# Cyclotomic derivatives of Beilinson–Flach classes and a new proof of a
Gross–Stark formula

Óscar Rivero  O. R.: Departament de Matemàtiques, Universitat Politècnica de
Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract.

We give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the
computation of the $\mathcal{L}$-invariant of the adjoint of a weight one
modular form in terms of units and $p$-units. While in our previous work with
Rotger the essential ingredient was the use of Galois deformations techniques
following the computations of Bellaïche and Dimitrov, we propose a new
approach exclusively using the properties of Beilinson–Flach classes. One of
the key ingredients is the computation of a cyclotomic derivative of a
cohomology class in the framework of Perrin-Riou theory, which can be seen as
a counterpart to the earlier work of Loeffler, Venjakob and Zerbes. We hope
that this method could be applied to other scenarios regarding exceptional
zeros, and illustrate how this could lead to a better understanding of this
setting by conjecturally introducing a new $p$-adic $L$-function whose special
values involve information just about the unit of the adjoint (and not also
the $p$-unit), in the spirit of the conjectures of Harris and Venkatesh.

###### 2010 Mathematics Subject Classification:

11F67; 11S40, 19F27

###### Contents

  1. 1 Introduction
  2. 2 Analogy with the case of circular units
  3. 3 Beilinson–Flach elements
    1. 3.1 The three variable cohomology classes
    2. 3.2 Derivatives of Beilinson–Flach elements
  4. 4 Cyclotomic derivatives and proof of the main theorem
    1. 4.1 Cyclotomic derivatives
    2. 4.2 Improved p-adic L-functions
  5. 5 A reinterpretation of the special value formula
    1. 5.1 Deformations of weight one modular forms
    2. 5.2 Spaces of generalized eigenvectors
  6. 6 A conjectural $p$-adic $L$-function

##  1\. Introduction

In our series of works with Rotger [RiRo1], [RiRo2], we proposed a systematic
study of the conjecture of Darmon, Lauder and Rotger [DLR] on $p$-adic
iterated integrals in terms of certain cohomology classes constructed from the
$p$-adic interpolation of Beilinson–Flach elements. This conjecture may be
subsumed in a broader programme comprising both the Gross–Stark conjectures
and also the celebrated Elliptic Stark Conjectures, which shed some light on
the arithmetic of elliptic curves of rank 2.

Let $\chi$ be a Dirichlet character of level $N\geq 1$, and let
$S_{1}(N,\chi)$ stand for the space of cuspidal modular forms of weight 1,
level $N$ and nebentype $\chi$. Let $g=\sum_{n\geq 1}a_{n}q^{n}\in
S_{1}(N,\chi)$ be a normalized newform and let $g^{*}=g\otimes\bar{\chi}$
denote its twist by the inverse of its nebentype. Let

$\varrho_{g}:\mathrm{Gal\,}(H_{g}/\mathbb{Q})\hookrightarrow\mathrm{GL}(V_{g})\simeq\mathrm{GL}_{2}(L),\quad\varrho_{\mathrm{ad}^{0}(g)}:\mathrm{Gal\,}(H/\mathbb{Q})\hookrightarrow\mathrm{GL}(\mathrm{ad}^{0}(g))\simeq\mathrm{GL}_{3}(L)$

denote the Artin representations associated to $g$ and its adjoint,
respectively. Here $H_{g}\supseteq H$ denote the finite Galois extensions of
$\mathbb{Q}$ cut out by these representations, and $L$ is a sufficiently large
finite extension of $\mathbb{Q}$ containing their traces.

Label and order the roots of the $p$-th Hecke polynomial of $g$ as
$X^{2}-a_{p}(g)X+\chi(p)=(X-\alpha)(X-\beta)$. We assume throughout that

  1. (H1)

The reduction of $\varrho_{g}$ mod $p$ is irreducible;

  2. (H2)

$g$ is $p$-distinguished, i.e. $\alpha\neq\beta\,(\mathrm{mod}\,p)$, and

  3. (H3)

$\varrho_{g}$ is not induced from a character of a real quadratic field in
which $p$ splits.

Hence, following [RiRo2], there are four (a priori distinct!) cohomology
classes

(1)
$\kappa(g_{\alpha},g_{1/\alpha}^{*}),\quad\kappa(g_{\beta},g_{1/\beta}^{*}),\quad\kappa(g_{\alpha},g_{1/\beta}^{*}),\quad\kappa(g_{\beta},g_{1/\alpha}^{*})\in
H^{1}(\mathbb{Q},\operatorname{ad}^{0}(g)(1)).$

However, and as it was proved in [RiRo1, Sections 3.2, 3.4], it happens that

(2) $\kappa(g_{\alpha},g_{1/\beta}^{*})=\kappa(g_{\beta},g_{1/\alpha}^{*})=0$

due to an exceptional zero phenomenon which can be explained by the vanishing
of an Euler factor. To overcome this situation, in Section 3.3 of loc. cit. we
had constructed certain derivatives of those classes, but it turns out that
the definition we had used there is not completely useful for our purposes.
Roughly speaking, we had taken the derivative along one of the weight
directions associated to the Hida family interpolating one of the modular
forms, while towards obtaining a more flexible and arithmetically interesting
setting we need to consider also the cyclotomic derivative. This is an
analogue situation to the scenario of [Buy2] and [Ven], where the computation
of the derivative of the Mazur–Kitagawa $p$-adic $L$-function along a certain
direction of the weight space was relatively easy using the classical theory
of Heegner points (and had already been carried out by Bertolini and Darmon
[BD]), but the computation of the cyclotomic derivative required new ideas.
Hence, this work may be thought as a counterpart to the approach of Büyükboduk
and Venerucci to the exceptional zero phenomenon, but in the easier case where
elliptic curves are replaced by unit groups (and hence one can circumvent the
technical complications introduced by the use of Nekovar’s height theory).
Similar results had been obtained by Loeffler, Venjakob and Zerbes [LVZ], and
one can see our computations as the dual of Proposition 2.5.5 and Theorem
3.1.2 of loc. cit. We refer also to the seminal works of Benois [Ben1], [Ben2]
where similar questions are addressed.

Our main result in [RiRo1] was the computation of a special value formula for
the Hida–Rankin $p$-adic $L$-function at weight one (alternatively, the
derivative of the adjoint of the modular form). This is specially intriguing
since that function, that we denote as $L_{p}(g,g^{*},s)$, cannot be directly
defined in terms of an interpolation property, and requires to consider the
$p$-adic variation of the modular forms $(g,g^{*})$ along a Hida family.
Indeed, it depends on the choice of a $p$-stabilization for $g$. We sometimes
write $L_{p}^{g_{\alpha}}(g,g^{*},s)$ to emphasize this dependence. In [DLR,
Section 1] it is shown that

$\dim_{L}({\mathcal{O}}_{H}^{\times}\otimes\mathrm{ad}^{0}(g))^{G_{\mathbb{Q}}}=1,\quad\dim_{L}({\mathcal{O}}_{H}[1/p]^{\times}/p^{\mathbb{Z}}\otimes\mathrm{ad}^{0}(g))^{G_{\mathbb{Q}}}=2.$

Fix a generator $u$ of
$({\mathcal{O}}_{H}^{\times}\otimes\mathrm{ad}^{0}(g))^{G_{\mathbb{Q}}}$ and
also an element $v$ of
$({\mathcal{O}}_{H}^{\times}[1/p]^{\times}\otimes\mathrm{ad}^{0}(g))^{G_{\mathbb{Q}}}$
in such a way that $\\{u,v\\}$ is a basis of
$({\mathcal{O}}_{H}[1/p]^{\times}/p^{\mathbb{Z}}\otimes\mathrm{ad}^{0}(g))^{G_{\mathbb{Q}}}$.
The element $v$ may be chosen to have $p$-adic valuation
${\mathrm{ord}}_{p}(v)=1$, and we do so.

Viewed as a $G_{\mathbb{Q}_{p}}$-module, $\mathrm{ad}^{0}(g)$ decomposes as
$\mathrm{ad}^{0}(g)=L\oplus L^{\alpha\otimes\bar{\beta}}\oplus
L^{\beta\otimes\bar{\alpha}}$, where each line is characterized by the
property that the arithmetic Frobenius $\mathrm{Fr}_{p}$ acts on it with
eigenvalue $1$, $\alpha/\beta$ and $\beta/\alpha$, respectively. Let $H_{p}$
denote the completion of $H$ in $\bar{\mathbb{Q}}_{p}$ and let

$u_{1},\,\,u_{\alpha\otimes\bar{\beta}},\,\,u_{\beta\otimes\bar{\alpha}},\,\,v_{1},\,\,v_{\alpha\otimes\bar{\beta}},\,\,v_{\beta\otimes\bar{\alpha}}\in
H_{p}^{\times}\otimes_{\mathbb{Q}}L\quad(\mathrm{mod}\,L^{\times})$

denote the projection of the elements $u$ and $v$ in
$(H_{p}^{\times}\otimes\mathrm{ad}^{0}(g))^{G_{\mathbb{Q}_{p}}}$ to the above
lines.

Then, we have the following:

######  Theorem 1.1.

Assume that hypothesis (H1)-(H3) hold. Then, the following equality holds up
to multiplication by a scalar in $L^{\times}$

$L_{p}^{g_{\alpha}}(g,g^{*},1)=\frac{\log_{p}(u_{\alpha\otimes\bar{\beta}})\log_{p}(v_{1})-\log_{p}(v_{\alpha\otimes\bar{\beta}})\log_{p}(u_{1})}{\log_{p}(u_{\alpha\otimes\bar{\beta}})}.$

The proof we had given in [RiRo1, Section 4] was lengthy and made use of the
results of Bellaiche–Dimitrov computing the tangent space of a deformation
problem, together with some elements taken from the earlier work [DLR2]. In a
certain way, that proof mimicked the approach of Greenberg–Stevens [GS] to the
exceptional zero phenomenon for elliptic curves with split multiplicative
reduction. However, the authors had observed a tantalizing connection with the
theory of Beilinson–Flach elements, that were affected by a similar
exceptional zero phenomenon. This allowed us to interpret derived classes of
Beilinson–Flach elements in terms of the units $\\{u,v\\}$, but does not give
any new insight into the proof of Theorem 1.1, as we had initially expected.
This work may be seen as a culmination of the purpose that the authors had
when they began to write both [RiRo1] and [RiRo2], that was proving the
Gross–Stark conjecture of Darmon, Lauder and Rotger using just the properties
of Beilinson–Flach elements and the flexibility provided by the notion of
derivatives. This is just another instance of the power of Euler systems when
dealing with arithmetic questions.

We can give, with these ideas at hand, a different proof of the main theorem
of [RiRo1]. This can be seen as the counterpart to the approach of Kobayashi
[Ko] to the Mazur–Tate–Teitelbaum conjecture in rank 0, since he reproves the
result of Greenberg and Stevens [GS] using the properties of Kato’s cohomology
classes.

Our proof is a combination of four main ideas (together with the same starting
point coming from Hida’s theory of improved $p$-adic $L$-functions):

  1. (0)

The results of Hida [Hi1], [Hi2], which reduce the conjecture to the
computation of the derivative of the Frobenius eigenvalue along the weight
direction. This part is common to our earlier work [RiRo1].

  2. (1)

The local properties at $p$ of Beilinson–Flach elements, which give an
expression, up to multiplication by a $p$-adic scalar, for the derived class
$\kappa^{\prime}(g_{\alpha},g_{1/\beta}^{*})$ in terms of the units $u$ and
$v$, where here the derivative is taken along any arbitrary direction of the
weight space.

  3. (2)

A comparison between the different reciprocity laws and the observation that
knowing two weight derivatives, together with the vanishing of the class
$\kappa({\mathbf{g}},{\mathbf{g}}^{*})$ along the line $(\ell,\ell,\ell-1)$,
allows us to determine the cyclotomic derivative of the class.

  4. (3)

An explicit reciprocity law for the $\Lambda$-adic class
$\kappa({\mathbf{g}},{\mathbf{g}}^{*})$, obtained when $g$ and $g^{*}$ vary
over Hida families ${\mathbf{g}}$ and ${\mathbf{g}}^{*}$, respectively. This
was first proved by Kings–Loeffler–Zerbes [KLZ1]. In our situation, there is
an exceptional vanishing, and hence we may consider a derived reciprocity law,
in the sense of [RiRo1]. This gives an expression for the weight derivative of
the Beilinson–Flach class in terms of an unknown $p$-adic period and involving
also the $\mathcal{L}$-invariant of the adjoint of $g_{\alpha}$.

  5. (4)

The results of Büyükboduk [Buy1], [Buy2] and Venerucci [Ven] around Coleman
maps, which allow us to relate the cyclotomic derivative of the
Beilinson–Flach class to the Hida–Rankin $p$-adic $L$-function. This part can
be also recast, by duality, in terms of the computations developed in [LVZ].
Comparing this result with (3), we get a formula for the
$\mathcal{L}$-invariant, and consequently for the special $p$-adic $L$-value.

Observe that the study of universal norms has also allowed Roset, Rotger and
Vatsal [RRV] to reinterpret the $\mathcal{L}$-invariant of Theorem 1.1 in
terms of an algebraic avatar initially defined by Greenberg [Gr].

However, Theorem 1.1 is not completely satisfactory towards the understanding
of the arithmetic of the adjoint of a weight one modular form, since it
involves both the unit and the $p$-unit attached to the Galois representation.
It is natural to expect a putative refinement of the previous result in the
spirit of the conjectures of Harris–Venkatesh [HV], with a $p$-adic
$L$-function whose special values encode information just about the unit $u$.
The last section of this note is devoted to discuss the following conjecture
in the framework provided by Perrin-Riou maps.

######  Conjecture 1.2.

There exists an analytic $p$-adic $L$-function
$L_{p}^{\operatorname{Eis}}(g,g^{*},s)$, appropriately related with the
cohomology class $\kappa(g_{\alpha},g_{1/\alpha}^{*})$ via the theory of
Perrin-Riou maps, and such that

$L_{p}^{\operatorname{Eis}}(g,g^{*},1)=\log_{p}(u_{1})\pmod{L^{\times}}.$

We finish this preliminary discussion by pointing out that the formalism we
discuss regarding exceptional zeros and derivatives at the level of Euler
systems suggests the possibility of developing an axiomatic treatment
mimicking the general conjectures of Greenberg and Benois. We believe that
this could be specially intriguing in the scenario of diagonal cycles or in
its recent generalizations to the Siegel case, where a cohomology class may
simultaneously encode information about the behavior of different, and a
priori unrelated, $p$-adic $L$-functions.

The organization of this note is as follows. Section 2 discusses the
motivational case of circular units, where these same phenomena arise and that
can serve as a motivation for our later work. Section 3 recalls the notations
and results of [RiRo1] around Beilinson–Flach elements which are needed in the
proof. Section 4 contains the main results of the article and discuss the new
proof of Theorem 1.1 using the notion of derived Belinson–Flach elements.
Section 5 proposes an alternative interpretation of the previous results in
terms of deformation theory. Finally, Section 6 discusses the $p$-adic
$L$-function of Conjecture 1.2 and its relationship with the periods of weight
one modular forms.

Acknowledgements. This research began during a research stay of the author at
McGill University in Fall 2019. It is a pleasure to sincerely thank Henri
Darmon, who organised a small seminar around the symmetric square $p$-adic
$L$-function where the author could explain his previous work with Rotger, and
was encouraged to try to find an alternative proof of those results in terms
of derived cohomology classes. I am indebted to Denis Benois, for many
insights and useful conversations around this work. Finally, I thank Victor
Rotger for many stimulating conversations around the topic of this note, which
is the natural continuation of our earlier works on this question. This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 682152). The author has also received financial support through
“la Caixa” Banking Foundation (grant LCF/BQ/ES17/11600010).

##  2\. Analogy with the case of circular units

The situation we want to deal with has a clear parallelism with the case of
circular units, that we now recall. We fix a Dirichlet character $\chi$, and
write $H$ for the field cut out by the Artin representation attached to it,
and $L$ for its coefficient field. The case where $\chi$ is odd gives rise to
an exceptional vanishing of the Deligne–Ribet $p$-adic $L$-function
$L_{p}(\chi\omega,s)$ at $s=0$ when $\chi(p)=1$, where $\omega$ is the
Teichmüller character. Under the assumption that $\chi$ is odd,
$(\mathcal{O}_{H}^{\times}\otimes L)^{\chi}$ is a zero-dimensional $L$-vector
space, while $(\mathcal{O}_{H}[1/p]^{\times}\otimes L)^{\chi}$ has dimension 1
if $\chi(p)=1$. In this case, choosing a non-zero element $v_{\chi}$ of the
latter space, we may define an $\mathcal{L}$-invariant

(3)
$\mathcal{L}(\chi)=-\frac{\log_{p}(v_{\chi})}{{\mathrm{ord}}_{p}(v_{\chi})},$

where we are implicitly choosing a prime of $H$ above $p$. Then,

(4) $L_{p}^{\prime}(\chi\omega,s)=\mathcal{L}(\chi)\cdot L(\chi,0).$

See [DDP] for more details on that and for a broader treatment in the setting
of totally real number fields.

In the case where $\chi$ is even, the situation is ostensibly different. Here,
$(\mathcal{O}_{H}^{\times}\otimes L)^{\chi}$ is one-dimensional and we may fix
a generator $c_{\chi}$ of it, that we call the circular unit associated to
$\chi$. We take it, as usual, as a weighted combination of cyclotomic units

$c_{\chi}=\prod_{a=1}^{N-1}(1-\zeta_{N}^{a})^{\chi^{-1}(a)},$

where $\zeta_{N}$ is a fixed primitive $N$-th root of unity and the notation
$(1-\zeta_{N}^{a})^{\chi^{-1}(a)}$ means
$(1-\zeta_{N}^{a})\otimes\chi^{-1}(a)$. Moreover, if we further assume that
$\chi(p)=1$, $(\mathcal{O}_{H}[1/p]^{\times}\otimes L)^{\chi}$ has dimension
2, and we may consider a basis of the form $\\{c_{\chi},v_{\chi}\\}$, with the
convention that ${\mathrm{ord}}_{p}(v_{\chi})=1$.

Given any even, non trivial Dirichlet character, one always has Leopoldt’s
formula, which asserts that

(5)
$L_{p}(\chi,1)=-\frac{(1-\chi(p)p^{-1})}{\mathfrak{g}(\bar{\chi})}\cdot\log_{p}(c_{\chi}).$

We may understand the previous result in the more general setting of
reciprocity laws. For that purpose, let
$\Lambda=\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$. By adding $p$-power
conductors to $\chi$ and considering a family of coherent units along the
cyclotomic tower, one may construct a $\Lambda$-adic class $\kappa(\chi,s)$,
whose bottom layer vanishes when $\chi(p)=1$. We keep this assumption on the
character $\chi$ all along this section and write

$\kappa(\chi,s)\in
H^{1}(\mathbb{Q},L_{p}(\bar{\chi})\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))$

for the $\Lambda$-adic class, where $\varepsilon_{\operatorname{cyc}}$ is the
usual cyclotomic character and $\underline{\varepsilon}_{\operatorname{cyc}}$
stands for the $\Lambda$-adic cyclotomic character. Given $s\in\mathbb{Z}$,
let
$\nu_{s}:\Lambda(\underline{\varepsilon}_{\operatorname{cyc}})\rightarrow\mathbb{Z}_{p}$
be the ring homomorphism sending the group-like element
$a\in\mathbb{Z}_{p}^{\times}$ to $a^{s}$. This induces a
$G_{\mathbb{Q}}$-equivariant specialization map

$\nu_{s}:\,\Lambda(\underline{\varepsilon}_{\operatorname{cyc}})\rightarrow\mathbb{Z}_{p}(s)$

and gives rise to a collection of global cohomology classes

$\kappa(\chi,s)\in H^{1}(\mathbb{Q},L_{p}(\bar{\chi})(s)).$

The Perrin-Riou formalism allows us to understand $L_{p}(\chi,s)$ as the image
under a Coleman map (also named as Perrin-Riou map, or Perrin-Riou regulator)
of the local class $\kappa_{p}(\chi,s)$

$\mathcal{L}_{\chi}\,:\,H^{1}(\mathbb{Q}_{p},L_{p}(\bar{\chi})\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))\longrightarrow
I^{-1}\Lambda,\quad\mathcal{L}_{\chi}(\kappa_{p}(\chi,s))=L_{p}(\chi,s),$

where $I$ is the ideal of $\Lambda$ corresponding to the specialization at
$s=1$ (see [KLZ1, Section 8.2] for the precise definition). This map
interpolates either the dual exponential map (for $s\leq 0$) or the Bloch–Kato
logarithm (for $s\geq 1$). Unfortunately, the bottom layer $\kappa(\chi,1)$
vanishes when $\chi(p)=1$. Following the construction of [Buy1, Section 3],
there is a derived class $\kappa^{\prime}(\chi,s)$, defined as the unique
class satisfying that

(6)
$\kappa(\chi,s)=\frac{\gamma-1}{\log_{p}(\gamma)}\cdot\kappa^{\prime}(\chi,s),$

where $\gamma$ is a fixed topological generator of
$\mathbb{Z}_{p}[[1+p\mathbb{Z}_{p}]]$. It is also proved in [Buy1] that
$\kappa^{\prime}(\chi,1)$ belongs to an extended Selmer group, which in this
case may be identified with the group of $p$-units where the Galois group acts
via $\chi$ (we insist that when $\chi$ is even this space is two-dimensional);
this is coherent with the fact that taking Iwasawa derivatives along a
$\mathbb{Z}_{p}$-extension can only introduce ramification at $p$. Hence, the
exceptional zero phenomenon does not appear at the level of $p$-adic
$L$-functions, since at least generically $L_{p}(\chi,1)\neq 0$, but at the
level of cohomology classes.

Moreover, in the cases where $\chi(p)=1$ one can also define an improved map

$\widetilde{\mathcal{L}_{\chi}}=\frac{\gamma-1}{\frac{1}{p}\log_{p}(\gamma)}\times\mathcal{L}_{\chi}:\,H^{1}(\mathbb{Q}_{p},L_{p}(\bar{\chi})(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))\longrightarrow
I^{-1}\Lambda,$

where $\varepsilon_{\operatorname{cyc}}$ is the usual cyclotomic character.
Therefore,

$\widetilde{\mathcal{L}_{\chi}}(\kappa_{p}^{\prime}(\chi,s))=p\cdot
L_{p}(\chi,s).$

The computations done in [Buy1, Section 6.2], in particular Remark 6.5, show
that the map $\widetilde{\mathcal{L}_{\chi}}$, when specialized at $s=1$, is
given by the order map (applied in this case to the derived class). The key
point is a computation of the universal norms over the cyclotomic tower, as
well as the use of Lemma 6.4 of loc. cit (see also [Ven, Section 3]). Hence,
we have the following (identifying as usual the cohomology classes with the
corresponding units via the standard Kummer map).

######  Proposition 2.1.

The element $\kappa^{\prime}(\chi,1)\in(\mathcal{O}_{H}[1/p]^{\times}\otimes
L)^{\chi}$ satisfies that

$L_{p}(\chi,1)=-\frac{1-p^{-1}}{\mathfrak{g}(\bar{\chi})}\cdot{\mathrm{ord}}_{p}(\kappa^{\prime}(\chi,1)).$

###### Proof.

This follows after combining the results of [Buy1, Section 6.2] on the
properties of the map $\tilde{\mathcal{L}}_{\chi}$ with Solomon’s
computations, showing that the $p$-adic valuation of the derived class
(sometimes referred as the wild cyclotomic unit) agrees with the logarithm of
the circular unit (see also Proposition 4.1 of loc. cit.). ∎

##  3\. Beilinson–Flach elements

###  3.1. The three variable cohomology classes

Let ${\mathbf{g}}\in\Lambda_{{\mathbf{g}}}[[q]]$ and
${\mathbf{g}}^{*}\in\Lambda_{{\mathbf{g}}}[[q]]$ be two Hida families of tame
conductor $N$ and tame nebentype $\chi$ and $\bar{\chi}$, where
$\Lambda_{{\mathbf{g}}}$ is a finite flat extension of the Iwasawa algebra
$\Lambda=\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$. Let
$\Lambda_{{\mathbf{g}}{\mathbf{g}}^{*}}=\Lambda_{{\mathbf{g}}}\hat{\otimes}\Lambda_{{\mathbf{g}}}\hat{\otimes}\Lambda$,
and consider also the $\Lambda_{{\mathbf{g}}}$-modules afforded by the Hida
families attached to ${\mathbf{g}}$ and ${\mathbf{g}}^{*}$, that we denote
$\mathbb{V}_{{\mathbf{g}}}$ and $\mathbb{V}_{{\mathbf{g}}^{*}}$, respectively.
Finally, consider the $\Lambda_{{\bf gg^{*}}}$-module

(7) $\mathbb{V}_{{\bf gg^{*}}}:=\mathbb{V}_{{\bf
g}}\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{V}_{{\bf
g^{*}}}\hat{\otimes}_{\mathbb{Z}_{p}}\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}^{-1}),$

where $\Lambda(\underline{\varepsilon}_{\operatorname{cyc}}^{-1})$ stands for
the twist of $\Lambda$ by the inverse of the $\Lambda$-adic cyclotomic
character. The formal spectrum of $\Lambda_{{\mathbf{g}}{\mathbf{g}}^{*}}$ is
endowed with certain distinguished points, the so-called crystalline points,
denoted as $\mathcal{W}_{{\bf gh}}^{\circ}$ and indexed by triples
$(y,z,\sigma)$; we refer the reader to Section 2 of loc. cit. for the
definitions.

The $\Lambda$-adic Galois representation
$\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}$ is characterized by the property
that for $(y,z,\sigma)\in\mathcal{W}_{{\bf gg^{*}}}^{\circ}$, with $\sigma$ of
weight $s\in\mathbb{Z}$, (7) specializes to

$\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}(y,z,\sigma)=V_{g_{y}}\otimes
V_{g_{z}^{*}}(1-s),$

the $(1-s)$-th Tate twist of the tensor product of the Galois representations
attached to $g_{y}$ and $g_{z}^{*}$.

Fix $c\in\mathbb{Z}_{>1}$ such that $(c,6pN)=1$. [KLZ1, Theorem A] yields a
three-variable $\Lambda$-adic global Galois cohomology class

$\kappa^{c}({\mathbf{g}},{\mathbf{g}}^{*})\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}})$

that is referred to as the Euler system of Beilinson–Flach elements associated
to ${\mathbf{g}}$ and ${\mathbf{g}}^{*}$. We denote by
$\kappa_{p}^{c}({\mathbf{g}},{\mathbf{g}}^{*})\in
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}})$ the image of
$\kappa^{c}({\mathbf{g}},{\mathbf{g}}^{*})$ under the restriction map. Since
$c$ is fixed throughout, we may sometimes drop it from the notation. This
constant does make an appearance in fudge factors accounting for the
interpolation properties satisfied by the Euler system, but in all cases we
are interested in these fudge factors do not vanish and hence do not pose any
problem for our purposes.

Given a crystalline arithmetic point
$(y,z,s)\in\mathcal{W}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{\circ}$ of weights
$(\ell,m,s)$, set for notational simplicity throughout this section
$g=g_{y}^{\circ}$, $g^{*}=(g_{z}^{*})^{\circ}$. With these notations, $g_{y}$
(resp. $g_{z}^{*}$) is the $p$-stabilization of $g$ (resp. $g^{*}$) with
$U_{p}$-eigenvalue $\alpha_{g}$ (resp. $\alpha_{g^{*}}$).

Define

(8) $\kappa(g_{y},g_{z}^{*},s):=\kappa({\bf g},{\bf g}^{*})(y,z,s)\in
H^{1}(\mathbb{Q},V_{g_{y}}\otimes V_{g_{z}^{*}}(1-s))$

as the specialisation of $\kappa({\bf g},{\bf g}^{*})$ at $(y,z,s)$.

As explained in [DR2, Section 2], the spaces $\mathbb{V}_{{\mathbf{g}}}$ and
$\mathbb{V}_{{\mathbf{g}}^{*}}$, as $G_{\mathbb{Q}_{p}}$-modules, are endowed
with a stable filtration

$0\longrightarrow\mathbb{V}_{{\mathbf{g}}}^{+}\longrightarrow\mathbb{V}_{{\mathbf{g}}}\longrightarrow\mathbb{V}_{{\mathbf{g}}}^{-}\longrightarrow
0,$

where $\mathbb{V}_{{\mathbf{g}}}^{+}$ and $\mathbb{V}_{{\mathbf{g}}}^{-}$ are
flat $\Lambda_{{\mathbf{g}}}$-modules with a $G_{\mathbb{Q}_{p}}$-action,
locally free of rank one over $\Lambda_{{\mathbf{g}}}$, and such that the
quotient $\mathbb{V}_{{\mathbf{g}}}^{-}$ is unramified. Define the
$G_{\mathbb{Q}_{p}}$-subquotient
$\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{-+}:=\mathbb{V}_{{\mathbf{g}}}^{-}\hat{\otimes}\mathbb{V}_{{\mathbf{g}}^{*}}^{+}$
of $\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}$, which is of rank one over the
two-variable Iwasawa algebra
$\Lambda_{{\mathbf{g}}}\hat{\otimes}\Lambda_{{\mathbf{g}}^{*}}$ (this quotient
makes sense because of [KLZ1, Proposition 8.1.7]).

Then, one may consider a Perrin Riou map

(9)
$\langle\mathcal{L}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{-+},\eta_{{\mathbf{g}}}\otimes\omega_{{\mathbf{g}}^{*}}\rangle:H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{-+}\hat{\otimes}\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}^{-1}))\longrightarrow\Lambda_{{\mathbf{g}}{\mathbf{g}}^{*}}\otimes\mathbb{Q}_{p}(\mu_{N}).$

This application satisfies an explicit reciprocity law, which is the content
of [KLZ1, Theorem B], and which asserts that

(10)
$\langle\mathcal{L}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{-+}(\kappa_{p}^{-+}({\mathbf{g}},{\mathbf{g}}^{*})),\eta_{{\mathbf{g}}}\otimes\omega_{{\mathbf{g}}^{*}}\rangle=\mathcal{A}({\mathbf{g}},{\mathbf{g}}^{*})\cdot
L_{p}({\mathbf{g}},{\mathbf{g}}^{*}),$

where $\mathcal{A}({\mathbf{g}},{\mathbf{g}}^{*})$ is the Iwasawa function of
[KLZ1, Theorem 10.2.2] and $\kappa_{p}^{-+}({\mathbf{g}},{\mathbf{g}}^{*})$
stands for the composition of the localization-at-$p$ map with the projection
$\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}\rightarrow\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{-+}$
in local cohomology.

The different specializations of the map
$\langle\mathcal{L}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{-+},\eta_{{\mathbf{g}}}\otimes\omega_{{\mathbf{g}}^{*}}\rangle$
can be expressed in terms of the Bloch–Kato logarithm or the dual exponential
map. In particular, we are interested in the specializations of the class
$\kappa({\mathbf{g}},{\mathbf{g}}^{*})$ at weights $(1,1,0)$, and more
generally, weights $(\ell,\ell,\ell-1)$, where the Perrin-Riou map
interpolates, up to some explicit Euler factors, the Bloch–Kato logarithm.
Unfortunately, these factors may vanish in the self-dual case, and one must
recast to the concept of derivatives of Euler systems.

###  3.2. Derivatives of Beilinson–Flach elements

We keep the notations fixed in the introduction regarding weight one modular
forms and units for the adjoint representation. Further, we fix a point of
weight one $y_{0}\in\operatorname{Spf}(\Lambda_{{\mathbf{g}}})$ such that
${\mathbf{g}}_{y_{0}}=g_{\alpha}$ and
${\mathbf{g}}^{*}_{y_{0}}=g_{1/\beta}^{*}$.

In [RiRo1, §3], we proved that the class
$\kappa({\mathbf{g}},{\mathbf{g}}^{*})$ was zero over the line corresponding
to the Zariski closure of points of weights $(\ell,\ell,\ell-1)$ due to the
vanishing of an Euler factor in the interpolation property for the Eisenstein
class. In loc. cit. we also constructed a derivative along the $y$-direction
(alternatively, keeping $y$ fixed and varying at a time the other two
variables). However, since the weight space is three-dimensional, it makes
sense to ask about the derivative along any other direction.

Since along the line $(\ell,\ell,\ell-1)$ the class is identically zero, the
derivative also vanishes. Hence, by an elementary argument in linear algebra,
it suffices to determine the derivative along any other two independent
directions to capture all the first-order information.

######  Remark 3.1.

Observe that we are using the results of [LZ] which assert that the
Beilinson–Flach elements lie in the part corresponding to the adjoint in the
decomposition $V_{gg^{*}}=\operatorname{ad}^{0}(V_{g})\oplus 1$.
Alternatively, one can consider the projection to the alternate part for
weight greater than one and then apply a limit argument. Moreover, and
following the discussion of [RiRo2, Section 4], one has that the projection of
the subspace $p^{\mathbb{Z}}$ to the adjoint component is trivial.

For the sake of simplicity, and since this suffices for our purposes, we
restrict to the local classes at $p$. We closely follow the ideas of [RiRo1,
Section 3], showing that the specialization at weight 1 of the $\Lambda$-adic
class is a linear combination of the unit $u$ and the $p$-unit $v$, as
described in the introduction.

######  Lemma 3.2.

The derivative of $\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})$ at
$(y_{0},y_{0},0)$ along the $y$-direction (keeping fixed both $z$ and $s$)
satisfies the following equality in $H^{1}(\mathbb{Q}_{p},V_{gg^{*}}(1))$

(11) $\frac{\partial\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})}{\partial
y}\Big{|}_{(y_{0},y_{0},0)}=\Omega\cdot(\log_{p}(v_{\alpha\otimes\bar{\beta}})u-\log_{p}(u_{\alpha\otimes\bar{\beta}})v),$

where $\Omega\in H_{p}$ and we have made use of the usual notations for
writing directional derivatives.

###### Proof.

According to the properties of the cohomology classes established in [RiRo1,
Section 3], the left hand side may be written as a combination of the units
$u$ and $v$. Then, the result follows by applying [KLZ1, Proposition 8.1.7] to
$\kappa({{\mathbf{g}}},{{\mathbf{g}}}^{*})$ in order to conclude that its
projection to $\mathbb{V}_{{\mathbf{g}}{\mathbf{g}}^{*}}^{--}$ is identically
zero, and therefore the same is true for its derivative. Specializing at
$(y_{0},y_{0},0)$, the result automatically follows. ∎

######  Lemma 3.3.

The derivative of $\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})$ at
$(y_{0},y_{0},0)$ along the $z$-direction (keeping fixed both $y$ and $s$)
satisfies the following equality in $H^{1}(\mathbb{Q}_{p},V_{gg^{*}}(1))$ up
to a factor in $L^{\times}$

(12) $\frac{\partial\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})}{\partial
z}\Big{|}_{(y_{0},y_{0},0)}=\Omega\cdot(\log_{p}(v_{\alpha\otimes\bar{\beta}})u-\log_{p}(u_{\alpha\otimes\bar{\beta}})v).$

###### Proof.

This follows by applying the same result to the $p$-adic $L$-value associated
to $g_{1/\beta}^{*}$, which agrees with the former since Dasgupta’s
factorization allows us to say that
$L_{p}^{g_{\alpha}}(g_{\alpha},g_{1/\beta}^{*},1)=L_{p}^{g_{1/\beta}^{*}}(g_{1/\beta}^{*},g_{\alpha},1)$
(alternatively, this follows after a direct computation with Hida’s
factorization, see Proposition 4.4). Here, the latter corresponds to the
function obtained by interpolating along the region where the Hida family
attached to $g_{1/\beta}^{*}$ is dominant. Further, the product of the periods
arising when pairing with the differentials is a rational quantity, as
discussed in [RiRo1, Section 5.2], so it does not affect the result. ∎

Therefore, we may determine the derivative along the direction cyclotomic
direction (keeping the weights fixed) by a linear algebra argument.

######  Proposition 3.4.

Assume that the derivative of $\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})$
along the cyclotomic derivative is non-vanishing. Then, up to multiplication
by scalar, the cyclotomic derivative of the Beilinson–Flach class
$\kappa({\mathbf{g}},{\mathbf{g}}^{*})$ at $(y_{0},y_{0},0)$ is

$\frac{\partial\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})}{\partial
s}\Big{|}_{(y_{0},y_{0},0)}=\Omega\cdot(\log_{p}(v_{\alpha\otimes\bar{\beta}})u-\log_{p}(u_{\alpha\otimes\bar{\beta}})v)\pmod{L^{\times}},$

where $\Omega\in H_{p}$ is the period of equation (11) and the equality holds
in $H^{1}(\mathbb{Q}_{p},V_{gg^{*}}(1))$.

###### Proof.

Recall that the class vanishes along the line $(\ell,\ell,\ell-1)$. Hence, the
result follows from equations (11) and (12) combined with the fact that

$(0,0,1)=(1,1,1)-(1,0,0)-(0,1,0).$

∎

We will see in the next section that the vanishing of the cyclotomic
derivative is equivalent to the vanishing of the special value
$L_{p}^{g_{\alpha}}(g,g^{*},1)$ and also of the regulator corresponding to the
$\mathcal{L}$-invariant. Observe that the previous results show that the
different derived classes, which are elements living in a two-dimensional
space, span the same line!

We can now mimic the approach of [RiRo1] when proving the derived reciprocity
law and obtain an expression for $L_{p}^{g_{\alpha}}(g,g^{*},1)$ involving the
period $\Omega$. In particular, considering the derivative along the analytic
direction $(1,0,0)$ we have the following.

######  Proposition 3.5.

Up to multiplication by an element in $L^{\times}$, the following equality
holds

$L_{p}^{g_{\alpha}}(g,g^{*},1)\cdot\Big{(}\frac{-\alpha_{{\mathbf{g}}}^{\prime}}{\alpha_{g}}\Big{)}_{|y_{0}}=\Omega\cdot(\log_{p}(u_{\alpha\otimes\bar{\beta}})\cdot\log_{p}(v_{1})-\log_{p}(v_{\alpha\otimes\bar{\beta}})\cdot\log_{p}(u_{1})).$

As usual $\alpha_{{\mathbf{g}}}$ stands for the derivative of the
$U_{p}$-eigenvalue along the weight direction.

###### Proof.

This follows from making explicit the Euler factors in the explicit
reciprocity law of [KLZ1, §10] and taking derivatives along the $y$-direction.
∎

In the next section, our aim is determining the value of the period $\Omega$
appearing in Proposition 3.4, which would complete the proof of our main
theorem.

##  4\. Cyclotomic derivatives and proof of the main theorem

###  4.1. Cyclotomic derivatives

Along this section, we assume that $g$ and $g^{*}$ do not move along Hida
families and we just consider the cyclotomic variation. As an abuse of
notation, write
$\kappa(g,g^{*},s):=\kappa({\mathbf{g}},{\mathbf{g}}^{*})(y_{0},y_{0},s)$ to
emphasize the dependence on $s$. The image of this class under the Perrin-Riou
map recovers the $p$-adic $L$-function $L_{p}^{g_{\alpha}}(g,g^{*},s)$, that
is,

(13)
$\langle\mathcal{L}_{gg^{*}}^{-+}(\kappa_{p}^{-+}(g,g^{*},s)),\eta_{g}\otimes\omega_{g^{*}}\rangle=L_{p}^{g_{\alpha}}(g,g^{*},s).$

Although we have shown that $\kappa_{p}(g,g^{*},0)$ is zero, we do not expect
that $L_{p}^{g_{\alpha}}(g,g^{*},0)=0$ in general. This is the same situation
we previously found in the setting of circular units: the Kubota–Leopoldt
$p$-adic $L$-function of a non-trivial, even Dirichlet character
$L_{p}(\chi,s)$ is seen as the image of a $\Lambda$-adic cohomology class
$\kappa(\chi,s)$ under a Perrin-Riou map; unfortunately, it happens that
$\kappa(\chi,1)=0$ when $\chi(p)=1$ and an Euler factor also vanishes, so we
cannot assert (and indeed it is false!) that $L_{p}(\chi,1)=0$.

Along this section, and since there is no possible confusion, we write
$\kappa^{\prime}(g,g^{*},s)$ for the cyclotomic derivative. Define the
improved Perrin-Riou map as

(14)
$\langle\widetilde{\mathcal{L}_{gg^{*}}^{-+}},\eta_{g}\otimes\omega_{g^{*}}\rangle=\frac{\gamma-1}{\frac{1}{p}\log_{p}(\gamma)}\times\langle\mathcal{L}_{gg^{*}}^{-+},\eta_{g}\otimes\omega_{g^{*}}\rangle\,:\,H^{1}(\mathbb{Q}_{p},V_{gg^{*}}^{-+}(1-s))\longrightarrow\Lambda.$

Therefore, we have

(15)
$\langle\widetilde{\mathcal{L}_{gg^{*}}^{-+}}(\kappa_{p}^{\prime-+}(g,g^{*},s)),\eta_{g}\otimes\omega_{g^{*}}\rangle=p\cdot
L_{p}^{g_{\alpha}}(g,g^{*},s).$

Hence, the value of
$\langle\mathcal{L}_{gg^{*}}^{-+}(\kappa_{p}^{-+}(g,g^{*},s)),\eta_{g}\otimes\omega_{g^{*}}\rangle$
agrees with

(16)
$\langle\widetilde{\mathcal{L}_{gg^{*}}^{-+}}(\kappa_{p}^{\prime-+}(g,g^{*},s)),\eta_{g}\otimes\omega_{g^{*}}\rangle.$

For the following result, consider as usual the identification

(17) $H^{1}(\mathbb{Q}_{p},\operatorname{ad}^{0}(V_{g})(1))\simeq
H_{p}^{\times}[\operatorname{ad}^{0}(g)]\otimes L_{p},$

and take the element $\kappa_{p}^{\prime}(g,g^{*},0)$, which belongs to the
latter space (and may be therefore identified with a local unit in
$H_{p}^{\times}$). The same study of [Buy1, Remark 6.5] works verbatim in our
setting, where he argues that the improved Perrin-Riou map is a multiple of
the order map applied to the derived class. However, we want to find out this
explicit multiple (at least, up to multiplication by a rational constant).
Compare for example this setting with the computations of [LVZ, Proposition
2.5.5] and the discussions of Section 3 of loc. cit., showing that the
improved exponential map they consider is indeed the order map (up to sign).

######  Proposition 4.1.

Identifying $\kappa_{p}^{\prime-+}(g,g^{*},0)$ with an element in
$(H_{p}^{\times}\otimes L)^{G_{\mathbb{Q}_{p}}}$, one has

$L_{p}^{g_{\alpha}}(g,g^{*},0)={\mathrm{ord}}_{p}(\kappa_{p}^{\prime-+}(g,g^{*},0))\pmod{L^{\times}}.$

###### Proof.

We can rephrase the statement in terms of the well-known theory of Coleman’s
power series. Then, $\kappa_{p}^{-+}(g,g^{*},s)$ may be seen as a compatible
system of units varying over the cyclotomic $p$-tower, but whose bottom layer
is trivial. Hence, we may use the properties of universal norms and Coleman
maps, and invoke the results developed in the proof of [Ven, Proposition 3.6],
and more precisely (via duality) equation (27).

Then, the $p$-adic $L$-value can be obtained applying to
$\kappa_{p}^{\prime}(g,g^{*},0)$ the map

${\mathrm{ord}}^{-+}\,:\,H^{1}(\mathbb{Q},V_{gg^{*}}\otimes
L_{p}(1))\xrightarrow{\operatorname{pr}^{-+}}H^{1}(\mathbb{Q}_{p},L_{p}(1))\xrightarrow{\widetilde{\mathcal{L}_{gg^{*},0}^{-+}}}\mathbb{Q}_{p},$

where arguing as in [RiRo1, Section 5.2], the map
$\widetilde{\mathcal{L}_{gg^{*},0}^{-+}}$ corresponds to the usual $p$-adic
order map multiplied by the $p$-adic period
$\Omega_{g_{\alpha}}\Xi_{g_{1/\beta}^{*}}$, which belongs to $L^{\times}$.

According to (16), we conclude that

$L_{p}(g,g^{*},0)={\mathrm{ord}}_{p}(\kappa_{p}^{\prime-+}(g,g^{*},0))\pmod{L^{\times}},$

as desired. ∎

Roughly speaking, the previous theorem says that the derivative of the
logarithm is the order (which can be seen as the dual of the result which
interprets the derivative of the dual exponential as a logarithm).

######  Corollary 4.2.

With the notations introduced along the previous section, and up to
multiplication by $L^{\times}$,

$L_{p}^{g_{\alpha}}(g,g^{*},1)=\Omega\cdot\log_{p}(u_{\alpha\otimes\bar{\beta}}).$

###### Proof.

This directly follows by combining Propositions 4.1 and 3.4. ∎

This can be connected again with the case of circular units, that is,
$L_{p}(g,g^{*},s)$ is also the order of the derivative of $\kappa(g,g^{*},s)$
along the $s$-direction.

As usual, let

(18)
$\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha})):=\frac{-\alpha^{\prime}_{{\mathbf{g}}}(y_{0})}{\alpha_{{\mathbf{g}}}(y_{0})},$

where recall
$\alpha_{{\mathbf{g}}}=a_{p}({\mathbf{g}})\in\Lambda_{{\mathbf{g}}}$ is the
Iwasawa function given by the eigenvalue of the Hecke operator $U_{p}$ acting
on ${\mathbf{g}}$, and $\alpha^{\prime}_{{\mathbf{g}}}$ is its derivative.

######  Proposition 4.3.

Assume that the $\mathcal{L}$-invariant
$\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))$ is non-zero. Then, it may be
written as

$\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))=\frac{\log_{p}(u_{\alpha\otimes\bar{\beta}})\cdot\log_{p}(v_{1})-\log_{p}(v_{\alpha\otimes\bar{\beta}})\cdot\log_{p}(u_{1})}{\log_{p}(u_{\alpha\otimes\bar{\beta}})}\pmod{L^{\times}}.$

###### Proof.

Combining Proposition 3.5 with Proposition 4.1, we have that

$\frac{\Omega}{\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))}\cdot\Big{(}\frac{\log_{p}(u_{\alpha\otimes\bar{\beta}})\cdot\log_{p}(v_{1})-\log_{p}(v_{\alpha\otimes\bar{\beta}})}{\log_{p}(u_{\alpha\otimes\bar{\beta}})}\Big{)}=\Omega\pmod{L^{\times}}.$

Dividing by $\Omega$ (provided that this value is non zero), the result
follows. ∎

###  4.2. Improved p-adic L-functions

We finish the proof with the same argument invoked in [RiRo1], involving
Hida’s improved $p$-adic $L$-function. Then, the main theorem is automatically
proved by virtue of the following result, which is Proposition 2.5 of loc.
cit.

######  Proposition 4.4.

For a crystalline classical point $y_{0}\in\mathcal{W}_{{\mathbf{g}}}^{\circ}$
of weight $\ell\geq 1$, we have

$\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))=L_{p}({\bf g},{\bf
g^{*}})(y_{0},y_{0},\ell)=L_{p}^{\prime}(\operatorname{ad}^{0}(g_{y_{0}}),\ell),$

up to a non-zero rational constant. Then, Theorem 1.1 in the introduction
holds.

##  5\. A reinterpretation of the special value formula

The result we have proven along the last two sections was presented in the
introduction as a special value formula for the Hida–Rankin $p$-adic
$L$-function. Alternatively, it can be regarded as the computation of the
$\mathcal{L}$-invariant for the adjoint of a weight one modular form, and
following the original formulation given by Darmon, Lauder, and Rotger, it
also admits a reinterpretation in terms of $p$-adic iterated integral (and
this point of view is specially useful towards computational experiments). In
order to give a more conceptual view of our results, and how they fit in the
theory of exceptional zeros and Galois deformations of modular forms, we would
like two discuss two other interpretations which were already behind the
scenes in our joint works with Rotger.

###  5.1. Deformations of weight one modular forms

As usual, fix a $p$-stabilization $g_{\alpha}$ of the weight one modular form
$g\in S_{1}(N,\chi)$. We discuss a reinterpretation of the main results in
terms of deformations of modular forms, in a striking analogy with the
different works around the Gross–Stark conjecture, and which may be useful
towards generalizations of the main results to totally real fields, following
the recent approach of Dasgupta, Kakde, and Ventullo.

Let $E_{k}$ denote the weight $k$ Eisenstein series, whose Fourier expansion
is given by

(19)
$E_{k}=\frac{\zeta(1-k)}{2}+\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^{n},\quad\text{
where }\sigma_{k-1}(n)=\sum_{d\,|\,n}d^{k-1}.$

There are two possible ways of considering its $p$-adic variation in families,
either by taking the ordinary $p$-stabilization, $E_{k}^{{\mathrm{ord}}}$, or
the critical one, $E_{k}^{\operatorname{crit}}$. For the sake of simplicity,
we restrict to the ordinary $p$-stabilization, and after further normalizing
by $\zeta(1-k)/2$, we have the usual Eisenstein series $G_{k}^{(p)}$, given by

$G_{k}^{(p)}=1+2\zeta_{p}(1-k)^{-1}\sum_{n=1}^{\infty}\sigma_{k-1}^{(p)}(n)q^{n}.$

Since $\zeta_{p}(1-k)$ has a pole at $k=0$, it turns out that we can formally
write $G_{0}^{(p)}=1$, and it makes sense to consider its derivative

$(G_{0}^{(p)})^{\prime}:=2(1-p^{-1})^{-1}\cdot\sum_{n=1}^{\infty}\sigma_{-1}^{(p)}q^{n}.$

Further, it is possible to take the infinitesimal deformation
$G_{0}^{(p)}+\varepsilon(G_{0}^{(p)})^{\prime}$, and then multiplying by
$g_{\alpha}$ we obtain

(20) $(G_{0}^{(p)}+\varepsilon(G_{0}^{(p)})^{\prime})\cdot
g_{\alpha}=g_{\alpha}+\varepsilon(G_{0}^{(p)})^{\prime}g_{\alpha}.$

We regard this expression as a modular form of weight $1+\varepsilon$
corresponding to an infinitesimal deformation of $g_{\alpha}$.

There is another natural deformation of $g_{\alpha}$ we want to consider,
which is precisely the one behind the scenes in [RiRo1] and which also
appeared in [DLR2]. This is defined as

$g_{\alpha}^{\prime}:=\Big{(}\frac{d}{dy}{\mathbf{g}}_{\alpha}\Big{)}_{|y=y_{0}}.$

Then, we may take a second deformation of the modular form $g_{\alpha}$, given
by

(21) $g_{\alpha}+\varepsilon g_{\alpha}^{\prime}.$

Let $e_{{\mathrm{ord}}}$ stand for the ordinary projector, and
$e_{g_{\alpha}}$ for the projector onto the $g_{\alpha}$-isotypic component.

######  Proposition 5.1.

Under the running assumptions,

$e_{g_{\alpha}}e_{{\mathrm{ord}}}(g_{\alpha}E_{0}^{[p]})=(1-\alpha_{g}U_{p}^{-1})g_{\alpha}^{\prime}\pmod{L^{\times}}.$

###### Proof.

Substracting the deformations in (20) and (21), we obtain
$g_{\alpha}(G_{0}^{(p)})^{\prime}-g_{\alpha}^{\prime}$. If we furthermore take
the ordinary projection and project to the $g_{\alpha}$-component, we obtain a
multiple of $g_{\alpha}$, that is

(22)
$e_{g_{\alpha}}e_{{\mathrm{ord}}}(g_{\alpha}(G_{0}^{(p)})^{\prime})=g_{\alpha}^{\prime}+\mathcal{L}\cdot
g_{\alpha}.$

Next, if we apply the operator $1-\alpha_{g}U_{p}^{-1}$ to both sides of the
previous equation, the left hand side becomes just the $p$-depletion

(23) $e_{g_{\alpha}}e_{{\mathrm{ord}}}(g_{\alpha}(G_{0}^{[p]})^{\prime}),$

while in the right hand side the operator $1-\alpha_{g}U_{p}^{-1}$ annihilates
$g_{\alpha}$. We have thus proved the result. ∎

Note that the left hand side is an explicit multiple of the $p$-adic
$L$-function $L_{p}^{g_{\alpha}}(g,g^{*},1)$, so the proposition asserts that
the $\mathcal{L}$-invariant which governs the arithmetic of the adjoint may be
read as a generalized eigenvalue attached to the deformation
$g_{\alpha}^{\prime}$, that is,

(24)
$(1-\alpha_{g}U_{p}^{-1})g_{\alpha}^{\prime}=\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))\cdot
g_{\alpha}\pmod{L^{\times}}.$

###  5.2. Spaces of generalized eigenvectors

This section is merely speculative and tries to shed some light into very
natural questions around the previously discussed phenomena. When discussing
circular units, we have seen that the condition $\chi(p)=1$ provides us with a
$p$-unit in an extended Selmer group, and we have discussed how to mimic this
approach in the case of Beilinson–Flach elements. But more generally, given
two modular forms $g$ and $h$ of weights $\ell$ and $m$, respectively, and an
integer $s$, there is a geometric construction of the so-called Eisenstein
classes $\operatorname{Eis}^{[g,h,s]}$ whenever the triple $(\ell,m,s)$
satisfies the weight condition of [KLZ1, Section 7], that is,

$1\leq s<\min\\{\ell,m\\}.$

This includes all the points of weights $(\ell,\ell,\ell-1)$ when $\ell\geq
2$, and in particular the classical weight two situation. This suggests an
interpretation of the derived class as a limit of Eisenstein classes (which
are generically non-vanishing) for weights $\ell\geq 2$.

The proof of the explicit reciprocity law for Beilinson–Flach classes rests on
an explicit connection between the $p$-depleted Eisenstein series (which
encodes values of the $p$-adic $L$-function) and the $p$-stabilized one (which
encodes values of the regulator of a geometric cycle). In this note we
recovered the expressions for the logarithm of the derived class in terms of
$p$-adic $L$-values, but it is natural to look for a reciprocity law involving
$\operatorname{Eis}^{[g,h,s]}$, whenever $h=g^{*}$ and $s=\ell-1$. Note that
this was the only case excluded by [KLZ2, Theorem 6.5.9]. Let us discuss the
limitations for a result like that and that one may find natural in this
framework.

According to [DR1, Lemma 4.10] (see also [KLZ2, Lemma 6.5.8]), one has

(25) $gE_{0}^{[p]}=(1-\alpha_{g}\cdot
U_{p}^{-1})\cdot(g_{\alpha}E_{0}^{(p)}),$

where $E_{0}^{[p]}$ (resp. $E_{0}^{(p)}$) stands for the $p$-depletion (resp.
$p$-stabilization) of the weight 0 Eisenstein series $E_{0}$. In the non self-
dual case, the corresponding operator acting on the space of generalized
eigenforms $S_{1}(Np)[[g_{\alpha}]]$ is invertible, and we obtain a
straightforward linear relation. But when $h=g^{*}$, the connection is more
involved. In this case, consider a generalized eigenbasis
$\\{e_{1},\ldots,e_{n}\\}$ for the $U_{p}$-operator acting on the space of
generalized (non-necessarily overconvergent) modular forms
$S_{1}(Np)[[g_{\alpha}]]$, that is,

$U_{p}\cdot e_{1}=\alpha_{g}\cdot e_{1},\quad U_{p}\cdot
e_{2}=e_{1}+\alpha_{g}\cdot e_{2},\quad\ldots,\quad U_{p}\cdot
e_{n}=e_{n-1}+\alpha_{g}\cdot e_{n}.$

Hence, the matrix corresponding to the operator $1-\alpha_{g}\cdot U_{p}^{-1}$
acting on this space has the quantity $-1/\alpha_{g}$ all over the upper
diagonal and zeros elsewhere. If we now apply this operator to
$E_{0}^{(p)}g_{\alpha}$, written in this basis as $\sum\lambda_{i}e_{i}$, what
we get in the first non-zero component is $-1/\alpha_{g}\cdot\lambda_{2}$.
That is, the second vector of the generalized eigenbasis is the one which
encodes the $p$-adic $L$-value. Therefore, one may consider two different
classes.

  1. (a)

The class $\operatorname{Eis}^{[g,g^{*},\ell-1]}$, where $\ell$ is the weight
of $\ell$, is related with the first coefficient in the expansion in the
generalized eigenbasis (see [KLZ1, Corollary 6.5.7]). This controls the
$p$-stabilization of the Eisenstein series.

  2. (b)

The derived class $\kappa^{\prime}(g,g^{*})$, constructed in [RiRo1], is
related with the $p$-adic $L$-value, and hence with second coefficient in the
generalized eigenbasis. This measures the $p$-depletion of the Eisenstein
series.

Hence, when $g\in S_{\ell}(N,\chi_{g})$ is an ordinary modular form of weight
$\ell\geq 2$, the two classes

$\\{\operatorname{Eis}^{[g,g^{*},\ell-1]},\kappa^{\prime}(g,g^{*})\\}$

are a priori unrelated.

######  Question 5.2.

Can we interpret the class $\operatorname{Eis}^{[g,g^{*},\ell-1]}$ in terms of
some $p$-adic $L$-value? And can we make sense of the limit of these classes
for weights $(\ell,\ell,\ell-1)$ when $\ell$ goes to 1 $p$-adically?

Note that while the $p$-depleted class is connected with the usual $p$-adic
$L$-value, a priori there is no natural $p$-adic avatar encoding the value of
the $p$-stabilized class.

Observe that in the setting of diagonal cycles of [BSV1], the authors take a
different approach to the vanishing phenomenon, defining an improved class
which is a putative geometric refinement to the analogue of the Eisenstein
class, and which agrees up to some $\mathcal{L}$-invariant with the derived
class.

##  6\. A conjectural $p$-adic $L$-function

It is a somewhat vexing fact that our computations regarding the
$\mathcal{L}$-invariant of the adjoint of a weight one modular form captures a
$2\times 2$ regulator encoding information about both a unit and a $p$-unit,
while the most natural object to work would be the unit itself, as it occurs
with the celebrated Gross–Stark conjecture. Similarly, one would expect to be
able to construct an Eisenstein $p$-adic $L$-function, in such a way that
appropriate special values of it also capture information about the
Beilinson–Flach classes, in a way that we now make precise. This section is
purely conjectural, and must be regarded as a failure in our current work,
where we have not succeeded in studying these aspects.

Consider the most general setting in which $g$ and $h$ are two weight one
modular forms. As we have already recalled, there are four Beilinson–Flach
classes attached to the choice of $p$-stabilizations of $g$ and $h$,

(26)
$\kappa(g_{\alpha},h_{\alpha}),\quad\kappa(g_{\alpha},h_{\beta}),\quad\kappa(g_{\beta},h_{\alpha}),\quad\kappa(g_{\beta},h_{\beta}).$

We know that different components of it are related to special values of
$p$-adic $L$-functions. Take for instance the case of
$\kappa(g_{\alpha},h_{\alpha})$. Considering its restriction to a
decomposition group at $p$, we may take a decomposition of
$\kappa_{p}(g_{\alpha},h_{\alpha})$ of the form

(27) $\kappa_{p}^{--}(g_{\alpha},h_{\alpha})\otimes
e_{\beta\beta}^{\vee}\oplus\kappa_{p}^{-+}(g_{\alpha},h_{\alpha})\otimes
e_{\beta\alpha}^{\vee}\oplus\kappa_{p}^{+-}(g_{\alpha},h_{\alpha})\otimes
e_{\alpha\beta}^{\vee}\oplus\kappa_{p}^{++}(g_{\alpha},h_{\alpha})\otimes
e_{\alpha\alpha}^{\vee},$

where
$\\{e_{\alpha\alpha}^{\vee},e_{\alpha\beta}^{\vee},e_{\beta\alpha}^{\vee},e_{\beta\beta}^{\vee}\\}$
is a basis for $V_{gh}^{\vee}=\operatorname{Hom}(V_{gh},L)$, where

$\mathrm{Fr}_{p}(e_{\alpha\alpha}^{\vee})=\chi_{gh}^{-1}(p)\beta_{g}\beta_{h}\cdot
e_{\alpha\alpha}^{\vee}\,,\,...\,,\,\mathrm{Fr}_{p}(e_{\beta\beta}^{\vee})=\chi_{gh}^{-1}(p)\alpha_{g}\alpha_{h}\cdot
e_{\beta\beta}^{\vee}.$

According to [KLZ1, Proposition 8.2.6], the component
$\kappa_{p}^{--}(g_{\alpha},h_{\alpha})=0$ vanishes. In the same way, the
components $\kappa_{p}^{-+}(g_{\alpha},h_{\alpha})$ and
$\kappa_{p}^{+-}(g_{\alpha},h_{\alpha})$ are related to the special values of
the Hida–Rankin $p$-adic $L$-functions ${\mathscr{L}_{p}}^{g_{\alpha}}$ and
${\mathscr{L}_{p}}^{h_{\alpha}}$, respectively. Hence, it is natural to expect
that the remaining component $\kappa_{p}^{++}(g_{\alpha},h_{\alpha})$ could
arise as the special value of some $p$-adic $L$-function.

Following the analogy with the case of diagonal cycles and triple product
$p$-adic $L$-functions, it would be attached to the triple
$(E_{2}(1,\chi_{gh}^{-1}),g,h)$, but varying over the region where the
Eisenstein family is dominant. Of course this is not possible, but let us work
formally recasting to the theory of Perrin-Riou maps. In particular, we may
consider the three-variable cohomology class
$\kappa({\mathbf{g}},{\mathbf{h}})$, take the restriction to the line where
both $g$ and $h$ are fixed and take the image under the Perrin-Riou map. That
way we would get an element over the Iwasawa algebra that we may denote
$L_{p}^{\operatorname{Eis}}(g,h,s)$. It may be instructive to compare this
with the scenario of triple products, where the existence of a third $p$-adic
$L$-function ${\mathscr{L}_{p}}^{f}$, which at points of weight $(2,1,1)$
interpolates classical $L$-values, provides a richer framework and draws a
more complete picture.

To simplify things, let us focus just on the case where both $g$ and $h$ are
self dual, that is $h=g^{*}$. Recall that this situation naturally splits in
two scenarios, namely $h_{\alpha}=g_{1/\beta}^{*}$ and
$h_{\alpha}=g_{1/\alpha}^{*}$. As we have mentioned before, we expect the
previous cyclotomic $p$-adic $L$-function to encode information about the
logarithm of the unit $u$, and not about the apparently complicated regulator
of our main result.

More concretely, assume that $\alpha_{g}\alpha_{h}=1$, and take the class
$\kappa(g_{\alpha},g_{1/\alpha}^{*})$, although the same works verbatim for
$\kappa(g_{\beta},g_{1/\beta}^{*})$. From the general theory of Perrin Riou
maps, we may consider the map

(28)
$\langle\mathcal{L}_{gg^{*}}^{++},\omega_{g}\otimes\omega_{g^{*}}\rangle\,:\,H^{1}(\mathbb{Q}_{p},V_{gg^{*}}^{++}(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}^{-1}))\longrightarrow
I^{-1}\Lambda_{{\mathbf{g}}},$

with specializations

$\nu_{s}(\langle\mathcal{L}_{gg^{*}}^{++},\omega_{g}\otimes\omega_{g^{*}}\rangle)\,:\,H^{1}(\mathbb{Q}_{p},V_{gg^{*}}^{++}(1-s))\longrightarrow\mathbb{C}_{p},$

where

$\nu_{s}(\langle\mathcal{L}_{gg^{*}}^{++},\omega_{g}\otimes\omega_{g^{*}}\rangle)=\frac{1-p^{s-1}}{1-p^{-s}}\cdot\begin{cases}\langle\frac{(-1)^{s}}{(-s)!}\cdot\langle\log_{\operatorname{BK}},\omega_{g}\otimes\omega_{g^{*}}\rangle&\text{
if }s<0\\\
(s-1)!\cdot\langle\exp_{\operatorname{BK}}^{*},\omega_{g}\otimes\omega_{g^{*}}\rangle&\text{
if }s>1.\end{cases}$

Observe that we have not said anything about the specializations at $s=0$ and
at $s=1$.

When $s=0$ (resp. $s=1$), we are still in the region of the Bloch–Kato
logarithm (resp. dual exponential map), but the expression $1-p^{-s}$ (resp.
$1-p^{s-1}$) vanishes. Hence, and following [LVZ, Proposition 2.5.5] (see also
the computations of [Ven, Section 3.1] and [Buy1, Section 6.3]), we have the
following.

######  Lemma 6.1.

The map

(29)
$\nu_{0}(\langle\mathcal{L}_{gg^{*}}^{++},\omega_{g}\otimes\omega_{g^{*}}\rangle)\,:\,H^{1}(\mathbb{Q}_{p},V_{gg^{*}}^{++}(1))\longrightarrow\mathbb{C}_{p}$

is given by

(30)
$\nu_{0}(\langle\mathcal{L}_{gg^{*}}^{++},\omega_{g}\otimes\omega_{g^{*}}\rangle)=(1-p^{-1})\cdot\langle{\mathrm{ord}}_{p},\omega_{g}\otimes\omega_{g^{*}}\rangle.$

######  Remark 6.2.

The situation is slightly different to that of circular units: there, the fact
of taking the derived class was related to the fact that the Coleman map was
connected to an imprimitive $p$-adic $L$-function, vanishing at the point of
interest and whose derivative there corresponds to the special value of the
Kubota–Leopoldt $p$-adic $L$-function.

Define

(31)
$L_{p}^{\operatorname{Eis}}(g,h,s)=\langle\mathcal{L}_{gg^{*}}^{++}(\kappa_{p}^{++}(g_{\alpha},g_{1/\alpha}^{*},s)),\omega_{g}\otimes\omega_{g^{*}}\rangle,$

where $\kappa(g_{\alpha},g_{1/\alpha}^{*},s)$ is the restriction of the
3-variable cohomology class to the cyclotomic line.

As a piece of notation for the following result, let
$\mathcal{L}_{g_{\alpha}}$ stand for the period ratio introduced in [DR2,
Section 2].

######  Proposition 6.3.

The special value of the derivative of
$L_{p}^{\operatorname{Eis}}(g_{\alpha},g_{1/\alpha}^{*},0)$ satisfies that

$L_{p}^{\operatorname{Eis}}(g_{\alpha},g_{1/\alpha}^{*},0)=\frac{\mathcal{L}_{g_{\alpha}}}{\log_{p}(u_{\alpha\otimes\bar{\beta}})}\times\log_{p}(u_{1})\pmod{L^{\times}}.$

###### Proof.

When $s=0$, the denominator of the Perrin-Riou map $\mathcal{L}_{gg^{*}}^{++}$
vanishes and we are in the setting discussed before. Then, the Perrin-Riou map
is given by the order followed by the pairing with the canonical
differentials, as in (30). Since according to the results of [RiRo2, Section
4]

$\kappa(g_{\alpha},g_{1/\alpha}^{*})=\frac{1}{\Xi_{g_{\alpha}}\Omega_{g_{1/\alpha}^{*}}}\frac{\log_{p}(u_{1})\cdot
v-\log_{p}(v_{1})\cdot
u}{\log_{p}(u_{\alpha\otimes\bar{\beta}})}\pmod{L^{\times}},$

the image of $\kappa_{p}(g_{\alpha},g_{1/\alpha}^{*})$ under the map (30)
agrees with

$\log_{p}(u_{1})\cdot\frac{\mathcal{L}_{g_{\alpha}}}{\log_{p}(u_{\alpha\otimes\bar{\beta}})}\pmod{L^{\times}}.$

∎

Further, recall that according to [DR2, Conjecture 2.3], we expect that
$\mathcal{L}_{g_{\alpha}}$ must agree with
$\log_{p}(u_{\alpha\otimes\bar{\beta}})$ up to multiplication by $L^{\times}$,
and this would give just $\log_{p}(u_{1})$ in the previous formula.

As a final comment, observe that the class
$\kappa(g_{\alpha},g_{1/\beta}^{*})$ vanishes, while neither the corresponding
numerator nor the denominator of the Perrin-Riou map do. Hence, we expect the
special value to be zero. However, it would be licit to take the derivative of
both the class and the $p$-adic $L$-function.

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