#  $\mathscr{L}$-invariants of Artin motives

Mladen Dimitrov  University of Lille, CNRS, UMR 8524 – Laboratoire Paul
Painlevé, 59000 Lille, France<EMAIL_ADDRESS>and  Alexandre
Maksoud  University of Luxembourg, Unité de Recherche en Mathématiques, Maison
du nombre, 6 avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg.
<EMAIL_ADDRESS>To Bernadette Perrin-Riou on the occasion of her 65th
birthday

###### Abstract.

We compute Benois $\mathscr{L}$-invariants of weight $1$ cuspforms and of
their adjoint representations and show how this extends Gross’ $p$-adic
regulator to Artin motives which are not critical in the sense of Deligne.
Benois’ construction depends on the choice of a regular submodule which is
well understood when the representation is $p$-regular, as it then amounts to
the choice of a “motivic” $p$-refinement. The situation is dramatically
different in the $p$-irregular case, where the regular submodules are
parametrized by a flag variety and thus depend on continuous parameters. We
are nevertheless able to show in some examples, how Hida theory and the
geometry of the eigencurve can be used to detect a finite number of choices of
arithmetic and “mixed-motivic” significance.

###### Contents

  1. 1 Introduction
  2. 2 Computation of Benois $\mathscr{L}$-invariants
    1. 2.1 Bloch–Kato Selmer groups
    2. 2.2 Regular submodules and $p$-refinements
    3. 2.3 The $\mathscr{L}$-invariant
    4. 2.4 Computation of the $\mathscr{L}$-invariant
    5. 2.5 The dual $\mathscr{L}$-invariant
  3. 3 Examples of computation of $\mathscr{L}$-invariant
    1. 3.1 Gross’ $p$-adic regulator
    2. 3.2 Weight $1$ cuspforms
    3. 3.3 Adjoint of a weight $1$ $p$-regular cuspform
    4. 3.4 Adjoint of a weight $1$ $p$-irregular CM cuspform

##  1\. Introduction

Let $p$ be a prime number and let
$\varrho:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\operatorname{GL}_{\overline{\mathbb{Q}}_{p}}(W)$
be a $d$-dimensional $p$-adic Galois representation having finite image and
not containing the trivial representation, i.e.
$\operatorname{H}^{0}(\mathbb{Q},W)=0$. After choosing an embedding
$\iota_{p}:\overline{\mathbb{Q}}\subseteq\overline{\mathbb{Q}}_{p}$ one may
think of $\varrho$ as the $p$-adic realization of an Artin motive $[\varrho]$
with $\overline{\mathbb{Q}}$-coefficients whose arithmetic dual
$[\varrho]^{*}(1)$ will be denoted by $M$. When $\varrho$ is unramified at
$p$, then the $p$-adic realization $V=W^{*}(1)$ of $M$ is crystalline at $p$
and, given any regular submodule $D\subset D_{\mathrm{crys}}(V)$ (see Def.
2.6), Perrin-Riou [29] predicts the existence of $p$-adic $L$-functions
$L_{p}(V,D,s)$ and $L_{p}(W,D^{\perp},s)$ interpolating the corrected at $p$
values of the Artin $L$-functions $L(\varrho^{*}(1),s)$ and $L(\varrho,s)$ at
rational integers. When the correcting Euler factor at $p$ vanishes, we say
that the $p$-adic $L$-function admits a trivial (or extra) zero. Under some
mild assumptions, which are satisfied by all $M$ as above, Benois has proposed
in [4, §5.1.2] a formula for the leading term of the Taylor’s expansion at
$s=0$ of $L_{p}(V,D,s)$. This formula, called the Trivial Zero Conjecture,
differs from the interpolation formula proposed by Perrin-Riou by a certain
scalar $\mathscr{L}(V,D)$ defined in terms of Bloch–Kato Selmer groups
$\operatorname{H}^{1}_{f}(\mathbb{Q},V)\subset\operatorname{H}^{1}_{f,p}(\mathbb{Q},V)$
(whose definitions are recalled in §2.1) and generalizing Greenberg’s
$\mathscr{L}$-invariant111The definition of $L_{p}(V,D,s)$, resp.
$L_{p}(W,D^{\perp},s)$, depends on the choice of a Galois stable lattice in
$V$, resp. $D$, but $\mathscr{L}(V,D)$, resp. $\mathscr{L}(W,D^{\perp})$, does
not, hence our abuse of notation..

Trivial zeros occur abundantly in our setting and were discovered by Gross
when $\varrho$ is induced from a totally odd Hecke character over a totally
real field. However the construction of cohomological $\mathscr{L}$-invariants
for a general Artin motive, presented in Section 2, was only recently
introduced by Benois. Rewriting Benois’ definition as a determinant involving
explicit linear combinations (with algebraic coefficients) of $p$-adic
logarithms of $p$-units (Theorem 2.15), allows us to connect
$\mathscr{L}(V,D)$ with Gross’ regulator (see §3.1) and Hida’s
$\mathscr{L}$-invariant for the adjoint of a $p$-regular weight one cuspform
(see §3.3).

Another fundamental aspect of the theory of $p$-adic $L$-functions is to
determine whether the $\mathscr{L}$-invariant vanishes. The Weak $p$-adic
Schanuel Conjecture from transcendental number theory strongly indicates that
$\mathscr{L}(V,D)\neq 0$ when the regular submodule $D$ is motivic, i.e. has a
$\overline{\mathbb{Q}}$-structure, which is always the case when $V$ is
regular at $p$. In contrast, in the $p$-irregular setting $D$ depends on
continuous parameters and $\mathscr{L}(V,D)$ could very well vanish for some
$D$. We illustrate the dependence of $\mathscr{L}(V,D)$ on $D$ in the baby
case of a weight one cuspform with complex multiplication (CM) which is
irregular at $p$ (see §3.2.3).

Constructing non-trivial classes in Bloch–Kato Selmer groups is a central yet
notoriously difficult problem in number theory as witnesses the long-standing
Birch and Swinnerton–Dyer Conjecture. The Euler Systems approach, that we will
not pursue here, sometimes succeeds in writing down such classes explicitly
(e.g. by using cyclotomic, resp. elliptic, units in the case of the
Kubota–Leopoldt, resp. Katz, p-adic $L$-function). While a motive $M$ as above
cannot be adjoint (as it is pure of motivic weight $-2$), a theorem of
Benois–Horte showing $\mathscr{L}(V,D)=(-1)^{e}\mathscr{L}(W,D^{\perp})$,
where $e$ is the order of the trivial zero, allows us to use its motivic dual
$M^{*}(1)$ which is pure of weight $0$ and thus can be adjoint. The advantage
then is that one can use Mazur’s deformation theory of Galois representations
to study first order deformations of the $p$-adic realization of a motive and
produce classes in the $\operatorname{H}^{1}$ of its adjoint. A particularly
interesting case, discussed in §3.4, is that of the adjoint of a weight $1$ CM
cuspform $f$ which is irregular at $p$, for which one encounters a double
trivial zero. Using some results in [7] and the explicit description of
$\mathscr{L}(W,D^{\perp})$ in §2.5, one deduces an intriguing expression (15)
for the $\mathscr{L}$-invariant in the direction of a non-CM Hida family
$\mathscr{F}$ containing $f$ (see §3.2.3 and §3.4.2 for notation):

$\mathscr{L}(\operatorname{ad}^{0}(\varrho_{f}),W_{\mathscr{F}}^{+})=\frac{2\mathscr{L}_{\mathfrak{p}}\cdot\left(\mathscr{L}_{\mbox{-}}(\bar{\varphi})\mathscr{L}(\varphi)+\mathscr{L}_{\mbox{-}}(\varphi)\mathscr{L}(\bar{\varphi})\right)}{\mathscr{L}_{\mbox{-}}(\bar{\varphi})+\mathscr{L}_{\mbox{-}}(\varphi)}.$

An interesting feature of this formula is the presence of the _anti-
cyclotomic_ $\mathscr{L}$-invariants $\mathscr{L}_{\mbox{-}}(\varphi)$ and
$\mathscr{L}_{\mbox{-}}(\bar{\varphi})$. Although the regular submodule $D$
associated with $\mathscr{F}$ has no natural
$\overline{\mathbb{Q}}$-structure, the $\mathscr{L}$-invariant is an algebraic
expression in $p$-adic logarithms of $p$-units, hence it should not vanish
according to the Weak $p$-adic Schanuel Conjecture.

Let us mention that, on the algebraic side, it was first shown by Greenberg
[23], Federer-Gross [20] and later proved in greater generality by Benois [4]
via the machinery of Selmer complexes that $\mathscr{L}$-invariants control
the fine structure of Iwasawa modules attached to $\varrho$. These control
theorems might be taken as an additional motivation for the computation of
Benois’ $\mathscr{L}$-invariant for arbitrary Artin motives, where a
construction of a Perrin-Riou’s $p$-adic $L$-function is at the moment totally
out of reach.

##  2\. Computation of Benois $\mathscr{L}$-invariants

Given a field $F$ of characteristic $0$, we let $\mathcal{O}_{F}$ denote its
ring of integers and $\operatorname{G}_{F}$ its absolute Galois group. When
$F$ is local, we let $\operatorname{I}_{F}\subset\operatorname{G}_{F}$ denote
the inertia subgroup.

For any rational prime $\ell$ we choose an embedding
$\iota_{\ell}:\overline{\mathbb{Q}}\subseteq\overline{\mathbb{Q}}_{\ell}$. We
also fix an embedding
$\iota_{\infty}:\overline{\mathbb{Q}}\subseteq\mathbb{C}$ which determines a
complex conjugation $\tau\in G_{\mathbb{Q}}$.

###  2.1. Bloch–Kato Selmer groups

####  2.1.1. Local and global Selmer groups

Let $V$ be a $p$-adic representation of $\operatorname{G}_{F}$.

If $F$ is a finite extension of $\mathbb{Q}_{\ell}$, the local Bloch–Kato
subgroup is defined as

$\operatorname{H}^{1}_{f}(F,V)=\begin{cases}\ker\left(\operatorname{H}^{1}(F,V)\to\operatorname{H}^{1}(\operatorname{I}_{F},V)\right)&\text{
, if } \ell\neq p,\\\
\ker\left(\operatorname{H}^{1}(F,V)\to\operatorname{H}^{1}(F,B_{\mathrm{crys}}\otimes_{\mathbb{Q}_{p}}V)\right)&\text{
, if } \ell=p,\end{cases}$

where $B_{\mathrm{crys}}$ is the Fontaine ring of crystalline periods (see
[10, §3]).

If $F$ is a number field and $S$ a finite set of finite places of $F$, then
the Selmer group is defined as

$\operatorname{H}^{1}_{S}(F,V)=\ker\left(\operatorname{H}^{1}(F,V)\to\bigoplus_{v\notin
S}\frac{\operatorname{H}^{1}(F_{v},V)}{\operatorname{H}^{1}_{f}(F_{v},V)}\right).$

Here, the map $\operatorname{H}^{1}(F,V)\to\operatorname{H}^{1}(F_{v},V)$ is
given by the restriction of cocycles to $\operatorname{G}_{F_{v}}$ and shall
be denoted by $\mathrm{loc}_{v}$. We will be particularly interested in the
Bloch–Kato Selmer group
$\operatorname{H}^{1}_{f}(F,V)=\operatorname{H}^{1}_{\varnothing}(F,V)$ and
the group
$\operatorname{H}^{1}_{f,p}(F,V)=\operatorname{H}^{1}_{\\{p\\}}(F,V)\supset\operatorname{H}^{1}_{f}(F,V)$
consisting of cocycles that are unramified outside $p$.

####  2.1.2. Kummer Theory

When $V=\overline{\mathbb{Q}}_{p}(1)$, Kummer theory provides an explicit
description of the Bloch–Kato Selmer groups. For all integers $n\geqslant 1$,
the Kummer map
$\kappa_{F,n}:F^{\times}/(F^{\times})^{p^{n}}\to\operatorname{H}^{1}(F,\mu_{p^{n}})$
is an isomorphism sending $a\in F^{\times}$ to the class of the cocycle given
by $g\mapsto g(a^{1/p^{n}})/a^{1/p^{n}}$, where
$a^{1/p^{n}}\in\overline{F}^{\times}$ is any $p^{n}$-th root of $a$ (see [33,
Chap. X, §3.b)]). It is clear from the definition that $\kappa_{F,n}$ is
functorial in $F$ and Galois equivariant. Taking an inverse limit on $n$ and
extending scalars to $\overline{\mathbb{Q}}_{p}$ yields a Galois equivariant
isomorphism

$\kappa_{F}:\widehat{F^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\xrightarrow{\sim}\operatorname{H}^{1}(F,\overline{\mathbb{Q}}_{p}(1)),$

where $\widehat{A}=\varprojlim_{n}A/A^{p^{n}}$ denotes the $p$-adic completion
of an abelian group $A$. Moreover, when $F$ is local, $\kappa_{F}$ restricts
to an isomorphism

$\kappa_{F}:\widehat{\mathcal{O}_{F}^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\xrightarrow{\sim}\operatorname{H}^{1}_{f}(F,\overline{\mathbb{Q}}_{p}(1)),$

(see e.g. [10, Example 3.9]). Note that if $F$ is a finite extension of
$\mathbb{Q}_{\ell}$ and $\ell\neq p$, then
$\operatorname{H}^{1}_{f}(F,\overline{\mathbb{Q}}_{p}(1))$ vanishes, as
$\widehat{\mathcal{O}_{F}^{\times}}$ is finite. In contrast, for a finite
extension $F$ of $\mathbb{Q}_{p}$, $\widehat{\mathcal{O}_{F}^{\times}}$ is the
subgroup $\mathcal{O}_{F}^{1}$ of principal units of
$\mathcal{O}_{F}^{\times}$. Finally, when $F$ is a number field and $S$ a
finite set of finite places of $F$, the global Kummer map restricts to a
Galois equivariant isomorphism

(1)
$\kappa_{F}:\widehat{\mathcal{O}_{F}\left[\tfrac{1}{S}\right]^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\xrightarrow{\sim}\operatorname{H}^{1}_{S}(F,\overline{\mathbb{Q}}_{p}(1)),$

where $\mathcal{O}_{F}\left[\tfrac{1}{S}\right]^{\times}$ denotes the group of
$S$-units of $F$.

Our next task is to give an explicit description of Bloch–Kato Selmer groups
for (duals of) Artin representations.

####  2.1.3. Units

Henceforth $V=W^{*}(1)$, where $W$ is a $\overline{\mathbb{Q}}_{p}$-valued
Artin representation unramified at $p$ as in the introduction. Denote by
$\operatorname{H}^{1}_{f}(V)\subseteq\operatorname{H}^{1}(V)=\operatorname{H}^{1}(\mathbb{Q},V)$
the global Bloch–Kato Selmer group of $V$ and by
$\operatorname{H}^{1}_{f,p}(V)$ the group consisting of cocycles that are
unramified outside $p$.

Let $H$ be the fixed field of the kernel of
$\varrho:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\operatorname{GL}_{\overline{\mathbb{Q}}_{p}}(W)$
and $G=\operatorname{Gal}(H/\mathbb{Q})$. Similarly, we let $E$ be the fixed
field of $\ker(\varrho_{|\operatorname{G}_{\mathbb{Q}_{p}}})$ and
$G_{p}=\operatorname{Gal}(E/\mathbb{Q}_{p})$. Thus, $E$ is the $p$-adic
completion of $H$ with respect to the place defined by $\iota_{p}$.

######  Definition 2.1.

We let
$U_{H}=\mathcal{O}_{H}^{\times}\otimes_{\mathbb{Z}}\overline{\mathbb{Q}}_{p}$
and
$U_{H}^{(p)}=\mathcal{O}_{H}\left[\frac{1}{p}\right]^{\times}\otimes_{\mathbb{Z}}\overline{\mathbb{Q}}_{p}$
and we denote by
$U_{H,\overline{\mathbb{Q}}}=\mathcal{O}_{H}^{\times}\otimes_{\mathbb{Z}}\overline{\mathbb{Q}}$
and
$U^{(p)}_{H,\overline{\mathbb{Q}}}=\mathcal{O}_{H}\left[\frac{1}{p}\right]^{\times}\otimes_{\mathbb{Z}}\overline{\mathbb{Q}}$
the natural $\overline{\mathbb{Q}}$-subspaces.

######  Lemma 2.2.

  1. (i)

The local Kummer map $\kappa_{E}$ yields a canonical identification:

$\operatorname{H}^{1}_{f}(\mathbb{Q}_{p},V)=\operatorname{Hom}_{G_{p}}(W,\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}).$

  2. (ii)

The global Kummer map $\kappa_{H}$ yields canonical identifications:

$\operatorname{H}^{1}_{f}(V)=\operatorname{Hom}_{G}(W,U_{H})\text{ and }
\operatorname{H}^{1}_{f,p}(V)=\operatorname{Hom}_{G}(W,{U_{H}^{(p)}}).$

  3. (iii)

The localization map

$\mathrm{loc}_{p}:\operatorname{H}^{1}_{f}(V)\longrightarrow\operatorname{H}^{1}_{f}(\mathbb{Q}_{p},V)$

is induced by the diagonal embedding
$U_{H}=\widehat{\mathcal{O}_{H}^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\to\widehat{(\mathcal{O}_{H}\otimes\mathbb{Z}_{p})^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}$
and Frobenius reciprocity isomorphism

$\operatorname{Hom}_{G}(W,\widehat{(\mathcal{O}_{H}\otimes\mathbb{Z}_{p})^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p})=\operatorname{Hom}_{G_{p}}(W,\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}).$

###### Proof.

By the inflation-restriction exact sequence,
$\operatorname{H}^{1}_{f}(\mathbb{Q}_{p},V)$ is canonically isomorphic with

$\operatorname{H}^{1}_{f}(E,V)^{G_{p}}=\left(\operatorname{H}^{1}_{f}(E,\overline{\mathbb{Q}}_{p}(1))\otimes
W^{*}\right)^{G_{p}}=\operatorname{Hom}_{G_{p}}\left(W,\operatorname{H}^{1}_{f}(E,\overline{\mathbb{Q}}_{p}(1))\right)$

and the first statement then follows form the fact that $\kappa_{E}$ is
$G_{p}$-equivariant. The second statement is obtained similarly using in
addition that
$\widehat{\mathcal{O}_{H}\left[\tfrac{1}{S}\right]^{\times}}=\mathcal{O}_{H}\left[\tfrac{1}{S}\right]^{\times}\otimes\mathbb{Z}_{p}$.
The third statement is a consequence of the other two (see also [31, Appendix
B, Cor. 5.3.(ii)]) ∎

######  Remark 2.3.

  1. (i)

By the finiteness of the ideal class group of $H$, we have
$\operatorname{H}^{1}_{f}(V^{*}(1))=\operatorname{H}^{1}_{f}(W)=\\{0\\}$, in
concordance with the non-vanishing of $L(\varrho,1)$.

  2. (ii)

Let $W_{\overline{\mathbb{Q}}}$ be a $\overline{\mathbb{Q}}$-structure in the
finite image $\operatorname{G}_{\mathbb{Q}}$-representation $W$ with respect
to which the motive $M^{*}(1)$ is defined. Let $\operatorname{H}^{1}_{f}(M)$,
resp. $\operatorname{H}^{1}_{f,p}(M)$, be the group classifying extensions of
the trivial motive by $M$ which are unramified everywhere, resp. everywhere
outside $p$, in the category of motives over $\mathbb{Q}$ with
$\overline{\mathbb{Q}}$-coefficient (see e.g. [15, §2.2]). It is expected that

$\displaystyle\operatorname{H}^{1}_{f}(M)$
$\displaystyle=\operatorname{H}^{1}_{f}(V)_{\overline{\mathbb{Q}}}:=\operatorname{Hom}_{G}(W_{\bar{\mathbb{Q}}},U_{H,\overline{\mathbb{Q}}}),\text{
and } $ $\displaystyle\operatorname{H}^{1}_{f,p}(M)$
$\displaystyle=\operatorname{H}^{1}_{f,p}(V)_{\overline{\mathbb{Q}}}:=\operatorname{Hom}_{G}(W_{\bar{\mathbb{Q}}},{U_{H,\overline{\mathbb{Q}}}^{(p)}}).$

Note that by Lemma 2.2(ii) this is consistent with Jannsen’s Conjecture (see
[15, Conj. 2.3]) identifying
$\operatorname{H}^{1}_{f}(M)_{\overline{\mathbb{Q}}_{p}}$ and
$\operatorname{H}^{1}_{f,p}(M)_{\overline{\mathbb{Q}}_{p}}$ with
$\operatorname{H}^{1}_{f}(V)$ and $\operatorname{H}^{1}_{f,p}(V)$,
respectively.

####  2.1.4. Dimensions

Let $d^{+}=\dim\operatorname{H}^{0}(\mathbb{R},W)$, resp. $d^{-}=d-d^{+}$, be
the dimension of the largest subspace of $W$ on which the complex conjugation
acts as $1$, resp. $-1$, and let
$f=\dim\operatorname{H}^{0}(\mathbb{Q}_{p},W)$. By Dirichlet’s Unit Theorem,
the Bloch–Kato Selmer groups $\operatorname{H}^{1}_{f}(V)$ and
$\operatorname{H}^{1}_{f,p}(V)$ have dimension $d^{+}$ and $d^{+}+f$,
respectively.

####  2.1.5. Transcendence Theory

The $\mathbb{Z}_{p}$-module
$\widehat{E^{\times}}=p^{\mathbb{Z}_{p}}\times\mathcal{O}_{E}^{1}$ is endowed
with a $p$-adic valuation map
$\mathrm{ord}_{p}:\widehat{E^{\times}}\to\mathbb{Z}_{p}$, factoring through
the first projection, and an Iwasawa determination of a $p$-adic logarithm
$\log_{p}:\widehat{E^{\times}}\to
p\mathcal{O}_{E}\subset\overline{\mathbb{Q}}_{p}$, factoring through the
second projection. We consider their $\overline{\mathbb{Q}}_{p}$-linear
extensions:

$\mathrm{ord}_{p}:\widehat{E^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\longrightarrow\overline{\mathbb{Q}}_{p}\quad\text{
and
}\quad\log_{p}:\widehat{E^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\longrightarrow\overline{\mathbb{Q}}_{p}.$

Pre-composing with $\iota_{p}$ allows one to apply both $\log_{p}$ and
$\mathrm{ord}_{p}$ to elements of $U_{H}^{(p)}\supset U_{H}$ and we shall do
so without further notice.

Let us recall a consequence of Brumer’s $p$-adic version of Baker’s Theorem on
the independence of logarithms of global units [11].

######  Theorem 2.4 (Baker–Brumer).

The map $\log_{p}:U_{H,\overline{\mathbb{Q}}}\to\overline{\mathbb{Q}}_{p}$ is
injective.

By the $G_{p}$-equivariance of the $p$-adic logarithm and Frobenius
reciprocity, we have:

(2)
$\operatorname{Hom}_{G_{p}}(W,\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p})\underset{\sim}{\xrightarrow{\log_{p}}}\operatorname{Hom}_{G_{p}}(W,E\otimes_{\mathbb{Q}_{p}}\overline{\mathbb{Q}}_{p})\xrightarrow{\sim}\operatorname{Hom}(W,\overline{\mathbb{Q}}_{p}).$

Composing this isomorphism with $\mathrm{loc}_{p}$ from Lemma 2.2(iii) yields
a map

$\log_{p}\circ\mathrm{loc}_{p}:\operatorname{Hom}_{G}(W,U_{H})\to\operatorname{Hom}_{G_{p}}(W,\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p})\xrightarrow{\sim}\operatorname{Hom}(W,\overline{\mathbb{Q}}_{p}),$

whose injectivity is the content of the $\varrho$-part of the Leopoldt
Conjecture.

Note that the Leopoldt Conjecture for $H$ at $p$, asserting the injectivity of
$\widehat{\mathcal{O}_{H}^{\times}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\to\widehat{(\mathcal{O}_{H}\otimes\mathbb{Z}_{p})^{\times}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$,
is equivalent to its $\varrho$-parts for $\varrho$ running over all
irreducible representations of $G$ contributing to $U_{H}$, justifying the
terminology.

######  Lemma 2.5.

The following are equivalent:

  1. (i)

the $\varrho$-part of the Leopoldt Conjecture holds;

  2. (ii)

$\mathrm{loc}_{p}:\operatorname{H}^{1}_{f}(V)\to\operatorname{H}^{1}_{f}(\mathbb{Q}_{p},V)$
is injective;

  3. (iii)

$\dim W^{\circ}=d^{-}$, where $W^{\circ}=\left\\{w\in
W\,\mid\,\log_{p}(\kappa(w))=0\text{ for all
}\kappa\in\operatorname{Hom}_{G}(W,U_{H})\right\\}$.

Note that, in general, the $\overline{\mathbb{Q}}_{p}$-linear subspace
$W^{\circ}$ of $W$ is _not_ $G_{p}$-stable.

By Theorem 2.4 the $\varrho$-part of the Leopoldt Conjecture holds when
$d^{+}\leqslant 1$.

###  2.2. Regular submodules and $p$-refinements

Let
$\chi:\operatorname{G}_{\mathbb{Q}_{p}}\twoheadrightarrow\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})\simeq\mathbb{Z}^{\times}_{p}\subseteq\bar{\mathbb{Q}}^{\times}_{p}$
be the $p$-adic cyclotomic character and let us fix an arithmetic Frobenius
element $\operatorname{Frob}_{p}$.

As $\varrho$ is unramified at $p$, its restriction to
$\operatorname{G}_{\mathbb{Q}_{p}}$ is a direct sum of unramified finite order
characters $\chi_{1},\ldots,\chi_{d}$, so that one has
$V_{|\operatorname{G}_{\mathbb{Q}_{p}}}\simeq\chi\cdot\chi_{1}^{-1}\oplus\ldots\oplus\chi\cdot\chi_{d}^{-1}$.

####  2.2.1. The Bloch-Kato logarithm

It follows from the definition of $B_{\mathrm{crys}}$ (see [22]) that it
contains an element $t$ such that $\left(\mathbb{Q}_{p}(1)\otimes
B_{\mathrm{crys}}\right)^{\operatorname{G}_{E}}=t^{-1}E$ as $G_{p}$-modules,
hence

$D_{\mathrm{crys}}(V)=\left(V\otimes_{\mathbb{Q}_{p}}B_{\mathrm{crys}}\right)^{\operatorname{G}_{\mathbb{Q}_{p}}}=\operatorname{Hom}_{G_{p}}\left(W,t^{-1}E\otimes_{\mathbb{Q}_{p}}\overline{\mathbb{Q}}_{p}\right)\overset{\mathrm{\cdot
t}}{=\joinrel=}\operatorname{Hom}(W,\overline{\mathbb{Q}}_{p}).$

Moreover, the crystalline Frobenius $\varphi$ acts semi-simply on
$D_{\mathrm{crys}}(V)$ with eigenvalues $(p^{-1}\alpha_{i})_{1\leqslant
i\leqslant d}$, where
$\alpha_{i}=\chi_{i}(\operatorname{Frob}_{p})\in\overline{\mathbb{Q}}^{\times}_{p}$.

Using Kummer theory, the Bloch–Kato logarithm (see [10, §3]) for the
$\operatorname{G}_{E}$-module $\overline{\mathbb{Q}}_{p}(1)$ is simply given
by:

$\log_{p}:\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\overset{\sim}{\longrightarrow}E\otimes_{\mathbb{Q}_{p}}\overline{\mathbb{Q}}_{p}\quad
x\otimes y\mapsto\log_{p}(x)\otimes y.$

As $V$ is crystalline and $\mathrm{Fil}^{0}D_{\mathrm{dR}}(V)=\\{0\\}$, the
$p$-adic tangent space of $V$ is given by

$t_{V}=D_{\mathrm{dR}}(V)/\mathrm{Fil}^{0}D_{\mathrm{dR}}(V)=D_{\mathrm{crys}}(V).$

Using the identifications of (2) and Lemma 2.2(i), the Bloch–Kato logarithm
$\log_{\mathrm{BK}}$ for $V_{|\operatorname{G}_{\mathbb{Q}_{p}}}$ fits in the
following commutative diagram

(3)
$\textstyle{\operatorname{H}^{1}_{f}(\mathbb{Q}_{p},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log_{\mathrm{BK}}}$$\scriptstyle{\sim}$$\textstyle{D_{\mathrm{crys}}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{G_{p}}(W,\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\log_{p}}$$\scriptstyle{\quad\sim}$$\textstyle{\operatorname{Hom}(W,\overline{\mathbb{Q}}_{p}).}$

Using the identification of Lemma 2.2(ii), the $p$-adic regulator is the map
$\log_{\mathrm{BK}}\circ\mathrm{loc}_{p}$ fitting in the following commutative
diagram

(4)
$\textstyle{\operatorname{H}^{1}_{f}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log_{\mathrm{BK}}\circ\mathrm{loc}_{p}}$$\textstyle{D_{\mathrm{crys}}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{G}\left(W,U_{H}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\log_{p}\circ\iota_{p}}$$\textstyle{\operatorname{Hom}(W,\overline{\mathbb{Q}}_{p}).}$

Note that $\operatorname{H}^{1}_{f}(V)$ and $\operatorname{H}^{1}_{f,p}(V)$
are endowed with canonical $\overline{\mathbb{Q}}$-structures given
respectively by
$\operatorname{Hom}_{G}\left(W_{\overline{\mathbb{Q}}},U_{H,\overline{\mathbb{Q}}}\right)$
and
$\operatorname{Hom}_{G}\left(W_{\overline{\mathbb{Q}}},U^{(p)}_{H,\overline{\mathbb{Q}}}\right)$.

####  2.2.2. Regular submodules

According to Perrin-Riou, the conjectural $p$-adic $L$-function of a motive
should depend on the choice of local data at $p$, namely a regular submodule
(see [29, §3.1.2] and [4, §4.1.3]).

######  Definition 2.6.

A submodule $D$ of $D_{\mathrm{crys}}(V)$, stable by $\varphi$, is regular if
the map

$r_{D}:\operatorname{H}^{1}_{f}(V)\longrightarrow D_{\mathrm{crys}}(V)/D$

induced by the $p$-adic regulator map
$\log_{\mathrm{BK}}\circ\mathrm{loc}_{p}$ is an isomorphism.

######  Remark 2.7.

The existence of a regular submodule of $D_{\mathrm{crys}}(V)$ implies, in
particular, that the localization map $\mathrm{loc}_{p}$ of Lemma 2.2(iii) is
injective.

####  2.2.3. $p$-refinements

Using (3), any submodule $D$ of $D_{\mathrm{crys}}(V)$, stable by $\varphi$,
can be written as

(5) $D=\operatorname{Hom}(W/W^{+},\overline{\mathbb{Q}}_{p}),$

for some $G_{p}$-stable linear subspace $W^{+}$ of $W$. Note that the action
of $\varphi$ on $D$ corresponds, under (5), to the action of
$p^{-1}\cdot\operatorname{Frob}^{-1}_{p}$ on
$\operatorname{Hom}(W/W^{+},\overline{\mathbb{Q}}_{p})$. Moreover, if $D$ is
regular (see Def. 2.6) then $W^{+}$ has to be of dimension $d^{+}$, motivating
the following definition.

######  Definition 2.8.

  1. (i)

A _$p$ -refinement_ of $W$ is a $d^{+}$-dimensional $G_{p}$-stable linear
subspace $W^{+}$ of $W$.

  2. (ii)

A $p$-refinement $W^{+}$ is _regular_ if the submodule $D$ given by (5) is
regular in the sense of Definition 2.6.

  3. (iii)

A $p$-refinement $W^{+}$ is _motivic_ if there exists a
$\overline{\mathbb{Q}}$-linear subspace $W^{+}_{\overline{\mathbb{Q}}}$ of
$W_{\overline{\mathbb{Q}}}$ such that
$W^{+}=W_{\overline{\mathbb{Q}}}^{+}\otimes_{\overline{\mathbb{Q}}}\overline{\mathbb{Q}}_{p}$.

  4. (iv)

Given any $p$-refinement $W^{+}$ one can attach a $p$-adic regulator of size
$d^{+}$ as follows. Choose a basis $(\kappa_{1},\ldots,\kappa_{d^{+}})$ of
$\operatorname{Hom}_{G}(W,U_{H})$ and a basis
$(w^{+}_{1},\ldots,w^{+}_{d^{+}})$ of $W^{+}$, and put

$\operatorname{Reg}_{p}(W,W^{+})=\det\left(\log_{p}\left(\kappa_{j}(w^{+}_{i})\right)\right)_{1\leqslant
i,j\leqslant d^{+}}\in\overline{\mathbb{Q}}_{p}.$

When $W^{+}$ is motivic one can use $\overline{\mathbb{Q}}$-bases of
$\operatorname{H}^{1}_{f}(V)_{\overline{\mathbb{Q}}}=\operatorname{Hom}_{G}\left(W_{\overline{\mathbb{Q}}},U_{H,\overline{\mathbb{Q}}}\right)$
and $W^{+}_{\overline{\mathbb{Q}}}$.

Recall the subspace
$W^{\circ}=\bigcap_{i=1}^{d^{+}}\ker\left(\log_{p}\circ\kappa_{i}\right)$ from
Lemma 2.5.

######  Proposition 2.9.

  1. (i)

If $W$ admits a regular $p$-refinement then the $\varrho$-part of the Leopoldt
Conjecture holds and $\dim W^{\circ}=d^{-}$.

  2. (ii)

A $p$-refinement $W^{+}$ of $W$ is regular if any (hence all) of the following
holds:

     * •

$\operatorname{Reg}_{p}(W,W^{+})\neq 0$,

     * •

$\theta_{D}\ \colon\
\operatorname{Hom}_{G}\left(W,U_{H}\right)\to\operatorname{Hom}_{G_{p}}\left(W^{+},\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right)$
is an isomorphism,

     * •

$W=W^{+}\oplus W^{\circ}$.

  3. (iii)

A motivic $p$-refinement is always regular if $d^{+}=1$, or if we assume the
Weak $p$-adic Schanuel Conjecture [13, Conj. 3.10].

###### Proof.

(i) follows from Lemma 2.5, as regularity implies the injectivity of
$\mathrm{loc}_{p}$.

(ii) By definition $W^{+}$ is regular if the map $r_{D}$, for $D$ as in (5),
is an isomorphism. Using (4) and the canonically isomorphism between
$D_{\mathrm{crys}}(V)/D$ and
$\operatorname{Hom}(W^{+},\overline{\mathbb{Q}}_{p})$, we are lead to study
the map

(6)
$r_{D}:\operatorname{Hom}_{G}(W,U_{H})\longrightarrow\operatorname{Hom}(W^{+},\overline{\mathbb{Q}}_{p}),$

whose determinant, computed with respect to our fixed bases, is precisely
$\operatorname{Reg}_{p}(W,W^{+})$. Thus, $r_{D}$ is an isomorphism if and only
if $\operatorname{Reg}_{p}(W,W^{+})\neq 0$.

As the map (6) is obtained by composing $\theta_{D}$ with the Frobenius
reciprocity isomorphism
$\operatorname{Hom}_{G_{p}}(W^{+},\mathcal{O}^{1}_{E}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p})\simeq\operatorname{Hom}(W^{+},\overline{\mathbb{Q}}_{p})$
induced by $\log_{p}$, it follows that $\theta_{D}$ is an isomorphism if and
only if $\operatorname{Reg}_{p}(W,W^{+})\neq 0$.

As the map (6) factors through
$\operatorname{Hom}(W/W^{\circ},\overline{\mathbb{Q}}_{p})$, if $W^{+}$ is
regular then necessarily $W^{+}\cap W^{\circ}=\\{0\\}$ and therefore
$W^{+}\oplus W^{\circ}=W$ (as $\dim W^{\circ}=d^{-}$ by (i)). Conversely, if
$W^{+}\oplus W^{\circ}=W$, then Lemma 2.5 implies that $\mathrm{loc}_{p}$ is
injective, hence $r_{D}$ is an isomorphism, i.e. $W^{+}$ is regular.

(iii) Suppose that $W^{+}$ is motivic. Then one can assume the basis
$w_{1}^{+},\ldots,w_{d^{+}}^{+}$ to be chosen in $W_{\overline{\mathbb{Q}}}$.
Moreover the $\kappa_{i}$’s can be chosen in
$\operatorname{H}^{1}_{f}(V)_{\overline{\mathbb{Q}}}$, so the entries of the
regulator matrix are $p$-adic logarithms of elements of
$U_{H,\overline{\mathbb{Q}}}$ which are easily seen to be linearly independent
over $\mathbb{Q}$. Thus $\operatorname{Reg}_{p}(W,W^{+})\neq 0$ by Theorem 2.4
if $d^{+}=1$, or else if the Weak $p$-adic Schanuel Conjecture holds, as the
entries of the determinant are linearly independent over $\mathbb{Q}$. ∎

######  Remark 2.10.

If the characters $(\chi_{i})_{1\leqslant i\leqslant d}$ are pairwise
distinct, then a choice of a $p$-refinement of $W$ amounts to a choice of
$d^{+}$ amongst these characters, and such a $p$-refinement is automatically
motivic. However, when the characters are _not_ pairwise distinct, then there
are infinitely many $p$-refinements some of which are not motivic.

####  2.2.4. Extra zeros

Let $W^{+}$ be a $p$-refinement of $W$ and put $W^{-}=W/W^{+}$. Let
$W_{1}^{-}=\operatorname{H}^{0}(\mathbb{Q}_{p},W^{-})$ be the subspace of
$W^{-}$ on which the Frobenius acts trivially and consider the (unique)
$G_{p}$-stable decomposition

$W^{-}=W/W^{+}=W_{1}^{-}\oplus W_{-1}^{-}.$

We assume that $\operatorname{Reg}_{p}(W,W^{+})\neq 0$, so that
$D=\operatorname{Hom}(W^{-},\overline{\mathbb{Q}}_{p})$ is a regular submodule
of $D_{\mathrm{crys}}(V)$ by Proposition 2.9. The resulting decomposition of
the $\varphi$-module $D$ as a direct sum of

(7) $D_{1}=\operatorname{Hom}(W_{1}^{-},\overline{\mathbb{Q}}_{p})\text{ and
}D_{-1}=\operatorname{Hom}(W_{-1}^{-},\overline{\mathbb{Q}}_{p})$

coincides with the one introduced in [3, §2.1.4] and [4, §4.1.3] in the sense
that $D_{1}=D^{\varphi=p^{-1}}$ and $D_{-1}=(1-p^{-1}\varphi^{-1})D$. As
explained in _loc. cit._ , the $p$-adic $L$-function attached to (lattices of)
$V$ and $D$ should have $e$ extra zeros at $s=0$, where $e=\dim D_{1}=\dim
W_{1}^{-}$.

###  2.3. The $\mathscr{L}$-invariant

We now unwind Benois’ definition of the $\mathscr{L}$-invariant in the setting
of Artin motives. Henceforth, we fix a regular submodule $D$ of
$D_{\mathrm{crys}}(V)$, stable by $\varphi$, and we let $W^{+}$ denote the
corresponding regular $p$-refinement of $W$ (see (5)).

####  2.3.1. $(\varphi,\Gamma)$-modules

Benois’ definition of the $\mathscr{L}$-invariant involves the
$(\varphi,\Gamma)$-modules D, $\textbf{D}_{1}$ and $\textbf{D}_{-1}$,
associated by Berger’s theory [6] with $D$, $D_{1}$ and $D_{-1}$ (see (7)). In
our setting, these $(\varphi,\Gamma)$-modules come from local Galois
representations, so their cohomology (in the sense of Herr [26]) is easily
computable by [4, Corollary 3]. From Lemma 2.2(i) one obtains:

$\displaystyle\operatorname{H}^{1}(\textbf{D})$
$\displaystyle=\operatorname{H}^{1}(\mathbb{Q}_{p},V^{-})=\operatorname{Hom}_{G_{p}}\left(W^{-},\widehat{E^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right),$
$\displaystyle\operatorname{H}_{f}^{1}(\textbf{D})$
$\displaystyle=\operatorname{H}_{f}^{1}(\mathbb{Q}_{p},V^{-})=\operatorname{Hom}_{G_{p}}\left(W^{-},\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right),$
$\displaystyle\operatorname{H}_{f}^{1}(\textbf{D}_{-1})$
$\displaystyle=\operatorname{H}^{1}(\textbf{D}_{-1})=\operatorname{H}^{1}(\mathbb{Q}_{p},V_{-1}^{-})=\operatorname{Hom}_{G_{p}}\left(W_{-1}^{-},\widehat{E^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right),$
$\displaystyle\operatorname{H}^{1}(\textbf{D}_{1})$
$\displaystyle=\operatorname{H}^{1}(\mathbb{Q}_{p},V_{1}^{-})=\operatorname{Hom}_{G_{p}}\left(W_{1}^{-},\widehat{E^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right),$
$\displaystyle\operatorname{H}_{f}^{1}(\textbf{D}_{1})$
$\displaystyle=\operatorname{H}_{f}^{1}(\mathbb{Q}_{p},V_{1}^{-})=\operatorname{Hom}_{G_{p}}\left(W_{1}^{-},\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right),$

where $V^{\pm}=(W^{\pm})^{*}(1)$ and where $V_{1}^{-}$ and $V_{-1}^{-}$ are
similarly defined.

As $\widehat{E^{\times}}=\mathcal{O}_{E}^{1}\times p^{\mathbb{Z}_{p}}$, the
map $U_{H}^{(p)}\to\widehat{E^{\times}}\otimes\overline{\mathbb{Q}}_{p}$
induced by $\iota_{p}$ gives rise to a map

$j_{D}:\operatorname{Hom}_{G}(W,U_{H}^{(p)})\to\dfrac{\operatorname{Hom}_{G_{p}}(W,\widehat{E^{\times}}\otimes\overline{\mathbb{Q}}_{p})}{\operatorname{Hom}_{G_{p}}(W^{-},\mathcal{O}_{E}^{1}\otimes\overline{\mathbb{Q}}_{p})}=\operatorname{Hom}_{G_{p}}\left(W^{+},\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right)\oplus\operatorname{Hom}_{G_{p}}\left(W,p^{\overline{\mathbb{Q}}_{p}}\right),$

which can be seen to coincide with the map $\kappa_{D}$ in [4, §4.1.3].

######  Lemma 2.11.

The map $j_{D}$ is an isomorphism.

###### Proof.

By Frobenius reciprocity one finds

$\operatorname{Hom}_{G}(W,U_{H}^{(p)})=\operatorname{Hom}_{G}(W,U_{H})\oplus\operatorname{Hom}_{G_{p}}(W,p^{\overline{\mathbb{Q}}_{p}}),$

and the restriction of $j_{D}$ on the first summand is given by the map
$\theta_{D}$ from Proposition 2.9(ii). Thus, the regularity of $D$ implies
that $j_{D}$ is an isomorphism. ∎

####  2.3.2. The space $\operatorname{H}^{1}(D,V)$

Let $\operatorname{H}^{1}(D,V)$ be the subspace of
$\operatorname{H}^{1}_{f,p}(V)=\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ (see
Lemma 2.2(ii)) consisting of $\kappa:W\to U_{H}^{(p)}$ such that
$\iota_{p}\circ\kappa$ is trivial on $W^{+}$, i.e.
$\iota_{p}\circ\kappa\in\operatorname{Hom}_{G_{p}}(W^{-},\widehat{E^{\times}}\otimes\overline{\mathbb{Q}}_{p})$.

We describe the isomorphisms $j_{D,f}$ and $j_{D,c}$ (denoted by
$\varrho_{D,f}$ and $\varrho_{D,c}$ in [4, §4.1.4]) following [3, §1.5.6]. As
$j_{D}$ is an isomorphism, its restriction to $\operatorname{H}^{1}(D,V)$
yields

$j_{D,c}:\operatorname{H}^{1}(D,V)\underset{\sim}{\xrightarrow{j_{D}}}\operatorname{Hom}_{G_{p}}\left(W^{-},p^{\overline{\mathbb{Q}}_{p}}\right)\underset{\sim}{\xrightarrow{\mathrm{ord}_{p}}}\operatorname{Hom}\left(W_{1}^{-},\overline{\mathbb{Q}}_{p}\right),$

Thus $\operatorname{H}^{1}(D,V)$ has dimension $e$. Denoting by
$\mathrm{res}_{W_{1}^{-}}$ the restriction to $W^{-}_{1}$, the map $j_{D,f}$
fits in the following commutative diagram:

$\textstyle{\operatorname{H}^{1}(D,V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{res}_{W_{1}^{-}}\circ\iota_{p}}$$\scriptstyle{j_{D,f}}$$\textstyle{\operatorname{Hom}_{G_{p}}\left(W_{1}^{-},\widehat{E^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{G_{p}}\left(W_{1}^{-},\mathcal{O}_{E}^{1}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\log_{p}}$$\textstyle{\operatorname{Hom}\left(W_{1}^{-},\overline{\mathbb{Q}}_{p}\right).}$

The following definition is taken from [4, §4.1.4].

######  Definition 2.12.

Benois’ $\mathscr{L}$-invariant is defined as the following $e\times e$
determinant:

$\mathscr{L}(V,D)=\det\left(j_{D,f}\circ j_{D,c}^{-1}\ |\
D_{1}\right)\in\overline{\mathbb{Q}}_{p}.$

###  2.4. Computation of the $\mathscr{L}$-invariant

Benois’ $\mathscr{L}$-invariant $\mathscr{L}(V,D)$ is described in the last
paragraph in terms of an $e$-dimensional space $\operatorname{H}^{1}(D,V)$
whose definition is not always practical. In this paragraph, we give an
alternate formula for $\mathscr{L}(V,D)$ involving a determinant of size
$(d^{+}+e)$ which turns out to be especially useful in the examples of §3.1,
§3.2.1 and §3.3.

######  Lemma 2.13.

Let $W_{1}^{+}=\operatorname{H}^{0}(\mathbb{Q}_{p},W^{+})$. The
$\overline{\mathbb{Q}}_{p}$-linear map

$\operatorname{Hom}_{G}\left(W,U_{H}^{(p)}\right)\longrightarrow\operatorname{Hom}_{G_{p}}(W^{+},\widehat{E^{\times}}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{Q}}_{p})\longrightarrow\operatorname{Hom}_{G_{p}}\left(W^{+},\overline{\mathbb{Q}}_{p}\right)=\operatorname{Hom}\left(W_{1}^{+},\overline{\mathbb{Q}}_{p}\right)$

obtained by localizing, restricting to $W^{+}$ and then composing with
$\mathrm{ord}_{p}$ is surjective. Its kernel contains
$\operatorname{Hom}_{G}\left(W,U_{H}\right)$ and has dimension $d^{+}+e$ over
$\overline{\mathbb{Q}}_{p}$.

###### Proof.

The first map is surjective because of the surjectivity of $j_{D}$, while the
second map is surjective because it is induced by
$\mathrm{ord}_{p}:E^{\times}\twoheadrightarrow\mathbb{Z}$. Thus the composed
map is surjective and its kernel has dimension $(d^{+}+f)-(f-e)=d^{+}+e$ (see
§2.1.4). ∎

######  Notation 2.14.

Let us fix a basis $(w^{+}_{1},\ldots,w^{+}_{f-e})$ of $W_{1}^{+}$ which we
complete to a basis $(w^{+}_{1},\ldots,w^{+}_{d^{+}})$ of $W^{+}$, and also to
a basis $(w^{+}_{1},\ldots,w^{+}_{f-e},w^{-}_{1},\ldots,w^{-}_{e})$ of
$W_{1}=\operatorname{H}^{0}(\mathbb{Q}_{p},W)$. Note that
$(w^{-}_{1},\ldots,w^{-}_{e})$ maps onto a basis of $W^{-}_{1}$ under the
natural projection map $W\twoheadrightarrow W^{-}$.

Fix also a basis $(\kappa_{1},\ldots,\kappa_{d^{+}})$ of
$\operatorname{Hom}_{G}\left(W,U_{H}\right)$ which we complete to a basis
$(\kappa_{1},\ldots,\kappa_{d^{+}},\kappa^{\prime}_{1},\ldots,\kappa^{\prime}_{e})$
of the kernel of the map considered in Lemma 2.13. Our choices allow us to
construct global units and $p$-units in the following way.

  * •

The global units $\varepsilon_{i,j}^{+}=\kappa_{j}(w^{+}_{i})$ and
$\varepsilon_{i,j}^{-}=\kappa_{j}(w^{-}_{i})$ belong to $U_{H}$, and taking
$p$-adic logarithms yields two matrices

$A^{+}=\left[\log_{p}(\varepsilon_{i,j}^{+})\right]_{1\leqslant i,j\leqslant
d^{+}}\textrm{ and
}A^{-}=\left[\log_{p}(\varepsilon_{i,j}^{-})\right]_{\begin{subarray}{c}1\leqslant
i\leqslant e\phantom{{}^{+}}\\\ 1\leqslant j\leqslant d^{+}\end{subarray}}$

whose entries are in $\overline{\mathbb{Q}}_{p}$.

  * •

The global $p$-units $u_{i,j}^{+}=\kappa^{\prime}_{j}(w^{+}_{i})$ and
$u_{i,j}^{-}=\kappa^{\prime}_{j}(w^{-}_{i})$ belong to $U_{H}^{(p)}$, and
taking $p$-adic logarithms yields two matrices

$B^{+}=\left[\log_{p}(u_{i,j}^{+})\right]_{\begin{subarray}{c}1\leqslant
i\leqslant d^{+}\\\ 1\leqslant j\leqslant
e\phantom{{}^{+}}\end{subarray}}\textrm{ and
}B^{-}=\left[\log_{p}(u_{i,j}^{-})\right]_{1\leqslant i,j\leqslant e}$

whose entries are in $\overline{\mathbb{Q}}_{p}$. We put
$O^{-}=\left[\mathrm{ord}_{p}(u_{i,j}^{-})\right]_{1\leqslant i,j\leqslant
e}$.

Note that $\det A^{+}$ is precisely the $p$-adic regulator
$\operatorname{Reg}_{p}(W,W^{+})$ from Definition 2.8(iv), and the regularity
of $W^{+}$ implies that $\det A^{+}\neq 0$ by Proposition 2.9(ii).

######  Theorem 2.15.

Assume $\operatorname{Reg}_{p}(W,W^{+})$ is non-zero, so that the
$\varphi$-submodule $D$ given by (5) is regular. The $\mathscr{L}$-invariant
associated with $V$ and $D$ is given by the following formula:

$\mathscr{L}(V,D)=(-1)^{e}\frac{\det\begin{pmatrix}A^{+}&B^{+}\\\
A^{-}&B^{-}\end{pmatrix}}{\det A^{+}\cdot\det O^{-}}$

where $A^{\pm}$, $B^{\pm}$ and $O^{-}$ are the matrices defined in Notation
2.14.

###### Proof.

We first define convenient bases
$\widetilde{\kappa}_{1},\ldots,\widetilde{\kappa}_{e}$ and
$\eta_{1},\ldots,\eta_{e}$ of $\operatorname{H}^{1}(D,V)$ and $D_{1}$
respectively and then express the matrices of $j_{D,f}$ and $j_{D,c}$ in these
bases in terms of $A^{\pm},B^{\pm}$ and $O^{-}$.

Let us complete
$(\kappa_{1},\ldots,\kappa_{d^{+}},\kappa^{\prime}_{1},\ldots,\kappa^{\prime}_{e})$
into a $\overline{\mathbb{Q}}_{p}$-basis
$(\kappa_{1},\ldots,\kappa_{d^{+}},\kappa^{\prime}_{1},\ldots,\kappa^{\prime}_{f})$
of $\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ such that the image of
$(\kappa^{\prime}_{e+1},\ldots,\kappa^{\prime}_{f})$ under the surjective map
in Lemma 2.13 is the dual basis of $(w^{+}_{1},\ldots,w^{+}_{f-e})$ of
$\left(W_{1}^{+}\right)^{*}$. In other words, for $1\leqslant i\leqslant
d^{+}$ and $e+1\leqslant j\leqslant f$, we have

$\mathrm{ord}_{p}(\kappa^{\prime}_{j}(w^{+}_{i}))=\begin{cases}1,\text{ if }
e+i=j,\\\ 0,\text{ otherwise}.\end{cases}$

Take $\kappa\in\operatorname{H}^{1}_{f,p}(V)$ and write
$\kappa=\sum_{i=1}^{d^{+}}c_{i}\kappa_{i}+\sum_{j=1}^{f}c^{\prime}_{j}\kappa^{\prime}_{j}$
for $c_{j},c^{\prime}_{j}\in\overline{\mathbb{Q}}_{p}$. We have
$\kappa\in\operatorname{H}^{1}(D,V)$ if and only if
$\log_{p}\left(\kappa(w^{+}_{i})\right)=\mathrm{ord}_{p}\left(\kappa(w^{+}_{i})\right)=0$,
i.e.

$c^{\prime}_{e+1}=\ldots=c^{\prime}_{f}=0,\text{ and } A^{+}\cdot
C_{\kappa}+B^{+}\cdot C_{\kappa}^{\prime}=0,$

where $C_{\kappa}$ and $C^{\prime}_{\kappa}$ are the column vectors whose
entries are $(c_{1},\ldots,c_{d^{+}})$ and
$(c^{\prime}_{1},\ldots,c^{\prime}_{e})$ respectively. As $A^{+}$ is
invertible, for all $\kappa\in\operatorname{H}^{1}(D,V)$ we have

$C_{\kappa}=-(A^{+})^{-1}B^{+}C_{\kappa}^{\prime}.$

It follows that $\operatorname{H}^{1}(D,V)$ admits a basis
$(\widetilde{\kappa}_{1},\ldots,\widetilde{\kappa}_{e})$ defined by

$\widetilde{\kappa}_{j}=\left(\sum_{i=1}^{d^{+}}c_{i,j}\kappa_{i}\right)-\kappa^{\prime}_{j},$

where $(c_{1,j},\ldots,c_{d^{+},j})$ is the $j$-th column of the matrix
$(A^{+})^{-1}B^{+}$, $1\leqslant j\leqslant e$. Letting
$(\eta_{1},\ldots,\eta_{e})$ denote the basis of
$D_{1}=\operatorname{Hom}_{G_{p}}\left(W_{1}^{-},\overline{\mathbb{Q}}_{p}\right)$
which is dual to $(w^{-}_{1},\ldots,w^{-}_{e})$, for $1\leqslant j\leqslant
e$, one has

$\begin{array}[]{l}j_{D,f}(\widetilde{\kappa}_{j})=\sum_{i=1}^{e}\left(\log_{p}(u_{i,j}^{-})-c_{1,j}\log_{p}(\varepsilon_{i,1}^{-})-\ldots-
c_{d^{+},j}\log_{p}(\varepsilon_{i,d^{+}}^{-})\right)\eta_{i},\\\
j_{D,c}(\widetilde{\kappa}_{j})=-\sum_{i=1}^{e}\mathrm{ord}_{p}(u_{i,j}^{-})\eta_{i}.\end{array}$

The matrices of $j_{D,f}$ and $j_{D,c}$ in our distinguished bases are thus
given by $B^{-}-A^{-}(A^{+})^{-1}B^{+}$ and $-O^{-}$ respectively, and the
claimed formula follows by a computation of a determinant of a block matrix
using the fact that $A^{+}$ is invertible. ∎

######  Remark 2.16.

  1. (i)

One easily checks that the formula given in Theorem 2.15 does not depend on
the choice of bases of $W^{+}$ and $W_{1}$. Furthermore, one always can find
suitable $\overline{\mathbb{Q}}_{p}$-bases in which $\det O^{-}=1$.

  2. (ii)

Assume that $\operatorname{Frob}_{p}$ acts trivially on $W$, i.e. $W_{1}=W$
and $e=d^{-}$, and choose $(w^{-}_{1},\ldots,w^{-}_{e})$ to be a basis of the
linear subspace $W^{\circ}$ from Lemma 2.5, which by Proposition 2.9 is a
supplement of $W^{+}$. Then $A^{-}$ is the zero matrix and the above formula
becomes $\mathscr{L}(V,D)=(-1)^{e}\frac{\det B^{-}}{\det O^{-}}$.

  3. (iii)

In view of the results in this subsection, the effective computation of
$\mathscr{L}(V,D)$ relies a priori on that of
$(\kappa_{1},\ldots,\kappa_{d^{+}},\kappa^{\prime}_{1},\ldots,\kappa^{\prime}_{e})$,
which amounts to computing the $\varrho$-isotypic components of $U_{H}$ and of
$U_{H}^{(p)}$. This could be achieved by applying an idempotent (with
algebraic coefficients) to a Minkowski unit $\varepsilon_{0}$ in $U_{H}$, or
more generally to any element $\varepsilon_{0}$ generating $U_{H}$ as a
$G$-module. In the same vein, since
$U_{H}^{(p)}/U_{H}\simeq\mathrm{Ind}_{G_{p}}^{G}\overline{\mathbb{Q}}_{p}$ by
Dirichlet’s Unit Theorem, any sufficiently generic element $u_{0}$ in
$U_{H}^{(p)}$ generates $U_{H}^{(p)}/U_{H}$ as a $G$-module and can be used to
determine the $\kappa^{\prime}_{j}$’s. In contrast, determining a basis of
$\operatorname{H}^{1}(D,V)$ would involve linear equations with possibly
transcendental coefficients.

The situation is particularly nice when $d^{+}=e=f=1$ and
$W^{+}=W^{\operatorname{Frob}_{p}=\beta}$. Let
$e_{\rho}\in\overline{\mathbb{Q}}[G]$ be the idempotent attached to $\rho$ and
define $e_{W^{+}},e_{W_{1}}\in\overline{\mathbb{Q}}[G_{p}]$ as

$e_{W^{+}}=\sum_{i=1}^{|G_{p}|}\beta^{-i}\cdot\operatorname{Frob}_{p}^{i}\quad\mbox{and}\quad
e_{W_{1}}=\sum_{i=1}^{|G_{p}|}\operatorname{Frob}_{p}^{i}.$

We can obtain $\varepsilon_{1,1}^{\pm}$ from $\varepsilon_{0}$ by letting
$\varepsilon_{1,1}^{+}=e_{W^{+}}\cdot e_{\rho}\cdot\varepsilon_{0}$ and
$\varepsilon_{1,1}^{-}=e_{W_{1}}\cdot e_{\rho}\cdot\varepsilon_{0}$ and
similarly we can take $u_{1,1}^{+}=e_{W^{+}}\cdot e_{\rho}\cdot u_{0}$ and
$u_{1,1}^{-}=e_{W_{1}}\cdot e_{\rho}\cdot u_{0}$. Such numerical computations
have been performed for certain $3$-dimensional representations in [16].

###  2.5. The dual $\mathscr{L}$-invariant

In this section, we give an alternative expression of the
$\mathscr{L}$-invariant that derives from a dual construction given in [5,
§2.3] which will be used in §3.4. Perrin-Riou’s definition of regularity
actually applies to $D_{\mathrm{crys}}(W)$ and one checks that a submodule
$D\subseteq D_{\mathrm{crys}}(V)$ is regular if and only if its orthogonal
complement $D^{\perp}$ for the natural pairing $D_{\mathrm{crys}}(V)\times
D_{\mathrm{crys}}(W)\to D_{\mathrm{crys}}(\overline{\mathbb{Q}}_{p}(1))$ is a
regular submodule of $D_{\mathrm{crys}}(W)$. We now fix a regular submodule
$D$ of $D_{\mathrm{crys}}(V)$ and we keep the same notations as in the
previous sections.

By Class Field Theory and Frobenius reciprocity one has

$\operatorname{H}^{1}_{f,p}(W)=\ker\left(\operatorname{Hom}_{G_{p}}\left(E^{\times},W\right)\longrightarrow\operatorname{Hom}_{G}({\mathcal{O}_{H}\left[\tfrac{1}{p}\right]^{\times}},W)\right).$

Consider the linear subspace $\operatorname{H}^{1}(D^{\perp},W)$ of
$\operatorname{H}^{1}_{f,p}(W)$, consisting of elements $\kappa$ such that
$\kappa(E^{\times})\subset W^{+}+W_{1}$. Computations dual to those in Section
2.3 show that $\operatorname{H}^{1}(D^{\perp},W)$ is the module introduced by
Benois-Horte in [5, §2.3.6] and that it has dimension $e$ over
$\overline{\mathbb{Q}}_{p}$. For any
$\kappa\in\operatorname{H}^{1}(D^{\perp},W)$ we denote by $\kappa^{-}$ its
composition with the projection $(W^{+}+W_{1})\twoheadrightarrow W_{1}^{-}$
and we put

$j_{D^{\perp},f}(\kappa)=\kappa^{-}(p),\qquad
j_{D^{\perp},c}(\kappa)=\frac{\kappa^{-}(1+p)}{\log_{p}(1+p)}.$

This defines two linear maps
$j_{D^{\perp},f},j_{D^{\perp},c}:\operatorname{H}^{1}(W,D^{\perp})\to
W_{1}^{-}$ such that $j_{D^{\perp},c}$ is an isomorphism. When specialized to
our setting, the construction of the dual $\mathscr{L}$-invariant
$\mathscr{L}(W,D^{\perp})$ given in [5, Definition 2.3.7], leads to the
following formula:

$\mathscr{L}(W,D^{\perp})=(-1)^{e}\det\left(j_{D^{\perp},f}\circ\left(j_{D^{\perp},c}\right)^{-1}|W_{1}^{-}\right).$

The following proposition is [5, Prop. 2.3.8]. Note that the sign $(-1)^{e}$
corresponds to the sign in the conjectural functional equation for $p$-adic
$L$-functions, as explained in detail in [3, §2.2.6, §2.3.5].

######  Proposition 2.17.

Assume that $D$ is a regular submodule of $D_{\mathrm{crys}}(V)$. Then

$\mathscr{L}(W,D^{\perp})=(-1)^{e}\mathscr{L}(V,D).$

##  3\. Examples of computation of $\mathscr{L}$-invariant

###  3.1. Gross’ $p$-adic regulator

We assume throughout this paragraph that $d^{+}=0$, i.e. any complex
conjugation in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on
$W$ as $-1$. Then the Galois extension $H/\mathbb{Q}$ cut out by $\varrho$ is
a CM field. Conversely, if $\varrho$ is irreducible and cuts out a CM field
$H$, then any complex conjugation defines a non-trivial central element in
$G=\operatorname{Gal}(H/\mathbb{Q})$ which is necessarily scalar, hence equal
to $-1$, by Schur’s lemma. The unique $p$-refinement $W^{+}=\\{0\\}$ of $W$ is
automatically regular and the corresponding regular submodule is
$D=D_{\mathrm{crys}}(V)$. Note that we have

$e=\dim\operatorname{H}^{0}(\mathbb{Q}_{p},W).$

One may attach to $\varrho$ a $p$-adic $L$-function by using Deligne-Ribet’s
construction [19] together with Brauer’s induction theorem (see [24]). Gross
[25] has proposed a conjectural formula for the leading term of this $p$-adic
$L$-function at $s=0$ in terms of a certain $\mathscr{L}$-invariant which is
nowadays referred to as Gross’s $p$-adic regulator. We briefly recall its
definition and relate it to Benois’ $\mathscr{L}$-invariant.

Let $Y$ be the $\overline{\mathbb{Q}}_{p}$-vector space
$\bigoplus_{\mathfrak{p}}\overline{\mathbb{Q}}_{p}\cdot\mathfrak{p}$, where
the direct sum runs over the set $S_{p}(H)$ of all primes $\mathfrak{p}$ of
$H$ above $p$. We denote by $X\subset Y$ the kernel of the augmentation map
$\sum_{\mathfrak{p}}a_{\mathfrak{p}}\cdot\mathfrak{p}\mapsto\sum_{\mathfrak{p}}a_{\mathfrak{p}}$.
Note that $G$ naturally acts on both $X$ and $Y$ (see e.g. [34, Chap. I §3]),
and $\operatorname{Hom}_{G}(W,X)=\operatorname{Hom}_{G}(W,Y)$ as
$Y=X\oplus\overline{\mathbb{Q}}_{p}$. For $\mathfrak{p}\in S_{p}(H)$, Gross
[25, (1.8)] defines the local absolute value of $\alpha\in H_{\mathfrak{p}}$
by letting
$||\alpha||_{\mathfrak{p},p}=p^{-\mathrm{ord}_{\mathfrak{p}}(\alpha)f_{p}}\textbf{N}_{H_{\mathfrak{p}}/\mathbb{Q}_{p}}(\alpha)$,
where $f_{p}$ is the residual degree of $H_{\mathfrak{p}}$ and
$\textbf{N}_{H_{\mathfrak{p}}/\mathbb{Q}_{p}}$ is the norm map for the field
extension $H_{\mathfrak{p}}/\mathbb{Q}_{p}$. Gross’ $p$-adic regulator map is
given by

$\lambda_{p}:U_{H}^{(p)}\longrightarrow Y,\qquad u\mapsto\sum_{\mathfrak{p}\in
S_{p}(H)}\log_{p}(||u||_{\mathfrak{p},p})\cdot\mathfrak{p},$

whose image lies in $X$ by the product formula. Gross introduces another
$G$-equivariant map

$o_{p}:U_{H}^{(p)}\longrightarrow Y,\qquad u\mapsto\sum_{\mathfrak{p}\in
S_{p}(H)}f_{p}\cdot\mathrm{ord}_{\mathfrak{p}}(u)\cdot\mathfrak{p},$

(see _loc. cit._ , (1.12) and (1.20)). Since
$\dim\operatorname{Hom}_{G}(W,U_{H})=d^{+}=0$, the map
$\operatorname{Hom}_{G}(W,U_{H}^{(p)})\to\operatorname{Hom}_{G}(W,Y)=\operatorname{Hom}_{G}(W,X)$
induced by $o_{p}$ is injective, and it is in fact an isomorphism of
$e$-dimensional $\overline{\mathbb{Q}}_{p}$-vector spaces. Gross’ regulator
for
$W^{*}=\operatorname{Hom}_{\overline{\mathbb{Q}}_{p}}(W,\overline{\mathbb{Q}}_{p})$
is then defined as

$R_{p}(W^{*})=\det\left(\lambda_{p}\circ o_{p}^{-1}\ \big{|}\
\operatorname{Hom}_{G}(W,X)\right).$

Here, we still denote by $\lambda_{p}$ and $o_{p}$ the maps obtained by post-
composing by $\lambda_{p}$ and $o_{p}$ respectively. Note that, under the
canonical identification $(W^{*}\otimes-)^{G}=\operatorname{Hom}_{G}(W,-)$,
these maps coincide with the maps $1\otimes\lambda_{p},1\otimes o_{p}\ \colon\
(W^{*}\otimes U_{H}^{(p)})^{G}\to(W^{*}\otimes X)^{G}$ appearing in Gross’
definition (loc. cit., (2.10)).

######  Proposition 3.1.

Let $D=D_{\mathrm{crys}}(V)$. One has $\mathscr{L}(V,D)=(-1)^{e}\cdot
R_{p}(W^{*})$.

###### Proof.

Let $\mathfrak{p}_{0}\in S_{p}(H)$ be the prime above $p$ determined by
$\iota_{p}$ and put $W_{1}=\operatorname{H}^{0}(\mathbb{Q}_{p},W)$. Consider
the following diagram

$\textstyle{\operatorname{Hom}_{G}(W,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\operatorname{Hom}_{G_{p}}(W,\overline{\mathbb{Q}}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}(W_{1},\overline{\mathbb{Q}}_{p}),}$$\textstyle{\operatorname{Hom}_{G}(W,U_{H}^{(p)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{p}}$$\scriptstyle{o_{p}}$$\scriptstyle{\log_{p}}$$\scriptstyle{\mathrm{ord}_{p}}$

where the map $\pi$ is induced by the projection
$\sum_{\mathfrak{p}}a_{\mathfrak{p}}\cdot\mathfrak{p}\in X\mapsto
a_{\mathfrak{p}_{0}}$. By Frobenius reciprocity $\pi$ is an isomorphism
because $W$ does not contain the trivial representation. We claim that the
above diagram becomes commutative when we multiply the maps $\log_{p}$ and
$\mathrm{ord}_{p}$ by the same factor $f_{p}$, that is, the maps
$\pi\circ\lambda_{p}$ and $\pi\circ o_{p}$ respectively coincide with
$f_{p}\cdot\log_{p}$ and $f_{p}\cdot\mathrm{ord}_{p}$. Indeed, if
$\kappa\in\operatorname{Hom}_{G}(W,U_{H}^{(p)})$, then $\kappa_{|W_{1}}$ takes
values in $(U_{H}^{(p)})^{G_{p}}$, hence $\iota_{p}\circ\kappa_{|W_{1}}$ lands
in
$(H_{\mathfrak{p}_{0}}^{\times}\otimes\overline{\mathbb{Q}}_{p})^{G_{p}}=\mathbb{Q}_{p}^{\times}\otimes\overline{\mathbb{Q}}_{p}$.
Our claim then follows from the fact that the map
$\log_{p}\circ||\cdot||_{\mathfrak{p}_{0},p}$ coincides with
$f_{p}\cdot\log_{p}$ on $\mathbb{Q}_{p}^{\times}$. This proves that
$R_{p}(W^{*})$ is equal to the determinant of the map
$\log_{p}\circ\mathrm{ord}_{p}^{-1}$ acting on
$\operatorname{Hom}(W_{1},\overline{\mathbb{Q}}_{p})$. In terms of Notation
2.14, it is then clear that $R_{p}(W^{*})=\det(B^{-})\cdot\det(O^{-})^{-1}$,
so $\mathscr{L}(V,D)=(-1)^{e}\cdot R_{p}(W^{*})$ by Theorem 2.15. ∎

###  3.2. Weight $1$ cuspforms

Given a weight $1$ newform $f$, we let $\varrho_{f}$ denote the corresponding
Artin representation constructed by Deligne and Serre, which ramifies
precisely at the primes dividing the level of $f$. Since $\varrho_{f}$ is two-
dimensional and odd, one has $d^{+}=d^{-}=1$, which can be thought of as a
regularity condition at $\infty$. It might be worth recalling that the
Khare–Wintenberger proof of Serre’s Modularity Conjecture implies that all
two-dimensional odd Artin representations are obtained in this way. Since
$\varrho_{f}$ has finite image, one can choose a
$\overline{\mathbb{Q}}$-rational basis for its image and consider
$\varrho_{f}$ as $\overline{\mathbb{Q}}_{p}$-valued, via the fixed embedding
$\iota_{p}:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$. It
is then crystalline at $p$ if and only if $p$ does not divide the level of
$f$, which we will henceforth assume.

We say that $f$ is $p$-regular if the eigenvalues $\alpha$ and $\beta$ of
$\varrho_{f}(\operatorname{Frob}_{p})$ are distinct, and $p$-irregular
otherwise.

####  3.2.1. The $p$-regular case

Assume that $f$ is $p$-regular. The choice of a $p$-refinement $W^{+}$ (see
Definition 2.8) amounts to choosing between $\alpha$ and $\beta$. We let
$f_{\alpha}$ denote the $p$-stabilization of $f$ having $U_{p}$-eigenvalue
$\alpha$ and let $W^{+}=W_{\beta}$ be the line on which
$\operatorname{Frob}_{p}$ acts by multiplication by
$\beta\in\overline{\mathbb{Q}}^{\times}$. As $\alpha\neq\beta$, $W^{+}$ admits
an $\overline{\mathbb{Q}}$-rational basis, i.e. it is a motivic $p$-refinement
in the sense of Definition 2.8. It is thus a regular $p$-refinement by
Proposition 2.9. The $\mathscr{L}$-invariant $\mathscr{L}(f_{\alpha})$ of
$f_{\alpha}$ is defined as $\mathscr{L}(V,D_{\alpha})$ where $D_{\alpha}$ is
the regular submodule attached to $W_{\beta}$ (see (5)). One checks that $e=0$
if $\alpha\neq 1$, in which case one has $\mathscr{L}(f_{\alpha})=1$, and that
$e=1$ if $\alpha=1$. Thus, the conjectural Perrin-Riou $p$-adic $L$-function
$L_{p}(f_{\alpha},s)$ attached to $f_{\alpha}$, i.e. to choices of lattices in
$V$ and in $D_{\alpha}$, has a trivial zero at $s=0$ if and only if
$\alpha=1$. A potential candidate for $L_{p}(f_{\alpha},s)$ is constructed (up
to a non-zero constant) by Bellaïche and the first author in [2] and some of
its properties are studied by the second author in [28]. It is namely shown
that it vanishes at $s=0$ when $\alpha=1$, as expected.

Assuming $e=1$, let $(w_{\alpha},w_{\beta})$ be a basis of $W$ consisting of
eigenvectors for $\operatorname{Frob}_{p}$ and let $(\kappa,\kappa^{\prime})$
be a $\overline{\mathbb{Q}}_{p}$-basis of
$\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ such that $\kappa$ takes values in
$U_{H}$. These choices pin down two elements
$\varepsilon_{\alpha}=\kappa(w_{\alpha})$ and
$\varepsilon_{\beta}=\kappa(w_{\beta})$ in $U_{H}$ and two elements
$u_{\alpha}=\kappa^{\prime}(w_{\alpha})$ and
$u_{\beta}=\kappa^{\prime}(w_{\beta})$ in $U^{(p)}_{H}$, and Theorem 2.15
yields:

$\mathscr{L}(f_{\alpha})=\frac{\log_{p}(\varepsilon_{\alpha})\log_{p}(u_{\beta})-\log_{p}(\varepsilon_{\beta})\log_{p}(u_{\alpha})}{\log_{p}(\varepsilon_{\beta})\
\mathrm{ord}_{p}(u_{\alpha})}.$

####  3.2.2. The $p$-irregular case

Assume that $f$ is $p$-irregular. Then $\operatorname{Frob}_{p}$ acts as the
scalar $\alpha=\beta$ on $W$ and any line $W^{+}\neq W^{\circ}$ of $W$ defines
a regular $p$-refinement. One has again $e=0$ (and hence,
$\mathscr{L}(V,D)=1$) if $\alpha\neq 1$ and $e=1$ if $\alpha=1$. Assuming
$e=1$, one can compute $\mathscr{L}(V,D)$ as in Section 2.4. We choose a basis
$w^{+}$ of $W^{+}$ that we complete into a basis $(w^{+},w^{-})$ of $W$. Given
a basis $\kappa$ of $\operatorname{Hom}_{G}(W,U_{H})$, Lemma 2.13 ensures the
existence of $\kappa^{\prime}\in\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ non-
proportional to $\kappa$ and such that
$\mathrm{ord}_{p}(\kappa^{\prime}(w^{+}))=0$. As in the $p$-regular case,
these choices pin down two elements $\varepsilon^{\pm}=\kappa(w^{\pm})$ in
$U_{H}$ and two elements $u^{\pm}=\kappa^{\prime}(w^{\pm})$ in $U^{(p)}_{H}$,
with the extra property that $\mathrm{ord}_{p}(u^{+})=0$. By Theorem 2.15, one
has the following expression of the $\mathscr{L}$-invariant:

$\mathscr{L}(V,D)=\frac{\log_{p}(\varepsilon^{-})\log_{p}(u^{+})-\log_{p}(\varepsilon^{+})\log_{p}(u^{-})}{\log_{p}(\varepsilon^{+})\
\mathrm{ord}_{p}(u^{-})}.$

####  3.2.3. The CM case

Let $f$ be a weight one cuspform having complex multiplication by an imaginary
quadratic field $K$. Then $\varrho_{f}=\mathrm{Ind}_{\mathbb{Q}}^{K}\psi$ with
$\psi:\operatorname{G}_{K}\to\overline{\mathbb{Q}}_{p}$ a finite order
character cutting out an abelian extension $H_{\psi}/K$. As $\varrho_{f}$ is
irreducible, $\psi$ is different from its conjugate
$\bar{\psi}=\psi(\tau\cdot\tau)$, allowing us to decompose $W$ as a direct sum
$\overline{\mathbb{Q}}_{p}$-lines $W_{\psi}$ and $W_{\bar{\psi}}$ on which
$\operatorname{G}_{K}$ acts by $\psi$ and ${\bar{\psi}}$, respectively. Fix a
basis $e_{1}$ of $W_{\psi}$ and put $e_{2}=\tau(e_{1})\in W_{\bar{\psi}}$. The
fixed field $H$ of $\ker(\varrho_{f})$ is the compositum of $H_{\psi}$ and
$H_{\bar{\psi}}$, and we let $G=\operatorname{Gal}(H/\mathbb{Q})$. For any
$\overline{\mathbb{Q}}_{p}[G]$-module $M$, Frobenius reciprocity allows us to
canonically identify $\operatorname{Hom}_{G}(W,M)$ with the $\psi$-isotypic
component $M[\psi]$ of $M$. We now give a simpler expression for the
$\mathscr{L}$-invariants computed in §3.2.1-3.2.2 when $p$ splits in $K$.

We parameterize the lines of $W$ by
$\mathbb{P}^{1}(\overline{\mathbb{Q}}_{p})$ by letting
$W^{+}_{s}=\overline{\mathbb{Q}}_{p}(e_{1}+se_{2})$ when
$s\in\overline{\mathbb{Q}}_{p}$ and
$W_{\infty}^{+}=\overline{\mathbb{Q}}_{p}e_{2}$. Write
$p\mathcal{O}_{K}=\mathfrak{p}\bar{\mathfrak{p}}$, where $\mathfrak{p}$ is the
prime determined by $\iota_{p}$ and $\bar{\mathfrak{p}}=\tau(\mathfrak{p})$.
Note that
$\\{\alpha,\beta\\}=\\{\psi(\mathfrak{p}),\psi(\overline{\mathfrak{p}})\\}$.

Given a basis $\varepsilon_{\psi}$ of the line $U_{H}[\psi]$ the slope of
$\psi$ is defined as (see [7, §1.1])

(8)
$\mathscr{S}_{\psi}=-\frac{\log_{p}(\varepsilon_{\psi})}{\log_{p}(\tau(\varepsilon_{\psi}))}\in\overline{\mathbb{Q}}_{p}^{\times}.$

As $G_{p}\subseteq\operatorname{Gal}(H/K)$, one can check that $W_{s}^{+}$ is
a regular $p$-refinement if and only if

  * •

either $\psi(\mathfrak{p})\neq\psi(\bar{\mathfrak{p}})$ and
$s\in\\{0,\infty\\}$ (the $p$-regular case),

  * •

or $\psi(\mathfrak{p})=\psi(\bar{\mathfrak{p}})$ and $s\neq\mathscr{S}_{\psi}$
(the $p$-irregular case).

Let $D_{s}$ be the regular submodule associated with $W^{+}_{s}$ and assume
that we are in presence of trivial zeros, i.e. $\psi(\bar{\mathfrak{p}})=1$ if
$s=0$, $\psi(\mathfrak{p})=1$ if $s=\infty$, or
$\psi(\mathfrak{p})=\psi(\bar{\mathfrak{p}})=1$ in the $p$-irregular setting.
Then $d^{+}=d^{-}=e=1$.

Assuming that $\psi(\bar{\mathfrak{p}})=1$, the cyclotomic
$\mathscr{L}$-invariant of $\psi$ is defined as

(9)
$\mathscr{L}(\psi)=-\frac{\log_{p}(\tau(u^{\circ}_{\psi}))}{\mathrm{ord}_{p}(\tau(u^{\circ}_{\psi}))},$

where $u^{\circ}_{\psi}$ a basis of the line in $U_{H}^{(p)}[\psi]$ defined by
the equations $\log_{p}(\cdot)=\mathrm{ord}_{p}(\cdot)=0$ (see [12, Rem.
1.5(iii)]). In order to avoid the presence of the possibly transcendental
equation $\log_{p}(\cdot)=0$, we consider the plane in $U_{H}^{(p)}[\psi]$ cut
out simply by the rational equation $\mathrm{ord}_{p}(\cdot)=0$ (the careful
reader might notice the analogy with §2.4 where
$\operatorname{H}^{1}(D_{s},V)$ was replaced by a more concrete space of a
larger dimension). It is easy to see that given any $\bar{\mathfrak{p}}$-unit
$u_{\psi}\in U_{H}^{(p)}[\psi]$ (which is not a unit), the set
$\\{\varepsilon_{\psi},u_{\psi}\\}$ forms a basis of this plane, and
furthermore $u^{\circ}_{\psi}$ from (9) can be taken to be
$u_{\psi}\otimes\log_{p}(\varepsilon_{\psi})-\varepsilon_{\psi}\otimes\log_{p}(u_{\psi})$.
Thus one retrieves the more explicit formula from [7, (6)]

(10)
$\mathscr{L}(\psi)=-\frac{\log_{p}(\tau(u_{\psi}))+\mathscr{S}_{\bar{\psi}}\log_{p}(u_{\psi})}{\mathrm{ord}_{p}(\tau(u_{\psi}))}.$

######  Proposition 3.2.

  1. (i)

In the $p$-irregular case, for all $s\neq\mathscr{S}_{\psi}$, we have the
following formula

$\mathscr{L}(V,D_{s})=\frac{s\cdot\mathscr{L}(\bar{\psi})-\mathscr{S}_{\psi}\cdot\mathscr{L}(\psi)}{s-\mathscr{S}_{\psi}}.$

  2. (ii)

In the $p$-regular case one has $\mathscr{L}(V,D_{0})=\mathscr{L}(\psi)$, if
$\psi(\bar{\mathfrak{p}})=1$, and
$\mathscr{L}(V,D_{\infty})=\mathscr{L}(\bar{\psi})$, if
$\psi(\mathfrak{p})=1$.

###### Proof.

We apply the formalism of §2.4 noting that a natural isomorphism between
$\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ and $U_{H}^{(p)}[\psi]$ is obtained by
evaluating at $e_{1}\in W_{\psi}$. Moreover the line $W^{\circ}$ from Lemma
2.5 has basis $e_{1}+\mathscr{S}_{\psi}e_{2}$.

(i) Assume that $\psi(\mathfrak{p})=\psi(\bar{\mathfrak{p}})=1$ and fix
$s\neq\mathscr{S}_{\psi}$. A supplement of the line
$\operatorname{Hom}_{G}\left(W,U_{H}\right)$ in the kernel of the map in Lemma
2.13 is generated by an element
$\kappa^{\prime}_{s}\in\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ such that
$\mathrm{ord}_{p}(\kappa^{\prime}_{s}(e_{1}+se_{2}))=0$. Rescaling the units
$u^{\circ}_{\psi}$ so that $\mathrm{ord}_{p}(\tau(u^{\circ}_{\psi}))=1$, one
finds that the element $\kappa^{\prime}_{s}$ determined by
$\kappa^{\prime}_{s}(e_{1})=u^{\circ}_{\psi}\cdot\tau(u^{\circ}_{\bar{\psi}})^{-s}\in
U_{H}^{(p)}$ has all the required properties, and of course
$\kappa^{\prime}_{s}(e_{2})=\tau(u^{\circ}_{\psi})\cdot(u^{\circ}_{\bar{\psi}})^{-s}$.
Moreover, as $G_{p}$ acts trivially on $W^{\circ}$, Remark 2.16(ii) applies
and we have

$\mathscr{L}(V,D_{s})=-\frac{\log_{p}\left(\kappa^{\prime}_{s}(e_{1}+\mathscr{S}_{\psi}e_{2})\right)}{\mathrm{ord}_{p}\left(\kappa^{\prime}_{s}(e_{1}+\mathscr{S}_{\psi}e_{2})\right)}=-\frac{-s\cdot\log(\tau(u^{\circ}_{\bar{\psi}}))+\mathscr{S}_{\psi}\log(\tau(u^{\circ}_{\psi}))}{-s+\mathscr{S}_{\psi}},$

so the formula follows from (9).

(ii) By symmetry, it suffices to compute $\mathscr{L}(V,D_{0})$ when
$\psi(\bar{\mathfrak{p}})=1\neq\psi(\mathfrak{p})$. Following Notation 2.14,
we have $W^{+}=\overline{\mathbb{Q}}_{p}e_{1}$,
$W_{1}=\overline{\mathbb{Q}}_{p}e_{2}$ and $W_{1}^{+}=\\{0\\}$, in particular
the map in Lemma 2.13 is the zero map. Let $\kappa$ be a basis of
$\operatorname{Hom}_{G}(W,U_{H})$ that we complete in a basis
$(\kappa,\kappa^{\prime}_{0})$ of $\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ such
that $\kappa_{0}^{\prime}(e_{1})=u_{\psi}^{\circ}$. Since
$\log_{p}(\kappa^{\prime}_{0}(e_{1}))=0$, i.e. $B^{+}=(0)$, Theorem 2.15
yields

$\mathscr{L}(V,D_{0})=-\frac{\log_{p}(\kappa^{\prime}_{0}(e_{2}))}{\mathrm{ord}_{p}(\kappa^{\prime}_{0}(e_{2}))}=\mathscr{L}(\psi),$

as claimed. ∎

###  3.3. Adjoint of a weight $1$ $p$-regular cuspform

Let $\varrho$ be the adjoint of the Artin representation $\varrho_{f}$
attached by Deligne and Serre to a weight one cuspform $f$ of level relatively
prime to $p$. As in the introduction, we assume that $\varrho_{f}$ has
$\overline{\mathbb{Q}}$-coefficients and we view it via $\iota_{p}$ as a
$p$-adic representation. Let $\alpha$ and $\beta$ be the eigenvalues of
$\varrho_{f}(\operatorname{Frob}_{p})$ and assume that $\alpha\neq\pm\beta$
(so $f$ is in particular $p$-regular). We let
$(w_{1},w_{\beta/\alpha},w_{\alpha/\beta})$ be a basis of $W$ on which
$\operatorname{Frob}_{p}$ acts with the eigenvalue prescribed in the index and
we denote by $(w^{*}_{1},w^{*}_{\beta/\alpha},w^{*}_{\alpha/\beta})$ its dual
basis. The $p$-adic realization $V$ of the motive
$M=(\operatorname{ad}^{0}(f))(1)$ has a unique decomposition as
$\operatorname{G}_{\mathbb{Q}_{p}}$-modules
$V=\overline{\mathbb{Q}}_{p}(1)\cdot
w^{*}_{1}\bigoplus\overline{\mathbb{Q}}_{p}(1)\cdot
w^{*}_{\beta/\alpha}\bigoplus\overline{\mathbb{Q}}_{p}(1)\cdot
w^{*}_{\alpha/\beta}$.

We now apply the results of §2.4. Since one has $d^{+}=e=f=1$ and
$\varrho(\operatorname{Frob}_{p})$ acts with pairwise distinct eigenvalues,
the line $W^{+}$ generated by $w_{\beta/\alpha}$ is a motivic $p$-refinement
(so it is regular by Proposition 2.9(iii)) and $W_{1}$ is a line with basis
$w_{1}$. Let $(\kappa,\kappa^{\prime})$ be a $\overline{\mathbb{Q}}_{p}$-basis
of $\operatorname{Hom}_{G}(W,U_{H}^{(p)})$ such that $\kappa$ takes values in
$U_{H}$. Without loss of generality, we may assume that
$w_{\beta/\alpha},w_{1}$ (resp. $\kappa,\kappa^{\prime}$) belong to the
canonical $\overline{\mathbb{Q}}$-structure of $W$ (resp. of
$\operatorname{H}^{1}_{f,p}(V)$ described in §2.2.1). By Theorem 2.15, Benois’
$\mathscr{L}$-invariant for the regular submodule
$D=\operatorname{Hom}(W/W^{+},\overline{\mathbb{Q}}_{p})$ is given by

$\mathscr{L}(V,D)=\frac{\log_{p}(\varepsilon_{1})\log_{p}(u_{\beta/\alpha})-\log_{p}(\varepsilon_{\beta/\alpha})\log_{p}(u_{1})}{\log_{p}(\varepsilon_{\beta/\alpha})\mathrm{ord}_{p}(u_{1})},$

where $\varepsilon_{1}=\kappa(w_{1})$,
$\varepsilon_{\beta/\alpha}=\kappa(w_{\beta/\alpha})$,
$u_{1}=\kappa^{\prime}(w_{1})$ and
$u_{\beta/\alpha}=\kappa^{\prime}(w_{\beta/\alpha})$.

Let us assume that $\varrho_{f}$ is residually irreducible and
$p$-distinguished, the latter condition meaning that $\alpha$ and $\beta$ are
distinct modulo the maximal ideal of $\overline{\mathbb{Z}}_{p}$. We suppose
that $\varrho_{f}$ is not induced by a character of a real quadratic field in
which $p$ splits, hence by [2] the eigencurve is étale over the weight space
at both $p$-stabilizations $f_{\alpha}$ and $f_{\beta}$ of $f$. Using the
automorphic deformation of $f_{\alpha}$, Hida defined an analytic
$\mathscr{L}$-invariant $\mathscr{L}(\operatorname{ad}^{0}(f_{\alpha}))$ for
which Darmon, Lauder and Rotger proposed in [16, Conj. 1.2] a conjectural
description in terms of the $p$-adic logarithms of
$\varepsilon_{1},\varepsilon_{\beta/\alpha},u_{1},u_{\beta/\alpha}$ (see also
[30, §5.1]). In light of our computation of Benois’ $\mathscr{L}$-invariant in
this setting, the main theorem of Rivero and Rotger [30, Thm. A’] can then be
recast as

$\mathscr{L}(V,D)=\mathscr{L}(\operatorname{ad}^{0}(f_{\alpha}))\mod\overline{\mathbb{Q}}^{\times}.$

###  3.4. Adjoint of a weight $1$ $p$-irregular CM cuspform

We consider the case of the Artin motive $W=\operatorname{ad}^{0}\varrho_{f}$,
where $f$ is a weight one cuspform irregular at $p$ and having CM by an
imaginary quadratic field $K$ in which $p$ splits. Letting
$\varrho_{f}=\mathrm{Ind}_{K}^{\mathbb{Q}}\psi$, where $\psi$ is a finite
order Hecke character of $K$, the above assumptions translate into the fact
that the anti-cyclotomic (or ring class) character $\varphi=\psi/\bar{\psi}$
is non-trivial, but is locally trivial at (places above) $p$. Let $H$ denote
the fixed field of $\ker(\varphi)$.

####  3.4.1. Regular submodules

Fix a dihedral basis of $\varrho_{f}$ as in §3.2.3, so that
$\varrho_{f|\operatorname{G}_{K}}=\psi\oplus\bar{\psi}$. This choice yields a
matrix representation $\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}$ of
$\operatorname{ad}^{0}\varrho_{f}$ with basis $\\{w_{1},w_{2},w_{3}\\}$ given
by $w_{1}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ and
$w_{2}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$,
$w_{3}=\tau(w_{2})=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$ Moreover, this
basis is adapted to the $\operatorname{G}_{\mathbb{Q}}$-decomposition
$\operatorname{ad}^{0}\varrho_{f}=\varepsilon_{K}\oplus\mathrm{Ind}_{K}^{\mathbb{Q}}\varphi$,
where $\varepsilon_{K}$ is the Dirichlet character associated with the
quadratic extension $K/\mathbb{Q}$. If $\varphi$ is quadratic then
$\mathrm{Ind}_{K}^{\mathbb{Q}}\varphi=\varepsilon_{K^{\prime}}\oplus\varepsilon_{F}$,
where $K^{\prime}$ and $F$ are the other two quadratic subfields of $H$.
Otherwise $\mathrm{Ind}_{K}^{\mathbb{Q}}\varphi$ is irreducible and we have
$(\mathrm{Ind}_{K}^{\mathbb{Q}}\varphi)_{|\operatorname{G}_{K}}=\varphi\oplus\bar{\varphi}$
in the dihedral basis $\\{w_{2},w_{3}\\}$.

By Lemma 2.2(ii) and Frobenius reciprocity one has a natural isomorphism

$\operatorname{H}^{1}_{f}(V)=\operatorname{Hom}_{G}(W,U_{H})=\operatorname{Hom}_{G}(\mathrm{Ind}_{K}^{\mathbb{Q}}\varphi,U_{H})\xrightarrow{\sim}U_{H}[\varphi],$

where $\kappa\in\operatorname{Hom}_{G}(W,U_{H})$ sends $w_{1}$ to $1$, sends
$w_{2}$ to a $\varphi$-unit $u_{\varphi}\in U_{H}[\varphi]$ and sends
$w_{3}=\tau(w_{2})$ to $\tau(u_{\varphi})\in U_{H}[\bar{\varphi}]$.

We have $d=3$, $d^{+}=1$ and any line $W^{+}\subset W$ is $G_{p}$-stable. By
Proposition 2.9(ii), $W^{+}$ is regular if and only if it is not contained in
the plane defined by the equation

$\log_{p}(u_{\varphi})b+\log_{p}(\tau(u_{\varphi}))c=0\text{, i.e.
}\mathscr{S}_{\varphi}\cdot b=c,$

where the slope $\mathscr{S}_{\varphi}$ is defined in (8). In particular, the
only non-regular submodule of $\mathrm{Ind}_{K}^{\mathbb{Q}}\varphi$ is the
line generated by $w_{2}+\mathscr{S}_{\varphi}w_{3}$.

####  3.4.2. A modular basis of
$\operatorname{H}^{1}(\mathbb{Q},\operatorname{ad}^{0}\varrho_{f})$

Letting
$\mathscr{L}_{\mathfrak{p}}=-\frac{\log_{p}(u_{\mathfrak{p}})}{\mathrm{ord}_{p}(u_{\mathfrak{p}})}$
for any non-torsion element $u_{\mathfrak{p}}$ of
$\mathcal{O}_{K}[\tfrac{1}{\mathfrak{p}}]^{\times}$, the anti-cyclotomic
$\mathscr{L}$-invariant is defined as
$\mathscr{L}_{\mbox{-}}(\varphi)=\mathscr{L}(\varphi)-2\mathscr{L}_{\mathfrak{p}}$
(see (9)).

The space
$\operatorname{H}_{f,p}^{1}(\mathbb{Q},\operatorname{ad}^{0}\varrho_{f})=\operatorname{H}^{1}(\mathbb{Q},\operatorname{ad}^{0}\varrho_{f})$
is two-dimensional (see [2, Lem. 3.2]). We will now give a modular description
of this space and use it to write down an explicit basis, under the assumption
that

(11)
$\mathscr{L}_{\mbox{-}}(\varphi)\cdot\mathscr{L}_{\mbox{-}}(\bar{\varphi})\cdot(\mathscr{L}_{\mbox{-}}(\varphi)+\mathscr{L}_{\mbox{-}}(\bar{\varphi}))\neq
0.$

The condition (11) holds for $\varphi$ quadratic, whereas in all other cases
the Weak $p$-adic Schanuel Conjecture predicts that
$\mathscr{L}_{\mbox{-}}(\varphi)$ and $\mathscr{L}_{\mbox{-}}(\bar{\varphi})$
are algebraically independent.

Fix a square root $\xi$ of
$\mathscr{L}_{\mbox{-}}(\bar{\varphi})\mathscr{L}_{\mbox{-}}(\varphi)^{-1}\mathscr{S}_{\bar{\varphi}}$.

It was conjectured in [17] and proven in [7] that there is a natural
isomorphism between
$\operatorname{H}^{1}(\mathbb{Q},\operatorname{ad}^{0}\varrho_{f})$ and the
space $S^{\dagger}_{1}\lsem f\rsem_{0}$ of genuine overconvergent modular
forms belonging to the generalized eigenspace of $f$. Furthermore one computes
in [8] the $q$-expansions of a basis
$\\{f_{\mathscr{F}}^{\dagger},f_{\Theta}^{\dagger}\\}$ of the latter space.
The corresponding basis $\\{\kappa_{\mathscr{F}},\kappa_{\Theta}\\}$ of
$\operatorname{H}^{1}(\mathbb{Q},\operatorname{ad}^{0}\varrho_{f})$ is
characterized by $\kappa_{\mathscr{F}}(\tau)=\kappa_{\Theta}(\tau)=0$ and

$\kappa_{\Theta|\operatorname{G}_{K}}=\begin{pmatrix}\eta_{\mathfrak{p}}-\eta_{\bar{\mathfrak{p}}}&0\\\
0&\eta_{\bar{\mathfrak{p}}}-\eta_{\mathfrak{p}}\end{pmatrix},\text{ and
}\kappa_{\mathscr{F}|\operatorname{G}_{K}}=\begin{pmatrix}0&\eta_{\varphi}\\\
\bar{\eta}_{\varphi}&0\end{pmatrix}.$

Here $\eta_{\varphi}:\operatorname{G}_{K}\to\bar{\mathbb{Q}}_{p}$ represents
the cocycle $[\eta_{\varphi}]\in\operatorname{H}^{1}(K,\varphi)$, normalized
so that
$\mathrm{loc}_{p}([\eta_{\varphi}])=\log_{p}\in\operatorname{H}^{1}(\mathbb{Q}_{p},\bar{\mathbb{Q}}_{p})$,
whereas $(\eta_{\mathfrak{p}},\eta_{\bar{\mathfrak{p}}})$ is a basis of
$\operatorname{H}^{1}(K,\bar{\mathbb{Q}}_{p})$ with $\eta_{\mathfrak{p}}$ is
unramified outside $\mathfrak{p}$ such that
$\mathrm{loc}_{p}(\eta_{\mathfrak{p}})=\log_{p}$, and
$\eta_{\bar{\mathfrak{p}}}=\eta_{\mathfrak{p}}\circ\tau$.

####  3.4.3. The $\mathscr{L}$-invariant

As in §3.2.3 we parameterize the lines of $W$ by
$s\in\mathbb{P}^{1}(\overline{\mathbb{Q}}_{p})\setminus{\mathscr{S}_{\varphi}}$
and $t\in\overline{\mathbb{Q}}_{p}$ by letting
$W^{+}_{s,t}=\overline{\mathbb{Q}}_{p}(tw_{1}+w_{2}+sw_{3})$ with the
convention that $W^{+}_{\infty,t}=\overline{\mathbb{Q}}_{p}(tw_{1}+w_{3})$.
The resulting projection $W\to W^{-}=W/W^{+}_{s,t}$ is given by
$\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}\mapsto\begin{pmatrix}a-tb\\\
c-sb\end{pmatrix}$.

It is proven in [7, Lem. 1.4] that
$\eta_{\bar{\mathfrak{p}}}(\operatorname{Frob}_{\mathfrak{p}})=\eta_{\mathfrak{p}}(\operatorname{Frob}_{\bar{\mathfrak{p}}})=-\mathscr{L}_{\mathfrak{p}}$.
Using Local Class Field Theory, the fact that $\eta_{\mathfrak{p}}$ is
unramified outside $\mathfrak{p}$ allows us to compute:

$\eta_{\mathfrak{p}|K_{\mathfrak{p}}^{\times}}(p)=\frac{\eta_{\mathfrak{p}|K_{\mathfrak{p}}^{\times}}(u_{\mathfrak{p}}\bar{u}_{\mathfrak{p}})}{\mathrm{ord}_{p}(u_{\mathfrak{p}})}=0+\frac{\log_{p}(\bar{u}_{\mathfrak{p}})}{\mathrm{ord}_{p}(u_{\mathfrak{p}})}=\mathscr{L}_{\mathfrak{p}}.$

Using also [7, Prop 1.5], one finds:

(12) $\displaystyle
j_{D^{\perp},f}(\kappa_{\Theta})=\begin{pmatrix}2\mathscr{L}_{\mathfrak{p}}\\\
0\end{pmatrix},\qquad$ $\displaystyle
j_{D^{\perp},f}(\kappa_{\mathscr{F}})=\begin{pmatrix}-t\mathscr{L}(\bar{\varphi})\\\
\mathscr{S}_{\varphi}\mathscr{L}(\varphi)-s\mathscr{L}(\bar{\varphi})\end{pmatrix},$
(13) $\displaystyle j_{D^{\perp},c}(\kappa_{\Theta})=\begin{pmatrix}1\\\
0\end{pmatrix},\qquad$ $\displaystyle
j_{D^{\perp},c}(\kappa_{\mathscr{F}})=\begin{pmatrix}-t\\\
\mathscr{S}_{\varphi}-s\end{pmatrix}.$

As $e=2$, applying Proposition 2.17 yields

(14)
$\begin{split}&\mathscr{L}(V,D_{s,t})=(-1)^{2}\mathscr{L}(W,W^{+}_{s,t})\\\
&=\frac{\det\begin{pmatrix}2\mathscr{L}_{\mathfrak{p}}&-t\mathscr{L}(\bar{\varphi})\\\
0&\mathscr{S}_{\varphi}\mathscr{L}(\varphi)-s\mathscr{L}(\bar{\varphi})\end{pmatrix}}{\det\begin{pmatrix}1&-t\\\
0&\mathscr{S}_{\varphi}-s\end{pmatrix}}=\frac{2\mathscr{L}_{\mathfrak{p}}\cdot\left(\mathscr{S}_{\varphi}\mathscr{L}(\varphi)-s\mathscr{L}(\bar{\varphi})\right)}{\mathscr{S}_{\varphi}-s}.\end{split}$

We would like to conclude by determining the regular submodules of arithmetic
significance. As already noted $\operatorname{ad}^{0}(\varrho_{f})$ being
locally trivial at $p$, it does not point out in any particular direction.
Nevertheless, according to [7] the form $f$ belongs to exactly $4$ Hida
families, i.e. the representation $\varrho_{f}$ admits four non-trivial
deformations to $\overline{\mathbb{Q}}_{p}\lsem X\rsem$, each determining an
ordinary line. If this line is given by
$\overline{\mathbb{Q}}_{p}(e_{1}+se_{2})$ then the line of the corresponding
full flag in $\operatorname{ad}^{0}(\varrho_{f})$ is given by
$\overline{\mathbb{Q}}_{p}\left(\begin{smallmatrix}-s&1\\\
-s^{2}&s\end{smallmatrix}\right)=\overline{\mathbb{Q}}_{p}(-sw_{1}+w_{2}-s^{2}w_{3})$.

When $s=0$ one finds
$\mathscr{L}(V,D_{0,0})=2\mathscr{L}_{\mathfrak{p}}\cdot\mathscr{L}(\varphi)$,
corresponding to a $p$-adic variation along the CM family
$\Theta=\mathrm{Ind}_{K}^{\mathbb{Q}}(\psi\chi_{\mathfrak{p}})$ containing $f$
whose ordinary line is given by
$W^{+}_{\Theta}=\overline{\mathbb{Q}}_{p}e_{1}$. Hence this
$\mathscr{L}$-invariant may be denoted by
$\mathscr{L}(\operatorname{ad}^{0}(\varrho_{f}),W^{+}_{\Theta})$.

Analogously, when $s=\infty$ one finds
$\mathscr{L}(V,D_{\infty,0})=2\mathscr{L}_{\mathfrak{p}}\cdot\mathscr{L}(\bar{\varphi})$,
corresponding to a $p$-adic variation along the CM family
$\bar{\Theta}=\mathrm{Ind}_{K}^{\mathbb{Q}}(\bar{\psi}\chi_{\mathfrak{p}})$
containing $f$ whose ordinary line is given by
$W^{+}_{\bar{\Theta}}=\overline{\mathbb{Q}}_{p}e_{2}$. Hence this
$\mathscr{L}$-invariant may be denoted by
$\mathscr{L}(\operatorname{ad}^{0}(\varrho_{f}),W^{+}_{\bar{\Theta}})$.

Finally, when $s=\xi^{-1}$ corresponds to the ordinary line in one of the the
non-CM families $\mathscr{F}$ containing $f$, then
$W_{\mathscr{F}}^{+}=W^{+}_{-s^{2},-s}$, and similarly
$W_{\mathscr{F}\otimes\varepsilon_{K}}^{+}=W^{+}_{-s^{2},s}$. Note that these
are regular submodules as
$-\xi^{-2}=-\mathscr{L}_{\mbox{-}}(\varphi)\mathscr{L}_{\mbox{-}}(\bar{\varphi})^{-1}\mathscr{S}_{\varphi}\neq\mathscr{S}_{\varphi}$
by (11). Using formula (14), we get

(15)
$\displaystyle\mathscr{L}(\operatorname{ad}^{0}(\varrho_{f}),W_{\mathscr{F}}^{+})=\mathscr{L}(\operatorname{ad}^{0}(\varrho_{f}),W_{\mathscr{F}\otimes\varepsilon_{K}}^{+})$
$\displaystyle=\frac{2\mathscr{L}_{\mathfrak{p}}\cdot\left(\mathscr{S}_{\varphi}\mathscr{L}(\varphi)+\frac{\mathscr{L}_{\mbox{-}}(\varphi)}{\mathscr{L}_{\mbox{-}}(\bar{\varphi})}\mathscr{S}_{\varphi}\mathscr{L}(\bar{\varphi})\right)}{\mathscr{S}_{\varphi}+\frac{\mathscr{L}_{\mbox{-}}(\varphi)}{\mathscr{L}_{\mbox{-}}(\bar{\varphi})}\mathscr{S}_{\varphi}}=\frac{2\mathscr{L}_{\mathfrak{p}}\cdot\left(\mathscr{L}_{\mbox{-}}(\bar{\varphi})\mathscr{L}(\varphi)+\mathscr{L}_{\mbox{-}}(\varphi)\mathscr{L}(\bar{\varphi})\right)}{\mathscr{L}_{\mbox{-}}(\bar{\varphi})+\mathscr{L}_{\mbox{-}}(\varphi)}.$

Note that although $W_{\mathscr{F}\otimes\varepsilon_{K}}^{+}$ differs from
$W_{\mathscr{F}}^{+}$, the resulting $\mathscr{L}$-invariant is the same.

####  3.4.4. Compatibility with $p$-adic $L$-functions

The most promising approach for the construction of $p$-adic $L$-functions for
Artin motives is via $p$-adic deformation on an eigenvariety containing
sufficiently many points of regular weight. This is explained by Bellaïche [1]
in the case of weight one forms using the modular symbol construction of the
Coleman–Mazur eigencurve. His construction of an adjoint $p$-adic $L$-function
on the eigencurve is very relevant to our context, except that it misses the
cyclotomic variable. The upshot is that the Perrin-Riou $p$-adic $L$-function
should be easier to construct and arithmetically more significant to study for
the regular submodules $W^{+}_{\Theta}$, $W^{+}_{\bar{\Theta}}$,
$W^{+}_{\mathscr{F}}$ and $W^{+}_{\mathscr{F}\otimes\varepsilon_{K}}$, as this
would allow them to vary in families.

We now discuss potential candidates for such $p$-adic $L$-functions and the
compatibility with our expression for the $\mathscr{L}$-invariants.

First consider the CM families $\Theta$ and $\bar{\Theta}$. Note that
$W^{+}_{\Theta}$ and $W^{+}_{\bar{\Theta}}$ actually lie in the first factor
of the Galois decomposition
$\operatorname{ad}^{0}(f)=\mathrm{Ind}_{K}^{\mathbb{Q}}\varphi\bigoplus\varepsilon_{K}$.
Since Perrin-Riou’s conjecture is compatible with the Artin formalism, a
natural candidate for the $p$-adic $L$-function in the direction
$W^{+}_{\Theta}$, resp. $W^{+}_{\bar{\Theta}}$, is

$L_{p}(\varphi,s)\cdot L_{p}(\varepsilon_{K}\cdot\omega_{p},s),\quad\text{
resp. }\quad L_{p}(\bar{\varphi},s)\cdot
L_{p}(\varepsilon_{K}\cdot\omega_{p},s),$

where $L_{p}(\varphi,s)$, resp. $L_{p}(\bar{\varphi},s)$, is the restriction
to the cyclotomic line of Katz’s $p$-adic $L$-function for $\varphi$, resp.
for $\bar{\varphi}$ (e.g. see [7, §4.3]) and
$L_{p}(\varepsilon_{K}\cdot\omega_{p},s)$ is the Kubota-Leopoldt $p$-adic
$L$-function attached to $\varepsilon_{K}$. Note that Katz’s construction
indeed allows us to vary this $p$-adic $L$-function along the families
$\Theta$ and $\bar{\Theta}$. The expression for
$\mathscr{L}(\operatorname{ad}^{0}(\varrho_{f}),W^{+}_{\Theta})$ also factors
as a product of two $\mathscr{L}$-invariants, namely, $\mathscr{L}(\varphi)$
and $\mathscr{L}(\varepsilon_{K})=2\cdot\mathscr{L}_{\mathfrak{p}}$, the
latter being the $\mathscr{L}$-invariant introduced by Ferrero and Greenberg
[21] in their proof of the Trivial Zero Conjecture for
$L_{p}(\varepsilon_{K}\cdot\omega_{p},s)$. The term $\mathscr{L}(\varphi)$ is
the $\mathscr{L}$-invariant appearing in the Trivial Zero Conjecture for
$L_{p}(\varphi,s)$ proposed in [7, (45)]. When $\varphi$ is quadratic, this
conjecture is deduced from Gross’s factorization formula in _loc. cit._ , end
of §4.3. A proof of this conjecture for general $\varphi$ (under mild
hypotheses on $p$) based on the equivariant Tamagawa Number Conjecture proven
by Bley [9] has been announced by Büyükboduk and Sakamoto [12]. Meanwhile
Chida and Hsieh [14] provided an entirely different proof removing those
assumptions.

Given a Hida family $\mathscr{G}$ and letting $\mathscr{G}^{*}$ be its dual
family, Hida [27] attaches to the pair $(\mathscr{G},\mathscr{G}^{*})$ a
three-variable $p$-adic meromorphic function
$L_{p}(\mathscr{G},\mathscr{G}^{*},s)$ interpolating the special values of the
Rankin-Selberg $L$-functions $L(g,h,s)$ of crystalline classical members
$g\in\mathscr{G}$, $h\in\mathscr{G}^{*}$ of weights $\ell,m\geqslant 2$ at
integers $m\leqslant s<\ell$. Let $L_{p}(g,h,s)$ be the specialization of
$L_{p}(\mathscr{G},\mathscr{G}^{*},s)$ at $(g,h)$. When $g=h^{*}$ is a
crystalline classical point of $\mathscr{G}$ of weight $\ell=m\geqslant 2$,
Dasgupta’s factorization theorem [18] asserts that
$L_{p}(g,g^{*},s)=\zeta_{p}(s-\ell+1)L_{p}(\operatorname{ad}^{0}(g),s)$ where
$\zeta_{p}(s)$ is the $p$-adic zeta function and
$L_{p}(\operatorname{ad}^{0}(g),s)$ is Hida-Schmidt’s $p$-adic $L$-function
[32] for the adjoint motive of $g$. It makes sense to apply the above
construction to
$\mathscr{G}\in\\{\Theta,\bar{\Theta},\mathscr{F},\mathscr{F}\otimes\varepsilon_{K}\\}$,
in which case the weight $1$ specialiazation of the $p$-adic meromorphic
function $L_{p}(\mathscr{G},\mathscr{G}^{*},s)\cdot\zeta_{p}(s-\ell+1)^{-1}$
should be related to Perrin-Riou’s $p$-adic $L$-function attached to
$\operatorname{ad}^{0}(\varrho_{f})$ and $W^{+}_{\mathscr{G}}$. Some evidence
of this fact for $\mathscr{G}=\Theta,\bar{\Theta}$ lies in the analogy with
[30, Theorem 6.2] which treats the case where $f$ is $p$-regular and the
weight map is étale at $f$. Note that the simple (resp. double) trivial zero
in the $p$-regular (resp. $p$-irregular) case is heuristically related to the
smoothness (resp. non-Gorensteinness) of the eigencurve at $f$.

Acknolwedgements: The authors are mostly indebted to D. Benois for numerous
discussions related to this project. We would like also to thank A. Betina and
J. Bellaïche for helpful discussions. The research leading to this article is
jointly funded by the Agence Nationale de Recherche ANR-18-CE40-0029 and the
Fonds National de Recherche Luxembourg INTER/ANR/18/12589973 in the project
Galois representations, automorphic forms and their L-functions (GALF).

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