# Motivic congruences and Sharifi’s conjecture

Óscar Rivero and Victor Rotger  O. R.: Mathematics Institute, University of
Warwick, Coventry CV4 7AL, United Kingdom<EMAIL_ADDRESS>V. R.:
IMTech, UPC and Centre de Recerca Matemàtiques, C. Jordi Girona 1-3, 08034
Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract.

Let $f$ be a cuspidal eigenform of weight two and level $N$, let $p\nmid N$ be
a prime at which $f$ is congruent to an Eisenstein series and let $V_{f}$
denote the $p$-adic Tate module of $f$. Beilinson [Bei] constructed a class
$\kappa_{f}\in H^{1}(\mathbb{Q},V_{f}(1))$ arising from the cup-product of two
Siegel units and proved a striking relationship with the first derivative
$L^{\prime}(f,0)$ at the near central point $s=0$ of the $L$-series of $f$,
which led him to formulate his celebrated conjecture. In this note we prove
two congruence formulae relating the “motivic part” of
$L^{\prime}(f,0)\,(\mathrm{mod}\,p)$ and
$L^{\prime\prime}(f,0)\,(\mathrm{mod}\,p)$ with circular units. The proofs
make use of delicate Galois properties satisfied by various integral lattices
within $V_{f}$ and exploits Perrin-Riou’s, Coleman’s and Kato’s work on the
Euler systems of circular units and Beilinson–Kato elements and, most
crucially, the work of Sharifi [Sha], Fukaya–Kato [FK] and Ohta [Oh1].

###### 2010 Mathematics Subject Classification:

11F33 (primary); 11F67, 11F80 (secondary)

###### Contents

  1. 1 Introduction
  2. 2 Eisenstein series, modular curves and lattices
  3. 3 First congruence relation
    1. 3.1 Mazur’s factorization formula
    2. 3.2 Dieudonné modules and congruences among Ohta’s periods
    3. 3.3 Coleman’s power series and the Kubota-Leopoldt $p$-adic $L$-function
    4. 3.4 Kato’s explicit reciprocity law
    5. 3.5 Proof of Theorem 3.1
  4. 4 Second congruence relation
    1. 4.1 Cohomology and Eisenstein quotients
    2. 4.2 Fukaya–Kato maps
    3. 4.3 Proof of Theorem 4.2

##  1\. Introduction

The aim of this paper is showing how the ideas underlying Sharifi’s conjecture
[Sha] and the work [FK] of Fukaya and Kato can be exploited to study
congruences among motivic classes that naturally arise from the Euler systems
of Beilinson-Kato elements and circular units.

In order to set the stage, let $N>1$ be a positive integer,
$\theta:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\bar{\mathbb{Q}}^{\times}$
an even Dirichlet character and $f\in S_{2}(N,\theta)$ a normalized cuspidal
eigenform of level $N$, weight $2$ and nebentype $\theta$.

Fix a prime $p\nmid 6N\varphi(N)$. Let $T_{f,X}$ and $T_{f,Y}$ denote the
integral $p$-adic Galois representations given as the $f$-isotypical quotient
of $H^{1}_{\operatorname{et}}(\bar{X},\mathbb{Z}_{p}(1))$, resp.
$H^{1}_{\operatorname{et}}(\bar{Y},\mathbb{Z}_{p}(1))$ of the closed, resp.
open modular curve111Cf. §2 for the particular models of these curves we
employ in this article and precise definitions.  of level $\Gamma_{1}(N)$. Set
as usual $V_{f}=T_{f,X}\otimes\mathbb{Q}=T_{f,Y}\otimes\mathbb{Q}$.

Beilinson [Bei] introduced a motivic element in the $K$-group $K_{2}(Y)$
giving rise to a global Galois cohomology class

(1) $\kappa_{f}=\kappa_{f}(\chi_{1},\chi_{2})\in H^{1}(\mathbb{Q},T_{f,Y}(1))$

that was later the basis of Kato’s Euler system in [Ka]. This class depends on
auxiliary data (cf. loc. cit. and [BD2], [Han], [KLZ, §9], [Sch] for several
presentations of the subject in the literature). With our normalizations,
$\kappa_{f}$ depends on the choice of two auxiliary Dirichlet characters
$\chi_{1}$ and $\chi_{2}$ of the same parity (see §3) and it is straight-
forward to relate it to other equivalent conventions adopted in loc. cit.

The main motivation of Beilinson’s construction of $\kappa_{f}$ was providing
evidence for his celebrated conjecture on values of $L$-functions of mixed
motives, encompassing Dirichlet’s unit theorem and the Birch and Swinnerton-
Dyer conjecture for elliptic curves as instances of it. This conjecture was
later refined by Bloch-Kato in [BK], and in the case at hand predicts that

(2) ${\mathrm{ord}}_{s=0}L(f,s)\,\stackrel{{\scriptstyle?}}{{=}}\dim
H^{1}_{\operatorname{f}}(\mathbb{Q},V_{f}(1)),$

where the left-hand side is the $L$-series associated to $f$ and the right-
hand side denotes the space of crystalline classes in
$H^{1}(\mathbb{Q},V_{f}(1))$. Since $L(f,0)$ vanishes (for innocent reasons,
due to the vanishing of a $\Gamma$-factor arising from analytic continuation),
(2) suggests that $H^{1}_{\operatorname{f}}(\mathbb{Q},V_{f}(1))$ should
contain non-trivial classes, and Beilinson proved in loc. cit. that
$\kappa_{f}$ is indeed crystalline and non-trivial, provided
$L^{\prime}(f,0)\neq 0$.

Let $F$ be the finite extension of $\mathbb{Q}$ generated by the field of
coefficients of $f$ and the values of all Dirichlet characters of conductor
$N$; let $\mathcal{O}$ be its ring of integers and
$\mathfrak{p}\subset{\mathcal{O}}$ a prime ideal above $p$.

In this note we assume $f$ is congruent to an Eisenstein series modulo
$\mathfrak{p}$. Up to replacing $f$ with a twist of it, we may assume without
loss of generality that

(3) $f\equiv E_{2}(\theta,1)\,\,\mathrm{mod}\,\mathfrak{p}^{t}$

for some $t\geq 1$, where $E_{2}(\theta,1)$ is the classical Eisenstein series
recalled in (7) below.222Congruence (3) is equivalent to $f^{*}\equiv
E_{2}(1,\bar{\theta})$, where $f^{*}=f\otimes\theta^{-1}$ is the dual form of
$f$, and this in turn implies that $\mathfrak{p}^{t}$ divides the generalized
Bernoulli number $B_{2}(\bar{\theta})$ (or equivalently the $L$-value
$L(\bar{\theta},-1)$). We take $t$ to be the largest power satisfying (3).

As is well-known, $T_{f,X}$ and $T_{f,Y}$ are finitely generated
${\mathcal{O}}_{\mathfrak{p}}[G_{\mathbb{Q}}]$-modules giving rise to the same
Galois representation $V_{f}$ over $F_{\mathfrak{p}}$. However, the lattices
$T_{f,X}$ and $T_{f,Y}$ are not isomorphic as $G_{\mathbb{Q}}$-modules, and in
general none of them are necessarily free as
${\mathcal{O}}_{\mathfrak{p}}$-modules. Setting
$\bar{T}:=T\otimes\mathcal{O}/\mathfrak{p}^{t}$ for any
${\mathcal{O}}_{\mathfrak{p}}$-module, congruence (3) does imply (cf. §2) that
one always has surjective homomorphisms of $G_{\mathbb{Q}}$-modules

(4)
$\bar{T}_{f,Y}\stackrel{{\scriptstyle\bar{\pi}_{1}}}{{\longrightarrow}}\mathcal{O}/\mathfrak{p}^{t}(\theta),\quad\bar{T}_{f,X}\stackrel{{\scriptstyle\bar{\pi}_{2}}}{{\longrightarrow}}\mathcal{O}/\mathfrak{p}^{t}(1).$

The maps $\bar{\pi}_{1}$ and $\bar{\pi}_{2}$ in (4) are non-canonical, but we
exploit the work of Ohta [Oh1], [Oh2], Sharifi [Sha] and Fukaya-Kato [FK] to
rigidify them in a canonical way, in the sense that both $\bar{\pi}_{1}$ and
$\bar{\pi}_{2}$ only depend on canonical periods naturally associated to $f$;
cf. (39) and (49).

Write $\bar{\kappa}_{f}=\kappa_{f}\,(\mbox{mod }\mathfrak{p}^{t})$ and define

$\bar{\kappa}_{f,1}=\bar{\pi}_{1*}(\bar{\kappa}_{f})\in
H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1)).$

Motivated by the above discussion of Beilinson’s conjecture,
$\bar{\kappa}_{f,1}$ may be regarded as a motivic avatar of the first
derivative $L^{\prime}(f,0)$ $(\mathrm{mod}\,\mathfrak{p}^{t})$. The first
main result of this note, Theorem 1.1 below, is an explicit formula for
$\bar{\kappa}_{f,1}$ in terms of algebraic $L$-values and circular units,
which in particular provides a criterion for this class to vanish. In the
parlance of [BD1], our Theorem 1.1 may be interpreted as a Jochnowitz
congruence

$L_{\mathrm{alg}}^{\prime}(f,0)\,\equiv\,L_{\mathrm{alg}}^{\prime}(\theta,0)\quad(\mathrm{mod}\,\mathfrak{p}^{t})$

between the “algebraic” or ”motivic parts” of the derivative at $s=0$ of the
Hasse-Weil $L$-function $L(f,s)$ and Dirichlet’s $L$-function $L(\theta,s)$.

We further show that $\kappa_{f}$ may be lifted to an element in
$H^{1}(\mathbb{Q},T_{f,X}(1))$ if and only if the
$(\mathrm{mod}\,\mathfrak{p}^{t})$ class $\bar{\kappa}_{f,1}$ vanishes. When
this happens, such a lift is unique and we thus continue to denote it
$\kappa_{f}$ by slight abuse of notation; we may then define

$\bar{\kappa}_{f,2}=\bar{\pi}_{2*}(\bar{\kappa}_{f})\in
H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2)),$

which we regard as the motivic counterpart of the second derivative
$L^{{}^{\prime\prime}}(f,0)$ $(\mathrm{mod}\,\mathfrak{p}^{t})$.

The second main result of this note, Theorem 1.2 below, provides an explicit
formula for $\bar{\kappa}_{f,2}$ as the cup-product of two circular units. We
find interesting that the circle of ideas appearing in [FK] and [Sha] can be
applied to the computation of second derivatives, a type of result which
appears to be quite novel.

In order to state our results precisely, let

(5)
$\mathbb{Z}[\mu_{N}]^{\times}[\theta]=(\mathbb{Z}[\mu_{N}]^{\times}\otimes\mathcal{O}_{\mathfrak{p}}(\bar{\theta}))^{\mathrm{Gal\,}(\mathbb{Q}(\mu_{N})/\mathbb{Q})}\simeq\operatorname{Hom}(\mathcal{O}_{\mathfrak{p}}(\theta),\mathbb{Z}[\mu_{N}]^{\times}\otimes\mathcal{O}_{\mathfrak{p}})$

denote the $\theta$-isotypic component of
$\mathbb{Z}[\mu_{N}]^{\times}\otimes\mathcal{O}_{\mathfrak{p}}$ on which
$\mathrm{Gal\,}(\mathbb{Q}(\mu_{N})/\mathbb{Q})$ acts through $\theta$, which
may be naturally identified with a $\mathcal{O}_{\mathfrak{p}}$-submodule of
$\mathbb{Z}[\mu_{N}]^{\times}\otimes\mathcal{O}_{\mathfrak{p}}$ of rank 1 when
$\theta\neq 1$ (resp. rank 0 when $\theta=1$). Kummer theory gives rise to an
injective homomorphism

$\mathbb{Z}[\mu_{N}]^{\times}[\bar{\theta}]\rightarrow\operatorname{Hom}(G_{\mathbb{Q}(\mu_{N})},\mathcal{O}_{\mathfrak{p}}(1))[\bar{\theta}]\rightarrow
H^{1}(\mathbb{Q},\mathcal{O}_{\mathfrak{p}}(\theta)(1)).$

Fix a primitive $N$-th root of unity $\zeta_{N}$ and define the circular unit

(6)
$c_{\theta}:=\prod_{a=1}^{N-1}(1-\zeta_{N}^{a})^{\theta(a)}\in\mathbb{Z}[\mu_{N}]^{\times}[\bar{\theta}].$

Let

$c_{\theta}\in H^{1}(\mathbb{Q},\mathcal{O}_{\mathfrak{p}}(\theta)(1))$

denote, with the same symbol by a slight abuse of notation, its image under
the identification provided by the Kummer map. Write
$\bar{c}_{\theta}=c_{\theta}\,(\mbox{mod }\mathfrak{p}^{t})\,\in
H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$.

Our main theorems are conditional on the following two hypotheses, that we
assume throughout this article:

  1. (H1)

Non-trivial zeroes mod $\mathfrak{p}$:

$\theta(p)-1,\,\chi_{1}\bar{\chi}_{2}(p)-1,\,\theta\chi_{1}\bar{\chi}_{2}(p)-1\neq
0\,\,(\mbox{mod}\,\,\mathfrak{p}).$

  2. (H2)

Letting $\Sigma_{X}$, $\Sigma_{Y}$ denote the torsion submodules of $T_{f,X}$
and $T_{f,Y}$ respectively, the $G_{\mathbb{Q}_{p}}$-module
$\mathcal{O}/\mathfrak{p}^{t}(\theta)$ does not show up as a quotient of
$\Sigma_{Y}/\Sigma_{X}$.

Note that (H1) implies that $\theta$ and in fact $\theta_{|\mathbb{Q}_{p}}$ is
non-trivial, even mod $\mathfrak{p}$; in particular, $c_{\theta}$ is a non-
trivial unit. Note that (H2) follows automatically if the localization of the
Hecke algebra acting on $M_{2}(\Gamma_{1}(N))$ at the Eisenstein ideal is
Gorenstein, as this implies that $T_{f,Y}$ is free as
$\mathcal{O}_{\mathfrak{p}}$-module. We wonder whether (H2) might be weaker
and more tractable than asking the Hecke algebra to be Gorenstein.

Define the algebraic $L$-value

$L^{\mathrm{alg}}(f^{*},\bar{\chi}_{1}\chi_{2},1)=L(f^{*},\bar{\chi}_{1}\chi_{2},1)/\Omega_{f}^{+}\in{\mathcal{O}}$

where $\Omega_{f}^{+}$ is Shimura’s complex period associated to $f^{*}$,
chosen in a specific way that we recall in §3.1. Let also

$\mathfrak{g}(\chi)=\sum_{a=1}^{N-1}\chi(a)\zeta_{N}^{a}$

denote the Gauss sum attached to a Dirichlet character $\chi$ of conductor
$N$.

######  Theorem 1.1.

In $H^{1}(\mathbb{Q}_{p},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$ we have

$\bar{\kappa}_{f,1}\equiv\frac{iN}{12}\cdot\frac{\mathfrak{g}(\theta\chi_{1}\bar{\chi}_{2})}{\mathfrak{g}(\theta)}\cdot
L^{\mathrm{alg}}(f^{*},\bar{\chi}_{1}\chi_{2},1)\times\bar{c}_{\theta}\pmod{\mathfrak{p}^{t}}.$

This equality takes place globally in
$H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$ if $p$ is
$\bar{\theta}$-regular as specified in (44).

Let us state now our second main result. As we describe in more detail in §3,
Kato’s class is constructed as

$\kappa_{f}=\pi_{f*}(u\cup v)\in H^{1}(\mathbb{Q},T_{f,Y}(1)),$

namely the push-forward to the $f$-isotypic component of the cup product of
two modular units

$u=u_{\chi_{1},\chi_{2}}\quad\mbox{and}\quad
v=u_{\bar{\chi}_{1},\theta\bar{\chi}_{2}}$

whose logarithmic derivative are respectively the classical Eisenstein series
$E_{2}(\chi_{1},\chi_{2})$ and $E_{2}(\bar{\chi}_{1},\theta\bar{\chi}_{2})$
given in (7). Note that $u_{\chi_{1},\chi_{2}}$ are determined by their
logarithmic derivative only up to a multiplicative constant, and therefore the
first non-vanishing coefficient in the Laurent expansion of
$u_{\chi_{1},\chi_{2}}$ at $\infty$, which we simply denote
$u_{\chi_{1},\chi_{2}}(\infty)$ as in [FK, §5], may be chosen arbitrarily.
Since $\mathrm{Gal\,}(\mathbb{Q}(\mu_{N})/\mathbb{Q})$ acts on
$E_{2}(\chi_{1},\chi_{2})$ via $\chi_{1}$ [St1, Theorem 1.3.1], it is natural
to normalize $u_{\chi_{1},\chi_{2}}$ likewise, so that
$u_{\chi_{1},\chi_{2}}(\infty)$ may be any power of the circular unit
$c_{\chi_{1}}$. In the literature one finds different normalizations,
typically either $u_{\chi_{1},\chi_{2}}(\infty)=1$ or $c_{\chi_{1}}$. In the
statement below we have chosen to normalize the modular units above so that

$u_{\chi_{1},\chi_{2}}(\infty)=c_{\chi_{1}},\quad
u_{\bar{\chi}_{1},\theta\bar{\chi}_{2}}(\infty)=c_{\bar{\chi}_{1}}$

but any other choice would be perfectly fine, upon replacing accordingly the
two circular units appearing in the cup-product below.

######  Theorem 1.2.

Assume $\theta$ is primitive of conductor $N$. Suppose further that
$\bar{\kappa}_{f,1}=0$. Let $L_{p}(\bar{\theta},s)$ denote the Kubota–Leopoldt
$p$-adic $L$-function attached to $\bar{\theta}$ and assume
$L_{p}^{\prime}(\bar{\theta},-1)$ is a $p$-adic unit. The following equality
holds in $H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2))$:

$\bar{\kappa}_{f,2}\equiv\frac{L_{p}^{\prime}(\bar{\theta},-1)}{1-p^{-1}}\cdot\frac{\bar{c}_{\bar{\chi}_{1}}\cup\bar{c}_{\chi_{1}}}{\cup\log_{p}(\varepsilon_{\operatorname{cyc}})}\pmod{\mathfrak{p}^{t}}.$

Here $\varepsilon_{\operatorname{cyc}}$ is the cyclotomic character and
$1/\cup\log_{p}(\varepsilon_{\operatorname{cyc}})$ denotes the inverse of the
map

$H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2))\rightarrow
H^{2}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2)),\quad\kappa\mapsto\kappa\cup\log_{p}(\varepsilon_{\operatorname{cyc}}),$

which is invertible under our assumptions.

In the above statement note that our running assumptions imply that
$L_{p}(\bar{\theta},-1)\equiv 0\pmod{{\mathfrak{p}}^{t}}$ and it is thus
natural that the first derivative of the Kubota–Leopoldt $p$-adic $L$-function
makes an appearance.

Theorems 1.1 and 1.2 are proved in §3 and §4 respectively. We hope the methods
introduced in this note may help to extend Sharifi’s conjectures to other
scenarios where the theory of Euler systems has experienced exciting progress
in recent years (cf. e.g. [Eu], [KLZ], [LPSZ], [LSZ]).

Acknowledgements. We thank P. Wake for guiding us through the delicate subject
of integral lattices in Galois representations. We also thank G. Gras for
pointing us to suitable references and providing careful explanations of his
work. We are also grateful to H. Darmon and R. Sharifi for their comments.
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 682152). The first author has also received financial
support through “la Caixa” Fellowship Grant for Doctoral Studies (grant
LCF/BQ/ES17/11600010). The second author is supported by Icrea through an
Icrea Academia Grant.

##  2\. Eisenstein series, modular curves and lattices

The aim of this section is recalling well-known facts and setting notations
concerning Eisenstein series, models of modular curves and various integral
lattices associated to them. We take the chance to prove some elementary
relationships among the latter, which are surely well-known to experts but
that we include because we failed to find precise references in the
literature.

Fix algebraic closures $\bar{\mathbb{Q}}$, $\bar{\mathbb{Q}}_{p}$ of
$\mathbb{Q}$ and $\mathbb{Q}_{p}$ respectively, and embeddings of
$\bar{\mathbb{Q}}$ into $\bar{\mathbb{Q}}_{p}$ and $\mathbb{C}$. The former
singles out a prime ideal $\mathfrak{p}$ of ${\mathcal{O}}$ lying above $p$
and we let $\mathcal{O}_{\mathfrak{p}}$ denote the completion of $\mathcal{O}$
at $\mathfrak{p}$. We also fix throughout an uniformizer $\varpi$ of
$\mathcal{O}_{\mathfrak{p}}$ and an isomorphism
$\mathbb{C}_{p}\simeq\mathbb{C}$.

Given a variety $Y/\mathbb{Q}$ and a field extension $F/\mathbb{Q}$, let
$Y_{F}=Y\times F$ denote the base change of $Y$ to $F$ and set
$\bar{Y}=Y_{\bar{\mathbb{Q}}}$. Fix an integer $N\geq 3$ and let
$Y_{1}(N)\subset X_{1}(N)$ denote the canonical models over $\mathbb{Q}$ of
the (affine and projective, respectively) modular curves classifying pairs
$(A,i)$ where $A$ is a (generalized) elliptic curve and $i:\mu_{N}\rightarrow
A$ is an embedding of group schemes. It is important to recall that this is
not the model used by Fukaya and Kato, as they consider the one which
classifies pairs $(A,P)$, where $A$ is a (generalized) elliptic curve and $P$
is an $N$-torsion point of it. In any case, the model of [FK] can be obtained
from ours just taking the twist by the cocyle

$\mathrm{Gal\,}(\mathbb{Q}(\mu_{N})/\mathbb{Q})\rightarrow\mathrm{Aut}(Y),\quad
s\mapsto\langle s^{-1}\rangle,$

where $\langle s\rangle$ stands for the diamond operator associated to
$s\in(\mathbb{Z}/N\mathbb{Z})^{\times}$.

Let $C_{N}:=X_{1}(N)\setminus Y_{1}(N)$ denote the finite scheme of cusps;
among them one may distinguish the cusp $\infty\in C_{N}(\mathbb{Q})$
associated to Tate’s elliptic curve over $\mathbb{Z}((q))$, which is rational
over $\mathbb{Q}$ in this choice of model (cf. e.g. [St1, §1.3], [St2]).
(Again, note that in the model of [FK, §1.3.3], cusp $\infty$ is not defined
over $\mathbb{Q}$ but over $\mathbb{Q}(\mu_{N})$.)

Assume now that $F$ contains the values of all Dirichlet characters of
conductor $N$. Then a basis of $\operatorname{Eis}_{2}(\Gamma_{1}(N),F)$ is
indexed by triples $(\chi_{1},\chi_{2},r)$ where $\chi_{1}$ and $\chi_{2}$ are
primitive Dirichlet characters of conductors $N_{1}$ and $N_{2}$ with
$N_{1}\cdot N_{2}\,\mid\,N$, $\chi_{1}(-1)=\chi_{2}(-1)$, and $r$ is a
positive integer with $1<rN_{1}N_{2}\mid N$, provided by the Eisenstein series
(cf. e.g. [DS, Theorem 4.6.2], [St1, Def. 3.4.1]):

(7)
$E_{2}(\chi_{1},\chi_{2},r)=a_{0}+\sum_{n=1}^{\infty}\Big{(}\sum_{d|n}\chi_{1}(n/d)\chi_{2}(d)d\Big{)}q^{rn},\quad
a_{0}=\begin{cases}\frac{L(\chi_{2},-1)}{2}&\mbox{ if }\chi_{1}=1\\\ 0&\mbox{
if }\chi_{1}\neq 1\end{cases}$

unless $\chi_{1}=\chi_{2}=1$, in which case
$E_{2}(1,1,r)=\sum_{n=1}^{\infty}\Big{(}\sum_{d|n}d\Big{)}q^{n}-r\sum_{n=1}^{\infty}\Big{(}\sum_{d|n}d\Big{)}q^{rn}$.

When $r=1$ we shall simply denote
$E_{2}(\chi_{1},\chi_{2}):=E_{2}(\chi_{1},\chi_{2},1)$.

When $\chi_{1}=1$, the constant term may also be recast as a generalized
Bernoulli number: setting $B_{2}(x)=x^{2}-x+1/6$, define

$B_{2}(\chi):=N\sum_{a=1}^{N-1}\chi(a)\cdot B_{2}(a/N)$

for any Dirichlet character $\chi$ of conductor $N$. One then has
$-2L(\chi_{2},-1)=B_{2}(\chi_{2})$.

Define the group of modular units $U(N)$ as the subgroup of rational functions
of $X_{1}(N)_{\mathbb{Q}(\mu_{N})}$ with zeroes and poles concentrated at the
cusps, that is to say

$U(N)={\mathcal{O}}(Y_{1}(N)_{\mathbb{Q}(\mu_{N})})^{\times}.$

Similarly as in (5), let $U(N)[\chi]$ denote the $\chi$-isotypic component of
$U(N)\otimes{\mathcal{O}}_{\mathfrak{p}}$ on which
$\mathrm{Gal\,}(\mathbb{Q}(\mu_{N})/\mathbb{Q})$ acts through the character
$\chi$. In light of [St1, Theorem 1.3.1], there exists a modular unit
$u_{\chi_{1},\chi_{2}}\in U(N)[\chi_{1}]$ satisfying

(8)
$\operatorname{dlog}(u_{\chi_{1},\chi_{2}})=E_{2}(\chi_{1},\chi_{2})\frac{dq}{q}\quad\mbox{and}\quad
u_{\chi_{1},\chi_{2}}(\infty)=c_{\chi_{1}}.$

The $q$-expansion of the modular units $u_{\chi_{1},\chi_{2}}$ can be written
down explicitly. Given a pair of integers $(a,b)$ between $0$ and $N-1$, not
both equal to 0, define the Siegel unit

$u_{a,b;N}=q^{w}\prod_{n\geq
0}\Big{(}1-q^{n+a/N}\zeta_{N}^{b}\Big{)}\prod_{n\geq
1}\Big{(}1-q^{n-a/N}\zeta_{N}^{-b}\Big{)},$

where $w=\frac{1}{12}-\frac{a}{N}+\frac{a^{2}}{2N^{2}}$. Then the
$q$-expansion of the modular unit $u_{\chi_{1},1}$ is given by

(9)
$u_{\chi_{1},1}=\frac{-1}{2\mathfrak{g}(\bar{\chi}_{1})}\sum_{b=1}^{N-1}\bar{\chi}_{1}(b)\otimes
u_{0,b;N},$

where here $N$ stands for the conductor of $\chi_{1}$. Although we will not
use them here in this note, similar expressions can be given for
$u_{\chi_{1},\chi_{2}}$ for arbitrary $\chi_{2}$ by averaging $u_{a,b;N}$ and
choosing an appropriate uniformizer.

Kummer theory induces a morphism

(10) $\delta:U(N)[\chi]\rightarrow
H_{\operatorname{et}}^{1}(Y_{1}(N),{\mathcal{O}}_{\mathfrak{p}}(\bar{\chi})(1)).$

Let
$\mathbb{T}\subset\mathrm{End}\,H_{\operatorname{et},c}^{1}(\overline{Y}_{1}(N),\mathcal{O}_{\mathfrak{p}}(1))$
denote the Hecke algebra acting on the compactly-supported cohomology of the
open modular curve generated by the standard Hecke operators $T_{\ell}$ for
every (good or bad) prime $\ell$ let–commonly denoted $U_{\ell}$ at primes
$\ell\mid N$. Let also
$\mathbb{T}^{*}\subset\mathrm{End}\,H_{\operatorname{et}}^{1}(\overline{Y}_{1}(N),\mathcal{O}_{\mathfrak{p}}(1))$
denote the Hecke algebra acting on the cohomology of the open modular curve
generated by the dual Hecke operators $T_{\ell}^{*}$ as defined in [Oh1,
§3.4], [KLZ, Def. 2.4.3] for every prime $\ell$.

Recall from the introduction the newform $f\in S_{2}(N,\theta)$ satisfying the
congruence $f\equiv E_{2}(\theta,1)\,\,\mathrm{mod}\,\mathfrak{p}^{t}$. Let
$f(q)=\sum a_{n}(f)q^{n}$ denote its $q$-expansion at the cusp $\infty$.
Enlarge $F$ so that it also contains the eigenvalues $\\{a_{n}(f)\\}_{n\geq
1}$, let ${\mathcal{O}}$ denote its ring of integers and let
${\mathcal{O}}_{{\mathfrak{p}}}$ denote its completion at the prime ideal
$\mathfrak{p}$ over $p$ fixed in the introduction.

Let $I_{f}^{*}=(T^{*}_{\ell}-a_{\ell}(f))\subset\mathbb{T}^{*}$ denote the
ideal associated to the system of eigenvalues of $f$ with respect to the dual
Hecke operators. Define the ${\mathcal{O}}_{{\mathfrak{p}}}$-modules

(11)
$T_{f,X}=H_{\operatorname{et}}^{1}(\overline{X}_{1}(N),\mathcal{O}_{\mathfrak{p}}(1))/I^{*}_{f},\quad
T_{f,Y}=H_{\operatorname{et}}^{1}(\overline{Y}_{1}(N),\mathcal{O}_{\mathfrak{p}}(1))/I^{*}_{f}.$

Note that the two lattices $T_{f,X}$ and $T_{f,Y}$ may give rise to completely
different ${\mathcal{O}}_{{\mathfrak{p}}}[G_{\mathbb{Q}}]$-modules in spite of
the fact that the associated rational Galois representations

$V_{f}:=T_{f,X}\otimes F_{{\mathfrak{p}}}\simeq T_{f,Y}\otimes
F_{{\mathfrak{p}}}$

are isomorphic.

######  Proposition 2.1.

The natural inclusion $X\hookrightarrow Y$ induces by push-forward in
cohomology a map $T_{f,X}\rightarrow T_{f,Y}$ that sits in an exact sequence
of ${\mathcal{O}}_{\mathfrak{p}}[G_{\mathbb{Q}}]$-modules

$0\rightarrow T_{f,X}\longrightarrow
T_{f,Y}\stackrel{{\scriptstyle\pi_{1}}}{{\longrightarrow}}\mathcal{O}/\mathfrak{p}^{t}(\theta)\rightarrow
0.$

###### Proof.

Write for short
$H^{1}(X)=H_{\operatorname{et}}^{1}(\bar{X}_{1}(N),{\mathcal{O}}_{\mathfrak{p}}(1))$
and
$H^{1}(Y)=H_{\operatorname{et}}^{1}(\bar{Y}_{1}(N),{\mathcal{O}}_{\mathfrak{p}}(1))$.
As it is shown for instance in [St1, §1.8], there is a short exact sequence of
the form

(12) $0\longrightarrow H^{1}(X)\longrightarrow
H^{1}(Y)\longrightarrow\mathrm{Div}^{0}[C_{N}]\longrightarrow 0,$

where $\mathrm{Div}^{0}[C_{N}]$ is the free
${\mathcal{O}}_{\mathfrak{p}}$-module of degree $0$ divisors supported on
$C_{N}$ with coefficients in ${\mathcal{O}}_{\mathfrak{p}}$. Since the map
$H^{1}(X)\hookrightarrow H^{1}(Y)$ induces an injection

$H^{1}(X)/(I_{f}^{*}\cdot H^{1}(Y)\cap H^{1}(X))\hookrightarrow
H^{1}(Y)/I_{f}^{*}\cdot H^{1}(Y),$

the exactness on the left follows once we show that

(13) $I_{f}^{*}\cdot H^{1}(Y)\cap H^{1}(X)=I_{f}^{*}\cdot H^{1}(X).$

The inclusion $I_{f}^{*}H^{1}(X)\subset I_{f}^{*}H^{1}(Y)\cap H^{1}(X)$ is
clear. As for the opposite one, take an element $t\alpha\in
I_{f}^{*}H^{1}(Y)\cap H^{1}(X)$ with $t\in I_{f}^{*}$ and $\alpha\in
H^{1}(Y)$. After inverting $p$, there is an isomorphism

$H^{1}(Y)[1/p]\simeq H^{1}(X)[1/p]\oplus\mathrm{Div}^{0}[C_{N}][1/p],$

since (12) is split after tensoring with $\mathbb{Q}_{p}$ by Manin-Drinfeld’s
theorem. We may thus write $\alpha=\beta+\gamma$, where $\beta\in
H^{1}(X)[1/p]$ and $\gamma\in\mathrm{Div}^{0}[C_{N}][1/p]$. Since $t\alpha\in
H^{1}(X)$, we have $t\gamma=0$, and thus $t\alpha=t\beta$ in
$I_{f}^{*}(H^{1}(Y)\cap H^{1}(X)[1/p])$.

Note that $H^{1}(Y)\cap H^{1}(X)[1/p]=H^{1}(X)$. Indeed, otherwise there would
exist an element $y\in H^{1}(Y)\setminus H^{1}(X)$ such that $p^{u}y\in
H^{1}(X)$ for some $u$. This would imply that $H^{1}(Y)/H^{1}(X)$ contains
non-trivial torsion, but this quotient is isomorphic to
$\mathrm{Div}^{0}[C_{N}]$, which is free. It thus follows that $t\alpha$ lies
in $I_{f}^{*}H^{1}(X)$, and this proves exactness on the left.

In order to conclude the proof of the proposition, let us now show that
$T_{f,Y}/T_{f,X}\simeq\mathcal{O}/\mathfrak{p}^{t}(\theta)$. As it follows
from the above, (12) induces an exact sequence

(14) $0\longrightarrow T_{f,X}\longrightarrow
T_{f,Y}\longrightarrow\mathrm{Div}^{0}[C_{N}]/I_{f}^{*}\mathrm{Div}^{0}[C_{N}]\longrightarrow
0.$

The Hecke action on $\mathrm{Div}^{0}[C_{N}]$ is Eisenstein, and the
eigenvalues of $T^{*}_{\ell}$ are $\chi_{1}(\ell)+\ell\cdot\chi_{2}(\ell)$,
where $(\chi_{1},\chi_{2})$ range through pairs of Dirichlet characters of the
same parity, not both trivial, whose product of conductors is $N$ (as
described e.g. in [St1, §1.3]).

Our running assumptions imply that $E_{2}(\theta,1)$ occurs with multiplicity
one in $\mathrm{Div}^{0}[C_{N}]/\mathfrak{p}^{t}$. Indeed, otherwise there
would exist a pair of Dirichlet characters $(\xi_{1},\xi_{2})$ as above such
that $E_{2}(\theta,1)\equiv E_{2}(\xi_{1},\xi_{2})\pmod{\mathfrak{p}}$. This
amounts to saying that $\bar{\theta}\xi_{1}\equiv\xi_{2}\equiv 1$ (mod
$\mathfrak{p}^{t}$), and this is only possible when
$(\xi_{1},\xi_{2})=(\theta,1)$ because $p\nmid\varphi(N)$.

Since $f\equiv E_{2}(\theta,1)$ (mod $\mathfrak{p}^{t}$), it follows that
$T_{f,Y}/T_{f,X}\simeq{\mathcal{O}}/\mathfrak{p}^{t}$ as
${\mathcal{O}}_{\mathfrak{p}}$-modules. The action of $G_{\mathbb{Q}}$ on
$T_{f,Y}/T_{f,X}$ is given by the character $\theta$ by [St1, Theorem 1.3.1].
∎

Since $f$ is ordinary at $\mathfrak{p}$, it is well-known (cf. e.g. [FK, 1.7]
for a twisted version that boils down to the one below with our
normalizations) that there are exact sequences of finitely generated
${\mathcal{O}}_{\mathfrak{p}}[G_{\mathbb{Q}_{p}}]$-modules

(15) $\displaystyle 0\rightarrow T_{f,X}^{\operatorname{sub}}\rightarrow
T_{f,X}\rightarrow T_{f,X}^{\operatorname{quo}}\rightarrow 0$ $\displaystyle
0\rightarrow T_{f,Y}^{\operatorname{sub}}\rightarrow T_{f,Y}\rightarrow
T_{f,Y}^{\operatorname{quo}}\rightarrow 0$

such that

  1. (i)

$T_{f,X}^{\operatorname{quo}}$ and $T_{f,Y}^{\operatorname{quo}}$ are
unramified as $G_{\mathbb{Q}_{p}}$-modules.

  2. (ii)

The map $T_{f,X}\rightarrow T_{f,Y}$ induces an isomorphism
$T_{f,X}^{\operatorname{sub}}=T_{f,Y}^{\operatorname{sub}}$ of free
${\mathcal{O}}_{{\mathfrak{p}}}$-modules of rank $1$ on which
$G_{\mathbb{Q}_{p}}$ acts through the cyclotomic character.

Note that $T_{f,X}^{\operatorname{quo}}$ and $T_{f,Y}^{\operatorname{quo}}$
are finitely generated ${\mathcal{O}}_{\mathfrak{p}}$-modules such that
$V_{f}^{\operatorname{quo}}:=T_{f,X}^{\operatorname{quo}}\otimes
F_{\mathfrak{p}}\simeq T_{f,Y}^{\operatorname{quo}}\otimes F_{\mathfrak{p}}$
are $1$-dimensional over $F_{\mathfrak{p}}$. In general though
$T_{f,X}^{\operatorname{quo}}$ and $T_{f,Y}^{\operatorname{quo}}$ are not
necessarily free and we let $\Sigma_{X}$ and $\Sigma_{Y}$, respectively, their
torsion submodules.

Let $\alpha_{f}\in{\mathcal{O}}_{\mathfrak{p}}^{\times}$ denote the unit root
of the $p$-th Hecke polynomial of $f$. Let
$\psi_{f}:G_{\mathbb{Q}_{p}}\longrightarrow{\mathcal{O}}_{\mathfrak{p}}^{\times}$
denote the unramified character characterized by
$\psi_{f}(\mathrm{Fr}_{p})=\alpha_{f}$, so that there is an isomorphism of
$F_{\mathfrak{p}}[G_{\mathbb{Q}_{p}}]$-modules

(16) $V_{f}^{\operatorname{quo}}\simeq F_{\mathfrak{p}}(\psi_{f}).$

Given a ${\mathcal{O}}_{\mathfrak{p}}$-module $T$, set
$\bar{T}=T\otimes{\mathcal{O}}/\mathfrak{p}^{t}$. For a
$G_{\mathbb{Q}}$-module $T$ we let $T^{\pm}$ denote the submodule on which
complex conjugation acts as $\pm 1$. As shown by Sharifi in [Sha, Theorem 4.3]
and by Fukaya-Kato in [FK, §6.3.1, §7.1.11], there is an exact sequence of
${\mathcal{O}}/\mathfrak{p}^{t}[G_{\mathbb{Q}}]$-modules

(17)
$0\longrightarrow\bar{T}_{f,X}^{+}\longrightarrow\bar{T}_{f,X}\longrightarrow\bar{T}_{f,X}^{-}\longrightarrow
0,$

Note the switch of signs between (17) and [FK, §6.3.1, §7.1.11], which is due
to the different Tate twist adopted in the definition of $T_{f,X}$. As
explained e.g. in [FKS, §2.5.5] there are isomorphisms of
${\mathcal{O}}/\mathfrak{p}^{t}[G_{\mathbb{Q}_{p}}]$-modules

(18)
$\bar{T}_{f,X}^{+}\simeq\bar{T}_{f,X}^{\operatorname{quo}},\quad\bar{T}_{f,X}^{\operatorname{sub}}\simeq\bar{T}_{f,X}^{-}.$

The first isomorphism is given by the composition of the inclusion in (17)
with the projection map in (15) mod $\mathfrak{p}^{t}$; the second one is
given by the analogous composition obtained by switching the roles of (17) and
(15). In particular $\bar{T}_{f,X}^{-}$ is free of rank $1$ over
${\mathcal{O}}/\mathfrak{p}^{t}$.

As for the lattice associated to the open modular curve, let

(19)
$T_{f,Y,\circ}^{\operatorname{quo}}:=T_{f,Y}^{\operatorname{quo}}/\Sigma_{Y}\simeq{\mathcal{O}}_{\mathfrak{p}}(\psi_{f})$

denote the free quotient. The latter isomorphism follows from (16).

Since $f\equiv E_{2}(\theta,1)\,\mbox{mod}\,\mathfrak{p}^{t}$, we have

(20)
$T_{f,Y,\circ}^{\operatorname{quo}}\otimes\mathcal{O}/\mathfrak{p}^{t}\simeq\mathcal{O}/\mathfrak{p}^{t}(\theta),$

which amounts to the congruence
$\psi_{f}\equiv\theta\,(\mbox{mod}\,\mathfrak{p}^{t})$ as unramified
characters of $G_{\mathbb{Q}_{p}}$.

Note that $\pi_{1}$ factors through $T_{f,Y}^{\operatorname{quo}}$ because the
latter is the maximal unramified $G_{\mathbb{Q}_{p}}$-quotient of $T_{f,Y}$.
Assuming also hypothesis (H2) from the introduction, it follows that $\pi_{1}$
factors further through the natural projection $T_{f,Y}\longrightarrow
T_{f,Y,\circ}^{\operatorname{quo}}$ and may thus be written as the following
composition of ${\mathcal{O}}_{\mathfrak{p}}[G_{\mathbb{Q}_{p}}]$-modules

(21) $\pi_{1}:\quad T_{f,Y}\longrightarrow
T_{f,Y,\circ}^{\operatorname{quo}}\longrightarrow\bar{T}_{f,Y,\circ}^{\operatorname{quo}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}/\mathfrak{p}^{t}(\theta).$

While the first and second arrows are canonical projection maps, the latter
isomorphism is non-canonical and depends on the choice of a unit of
${\mathcal{O}}/\mathfrak{p}^{t}$.

##  3\. First congruence relation

Keep the notations and assumptions fixed in the introduction concerning the
first congruence relation. Here we shall mainly work with the integral Galois
representation associated to the open modular curve, and hence throughout this
section we abbreviate

$T_{f}:=T_{f,Y}.$

We begin by recalling more precisely the definition of Kato classes. Choose
auxiliary Dirichlet characters $\chi_{1}$, $\chi_{2}$ as in the introduction
and set $\xi_{1}=\bar{\chi}_{1}$, $\xi_{2}=\theta\bar{\chi}_{2}$. By [Nek,
(1.2)], together with the fact that
$H_{\operatorname{et}}^{j}(\bar{V},\mathcal{O}_{\mathfrak{p}})$ vanishes for
any smooth affine variety $V$ of dimension $d$ and any $j>d$, the
Hochschild–Serre’s spectral sequence gives rise to an isomorphism

(22) $H_{\operatorname{et}}^{2}(Y_{1}(N),\mathcal{O}_{\mathfrak{p}}(2))\simeq
H^{1}(\mathbb{Q},H_{\operatorname{et}}^{1}(\overline{Y_{1}(N)},\mathcal{O}_{\mathfrak{p}}(2))).$

In view of (11) there is a $G_{\mathbb{Q}}$-equivariant projection

$\pi_{f}:H_{\operatorname{et}}^{1}(\overline{Y}_{1}(N),\mathcal{O}_{\mathfrak{p}}(1))\rightarrow
T_{f},$

which in turn gives rise to a homomorphism

(23)
$\pi_{f*}:H_{\operatorname{et}}^{2}(Y_{1}(N),\mathcal{O}_{\mathfrak{p}}(2))\longrightarrow
H^{1}(\mathbb{Q},T_{f}(1)).$

It thus makes sense to define

$\kappa_{f}:=\pi_{f*}(\delta(u_{\xi_{1},\xi_{2}})\cup\delta(u_{\chi_{1},\chi_{2}}))\in
H^{1}(\mathbb{Q},T_{f}(1)).$

Note that $f$ is ordinary at $p$ because of (3) and label the roots of the
$p$-th Hecke polynomial of $f$ as $\alpha_{f}$, $\beta_{f}$ so that
${\mathrm{ord}}_{p}(\alpha_{f})=0$ and ${\mathrm{ord}}_{p}(\beta_{f})=1$.

Let $\mathcal{W}$ denote the weight space defined as the formal spectrum of
the Iwasawa algebra
$\Lambda=\mathcal{O}_{\mathfrak{p}}[[\mathbb{Z}_{p}^{\times}]]$. The set of
classical points in $\mathcal{W}$ is given by characters $\nu_{s,\xi}$ of the
form $\xi\varepsilon_{\operatorname{cyc}}^{s-1}$ where $\xi$ is a Dirichlet
character of $p$-power conductor and $s$ is an integer; this forms a dense
subset in $\mathcal{W}$ for the Zariski topology. Let $\mathcal{W}^{\circ}$
further denote the set of those points with $\xi=1$; we shall often write $s$
in place of $\nu_{s}=\nu_{s,1}$. Let $\mathcal{W}^{\pm}\subset\mathcal{W}$
denote the topological closure of the set of points
$\xi\varepsilon_{\operatorname{cyc}}^{s-1}$ with $(-1)^{s-1}\xi(-1)=\pm 1$. We
have $\mathcal{W}=\mathcal{W}^{+}\sqcup\mathcal{W}^{-}$ and we write
$\Lambda=\Lambda^{+}\oplus\Lambda^{-}$ for the corresponding decomposition of
the Iwasawa algebra.

Let $\psi$ be an even Dirichlet character and let $L_{p}(f^{*},\psi)$ denote
the Mazur–Tate–Teitelbaum $p$-adic $L$-function associated to $(f^{*},\psi)$.
As discussed in [FK, §4.5], $L_{p}(f^{*},\psi)$ depends on the choice of two
complex periods $(\Omega_{f}^{+},\Omega_{f}^{-})$, which in turn are
determined by the choice of generators $\delta^{\pm}$ of
$V_{f}^{\pm}=T_{f,X}^{\pm}\otimes\mathbb{Q}$ as

(24) $\Omega_{f}^{\pm}=\int_{\delta^{\pm}}\omega_{f}.$

The $p$-adic $L$-function $L_{p}(f^{*},\psi)$ is a rigid-analytic function on
$\mathcal{W}$ characterized by the following formula interpolating classical
$L$-values: let
$\xi\,:\,(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}\longrightarrow\bar{\mathbb{Q}}^{\times}$
be a homomorphism which does not factor through
$(\mathbb{Z}/p^{n-1}\mathbb{Z})^{\times}$. Then

(25)
$L_{p}(f^{*},\psi)(\xi\varepsilon_{\operatorname{cyc}})=\begin{cases}\mathfrak{g}(\bar{\psi})(1-\psi(p)\alpha_{f}^{-1})(1-\theta\bar{\psi}(p)\alpha_{f}^{-1})\frac{L(f^{*},\psi,1)}{\Omega_{f}^{+}}&\mbox{
if }\,\xi=1\\\
\alpha_{f}^{-n}\mathfrak{g}(\bar{\psi}\bar{\xi})\frac{L(f^{*},\psi\xi,1)}{\Omega_{f}^{\pm}}\mbox{
with }\pm=\xi(-1)&\mbox{ if }\,n\geq 1.\end{cases}$

The set of characters of the form $\xi\varepsilon_{\operatorname{cyc}}$ as
$\xi$ ranges through all Dirichlet characters of arbitrary $p$-power conductor
is dense within $\mathcal{W}$ and hence $L_{p}(f^{*},\psi)$ is uniquely
determined by (25). More classically, one can also view $L_{p}(f^{*},\psi)$ as
a one-variable $p$-adic $L$-function by setting
$L_{p}(f^{*},\psi,s)=L_{p}(f^{*},\psi)(\varepsilon_{\operatorname{cyc}}^{s})$.

Define the Euler-like factor

(26)
$\displaystyle\mathcal{E}_{f}=(1-\alpha_{f})(1-\beta_{f})(1-\bar{\theta}\bar{\chi}_{1}\chi_{2}(p)\beta_{f}p^{-1})(1-\chi_{1}\bar{\chi}_{2}(p)\beta_{f}p^{-1}).$

Define also the $p$-adic $L$-value

$\ell=-iN\cdot\frac{\mathfrak{g}(\theta\chi_{1}\bar{\chi}_{2})}{\mathfrak{g}(\theta)\mathfrak{g}(\chi_{1}\bar{\chi}_{2})}\cdot(1-\theta(p))\cdot\zeta_{p}(-1)\cdot
L_{p}(f^{*},\bar{\chi}_{1}\chi_{2},1),$

Thanks to Proposition 2.1, we can define as in the introduction the class

$\bar{\kappa}_{f,1}:=\pi_{1*}(\kappa_{f})=\bar{\pi}_{1*}(\bar{\kappa}_{f})\in
H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1)).$

######  Theorem 3.1.

(First congruence relation) Assuming (H1)-(H2) we have

$\mathcal{E}_{f}\cdot\bar{\kappa}_{f,1}\equiv\ell\cdot\bar{c}_{\theta}\quad\mbox{
in }\,H^{1}(\mathbb{Q}_{p},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1)).$

If $p$ is $\bar{\theta}$-regular as described in (44), this equality takes
place in $H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$.

Theorem 1.1 in the introduction readily follows from the above statement.
Indeed, it follows from (25) that

$L_{p}(f^{*},\bar{\chi}_{1}\chi_{2},1)=(1-\bar{\theta}\bar{\chi}_{1}\chi_{2}(p)\beta_{f}p^{-1})(1-\chi_{1}\bar{\chi}_{2}(p)\beta_{f}p^{-1})\times\mathfrak{g}(\chi_{1}\bar{\chi}_{2})\times\frac{L(f^{*},\bar{\chi}_{1}\chi_{2},1)}{\Omega_{f}^{+}}.$

Since
$\zeta_{p}(-1)=(1-p)\cdot\zeta(-1)=(p-1)\cdot\frac{B_{2}}{2}=\frac{p-1}{12}$,
the Euler factors in the above theorem cancel out in light of (H1) and the
congruence $(\alpha_{f},\beta_{f})\equiv(\theta(p),p)\pmod{\mathfrak{p}^{t}}$.
Then, Theorem 1.1 follows.

The remainder of this section is devoted to the proof of Theorem 3.1.

###  3.1. Mazur’s factorization formula

The aim of this section is to recall a mod $\mathfrak{p}$ factorization
formula for $L_{p}(f^{*},\psi)$ in terms of two Kubota–Leopoldt $p$-adic
$L$-functions. The first result in this direction was proved by Mazur in
[Maz], and further generalizations were provided by Stevens [St1] and
Greenberg–Vatsal [GV, Theorem 3.12]. As explained before, this formula
actually depends on the choice of periods $\Omega_{f}^{\pm}$ and we make this
choice by invoking the work of Sharifi [Sha] and Fukaya-Kato [FK], which gives
rise to a rather precise formula that turns out to be crucial for our
purposes.

Note firstly that with the conventions adopted above, character
$\xi\varepsilon_{\operatorname{cyc}}$ lies in $\mathcal{W}^{+}$ if and only if
$\xi$ is odd, and in that case the period appearing in (25) is
$\Omega_{f}^{-}$. The main result we aim to prove in this section is concerned
with the value of $L_{p}(f^{*},\psi,s)$ at $s=2$ (and $\xi=1$), which again
lies in $\mathcal{W}^{+}$. For this reason it will suffice to work with the
restriction of $L_{p}(f^{*},\psi)$ to $\mathcal{W}^{+}$, and accordingly we
only need to choose period $\Omega_{f}^{-}$.

The work of Sharifi [Sha] and Fukaya-Kato [FK] allows to take a canonical
choice of $\Omega_{f}^{-}$. Namely, recall from (18) that $T_{f,X}^{-}$ is a
free ${\mathcal{O}}_{\mathfrak{p}}$-module of rank $1$. Take any generator
$\delta^{-}$ of $T_{f,X}^{-}$ such that $\delta^{-}$ (mod $\mathfrak{p}^{t}$)
is the basis of $\bar{T}_{f,X}^{-}$ specified in [FK, 6.3.18, 7.1.11, 8.2.4],
which in turn builds on the work of Sharifi [Sha, §4] and Ohta [Oh1], [Oh2].
Define $\Omega_{f}^{-}$ from $\delta^{-}$ as in (24).

It is readily verified that this choice of $\Omega_{f}^{-}$ satisfies the
defining properties imposed by Vatsal in [Va]. Beware however that in loc.
cit. these periods are only well-defined up to $p$-adic units and this is not
enough for our aims in this paper.

With our canonical definition of $\Omega_{f}^{-}$ at hand, we can now define
$\Omega_{f}^{+}$ as in [Va2, Remark 2.7], namely the one satisfying

(27) $\Omega_{f}^{+}\Omega_{f}^{-}=\frac{4\pi^{2}}{\varpi^{r}}\langle
f,f\rangle.$

Given an even Dirichlet character $\chi$, let $L_{p}(\chi)\in\Lambda$ denote
the associated Kubota-Leopoldt $p$-adic $L$-function, which again we may
regard as a one-variable function by setting
$L_{p}(\chi,s)=L_{p}(\chi)(\varepsilon_{\operatorname{cyc}}^{s})$.

######  Proposition 3.2.

The following congruence relation holds in $\Lambda^{+}$:

(28) $L_{p}(f^{*},\psi,s)\equiv 2\cdot L_{p}(\bar{\psi},1-s)\cdot
L_{p}(\bar{\theta}\psi,s-1)\pmod{\mathfrak{p}^{t}}.$

###### Proof.

This follows from [FK, Proposition 8.2.4]. ∎

In particular, since $\psi$ is even, it holds that

(29) $L_{p}(f^{*},\psi,2)=2\cdot L_{p}(\bar{\psi},-1)\cdot
L_{p}(\bar{\theta}\psi,1)\equiv-B_{2}(\bar{\psi})\cdot
L_{p}(\bar{\theta}\psi,1)\pmod{\mathfrak{p}^{t}}.$

Let $C_{f^{*}}\subseteq O_{\mathfrak{p}}$ denote the congruence ideal attached
to $f^{*}$ as defined e.g. in [Oh3]. In Hida’s terminology, the appropriate
power of the uniformizer $\varpi$ generating $C_{f^{*}}$, say $\varpi^{r}$, is
sometimes called a congruence divisor.

###  3.2. Dieudonné modules and congruences among Ohta’s periods

Given a $p$-adic de Rham representation $V$ of $G_{\mathbb{Q}_{p}}$ with
coefficients in $F_{\mathfrak{p}}$, let
$D_{\operatorname{\mathrm{dR}}}(V)=(V\otimes_{\mathbb{Q}_{p}}B_{\operatorname{\mathrm{dR}}})^{G_{\mathbb{Q}_{p}}}$
denote its de Rham Dieudonné module and let $\log_{\operatorname{BK}}$ (resp.
$\exp_{\operatorname{BK}}^{*}$) stand for the Bloch–Kato logarithm (resp. dual
exponential map) attached to $V$ as defined in [BK], [Bel].

In this note we shall work with the Dieudonné modules associated to the
following two basic types of representations. Firstly, if
$\chi:G_{\mathbb{Q}_{p}}\longrightarrow{\mathcal{O}}^{\times}_{\mathfrak{p}}$
is a finite order character and $s$ is an integer, let
$F_{\mathfrak{p}}(\chi)(s)$ denote the $1$-dimensional representation on which
$G_{\mathbb{Q}_{p}}$ acts via $\chi\varepsilon_{\operatorname{cyc}}^{s}$. Then
$D_{\operatorname{\mathrm{dR}}}(F_{\mathfrak{p}}(\chi)(s)$ is again
$1$-dimensional and a canonical generator of
$D_{\operatorname{\mathrm{dR}}}(F_{\mathfrak{p}}(\bar{\chi})(-s))$ is given by
$t^{s}\mathfrak{g}(\chi)^{-1}$, where $t$ is Fontaine’s $p$-adic analogue of
$2\pi i$. Moreover, there is a perfect pairing

$\langle\,,\rangle\,:\,D_{\operatorname{\mathrm{dR}}}(F_{\mathfrak{p}}(\chi)(s))\times
D_{\operatorname{\mathrm{dR}}}(F_{\mathfrak{p}}(\bar{\chi})(-s))\longrightarrow
F_{\mathfrak{p}}$

which gives rise to the isomorphism

(30) $D_{\operatorname{\mathrm{dR}}}(F_{\mathfrak{p}}(\chi)(s))\rightarrow
F_{\mathfrak{p}}\quad c\mapsto\langle
c,\frac{t^{s}}{\mathfrak{g}(\chi)}\rangle.$

If $s\geq 1$ and $\chi\neq 1$ when $s=1$, Bloch-Kato’s logarithm gives rise to
an isomorphism

(31)
$\log_{\operatorname{BK}}:H^{1}(\mathbb{Q}_{p},F_{\mathfrak{p}}(\chi)(s))\rightarrow
D_{\operatorname{\mathrm{dR}}}(F_{\mathfrak{p}}(\chi)(s)).$

Secondly, the de Rham Dieudonné module $D_{\operatorname{\mathrm{dR}}}(V_{f})$
associated to the eigenform $f$ is a $F_{\mathfrak{p}}$-filtered vector space
of dimension $2$. As discussed in e.g. [KLZ, §2.8], Poincaré duality yields a
perfect pairing

$\langle\,,\,\rangle:D_{\operatorname{\mathrm{dR}}}(V_{f}(-1))\times
D_{\operatorname{\mathrm{dR}}}(V_{f^{*}})\rightarrow F_{\mathfrak{p}}$

and (15) induces an exact sequence of Dieudonné modules

(32) $0\rightarrow
D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{sub}})\rightarrow
D_{\operatorname{\mathrm{dR}}}(V_{f})\rightarrow
D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{quo}})\rightarrow 0,$

where $D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{sub}})$ and
$D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{quo}})$ have both
dimension $1$.

Falting’s theorem associates to $f$ a regular differential form
$\omega_{f}\in\mathrm{Fil}(D_{\operatorname{\mathrm{dR}}}(V_{f}))$, which
gives rise to an element in
$D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{quo}})$ via the right-
most map in (32) and in turn induces a linear form

(33)
$\omega_{f}\,:\,D_{\operatorname{\mathrm{dR}}}(V_{f^{*}}^{\operatorname{sub}}(-1))\rightarrow
F_{\mathfrak{p}},\quad\eta\mapsto\langle\omega_{f},\eta\rangle$

that we continue to denote with same symbol by a slight abuse of notation.

There is also a differential $\eta_{f}$, which is characterized by the
property that it spans the line
$D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{sub}}(-1))$ and is
normalized so that

(34) $\langle\eta_{f},\omega_{f^{*}}\rangle=1.$

Again, it induces a linear form

(35)
$\eta_{f}\,:\,D_{\operatorname{\mathrm{dR}}}(V_{f^{*}}^{\operatorname{quo}})\rightarrow
F_{\mathfrak{p}},\quad\omega\mapsto\langle\eta_{f},\omega\rangle$

We turn now to the more delicate $p$-adic Hodge theory of integral Galois
representations. Let $T$ be an unramified
${\mathcal{O}}_{\mathfrak{p}}[\mathrm{Gal\,}(\bar{\mathbb{Q}}_{p}/\mathbb{Q}_{p})]$-module
and set $V=T\otimes F_{\mathfrak{p}}$. Let
$\hat{\mathbb{Z}}_{p}^{\mathrm{ur}}$ denote the completion of the ring of
integers of the maximal unramified extension of $\mathbb{Q}_{p}$, and define
as in e.g. [Oh2, Theorem 2.1.11] the integral Dieudonné module

$D(T):=(T\,\hat{\otimes}_{\mathbb{Z}_{p}}\,\hat{\mathbb{Z}}_{p}^{\mathrm{ur}})^{\mathrm{Fr}_{p}=1}.$

As shown in loc. cit. we have $D_{\operatorname{\mathrm{dR}}}(V)=D(T)\otimes
F_{\mathfrak{p}}$.

As explained e.g. in [FK, Prop. 1.7.6], there is a functorial isomorphism of
${\mathcal{O}}_{\mathfrak{p}}$-modules (forgetting the Galois structure) given
by

(36) $T\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}D(T).$

This map is not canonical as it depends on a choice of root of unity; for
$T={\mathcal{O}}_{\mathfrak{p}}(\chi)$ we take it to be given by the rule
$1\mapsto\,\mathfrak{g}(\chi)$.

Recall from (19) the free ${\mathcal{O}}_{\mathfrak{p}}$-quotient

$T_{f,\circ}^{\operatorname{quo}}\simeq{\mathcal{O}}_{\mathfrak{p}}(\psi_{f});$

note in particular that $T_{f,\circ}^{\operatorname{quo}}$ is unramified.

According to [KLZ, §10.1.2], the image of
$\eta_{f^{*}|D(T_{f}^{\operatorname{quo}})}$ is precisely the inverse
$C_{f^{*}}^{-1}$ of the congruence ideal introduced above, and hence there is
an isomorphism

$\eta_{f^{*}}\,:\,D(T_{f,\circ}^{\operatorname{quo}})\longrightarrow
C_{f^{*}}^{-1},\quad\omega\mapsto\langle\eta_{f^{*}},\omega\rangle.$

Writing $C_{f}=(\varpi^{r})$ for the appropriate power $r\geq 1$ and setting
$\tilde{\eta}_{f^{*}}:=\varpi^{r}\cdot\eta_{f^{*}}$, the above map gives rise
to an isomorphism of $O_{\mathfrak{p}}$-modules

(37)
$\tilde{\eta}_{f^{*}}\,:\,D(T_{f,\circ}^{\operatorname{quo}})\longrightarrow\mathcal{O}_{\mathfrak{p}},\quad\omega\mapsto\langle\tilde{\eta}_{f^{*}},\omega\rangle.$

The map $\pi_{1}$ appearing in Proposition 2.1

$\pi_{1}\,:\,T_{f}\longrightarrow\mathcal{O}/\mathfrak{p}^{t}(\theta)$

is only well-defined up to units in $\mathcal{O}/\mathfrak{p}^{t}$. We
rigidify it by invoking diagram (21), which tells us that $\pi_{1}$ is fixed
once we take a choice of an isomorphism of $G_{\mathbb{Q}_{p}}$-modules

(38)
$\bar{\iota}:\bar{T}_{f,\circ}^{\operatorname{quo}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}/\mathfrak{p}^{t}(\theta).$

Fixing such a map amounts to choosing the class
$(\mathrm{mod}\,{\mathfrak{p}^{t}})$ of an isomorphism of local modules
$\iota:T_{f,\circ}^{\operatorname{quo}}\simeq\mathcal{O}_{\mathfrak{p}}(\psi_{f})$.
In light of the functoriality provided by (36) this determines and is
determined by the class $(\mathrm{mod}\,{\mathfrak{p}^{t}})$ of an isomorphism
$D(\iota):D(T_{f,\circ}^{\operatorname{quo}})\simeq
D({\mathcal{O}}_{\mathfrak{p}}(\psi_{f}))$.

We choose $\iota$ as the single isomorphism making the following diagram
commutative:

(39)
$\textstyle{D(T_{f,\circ}^{\operatorname{quo}})\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\langle\,,\tilde{\eta}_{f^{*}}\rangle}$$\scriptstyle{D(\iota)}$$\textstyle{\quad\mathcal{O}_{\mathfrak{p}}}$$\textstyle{D({\mathcal{O}}_{\mathfrak{p}}(\psi_{f}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
1/\mathfrak{g}(\theta)}$

Indeed, since both $\langle\,,\tilde{\eta}_{f^{*}}\rangle$ and $\cdot
1/\mathfrak{g}(\theta)$ are isomorphisms, it follows that such a map
$D(\iota)$ exists and is unique, and this in turn pins down $\iota$ and
$\bar{\iota}$ in light of (36).

###  3.3. Coleman’s power series and the Kubota-Leopoldt $p$-adic $L$-function

Let

$\underline{\varepsilon}_{\operatorname{cyc}}:G_{\mathbb{Q}}\rightarrow\Lambda^{\times}$

denote the $\Lambda$-adic cyclotomic character which sends a Galois element
$\sigma$ to the group-like element
$[\varepsilon_{\operatorname{cyc}}(\sigma)]$. It interpolates the powers of
the $\mathbb{Z}_{p}$-cyclotomic character, in the sense that for any
arithmetic point $\nu_{s,\xi}\in\mathcal{W}^{\mathrm{cl}}$,

(40)
$\nu_{s,\xi}\circ\underline{\varepsilon}_{\operatorname{cyc}}=\xi\cdot\varepsilon_{\operatorname{cyc}}^{s-1}.$

The following result follows from the general theory of Perrin-Riou maps (see
for instance [KLZ, §8]).

######  Proposition 3.3.

There exists a morphism of $\Lambda$-modules

$\mathcal{L}_{\chi}:H^{1}(\mathbb{Q}_{p},{\mathcal{O}}_{\mathfrak{p}}(\chi)\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))\rightarrow\Lambda$

satisfying that for all integers $r$, the specialization of
$\mathcal{L}_{\chi}$ at $s\in\mathcal{W}^{\circ}$ is the homomorphism

$\mathcal{L}_{\chi,s}:H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\chi)(s))\rightarrow\mathcal{O}_{\mathfrak{p}}$

given by

$\mathcal{L}_{\chi,s}=\frac{1-\bar{\chi}(p)p^{-s}}{1-\chi(p)p^{s-1}}\cdot\begin{cases}\frac{(-1)^{s}}{(s-1)!}\cdot\langle\log_{\operatorname{BK}},\frac{t^{s}}{\mathfrak{g}(\chi)}\rangle&\text{
if }s\geq 1\\\
(-s)!\cdot\langle\exp_{\operatorname{BK}}^{*},\frac{t^{s}}{\mathfrak{g}(\chi)}\rangle&\text{
if }s<1,\end{cases}$

As a piece of notation, and for any $p$-adic representation $V$, we write
$H_{\operatorname{f}}^{1}(\mathbb{Q},V)$ for the finite Bloch–Kato Selmer
group, which is the subspace of $H^{1}(\mathbb{Q},V)$ which consists on those
classes which are crystalline at $p$ and unramified at $\ell\neq p$.

The following result is a reformulation of Coleman and Perrin-Riou’s
reciprocity law ([Co], [PR]), with the normalizations used for instance in
[Eu].

######  Proposition 3.4.

There exists a $\Lambda$-adic cohomology class

$\kappa_{\chi,\infty}\in
H^{1}(\mathbb{Q},\mathcal{O}_{\mathfrak{p}}(\chi)\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))$

such that:

  1. (a)

Its image under restriction at $p$ followed by the Perrin–Riou regulator gives
the Kubota–Leopoldt $p$-adic $L$-function:

$\mathcal{L}_{\chi}(\mathrm{res}_{p}(\kappa_{\chi,\infty}))=L_{p}(\bar{\chi}).$

  2. (b)

The bottom layer $\kappa_{\chi}(1):=\nu_{1}(\kappa_{\chi,\infty})$ lies in
$H_{\operatorname{f}}^{1}(\mathbb{Q},\mathcal{O}_{\mathfrak{p}}(\chi)(1))$ and
satisfies

$\kappa_{\chi}(1)=(1-\chi(p))\cdot c_{\chi}.$

###  3.4. Kato’s explicit reciprocity law

######  Proposition 3.5.

There exists a homomorphism of $\Lambda$-modules

$\mathcal{L}_{f}^{-}:H^{1}(\mathbb{Q}_{p},T_{f,\circ}^{\operatorname{quo}}\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))\rightarrow\Lambda$

satisfying the following interpolation property: for
$s\in\mathcal{W}^{\circ}$, the specialization of $\mathcal{L}_{f}^{-}$ at $s$
is the homomorphism

$\mathcal{L}_{f,s}^{-}:H^{1}(\mathbb{Q}_{p},T_{f,\circ}^{\operatorname{quo}}(s))\longrightarrow\mathcal{O}_{\mathfrak{p}}$

given by

$\mathcal{L}_{f,s}^{-}=\frac{1-\bar{\theta}(p)\beta_{f}p^{-s-1}}{1-\theta(p)\beta_{f}^{-1}p^{s}}\times\begin{cases}\frac{(-1)^{s}}{(s-1)!}\times\langle\log_{\operatorname{BK}},t^{s}\tilde{\eta}_{f^{*}}\rangle&\text{
if }s\geq 1\\\
(-s)!\times\langle\exp_{\operatorname{BK}}^{*},t^{s}\tilde{\eta}_{f^{*}}\rangle&\text{
if }s<1,\end{cases}$

where $\log_{\operatorname{BK}}$ is the Bloch-Kato logarithm and
$\exp_{\operatorname{BK}}^{*}$, the dual exponential map.

###### Proof.

This follows from Coleman and Perrin-Riou’s theory of $\Lambda$-adic logarithm
maps as extended by Loeffler and Zerbes in [LZ]. This is recalled for instance
in [KLZ, §8,9]. More precisely, [KLZ, Theorem 8.2.3] and, more particularly,
the second displayed equation in [KLZ, p. 82] yields an injective map

$H^{1}(\mathbb{Q}_{p},T_{f,\circ}^{\operatorname{quo}}\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))\longrightarrow
D(T_{f}^{\operatorname{quo}})\otimes\Lambda,$

since $H^{0}(\mathbb{Q}_{p},T_{f,\circ}^{\operatorname{quo}}(1))=0$ because of
the assumption that $\alpha_{f}\equiv\theta(p)\neq 1$ modulo $\mathfrak{p}$.

This map is characterized by the interpolation property formulated in [LZ,
Appendix B]. Next we apply the pairing of (37) and the result follows. ∎

######  Theorem 3.6.

There exists a $\Lambda$-adic cohomology class

$\kappa_{f,\infty}\in
H^{1}(\mathbb{Q},T_{f}\otimes\Lambda^{-}(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))$

such that:

  1. (a)

There is an explicit reciprocity law

$\mathcal{L}_{f}^{-}(\mathrm{res}_{p}(\kappa_{f,\infty})^{-})=\frac{N\mathfrak{g}(\theta\chi_{1}\bar{\chi}_{2})\cdot
L_{p}(f^{*},\bar{\chi}_{1}\chi_{2},1)}{2i\mathfrak{g}(\chi_{1}\bar{\chi}_{2})}\times
L_{p}(f^{*},1+s),$

where $\mathrm{res}_{p}$ stands for the map corresponding to localization at
$p$ and $\mathrm{res}_{p}(\kappa_{f,\infty})^{-}$ is the map induced in
cohomology from the projection map $T_{f}\rightarrow
T_{f,\circ}^{\operatorname{quo}}$ of (15).

  2. (b)

The bottom layer $\kappa_{f}(1)$ lies in
$H_{\operatorname{f}}^{1}(\mathbb{Q},T_{f}(1))$ and satisfies

$\kappa_{f}(1)=\mathcal{E}_{f}\cdot\kappa_{f},$

where $\mathcal{E}_{f}$ is the Euler factor introduced in (26).

###### Proof.

This is due to Kato [Ka] and has been reported in many other places in the
literature. See [Och] and, more specifically, [BD2, Theorems 4.4 and 5.1]
combined with Besser’s [Bes, Proposition 9.11 and Corollary 9.10] showing that
the $p$-adic regulator can be recast as the composition of the $p$-adic étale
regulator followed by the Bloch–Kato logarithm.

Note however that the normalizations adopted in loc. cit. are slightly
different from ours. More precisely, in [BD2] employ the functional $\eta_{f}$
instead of $\tilde{\eta}_{f}$, but note that this discrepancy is compensated
by our choice of periods $\Omega_{f}^{+}$ and $\Omega_{f}^{-}$, which we have
normalized according to (27). Taking this into account, the theorem holds as
stated. ∎

Recall our running assumption that $f\equiv
E_{2}(\theta,1)\,\pmod{\mathfrak{p}^{t}}$.

######  Corollary 3.7.

The following equality holds in $\Lambda^{-}/\mathfrak{p}^{t}\Lambda^{-}$:

$\mathcal{L}_{f}^{-}(\mathrm{res}_{p}\,\kappa_{f,\infty}^{-})\equiv(-i)N\frac{\mathfrak{g}(\theta\chi_{1}\bar{\chi}_{2})}{\mathfrak{g}(\chi_{1}\bar{\chi}_{2})}\zeta_{p}(-1)\cdot
L_{p}(f^{*},\bar{\chi}_{1}\chi_{2},1)\cdot\mathcal{L}_{\theta}(\mathrm{res}_{p}\,\kappa_{\theta,\infty})\pmod{\mathfrak{p}^{t}}.$

###### Proof.

By Proposition 3.2,

$L_{p}(f^{*},1+s)\equiv 2\cdot\zeta_{p}(-s)\cdot
L_{p}(\bar{\theta},s)\pmod{\mathfrak{p}^{t}}.$

Applying now part (a) of Theorem 3.6 and Proposition 3.4 to the left and right
hand sides respectively, the result follows. ∎

###  3.5. Proof of Theorem 3.1

We now prove Theorem 3.1. Note that since $\theta_{|G_{\mathbb{Q}_{p}}}$ is a
non-trivial unramified character, it follows from e.g. [Bel, §2.2] that

$H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1))=H_{\operatorname{f}}^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1)).$

We have seen in Proposition 3.3 that there is a homomorphism

(41)
$\mathcal{L}_{\theta,1}:H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1))\rightarrow
L_{\mathfrak{p}}.$

######  Lemma 3.8.

The map $\mathcal{L}_{\theta,1}$ induces an isomorphism

$\mathcal{L}_{\theta,1}:H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1))\rightarrow\mathcal{O}_{\mathfrak{p}}.$

###### Proof.

According to Proposition 3.3, the map (41) is given by

$\mathcal{L}_{\theta,1}=\frac{\bar{\theta}(p)p^{-1}-1}{1-\theta(p)}\cdot\langle\log_{\operatorname{BK}},\frac{t}{\mathfrak{g}(\theta)}\rangle.$

Given a place $v$ of $\mathbb{Q}(\mu_{N})$ above $p$, let
$\mathbb{Z}[\mu_{N}]_{v}$ denote the completion of $\mathbb{Z}[\mu_{N}]$ at
$v$. Define the module of local units
$U_{p}(N)=\prod_{v|p}\mathbb{Z}[\mu_{N}]_{v}^{\times}$, where
$v=v_{1},\ldots,v_{r}$ ranges over all places of $\mathbb{Q}(\mu_{N})$ above
$p$. Note that $G=\mathrm{Gal\,}(\mathbb{Q}(\mu_{N})/\mathbb{Q})$ acts on
$U_{p}(N)$ by permuting the places $v$, and hence it makes sense to pick the
eigen-component of $U_{p}(N)$ with respect to a character of $G$. In
particular, we have

$U_{p}(N)[\bar{\theta}]:=(U_{p}(N)\otimes\mathcal{O}_{\mathfrak{p}}(\theta))^{G}.$

Kummer theory identifies
$H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1))$ with
$U_{p}(N)[\bar{\theta}]$, which is a $\mathcal{O}_{\mathfrak{p}}$-module of
rank one.

Since $\mathbb{Q}(\mu_{N})_{v}$ is an unramified extension of
$\mathbb{Q}_{p}$, the maximal ideal of $\mathbb{Z}[\mu_{N}]_{v}$ is
$p\mathbb{Z}[\mu_{N}]_{v}$ and the logarithm defines an isomorphism, as
recalled for instance in [Con, §8]

(42)
$\log_{v}\,:\,\mathbb{Z}[\mu_{N}]_{v}^{\times}\otimes\mathcal{O}_{\mathfrak{p}}\longrightarrow
p\mathbb{Z}[\mu_{N}]_{v}\otimes\mathcal{O}_{\mathfrak{p}}.$

Note that $\prod_{v}\mathbb{Z}[\mu_{N}]_{v}$ is naturally a $G$-module
isomorphic to the regular representation and hence
$(\prod_{v}\mathbb{Z}[\mu_{N}]_{v})[\bar{\theta}]$ is again a free module of
rank $1$ over $\mathcal{O}_{\mathfrak{p}}$. Define

$\log_{\bar{\theta}}:=\sum_{\sigma\in
G}\theta(\sigma)\log_{\sigma(v_{1})}\,:\,U_{p}(N)[\bar{\theta}]\longrightarrow
p(\prod_{v}\mathbb{Z}[\mu_{N}]_{v})[\bar{\theta}].$

A natural generator of the target may be taken to be the Gauss sum
$\mathfrak{g}(\theta)$ diagonally embedded in
$\prod_{v}\mathbb{Z}[\mu_{N}]_{v}$ and this yields an identification
$\frac{1}{\mathfrak{g}(\theta)}(\prod_{v}\mathbb{Z}[\mu_{N}]_{v})[\bar{\theta}]=\mathcal{O}_{\mathfrak{p}}$.
Under these identifications, Bloch-Kato’s logarithm may be recast classically
as

$\langle\log_{\operatorname{BK}},\frac{t}{\mathfrak{g}(\theta)}\rangle=\frac{1}{\mathfrak{g}(\theta)}\log_{\bar{\theta}}:H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1))=U_{p}(N)[\bar{\theta}]\longrightarrow\mathcal{O}_{\mathfrak{p}},$

and we already argued that this yields an isomorphism onto
$p\mathcal{O}_{\mathfrak{p}}$.

Since ${\mathrm{ord}}_{p}(\frac{\bar{\theta}(p)p^{-1}-1}{1-\theta(p)})=-1$, it
follows that $\mathcal{L}_{\theta,1}$ is an isomorphism onto
$\mathcal{O}_{\mathfrak{p}}$, as claimed. ∎

Recall from Proposition 3.5 the map

$\mathcal{L}_{f,1}^{-}:H^{1}(\mathbb{Q}_{p},T_{f,\circ}^{\operatorname{quo}}(1))\,\rightarrow
D(T_{f,\circ}^{\operatorname{quo}}(1))\,\stackrel{{\scriptstyle\cdot
t\tilde{\eta}_{f^{*}}}}{{\rightarrow}}\,\mathcal{O}_{\mathfrak{p}}.$

Recall the isomorphism
$T_{f}^{\operatorname{quo}}\otimes\mathcal{O}/\mathfrak{p}^{t}\simeq\mathcal{O}/\mathfrak{p}^{t}(\theta)$
of (38) fixed as in (39) above and use it to identify the source of
$\mathcal{L}_{f,1}^{-}\otimes{\mathcal{O}}/\mathfrak{p}^{t}$ with
$H^{1}(\mathbb{Q}_{p},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$.

######  Lemma 3.9.

As homomorphisms
$H^{1}(\mathbb{Q}_{p},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))\longrightarrow{\mathcal{O}}/\mathfrak{p}^{t}$
we have the congruence

$\mathcal{L}_{f,1}^{-}\equiv\mathcal{L}_{\theta,1}\pmod{\mathfrak{p}^{t}}.$

###### Proof.

This follows by comparing the maps $\mathcal{L}_{\theta,1}$ and
$\mathcal{L}_{f,1}^{-}$ described respectively in Proposition 3.3 and 3.5.
Note firstly that the Euler factors involved in the latter agree modulo
$\mathfrak{p}^{t}$ with those of the former, since
$\alpha_{f}\beta_{f}=\theta(p)p$.

Next, observe that in Proposition 3.3 the pairing takes place against
$t\mathfrak{g}(\chi)^{-1}$, while in Proposition 3.5 this pairing is with
$t\tilde{\eta}_{f^{*}}$. The lemma follows from the commutativity of the
diagram (39). ∎

We are finally in position to provide the proof of Theorem 3.1: After
specializing Corollary 3.7 at $s=1$ we obtain

$\mathcal{L}_{f,1}^{-}(\mathrm{res}_{p}\,\kappa_{f}(1)^{-})\equiv-
iN\frac{\mathfrak{g}(\theta\chi_{1}\bar{\chi}_{2})}{\mathfrak{g}(\chi_{1}\bar{\chi}_{2})}\zeta_{p}(-1)\cdot
L_{p}(f^{*},\bar{\chi}_{1}\chi_{2},1)\cdot\mathcal{L}_{\theta,1}(\mathrm{res}_{p}\,\kappa_{\theta}(1))\pmod{\mathfrak{p}^{t}}$

Recall that Proposition 3.4 and Theorem 3.6 assert that

$\kappa_{\theta}(1)=(1-\theta(p))\cdot
c_{\theta},\quad\kappa_{f}(1)=\mathcal{E}_{f}\cdot\kappa_{f}$

and hence

$\mathcal{E}_{f}\mathcal{L}_{f,1}^{-}(\mathrm{res}_{p}\,\kappa_{f}^{-})\equiv-
iN\frac{\mathfrak{g}(\theta\chi_{1}\bar{\chi}_{2})}{\mathfrak{g}(\chi_{1}\bar{\chi}_{2})}\zeta_{p}(-1)(1-\theta(p))\cdot
L_{p}(f^{*},\bar{\chi}_{1}\chi_{2},1)\cdot\mathcal{L}_{\theta,1}(\mathrm{res}_{p}\,c_{\theta})\pmod{\mathfrak{p}^{t}}$

Recall we have set

$\ell=-iN\cdot\frac{\mathfrak{g}(\theta\chi_{1}\bar{\chi}_{2})}{\mathfrak{g}(\theta)\mathfrak{g}(\chi_{1}\bar{\chi}_{2})}\cdot(1-\theta(p))\cdot\zeta_{p}(-1)\cdot
L_{p}(f^{*},\bar{\chi}_{1}\chi_{2},1).$

Using Lemma 3.9 together with Lemma 3.8, we deduce the equality of local
classes

(43)
$\mathcal{E}_{f}\cdot\mathrm{res}_{p}\,\kappa_{f}^{-}\equiv\ell\cdot\mathrm{res}_{p}\,c_{\theta}\,\pmod{\mathfrak{p}^{t}}$

in $H^{1}(\mathbb{Q}_{p},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$. Observe
that $\mathrm{res}_{p}(\kappa_{f})^{-}$ is the local class obtained in
cohomology by push-forward under the map induced by the projection
$\bar{T}_{f}\rightarrow\bar{T}_{f,\circ}^{\operatorname{quo}}$ of (15), as
already introduced in Theorem 3.6. This corresponds, modulo
$\mathfrak{p}^{t}$, to what we have called $\bar{\kappa}_{f,1}$. The first
(local) part of Theorem 3.1 follows.

In order to state more precisely and prove the second (global) assertion of
Theorem 3.1, let us introduce some notation. Let
$k=\mathbb{Q}(\mu_{N})^{+}=\mathbb{Q}(\zeta_{N}+\zeta_{N}^{-1})$ denote the
maximal totally real subfield of $\mathbb{Q}(\mu_{N})$ and set
$d_{N}=[k:\mathbb{Q}]$. Let $\mathrm{Cl}(k)$ denote its class group. As in (5)
let $\mathrm{Cl}(k)[\theta]$ denote its $\theta$-eigencomponent. It follows333
[Gr, Théorème I2] applies because Leopoldt’s conjecture is known for $(k,p)$
by the work of Brumer, primes in $k$ above $p$ are totally ramified in
$k(\mu_{p})$ and therefore the $\omega$-component of the
$\mathrm{Gal\,}(k(\mu_{p})/k)$-submodule of $\mathrm{Cl}(k(\mu_{p}))$
generated by ideals above $p$ is trivial. [Gr, Théorème I2] thus asserts that
$\mathrm{rank}_{\mathbb{Z}/p\mathbb{Z}}\,\mathrm{Cl}(k(\mu_{p}))[\bar{\theta}\omega]\otimes\mathbb{Z}/p\mathbb{Z}$
is equal to the rank of the $\bar{\theta}$-component of the $p$-torsion of the
Galois group $\mathrm{Gal\,}(H_{p}/k)$ of the maximal $p$-abelian extension of
$k$ unramified away from $p$. Hence $(\mathrm{Gr})$ follows because the
Hilbert class field $H/k$ is contained in $H_{p}$ and
$\mathrm{Gal\,}(H/k)=\mathrm{Cl}(k)$. from the work of G. Gras [Gr, Théorème
I2] that

(44)
$\mathrm{rank}_{\mathbb{Z}/p\mathbb{Z}}\,\mathrm{Cl}(k)[\bar{\theta}]\otimes\mathbb{Z}/p\mathbb{Z}\leq\mathrm{rank}_{\mathbb{Z}/p\mathbb{Z}}\,\mathrm{Cl}(k(\mu_{p}))[\bar{\theta}\omega]\otimes\mathbb{Z}/p\mathbb{Z},$

where
$\omega:(\mathbb{Z}/p\mathbb{Z})^{\times}\rightarrow\bar{\mathbb{Q}}^{\times}$
is the Teichmüller character. This inequality may be regarded as an instance
of Leopoldt’s spiegelungssatz.

Next lemma is conditional on the assumption, that we call
$\bar{\theta}$-regularity, that (44) is an equality.

To place in context this hypothesis, let $R_{p}(k)$ denote the $p$-adic
regulator of $k$. As explained in e.g. [Gr2, Def. 2.3], one always has
${\mathrm{ord}}_{p}R_{p}(k)\geq d_{N}-1$. It is shown in loc. cit. that
$(\mathrm{Gr})$ is an equality for all non-trivial even Dirichlet characters
of conductor $N$ if and only if ${\mathrm{ord}}_{p}R_{p}(k)=d_{N}-1$. We refer
to [Gr2, §7.3] for conjectures predicting that such an equality is expected to
hold for all primes $p$ away from a set of density $0$.

######  Lemma 3.10.

Assuming $p$ is $\bar{\theta}$-regular, the global-to-local restriction map

$H_{\operatorname{f}}^{1}(\mathbb{Q},\mathcal{O}_{\mathfrak{p}}(\theta)(1))\rightarrow
H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1))$

is an isomorphism.

###### Proof.

Recall from the proof of Lemma 3.8 the definition of the group $U_{p}(N)$ of
local units. Consider the following commutative diagram, where vertical arrows
are isomorphisms induced from Kummer theory and the upper horizontal arrow
stands for the map corresponding to localization at $p$:

$\textstyle{H_{\operatorname{f}}^{1}(\mathbb{Q},\mathcal{O}_{\mathfrak{p}}(\theta)(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q}_{p},\mathcal{O}_{\mathfrak{p}}(\theta)(1))}$$\textstyle{\mathbb{Z}[\mu_{N}]^{\times}[\bar{\theta}]\otimes\mathcal{O}_{\mathfrak{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{p}(N)[\bar{\theta}]\otimes\mathcal{O}_{\mathfrak{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The bottom horizontal arrow is injective because it is induced by the natural
inclusion $\mathbb{Z}[\mu_{N}]^{\times}\hookrightarrow U_{p}(N)$. Moreover,
since $\theta$ is even and nontrivial, both
$\mathbb{Z}[\mu_{N}]^{\times}[\bar{\theta}]\otimes\mathcal{O}_{\mathfrak{p}}$
and $U_{p}(N)[\bar{\theta}]$ are $\mathcal{O}_{\mathfrak{p}}$-modules of rank
$1$. The cokernel

$Q[\bar{\theta}]=U_{p}(N)[\bar{\theta}]\otimes\mathcal{O}_{\mathfrak{p}}/\mathbb{Z}[\mu_{N}]^{\times}[\bar{\theta}]\otimes\mathcal{O}_{\mathfrak{p}}$

is thus a finite group.

In order to prove the lemma it thus suffices to show that $Q[\bar{\theta}]$ is
trivial. Write $k=\mathbb{Q}(\mu_{N})^{+}$ (resp. $\mathbb{Z}[\mu_{N}]^{+}$)
for the maximal totally real subfield of $\mathbb{Q}(\mu_{N}$) (resp. its ring
of integers). Let $U_{p}^{1}(N)=\prod_{v|p}(\mathbb{Z}[\mu_{N}]_{v}^{+})^{1}$,
where $(\mathbb{Z}[\mu_{N}]_{v}^{+})^{1}$ stands for the set of local units in
$\mathbb{Z}_{p}[\mu_{N}]_{v}^{+}$ which are congruent to 1 modulo $v$. Let
$U^{+}(N)$ denote the closure of the set of units of $\mathbb{Z}[\mu_{N}]^{+}$
congruent to 1 modulo each place above $p$, diagonally embedded in
$U_{p}^{1}(N)$. Note that $Q[\bar{\theta}]=U_{p}^{1}(N)/U^{+}(N)$.

According to [Neu, Chapter 4, Theorem 7.8],
$Q[\bar{\theta}]\simeq\mathrm{Gal\,}(H_{p}/H)[\bar{\theta}]$, where $H_{p}$
(resp. $H$) is the maximal $p$-abelian extension of $k$ unramified away from
primes above $p$ (resp. everywhere unramified). Here the
$\theta$-eigencomponent on the Galois group is taken with respect to the
natural action of $\mathrm{Gal\,}(k/\mathbb{Q})$ by conjugation on
$\mathrm{Gal\,}(H_{p}/H)$. The lemma hence follows from the running
hypothesis. ∎

Assuming $\bar{\theta}$-regularity, Lemma 3.10 allows us to upgrade (43) to an
equality of global classes in
$H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$, namely

$\mathcal{E}_{f}\cdot\kappa_{f,1}\equiv\ell\cdot
c_{\theta}\pmod{\mathfrak{p}^{t}}.$

Theorem 3.1 follows.

##  4\. Second congruence relation

As in the introduction, let $N>1$ be a positive integer and
$\theta:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\bar{\mathbb{Q}}^{\times}$
an even Dirichlet character. Let $f\in S_{2}(N,\theta)$ a normalized cuspidal
eigenform of level $N$, weight $2$ and nebentype $\theta$. Fix a prime $p\nmid
6N\varphi(N)$ and assume as in (3) that $f\equiv
E_{2}(\theta,1)\,\,\mathrm{mod}\,\mathfrak{p}^{t}$ for some $t\geq 1$. This
implies that $L_{p}(\bar{\theta},-1)\equiv 0\pmod{{\mathfrak{p}}^{t}}$.

We keep the notations introduced along §2 and §3. Recall that the value of the
modular unit $u_{\chi_{1},\chi_{2}}$ at $\infty$ is some power of the circular
unit $c_{\chi_{1}}$, and likewise for $u_{\xi_{1},\xi_{2}}$. For the sake of
concreteness, in the statement below we normalize them so that
$u_{\chi_{1},\chi_{2}}(\infty)=c_{\chi_{1}}$ and
$u_{\xi_{1},\xi_{2}}(\infty)=c_{\xi_{1}}$ although any other normalization
would work.

######  Proposition 4.1.

The Beilinson–Kato class $\kappa_{f}\in H^{1}(\mathbb{Q},T_{f,Y}(1))$ may be
lifted to a class in $H^{1}(\mathbb{Q},T_{f,X}(1))$ if and only if
$\bar{\kappa}_{f,1}=0$.

###### Proof.

Recall from Proposition 2.1 the short exact sequence of
$G_{\mathbb{Q}}$-modules

(45) $0\longrightarrow T_{f,X}\longrightarrow
T_{f,Y}\longrightarrow\mathcal{O}/\mathfrak{p}^{t}(\theta)\longrightarrow 0.$

After taking a Tate twist, it gives rise to the long exact sequence in
cohomology

(46) $0\longrightarrow H^{1}(\mathbb{Z}[1/Np],T_{f,X}(1))\longrightarrow
H^{1}(\mathbb{Z}[1/Np],T_{f,Y}(1))\longrightarrow
H^{1}(\mathbb{Z}[1/Np],\mathcal{O}/\mathfrak{p}^{t}(\theta)(1))$

The last map sends $\kappa_{f}$ to $\bar{\kappa}_{f,1}$. Hence the latter
vanishes if and only if $\kappa_{f}$ belongs to
$H^{1}(\mathbb{Z}[1/Np],T_{f,X}(1))$. ∎

Assume for the remainder of this section that $\bar{\kappa}_{f,1}=0$ and hence
$\kappa_{f}\in H^{1}(\mathbb{Q},T_{f,Y}(1))$ lifts to a class in
$H^{1}(\mathbb{Q},T_{f,X}(1))$, that by a slight abuse of notation we continue
to denote with the same symbol.

As in the introduction, we may define the global class

$\bar{\kappa}_{f,2}=\bar{\pi}_{2*}(\bar{\kappa}_{f})\in
H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2)).$

######  Theorem 4.2.

(Second congruence relation) Assume $\theta$ is primitive and
$L_{p}^{\prime}(\bar{\theta},-1)\not\equiv 0\pmod{{\mathfrak{p}}}$. Then the
following equality holds in
$H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2))$:

$\bar{\kappa}_{f,2}=\frac{L_{p}^{\prime}(\bar{\theta},-1)}{1-p^{-1}}\cdot\frac{\bar{c}_{\xi_{1}}\cup\bar{c}_{\chi_{1}}}{\cup\log_{p}(\varepsilon_{\operatorname{cyc}})}\pmod{\mathfrak{p}^{t}}.$

Here, $1/\cup\log_{p}(\varepsilon_{\operatorname{cyc}})$ denotes the inverse
of the map

$H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2))\rightarrow
H^{2}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2)),\quad\kappa\mapsto\kappa\cup\log_{p}(\varepsilon_{\operatorname{cyc}}),$

which is invertible under our running assumptions.

###  4.1. Cohomology and Eisenstein quotients

For any $r\geq 1$ and $j\in\mathbb{Z}$ let

$H_{r}(j)=H_{\operatorname{et}}^{1}(\bar{X}_{1}(Np^{r}),\mathcal{O}_{\mathfrak{p}}(j))^{{\mathrm{ord}}}$

denote the ordinary component of the étale cohomology group
$H_{\operatorname{et}}^{1}(\bar{X}_{1}(Np^{r}),\mathcal{O}_{\mathfrak{p}}(j))$
with respect to the Hecke operator $U_{p}$. This is naturally an
${\mathcal{O}}_{{\mathfrak{p}}}[G_{\mathbb{Q}}]$-module and we may simply
denote it $H_{r}$ when the Galois action is understood or irrelevant. Let
$\mathfrak{h}_{r}$ be the subring of
$\mathrm{End}_{\mathcal{O}_{\mathfrak{p}}}(H_{r})$ spanned over
$\mathcal{O}_{\mathfrak{p}}$ by the Hecke operators $T_{n}$, $(n,N)=1$. The
Eisenstein ideal $I_{r}=I_{\operatorname{Eis},r}\subset\mathfrak{h}_{r}$ is
the ${\mathcal{O}}_{{\mathfrak{p}}}$-submodule of $\mathfrak{h}_{r}$ generated
by $U_{p}-1$ and $T_{\ell}-\ell\langle\ell\rangle-1$, for primes $\ell\nmid
Np$; here, $\langle\ell\rangle$ stands for the usual diamond operator.

Passing to the projective limit we may define:

$H(j):=H_{\operatorname{et}}^{1}(\bar{X}_{1}(Np^{\infty}),\mathcal{O}_{\mathfrak{p}}(j))^{\mathrm{ord}}=\lim_{\leftarrow}H_{r}(j),\quad\mathfrak{h}=\lim_{\leftarrow}\mathfrak{h}_{r},\quad
I=\lim_{\leftarrow}I_{r}\subset\mathfrak{h}.$

The ideal $I$ is a height one ideal contained in the maximal ideal
${\mathfrak{M}}=(I,{\mathfrak{p}})$; for any $t\geq 1$ we shall denote
${\mathfrak{M}}^{(t)}=(I,{\mathfrak{p}}^{t})$, so that
${\mathfrak{M}}={\mathfrak{M}}^{(1)}$. The ideal $I$ is the intersection of a
finite number of height one prime ideals $P\subset{\mathfrak{M}}$, each of
which corresponds to a weight two eigenform that is congruent to an Eisenstein
series mod ${\mathfrak{p}}$, like the modular form $f$ of the introduction.

Let
$\Lambda_{N}:=\lim_{\leftarrow}\mathcal{O}_{\mathfrak{p}}[(\mathbb{Z}/Np^{r}\mathbb{Z})^{\times}]$
denote the Iwasawa algebra of tame level $N$. Any Dirichlet character
$\psi:(\mathbb{Z}/Np\mathbb{Z})^{\times}\rightarrow{\mathcal{O}}_{\mathfrak{p}}^{\times}$
may be extended by linearity to yield a homomorphism

$\psi:\Lambda_{N}\longrightarrow\mathcal{O}_{\mathfrak{p}}[(\mathbb{Z}/Np\mathbb{Z})^{\times}]\longrightarrow\mathcal{O}_{\mathfrak{p}}$

that we continue to denote with the same symbol.

For any $\Lambda_{N}$-module $M$ let
$M_{\psi}=M\otimes_{\Lambda_{N},\psi}{\mathcal{O}}_{\mathfrak{p}}$ stand for
the associated $\psi$-isotypical component. Note that
$\Lambda_{N}=\oplus\Lambda_{N,\psi}$ where $\psi$ ranges over all characters
of $(\mathbb{Z}/Np\mathbb{Z})^{\times}$ and
$\Lambda_{N,\psi}\simeq\mathcal{O}_{\mathfrak{p}}[[1+p\mathbb{Z}_{p}]]$.

We begin by rephrasing the results of [FK, §9] on Sharifi’s conjecture in a
convenient way for our purposes. As we have seen at the beginning of §3, and
more precisely in the discussion before equation (22), the Hochschild–Serre
spectral sequence in étale cohomology yields the commutative diagram of
$\Lambda_{N,\theta}[G_{\mathbb{Q}}]$-modules:

(47)
$\textstyle{H_{\operatorname{et}}^{2}(X_{1}(Np^{\infty}),\mathcal{O}_{\mathfrak{p}}(2))^{{\mathrm{ord}}}_{\theta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\operatorname{et}}^{2}(X_{1}(Np),\mathcal{O}_{\mathfrak{p}}(2))^{{\mathrm{ord}}}_{\theta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},H(2)_{\theta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},H_{\operatorname{et}}^{1}(\bar{X}_{1}(Np),\mathcal{O}_{\mathfrak{p}}(2))_{\theta}^{{\mathrm{ord}}}).}$

The module $H(2)$ is endowed with an action of complex conjugation, yielding
the decomposition $H(2)=H(2)^{+}\oplus H(2)^{-}$; in the sequel we shall
employ a similar notation for any ${\mathcal{O}}_{{\mathfrak{p}}}$-module
acted on by complex conjugation. It follows from [FK, Prop. 6.3.2] that the
quotient

(48) $(H(2)/{\mathfrak{M}}^{(t)})_{\theta}^{+}$

is still endowed with a compatible action of $G_{\mathbb{Q}}$.

Our running assumptions imply that the (mod $\mathfrak{p}^{t}$) Galois
representation
$\bar{T}_{f,X}(1)=T_{f,X}(1)\otimes\mathcal{O}/\mathfrak{p}^{t}$ arises as a
quotient of $H(2)/{\mathfrak{M}}^{(t)}$ as
$\mathcal{O}/\mathfrak{p}^{t}[G_{\mathbb{Q}}]$-modules. Since the nebentype of
$f$ is $\theta$, it belongs to the $\theta$-isotypical component of the
latter. Denote

$\pi_{f}:(H(2)/{\mathfrak{M}}^{(t)})_{\theta}^{+}\longrightarrow\bar{T}_{f,X}(1)^{+}$

the resulting projection.

Recall also from (15) that there is a short exact sequence for
$\bar{T}_{f,X}$, which according to (17) and (18) is compatible with the
action of complex conjugation. In particular,
$(\bar{T}_{f,X}^{-})(1)=(\bar{T}_{f,X}(1))^{+}\simeq\mathcal{O}/\mathfrak{p}^{t}(2)$
and this is naturally a quotient of (48). Henceforth we fix the canonical
isomorphism provided by [FK, 6.3.18 and 7.1.11] in order to identify

(49) $\bar{T}_{f,X}(1)^{+}=\mathcal{O}/\mathfrak{p}^{t}(2)$

as $\mathcal{O}/\mathfrak{p}^{t}[G_{\mathbb{Q}}]$-modules.

Summing up there is a commutative diagram of $G_{\mathbb{Q}}$-modules, where
the horizontal arrows arise from specializing to $r=1$, i.e. level $Np$:

(50)
$\textstyle{H^{1}(\mathbb{Q},H(2)_{\theta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},H_{\operatorname{et}}^{1}(\bar{X}_{1}(Np),\mathcal{O}_{\mathfrak{p}}(2))_{\theta}^{{\mathrm{ord}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},(H(2)/{\mathfrak{M}}^{(t)})_{\theta}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},(H_{\operatorname{et}}^{1}(\bar{X}_{1}(Np),\mathcal{O}_{\mathfrak{p}}(2))^{{\mathrm{ord}}}/(I_{1},{\mathfrak{p}}^{t}))_{\theta}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},\bar{T}_{f,X}(1)^{+})=H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2))}$

###  4.2. Fukaya–Kato maps

Define the module $\mathcal{Q}$ as in [FK, §6.3.1], with a twist by the square
of the cyclotomic character, namely

$\mathcal{Q}:=(H(2)/IH(2))_{\theta}^{+}.$

Recall from §3 (cf. e.g. Prop. 3.4) the Kubota–Leopoldt $p$-adic $L$-function
$L_{p}(\bar{\theta})\in\Lambda_{N,\theta}$ attached to the Dirichlet character
$\bar{\theta}$. As shown in [FK, §6.1.7], there is an isomorphism of Galois
modules

$(\mathfrak{h}/I)_{\theta}\simeq\Lambda_{N,\theta}/(L_{p}(\bar{\theta}))(\underline{\varepsilon}_{\operatorname{cyc}}),$

where $\underline{\varepsilon}_{\operatorname{cyc}}$ is the $\Lambda$-adic
cyclotomic character introduced in (40) and the identification follows from
[FK, §6.3], with the conventions recalled in §9.1.2 and 9.1.4. Moreover, and
as a consequence of the proof of the Iwasawa main conjecture by Mazur and
Wiles, in §6.3.18 of loc. cit. the authors show that there are isomorphisms of
Galois modules

(51)
$\mathcal{Q}\xrightarrow{\simeq}\Lambda_{N,\theta}/(L_{p}(\bar{\theta}))(\underline{\varepsilon}_{\operatorname{cyc}})(2)\simeq(\mathfrak{h}(2)/I\mathfrak{h}(2))_{\theta}.$

Recall that $H^{i}(\mathbb{Z}[1/Np],M)\subset H^{i}(\mathbb{Q},M)$ stands for
the set of classes which are unramified at primes dividing $Np$. Shapiro’s
lemma gives an isomorphism

$\lim_{\leftarrow}H^{2}(\mathbb{Z}[1/Np,\zeta_{Np^{r}}],\mathcal{O}_{\mathfrak{p}}(2))\simeq
H^{2}(\mathbb{Z}[1/Np],\Lambda_{N}(\underline{\varepsilon}_{\operatorname{cyc}})(2)).$

As a piece of notation, and following the definition of [FK, §5.2.6], set

$\mathcal{S}=\lim_{\leftarrow}H^{2}(\mathbb{Z}[1/Np,\zeta_{Np^{r}}],\mathcal{O}_{\mathfrak{p}}(2))^{+}\simeq
H^{2}(\mathbb{Z}[1/Np],\Lambda_{N}(\underline{\varepsilon}_{\operatorname{cyc}})(2))^{+}.$

In [FK, §9.1] Fukaya and Kato established the existence of isomorphisms

$\operatorname{FK}_{1}:H^{1}(\mathbb{Z}[1/Np],\mathcal{Q})\simeq\mathcal{S}_{\theta},\quad\operatorname{FK}_{2}:\mathcal{S}_{\theta}\simeq
H^{2}(\mathbb{Z}[1/Np],\mathcal{Q})$

arising from the long exact sequence in cohomology induced by the short exact
sequence

$0\rightarrow\Lambda_{N,\theta}(\underline{\varepsilon}_{\operatorname{cyc}})(2)\xrightarrow{\cdot
L_{p}(\bar{\theta})}\Lambda_{N,\theta}(\underline{\varepsilon}_{\operatorname{cyc}})(2)\rightarrow\mathcal{Q}\rightarrow
0.$

stemming from (51).

In particular, the map we have denoted as $\operatorname{FK}_{2}$ is just the
$+$-component of the homomorphism

(52)
$H^{2}(\mathbb{Z}[1/Np],\Lambda_{N,\theta}(\underline{\varepsilon}_{\operatorname{cyc}})(2))\longrightarrow
H^{2}(\mathbb{Z}[1/Np],\Lambda_{N,\theta}/(L_{p}(\bar{\theta}))(\underline{\varepsilon}_{\operatorname{cyc}})(2))$

induced by (a twist of) the natural projection
$\Lambda_{N,\theta}\longrightarrow\Lambda_{N,\theta}/L_{p}(\bar{\theta})$. For
an explicit description of $\operatorname{FK}_{1}$, see [FK, §9.1].

The main result of §9.2 of loc. cit. asserts that the map

(53)
$\mathrm{ev}_{\infty}:H_{\operatorname{et}}^{2}(X_{1}(Np^{\infty}),\mathcal{O}_{\mathfrak{p}}(2))_{\theta}\longrightarrow\lim_{\leftarrow}H^{2}(\mathbb{Z}[1/Np,\zeta_{Np^{r}}],\mathcal{O}_{\mathfrak{p}}(2))_{\theta}$

induced by evaluation at the cusp $\infty$ factors through the Eisenstein
quotient, as stated below.

######  Proposition 4.3 (Fukaya–Kato).

The map $\mathrm{ev}_{\infty}$ of (53) agrees with the composition

$H_{\operatorname{et}}^{2}(X_{1}(Np^{\infty}),\mathcal{O}_{\mathfrak{p}}(2))_{\theta}\rightarrow
H^{1}(\mathbb{Z}[1/Np],H(2)_{\theta})\rightarrow
H^{1}(\mathbb{Z}[1/Np],\mathcal{Q})\simeq\mathcal{S}_{\theta}$

where:

  * •

the first map is the composition of
$H^{2}(X_{1}(Np^{\infty}),\mathcal{O}_{\mathfrak{p}}(2))\rightarrow
H^{2}(X_{1}(Np^{\infty}),\mathcal{O}_{\mathfrak{p}}(2))^{\mathrm{ord}}_{\theta}$
and the left vertical arrow in (47), both restricted to the subspace of
classes unramified at the primes dividing $Np$;

  * •

the second map is induced by the projection
$H(2)_{\theta}\rightarrow\mathcal{Q}$;

  * •

the last isomorphism is $\operatorname{FK}_{1}$.

In [FK, §9.3] Fukaya and Kato further introduced two distinguished morphisms

(54) $\displaystyle a,b:H^{1}(\mathbb{Z}[1/Np],\mathcal{Q})\rightarrow
H^{2}(\mathbb{Z}[1/Np],\mathcal{Q})$ $\displaystyle
a=\operatorname{FK}_{2}\circ\operatorname{FK}_{1},$ $\displaystyle
b=\cup\,(1-p^{-1})\log_{p}(\varepsilon_{\operatorname{cyc}})$

where $\varepsilon_{\operatorname{cyc}}\in
H^{1}(\mathbb{Q},\mathcal{O}^{\times}_{\mathfrak{p}})$ stands for the
cyclotomic character. Note that $(1-p^{-1})\log_{p}$ takes values in
$\mathbb{Z}_{p}$ and hence $b$ is indeed well-defined. Under these conditions,
they show the following.

######  Proposition 4.4 (Fukaya–Kato).

Let $L_{p}^{\prime}(\bar{\theta})\in\Lambda_{N,\theta}$ denote the derivative
of $L_{p}(\bar{\theta})$. Then

(55) $b=L_{p}^{\prime}(\bar{\theta})\cdot a.$

###### Proof.

This follows from [FK, Proposition 9.3.1]; in particular, we just need the
restriction of $\theta$ to $(\mathbb{Z}/N\mathbb{Z})^{\times}$ being
primitive, since Lemma 9.1.3 of loc. cit. also works in this setting. ∎

######  Corollary 4.5.

Assume $L_{p}^{\prime}(\bar{\theta},-1)$ is a $p$-adic unit. Then the map

$H^{1}(\mathbb{Z}[1/Np],\mathcal{O}/\mathfrak{p}^{t}(2))\longrightarrow
H^{2}(\mathbb{Z}[1/Np],\mathcal{O}/\mathfrak{p}^{t}(2)),\quad\kappa\mapsto\kappa\cup(1-p^{-1})\log_{p}(\varepsilon_{\operatorname{cyc}})$

is invertible.

###### Proof.

Observe that $\operatorname{FK}_{1}$ and $\operatorname{FK}_{2}$ are
$\Lambda$-adic isomorphisms, as it has been proved in [FK, §9.1]. Hence, once
we consider the specialization map at the trivial character, we still have
isomorphisms of $\mathcal{O}_{\mathfrak{p}}$-modules. The same must be true
for their composition multiplied by the $p$-adic unit
$L_{p}^{\prime}(\bar{\theta},-1)$, and according to Proposition 4.4 and the
definitions provided in [FK, §4.1.3], this is precisely the above map. ∎

After applying the Fukaya–Kato map $\operatorname{FK}_{1}$ to the bottom row
of diagram (50), restricting to the subspace of unramified classes at primes
dividing $Np$, we reach the commutative diagram

(56)
$\textstyle{H^{1}(\mathbb{Z}[1/Np],(H(2)/{\mathfrak{M}}^{(t)})_{\theta}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\operatorname{FK}}_{1}}$$\textstyle{H^{1}(\mathbb{Z}[1/Np],\bar{T}_{f,X}(1)^{+})=H^{1}(\mathbb{Z}[1/Np],\mathcal{O}/\mathfrak{p}^{t}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\operatorname{FK}}_{1}(r=1)}$$\textstyle{H^{2}(\mathbb{Z}[1/Np],\Lambda_{N,\theta}/{\mathfrak{p}}^{t}(\underline{\varepsilon}_{\operatorname{cyc}})(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(\mathbb{Z}[1/Np],\mathcal{O}/{\mathfrak{p}^{t}}(2)),}$

where the left-most vertical map is
$\bar{\operatorname{FK}}_{1}=\operatorname{FK}_{1}\pmod{\mathfrak{p}^{t}}$. As
in (50), the horizontal arrows are specialization at $r=1$, and the right-most
vertical arrow is accordingly the specialization of
$\bar{\operatorname{FK}}_{1}$ at $r=1$.

We may further apply now Fukaya–Kato’s map
$\bar{\operatorname{FK}}_{2}=\operatorname{FK}_{2}\pmod{\mathfrak{p}^{t}}$ to
the above diagram and obtain the following one:

(57)
$\textstyle{H^{2}(\mathbb{Z}[1/Np],\Lambda_{N,\theta}/{\mathfrak{p}}^{t}(\underline{\varepsilon}_{\operatorname{cyc}})(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\operatorname{FK}}_{2}}$$\textstyle{H^{2}(\mathbb{Z}[1/Np],\mathcal{O}/\mathfrak{p}^{t}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{||}$$\textstyle{H^{2}(\mathbb{Z}[1/Np],(H(2)/{\mathfrak{M}}^{(t)})_{\theta}^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(\mathbb{Z}[1/Np],\mathcal{O}/\mathfrak{p}^{t}(2))}$

Again the horizontal maps are specialization in level $Np$ at $r=1$ and the
right-most vertical map is the specialization of $\bar{\operatorname{FK}}_{2}$
at $r=1$. In view of (52) the latter may be identified with the identity map:
according to the definitions provided in [FK, §6.1.6, 4.1.3] and with our
current conventions, the specialization of $L_{p}(\bar{\theta})$ at $r=1$ is

$L_{p}(\bar{\theta},-1)=(1-\bar{\theta}(p)p)\cdot
L(\bar{\theta},-1)=-(1-\bar{\theta}(p)p)\cdot\frac{B_{2}(\bar{\theta})}{2},$

which vanishes$\pmod{\mathfrak{p}^{t}}$ in light of our assumptions.

###  4.3. Proof of Theorem 4.2

We can finally prove Theorem 4.2. With a slight abuse of notation, we identify
global units with their image in cohomology under the Kummer map.

According to Proposition 4.3, we have

(58)
$\operatorname{FK}_{1}(\bar{\kappa}_{f,2})=\mathrm{ev}_{\infty}(u_{\xi_{1},\xi_{2}}\cup
u_{\chi_{1},\chi_{2}})=\bar{c}_{\xi_{1}}\cup\bar{c}_{\chi_{1}}\pmod{\mathfrak{p}^{t}},$

where the circular units involved in the cup product are those resulting from
the evaluation at infinity of the modular units $u_{\xi_{1},\xi_{2}}$ and
$u_{\chi_{1},\chi_{2}}$, respectively. Recall we are assuming that
$\xi_{1}=\bar{\chi}_{1}$, and it was proved in Theorem 3.6 that $\kappa_{f}$
is unramified everywhere.

Next, we apply $\operatorname{FK}_{2}$ to both sides of (58). Proposition 4.4
together with the commutativity of (57) allow us to establish that

$\bar{\kappa}_{f,2}\cup(1-p^{-1})\log_{p}(\varepsilon_{\operatorname{cyc}})=L_{p}^{\prime}(\bar{\theta},-1)\cdot(\bar{c}_{\xi_{1}}\cup\bar{c}_{\chi_{1}}).$

Theorem 4.2 finally follows from Corollary 4.5.

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