# A canonical generator for congruence ideals of Hida families

Alexandre Maksoud  Universität Paderborn, Warburger Str. 100, 33098 Paderborn,
Germany<EMAIL_ADDRESS>https://sites.google.com/view/alexandre-
maksoud/

###### Abstract.

We construct canonical adjoint $p$-adic $L$-functions generating the
congruence ideal attached to Hida families using Ohta’s pairing. We show that
these $p$-adic $L$-functions, suitably modified by certain Euler factors, are
interpolated by a regular element of Hida’s universal ordinary Hecke algebra.
We also relate them to characteristic series of primitive adjoint Selmer
groups.

###### Key words and phrases:

$p$-adic $L$-functions, Modular forms, Selmer groups

###### 2020 Mathematics Subject Classification:

11R27.

##  1\. Introduction

In the context of $L$-functions, the notion of period often refers to
Deligne’s period for the value of a motivic $L$-function at a critical
integer. When the $L$-function has an automorphic origin, its special values
can sometimes be related to automorphic periods. Consider an elliptic cuspidal
newform $f$ of weight $k\geq 2$ and level $N$, and let
$L(\operatorname{ad}f,s)$ be the $L$-function attached to the adjoint lift of
the automorphic representation of $\operatorname{GL}_{2}(\mathbb{A})$
generated by $f$. A theorem of Sturm [Stu80] implies that the ratio

(1) $\dfrac{L(\operatorname{ad}f,j)}{\pi^{2j+k-1}\langle f,f\rangle}$

is an algebraic number for all odd integers $1\leq j\leq k-1$, where $\langle
f,f\rangle$ is the Petersson norm of $f$ for $\Gamma_{1}(N)$. The same
automorphic period $\langle f,f\rangle$ is involved in algebraicity statements
on special values of the Rankin-Selberg $L$-function $L(f\times g,s)$, where
$g$ is another newform of weight smaller than $k$.

Let $p\geq 5$ be a prime number. When $f$ is $p$-ordinary, Schmidt [Sch88]
constructed a $p$-adic $L$-function $\mathcal{L}_{p}(\operatorname{ad}f,s)$
interpolating (up to simple fudge factors) quantities like (1) or twisted
variants thereof. Hida showed in [Hid90] that this construction could be
realized for the primitive Hida family $F$ of $f$, yielding a two-variable
$p$-adic $L$-function $\mathcal{L}_{p}(\operatorname{ad}F,\kappa,s)$, $\kappa$
being the weight variable. However, Schmidt-Hida’s construction does not quite
fit into Coates-Perrin-Riou’s framework of $p$-adic $L$-functions of motives
developed in [CPR89]. Indeed, the motivic periods for $\operatorname{ad}f$ and
its critical twists should involve the product $\Omega_{f}^{+}\Omega_{f}^{-}$
of $p$-normalized Shimura periods of $f$ rather than $\langle f,f\rangle$.
This problem of normalizations of periods is perhaps better directly seen on
(1) for $j=1$, which turns out to be an explicit number depending on $f$ in an
elementary way and does not seem to capture deep algebraic invariants such as
the size of a Bloch-Kato Selmer group. This observation suggests that
$\mathcal{L}_{p}(\operatorname{ad}F,\kappa,s)$ is the ratio of two ”genuine“
(or ”motivic“) $p$-adic $L$-functions $L_{p}(\operatorname{ad}F,\kappa,s)$ and
$L_{p}(\operatorname{ad}F,\kappa)$ interpolating (up to simple fudge factors)

(2)
$\dfrac{L(\operatorname{ad}f,j)}{\pi^{2j+k-1}\Omega_{f}^{+}\Omega_{f}^{-}}\quad\mbox{and}\quad\dfrac{\langle
f,f\rangle}{\Omega_{f}^{+}\Omega_{f}^{-}}$

respectively. The second one should also generate the congruence ideal of $F$
in most cases. By Hida’s formula mentioned above, $\pi^{k+1}\langle
f,f\rangle$ is essentially equal to $L(\operatorname{ad}f,1)$, and it is
therefore reasonable to call $L_{p}(\operatorname{ad}F,\kappa)$ the (weight-
variable) adjoint $p$-adic $L$-function of $F$. These expectations are not
new; a similar picture for Rankin-Selberg $L$-series and their $p$-adic
counterparts is drawn by Hida in his monograph [Hid96] where many evidences
are provided in the CM case.

This article focuses on the construction of $L_{p}(\operatorname{ad}F,\kappa)$
and the study of its arithmetic properties. Hida [Hid88a] gave first evidences
for the existence of $L_{p}(\operatorname{ad}F,\kappa)$ by showing in certain
cases that any generator of the congruence ideal of $F$, evaluated at $f$, is
essentially equal to the second ratio of (2) (see also [Hid16, Corollary
6.29]). A construction of an adjoint $p$-adic $L$-function (or, rather, an
invertible sheaf) on the Eigencurve of tame level $N$ was carried out by
Bellaïche [Bel21] using Kim’s scalar product on overconvergent modular
symbols. Its vanishing locus is intimately related to the ramification locus
of the weight map, which constitutes a geometric counterpart to the support of
the congruence module. Note that Bellaïche’s interpolation formula is only
given at crystalline points of trivial Nebentype. Locally on the Eigencurve,
the Shimura periods can be normalized in a coherent way at all arithmetic
points, with a controllable $p$-adic error term. This approach has been
generalized to other settings in recent works by Balasubramanyam-Longo, Lee,
Wu and Lee-Wu [BL20, Lee21, Wu21, LW23].

Going back to the ordinary setting, we now set up some notations and we state
our main results.

Let $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}$ be an algebraic closure of ${\mathbb{Q}_{p}}$ and fix an
isomorphism $\mathbb{C}\simeq\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$. Let
$\Lambda={\mathbb{Z}_{p}}[[1+p{\mathbb{Z}_{p}}]]$ be the Iwasawa algebra.
Assume $F$ is a primitive Hida family of tame level $N$ with coefficients in
an integrally closed finite flat $\Lambda$-algebra $\mathbb{I}$. Denote by
$\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{F}$ residual
$p$-ordinary Galois representation over
$G_{\mathbb{Q}}=\operatorname{Gal}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu/\mathbb{Q})$
attached to $F$ and assume the following condition:

(CR) $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu_{F}\textrm{ is absolutely irreducible and $p$-distinguished.}$

Here, $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{F}$ is
called $p$-distinguished if the semi-simplification of its restriction to
$G_{\mathbb{Q}_{p}}=\operatorname{Gal}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}/{\mathbb{Q}_{p}})$ is non-scalar.

######  Theorem A (=Theorem 2.3.7).

Assume $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{F}$
satisfies (CR). There exists a generator
$L_{p}(\operatorname{ad}F)\in\mathbb{I}$ of the congruence ideal of $F$
satisfying the following interpolation property. Let $f$ be an arithmetic
specialization of $F$ of weight $k\geq 2$ and level $Np^{r}$, $r\geq 1$, and
denote by $f^{\circ}$ its associated newform. Then the specialization
$L_{p}(\operatorname{ad}f)$ of $L_{p}(\operatorname{ad}F)$ at $f$ is given by

$L_{p}(\operatorname{ad}f)=p^{r-1}\alpha^{r}w(f^{\circ})\mathcal{E}_{p}(\operatorname{ad}f^{\circ})\cdot(-2i)^{k}\dfrac{\langle
f^{\circ},f^{\circ}\rangle}{\Omega^{+}_{f^{\circ}}\Omega^{-}_{f^{\circ}}}\in\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}.$

In the above formula, $\alpha$ and $\beta$ are the roots of the $p$-th Hecke
polynomial of $f^{\circ}$ with $|\alpha|_{p}=1$, $w(f^{\circ})\in\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu^{\times}$ is
the Atkin-Lehner pseudo-eigenvalue of $f^{\circ}$, and
$\mathcal{E}_{p}(\operatorname{ad}f^{\circ})$ is a modified Euler factor at
$p$ given by $\mathcal{E}_{p}(\operatorname{ad}f^{\circ})=1$ if $f^{\circ}=f$
and by

$\mathcal{E}_{p}(\operatorname{ad}f^{\circ})=\alpha(p-1)(1-\frac{\beta}{\alpha})(1-p^{-1}\frac{\beta}{\alpha})$

if $f^{\circ}\neq f$.

Let $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu=\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{F}$ and let
$\mathbb{T}$ be the local component of Hida’s big ordinary Hecke algebra of
tame level $N$ associated with $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$. To $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ is also attached a
piece $e\mathrm{H}_{1}(Np^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}$ of the singular
homology of the tower of compactified modular curves $(X_{1}(Np^{r}))_{r\geq
1}$, where $e$ denotes Hida’s ordinary projector. Under (CR), the work of
Emerton-Pollack-Weston [EPW06] shows that
$e\mathrm{H}_{1}(Np^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}^{\pm}$ is
$\mathbb{T}$-free of rank 1, where the superscript $\pm$ stands for the $\pm
1$-eigenspace for the complex conjugation. The choice of a $\mathbb{T}$-basis
$\xi^{\pm}$ of $e\mathrm{H}_{1}(Np^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}^{\pm}$ and Hida’s
control theorem then simultaneously pin down Shimura periods for all classical
eigenforms $f$ with residual Galois representation isomorphic to $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$. Shimura periods for
the newform $f^{\circ}$ associated to $f$ are defined in Remark 2.3.6.

In order to construct $L_{p}(\operatorname{ad}F)$, we first pass, via
intersection of cycles, from $e\mathrm{H}_{1}(Np^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}^{\pm}$ to a
cohomology space $e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}^{\pm}$ where $e^{*}$
is an _anti-ordinary_ projector. We then use a work of Ohta [Oht95] to obtain
a perfect $\Lambda$-adic pairing on $e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}^{+}\times
e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}^{-}$ which
interpolates Petersson norms of anti-ordinary eigenforms at finite level.
Since members of $F$ contribute to ordinary cohomology instead of anti-
ordinary cohomology, we make use of the Atkin-Lehner involution, which is
known by Hida to transports integral structures between ordinary and anti-
ordinary cohomology (see Proposition 2.2.1).

Fix a modular $p$-ordinary residual representation $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu\colon
G_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{F}\mkern-1.5mu}\mkern 1.5mu_{p})$
satisfying (CR) and a finite set $\Sigma$ of primes not containing $p$. The
$p$-ordinarity means that $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ admits a
$G_{{\mathbb{Q}_{p}}}$-unramified one-dimensional quotient which is fixed.
These choices give rise to a universal $\Lambda$-adic Hecke algebra
$\mathbb{T}_{\Sigma}(\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu)=\mathbb{T}_{\Sigma}$ whose spectrum parameterizes Hida families with
residual $p$-ordinary Galois representation isomorphic to $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ that are minimally
ramified outside of $\Sigma$. In the next theorem, we choose $\mathbb{I}$ big
enough so that it contains the coefficients of any Hida family appearing in
$\operatorname{Spec}(\mathbb{T}_{\Sigma})$.

######  Theorem B.

There exists a regular element $L_{\Sigma}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu)\in\mathbb{T}_{\Sigma}\otimes_{\Lambda}\mathbb{I}$ whose specialization
at any primitive Hida family $F$ minimally ramified outside $\Sigma$ with
$\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu_{F}\simeq\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu$ is given by $E_{\Sigma}(\operatorname{ad}F)L_{p}(\operatorname{ad}F)$
up to $\mathbb{I}$-units, where $E_{\Sigma}(\operatorname{ad}F)\in\mathbb{I}$
is a product of Euler factors associated with the adjoint of $F$ at certain
primes of $\Sigma$.

See Theorem 3.1.5 for a precise statement of this theorem, and in particular,
(20) for the definition of the level $N_{\Sigma}$ of $\mathbb{T}_{\Sigma}$ and
(18) for the definition of the Euler factors. Theorems A and B together can be
interpreted as the existence of congruences between Petersson norms of
different $p$-ordinary newforms sharing the same residual Galois
representation. As for $L_{p}(\operatorname{ad}F)$, the element
$L_{\Sigma}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu)$ only depends on the
choice of a $\mathbb{T}_{\Sigma}$-basis of
$e\mathrm{H}_{1}(N_{\Sigma}p^{\infty})_{\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu}^{\pm}$. We use here
Fukaya-Kato’s variant of Ohta’s pairing with values in a space of
$\Lambda$-adic cuspforms, which turns out to be $\mathbb{T}_{\Sigma}$-free
since $\mathbb{T}_{\Sigma}$ is Gorenstein. The appearance of Euler factors
comes from the comparison of cohomology at two different levels (as in
classical level raising arguments) for which we make use of the main technical
result of [EPW06]. The author hopes in near future to apply Theorem B to the
study of the variation of Iwasawa invariants associated with primitive Rankin-
Selberg $p$-adic $L$-functions, as constructed in [CH20].

We next relate adjoint $p$-adic $L$-functions to their algebraic counterparts
called Selmer groups. Let $F$ be a primitive Hida family whose residual Galois
representation satisfies (CR) and let
$V_{\operatorname{ad}}=\mathbb{I}^{\oplus 3}$ be the adjoint Galois
representation associated with $F$. By ordinarity, $V_{\operatorname{ad}}$ has
a two-dimensional quotient $V_{\operatorname{ad}}^{-}$ which is stable under
the action of $G_{\mathbb{Q}_{p}}$. We let
$\mathbb{I}^{\vee}=\operatorname{Hom}(\mathbb{I},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$
be the Pontryagin dual of $\mathbb{I}$, and set
$D_{\operatorname{ad}}=V_{\operatorname{ad}}\otimes_{\mathbb{I}}\mathbb{I}^{\vee}$,
$D_{\operatorname{ad}}^{-}=V_{\operatorname{ad}}^{-}\otimes_{\mathbb{I}}\mathbb{I}^{\vee}$.
The (primitive) adjoint Selmer group $\operatorname{Sel}(\operatorname{ad}F)$
of $F$ is defined as the kernel of the global-to-local restriction map

$\mathrm{H}^{1}(\mathbb{Q},D_{\operatorname{ad}})\longrightarrow\prod_{\ell\neq
p}\mathrm{H}^{1}(I_{\ell},D_{\operatorname{ad}})\times\mathrm{H}^{1}(I_{p},D^{-}_{\operatorname{ad}}),$

where $I_{\ell}$ denotes the inertia subgroup at a prime $\ell$. Its
Pontryagin dual $\operatorname{Sel}(\operatorname{ad}F)^{\vee}$ is a finitely
generated $\mathbb{I}$-module. Since $\mathbb{I}$ is integrally closed, it
makes sense to consider the characteristic ideal of a finitely generated
torsion $\mathbb{I}$-module $M$ which we denote by
$\textrm{char}_{\mathbb{I}}M$.

In the next theorem, we make a slightly stronger assumption than (CR) on
residual Galois representations:

(CR’) $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ is
$p$-distinguished, and $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu_{|G_{\mathbb{Q}(\mu_{p})}}$ is absolutely irreducible.

We also need a certain finite set of exceptional primes, defined as

(3) $\Sigma_{e}:=\left\\{\ell\colon\pi_{f,\ell}\textrm{ is supercuspidal and
}\pi_{f,\ell}\simeq\pi_{f,\ell}\otimes\tau_{\mathbb{Q}_{\ell}^{2}}\right\\},$

where $\pi_{f,\ell}$ is the automorphic representation of
$\operatorname{GL}_{2}(\mathbb{Q}_{\ell})$ of any arithmetic specialization
$f$ of $F$ and $\tau_{\mathbb{Q}_{\ell}^{2}}$ is the unramified quadratic
character of $\mathbb{Q}_{\ell}^{\times}$. The definition of $\Sigma_{e}$ does
not depend on the choice of $f$.

######  Theorem C.

Assume that $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu_{F}$ satisfies (CR’). The Selmer group
$\operatorname{Sel}(\operatorname{ad}F)^{\vee}$ is a torsion
$\mathbb{I}$-module. If $F$ is twist-minimal, then the equality

$\textrm{char}_{\mathbb{I}}\operatorname{Sel}(\operatorname{ad}F)^{\vee}=\prod_{\ell\in\Sigma_{e}}(1+\ell^{-1})^{-1}L_{p}(\operatorname{ad}F).\mathbb{I}$

holds.

Recall that $F$ is twist-minimal if it has minimal level among its twists.
Note that $\operatorname{Sel}(\operatorname{ad}F)$ is invariant under twists,
but $L_{p}(\operatorname{ad}F)$ isn’t, so twist-minimality is a necessary
hypothesis in Theorem C. However, the relation between adjoint $p$-adic
$L$-functions of Hida families and their twists is well-understood (see
Proposition 3.1.4), so this is not a restricting hypothesis. We note that a
version of Theorem C is stated in [Hid16, Coro. 6.29] under rather restrictive
assumptions, forcing $F$ to be minimally ramified.

Under assumption (CR’), the ring $\mathbb{T}_{\Sigma}$ turns out to be the
universal ring $R_{\Sigma}$ for $p$-ordinary Galois deformations of $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ that are minimally
ramified outside $\Sigma$. As noticed by Mazur and Tilouine [MT90], the module
of continuous Kähler differentials of $R_{\Sigma}$ as a $\Lambda$-algebra is
related to both adjoint Selmer groups and congruence ideals of systems of
eigenvalues appearing in $\mathbb{T}_{\Sigma}$. This observation is a crucial
input in works of Wiles, Taylor-Wiles, and Diamond-Flach-Guo among others and
has applications to modularity and to the Bloch-Kato conjecture [Wil95, TW95,
DFG04]. We obtain a relation between $\Sigma$-imprimitive Selmer groups and
$p$-adic $L$-functions, from which we deduce the equality in Theorem C via a
generalization of Greenberg-Vatsal’s argument [GV00].

The plan of this article is as follows. In Section 2 we give the construction
of an adjoint $p$-adic $L$-function as in Theorem A and we introduce $p$-adic
$L$-functions with values in big Hecke algebras. We analyze in Section 3 how
these objects change when we vary the tame level, yielding the proof of
Theorem B. We then study Selmer groups and prove Theorem C in the end of
Section 3. Appendix A recalls definitions and standard facts related to
congruence modules and congruence ideals.

### Acknowledgments

The author would like to thank Ming-Lun Hsieh for his comments on this work.
This research is partially supported by the Luxembourg National Research Fund,
Luxembourg, INTER/ANR/ 18/12589973 GALF.

### Notations

We adopt the convention that the usual Hecke operators (acting on modular
forms, cohomology, etc.) in level $M$ are denoted $T_{n}$, even in the case
when $n$ is not coprime to $M$. In particular, for a prime $\ell$ dividing
$M$, $T_{\ell}$ is Atkin-Lehner’s operator $U_{\ell}$. We use the letter
$\mathfrak{h}$ for Hecke algebras, and $\mathbb{T}$ for local components of
Hecke algebras (i.e. their localizations at maximal ideals). Adjoint Hecke
operators for the Petersson scalar product and adjoint Hecke algebras are
typically denoted with an asterisk ($T_{n}^{*}$, $\mathfrak{h}^{*}$,
$\mathbb{T}^{*}$, etc.).

##  2\. Construction of a $p$-adic $L$-function

###  2.1. Adjoint $L$-values

We fix in this paragraph an integer $M\geq 3$. For $\Gamma=\Gamma_{1}(M)$, we
let $S_{k}(M)$ be the space of modular cusp forms of weight $k\geq 1$ and
level $\Gamma$. Recall the usual slash operator in weight $k$ defined by

$(f|s)(z)=\det(s)^{k-1}(cz+d)^{-k}f(\frac{az+b}{cz+d})$

for every function $f\colon\mathscr{H}\to\mathbb{C}$ on the upper half-plane
and every $s=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in
S^{+}=M_{2}(\mathbb{Z})\cap\operatorname{GL}_{2}(\mathbb{Q})^{+}$.

The abstract Hecke ring $\mathcal{H}(S^{+},\Gamma)$ consisting of double
cosets $[\Gamma s\Gamma]$, $s\in S^{+}$ acts on $S_{k}(M)$. For $n,d\geq 1$
with $(d,M)=1$, we denote by $T_{n}$, $T^{*}_{n}$, $\langle d\rangle$ the
usual Hecke operators, adjoint Hecke operators and diamond operators
respectively. The Atkin-Lehner involution is $W_{M}=[\Gamma\tau_{M}\Gamma]$,
where

$\tau_{M}=\begin{pmatrix}0&-1\\\ M&0\end{pmatrix}.$

Note that $\tau_{M}$ normalizes $\Gamma$, and $T_{n}W_{M}=W_{M}T^{*}_{n}$,
$T^{*}_{n}W_{M}=W_{M}T_{n}$ and $\langle d\rangle W_{M}=W_{M}\langle
d\rangle^{-1}$.

The Petersson scalar product is defined by the formula

$\langle f,g\rangle_{\Gamma}=\int_{X(\mathbb{C})}f(z)\mkern
1.5mu\overline{\mkern-1.5mug(z)\mkern-1.5mu}\mkern
1.5muy^{k-2}dxdy\qquad(z=x+iy),$

where $f,g\in S_{k}(M)$ and $X=X_{1}(M)_{/\mathbb{C}}$ is the compactified
modular curve of level $\Gamma$. The adjoint of $[\Gamma s\Gamma]$ for
$\langle\cdot,\cdot\rangle_{\Gamma}$ is $[\Gamma s^{\prime}\Gamma]$, where
$s^{\prime}=\det(s)s^{-1}$, i.e.,

$s^{\prime}=\begin{pmatrix}d&-b\\\ -c&a\end{pmatrix}\qquad\mbox{if
}s=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

The operators $T_{n}$ and $T_{n}^{*}$ (resp. $\langle d\rangle$, and $\langle
d\rangle^{-1}$) are adjoint to each other under $(\cdot,\cdot)_{\Gamma}$.

The Nebetype action decomposes $S_{k}(M)$ as a direct sum $\oplus
S_{k}(M,\chi)$ indexed on the set of Dirichlet characters
$\chi\colon(\mathbb{Z}/M\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$ and which
is Hecke-compatible.

Let $f(q)=\sum_{n\geq 1}a_{n}q^{n}\in S_{k}(M,\chi)$ be a normalized eigenform
(i.e., $a_{1}=1$ and $f|T_{n}=a_{n}f$ for all $n\geq 1$). For every prime
$\ell$, let $\alpha_{\ell}$ and $\beta_{\ell}$ be such that

$(1-\alpha_{\ell}X)(1-\beta_{\ell}X)=1-a_{\ell}X+\chi(\ell)\ell^{k-1}X^{2},$

where $\chi(\ell)=0$ if $\ell|M$. Define

$L^{\text{imp}}(s,\operatorname{ad}f)=\prod_{\ell}(1-\frac{\alpha_{\ell}^{2}}{\chi(\ell)\ell^{k-1}}\ell^{-s})(1-\frac{\alpha_{\ell}\beta_{\ell}}{\chi(\ell)\ell^{k-1}}\ell^{-s})(1-\frac{\beta_{\ell}^{2}}{\chi(\ell)\ell^{k-1}}\ell^{-s}),$

the product being taken over all prime numbers $\ell$. The product converges
for $\Re(s)>>0$ and it has a meromorphic continuation to $\mathbb{C}$. When
$f$ is new, $L^{\textrm{imp}}(s,\operatorname{ad}f)$ coincides, up to Euler
factors at certain primes $\ell|M$, with the $L$-function
$L(\operatorname{ad}f,s)$ attached to the adjoint lift of the automorphic
representation of $\operatorname{GL}_{2}(\mathbb{A})$ generated by $f$.

######  Proposition 2.1.1.

Assume $f$ is a newform of level $M$ and Nebentype $\chi$, and let $M_{\chi}$
be the conductor of $\chi$. Then

$L^{\text{imp}}(1,\operatorname{ad}f)=\frac{2^{2k}\pi^{k+1}}{(k-1)!MM_{\chi}\varphi(M/M_{\chi})}\langle
f,f\rangle_{\Gamma},$

where $\varphi$ is Euler’s totient function.

###### Proof.

This follows from [Hid81, Thm. 5.1], using that $M\geq 3$ by assumption. ∎

######  Remark 2.1.2.

Assume $f$ is twist-minimal. It is perhaps nicer to reformulate Proposition
2.1.1 in terms of primitive adjoint $L$-functions as

$\displaystyle\langle f,f\rangle_{\Gamma}$
$\displaystyle=\dfrac{(k-1)!M\varphi(M)}{2^{2k}\pi^{k+1}}\prod_{\ell\in\Sigma_{e}}(1+\ell^{-1})L(\operatorname{ad}f,1)$
$\displaystyle=M\varphi(M)2^{1-k}\prod_{\ell\in\Sigma_{e}}(1+\ell^{-1})\Gamma(\operatorname{ad}f,1)L(\operatorname{ad}f,1),$

where $\Sigma_{e}$ is the set of exceptional primes for $f$ of the
introduction and
$\Gamma(\operatorname{ad}f,s)=\Gamma_{\mathbb{C}}(s+k-1)\Gamma_{\mathbb{R}}(s)$
(and $\Gamma_{\mathbb{C}}(s)=2(2\pi)^{-s}\Gamma(s)$,
$\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)$) is the Gamma factor of the
motive attached to $\operatorname{ad}f$ (see [DFG04, Lemma 2.12]).

We recall the definition of Shimura periods attached to Hecke eigenforms. For
a subring $A$ of $\mathbb{C}$, consider the local system $V_{k}(A)$ on $X$
associated with the $A$-module $\textrm{TSym}^{k-2}(A^{\oplus 2})$. This
latter module consists of homogeneous polynomials of degree $k-2$ of the form
$P(X,Y)=\sum_{i}\binom{k-2}{i}a_{i}X^{i}Y^{k-2-i}$ with $a_{i}\in A$ for all
$0\leq i\leq k-2$. For such a polynomial $P$, we let $s\in
S=M_{2}(\mathbb{Z})\cap\operatorname{GL}_{2}(\mathbb{Q})$ act on $P$ via
$(P|s)(X,Y)=P((X\ Y)(s^{\prime})^{\textrm{tr}})$, i.e.,

$(P|s)(X,Y)=P(dX-bY,-cX+aY)\qquad\mbox{if }s=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}.$

As $\Gamma$ is torsion-free, $\mathrm{H}^{1}(X(\mathbb{C}),V_{k}(A))$ can be
identified with the parabolic cohomology group
$\mathrm{H}_{P}^{1}(\Gamma,\textrm{TSym}^{k-2}(A^{\oplus 2}))$. It comes
equipped with an action of the abstract Hecke algebra $\mathcal{H}(S,\Gamma)$
([Hid93, §6.3]).

For a given $f\in S_{k}(M)$, we define classes in
$\mathrm{H}^{1}(X(\mathbb{C}),V_{k}(\mathbb{C}))$ by the formulae

$\delta(f)(\gamma)=\int_{z_{0}}^{\gamma
z_{0}}(X-zY)^{k-2}f(z)\operatorname{d}\\!z,\ \
\delta^{c}(f)=-\int_{z_{0}}^{\gamma z_{0}}(X-\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5muY)^{k-2}\mkern
1.5mu\overline{\mkern-1.5muf^{c}(z)\mkern-1.5mu}\mkern
1.5mu\operatorname{d}\\!\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu,$

where $z_{0}\in X(\mathbb{C})$, $\gamma\in\Gamma$ and $f^{c}(z)=\mkern
1.5mu\overline{\mkern-1.5muf(-\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)\mkern-1.5mu}\mkern
1.5mu$. The cohomology classes do not depend on the choice of $z_{0}$. The
action of the complex conjugation $[\Gamma\iota\Gamma]$, where
$\iota=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right)$, sends
$\delta(f)$ to $\delta^{c}(f)$. The Eichler-Shimura period map is the
$\mathbb{C}$-linear isomorphism

(4) $\begin{array}[]{ccc}S_{k}(M)\oplus
S_{k}(M)&\overset{\sim}{\longrightarrow}&\mathrm{H}^{1}(X(\mathbb{C}),V_{k}(\mathbb{C}))\\\
(f,g)&\mapsto&\delta(f)+\delta^{c}(g).\end{array}$

Take $f\in S_{k}(M)$ and $s=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in S^{+}$. From

$(X-(s\cdot z)Y)^{k-2}|s^{\prime}=\left(\frac{\det
s}{cz+d}\right)^{k-2}(X-zY)^{k-2}\quad\mbox{and}\quad f^{c}|s=(f|\iota
s\iota)^{c},$

one sees that

(5) $\delta(f)|[\Gamma s\Gamma]=\delta(f|[\Gamma
s\Gamma])\qquad\mbox{and}\qquad\delta^{c}(f)|[\Gamma
s\Gamma]=\delta^{c}(f|[\Gamma\iota s\iota\Gamma]).$

In particular, (4) is equivariant with respect to the actions of
$T_{n},T^{*}_{n}$ and $\langle d\rangle$ for $n,d\geq 1$, $(M,d)=1$.

For any ring $A\subseteq\mathbb{C}$, we put

$\mathrm{H}^{1}_{k}(M)_{A}:=\mathrm{H}^{1}(X(\mathbb{C}),V_{k}(A))$

and we consider the Hecke algebras

$\mathfrak{h}_{k}(M)_{A}=A[T_{n},\langle\ell\rangle]_{n,\ell\geq
1,(\ell,M)=1},\qquad\mathfrak{h}^{*}_{k}(M)_{A}=A[T^{*}_{n},\langle\ell\rangle]_{n,\ell\geq
1,(\ell,M)=1},$

as subspaces of $\operatorname{End}_{A}(\mathrm{H}^{1}_{k}(M)_{A})$. They are
finitely generated commutative $A$-algebras. If $2$ is invertible in $A$, we
denote by $\mathrm{H}^{1}_{k}(M)_{A}^{\pm}$ the subspaces of
$\mathrm{H}^{1}_{k}(M)_{A}$ where the complex conjugation acts by $\pm 1$. For
$A=\mathbb{C}$, the map $f\mapsto\delta^{\pm}(f)$ identifies $S_{k}(M)$ with
$\mathrm{H}^{1}_{k}(M)_{A}^{\pm}$, where we have put

(6) $\delta^{\pm}(f)=\delta(f)\pm\delta^{c}(f).$

As $S_{k}(M)$ admits a basis consisting of modular forms with Fourier
coefficients in $\mathbb{Z}$, for every subring $A\subseteq\mathbb{C}$ we have
$S_{k}(M)=\mathbb{C}\otimes_{A}S_{k}(M)_{A}$ as
$\mathfrak{h}_{k}(M)_{\mathbb{C}}=\mathbb{C}\otimes_{A}\mathfrak{h}_{k}(M)_{A}$-modules,
where

$S_{k}(M)_{A}=\\{f\in S_{k}(M)\ \colon\ f(q)=\sum_{n\geq 1}a_{n}q^{n}\in
A\otimes_{\mathbb{Z}}\mathbb{Z}[[q]]\\}.$

Recall that $\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu\subset\mkern 1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\simeq\mathbb{C}$. For a normalized eigenform $f(q)=\sum_{n\geq
1}a_{n}q^{n}\in S_{k}(M,\chi)$, the ring
$\mathbb{Z}_{f}=\mathbb{Z}[a_{n}]_{n\geq 1}$ is a subring of $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$. We denote by

$\lambda_{f}\colon\mathfrak{h}_{k}(M)_{\mathbb{Z}_{f}}\to\mathbb{Z}_{f}$

the system of Hecke eigenvalues of $f$ given by $\lambda_{f}(T_{n})=a_{n}$ and
$\lambda_{f}(\langle d\rangle)=\chi(d)$ for all $n,d\geq 1$, $(d,M)=1$.

For any field $K\subset\mathbb{C}$ containing $\mathbb{Z}_{f}$ and any
$\mathfrak{h}_{k}(M)_{K}$-module $M$, let $M[\lambda_{f}]$ be the
$\lambda_{f}$-eigenspace of $M$. Using the maps $\delta^{\pm}$ in (6), it is
clear that
$\dim_{\mathbb{C}}\mathrm{H}^{1}_{k}(M)^{\pm}_{\mathbb{C}}[\lambda_{f}]=1$.

Let $\mathcal{O}_{f}$ be the completion of $\mathbb{Z}_{f}$ inside $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}\simeq\mathbb{C}$. The image of
$\mathrm{H}^{1}_{k}(M)_{\mathcal{O}_{f}}$ inside
$\mathrm{H}^{1}_{k}(M)_{\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}}$ is a
$\mathcal{O}_{f}$-lattice, which we still denote by
$\mathrm{H}^{1}_{k}(M)_{\mathcal{O}_{f}}$ with a slight abuse of notations. As
long as we deal with ordinary eigenforms, this won’t cause any problem as the
ordinary part of $\mathrm{H}^{1}_{k}(M)_{\mathcal{O}_{f}}$ is
$\mathcal{O}_{f}$-free (see Proposition 2.2.1 below).

######  Definition 2.1.3.

Let $f\in S_{k}(M)$ be a normalized eigenform. Let $\xi_{f}^{\pm}$ be a
$\mathcal{O}_{f}$-basis of

(7)
$\mathrm{H}^{1}_{k}(M)_{\mathcal{O}_{f}}\cap\mathrm{H}^{1}_{k}(M)^{\pm}_{\mathbb{C}}[\lambda_{f}].$

The $p$-normalized periods of $f$ with respect to $\xi^{\pm}$ are the complex
numbers $\Omega_{f}^{\pm}\in\mathbb{C}^{\times}$ defined by the relation

$\delta^{\pm}(f)=\Omega^{\pm}_{f}\cdot\xi_{f}^{\pm}.$

######  Remark 2.1.4.

One can actually replace the coefficient ring $\mathcal{O}_{f}$ in the above
definition with any PID $A\subseteq\mathbb{C}$ (see e.g. [Bel21, Rem.
5.4.10]). When $A$ is a subring of $\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$, one retrieves
the usual definition of complex periods attached to normalized eigenforms.

We now recall the cohomological interpretation of the Petersson norm of
normalized eigenforms. Let $A\subseteq\mathbb{C}$ be a ring, and consider the
nondegenerate $A$-linear pairing $[\cdot,\cdot]$ on
$\textrm{TSym}^{k-2}(A^{\oplus 2})$ defined in [Oht95, (4.2.1)]. It is
characterized by the identity

$[(uX+vY)^{k-2},(u^{\prime}X+v^{\prime}Y)^{k-2}]=(uv^{\prime}-u^{\prime}v)^{k-2}\qquad(u,u^{\prime},v,v^{\prime}\in
A).$

For $s\in S$ and $P,Q\in\textrm{TSym}^{k-2}(A^{\oplus 2})$, one checks that
$[P|s,Q|s]=(\det s)^{k-2}[P,Q]$. Combining $[\cdot,\cdot]$ with the cup-
product in cohomology we obtain a new $A$-linear pairing

(8)
$(\cdot,\cdot)_{\pi}\colon\mathrm{H}^{1}_{k}(M)_{A}\times\mathrm{H}^{1}_{k}(M)_{A}\to\mathrm{H}^{2}(X(\mathbb{C}),A)\simeq
A,$

where the last isomorphism is the cap-product with the fundamental class of
$X(\mathbb{C})$, endowed with its natural orientation of complex manifold. For
$s\in S$, the adjoint of $[\Gamma s\Gamma]$ under $(\cdot,\cdot)_{\pi}$ is
$[\Gamma s^{\prime}\Gamma]$, so for the modified anti-symmetric pairing

(9) $(x,y)\mapsto(x,y|W_{M})_{\pi},$

the Hecke algebras $\mathfrak{h}_{k}(M)_{A}$ and $\mathfrak{h}^{*}_{k}(M)_{A}$
become autoadjoint.

######  Proposition 2.1.5.

Let $f,g\in S_{k}(M)$. We have

$\displaystyle(\delta(f),\delta^{c}(g))_{\pi}$
$\displaystyle=(2i)^{k-1}\langle f,g^{c}\rangle_{\Gamma}$
$\displaystyle(\delta^{+}(f),\delta^{-}(g)|W_{M})_{\pi}$
$\displaystyle=-2(2i)^{k-1}\langle f,g^{c}|W_{M}\rangle_{\Gamma}.$

###### Proof.

We start with the first equality. Over $\mathbb{C}$, the cap-product with the
fundamental class of $X(\mathbb{C})$ is given by the integration over
$X(\mathbb{C})$ of cohomology classes seen as $2$-differential forms.
Therefore,

$\displaystyle(\delta(f),\delta^{c}(g))_{\pi}$
$\displaystyle=-\int_{X(\mathbb{C})}f(z)\mkern
1.5mu\overline{\mkern-1.5mug^{c}(z)\mkern-1.5mu}\mkern
1.5mu[(X-zY)^{k-2},(X-\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5muY)^{k-2}]dz\wedge d\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu$ $\displaystyle=2i\int_{X(\mathbb{C})}f(z)\mkern
1.5mu\overline{\mkern-1.5mug^{c}(z)\mkern-1.5mu}\mkern 1.5mu(z-\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)^{k-2}dxdy$
$\displaystyle=(2i)^{k-1}\int_{X(\mathbb{C})}f(z)\mkern
1.5mu\overline{\mkern-1.5mug^{c}(z)\mkern-1.5mu}\mkern 1.5muy^{k-2}dxdy.$

This proves the first formula. A similar computation as above yields

$\displaystyle(\delta(h),\delta(h^{\prime}))_{\pi}$
$\displaystyle=0=(\delta^{c}(h),\delta^{c}(h^{\prime}))_{\pi},$
$\displaystyle(\delta(h),\delta^{c}(h^{\prime}))_{\pi}$
$\displaystyle=(-1)^{k-1}(\delta^{c}(h^{\prime}),\delta(h))_{\pi},$

for any $h,h^{\prime}\in S_{k}(M)$, and from (5) we find
$\delta^{-}(g)|W_{M}=\delta(g|W_{M})+(-1)^{k-1}\delta^{c}(g|W_{M})$. Hence, by
linearity,

$\displaystyle(\delta^{+}(f),\delta^{-}(g)|W_{M})_{\pi}$
$\displaystyle=(-1)^{k-1}\left((\delta(f),\delta^{c}(g|W_{M}))_{\pi}+(\delta(g|W_{M}),\delta^{c}(f))_{\pi}\right)$
$\displaystyle=(-2i)^{k-1}\left(\langle
f,(g|W_{M})^{c}\rangle_{\Gamma}+\langle g|W_{M},f^{c}\rangle_{\Gamma}\right)$
$\displaystyle=2(-2i)^{k-1}\langle f,(g|W_{M})^{c}\rangle_{\Gamma}$
$\displaystyle=-2(2i)^{k-1}\langle f,g^{c}|W_{M}\rangle_{\Gamma},$

since $(g|W_{M})^{c}=(-1)^{k}g^{c}|W_{M}$. This proves the second formula. ∎

###  2.2. Hida families

Let $p$ be an odd prime and let $M$ be of the form $Np^{r}$ with $(N,p)=1$ and
$r\geq 1$. We let $X_{r}=X(\Gamma_{1}(Np^{r}))_{/\mathbb{C}}$. We shall
consider the ordinary and anti-ordinary parts of the (co-)homology of $X_{r}$,
defined with the help of Hida’s projectors

$e=\lim_{r}(T_{p})^{r!},\qquad e^{*}=\lim_{r}(T^{*}_{p})^{r!}.$

In what follows, we drop for convenience the subscript ${\mathbb{Z}_{p}}$ when
we work with cohomology with ${\mathbb{Z}_{p}}$-coefficients.

######  Proposition 2.2.1 (Hida).

The Atkin-Lehner map

$\cdot|W_{Np^{r}}\colon e\mathrm{H}^{1}_{k}(Np^{r})\to
e^{*}\mathrm{H}^{1}_{k}(Np^{r})$

is a ${\mathbb{Z}_{p}}$-linear isomorphism, and $e\mathrm{H}^{1}_{k}(Np^{r})$
and $e^{*}\mathrm{H}^{1}_{k}(Np^{r})$ are ${\mathbb{Z}_{p}}$-free.

###### Proof.

See [Hid88a, Corollary 3.2] and [Hid88a, (2.4) p.347]. ∎

In particular, we have $e\mathfrak{h}(Np^{r})=e^{*}\mathfrak{h}^{*}(Np^{r})$
via $T_{n}\leftrightarrow T^{*}_{n}$. Poincaré duality yields a canonical
identification

$\mathrm{H}_{1}(X_{r}(\mathbb{C}),{\mathbb{Z}_{p}})=\mathrm{H}^{1}(X_{r}(\mathbb{C}),{\mathbb{Z}_{p}}),$

which is compatible with the Hecke action, in the sense that the action of
$T_{n}$ and $\langle d\rangle$ on the left hand side corresponds to that of
$T_{n}^{*}$ and $\langle d\rangle^{-1}$ on the right hand side respectively.
By functoriality of (co)homology, we obtain

$\varprojlim_{r}e\mathrm{H}_{1}(X_{r}(\mathbb{C}),{\mathbb{Z}_{p}})=\varprojlim_{r}e^{*}\mathrm{H}^{1}(X_{r}(\mathbb{C}),{\mathbb{Z}_{p}})=:e^{*}\mathrm{H}^{1}(Np^{\infty}),$

as Hecke-modules, where the transition maps on cohomology are the trace maps.
Since the adjoint Hecke operators are compatible with trace maps in
cohomology, $e^{*}\mathrm{H}^{1}(Np^{\infty})$ inherits a structure of module
over

$e^{*}\mathfrak{h}^{*}(Np^{\infty})=\varprojlim_{r}e^{*}\mathfrak{h}_{2}^{*}(Np^{r}).$

We will primarily be interested in $e^{*}\mathfrak{h}^{*}(Np^{\infty})$, but
note that it is canonically isomorphic to
$e\mathfrak{h}(Np^{\infty})=\varprojlim_{r}e\mathfrak{h}_{2}(Np^{r})$.

The action of inverse diamond operators endows the ring
$e^{*}\mathfrak{h}(Np^{\infty})$ with a structure of algebra over
$\tilde{\Lambda}=\varprojlim_{r}{\mathbb{Z}_{p}}[(\mathbb{Z}/Np^{r})^{\times}]$.
Letting $U_{r}=1+p^{r}{\mathbb{Z}_{p}}$ for all $r\geq 1$, we have
$\tilde{\Lambda}=\Lambda[(\mathbb{Z}/Np)^{\times}]$ where we put

$\Lambda={\mathbb{Z}_{p}}[[U_{1}]]=\varprojlim_{r}{\mathbb{Z}_{p}}[U_{1}/U_{r}],$

and $e^{*}\mathfrak{h}(Np^{\infty})$ is known to be finite free over $\Lambda$
by [Hid86, Thm. 3.1].

Let $\mathbb{I}$ be a finite flat extension of $\Lambda$ and let $\gamma$ be a
topological generator of $U_{1}$. We identify via $\mathbb{C}\simeq\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p}$ Dirichlet
characters of $p$-power order and conductor with elements of
$\widehat{U_{1}}:=\operatorname{Hom}_{\textrm{cts}}(U_{1},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{p}^{\times})=\operatorname{Hom}_{{\mathbb{Z}_{p}}-\textrm{alg}}(\Lambda,\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$. The set
of arithmetic points of $\mathbb{I}$ is defined as

$\mathcal{X}(\mathbb{I})=\left\\{\nu\in\operatorname{Hom}_{\Lambda}(\mathbb{I},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})\ |\
\exists k\geq 2,\ \exists\varepsilon\in\widehat{U_{1}},\
\nu(\gamma)=\gamma^{k-2}\epsilon(\gamma)\right\\}.$

We refer to the pair $(k,\epsilon)$ in the definition of
$\nu\in\mathcal{X}(\mathbb{I})$ as the weight of $\nu$, and we define a finite
flat extension of ${\mathbb{Z}_{p}}$ by letting
$\mathcal{O}_{\nu}=\nu(\mathbb{I})$. We further denote by $S(N,\mathbb{I})$
the space of ordinary $\mathbb{I}$-adic cusp forms of tame level $N$. Recall
that $S(N,\mathbb{I})$ is spanned over $\mathbb{I}$ by formal $q$-expansions
$F(q)=\sum_{n\geq 1}a_{n}q^{n}\in\mathbb{I}[[q]]$ with the property that there
exists a Dirichlet character $\theta$ of level dividing $Np$ such that, for
all $\nu\in\mathcal{X}(\mathbb{I})$ of weight $(k,\varepsilon)$, the
specialization of $F$ at $\nu$ yields

(10) $F_{\nu}:=\sum_{n\geq 1}\nu(a_{n})q^{n}\in
eS_{k}(Np^{r_{\nu}},\theta\varepsilon\omega^{2-k}).$

Here, $\omega$ is the Teichmüller character,
$p^{r_{\nu}}=\max(p,C(\varepsilon))$ (and $C(\varepsilon)$ is the conductor of
$\varepsilon$), and $\mathcal{O}_{\nu}$ is seen inside $\mathbb{C}$.

On $S(N,\mathbb{I})$ there is an action of Hecke operators $T_{n}$ which is
compatible with specialization maps. More precisely,
$S(N,\mathbb{I})=S(N,\Lambda)\otimes\mathbb{I}$ and the $\Lambda$-subalgebra
of $\operatorname{End}_{\Lambda}(S(N,\Lambda))$ generated by the $T_{n}$’s is
canonically isomorphic to $e\mathfrak{h}(Np^{\infty})_{{\mathbb{Z}_{p}}}$.
Furthermore, for all $k\geq 2$ and $r\geq 1$,

$\displaystyle
e\mathfrak{h}(Np^{\infty})\otimes_{\Lambda}\Lambda/(\omega_{k,r})$
$\displaystyle\simeq e\mathfrak{h}_{k}(Np^{r}),$

by Hida’s control theorem [Hid86, Thm. 1.2], where $\omega_{k,r}\in\Lambda$ is
defined by $\omega_{k,r}(\gamma)=[\gamma]^{p^{r-1}}-\gamma^{(k-2)p^{r-1}}$. In
particular,
$e\mathfrak{h}(Np^{\infty})\simeq\varprojlim_{r}e\mathfrak{h}_{k}(Np^{r})$ for
all integers $k\geq 2$.

A primitive Hida family is a $q$-expansion $F$ as above which is a normalized
eigenform (i.e., $a_{1}=1$, $T_{n}F=a_{n}F$ for all $n\geq 1$) and such that
$F_{\nu_{0}}$ is a primitive newform for at least one
$\nu_{0}\in\mathcal{X}(\mathbb{I})$. The character $\theta$ in (10), called
the tame character of $F$, is necessarily even, and we have
$a_{p}\in\mathbb{I}^{\times}$. Moreover, given $\nu\in\mathcal{X}(\mathbb{I})$
of weight $(k,\varepsilon)$, if $F_{\nu}^{\circ}$ denotes the newform
associated with $F_{\nu}$, then the following is known. If $p$ divides
$C(\theta\varepsilon\omega^{2-k})$, then $F_{\nu}=F_{\nu}^{\circ}$. Otherwise,
$\varepsilon=1$, $\theta\omega^{2-k}$ is of prime-to-$p$ conductor and either
$F_{\nu}=F_{\nu}^{\circ}$ and $k=2$, or

$F_{\nu}(q)=F_{\nu}^{\circ}(q)-\beta_{\nu}F_{\nu}^{\circ}(q^{p}),$

where $\beta_{\nu}$ is the non-unit root of the polynomial
$X^{2}-\nu(a_{p})X+\theta\omega^{2-k}(p)p^{k-1}$ ([Hid88b, Thm. 4.1]).

Every normalized eigenform $F(q)=\sum_{n\geq 1}a_{n}q^{n}\in S(N,\mathbb{I})$
gives rise to a homomorphism $\lambda_{F}\colon
e\mathfrak{h}(Np^{\infty})\to\mathbb{I}$ sending $T_{n}$ to $a_{n}$. The
kernel $\mathfrak{a}$ of $\lambda_{F}$ is a minimal prime, and is contained in
a unique maximal prime $\mathfrak{m}$ of $e\mathfrak{h}(Np^{\infty})$. One may
assume that the ring of coefficients of $F$ is an integrally closed local
domain by taking $\mathbb{I}$ to be the normalization of
$\mathbb{T}/\mathfrak{a}$, where

$\mathbb{T}:=e\mathfrak{h}(Np^{\infty})_{\mathfrak{m}}.$

Let $\mathbb{T}^{\prime}$ be the subalgebra of $\mathbb{T}$ generated over
$\Lambda$ by $T_{n}$ for all $n\geq 1$ coprime with $Np$. Then
$\mathbb{T}^{\prime}$ is reduced and it is finitely generated and torsion-free
over $\Lambda$. Let

$\rho_{\mathfrak{m}}\colon
G_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathbb{T}^{\prime}\otimes_{\Lambda}\operatorname{Frac}(\Lambda))$

be Hida’s big ordinary representation and let $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{\mathfrak{m}}$ be
(the semisimplification of) its residual representation modulo $\mathfrak{m}$.
When $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu_{\mathfrak{m}}$ is absolutely irreducible, it is known by [Nys96,
Théorème 1] that $\rho_{\mathfrak{m}}$ is conjugate to a
$\mathbb{T}^{\prime}$-valued representation.

In what follows it is more convenient to work with the anti-ordinary big Hecke
algebra $e^{*}\mathfrak{h}^{*}(Np^{\infty})$. Since
$e^{*}\mathfrak{h}^{*}(Np^{\infty})=e\mathfrak{h}(Np^{\infty})$ canonically,
$F$ gives rise to a triple
$(\lambda_{F}^{*},\mathfrak{a}^{*},\mathfrak{m}^{*})$ defined in the obvious
way.

######  Proposition 2.2.2.

Let $\mathbb{T}^{*}=e^{*}\mathfrak{h}^{*}(Np^{\infty})_{\mathfrak{m}^{*}}$ be
a local component of the big Hecke algebra such that $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{\mathfrak{m}}$
satisfies (CR). Then
$e^{*}\mathrm{H}^{1}(Np^{\infty})^{\pm}_{\mathfrak{m}^{*}}$ is free of rank
$1$ over $\mathbb{T}^{*}$.

###### Proof.

This is a reformulation in terms of cohomology groups of [EPW06, Prop. 3.3.1].
∎

###  2.3. Interpolation of Petersson norms

We keep the notations of the preceding sections. For any $k\geq 2$ and $r\geq
1$, we see $e^{*}\mathrm{H}_{k}^{1}(Np^{r})$ as a $\Lambda$-module by letting
$d\in U_{1}$ act by $d^{k-2}\langle d\rangle^{-1}$. Then $\omega_{k,r}$ acts
trivially on it, and [Oht95, Thm. 1.4.3 (ii)] yields a
$\Lambda/\omega_{k,r}\Lambda$-linear isomorphism

(11)
$e^{*}\mathrm{H}^{1}(Np^{\infty})\otimes_{\Lambda}\Lambda/(\omega_{k,r})\simeq
e^{*}\mathrm{H}_{k}^{1}(Np^{r}).$

In [Oht95, Thm. 4.2.5], Ohta constructed a $\Lambda$-linear pairing on
$e^{*}\mathrm{H}^{1}(Np^{\infty})$ which interpolates the Poincaré pairings at
level $Np^{r}$ for all $r\geq 1$.

Recall that $e^{*}\mathfrak{h}^{*}(Np^{\infty})$ is a complete semi-local
ring, hence it is the finite product of its localizations at maximal ideals.
It is enough for our purpose to fix one of its local components
$\mathbb{T}^{*}=e^{*}\mathfrak{h}^{*}(Np^{\infty})_{\mathfrak{m}^{*}}$ which
we see as a $\Lambda$-algebra.

######  Theorem 2.3.1 (Ohta).

There exists a pairing

(12) $(\cdot,\cdot)_{\Lambda}\colon
e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mathfrak{m}^{*}}\times
e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mathfrak{m}^{*}}\to\Lambda$

fitting, for all $k\geq 2$ and $r\geq 1$, in the commutative diagram

${e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mathfrak{m}^{*}}\times
e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mathfrak{m}^{*}}}$${\Lambda}$${e^{*}\mathrm{H}_{k}^{1}(Np^{r})\times
e^{*}\mathrm{H}_{k}^{1}(Np^{r})}$${{\mathbb{Z}_{p}}[U_{1}/U_{r}],}$

where the bottom horizontal map is given by

(13) $(x,y)\mapsto(-1)^{k}\cdot\sum_{d\in
U_{1}/U_{r}}(dNp^{r})^{2-k}\cdot(x,y|(T_{p}^{*})^{r}|W_{Np^{r}}|\langle
d\rangle^{-1})_{\pi}\cdot[d],$

the left vertical map is induced by (11) and the right vertical map sends
$d\in U_{1}$ to $d^{k-2}\cdot[d]$. Moreover, $(\cdot,\cdot)_{\Lambda}$ is a
$\Lambda$-bilinear perfect pairing satisfying
$(x|T_{n}^{*},y)_{\Lambda}=(x,y|T_{n}^{*})_{\Lambda}$ for all integers $n\geq
1$.

###### Proof.

See [Oht95, Thm. 4.2.5 and Cor. 4.2.8]. The factor $(Np^{r})^{2-k}$ comes from
the difference of normalizations in the definition of $W_{Np^{r}}$ (see
[Oht95, (4.1.3)]). The perfectness of Ohta’s pairing follows from that of
Poincaré’s pairing on
$\mathrm{H}^{1}_{2}(Np)=\mathrm{H}^{1}(X_{1}(Np)(\mathbb{C}),{\mathbb{Z}_{p}})$
and Hida’s control theorem. ∎

Choose a finite flat extension $\mathbb{I}\subset\mkern
1.5mu\overline{\mkern-1.5mu\operatorname{Frac}(\Lambda)\mkern-1.5mu}\mkern
1.5mu$ of $\Lambda$ containing the Fourier coefficients of all the normalized
eigenforms in $S(N,\mkern
1.5mu\overline{\mkern-1.5mu\operatorname{Frac}(\Lambda)\mkern-1.5mu}\mkern
1.5mu)$ with residual Galois representation isomorphic to $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{\mathfrak{m}}$.
Replacing $\mathbb{I}$ with its integral closure in $\mkern
1.5mu\overline{\mkern-1.5mu\operatorname{Frac}(\Lambda)\mkern-1.5mu}\mkern
1.5mu$ if necessary, we assume that $\mathbb{I}$ is normal. Let

$\mathbb{T}_{\mathbb{I}}^{*}=e^{*}\mathfrak{h}^{*}(Np^{\infty})_{\mathfrak{m}^{*}}\otimes_{\Lambda}\mathbb{I},\qquad\mathcal{M}^{\pm}:=e^{*}\mathrm{H}^{1}(Np^{\infty})^{\pm}_{\mathfrak{m}^{*}}\otimes_{\Lambda}\mathbb{I}.$

Given a normalized eigenform $F(q)=\sum_{n\geq 1}a_{n}q^{n}\in
S(N,\mathbb{I})$ with residual Galois representation $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{\mathfrak{m}}$, the
extension of scalars of $\lambda_{F}^{*}$ to $\mathbb{I}$ induces a
$\mathbb{I}$-linear surjection
$\mathbb{T}^{*}_{\mathbb{I}}\twoheadrightarrow\mathbb{I}$ which we still
denote by $\lambda_{F}^{*}$. Let $\mathcal{K}$ be the fraction field of
$\mathbb{I}$ and let
$\mathbb{T}_{\mathcal{K}}^{*}=\mathbb{T}_{\mathbb{I}}^{*}\otimes_{\mathbb{I}}\mathcal{K}$.
We make the following assumption on $F$: there exists an $\mathcal{K}$-algebra
decomposition

(pr) $\mathbb{T}_{\mathcal{K}}^{*}\simeq\mathcal{K}\times X,$

where the first projection, say $e_{F}$, is induced by
$\lambda_{F}^{*}\otimes\mathcal{K}$. We further assume that

(fr) $\mathcal{M}^{\pm}\text{ is $\mathbb{T}_{\mathbb{I}}^{*}$-free of rank
1},$

and, following the notations of Appendix A, we let
$\mathcal{M}^{\pm}_{\lambda_{F}^{*}}:=\mathcal{M}^{\pm}\cap
e_{F}(\mathcal{M}^{\pm}\otimes_{\mathbb{I}}\mathcal{K})$. In particular,
$\mathcal{M}^{\pm}_{\lambda_{F}^{*}}$ is $\mathbb{I}$-free of rank $1$ by
Lemma A.1.

######  Remark 2.3.2.

Condition (pr) holds if $F$ is primitive of level $N$ by [Hid88b, Thm. 4.2].
It also holds if $\mathbb{T}_{\mathcal{K}}^{*}$ is semi-simple (i.e., if
$\mathbb{T}^{*}$ is reduced), which happens for instance when $N$ is cube-free
[Hid13, Cor. 1.3]. Condition (fr) holds if $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{\mathfrak{m}}$
satisfies (CR) by Proposition 2.2.2.

In the following definition, we write
$(\cdot,\cdot)_{\mathbb{I}}\colon\mathcal{M}^{+}\times\mathcal{M}^{-}\to\mathbb{I}$
for the $\mathbb{I}$-linear extension of Ohta’s pairing (12).

######  Definition 2.3.3.

Assume (pr) and (fr) and let $\xi_{F}^{\pm}$ be a $\mathbb{I}$-basis of
$\mathcal{M}^{\pm}_{\lambda_{F}^{*}}$. We define the $p$-adic adjoint
$L$-function of $F$ by letting

$L_{p}(\operatorname{ad}F)=(\xi^{+}_{F},\xi^{-}_{F})_{\mathbb{I}}\in\mathbb{I}.$

For any $\nu\in\operatorname{Hom}(\mathbb{I},\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu_{p})$, we shall
write $L_{p}(\operatorname{ad}F,\nu):=\nu(L_{p}(\operatorname{ad}F))$.

######  Proposition 2.3.4.

Assume (pr) and (fr). Then the congruence module of $F$ is isomorphic to
$\mathbb{I}/L_{p}(\operatorname{ad}F)\mathbb{I}$.

###### Proof.

The congruence module of $F$ is, by definition, the module
$C_{0}(\lambda_{F})$ over
$\mathbb{T}_{\mathbb{I}}=e\mathfrak{h}(Np^{\infty})_{\mathfrak{m}}\otimes_{\Lambda}\mathbb{I}$
defined in Appendix A, where
$\lambda_{F}\colon\mathbb{T}_{\mathbb{I}}\twoheadrightarrow\mathbb{I}$ is the
$\mathbb{I}$-algebra homomorphism sending $T_{n}$ to $a_{n}$. Under the
identification $\mathbb{T}=\mathbb{T}^{*}$, we then have
$C_{0}(\lambda_{F})=C_{0}(\lambda_{F}^{*})$, which, in turn, is isomorphic to
$\mathbb{I}/L_{p}(\operatorname{ad}F)\mathbb{I}$ by Corollary A.1. ∎

Given a primitive Hida family $F$, we now express the special values of
$L_{p}(\operatorname{ad}F)$ at arithmetic weights in terms of adjoint
$L$-values. We still assume (fr).

We first define $p$-normalized periods as follows. Let
$\nu\in\mathcal{X}(\mathbb{I})$ be of weight $(k,\varepsilon)$, let
$r=r_{\nu}$ and fix a sign $\pm$. Let $\mathbb{T}^{*}_{\nu}$ be the local
factor of $e^{*}\mathfrak{h}_{k}(Np^{r})_{\mathcal{O}_{\nu}}$ through which
$\lambda^{*}_{F_{\nu}}$ factors, and let $\mathcal{M}^{\pm}_{\nu}$ be the
corresponding direct summand of
$e^{*}\mathrm{H}^{1}_{k}(Np^{r})_{\mathcal{O}_{\nu}}$. Let also $a$ be the
sign of $(-1)^{k}$ and $\pm a$ be the sign of $\pm(-1)^{k}$. We have

(14)
$\mathcal{M}^{\pm}_{\lambda^{*}_{F}}\otimes_{\mathbb{I},\nu}\mathcal{O}_{\nu}\simeq(\mathcal{M}^{\pm}_{\nu})_{\lambda^{*}_{F_{\nu}}}\simeq(e\mathrm{H}^{1}_{k}(Np^{r})^{\pm
a}_{\mathcal{O}_{\nu}})_{\lambda_{F_{\nu}}},$

where the first map is the specialization map and the second one is the map
$W_{Np^{r}}^{-1}$. The first isomorphism is Hida’s control theorem [Hid88b,
Thm. 4.2], while the second follows from Proposition 2.2.1. Note that the last
module in (14) coincides with that in (7) with $M=Np^{r}$ and $f=F_{\nu}$.

######  Definition 2.3.5.

For each $\nu\in\mathcal{X}(\mathbb{I})$, we define $\xi_{F_{\nu}}^{\pm}$ as
the image of $\xi^{\pm}_{F}$ under the map in (14), and we let
$\Omega^{\pm}_{F_{\nu}}\in\mathbb{C}^{\times}$ be the associated
$p$-normalized canonical period of Definition 2.1.3. In other words,
$\xi_{F_{\nu}}^{\pm}|W_{Np^{r_{\nu}}}$ is the specialization of $\xi_{F}^{\pm
a}$ at $\nu$ (where $a$ is the sign of $(-1)^{k}$), and
$\delta^{\pm}(F_{\nu})=\Omega^{\pm}_{F_{\nu}}\cdot\xi_{F_{\nu}}^{\pm}$.

######  Remark 2.3.6.

For a weight $\nu\in\mathcal{X}(\mathbb{I})$ such that $F_{\nu}\in S_{k}(Np)$
is $p$-old, the $p$-normalized canonical period of $F_{\nu}^{\circ}$ can be
defined by simply letting
$\Omega^{\pm}_{F_{\nu}^{\circ}}=\Omega^{\pm}_{F_{\nu}}$. Indeed, by [KLZ17,
Prop. 7.3.1] there is a “$p$-stabilization isomorphism”
$(-)^{\circ}\colon\mathrm{H}^{1}_{k}(Np)_{\mathbb{C}}\simeq\mathrm{H}^{1}_{k}(N)_{\mathbb{C}}$
sending $\delta(F_{\nu})$ to $\delta(F_{\nu}^{\circ})$ and identifying the
lattices $(\mathrm{H}^{1}_{k}(Np)_{\mathcal{O}_{\nu}})_{\lambda_{F_{\nu}}}$
and $(\mathrm{H}^{1}_{k}(N)_{\mathcal{O}_{\nu}})_{\lambda_{F_{\nu}^{\circ}}}$.
Therefore, the periods of $F_{\nu}$ and of $F_{\nu}^{\circ}$ with respect to
$\xi_{F_{\nu}}^{\pm}$ and $(\xi_{F_{\nu}}^{\pm})^{\circ}$ respectively are
equal.

######  Theorem 2.3.7.

Assume (fr) holds and suppose $F$ is primitive as before. Let
$\nu\in\mathcal{X}(\mathbb{I})$ be of weight $(k,\varepsilon)$, let
$r=r_{\nu}$ and write the conductor of $F_{\nu}^{\circ}$ as $Np^{r_{0}}$. Put
$\alpha_{\nu}=a_{p}(F_{\nu})$,
$\beta_{\nu}=\frac{(\theta\omega^{2-k}\varepsilon)(p)p^{k-1}}{\alpha_{\nu}}$
and

$\mathcal{E}_{p}(\operatorname{ad}F_{\nu}^{\circ})=\alpha_{\nu}(p-1)(1-\frac{\beta_{\nu}}{\alpha_{\nu}})(1-p^{-1}\frac{\beta_{\nu}}{\alpha_{\nu}})$

if $F_{\nu}\neq F_{\nu}^{\circ}$ and
$\mathcal{E}_{p}(\operatorname{ad}F_{\nu}^{\circ})=1$ otherwise. We have

(15)
$L_{p}(\operatorname{ad}F,\nu)=p^{r-1}\alpha_{\nu}^{r}w_{\nu}\mathcal{E}_{p}(\operatorname{ad}F_{\nu}^{\circ})\cdot
i(-2i)^{k}\dfrac{\langle
F_{\nu}^{\circ},F_{\nu}^{\circ}\rangle_{\Gamma_{1}(Np^{r_{0}})}}{\Omega^{+}_{F^{\circ}_{\nu}}\Omega^{-}_{F^{\circ}_{\nu}}},$

where $w_{\nu}\in\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu^{\times}$ is
given by $F_{\nu}^{\circ}|W_{Np^{r_{0}}}=w_{\nu}(F_{\nu}^{\circ})^{c}$ and
$\Omega_{F^{\circ}_{\nu}}^{\pm}$ is as in Definition 2.3.5 and Remark 2.3.6.

###### Proof.

Let $W=W_{Np^{r}}$ for simplicity. Let $a$ be the sign of $(-1)^{k}$. Since
$\nu$ factors through $\mathbb{I}/\omega_{k,r}\mathbb{I}$,
$L_{p}(\operatorname{ad}F,\nu)$ equals

$(-1)^{k}\cdot\sum_{d\in
U_{1}/U_{r}}(dNp^{r})^{2-k}\cdot(\xi_{F_{\nu}}^{+a}|W,\xi_{F_{\nu}}^{-a}|W|(T_{p}^{*})^{r}|W|\langle
d\rangle^{-1})_{\pi}\cdot d^{k-2}\varepsilon(d).$

Noticing that $\xi_{F_{\nu}}^{-a}|T_{p}=\alpha_{\nu}\xi_{F_{\nu}}^{-a}$ and
$\xi_{F_{\nu}}^{-a}|\langle
d\rangle^{-1}=\varepsilon^{-1}(d)\xi_{F_{\nu}}^{-a}$, we obtain

$\displaystyle L_{p}(\operatorname{ad}F,\nu)$
$\displaystyle=(-1)^{k}(Np^{r})^{2-k}p^{r-1}\alpha_{\nu}^{r}\cdot(\xi_{F_{\nu}}^{+a}|W,\xi_{F_{\nu}}^{-a}|W^{2})_{\pi}$
$\displaystyle=(-1)^{k}p^{r-1}\alpha_{\nu}^{r}\cdot(\xi_{F_{\nu}}^{+a},\xi_{F_{\nu}}^{-a}|W)_{\pi}$
$\displaystyle=p^{r-1}\alpha_{\nu}^{r}\cdot(\xi_{F_{\nu}}^{+},\xi_{F_{\nu}}^{-}|W)_{\pi},$

since the adjoint of $W$ for $(\cdot,\cdot)_{\pi}$ is $Np^{r}\cdot W^{-1}$ and
(9) is anti-symmetric. Hence,

$L_{p}(\operatorname{ad}F,\nu)=-2(2i)^{k-1}p^{r-1}\alpha_{\nu}^{r}\cdot\dfrac{\langle
F_{\nu},F^{c}_{\nu}|W\rangle_{\Gamma_{1}(Np^{r})}}{\Omega^{+}_{F^{\circ}_{\nu}}\Omega^{-}_{F^{\circ}_{\nu}}},$

by Proposition 2.1.5. We are reduced to compute $\frac{\langle
F_{\nu},F^{c}_{\nu}|W\rangle_{\Gamma_{1}(Np^{r})}}{\langle
F^{\circ}_{\nu},F^{\circ}_{\nu}\rangle_{\Gamma_{1}(Np^{r_{0}})}}=c\cdot\frac{\langle
F_{\nu},F^{c}_{\nu}|W\rangle_{\Gamma_{0}(Np^{r})}}{\langle
F^{\circ}_{\nu},F^{\circ}_{\nu}\rangle_{\Gamma_{0}(Np^{r_{0}})}}$, where $c=1$
if $F_{\nu}=F_{\nu}^{\circ}$ and $c=p-1$ otherwise. This is already done in
[Hid88b, (9.5) p.79], which yields:

$\frac{\langle F_{\nu},F^{c}_{\nu}|W\rangle_{\Gamma_{0}(Np^{r})}}{\langle
F^{\circ}_{\nu},F^{\circ}_{\nu}\rangle_{\Gamma_{0}(Np^{r_{0}})}}=\left\\{\begin{array}[]{ll}(-1)^{k}w_{\nu}&\textrm{if
$F_{\nu}=F_{\nu}^{\circ}$}\\\
(-1)^{k}w_{\nu}\alpha_{\nu}(1-\frac{\beta_{\nu}}{\alpha_{\nu}})(1-p^{-1}\frac{\beta_{\nu}}{\alpha_{\nu}})&\textrm{if
$F_{\nu}\neq F_{\nu}^{\circ}$}.\end{array}\right.$

∎

###  2.4. A $p$-adic $L$-function for varying branches

We keep the notations of the preceding sections and we fix in particular a
local component $\mathbb{T}$ of $e\mathfrak{h}(Np^{\infty})$ with residual
Galois representation $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu=\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{\mathfrak{m}}$. We
let as before $\mathbb{T}^{*}$ denote the corresponding local component of
$e^{*}\mathfrak{h}^{*}(Np^{\infty})$ under $e\mathfrak{h}(Np^{\infty})\simeq
e^{*}\mathfrak{h}^{*}(Np^{\infty})$, and
$\mathbb{T}^{*}_{\mathbb{I}}=\mathbb{T}^{*}\otimes_{\Lambda}\mathbb{I}$. Put
also
$\mathcal{M}=e^{*}\mathrm{H}^{1}(Np^{\infty})_{\mathfrak{m}^{*}}\otimes_{\Lambda}\mathbb{I}$.

Assuming (fr), we discuss in this paragraph the existence of an element
$L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu)\in\mathbb{T}_{\mathbb{I}}$ such that, for every normalized eigenform
$F$ of level $N$ with residual representation $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ and satisfying (pr),
we have $\lambda_{F}(L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu))=L_{p}(\operatorname{ad}F)$. We shall in particular explain how to make
a consistent choice of canonical periods for every such eigenform $F$.

Following Fukaya and Kato [FK12, §1.6], we define a $\mathbb{I}$-linear
pairing on $\mathcal{M}\times\mathcal{M}$ by letting

$((x,y))_{\mathbb{I}}=\sum_{n\geq 1}(x,y|T_{n}^{*})_{\mathbb{I}}\cdot q^{n}\in
S(N,\mathbb{I})_{\mathfrak{m}}$

for all $(x,y)\in\mathcal{M}\times\mathcal{M}$.

######  Proposition 2.4.1.

The pairing $((\cdot,\cdot))_{\mathbb{I}}\colon\mathcal{M}\times\mathcal{M}\to
S(N,\mathbb{I})_{\mathfrak{m}}$ is perfect and it satisfies

$((x|T_{m}^{*},y))_{\mathbb{I}}=((x,y|T_{m}^{*}))_{\mathbb{I}}=((x,y))_{\mathbb{I}}|T_{m}$

for all $m\geq 1$ and $(x,y)\in\mathcal{M}\times\mathcal{M}$.

###### Proof.

See [FK12, Prop. 1.6.6]. ∎

Hida duality for $\Lambda$-adic cusp forms [Hid88b, Thm. 1.3] implies
$S(N,\Lambda)_{\mathfrak{m}}\simeq\operatorname{Hom}_{\Lambda}(\mathbb{T},\Lambda)$.
Hence, under the assumption that $\mathbb{T}$ is Gorenstein, we see that
$S(N,\mathbb{I})_{\mathfrak{m}}$ is $\mathbb{T}_{\mathbb{I}}$-free of rank
$1$. This Gorensteinness condition is implied by (fr). Indeed, Ohta’s pairing
provides an isomorphism
$\mathcal{M}^{+}\simeq\operatorname{Hom}_{\mathbb{I}}(\mathcal{M}^{-},\mathbb{I})$,
hence
$\mathbb{T}_{\mathbb{I}}\simeq\operatorname{Hom}_{\mathbb{I}}(\mathbb{T}_{\mathbb{I}},\mathbb{I})$.
Therefore, the perfectness of $((\cdot,\cdot))_{\mathbb{I}}$ shows that
$((\xi^{+}_{\mathfrak{m}},\xi^{-}_{\mathfrak{m}}))_{\mathbb{I}}$ generates
$S(N,\mathbb{I})_{\mathfrak{m}}$ as a $\mathbb{T}_{\mathbb{I}}$-module if
$\xi_{\mathfrak{m}}^{\pm}$ generates $\mathcal{M}^{\pm}$ as a
$\mathbb{T}_{\mathbb{I}}^{*}$-module.

######  Definition 2.4.2.

Assume (fr) and let $\xi_{\mathfrak{m}}^{\pm}$ be a generator of
$\mathcal{M}^{\pm}$ as a $\mathbb{T}_{\mathbb{I}}^{*}$-module. Put
$G_{\mathfrak{m}}:=((\xi^{+}_{\mathfrak{m}},\xi^{-}_{\mathfrak{m}}))_{\mathbb{I}}$.
We define the adjoint $p$-adic $L$-function of $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ as the unique
element $L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu)\in\mathbb{T}_{\mathbb{I}}$ such that

(16) $\sum F=L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu)\cdot
G_{\mathfrak{m}},$

where the sum runs over the set of all normalized eigenforms with ordinary
residual representation isomorphic to $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$.

######  Theorem 2.4.3.

Assume (fr) and let $\xi_{\mathfrak{m}}^{\pm}$ be a generator of
$\mathcal{M}^{\pm}$ as a $\mathbb{T}_{\mathbb{I}}^{*}$-module. For any
normalized eigenform $F$ satisfying (pr), we let
$\xi_{F}^{\pm}=e_{F}\cdot(L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu)\xi^{\pm}_{\mathfrak{m}})$, where $e_{F}$ is the projector associated
with $F$. Then $\xi_{F}^{\pm}$ is a generator of
$\mathcal{M}^{\pm}_{\lambda^{*}_{F}}$, and

$\lambda_{F}(L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu))=L_{p}(\operatorname{ad}F),$

where $L_{p}(\operatorname{ad}F)$ is computed with respect to $\xi^{\pm}_{F}$.

###### Proof.

Assume $F(q)=\sum_{n\geq 1}a_{n}q^{n}$ satisfies (pr). Let
$e_{F},e^{\prime}_{F}\in\mathbb{T}_{\mathcal{K}}=\mathbb{T}_{\mathbb{I}}\otimes_{\mathbb{I}}\mathcal{K}$
be the projectors of the algebra decomposition
$\mathbb{T}_{\mathcal{K}}\simeq\mathcal{K}\times X$, where
$\mathcal{K}=\operatorname{Frac}(\mathbb{I})$. We put
$L_{F}=\lambda_{F}(L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu))\in\mathbb{I}$ for
simplicity. Hence, $\xi_{F}^{\pm}=L_{F}e_{F}\cdot\xi^{\pm}_{\mathfrak{m}}$. By
multiplicity one, $e_{F}S(N,\mathcal{K})_{\mathfrak{m}}=\mathcal{K}.F$, so

(17) $F=e_{F}\cdot L_{p}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu)G_{\mathfrak{m}}=L_{F}e_{F}\cdot G_{\mathfrak{m}}.$

Since $S(N,\mathbb{I})_{\mathfrak{m}}$ is $\mathbb{T}_{\mathbb{I}}$-free and
generated by $G_{\mathfrak{m}}$, we thus have

$C_{0}(\lambda_{F})\simeq(S(N,\mathbb{I})_{\mathfrak{m}})^{\lambda_{F}}/(S(N,\mathbb{I})_{\mathfrak{m}})_{\lambda_{F}}=(\mathbb{I}.e_{F}\cdot
G_{\mathfrak{m}})/(\mathbb{I}.F)=\mathbb{I}/(L_{F}).$

As
$C_{0}(\lambda_{F})\simeq(\mathcal{M}^{\pm})^{\lambda_{F}^{*}}/\mathcal{M}^{\pm}_{\lambda_{F}^{*}}$,
we deduce that $\xi^{\pm}_{F}$ generates $\mathcal{M}^{\pm}_{\lambda_{F}^{*}}$
since $e_{F}\cdot\xi^{\pm}_{\mathfrak{m}}$ generates
$(\mathcal{M}^{\pm})^{\lambda_{F}^{*}}$.

We now prove that
$L_{p}(\operatorname{ad}F):=(\xi^{+}_{F},\xi^{-}_{F})_{\mathbb{I}}$ equals
$L_{F}$. Denote by a subscript $\mathcal{K}$ the extension of scalars to
$\mathcal{K}$ of the pairings $(\cdot,\cdot)_{\mathbb{I}}$ and
$((\cdot,\cdot))_{\mathbb{I}}$. Using $e_{F}^{2}=e_{F}$ and Proposition 2.4.1,
we see that

$\displaystyle e_{F}\cdot G_{\mathfrak{m}}$
$\displaystyle=((e_{F}\cdot\xi_{\mathfrak{m}}^{+},e_{F}\cdot\xi_{\mathfrak{m}}^{-}))_{\mathcal{K}}$
$\displaystyle=\sum_{n\geq
1}(e_{F}\cdot\xi_{\mathfrak{m}}^{+},e_{F}\cdot\xi_{\mathfrak{m}}^{-}|T_{n}^{*})_{\mathcal{K}}\cdot
q^{n}$ $\displaystyle=\sum_{n\geq
1}(e_{F}\cdot\xi_{\mathfrak{m}}^{+},e_{F}\cdot\xi_{\mathfrak{m}}^{-})_{\mathcal{K}}\cdot
a_{n}q^{n}=(e_{F}\cdot\xi_{\mathfrak{m}}^{+},e_{F}\cdot\xi_{\mathfrak{m}}^{-})_{\mathcal{K}}\cdot
F.$

Hence, multiplying the last equality by $L_{F}^{2}$ and combining it with
(17), one obtains

$L_{F}\cdot F=(\xi^{+}_{F},\xi^{-}_{F})_{\mathcal{K}}\cdot
F=L_{p}(\operatorname{ad}F)\cdot F,$

so $L_{F}=L_{p}(\operatorname{ad}F)$ as claimed. ∎

##  3\. The main conjecture

###  3.1. Twists and change of tame level

Fix a large enough integrally closed finite flat extension $\mathbb{I}$ of
$\Lambda$ as in §2.3. Let $F\in S(N,\mathbb{I})$ be a primitive Hida family of
tame character $\theta$ whose residual Galois representation satisfies (CR).
Then there exists a continuous Galois representation $\rho_{F}\colon
G_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathbb{I})$ unramified outside $Np$
such that

$\forall\ell\nmid
Np,\quad\left\\{\begin{array}[]{lcl}\operatorname{tr}\rho_{F}(\sigma_{\ell})&=&a_{\ell}(F)\\\
\det\rho_{F}(\sigma_{\ell})&=&\ell\cdot\theta_{\mathbb{I}}(\ell),\end{array}\right.$

where $\sigma_{\ell}$ is the arithmetic Frobenius at $\ell$ and
$\theta_{\mathbb{I}}\colon(\mathbb{Z}/Np)^{\times}\to\mathbb{I}^{\times}$ is
such that
$\nu(\theta_{\mathbb{I}}(\ell))=(\theta\omega^{2-k}\varepsilon)(\ell)\ell^{k-2}$
for any $\nu\in\mathcal{X}(\mathbb{I})$ of weight $(k,\varepsilon)$. In
particular, the specialization $\rho_{\nu}$ of $\rho_{F}$ at any
$\nu\in\mathcal{X}(\mathbb{I})$ is isomorphic to the usual Eichler-Shimura-
Deligne representation attached to $F_{\nu}$.

Given a character $\psi$ of prime-to-$p$ conductor, there exists a unique
primitive Hida family $F_{\psi}$ underlying $F\otimes\psi$. In particular,
$F_{\psi}$ has tame character $\theta\psi^{2}$ and if $\psi$ is unramified at
a prime $\ell\nmid N$, then $a_{\ell}(F_{\psi})=\psi(\ell)a_{\ell}(F)$. We say
that $F$ is twist-minimal if $F$ has minimal level among its twists.

Let $\pi=\otimes^{\prime}_{\ell}\pi_{\ell}$ be the automorphic representation
of $\operatorname{GL}_{2}(\mathbb{A})$ generated by $F^{\circ}_{\nu}$ for some
arithmetic weight $\nu\in\mathcal{X}(\mathbb{I})$. By [Dim14, §6.1], the
automorphic type of $\pi_{\ell}$ at a certain prime $\ell$ does not depend on
the choice of $\nu$. For any prime $\ell$, we then define

(18)
$E_{\ell}(\operatorname{ad}F)=\left\\{\begin{array}[]{ll}(\ell-1)(a_{\ell}(F)^{2}-\theta_{\mathbb{I}}(\ell)(1+\ell)^{2})&\textrm{if
$\ell\nmid N$}\\\ 1-\ell^{-1}&\textrm{if $\pi_{\ell}$ is a ramified principal
series}\\\ 1-\ell^{-2}&\textrm{if $\pi_{\ell}$ is unramified special}\\\
1&\textrm{if $\pi_{\ell}$ is supercuspidal}.\\\ \end{array}\right.$

Let $N_{0}$ be the tame conductor of the residual representation $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ of $F$ and let
$\Sigma$ be a finite set of primes containing $\\{\ell\ \colon\
\ell|(N/N_{0})\\}$ but not $p$. Denote by $\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu$ (resp. $V_{\nu}$) the
underlying space of $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu$ (resp. of $\rho_{\nu}$ over $\operatorname{Frac}(\mathcal{O}_{\nu})$),
by $\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu_{I_{\ell}}$
the co-invariants of $\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu$ under the action of the inertia group $I_{\ell}$ at $\ell$ and put
$m_{\ell}=\dim\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu_{I_{\ell}}$. Then, the invariance of the Swan conductor under reduction
([Liv89, §1] implies

(19) $\operatorname{cond}_{\ell}(\rho_{\nu})-\operatorname{cond}_{\ell}(\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu)=\dim\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu^{I_{\ell}}-\dim(V_{\nu})^{I_{\ell}}$

for every $\nu\in\mathcal{X}(\mathbb{I})$ and every prime $\ell\neq p$, where
$\operatorname{cond}_{\ell}$ refers to the local conductor at $\ell$ of Galois
representations. In particular, $N$ is a divisor of

(20) $N_{\Sigma}=N_{0}\prod_{\ell\in\Sigma}\ell^{m_{\ell}}.$

######  Lemma 3.1.1.

Let $\Sigma_{sc}$ be the set of supercuspidal primes of $F$ in $\Sigma$.

  1. (1)

If $F$ is twist-minimal, then
$\Sigma=\\{\ell\;\colon\;\ell|(N_{\Sigma}/N)\\}\cup\Sigma_{sc}$.

  2. (2)

Assume $F=F^{\prime}_{\psi}$ for some Dirichlet character $\psi$ unramified
outside $\Sigma$ and $F^{\prime}$ a twist-minimal newform. Then

$\\{\ell\colon\ell|(N_{\Sigma}/N)\\}\cup\Sigma_{sc}=\\{\ell\in\Sigma\colon\psi_{(\ell)}=1\mbox{
or }\psi_{(\ell)}=\theta_{(\ell)}\\}\cup\Sigma_{sc},$

where the index $(\ell)$ stands for the $\ell$-primary component of a
Dirichlet character. Moreover, for any $\ell|(N_{\Sigma}/N)$,
$E_{\ell}(\operatorname{ad}F)$ and $E_{\ell}(\operatorname{ad}F^{\prime})$ are
equal up to a unit in $\mathbb{I}$.

###### Proof.

Note that (1) follows from (2). Let $N^{\prime}$ and $\theta^{\prime}$ be the
exact level and the character of $F^{\prime}$ respectively. For
$\ell\in\Sigma\setminus S_{sc}$, (19) and the explicit description of the
local Galois representation attached to $F^{\prime}$ at $\ell$ ([Dim14, Lem.
6.3]) shows that $\ell$ divides $N_{\Sigma}/N$ if and only if $N$ and
$N^{\prime}$ share the same $\ell$-adic valuation. This condition is clearly
equivalent to $\psi_{(\ell)}=1$ or
$\psi_{(\ell)}=(\theta^{\prime}_{(\ell)})^{-1}=\theta_{(\ell)}$.

Regarding the last claim, if $\ell|(N_{\Sigma}/N)$, then
$E_{\ell}(\operatorname{ad}F^{\prime})=E_{\ell}(\operatorname{ad}F)$ unless
$\ell\nmid N^{\prime}$, in which case we have
$E_{\ell}(\operatorname{ad}F^{\prime})=\psi(\ell)^{2}E_{\ell}(\operatorname{ad}F)$.
∎

######  Theorem 3.1.2.

Assume $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$
satisfies (CR). Let $\mathfrak{h}^{\prime}(N_{\Sigma}p^{\infty})$ be the
$\Lambda$-subalgebra of $\mathfrak{h}(N_{\Sigma}p^{\infty})$ generated by the
Hecke operators $T_{n}$ for all $n\geq 1$ coprime to $N_{\Sigma}p$.

  1. (1)

There exists a unique local component
$\mathbb{T}^{\prime}_{\Sigma}=e\mathfrak{h}^{\prime}(N_{\Sigma}p^{\infty})_{\mathfrak{m}^{\prime}_{\Sigma}}$
of $e\mathfrak{h}^{\prime}(N_{\Sigma}p^{\infty})$ such that there exists an
isomorphism $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu\simeq\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu_{\mathfrak{m}^{\prime}_{\Sigma}}$ respecting the ordinary filtrations.

  2. (2)

There exists a unique maximal ideal $\mathfrak{m}_{\Sigma}\subset
e\mathfrak{h}(N_{\Sigma}p^{\infty})$ lifting $\mathfrak{m}^{\prime}_{\Sigma}$
and such that $\mathbb{T}^{\prime}_{\Sigma}\simeq\mathbb{T}_{\Sigma}$ for
$\mathbb{T}_{\Sigma}=e\mathfrak{h}(N_{\Sigma}p^{\infty})_{\mathfrak{m}_{\Sigma}}$.
Moreover, $T_{\ell}\in\mathfrak{m}_{\Sigma}$ for all $\ell\in\Sigma$ and the
image of $T_{\ell}$ in the localization $\mathbb{T}_{\Sigma}$ vanishes.

###### Proof.

This is the content of [EPW06, Thm. 2.4.1 and Prop. 2.4.2]. ∎

Denote by $F_{\Sigma}=\sum_{n}a_{n}(F_{\Sigma})$ the normalized eigenform of
$S(N_{\Sigma},\mathbb{I})$ sharing the same Hecke eigenvalues as $F$ at primes
not dividing $N_{\Sigma}/N$. We have $a_{n}(F_{\Sigma})=0$ for $n$ divisible
by a prime in $\Sigma$ and $a_{n}(F_{\Sigma})=a_{n}(F)$ otherwise. Note that
by Theorem 3.1.2 (2), $\mathbb{T}_{\Sigma}$ is reduced, so $F_{\Sigma}$
satisfies (pr) and it makes sense to consider its adjoint $p$-adic
$L$-function.

######  Proposition 3.1.3.

Fix a choice of canonical periods $\xi^{\pm}_{F}$ for $F$ as in Definition
2.3.3. There is a choice of periods $\xi^{\pm}_{F_{\Sigma}}$ for $F_{\Sigma}$
such that

$L_{p}(\operatorname{ad}F_{\Sigma})=L_{p}(\operatorname{ad}F)\prod_{\ell|(N_{\Sigma}/N)}E_{\ell}(\operatorname{ad}F),$

where $L_{p}(\operatorname{ad}F_{\Sigma})$ (resp. $L_{p}(\operatorname{ad}F)$)
is computed with respect to $\xi^{\pm}_{F_{\Sigma}}$ (resp. to
$\xi^{\pm}_{F}$).

###### Proof.

We keep the notations $\mathfrak{m}^{*}$, $\mathbb{T}^{*}$, $\mathcal{M}$ and
$\lambda^{*}_{F}$ of §2.3 and we denote by $\mathfrak{m}^{*}_{\Sigma}$,
$\mathbb{T}^{*}_{\Sigma}$, $\mathcal{M}_{\Sigma}$ and
$\lambda^{*}_{F_{\Sigma}}$ their counterpart for $F$ replaced with
$F_{\Sigma}$.

First note that $F$ is $\ell$-minimal at any divisor $\ell$ of $N_{\Sigma}/N$
by Lemma 3.1.1, yielding $a_{\ell}(F)\in\mathbb{I}^{\times}$. Write
$N_{\Sigma}/N$ as a product of prime powers $\prod_{i=1}^{d}\ell_{i}^{e_{i}}$
with $1\leq e_{i}\leq 2$ and $\ell_{i}\in\Sigma$. Then $e_{i}=2$ only if
$\ell_{i}\nmid N$. We define a $\mathbb{I}$-linear map

(21)
$\varepsilon=\varepsilon_{1}\circ\ldots\circ\varepsilon_{d}\colon\mathcal{M}_{\Sigma}\to\mathcal{M}$

as follows. Fix $1\leq i\leq d$, and let $\ell=\ell_{i}$, $e=e_{i}$ and
$M=N\ell_{1}^{e_{1}}\ldots\ell_{i-1}^{e_{i-1}}$. At level $Mp^{r}$, letting
$\Gamma=\Gamma_{1}(Mp^{r})$ and $\Gamma^{\prime}=\Gamma_{1}(M\ell^{e}p^{r})$
we have a map
$\varepsilon_{M,\ell}\colon\mathrm{H}^{1}(X_{1}(M\ell^{e}p^{r}),{\mathbb{Z}_{p}})\to\mathrm{H}^{1}(X_{1}(Mp^{r}),{\mathbb{Z}_{p}})$
given by

$\varepsilon_{M,\ell}(x)=\left\\{\begin{array}[]{ll}x\big{|}(\left[\Gamma^{\prime}\Gamma\right]-\ell^{-1}V_{\ell}^{*}T_{\ell}^{*})&\textrm{if
$e=1$}\\\
x\big{|}(\left[\Gamma^{\prime}\Gamma\right]-\ell^{-1}V_{\ell}^{*}T_{\ell}^{*}+\ell^{-1}V_{\ell^{2}}^{*}\langle\ell\rangle^{-1})&\textrm{if
$e=2$,}\end{array}\right.$

where $V_{\ell^{m}}^{*}=\left[\Gamma^{\prime}\begin{pmatrix}1&0\\\
0&\ell^{m}\end{pmatrix}\Gamma\right]$. Taking the limit over $r\geq 1$,
localizing and extending the scalars to $\mathbb{I}$, one obtains the map
which we denoted $\varepsilon_{i}$ in (21). Under Poincaré duality,
$\varepsilon$ coincides with (the extension of scalars of) the map on homology
introduced in [EPW06, §3.8] and denoted $\varepsilon_{\infty}$ therein.

Denote by $(-)^{\textbf{t}}$ the adjoint with respect to Ohta’s pairings (12)
or their specializations (13) at tame level $M$. For $\ell=\ell_{i}$ as
before, one finds that

$\varepsilon_{M,\ell}^{\textbf{t}}(x)=\left\\{\begin{array}[]{ll}x\big{|}(V_{\ell}-\ell^{-1}T_{\ell}^{*}\left[\Gamma\Gamma^{\prime}\right])&\textrm{if
$e=1$}\\\
x\big{|}(V_{\ell^{2}}-T_{\ell}^{*}V_{\ell}+\ell^{-1}\langle\ell\rangle^{-1}\left[\Gamma\Gamma^{\prime}\right])&\textrm{if
$e=2$,}\end{array}\right.$

where $V_{\ell^{m}}=\left[\Gamma\begin{pmatrix}\ell^{m}&0\\\
0&1\end{pmatrix}\Gamma^{\prime}\right]$. In particular, we obtain

$(\varepsilon\circ\varepsilon^{\textbf{t}})_{|\mathcal{M}_{\lambda^{*}_{F}}}=\prod_{\ell|(N_{\Sigma}/N)}E^{\prime}_{\ell}(\operatorname{ad}F)\operatorname{id},$

where $E^{\prime}_{\ell}(\operatorname{ad}F)=-\ell
a_{\ell}(F)E_{\ell}(\operatorname{ad}F)$ if $\pi_{\ell}$ is special or a
ramified principal series and $E^{\prime}_{\ell}(\operatorname{ad}F)=-\ell
E_{\ell}(\operatorname{ad}F)$ if $\pi_{\ell}$ is spherical.

Passing to the $\pm$-part, the map $\varepsilon^{\textbf{t}}$ injects
$\mathcal{M}_{\lambda^{*}_{F}}^{\pm}$ into
$(\mathcal{M}^{\pm}_{\Sigma})_{\lambda^{*}_{F_{\Sigma}}}$. Moreover, [EPW06,
Thm. 3.6.2] implies that $\varepsilon$ is injective on
$(\mathcal{M}^{\pm}_{\Sigma})_{\lambda^{*}_{F_{\Sigma}}}$ so, in fact,
$\varepsilon^{\textbf{t}}(\mathcal{M}^{\pm}_{\lambda^{*}_{F}})=(\mathcal{M}^{\pm}_{\Sigma})_{\lambda^{*}_{F_{\Sigma}}}$
by duality. As the element
$u=\prod_{\ell|(N_{\Sigma}/N)}E_{\ell}(\operatorname{ad}F)/E^{\prime}_{\ell}(\operatorname{ad}F)$
belongs to $\mathbb{I}^{\times}$, we may take
$\xi_{F_{\Sigma}}^{+}=u\cdot\varepsilon^{\textbf{t}}(\xi^{+}_{F})$ (resp.
$\xi_{F_{\Sigma}}^{-}=\varepsilon^{\textbf{t}}(\xi^{-}_{F})$) as a basis of
$(\mathcal{M}^{+}_{\Sigma})_{\lambda^{*}_{F_{\Sigma}}}$ (resp. of
$(\mathcal{M}^{-}_{\Sigma})_{\lambda^{*}_{F_{\Sigma}}}$). Letting
$(\cdot,\cdot)_{N,\mathbb{I}}$ (resp. $(\cdot,\cdot)_{N_{\Sigma},\mathbb{I}}$)
be Ohta’s pairing in tame level $N$ (resp. tame level $N_{\Sigma}$), we
therefore obtain

$\displaystyle L_{p}(\operatorname{ad}F_{\Sigma})$
$\displaystyle=(u\cdot\varepsilon^{\textbf{t}}(\xi^{+}_{F}),\varepsilon^{\textbf{t}}(\xi^{-}_{F}))_{N_{\Sigma},\mathbb{I}}$
$\displaystyle=u\cdot((\varepsilon\circ\varepsilon^{\textbf{t}})(\xi^{+}_{F}),\xi^{-}_{F})_{N,\mathbb{I}}$
$\displaystyle=L_{p}(\operatorname{ad}F)\prod_{\ell|(N_{\Sigma}/N)}E_{\ell}(\operatorname{ad}F),$

as wanted. ∎

An easy adaptation of the proof of Proposition 3.1.3 (arguing as in the proof
of [Hsi21, Lem. 7.4]) yields the following proposition.

######  Proposition 3.1.4.

Assume $F\in S(N,\mathbb{I})$ is a twist-minimal primitive Hida family, and
let $F_{\psi}$ be the primitive twist of $F$ by a Dirichlet character $\psi$
of conductor $M$ coprime with $p$. Then we can make a choice of canonical
periods for $F$ and $F_{\psi}$ such that

$L_{p}(\operatorname{ad}F_{\psi})=L_{p}(\operatorname{ad}F)\prod_{\ell}E_{\ell}(\operatorname{ad}F),$

where $\ell$ runs over the set of primes dividing $M$ such that
$\psi_{(\ell)}\neq\mathds{1},\theta^{-1}_{(\ell)}$.

In the next theorem, we let vary $F$ through the set of Hida families with
fixed residual representation $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ which are minimally
ramified outside a fixed finite set $\Sigma$.

######  Theorem 3.1.5.

Assume $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$
satisfies (CR). There exists a regular element
$L_{\Sigma}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu)\in\mathbb{T}_{\Sigma}$ such that, for all primitive Hida families $F$
with residual representation $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ which are minimally
ramified outside $\Sigma$, we have

$\lambda_{F_{\Sigma}}(L_{\Sigma}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu))=L_{p}(\operatorname{ad}F)\prod_{\ell|(N_{\Sigma}/N)}E_{\ell}(\operatorname{ad}F)$

up to a unit in $\mathbb{I}$, where
$\lambda_{F_{\Sigma}}\colon\mathbb{T}_{\Sigma}\to\mathbb{I}$ is the system of
eigenvalues associated with $F_{\Sigma}$.

###### Proof.

We simply define $L_{\Sigma}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu)$ as the adjoint
$p$-adic $L$-function of Definition 2.4.2 for $\mathbb{T}_{\Sigma}$ and
relative to some choice of canonical periods. Any primitive Hida family $F$ of
level $N$ with $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu_{F}\simeq\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu$ which is minimally ramified outside $\Sigma$ satisfies
$\\{\ell|(N/N_{0})\\}\subset\Sigma$, so Theorem 2.4.3 and Proposition 3.1.3
apply to $F_{\Sigma}$, yielding

$\lambda_{F_{\Sigma}}(L_{\Sigma}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu))=L_{p}(\operatorname{ad}F_{\Sigma})=L_{p}(\operatorname{ad}F)\prod_{\ell|(N_{\Sigma}/N)}E_{\ell}(\operatorname{ad}F)$

up to units in $\mathbb{I}$. The fact that $L_{\Sigma}(\operatorname{ad}\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu)$ is regular is
clear, as $L_{p}(\operatorname{ad}F_{\Sigma})$ does not identically vanish and
$\mathbb{T}_{\Sigma}$ is reduced. ∎

###  3.2. Selmer groups

Let $F(q)=\sum_{n\geq 1}a_{n}(F)q^{n}\in S(N,\mathbb{I})$ be a twist-minimal
primitive Hida family of tame level $N$ and character $\theta$, and let
$V_{F}$ be the free $\mathbb{I}$-module of rank $2$ on which $\rho_{F}$ acts.
We write $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ for
$\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu_{F}$, that
is, $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern
1.5mu=\rho_{F}\otimes_{\mathbb{I}}k$, where
$k=\mathbb{I}/\mathfrak{m}_{\mathbb{I}}$ is the residue field of the local
$\Lambda$-algebra $\mathbb{I}$.

Recall that $V_{F}$ is ordinary in the sense that there exists an exact
sequence of free $\mathbb{I}[G_{{\mathbb{Q}_{p}}}]$-modules

${0}$${V_{F}^{+}}$${V_{F}}$${V_{F}^{-}}$${0,}$

where $G_{{\mathbb{Q}_{p}}}$ acts on $V_{F}^{-}$ via the unramified character
$\mathrm{ur}_{a_{p}(F)}$ sending $\sigma_{p}$ on $a_{p}(F)$. On the traceless
adjoint $V_{\operatorname{ad}}=\operatorname{Hom}^{0}(V_{F},V_{F})$ of
$V_{F}$, there is a three-step $G_{{\mathbb{Q}_{p}}}$-filtration

(22) $0\subset V_{\operatorname{ad}}^{+}\subset
V_{\operatorname{ad}}^{\dagger}\subset V_{\operatorname{ad}},$

where $V_{\operatorname{ad}}^{+}=\operatorname{Hom}(V_{F}^{-},V_{F}^{+})$ is
the subspace of “upper-triangular” nilpotent endomorphisms and
$V_{\operatorname{ad}}^{\dagger}$ consists of the space of endomorphisms in
$V_{\operatorname{ad}}$ fixing $V_{F}^{+}$. We put
$V_{\operatorname{ad}}^{-}=V_{\operatorname{ad}}/V_{\operatorname{ad}}^{+}$.

Let
$M^{\vee}=\operatorname{Hom}_{{\mathbb{Z}_{p}}}(M,{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$
for any profinite ${\mathbb{Z}_{p}}$-module $M$ and let $\Sigma$ be a set of
finite places of $\mathbb{Q}$ not containing $p$. We define Selmer groups à la
Greenberg, using the discrete modules
$D_{\operatorname{ad}}=V_{\operatorname{ad}}\otimes_{\mathbb{I}}\mathbb{I}^{\vee}$
and
$D^{\pm}_{\operatorname{ad}}=V^{\pm}_{\operatorname{ad}}\otimes_{\mathbb{I}}\mathbb{I}^{\vee}$,
by letting

(23)
$\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)=\ker\left[\mathrm{H}^{1}(\mathbb{Q},D_{\operatorname{ad}})\to\prod_{\ell\notin\Sigma\cup\\{p\\}}\mathrm{H}^{1}(I_{\ell},D_{\operatorname{ad}})\times\mathrm{H}^{1}(I_{p},D^{-}_{\operatorname{ad}})\right],$

and
$\operatorname{Sel}(\operatorname{ad}F)=\operatorname{Sel}^{\emptyset}(\operatorname{ad}F)$.
Standard arguments show that
$\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)^{\vee}$ is a finitely
generated $\mathbb{I}$-module.

######  Theorem 3.2.1.

Assume (CR’) holds for $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$. Then
$\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)^{\vee}$ is a finitely
generated torsion $\mathbb{I}$-module. If $\Sigma$ contains all the places
dividing $N$, then

$\textrm{char}_{\mathbb{I}}\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)^{\vee}=L_{p}(\operatorname{ad}F_{\Sigma})\cdot\mathbb{I}.$

###### Proof.

We may assume in the the first claim that $\Sigma$ contains the primes
dividing $N$. As $L_{p}(\operatorname{ad}F_{\Sigma})$ is not identically $0$
(it does not vanish at any $\nu\in\mathcal{X}(\mathbb{I})$ such that
$F_{\nu}=F_{\nu}^{\circ}$ by Thm. 2.3.7), the torsionness of
$\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)^{\vee}$ will follow from the
second claim which we now prove.

Let $\mathcal{O}=W(k)$, $\Lambda_{\mathcal{O}}=\mathcal{O}[[U_{1}]]$ and let
$\mathbb{T}_{\Sigma}$ be the local component associated with $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ and $\Sigma$ as in
Theorem 3.1.2. Then $F_{\Sigma}$ gives rise to a homomorphism of
$\Lambda_{\mathcal{O}}$-algebras
$\lambda_{\Sigma}\colon\mathbb{T}_{\Sigma}\to\mathbb{I}$.

Consider the following deformation problem. Let $\Phi$ be the functor
classifying $p$-ordinary lifts of $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ unramified outside
$S:=\Sigma\cup\\{p\\}$ modulo strict equivalence. More precisely, let
$\mathbb{Q}_{S}\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern 1.5mu$ be the maximal
extension of $\mathbb{Q}$ unramified outside $S$ and let $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu^{\pm}$ be the
$k$-valued $G_{{\mathbb{Q}_{p}}}$-representation acting on
$V_{F}^{\pm}\otimes_{\mathbb{I}}k$. For any local complete noetherian
$\mathcal{O}$-algebra $A$ with residue field $k$, $\Phi(A)$ consists of
classes $[\rho]_{\sim}$, where $\rho\colon
G_{\mathbb{Q},S}=\operatorname{Gal}(\mathbb{Q}_{S}/\mathbb{Q})\to\operatorname{GL}_{2}(A)$
is such that $\rho\otimes_{A}k\simeq\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ and such that there
exists a $G_{\mathbb{Q}_{p}}$-equivariant exact sequence of free $A$-modules
$0\to\rho^{+}\to\rho\to\rho^{-}\to 0$ with $\rho^{\pm}\otimes_{A}k\simeq\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu^{\pm}$ and $\rho^{-}$
unramified. The equivalence relation $\sim$ here means that
$\rho\sim\rho^{\prime}$ if $\rho^{\prime}$ can be obtained from $\rho$ by
conjugation by a matrix in
$\ker(\operatorname{GL}_{2}(A)\to\operatorname{GL}_{2}(k))$. Since $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$ is absolutely
irreducible, $\Phi$ is representable by a universal pair
$(\rho_{\Sigma},R_{\Sigma})$, $[\rho_{\Sigma}]_{\sim}\in\Phi(R_{\Sigma})$ by
Mazur’s theory [Maz97]. Under our running hypotheses on $\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$, by the work of
Wiles, Taylor-Wiles and Diamond [Wil95, TW95, Dia96] (see also [EPW06, p.
539]), we then have an isomorphism

(24) $R_{\Sigma}\simeq\mathbb{T}_{\Sigma}.$

In fact, $\Lambda_{\mathcal{O}}$ can be interpreted as the universal
deformation ring associated with $\det(\mkern
1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu)$, so $R_{\Sigma}$
comes equipped with a structure of $\Lambda_{\mathcal{O}}$-algebra, and (24)
is an isomorphism of $\Lambda_{\mathcal{O}}$-algebras. A deep consequence of
(24) is that $\mathbb{T}_{\Sigma}$ is a local complete intersection
$\Lambda_{\mathcal{O}}$-algebra. This provides us, by a theorem of Tate
[Hid16, Thm. 6.8], with an equality

$\textrm{char}_{\mathbb{I}}C_{0}(\lambda_{\Sigma})=\textrm{char}_{\mathbb{I}}C_{1}(\lambda_{\Sigma}),$

where
$C_{1}(\lambda_{\Sigma})=\Omega_{\mathbb{T}_{\Sigma}/\Lambda_{\mathcal{O}}}\otimes_{\mathbb{T}_{\Sigma},\lambda_{\Sigma}}\mathbb{I}$
(and the symbol $\Omega$ here refers to continuous Kähler differentials). In
light of Proposition 2.3.4, it is enough to show that
$C_{1}(\lambda_{\Sigma})^{\vee}\simeq\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)$
as $\mathbb{I}$-modules, which is a standard fact from Mazur’s theory (see
e.g. [MT90, §6] and [Maz97, §30]), but we explain the main lines for sake of
completeness. We have:

$\displaystyle
C_{1}(\lambda_{\Sigma})^{\vee}=\operatorname{Hom}_{{\mathbb{Z}_{p}}}(\Omega_{\mathbb{T}_{\Sigma}/\Lambda_{\mathcal{O}}}\otimes_{\mathbb{T}_{\Sigma},\lambda_{\Sigma}}\mathbb{I},{\mathbb{Q}_{p}}/{\mathbb{Z}_{p}})$
$\displaystyle\simeq\operatorname{Hom}_{\mathbb{T}_{\Sigma}}(\Omega_{\mathbb{T}_{\Sigma}/\Lambda_{\mathcal{O}}},\mathbb{I}^{\vee})$
$\displaystyle\simeq\operatorname{Der}_{\Lambda_{\mathcal{O}}}(\mathbb{T}_{\Sigma},\mathbb{I}^{\vee}),$

the last term referring to the module of $\Lambda_{\mathcal{O}}$-linear
continuous derivations (and $\mathbb{I}^{\vee}$ being seen as a discrete
$\mathbb{T}_{\Sigma}$-module via $\lambda_{\Sigma}$). For $n\geq 1$, let
$A_{n}=(\mathbb{I}/\mathfrak{m}_{\mathbb{I}}^{n})^{\vee}$ which we see as a
$\mathbb{T}_{\Sigma}$-module, and note that
$\operatorname{Der}_{\Lambda_{\mathcal{O}}}(\mathbb{T}_{\Sigma},\mathbb{I}^{\vee})=\cup_{n\geq
1}\operatorname{Der}_{\Lambda_{\mathcal{O}}}(\mathbb{T}_{\Sigma},A_{n})$. The
free $\mathbb{I}$-module $\mathbb{I}[A_{n}]$ generated by elements of $A_{n}$
also has a structure of $\mathbb{T}_{\Sigma}$-algebra if we let $t\cdot
a=\lambda_{\Sigma}(t)a$ and $a\cdot b=0$ for all $t\in\mathbb{T}_{\Sigma}$ and
$a,b\in A_{n}$. There is a natural injective map

(25)
$\operatorname{Der}_{\Lambda_{\mathcal{O}}}(\mathbb{T}_{\Sigma},A_{n})\hookrightarrow\operatorname{Hom}_{\Lambda_{\mathcal{O}}}(\mathbb{T}_{\Sigma},\mathbb{I}[A_{n}])$

sending a derivation $\operatorname{d}\colon\mathbb{T}_{\Sigma}\to A_{n}$ to
$\phi_{\operatorname{d}}=\lambda_{\Sigma}+\iota\circ\operatorname{d}$, where
$\iota\colon A_{n}\hookrightarrow\mathbb{I}[A_{n}]$ is the natural injection.
Further, the map (25) has image
$\mathcal{H}_{n}:=\\{\phi\colon\phi\equiv\lambda_{\Sigma}\mod
A_{n}\\}\subseteq\operatorname{Hom}_{\Lambda_{\mathcal{O}}}(\mathbb{T}_{\Sigma},\mathbb{I}[A_{n}])$.
By the universal property of $\mathbb{T}_{\Sigma}$, it holds that

$\mathcal{H}_{n}=\left\\{[\rho]_{\sim}\in\Phi(\mathbb{I}[A_{n}])\ :\
\rho\equiv\rho_{F}\mod A_{n}\right\\}.$

Now, given a representative $\rho$ of a class
$[\rho]_{\sim}\in\Phi(\mathbb{I}[A_{n}])$, the condition $\rho\equiv\rho_{F}$
mod $A_{n}$ shows that the map $c\colon
G_{\mathbb{Q},S}\to\operatorname{End}(V_{F})\otimes_{\mathbb{I}}A_{n}$ defined
by the relation $\rho=\rho_{F}\cdot(1+c)$ is a cocycle in
$\mathrm{H}^{1}(\mathbb{Q}_{S}/\mathbb{Q},V_{\operatorname{ad}}\otimes_{\mathbb{I}}A_{n})$.
Moreover, $c$ is replaced by a cohomologous cocycle if $\rho$ is replaced by a
representation strict-equivalent to it. Therefore, the map

$\mathcal{H}_{n}\to\mathrm{H}^{1}(\mathbb{Q}_{S}/\mathbb{Q},V_{\operatorname{ad}}\otimes_{\mathbb{I}}A_{n})$

induced by $\rho\mapsto c_{\rho}=\rho\cdot\rho_{F}^{-1}-1$ is well-defined and
injective, and its image consists of classes of cocycles $c$ such that
$\rho=\rho_{F}\cdot(1+c)$ is $p$-ordinary. We check that this condition on
$c_{\rho}$ amounts to $c_{\rho}(I_{p})\subseteq
V_{\operatorname{ad}}^{+}\otimes A_{n}$, that is, $c_{\rho}(I_{p})$ consists
of upper-nilpotent matrices in an ordinary basis for $\rho$.

If $\rho$ is $p$-ordinary, then $\rho_{|I_{p}}$ and $\rho_{F|I_{p}}$ look like
$\left(\begin{smallmatrix}*&*\\\ 0&1\end{smallmatrix}\right)$ in their common
ordinary basis, and hence, it is clear that
$c_{\rho}(I_{p})\subseteq\left(\begin{smallmatrix}*&*\\\
0&0\end{smallmatrix}\right)\cap(V_{\operatorname{ad}}\otimes
A_{n})=V^{+}_{\operatorname{ad}}\otimes A_{n}$. Conversely, we obtain a
$p$-ordinary basis for $\rho$ by tensoring that of $\rho_{F}$ by
$\mathbb{I}[A_{n}]$ over $\mathbb{I}$. This proves that

$\mathcal{H}_{n}\simeq\ker\left[\mathrm{H}^{1}(\mathbb{Q}_{S}/\mathbb{Q},V_{\operatorname{ad}}\otimes_{\mathbb{I}}(\mathbb{I}/\mathfrak{m}_{\mathbb{I}}^{n})^{\vee})\to\mathrm{H}^{1}(I_{p},V_{\operatorname{ad}}^{-}\otimes_{\mathbb{I}}(\mathbb{I}/\mathfrak{m}_{\mathbb{I}}^{n})^{\vee}))\right]$

for all $n$, and by taking the direct limit over $n\geq 1$, one sees that
$C_{1}(\lambda_{\Sigma})^{\vee}\simeq\varinjlim_{n}\mathcal{H}_{n}\simeq\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)$
as claimed. ∎

The aim of the rest of the section is to relax the assumption on $\Sigma$ in
Thm. 3.2.1. We introduce more notations and we let, for any prime $\ell$,

(26)
$\mathrm{H}^{1}_{\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})=\left\\{\begin{array}[]{ll}\ker\left[\mathrm{H}^{1}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})\to\mathrm{H}^{1}(I_{\ell},D_{\operatorname{ad}})\right]&\mbox{if
$\ell\neq p$}\\\
\ker\left[\mathrm{H}^{1}(\mathbb{Q}_{p},D_{\operatorname{ad}})\to\mathrm{H}^{1}(I_{p},D^{-}_{\operatorname{ad}})\right]&\mbox{if
$\ell=p$,}\end{array}\right.$

and we put

(27)
$\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})=\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})}{\mathrm{H}^{1}_{\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})}.$

In particular, for any finite sets of primes $\Theta\subseteq\Sigma$ not
containing $p$, we have an exact sequence

${0}$${\operatorname{Sel}^{\Theta}(\operatorname{ad}F)}$${\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)}$${\prod_{\ell\in\Sigma-\Theta}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}}).}$

######  Proposition 3.2.2.

Let $\ell\neq p$ be a rational prime. Then
$\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})$
is of $\mathbb{I}$-cotorsion. Moreover,
$\textrm{char}_{\mathbb{I}}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})^{\vee}=E_{\ell}(\operatorname{ad}F)\cdot\mathbb{I}$,
where $E_{\ell}(\operatorname{ad}F)$ is as in (18), unless $\ell$ belongs to
the set $\Sigma_{e}$ defined in (3), in which case
$\textrm{char}_{\mathbb{I}}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})^{\vee}=(1+\ell)^{-1}\cdot\mathbb{I}$.

###### Proof.

Let $G_{\ell}=G_{\mathbb{Q}_{\ell}}$ and let $\sigma_{\ell}\in
G_{\ell}/I_{\ell}$ be the arithmetic Frobenius. By the inflation-restriction
exact sequence,
$\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})\simeq\mathrm{H}^{1}(I_{\ell},D_{\operatorname{ad}})^{G_{\ell}/I_{\ell}}$
as $\mathbb{I}$-modules. Let $\psi\colon G_{\ell}\to\mathbb{I}^{\times}$ be a
continuous character and let $D=\mathbb{I}^{\vee}(\psi)$ be the cofree
$\mathbb{I}$-module of rank one endowed with a $G_{\ell}$-action given by
$\psi$. We claim that

$\mathcal{H}:=\mathrm{H}^{1}(I_{\ell},D)^{G_{\ell}/I_{\ell}}=\left\\{\begin{array}[]{ll}(\mathbb{I}/(1-\psi(\sigma_{\ell})\ell^{-1})\mathbb{I})^{\vee}&\mbox{if
$\psi$ is unramified,}\\\ 0&\mbox{if $\psi$ is ramified.}\end{array}\right.$

Recall that $I_{\ell}$ fits into an exact sequence of
$G_{\ell}/I_{\ell}$-modules

${0}$${J_{\ell}}$${I_{\ell}}$${{\mathbb{Z}_{p}}(1)}$${0}$

with $J_{\ell}$ of prime-to-$p$ order. This property of $J_{\ell}$ shows that
the inflation map induces an isomorphism
$\mathrm{H}^{1}(I_{\ell}/J_{\ell},D^{J_{\ell}})\simeq\mathrm{H}^{1}(I_{\ell},D)$,
and moreover $D^{J_{\ell}}$ is $\mathbb{I}$-divisible. Since $D$ is
$\mathbb{I}$-cofree of corank $1$, either $D^{J_{\ell}}=0$ or $D^{J_{\ell}}=D$
holds. In the first case, we then have $\mathcal{H}=0$ and $\psi$ is ramified,
so we may assume $D^{J_{\ell}}=D$. This means that $\psi_{|I_{\ell}}$ factors
through $I_{\ell}/J_{\ell}$, which is procyclic, so
$\mathrm{H}^{1}(I_{\ell}/J_{\ell},D^{J_{\ell}})\simeq D/(t-1)D$, where
$t=\psi(g)$ for a topological generator $g$ of $I_{\ell}/J_{\ell}$. Of course,
$t\neq 1$ if $\psi$ is ramified, in which case $\mathcal{H}\hookrightarrow
D/(t-1)D=0$ as $D$ is divisible. In the unramified case, one has

$\displaystyle\mathcal{H}=\operatorname{Hom}(I_{\ell},D)^{G_{\ell}/I_{\ell}}\simeq\operatorname{Hom}({\mathbb{Z}_{p}}(1),D)^{G_{\ell}/I_{\ell}}=$
$\displaystyle D(-1)^{G_{\ell}/I_{\ell}}$ $\displaystyle=$ $\displaystyle
D[1-\psi(\sigma_{\ell})\ell^{-1}]$ $\displaystyle=$
$\displaystyle(\mathbb{I}/(1-\psi(\sigma_{\ell})\ell^{-1})\mathbb{I})^{\vee},$

as claimed.

Now we turn back to the proof of the lemma. If $\ell\nmid N$, then

(28)
$D_{\operatorname{ad}}\simeq\mathbb{I}^{\vee}(\chi_{1}\chi_{2}^{-1})\oplus\mathbb{I}^{\vee}\oplus\mathbb{I}^{\vee}(\chi_{2}\chi_{1}^{-1})$

as $\mathbb{I}[G_{\ell}]$-modules, where $\chi_{1},\chi_{2}$ are the
unramified characters such that $1-a_{\ell}(F)X+\theta_{\mathbb{I}}(\ell)\ell
X^{2}=(1-\chi_{1}(\sigma_{\ell})X)(1-\chi_{2}(\sigma_{\ell})X)$. Up to a unit
of $\mathbb{I}$, $E_{\ell}(\operatorname{ad}F)$ is equal to
$(1-\chi_{1}\chi_{2}^{-1}(\sigma_{\ell})\ell^{-1})(1-\ell^{-1})(1-\chi_{2}\chi_{1}^{-1}(\sigma_{\ell})\ell^{-1})$,
so the first part of the proof shows that
$\textrm{char}_{\mathbb{I}}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})^{\vee}=E_{\ell}(\operatorname{ad}F)\cdot\mathbb{I}$,
as wanted.

In the case where every classical specialization of $F$ is a ramified
principal series at $\ell$, a similar isomorphism as in (28) with $\chi_{1}$
ramified and $\chi_{2}$ unramified holds. In particular, both
$\chi_{1}\chi_{2}^{-1}$ and $\chi_{2}\chi_{1}^{-1}$ are ramified, so
$\textrm{char}_{\mathbb{I}}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})^{\vee}=(1-\ell^{-1})\cdot\mathbb{I}=E_{\ell}(\operatorname{ad}F)\cdot\mathbb{I}$.

Assume now that $F$ is a special at $\ell$. By [Dim14, Lem. 6.3], one has
$\rho_{F|G_{\ell}}\sim\begin{pmatrix}\psi&*\\\ 0&1\end{pmatrix}$, where $\psi$
is the unramified character of $G_{\ell}$ sending $\sigma_{\ell}$ to $\ell$
and $*$ is nontrivial on $I_{\ell}$. In particular,
$\rho_{F}(J_{\ell})=\\{1\\}$ and
$\rho_{F}(I_{\ell})=\begin{pmatrix}1&{\mathbb{Z}_{p}}.a\\\ 0&1\end{pmatrix}$
for some $a\in\mathbb{I}\setminus\\{0\\}$. Let $g\in I_{\ell}$ be such that
$\rho_{F}(g)=\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}$ and consider the adjoint
action of $g$ on $V_{\operatorname{ad}}$. In the basis
$\\{e_{1},e_{2},e_{3}\\}=\\{\begin{pmatrix}0&1\\\
0&0\end{pmatrix},\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\\}$, the matrix of $\operatorname{ad}\rho_{F}(g)$ is

$M=\begin{pmatrix}1&-2a&-a^{2}\\\ 0&1&a\\\ 0&0&1\end{pmatrix}.$

Hence,

$\mathrm{H}^{1}(I_{\ell},D_{\operatorname{ad}})=(D_{\operatorname{ad}})_{I_{\ell}}=D_{\operatorname{ad}}/(M-\operatorname{id})D_{\operatorname{ad}}=\dfrac{\mathbb{I}^{\vee}.e_{1}\oplus\mathbb{I}^{\vee}.e_{2}\oplus\mathbb{I}^{\vee}.e_{3}}{a\mathbb{I}^{\vee}.e_{1}\oplus
a\mathbb{I}^{\vee}.e_{2}\oplus 0.e_{3}}=\mathbb{I}^{\vee}.e_{3},\\\ $

as $M-\operatorname{id}$ has the same image as that of
$\left(\begin{smallmatrix}0&a&0\\\ 0&0&a\\\ 0&0&0\end{smallmatrix}\right)$ as
an endomorphism of $V_{\operatorname{ad}}$. Finally, the action of
$\operatorname{ad}\rho_{F}$ on
$D_{\operatorname{ad}}/(\mathbb{I}^{\vee}.e_{1}\oplus\mathbb{I}^{\vee}.e_{2})$
is given by $\psi^{-1}$, so we obtain

$\mathrm{H}^{1}(I_{\ell},D_{\operatorname{ad}})^{G_{\ell}/I_{\ell}}=\mathrm{H}^{1}(I_{\ell},\mathbb{I}^{\vee}(\psi^{-1}))^{G_{\ell}/I_{\ell}}=(\mathbb{I}/(1-\ell^{-2})\mathbb{I})^{\vee}=(\mathbb{I}/E_{\ell}(\operatorname{ad}F)\mathbb{I})^{\vee}.$

Consider now the last case where every specialization of $F$ is supercuspidal
at $\ell$. First notice that, as
$D_{\operatorname{ad}}^{J_{\ell}}\simeq(D_{\operatorname{ad}})_{J_{\ell}}$
canonically,
$\mathrm{H}^{1}(I_{\ell},D_{\operatorname{ad}})\simeq\mathrm{H}^{1}(I_{\ell}/J_{\ell},D_{\operatorname{ad}}^{J_{\ell}})\simeq(D_{\operatorname{ad}})_{I_{\ell}}$,
which is $\mathbb{I}$-divisible. Assume now $\ell\notin\Sigma_{e}$. For any
$\nu\in\mathcal{X}(\mathbb{I})$, the local $L$-factor at $\ell$ of the adjoint
lift of $F^{\circ}_{\nu}$ is $1$ by [GJ78, Corollary (1.3)]. Thus,
$(V_{\operatorname{ad}})_{I_{\ell}}=0$ by the local Langlands correspondence
and Nakayama’s lemma. It follows that $(D_{\operatorname{ad}})_{I_{\ell}}$ is
cotorsion, hence trivial, and
$\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})$
has characteristic ideal generated by $E_{\ell}(\operatorname{ad}F)=1$.

Assume finally $\ell\in\Sigma_{e}$ and let
$G_{\ell^{2}}=\operatorname{Gal}(\mkern
1.5mu\overline{\mkern-1.5mu\mathbb{Q}\mkern-1.5mu}\mkern
1.5mu_{\ell}/\mathbb{Q}_{\ell^{2}})$. Then
$\rho_{F|G_{\ell}}\simeq\mathrm{Ind}_{G_{\ell^{2}}}^{G_{\ell}}\chi$ for some
ramified character $\chi$ of $G_{\ell^{2}}$. In particular,
$\operatorname{ad}\rho_{F|G_{\ell}}\simeq\mathrm{Ind}_{G_{\ell^{2}}}^{G_{\ell}}\chi/\chi^{\prime}\oplus\eta$,
where $\chi^{\prime}$ is the conjugate of $\chi$ under
$\operatorname{Gal}(\mathbb{Q}_{\ell^{2}}/\mathbb{Q}_{\ell})$ and
$\eta\colon\operatorname{Gal}(\mathbb{Q}_{\ell^{2}}/\mathbb{Q}_{\ell})\simeq\\{\pm
1\\}$ is the unramified quadratic character. Note that $\chi/\chi^{\prime}$ is
ramified, as every classical specialization of $\chi/\chi^{\prime}$ is. Using
that $I_{\ell}\subset G_{\ell^{2}}$, we obtain
$\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})=\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},\mathbb{I}^{\vee}(\eta))$,
hence
$\textrm{char}_{\mathbb{I}}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})=(1-\eta(\sigma_{\ell})\ell^{-1}).\mathbb{I}=(1+\ell^{-1}).\mathbb{I}$.
∎

######  Proposition 3.2.3.

Let $S=\Sigma\cup\\{p,\infty\\}$ and let $\mathbb{Q}_{S}$ be the maximal
extension of $\mathbb{Q}$ unramified outside $S$. The localization map

$\phi\colon\mathrm{H}^{1}(\mathbb{Q}_{S}/\mathbb{Q},D_{\operatorname{ad}})\to\prod_{\ell\in\Sigma\cup\\{p\\}}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})$

is surjective.

###### Proof.

Without loss of generality, we may assume that $\Sigma$ contains all the
places dividing $N$. We apply Greenberg’s result [Gre16, Prop. 2.6.3] which
proves the surjectivity of $\phi$ under certain hypotheses that we now check.
By assumption, $\rho_{F}$ is residually irreducible when restricted to
$G_{\mathbb{Q}(\mu_{p})}$, so $\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu_{\operatorname{ad}}=D_{\operatorname{ad}}[\mathfrak{m}_{\mathbb{I}}]$
has trivial $G_{\mathbb{Q}(\mu_{p})}$-invariants by Schur’s lemma. Since
$\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu_{\operatorname{ad}}$ is self-dual, $(\mkern
1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu_{\operatorname{ad}})_{G_{\mathbb{Q}(\mu_{p})}}=0$, and in particular,
$\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern
1.5mu_{\operatorname{ad}}$ has no quotient isomorphic to $\mu_{p}$. Moreover,
$\mathbb{I}$ being an integrally closed domain which is finite over $\Lambda$,
it is reflexive as a $\Lambda$-module. But $\Lambda$ has Krull dimension $2$,
so $\mathbb{I}$ and $T_{\operatorname{ad}}$ are $\Lambda$-free. This checks
the validity of (b) of loc. cit..

We now check the weak Leopoldt condition (LEO) and the cotorsionness of
$\operatorname{coker}\phi$ (CRK). For (LEO), it suffices to prove that
$\mathrm{H}^{2}(\mathbb{Q}_{S}/\mathbb{Q},D_{\operatorname{ad}})$ is of
$\mathbb{I}$-cotorsion. Since $\rho_{F}$ is odd,
$\operatorname{corank}_{\mathbb{I}}((D_{\operatorname{ad}})^{G_{\mathbb{R}}})=1$,
and the computation of global Euler characteristics ([Gre16, §2.3]) yields

$\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{1}(\mathbb{Q}_{S}/\mathbb{Q},D_{\operatorname{ad}}))=\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{2}(\mathbb{Q}_{S}/\mathbb{Q},D_{\operatorname{ad}}))+2.$

Note that $\ker\phi=\operatorname{Sel}(\operatorname{ad}F)$ is cotorsion by
Thm. 3.2.1. Hence, if we let $C$ be the codomain of $\phi$, then the equality
$\operatorname{corank}_{\mathbb{I}}(C)=2$, which we show now, will imply (LEO)
and (CRK). By Proposition 3.2.2, we have to show that
$\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{1}_{/\operatorname{Gr}}({\mathbb{Q}_{p}},D_{\operatorname{ad}}))=2$.

Recall the filtration (22) which comes from the ordinary filtration on
$V_{F}$. Letting $\chi_{1},\chi_{2}\colon
G_{{\mathbb{Q}_{p}}}\to\mathbb{I}^{\times}$ with $\chi_{2}$ unramified be such
that $\rho_{F|G_{{\mathbb{Q}_{p}}}}$ is an extension of $\chi_{2}$ by
$\chi_{1}$, we have short exact sequences of
$\mathbb{I}[G_{{\mathbb{Q}_{p}}}]$-modules

${0}$${\mathbb{I}^{\vee}(\chi_{1}\chi_{2}^{-1})}$${D_{\operatorname{ad}}}$${D_{\operatorname{ad}}^{-}}$${0,}$

and

${0}$${\mathbb{I}^{\vee}}$${D_{\operatorname{ad}}^{-}}$${\mathbb{I}^{\vee}(\chi_{2}\chi_{1}^{-1})}$${0,}$

From the explicit description of $\det(\rho_{F})$, one sees that
$\mathbb{I}(\chi_{1}\chi_{2}^{-1})$ is not isomorphic to $\mathbb{I}(n)$ for
any $n\in\mathbb{Z}$. Thus, the cokernel of the map
$\mathrm{H}^{1}({\mathbb{Q}_{p}},D_{\operatorname{ad}})\to\mathrm{H}^{1}({\mathbb{Q}_{p}},D_{\operatorname{ad}}^{-})$
is of cotorsion, as
$\mathrm{H}^{2}({\mathbb{Q}_{p}},\mathbb{I}^{\vee}(\chi_{1}\chi_{2}^{-1}))\simeq\mathbb{I}^{\vee}(\chi_{2}\chi_{1}^{-1})(1)^{G_{{\mathbb{Q}_{p}}}}$
is itself cotorsion. This shows that
$\mathrm{H}^{1}_{/\operatorname{Gr}}({\mathbb{Q}_{p}},D_{\operatorname{ad}})$
has the same corank as
$\operatorname{im}(\mathrm{H}^{1}({\mathbb{Q}_{p}},D_{\operatorname{ad}^{-}})\to\mathrm{H}^{1}(I_{p},D^{-}_{\operatorname{ad}}))=\mathrm{H}^{1}(I_{p},D^{-}_{\operatorname{ad}})^{G_{{\mathbb{Q}_{p}}}/I_{p}}$.
In particular,

$\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{1}_{/\operatorname{Gr}}({\mathbb{Q}_{p}},D_{\operatorname{ad}}))=\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{1}(\mathbb{Q}_{p},D^{-}_{\operatorname{ad}}))-\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{1}_{\mathrm{ur}}(G_{{\mathbb{Q}_{p}}},D^{-}_{\operatorname{ad}})),$

where
$\mathrm{H}^{1}_{\mathrm{ur}}(G_{{\mathbb{Q}_{p}}},D^{-}_{\operatorname{ad}})=\mathrm{H}^{1}(G_{{\mathbb{Q}_{p}}}/I_{p},(D^{-}_{\operatorname{ad}})^{I_{p}})$.
Using the second exact sequence, one finds
$\mathrm{H}^{1}_{\mathrm{ur}}(G_{{\mathbb{Q}_{p}}},D^{-}_{\operatorname{ad}})$
has the same corank as
$\mathrm{H}^{1}(G_{{\mathbb{Q}_{p}}}/I_{p},\mathbb{I}^{\vee})$ which is $1$,
whereas the Euler-Poincaré formula yields

$\displaystyle\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{1}(\mathbb{Q}_{p},D^{-}_{\operatorname{ad}}))$
$\displaystyle=\operatorname{corank}_{\mathbb{I}}((D^{-}_{\operatorname{ad}})^{G_{{\mathbb{Q}_{p}}}})+\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{2}({\mathbb{Q}_{p}},D^{-}_{\operatorname{ad}}))+2$
$\displaystyle=1+0+2=3.$

This proves that
$\operatorname{corank}_{\mathbb{I}}(\mathrm{H}^{1}_{/\operatorname{Gr}}({\mathbb{Q}_{p}},D_{\operatorname{ad}}))=2$,
and that $\phi$ is surjective. ∎

######  Theorem 3.2.4.

Assume $\mkern 1.5mu\overline{\mkern-1.5mu\rho\mkern-1.5mu}\mkern 1.5mu$
satisfies (CR’) and that $F$ is twist-minimal. Then the following equality of
$\mathbb{I}$-ideals holds:

$\textrm{char}_{\mathbb{I}}\operatorname{Sel}(\operatorname{ad}F)^{\vee}=\prod_{\ell\in\Sigma_{e}}(1+\ell^{-1})^{-1}L_{p}(\operatorname{ad}F)\cdot\mathbb{I}.$

###### Proof.

Choose a finite set $\Sigma$ containing the primes dividing $N$ (but not $p$),
so $\Sigma_{e}\subseteq\Sigma$. Note that
$\prod_{\ell\in\Sigma}E_{\ell}(\operatorname{ad}F)=\prod_{\ell|(N_{\Sigma}/N)}E_{\ell}(\operatorname{ad}F)$
by Lemma 3.1.1 (1). Proposition 3.2.3 provides us with an exact sequence

${0}$${\operatorname{Sel}(\operatorname{ad}F)}$${\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)}$${\prod_{\ell\in\Sigma}\mathrm{H}^{1}_{/\operatorname{Gr}}(\mathbb{Q}_{\ell},D_{\operatorname{ad}})}$${0.}$

The multiplicativity of characteristic ideals with exact sequences, together
with Proposition 3.2.2, Theorem 3.2.1 and Proposition 3.1.3, then yields:

$\displaystyle\textrm{char}_{\mathbb{I}}\operatorname{Sel}(\operatorname{ad}F)^{\vee}$
$\displaystyle=\textrm{char}_{\mathbb{I}}\operatorname{Sel}^{\Sigma}(\operatorname{ad}F)^{\vee}\cdot\prod_{\ell\in\Sigma_{e}}(1+\ell^{-1})^{-1}\prod_{\ell\in\Sigma}E_{\ell}(\operatorname{ad}F)^{-1}$
$\displaystyle=L_{p}(\operatorname{ad}F_{\Sigma})\cdot\prod_{\ell\in\Sigma_{e}}(1+\ell^{-1})^{-1}\prod_{\ell\in\Sigma}E_{\ell}(\operatorname{ad}F)^{-1}\cdot\mathbb{I}$
$\displaystyle=\prod_{\ell\in\Sigma_{e}}(1+\ell^{-1})^{-1}L_{p}(\operatorname{ad}F)\cdot\mathbb{I},$

as wanted. ∎

##  Appendix A Congruence modules

We recall some standard definitions and results on congruence modules in a
slightly different setting than in [Bel21, §2.7.5], [Hid16, §6.2.2], or in
[TU22, §2]. Let $\Lambda$ be a local domain, and let $\mathcal{T}$ be a
commutative ring containing $\Lambda$. We further assume that $\mathcal{T}$ is
a finite flat $\Lambda$-algebra. We put $K=\operatorname{Frac}(\Lambda)$ and
we write $M_{K}=M\otimes_{\Lambda}K$ for any $\Lambda$-module $M$. Assume that
we are given

  * •

$M,N$ two free $\mathcal{T}$-modules of rank $1$, and

  * •

$\lambda\colon\mathcal{T}\twoheadrightarrow\Lambda$ a surjective
$\Lambda$-homomorphism.

Assume further that we have a decomposition as $K$-algebras of
$\mathcal{T}_{K}$ as $K\times X$, where the projection of $\mathcal{T}_{K}$
onto the first factor is induced by $\lambda$. We let $e_{1},e_{2}$ be the
corresponding idempotents of $\mathcal{T}_{K}$, and we let

$M^{\lambda}=e_{1}M\subset e_{1}M_{K}\subset M_{K},\qquad M_{\lambda}=M\cap
e_{1}M_{K}\subset e_{1}M_{K}\subset M_{K},$

and similarly for $N$. Note that $M_{\lambda}\subseteq M^{\lambda}$, as
$m=e_{1}\cdot m\in M^{\lambda}$ for any $m\in M_{\lambda}$. We further define
the congruence module associated to $M$ and $\lambda$ as

$C_{0}^{M}(\lambda)=M^{\lambda}/M_{\lambda}.$

The congruence ideal associated to $M$ and $\lambda$ is the annihilator of
$C^{M}_{0}(\lambda)$ as a $\Lambda$-module. Letting
$C_{0}(\lambda):=C_{0}^{\mathcal{T}}(\lambda)$, we then have
$C_{0}^{M}(\lambda)\simeq C_{0}(\lambda)$, as $M\simeq\mathcal{T}$ as
$\mathcal{T}$-modules.

######  Lemma A.1.

$M_{\lambda}$ is $\Lambda$-free of rank $1$, and it is a direct summand of
$M$.

###### Proof.

Fix an isomorphism $M\simeq\mathcal{T}$. We see $\mathcal{T}$ as a
sub-$\Lambda$-module of $\mathcal{T}_{K}=e_{1}\mathcal{T}_{K}\oplus
e_{2}\mathcal{T}_{K}$. The projection $\mathcal{T}_{K}\twoheadrightarrow
e_{1}\mathcal{T}_{K}$ is $\lambda\otimes K$, so we have
$\mathcal{T}_{\lambda}=\operatorname{im}(\mathcal{T}\to
e_{1}\mathcal{T}_{K})=\operatorname{im}(\lambda)=\Lambda$ as
$\Lambda$-modules. Hence, $M_{\lambda}$ is $\Lambda$-free of rank $1$.

Since $\lambda_{|\Lambda}=\operatorname{id}$, we have
$\mathcal{T}=\operatorname{im}(\lambda)\oplus\ker(\lambda)$, i.e.,
$\mathcal{T}=\mathcal{T}_{\lambda}\oplus(\mathcal{T}\cap
e_{2}\mathcal{T}_{K})$ inside $\mathcal{T}_{K}$. Therefore, we obtain
$M=M_{\lambda}\oplus(M\cap e_{2}M_{K})$. ∎

Suppose we are given a $\Lambda$-linear pairing $(\cdot,\cdot)\colon M\times
N\to\Lambda$ which is perfect (i.e., it induces isomorphisms
$N\simeq\operatorname{Hom}_{\Lambda}(M,\Lambda)$ and
$M\simeq\operatorname{Hom}_{\Lambda}(N,\Lambda)$) and such that

(29) $\forall t\in\mathcal{T},\ \forall(m,n)\in M\times
N,\qquad(tm,n)=(m,tn).$

Note that the orthogonal of $e_{1}M_{K}$ (resp. of $e_{2}M_{K}$) is
$e_{2}N_{K}$ (resp. $e_{1}N_{K}$). As a last piece of notation, we define for
any $M^{\prime}\subseteq M_{K}$

$(M^{\prime})^{*}:=\\{n\in N_{K}\ \colon\ \forall m\in M^{\prime},\
(m,n)\in\Lambda\\},$

and we define similarly $(N^{\prime})^{*}$ for $N^{\prime}\subseteq N_{K}$.

######  Proposition A.1.

We have $(M_{\lambda})^{*}=N^{\lambda}\oplus e_{2}N_{K}$, and
$(M^{\lambda})^{*}=N_{\lambda}\oplus e_{2}N_{K}$. Moreover, $(\cdot,\cdot)$
induces an isomorphism
$M^{\lambda}\simeq\operatorname{Hom}_{\Lambda}(N_{\lambda},\Lambda)$ (and also
$N^{\lambda}\simeq\operatorname{Hom}_{\Lambda}(M_{\lambda},\Lambda)$).

###### Proof.

We first check that $(M_{\lambda})^{*}\supseteq N^{\lambda}\oplus e_{2}N_{K}$.
Take $n_{1}\in N$, $n_{2}\in N_{K}$ and let $n=e_{1}n_{1}+e_{2}n_{2}$. For any
$m\in M_{\lambda}$, we have $m=e_{1}m$, so
$(m,e_{2}n_{2})=(m,e_{1}e_{2}n_{2})=0$ by (29). Hence,

$(m,n)=(m,e_{1}n_{1})=(e_{1}m,n_{1})=(m,n_{1})\in\Lambda,$

as $m\in M$ and $n_{1}\in N$. This shows that $n\in(M_{\lambda})^{*}$.

We now prove $(M_{\lambda})^{*}\subseteq N^{\lambda}\oplus e_{2}N_{K}$ and we
take $n\in(M_{\lambda})^{*}$. The map $m\mapsto(m,n)$ defines a
$\Lambda$-linear form $M_{\lambda}\to\Lambda$, which can be extended to a
linear form $M\to\Lambda$ by Lemma A.1. By perfectness of $(\cdot,\cdot)$,
there exists $p\in N$ such that $(\cdot,n)=(\cdot,p)_{|M_{\lambda}}$.
Moreover, as $\Lambda\simeq M_{\lambda}\subset e_{1}M_{K}\simeq K$ by Lemma
A.1, we have $M_{\lambda}\otimes_{\Lambda}K=e_{1}M_{K}$. In particular, $n-p$
is orthogonal to $e_{1}M_{K}$ by linearity of the pairing, that is, $n-p\in
e_{2}N_{K}$. Therefore, $n\in e_{1}p+e_{2}N_{K}\subset N^{\lambda}\oplus
e_{2}N_{K}$, as wanted.

Since $e_{1}N_{\lambda}=N_{\lambda}\subset N$, we have (as sets)
$(M^{\lambda},N_{\lambda}\oplus
e_{2}N_{K})=(M^{\lambda},N_{\lambda})=(e_{1}M,N_{\lambda})=(M,N_{\lambda})\subset\Lambda$,
which implies $(M^{\lambda})^{*}\supseteq N_{\lambda}\oplus e_{2}N_{K}$. We
check the reverse inclusion and we let $n\in(M^{\lambda})^{*}$. To prove that
$e_{1}n\in N$, we remark that, for all $m\in M$, we have
$(m,e_{1}n)=(e_{1}m,n)\in\Lambda$, so the map $m\mapsto(m,e_{1}n)$ belongs to
$\operatorname{Hom}_{\Lambda}(M,\Lambda)$. By the perfectness of
$(\cdot,\cdot)$, this means that $e_{1}n\in N$, as claimed.

It remains to prove that $m\mapsto(n\mapsto(m,n))$ yields an isomorphism
$M^{\lambda}\simeq\operatorname{Hom}_{\Lambda}(N_{\lambda},\Lambda)$. We
already know that $N_{\lambda}\otimes_{\Lambda}K=e_{1}N_{K}$, and it is clear
from the above arguments that $(\cdot,\cdot)$ restricts to a perfect pairing
$e_{1}M_{K}\times e_{1}N_{K}\to K$. We thus only have to check that the
inverse image of $\operatorname{Hom}_{\Lambda}(N_{\lambda},\Lambda)$ under
$e_{1}M_{K}\simeq\operatorname{Hom}_{\Lambda}(e_{1}N_{K},K)$, say
$M^{\prime}$, is equal to $M^{\lambda}$. But we have
$M^{\prime}=(N_{\lambda})^{*}\cap e_{1}M_{K}$ by definition, so we conclude
that $M^{\prime}=(M^{\lambda}\oplus e_{2}M_{K})\cap e_{1}M_{K}=M_{\lambda}$. ∎

######  Corollary A.1.

Let $m_{0}$ and $n_{0}$ be generators of $M_{\lambda}$ and $N_{\lambda}$ over
$\Lambda$ respectively (see Lemma A.1). The $\Lambda$-linear map

$\varphi\colon\left\\{\begin{array}[]{rcl}M^{\lambda}&\to&\Lambda\\\
m&\mapsto&(m,n_{0})\end{array}\right.$

is an isomorphism. Consequently, $C_{0}(\lambda)\simeq
C_{0}^{M}(\lambda)\simeq\Lambda/(m_{0},n_{0})\Lambda$.

###### Proof.

The proof of Proposition A.1 shows that $(\cdot,\cdot)$ restricts to a non-
degenerate pairing $M^{\lambda}\times N_{\lambda}\to\Lambda$, so $\varphi$ is
well-defined and injective. Again by Proposition A.1, the element of
$\operatorname{Hom}_{\Lambda}(N_{\lambda},\Lambda)$ sending $n_{0}$ to $1$ is
of the form $(m,\cdot)$ for some $m\in M^{\lambda}$, and this element thus
satisfies $\varphi(m)=1$. This shows that $\varphi$ is an isomorphism. It is
then clear that $\varphi$ induces an isomorphism
$M^{\lambda}/M_{\lambda}\simeq\Lambda/\varphi(m_{0})\Lambda=\Lambda/(m_{0},n_{0})\Lambda$.
∎

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