\mathfrak{m}_{\rho}}\oplus(k\otimes_{\mathbb{F}_{p}}J_{1}(N)[p])_{\mathfrak{m}_{\rho}}\longrightarrow(k\otimes_{\mathbb{F}_{p}}J_{1}(N,p)[p])_{\mathfrak{m}_{\rho}}$

induced by the morphisms between modular curves determined by
$\tau\longmapsto\tau$ and $\tau\longmapsto p\tau$ on the upper-half plane. The
image of $(P,0)$ is then one of the $p$-stabilisations of $P$: it is an
eigenvector for all Hecke operators $T_{\ell}$ for $\ell\nmid pN$ (with
eigenvalues determined by $\rho$), as well as the operator $U_{p}$ (resp.
$\langle n\rangle$ for any $n\in\mathbb{Z}$ with $n\equiv p\mod N$ and
$n\equiv 1\mod p$) with eigenvalue $\alpha$ (resp. $\chi(p)$). It is also non-
zero (for example, by Proposition 4.10 and its proof below, or by the
injectivity of [Wil95, (2.10)]). Hence, by [Wil95, Lemma 2.3], it defines
(under the isomorphisms above between group homology and $p$-torsion points in
Jacobians) a non-zero element of the localisation at $\mathfrak{m}_{\rho}$ of
the kernel of 4.8. ∎

Finally, we turn to the map from Section 4.2.

######  Proposition 4.10.

Let $R=k[\tau,\sigma,\sigma^{-1}]$ be as in Remark 3.2 and $s\in\mathbb{Z}$.
Let $\rho$ be a 2-dimensional odd irreducible representation. Then, the map

$H_{*}(\Gamma_{1}(N),\omega^{s}\circ\det)_{\rho}\longrightarrow
H_{*}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee}\otimes_{k}\omega^{s})_{\rho}$

induced from the map
$\widetilde{\pi}(0,\tau,\sigma,s)_{R}\longrightarrow\widetilde{\pi}(p-1,\tau,\sigma,s)_{R}$
is injective in degree 1. If $N(\rho)$ divides $N$ and
$\rho|_{\mathcal{G}_{p}}\simeq\begin{pmatrix}1&*\\\
0&\omega^{-1}\end{pmatrix}\otimes\omega^{s}\mathrm{unr}_{{b}}$, then the
cokernel is non-zero.

Proof. The proposition follows from Proposition 4.6 and Proposition 4.9 by
Poincaré duality. Let us spell out the details. Again, we may assume that
$s=0$. Given a $\mathbb{T}(pN)$-module $M$, let us write $M^{*}$ for the base
change of $M$ along the ring isomorphism
$\mathbb{T}(pN)\longrightarrow\mathbb{T}(pN)$ mapping $T_{\ell}\longmapsto
S_{\ell}^{-1}T_{\ell}$ and $S_{\ell}\longmapsto S_{\ell}^{-1}$. Thus, $\rho$
contributes to $M$ if and only if $\rho^{\vee}\otimes\varepsilon^{-1}$
contributes to $M^{*}$. Let us also write $M^{*}_{\rho}:=(M^{*})_{\rho}$. As
$\rho$ is irreducible, there are Poincaré duality isomorphisms

$\displaystyle H_{1}(\Gamma_{1}(N),k)_{\rho}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\Gamma_{1}(N),k)^{*}_{\rho}$
$\displaystyle
H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee})_{\rho}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee})^{*}_{\rho}$
$\displaystyle H_{1}(\Gamma_{1}(N)\cap\Gamma_{0}(p),k)_{\rho}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\Gamma_{1}(N)\cap\Gamma_{0}(p),k)^{*}_{\rho}.$

On the one hand, the $k$-linear dual of 4.8 is the direct sum of the pullbacks
in cohomology of the maps from the modular curve of level
$\Gamma_{1}(N)\cap\Gamma_{0}(p)$ to the modular curve of level $\Gamma_{1}(N)$
determined by $\tau\longmapsto\tau$ and $\tau\longmapsto p\tau$ on the upper-
half plane. On the other hand, the map 4.4 induces on $p$-arithmetic homology
a map

$H_{1}(\Gamma_{1}(N),k)\oplus H_{1}(\Gamma_{1}(N),k)\longrightarrow
H_{1}(\Gamma_{1}(N)\cap\Gamma_{0}(p),k),$

that is (by construction) the direct sum of the transfer homomorphisms in the
homology of the modular curves above induced by the same pair of maps. Now,
under Poincaré duality, pullbacks in cohomology correspond to transfer
homomorphisms in homology, and thus the two maps correspond (after localising
at $\rho$) under the above Poincaré duality isomorphism.

Fix an isomorphism
$\operatorname{Sym}^{p-1}(k^{2})^{\vee}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Sym}^{p-1}(k^{2})$.
It determines also an isomorphism between
$\operatorname{Map}(\mathbb{P}^{1}(\mathbb{F}_{p}),k)$ and its dual, and by
Shapiro’s lemma an isomorphism

$H^{1}(\Gamma_{1}(N)\cap\Gamma_{0}(p),k)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{1}(\Gamma_{1}(N),k)\oplus
H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee})$

that is compatible under Poincaré duality with the similar isomorphism for
homology. In conclusion, we have a commutative square

${H_{1}(\Gamma_{1}(N),k)_{\rho}\oplus
H_{1}(\Gamma_{1}(N),k)_{\rho}}$${H_{1}(\Gamma_{1}(N),k)_{\rho}\oplus
H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee})_{\rho}}$${H^{1}(\Gamma_{1}(N),k)^{*}_{\rho}\oplus
H^{1}(\Gamma_{1}(N),k)^{*}_{\rho}}$${H^{1}(\Gamma_{1}(N),k)^{*}_{\rho}\oplus
H^{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee})^{*}_{\rho}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the top horizontal map is obtained from taking the $p$-arithmetic
homology of 4.4 and the bottom horizontal map is (the Poincaré dual of) the
$k$-linear dual of 4.8. Thus, the map
$H_{1}(\Gamma_{1}(N),k)_{\rho}\longrightarrow
H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee})_{\rho}$ from the
statement of the lemma corresponds under these isomorphisms to the dual of the
map from Proposition 4.6. Thus, dualising Proposition 4.6 and Proposition 4.9
gives the result. ∎

##  5\. Proof of Theorem 1.1

In Section 3.2 we have proven the generic case of Theorem 1.1, in this section
we are going to deal with the non-generic cases of twists of the trivial and
Steinberg representations. We assume throughout that $p\geq 5$.

###  5.1. The Steinberg case

The case of twists of the Steinberg representation will follow from
Proposition 3.5 and Proposition 4.10 and some formal algebraic manipulations.
Consider a two-term complex $C_{1}\longrightarrow C_{0}$ of representations of
$G$ such that $H_{0}(C_{\bullet})\simeq H_{1}(C_{\bullet})$. Taking the
$p$-arithmetic hyperhomology of this complex induces two spectral sequences
converging to the same abutment, one $E$ with $E^{2}$ page given by

$E^{2}_{i,j}=H_{i}(\Gamma^{p}_{1}(N),H_{j}(C_{\bullet}))$

and the other ${}^{\prime}E$ with ${}^{\prime}E^{1}$ page given by

${}^{\prime}E^{1}_{i,j}=H_{j}(\Gamma^{p}_{1}(N),C_{i}).$

This spectral sequence degenerates at the ${}^{\prime}E^{2}$ page, so the
systems of Hecke eigenvalues appearing in this page are the same as those in
the abutment. A spectral sequence argument shows that these systems of Hecke
eigenvalues are the same as those appearing in $E^{2}$. Indeed, if $i_{0}$ is
the smallest degree for which a fixed system of eigenvalues appears in
$H_{i_{0}}(\Gamma^{p}_{1}(N),H_{0}(C_{\bullet}))$, then the localisation at
this system of eigenvalues of the $E_{i_{0},0}$ term is stable in the
localised spectral sequence, so the localisation of the abutment in degree
$i_{0}$ will be non-zero. Moreover, if the homologies
$H_{j}(\Gamma^{p}_{1}(N),C_{i})$ are finite-dimensional, then so is the
abutment, and the same type of spectral sequence argument shows that so are
the terms in $E^{2}_{i,j}$.

Let us now specialise to our case of interest. The complex we will be
considering is given by the map

$C_{1}:=\pi(0,1,\chi)\longrightarrow\pi(p-1,1,\chi):=C_{0}$

from Lemma 4.5, so that $H_{0}(C_{\bullet})\simeq
H_{1}(C_{\bullet})\simeq\operatorname{St}\otimes\chi$. Thus, the previous
paragraph and Proposition 3.5 show that Theorem 1.1 (i) and (ii) are satisfied
for twists of the Steinberg representation. Let us analyse the systems of
eigenvalues appearing in homology. Proposition 3.5 shows that the only odd
irreducible Galois representations $\rho$ which can contribute to the
${}^{\prime}E^{1}$ page above, and hence to
$H_{*}(\Gamma^{p}_{1}(N),\operatorname{St}\otimes\chi)$, are those such that
$N(\rho)$ divides $N$ and $\rho|_{\mathcal{G}_{p}}$ is an extension of
$\chi\omega^{-1}$ by $\chi$. Fix such a $\rho$ and write
$\chi=\omega^{a}\mathrm{unr}_{{b}}$. By the argument in [Tar22, Proposition
5.8], and using that derived tensor products commute with mapping cones, the
$p$-arithmetic hyperhomology of $C_{\bullet}$ is isomorphic (in the derived
category of $\mathbb{T}(pN)\otimes_{\mathbb{Z}}k[T,S]$-modules) to the derived
tensor product over $k[T,S]$ of $k[T,S]/(T-b,S-b^{2})$ and

$\displaystyle\left[C_{\bullet}(\Gamma_{1}^{p}(N),\operatorname{c-Ind}_{K}^{G}(\omega^{a}))\longrightarrow
C_{\bullet}(\Gamma_{1}^{p}(N),\operatorname{c-Ind}_{K}^{G}(\operatorname{Sym}^{p-1}(k^{2})^{\vee}\otimes\omega^{a}))\right]$
$\displaystyle\simeq\left[C_{\bullet}(\Gamma_{1}(N),\omega^{a}\circ\det)\longrightarrow
C_{\bullet}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee}\otimes\omega^{a})\right],$

where $\left[{-}\right]$ denotes mapping cones and the isomorphism follows
from Shapiro’s lemma [Tar22, Proposition 5.3]. Using that $\rho$ contributes
to arithmetic homology only in degree 1, we see that the localisation of the
above hyperhomology at $\rho$ is

$\displaystyle\frac{k[T,S]}{(T-b,S-b^{2})}\otimes^{\mathbb{L}}_{k[T,S]}\left[H_{1}(\Gamma_{1}(N),\omega^{a}\circ\det)_{\rho}[1]\longrightarrow
H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee}\otimes\omega^{a})_{\rho}[1]\right]$
$\displaystyle\simeq\frac{k[T,S]}{(T-b,S-b^{2})}\otimes^{\mathbb{L}}_{k[T,S]}U_{a,\rho}[1]$

where the map in the first line is that of Proposition 4.10 and

$U_{a}=\operatorname{coker}\left(H_{1}(\Gamma_{1}(N),\omega^{a}\circ\det)\longrightarrow
H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee}\otimes\omega^{a})\right).$

In conclusion, the abutment of the localisation spectral sequences $E$ and
${}^{\prime}E$ is given in degree $i$ by

$\text{Tor}^{k[T,S]}_{i-1}\left(\frac{k[T,S]}{(T-b,S-b^{2})},U_{a,\rho}\right)$

In particular, analysing the $E^{2}$ page shows that

$\displaystyle H_{0}(\Gamma^{p}_{1}(N),\operatorname{St}\otimes\chi)_{\rho}$
$\displaystyle=H_{3}(\Gamma^{p}_{1}(N),\operatorname{St}\otimes\chi)_{\rho}=0,$
$\displaystyle H_{1}(\Gamma^{p}_{1}(N),\operatorname{St}\otimes\chi)_{\rho}$
$\displaystyle\simeq\frac{k[T,S]}{(T-b,S-b^{2})}\otimes_{k[T,S]}U_{a,\rho},$
$\displaystyle H_{2}(\Gamma^{p}_{1}(N),\operatorname{St}\otimes\chi)_{\rho}$
$\displaystyle\simeq\text{Hom}_{k[T,S]}\left(\frac{k[T,S]}{(T-b,S-b^{2})},U_{a,\rho}\right).$

Moreover, Proposition 4.10 shows that the homology in degrees 1 and 2 is
always non-zero, since $\rho$ contributes to $U_{a}$ and can only contribute
to its $(T=b,S=b^{2})$-eigenspace by [Edi92, Theorem 2.5]. Together with
Proposition 2.4, this completes the proof of Theorem 1.1 (iii) in this case.

###  5.2. The trivial case

The case of twists of the trivial representation is completely analogous to
the case of twists of the Steinberg representation, this time applying the
above analysis to

$C_{1}:=\pi(p-1,1,\chi)\longrightarrow\pi(0,1,\chi):=C_{0},$

where the map is that of Lemma 4.2. It satisfies $H_{0}(C_{\bullet})\simeq
H_{1}(C_{\bullet})\simeq\chi\circ\det$, so Theorem 1.1 (i) and (ii) hold for
these representations. The corresponding spectral sequences and Proposition
4.6 show that if $\chi=\omega^{a}\mathrm{unr}_{{b}}$ and $\rho$ is an odd
irreducible representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$,

$\displaystyle H_{0}(\Gamma^{p}_{1}(N),\chi\circ\det)_{\rho}$
$\displaystyle=H_{1}(\Gamma^{p}_{1}(N),\chi\circ\det)_{\rho}=0,$
$\displaystyle H_{2}(\Gamma^{p}_{1}(N),\chi\circ\det)_{\rho}$
$\displaystyle\simeq\frac{k[T,S]}{(T-b,S-b^{2})}\otimes_{k[T,S]}V_{a,\rho},$
$\displaystyle H_{3}(\Gamma^{p}_{1}(N),\chi\circ\det)_{\rho}$
$\displaystyle\simeq\text{Hom}_{k[T,S]}\left(\frac{k[T,S]}{(T-b,S-b^{2})},V_{a,\rho}\right).$

where

$V_{a}:=\ker\left(H_{1}(\Gamma_{1}(N),\operatorname{Sym}^{p-1}(k^{2})^{\vee}\otimes\omega^{a})\longrightarrow
H_{1}(\Gamma_{1}(N),\omega^{a}\circ\det)\right).$

By Proposition 4.9 (and [BDJ10, Corollary 2.11]), the odd irreducible Galois
representations $\rho$ contributing to the $(T=b,S=b^{2})$-eigenspace of
$V_{a}$ are those such that $N(\rho)$ divides $N$ and
$\rho|_{\mathcal{G}_{p}}$ is an extension of $\chi\omega^{-1}$ by $\chi$.
These representations therefore appear in the $p$-arithmetic homology of
$\chi\circ\det$ exactly in degrees 2 and 3. There are no Galois
representations satisfying condition (iii)(a) of Theorem 1.1 (iii), as finite-
dimensional representations never appear in the socle of representations in
the image of the mod $p$ local Langlands correspondence for
$\text{GL}_{2}(\mathbb{Q}_{p})$. Therefore, Theorem 1.1 (iii) holds, which
completes the proof of Theorem 1.1.

######  Remark 5.1.

In the proof of Proposition 4.10 we showed using Poincaré duality for
arithmetic cohomology that $U_{a}=(V_{-a}^{\vee})^{*}$ (with the notation
introduced in that proof). The Poincaré duality isomorphisms in that proof
intertwine $T$ with $S^{-1}T$ and $S$ with $S^{-1}$. This, together with the
above computations, implies that for $\rho$ as above there are “Poincaré
duality” isomorphisms

$\displaystyle H^{i}(\Gamma_{1}^{p}(N),\chi\circ\det)_{\rho}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{4-i}(\Gamma_{1}^{p}(N),\operatorname{St}\otimes\chi)^{*}_{\rho}.$

######  Remark 5.2.

Assume that $\chi$ is the trivial character for simplicity. Then, the fact
that Galois representations as above contribute to
$H_{*}(\Gamma_{1}^{p}(N),k)$ and $H^{*}(\Gamma_{1}^{p}(N),k)$ but not in
degree 1 (unlike in the other cases) is related to the fact that, if $\pi$ is
the smooth representation of $\text{GL}_{2}(\mathbb{Q}_{p})$ corresponding to
$\rho|_{\mathcal{G}_{p}}$ under the mod $p$ Langlands correspondence, then
$\text{Hom}_{k[G]}(k,\pi)=0$, but $\text{Ext}^{1}_{k[G]}(k,\pi)\neq 0$ (at
least when $\rho|_{\mathcal{G}_{p}}$ is non-split). This can be made precise
by relating $p$-arithmetic cohomology to completed cohomology and taking into
account Emerton’s local-global compatibility results [Eme], but we do not
pursue this here.

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