# Generalized Kato classes and exceptional zero conjectures

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

The primary objective of this paper is the study of different instances of the
elliptic Stark conjectures of Darmon, Lauder and Rotger, in a situation where
the elliptic curve attached to the modular form $f$ has split multiplicative
reduction at $p$ and the arithmetic phenomena are specially rich. For that
purpose, we resort to the principle of improved $p$-adic $L$-functions and
study their $\mathcal{L}$-invariants. We further interpret these results in
terms of derived cohomology classes coming from the setting of diagonal
cycles, showing that the same $\mathcal{L}$-invariant which arises in the
theory of $p$-adic $L$-functions also governs the arithmetic of Euler systems.
Thus, we can reduce, in the split multiplicative situation, the conjecture of
Darmon, Lauder and Rotger to a more familiar statement about higher order
derivatives of a triple product $p$-adic $L$-function at a point lying inside
the region of classical interpolation, in the realm of the more well-known
exceptional zero conjectures.

###### 1991 Mathematics Subject Classification:

11G05, 11G40

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Hsieh’s triple product p-adic L-function
    2. 2.2 Improved p-adic L-functions
    3. 2.3 Exceptional zeros and L-invariants
  3. 3 Derived diagonal cycles and an explicit reciprocity law
    1. 3.1 Diagonal cycles and an explicit reciprocity law
    2. 3.2 Vanishing of cohomology classes
    3. 3.3 Derived classes and reciprocity laws
  4. 4 Derivatives of triple product p-adic L-functions
    1. 4.1 A conjecture for the second derivative
    2. 4.2 Some reducible cases
    3. 4.3 The conjecture in other settings
  5. 5 Applications to the elliptic Stark conjecture
    1. 5.1 Interplay between both settings and a conjecture of Darmon–Rotger
    2. 5.2 Case (a)
    3. 5.3 Case (b)

##  1\. Introduction

The elliptic Stark conjecture was first formulated by Darmon, Lauder and
Rotger in [DLR1] as a “more constructive alternative to the Birch and
Swinnerton-Dyer conjecture, since it often allows the efficient analytic
computation of $p$-adic logarithms of global points”. The conjecture relates a
$p$-adic iterated integral attached to a triple $(f,g,h)$ of cuspidal modular
forms with a regulator given in terms of points in an elliptic curve, in a
rank 2 situation. Until the moment, not too much work towards the proof of the
conjecture has been done: most of the results are restricted to situations
where there exists a factorization of $p$-adic $L$-functions, which allows to
interpret the conjecture in terms of the more familiar objects of
Bertolini–Darmon–Prasanna [BDP].

However, recent works of Bertolini–Seveso–Veneruci [BSV1], [BSV2] and
Darmon–Rotger [DR4], [DR3] suggest an alternative conjecture also in terms of
triple product $p$-adic $L$-functions: while the first formulation of [DLR1]
is concerned with the $p$-adic value at a point lying outside the region of
classical interpolation, the new version we discuss is about higher order
derivatives at a point which belongs to the classical interpolation region.
This setting is germane to that explored firstly by Greenberg–Stevens [GS] and
then by Bertolini–Darmon [BD2] or Venerucci [Ven]. We propose an alternative
conjecture in the split multiplicative setting, and one of the main results of
this note is the discussion of the equivalence between both formulations,
using for that purpose the setting of generalized cohomology classes. This
relies, however, on an apparently deep fact about periods of weight one
modular forms, stated in [DR2] as Conjecture 2.1. We believe that this
translation of the conjecture to a more well understood setting provides new
evidence for a better understanding of the problem.

The genesis of this project comes from a parallel story where a new
conjecture, formulated in [DLR2], arises; this gives a formula for the
$p$-adic iterated integral when the modular form $f$ is no longer cuspidal,
but an Eisenstein series. In [RiRo1], the authors envisaged a method of proof
for this conjecture when the two modular forms $(g,h)$ are self-dual: this was
based on Hida’s improved factorization theorem for the Hida–Rankin $p$-adic
$L$-function and allowed us to study the conjecture in terms of a question
concerning Galois deformations.

The discussion of our results in this paper also leads us to the study of an
exceptional vanishing of the generalized cohomology classes of [DR2] and
[CaHs], proposing a putative refinement in terms of some derived generalized
cohomology classes.

Setting and notations. Fix once for all a prime number $p\geq 3$ and three
positive integers $N_{f}$, $N_{g}$, $N_{h}$. Let
$N=\operatorname{lcm}(N_{f},N_{g},N_{h})$ and assume that $p\nmid N$. Let
$\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\mathbb{C}^{\times}$ be a
Dirichlet character. Let

$f\in S_{2}(pN_{f}),\quad g\in M_{1}(N_{g},\chi),\quad h\in
M_{1}(N_{h},\bar{\chi})$

be a triple of newforms of weights $(2,1,1)$, levels $(pN_{f},N_{g},N_{h})$
and nebentype characters $(1,\chi,\bar{\chi})$, where $\bar{\chi}$ stands for
the character obtained by composing $\chi$ with complex conjugation. Further,
we denote by $V_{g}$ and by $V_{h}$ the Artin representations attached to $g$
and $h$, respectively, and write $V_{gh}:=V_{g}\otimes V_{h}$. Let $H$ be the
number field cut out by this representation, and $L$ for the field over which
it is defined. To simplify the exposition, we assume that $f$ has rational
Fourier coefficients and that is attached via modularity to an elliptic curve
$E$ with split multiplicative reduction at $p$. Under the assumption that
$(pN_{f},N_{g}N_{h})=1$, the global sign of the functional equation of
$L(E,V_{gh},s)$ is $+1$. We keep this assumption from now on.

Label and order the roots of the $p$-th Hecke polynomial of $g$ as

$X^{2}-a_{p}(g)X+\chi(p)=(X-\alpha_{g})(X-\beta_{g})$

and do the same for those of $h$. Let $g_{\alpha}(q)=g(q)-\beta_{g}(q^{p})$
denote the $p$-stabilization of $g$ with $U_{p}$-eigenvalue $\alpha_{g}$; it
is defined by the $q$-expansion $g_{\alpha}(q)=g(q)-\beta_{g}g(q^{p})$. We
want to deal with a situation of exceptional zeros, that is, where one or
several of the Euler factors involved in the interpolation formula of the
$p$-adic $L$-function vanish (alternatively, and as we will see later on, this
can be understood in terms of the eigenvalues for the Frobenius action). This
naturally splits into two different settings, namely

  1. (a)

the case where $\alpha_{g}\alpha_{h}=1$ (and therefore
$\beta_{g}\beta_{h}=1$); and

  2. (b)

the case where $\alpha_{g}\beta_{h}=1$ (and therefore
$\beta_{g}\alpha_{h}=1$).

In both cases, if we denote the roots of the $p$-th Hecke polynomial of $g$ by
$\\{\alpha_{g},\beta_{g}\\}$, those of $h$ are
$\\{1/\alpha_{g},1/\beta_{g}\\}$. As a piece of notation, we write
$h_{1/\alpha}$ and $h_{1/\beta}$ for the $p$-stabilizations of $h$ with
eigenvalues $1/\alpha_{g}$ and $1/\beta_{g}$, respectively. Along this work,
we refer to these settings as Case (a) and Case (b). In the framework of
Beilinson–Flach elements and Hida–Rankin $p$-adic $L$-functions, the second
case has been studied in [RiRo1], and the former has been worked out in
[RiRo2, Section 5].

To prove our main results, we also need a classicality property for $g$.
Hence, we assume throughout that

  1. (H1)

the reduction of both $V_{g}$ and $V_{h}$ modulo $p$ is irreducible (this
requires the choice of integral lattices $T_{g}$ and $T_{h}$, but the fact of
being irreducible or not is independent of this choice);

  2. (H2)

$g$ and $h$ are $p$-distinguished, i.e,
$\alpha_{g}\neq\beta_{g},\alpha_{h}\neq\beta_{h}\pmod{p}$; and

  3. (H3)

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

Enlarge $L$ if necessary so that it contains all Fourier coefficients of
$g_{\alpha}$. As shown in [DLR1], the above hypotheses ensure that any
generalized overconvergent modular form with the same generalized eigenvalues
as $g_{\alpha}$ is classical, and hence simply a multiple of $g_{\alpha}$.

In order to describe our results more precisely, let
$\Lambda=\mathbb{Z}_{p}[[\mathbb{Z}^{\times}_{p}]]$ be the Iwasawa algebra and
denote by $\mathcal{W}=\operatorname{Spf}(\Lambda)$ the weight space. Hida’s
theory associates the following data to $f$:

  * •

a finite flat extension $\Lambda_{{\mathbf{f}}}$ of $\Lambda$, giving rise to
a covering $\mathrm{w}:\mathcal{W}_{{\bf
f}}=\operatorname{Spf}(\Lambda_{{\mathbf{f}}})\longrightarrow\mathcal{W}$;

  * •

a family of overconvergent $p$-adic ordinary modular forms ${\bf
f}\in\Lambda_{{\mathbf{f}}}[[q]]$ specializing to $f$ at some point
$x_{0}\in\mathcal{W}_{{\bf f}}$ of weight $\mathrm{w}(x_{0})=2$.

  * •

a representation of the absolute Galois group $G_{\mathbb{Q}}$,
$\varrho_{{\mathbf{f}}}:G_{\mathbb{Q}}\longrightarrow\mathrm{GL}(\mathbb{V}_{{\mathbf{f}}})\simeq\mathrm{GL}_{2}(\Lambda_{{\mathbf{f}}})$
characterized by the property that all its classical specializations coincide
with the Galois representation associated by Deligne to the corresponding
specialization of the Hida family.

The same occurs with $g_{\alpha}$ and $h_{\alpha}$ thanks to the work of
Bellaiche and Dimitrov [BeDi] on the geometry of the eigencurve for points of
weight one; we denote by $\Lambda_{{\mathbf{g}}}$ and $\Lambda_{{\mathbf{h}}}$
the corresponding extensions of $\Lambda$ over which the Hida families
${\mathbf{g}}$ and ${\mathbf{h}}$ are defined, and by
$y_{0}\in\mathcal{W}_{{\mathbf{g}}}$, $z_{0}\in\mathcal{W}_{{\mathbf{h}}}$ the
weight one points for which the corresponding specializations agree with
$g_{\alpha}$ and $h_{\alpha}$, respectively.

For each of the settings (a) and (b) presented above, we discuss three
different objects which are expected to encode arithmetic information
regarding the convolution of the three Galois representations attached to the
modular forms $f$, $g$ and $h$. We denote by $(x,y,z)$ a triple of points in
$\mathcal{W}_{{\mathbf{f}}}\times\mathcal{W}_{{\mathbf{g}}}\times\mathcal{W}_{{\mathbf{h}}}$,
whose weights are referred as $(k,\ell,m)$.

  1. (i)

The cohomology classes $\kappa(f,g_{\alpha},h_{\alpha})$ studied for instance
in [DR2] and [CaHs], arising as the specialization at weights $(2,1,1)$ of the
three-variable family $\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$
constructed as the image under a $p$-adic Abel-Jacobi map of certain diagonal
cycles. In general, one may construct four different classes

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

one for each $p$-stabilization of $g$ and $h$. Further, when some of these
classes vanish, we are lead to consider their derivatives.

  2. (ii)

The special value
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ at weights
$(2,1,1)$ and its derivatives. Here,
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ stands for the
three-variable $p$-adic $L$-function attached to three Hida families,
characterized by an interpolation property regarding the classical values of
the triple product $L$-function at the region where $k\geq\ell+m$. When this
function vanishes at the point $(2,1,1)$, the derivatives along different
directions of the weight space may encode interesting arithmetic information.

  3. (iii)

The special value
${\mathscr{L}_{p}}^{g_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ at
weights $(2,1,1)$, denoted ${\mathscr{L}_{p}}^{g_{\alpha}}$. This $p$-adic
$L$-function is defined in an analogue way to the previous one, but now the
region of interpolation is characterized by the inequality $\ell\geq k+m$ so
the point $(2,1,1)$ is outside the region of classical interpolation.
Similarly, we may also take
${\mathscr{L}_{p}}^{h_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$,
whose region of interpolation concerns those points for which $m\geq k+\ell$.
Observe that the first value depends on the choice of $p$-stabilizations for
the weight one form $g_{\alpha}$.

(i) Cohomology classes coming from the theory of diagonal cycles. We begin by
recalling the results concerning cohomology classes. Results of this kind had
already been explored in [BSV2] and [DR3] when $\alpha_{g}\alpha_{h}=1$. In
that case, the cohomology class is not expected to vanish, but the numerator
of the (Perrin-Riou) regulator in the reciprocity law for
${\mathscr{L}_{p}}^{f}$ does, which is coherent with the fact that the
$p$-adic $L$-function ${\mathscr{L}_{p}}^{f}(f,g,h)$ is zero (this can be
seen, of course, as an exceptional zero coming from the vanishing of an Euler
factor).

Here we are mostly interested in the case where the denominator of the Perrin-
Riou regulator in the reciprocity law for ${\mathscr{L}_{p}}^{g_{\alpha}}$
vanishes due to another exceptional zero phenomenon. This occurs when
$\alpha_{g}\beta_{h}=1$ and leads us to recover the ideas of [Cas], [RiRo1]
and [Ri1], where this same phenomenon was studied for Heegner points,
Beilinson–Flach elements and elliptic units, respectively. In those cases, the
reciprocity laws linking Euler systems and $p$-adic $L$-functions were updated
to derived reciprocity laws. A different approach is taken also in [BSV1,
Section 8], where the authors introduce certain improved cohomology classes,
which in this case we may compare in an explicit way with appropriate derived
elements.

Define the three-variable Iwasawa algebra $\Lambda_{{\bf fgh}}:=\Lambda_{{\bf
f}}\hat{\otimes}_{\mathbb{Z}_{p}}\Lambda_{{\bf
g}}\hat{\otimes}_{\mathbb{Z}_{p}}\Lambda_{{\bf h}}$ and the $\Lambda_{{\bf
fgh}}[G_{\mathbb{Q}}]$-module

$\mathbb{V}_{{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}}:=\mathbb{V}_{{\mathbf{f}}}\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{V}_{{{\mathbf{g}}}}\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{V}_{{{\mathbf{h}}}}.$

We work with $\mathbb{V}_{{\bf fgh}}^{{\dagger}}$, a certain twist of it by an
appropriate power of the $\Lambda$-adic cyclotomic character defined for
instance in [DR3, Section 5.1] and that is needed to satisfy the self-dual
assumption.

The works [BSV1] and [DR3] attach to
$({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ a $\Lambda$-adic global cohomology
class

$\kappa({\bf f},{\bf g},{\bf h})\in H^{1}(\mathbb{Q},\mathbb{V}_{\bf
fgh}^{{\dagger}})$

parameterized by the triple product of the weight space
$\mathcal{W}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}:=\mathcal{W}_{{\bf
f}}\times\mathcal{W}_{{\bf g}}\times\mathcal{W}_{{\bf h}}$.

Consider the specialization of the class at weights $(x_{0},y_{0},z_{0})$,

$\kappa(f,g_{\alpha},h_{1/\beta})\in H^{1}(\mathbb{Q},V_{fgh}),$

where $V_{fgh}$ is the tensor product $V_{f}\otimes V_{g}\otimes V_{h}$ of the
Galois representations attached to the modular forms $f$, $g$ and $h$. This
class can be shown to be trivial and hence we are placed to work with an
appropriate derived class $\kappa^{\prime}(f,g_{\alpha},h_{1/\beta})$.

As it occurred in the setting of Beilinson–Flach classes, the notion of
derivative is rather flexible. Following [RiRo1], we consider here a
derivative along an analytic direction, and keeping fixed the weight of $h$.
Rather informally, this may be thought as the line $(\ell+1,\ell,1)$ of the
weight space. Note that at least in the self-dual case, where we may argue
that the corresponding class vanishes all along the line $(2,\ell,\ell)$, we
may consider the derivative along any direction of the weight space.

Let $\alpha_{{\mathbf{f}}}$ (resp. $\alpha_{{\mathbf{g}}}$,
$\alpha_{{\mathbf{h}}}$) stand for the Iwasawa function corresponding to the
root of the $p$-th Hecke polynomial of ${\mathbf{f}}$ (resp. ${\mathbf{g}}$,
${\mathbf{h}}$) with smallest $p$-adic valuation. As an additional piece of
notation, let

(1)
$\mathcal{L}:=\frac{\alpha_{g}^{\prime}}{\alpha_{g}}-\frac{\alpha_{f}^{\prime}}{\alpha_{f}},$

where $\alpha_{f}^{\prime}$ (resp. $\alpha_{g}^{\prime}$,
$\alpha_{h}^{\prime}$) stands for the derivative of the Frobenius eigenvalues
at $x_{0}$ (resp. $y_{0}$, $z_{0}$) when seen as an Iwasawa function along the
Hida family $\Lambda_{{\mathbf{f}}}$ (resp. $\Lambda_{{\mathbf{g}}}$,
$\Lambda_{{\mathbf{h}}}$). Observe that we can give explicit formulas for
$\mathcal{L}$, involving both some units and $p$-units in the field cut out by
the representation $V_{gh}$ and the Tate uniformizer of the elliptic curve
$E$. Hence, the $\mathcal{L}$-invariant governing the arithmetic of the triple
$(f,g,h)$ is related both with the $\mathcal{L}$-invariant of the elliptic
curve (the logarithm of the Tate uniformizer) and also with the regulator
attached to the adjoint representation $\operatorname{ad}^{0}(V_{g})$,
expressed in [RiRo1] as a combination of logarithms of units and $p$-units.
Compare for instance this result with the main theorem of [Cas], where he
interprets the $\mathcal{L}$-invariant attached to a modular form $f$ and an
anticyclotomic character as the sum of the two $\mathcal{L}$-invariants. Our
first main result is the following (see Theorem 3.9 for the precise
formulation), relating an appropriate logarithm of the derived class with the
special value ${\mathscr{L}_{p}}^{g_{\alpha}}$.

######  Theorem 1.1.

The derived cohomology class satisfies

$\langle\log_{\operatorname{BK}}(\kappa_{p}^{\prime}(f,g_{\alpha},h_{1/\beta})^{g}),\eta_{f}\otimes\omega_{g_{\alpha}}\otimes\omega_{h_{1/\beta}}\rangle=\mathcal{L}\cdot{\mathscr{L}_{p}}^{g_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})(x_{0},y_{0},z_{0})\pmod{L^{\times}},$

where the superindex $g$ stands for an appropriate projection of
$\kappa_{p}^{\prime}$ that we later introduce, and $\log_{\operatorname{BK}}$
refers to the Bloch–Kato logarithm, followed by the pairing
$\langle\,-,-\,\rangle$ with certain canonical differentials.

######  Remark.

In [BSV1] the authors take a different approach to this exceptional zero
phenomenon, and construct an improved cohomology class
$\kappa_{g}^{*}(f,g_{\alpha},h_{1/\beta})$. As we will later show, there is a
connection between both constructions and one may prove (under mild
conditions!) that the following equality holds in
$H^{1}(\mathbb{Q},(V_{f}\otimes V_{g}\otimes V_{h})_{|\mathcal{S}})$, where
$\mathcal{S}$ stands for the subvariety of the weight space along which the
derived and the improved class are defined, corresponding to the set of
weights $k+m=\ell+2$:

(2)
$\kappa^{\prime}(f,g_{\alpha},h_{1/\beta})=\mathcal{L}\cdot\kappa_{g}^{*}(f,g_{\alpha},h_{1/\beta}).$

(ii) The special value ${\mathscr{L}_{p}}^{f}$ and derivatives of the triple
product $p$-adic $L$-function. In subsequent parts of the article we use the
previous cohomology classes to study different instances of the elliptic Stark
conjecture. Section 4 is devoted to analyze higher order derivatives of
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ at
$(x_{0},y_{0},z_{0})$. The presence of an Euler factor which vanishes at
weights $(2,1,1)$ automatically forces the vanishing of that value. Therefore,
it is natural to formulate several conjectures for the value of the
derivatives of
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$.

When $\alpha_{g}\alpha_{h}=1$ and $L(f\otimes g\otimes h,1)\neq 0$, the
results of [BSV1] relying on the existence of an improved $p$-adic
$L$-function allow us to state the following result. Although this can be seen
as a straightforward corollary of the results developed in loc. cit., we want
to point out that the $\mathcal{L}$-invariants attached to both $g$ and $h$
have a strong connection with the arithmetic of number fields. This reveals
that in the rank 0 situation the quantity ${\mathscr{L}_{p}}^{f}$ is also a
putative refinement of the more well-known $\mathcal{L}$-invariants of
Greenberg–Stevens, where not only the Tate period $q_{E}$ appears. This result
follows from [BSV1, Proposition 8.2].

######  Proposition (Bertolini–Seveso–Venerucci).

Let $I$ denote the ideal of functions in
$\Lambda_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}}$ which vanish at
$(x_{0},y_{0},z_{0})$. Assume that $L(f\otimes g\otimes h,1)\neq 0$, and let
$\mathcal{L}_{\xi}:=\alpha_{\xi}^{\prime}/\alpha_{\xi}$, for
$\xi\in\\{f,g_{\alpha},h_{\alpha}\\}$. Then, up to a constant in $L^{\times}$,

${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})=(\mathcal{L}_{g_{\alpha}}-\mathcal{L}_{f})(\ell-1)+(\mathcal{L}_{h_{\alpha}}-\mathcal{L}_{f})(m-1)\pmod{I^{2}}.$

Moreover, the quantities $\mathcal{L}_{\chi}$ are explicitly computable in
terms of the arithmetic of number fields and elliptic curves.

Observe for example that the derivative along the $y$-direction agrees with
the $\mathcal{L}$-invariant that also arises as the derivative of the diagonal
class discussed before.

However, the most interesting case appears when $L(f\otimes g\otimes h,1)=0$.
Let us put ourselves in the setting of [DLR1] and assume that $(E(H)\otimes
V_{gh}^{\vee})^{\mathrm{Gal\,}(H/\mathbb{Q})}$ is two-dimensional, where
$V_{gh}^{\vee}$ stands for the contragradient representation of $V_{gh}$. This
group is equipped with an inclusion in the $p$-adic Selmer group corresponding
to the group of extensions of $\mathbb{Q}_{p}$ by $V_{fgh}$ in the category of
$\mathbb{Q}_{p}$-linear representations of $G_{\mathbb{Q}}$ that are
crystalline at $p$. This group is denoted by
$H_{\operatorname{f}}^{1}(\mathbb{Q},V_{fgh})$, and we also assume that is
two-dimensional (the latter would follow from the Birch and Swinnerton-Dyer
conjecture for the pair $(E,V_{gh})$ and the finiteness of the corresponding
Tate–Shafarevich group).

Let $\\{P,Q\\}$ denote generators of $(E(H)\otimes
V_{gh}^{\vee})^{G_{\mathbb{Q}}}$, and fix a basis
$\\{e_{\alpha\alpha}^{\vee},e_{\alpha\beta}^{\vee},e_{\beta\alpha}^{\vee},e_{\beta\beta}^{\vee}\\}$
of $V_{gh}^{\vee}$ as a $G_{\mathbb{Q}_{p}}$-module with the Frobenius action.
This allows us to write

$P=P_{\beta\beta}\otimes e_{\beta\beta}^{\vee}+P_{\beta\alpha}\otimes
e_{\beta\alpha}^{\vee}+P_{\alpha\beta}\otimes
e_{\alpha\beta}^{\vee}+P_{\alpha\alpha}\otimes e_{\alpha\alpha}^{\vee},$

and similarly for $Q$. Here, the arithmetic Frobenius $\mathrm{Fr}_{p}$ acts
on $P_{\beta\beta}$ with eigenvalue $\beta_{g}\beta_{h}$ and analogously for
the remaining components. In this scenario, we can conjecture the following
result, that we extensively discuss in Section 5.

######  Conjecture 1.2.

Assume that the $L$-dimension of $(E(H)\otimes
V_{gh}^{\vee})^{G_{\mathbb{Q}}}$ is two. Then, under the running assumptions,
the $p$-adic $L$-function
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ satisfies

$\frac{\partial^{2}{\mathscr{L}_{p}}^{f}({\mathbf{f}}_{x},g_{\alpha},h_{1/\alpha})}{\partial
x^{2}}\Big{|}_{x=x_{0}}=\log_{p}(P_{\beta\beta})\cdot\log_{p}(Q_{\alpha\alpha})-\log_{p}(Q_{\beta\beta})\cdot\log_{p}(P_{\alpha\alpha})\pmod{L^{\times}}.$

If the $L$-dimension of $(E(H)\otimes V_{gh}^{\vee})^{G_{\mathbb{Q}}}$ is
greater than two, then the left hand side vanishes.

There are other interesting lines along weight space to take derivatives. For
example, in [CaHs] the study is concerned with the line $(2,\ell,\ell)$, where
the derivatives are connected with appropriate derived heights of the points
$P$ and $Q$.

The work of Bertolini–Seveso–Venerucci and Darmon–Rotger establishes the
conjecture for the case where $g$ and $h$ are theta series of a quadratic
field where $p$ is inert, which leads to a decomposition
$V_{gh}=V_{\psi_{1}}\oplus V_{\psi_{2}}$. In the imaginary case, we can extend
their computations to the split case, observing that here one has a trivial
equality of the form $0=0$.

Therefore, we may establish that Conjecture 1.2 holds in some dihedral cases.
The first part of this Proposition follows from [BSV2, Theorem A], and the
second is established as part of Proposition 4.7.

######  Proposition 1.3.

Conjecture 1.2 holds in the following cases:

  1. (a)

CM or RM series with $p$ inert in $K$ and at least one of $\psi_{1}$ or
$\psi_{2}$ being a genus characters;

  2. (b)

CM series with $p$ split in $K$, $V_{gh}=V_{\psi_{1}}\oplus V_{\psi_{2}}$,
with each component of rank one.

We must say that in all these cases the proof is based on a factorization
formula, so we expect that new ideas would be required for the proof in the
general case.

(iii) The special value ${\mathscr{L}_{p}}^{g_{\alpha}}$. In the last section,
we discuss a way to connect the previous conjecture with the elliptic Stark
conjecture of [DLR1] when $\alpha_{g}\alpha_{h}=1$. Recall that the conjecture
predicts that

(3)
${\mathscr{L}_{p}}^{g_{\alpha}}(f,g_{\alpha},h_{\alpha})=\frac{\log_{p}(P_{\beta\alpha})\log_{p}(Q_{\beta\beta})-\log_{p}(P_{\beta\beta})\log_{p}(Q_{\beta\alpha})}{\log_{p}(u_{g_{\alpha}})}\pmod{L^{\times}},$

with $u_{g_{\alpha}}$ being a Gross–Stark unit whose characterization we later
recall. In particular, it is expected that this unit could be expressed as a
ratio of periods attached to weight one forms. These two periods, denoted by
$\Omega_{g_{\alpha}}$ and $\Xi_{g_{\alpha}}$, will play a prominent role in
the last part of the work. More precisely, in [DR2, eq. (9)], the authors
introduce a $p$-adic period,
$\mathcal{L}_{g_{\alpha}}=\Omega_{g_{\alpha}}/\Xi_{g_{\alpha}}$ and conjecture
(see Conjecture 2.1 of loc. cit.)

(4) $\mathcal{L}_{g_{\alpha}}=\log_{p}(u_{g_{\alpha}}).$

In Section 5 we consider the following three conjectures:

  1. (i)

the elliptic Stark conjecture for ${\mathscr{L}_{p}}^{g_{\alpha}}$;

  2. (ii)

the conjecture for the second derivative along the $f$-direction for
${\mathscr{L}_{p}}^{f}$, i.e., Conjecture 1.2;

  3. (iii)

[DR2, Conjecture 2.1] about periods of weight one modular forms. Proposition
5.1 can be seen as an extra piece of theoretical evidence towards this
conjecture, showing that

$\frac{\mathcal{L}_{g_{\alpha}}}{\mathcal{L}_{g_{\beta}}}=\frac{\log_{p}(u_{g_{\alpha}})}{\log_{p}(u_{g_{\beta}})}.$

Under certain non-vanishing hypothesis, we prove that if two of the previous
conjectures are true, the third one automatically holds. In particular, we
establish the following in Corollary 5.5.

######  Theorem 1.4.

Let $g$ and $h$ be theta series of a quadratic field (either real or
imaginary) where $p$ is inert. Write $V_{gh}=V_{\psi_{1}}\oplus V_{\psi_{2}}$,
and assume that either $\psi_{1}$ or $\psi_{2}$ is a genus character. Then,
under the given assumptions, the equality (4) is equivalent to the elliptic
Stark conjecture of Darmon, Lauder and Rotger (3).

All the previous results are based on the interaction of the different
arithmetic actors when $\alpha_{g}\alpha_{h}=1$. The case where
$\alpha_{g}\beta_{h}=1$ is more subtle, since here the cohomology class
$\kappa(f,g_{\alpha},h_{1/\beta})$ vanishes and we cannot extract the same
arithmetic information. In any case, we expect that a similar result must hold
in this setting. The reason is that the value of
${\mathscr{L}_{p}}^{g_{\alpha}}$ does not depend on the choice of a
$p$-stabilization for $h$, and hence we can also give a conjectural expression
for the derived cohomology class in terms of points, in complete analogy with
[RiRo1, Theorem B].

######  Conjecture 1.5.

The following equality holds in
$H_{\operatorname{f}}^{1}(\mathbb{Q},V_{fgh})$:

(5)
$\kappa^{\prime}(f,g_{\alpha},h_{1/\beta})=\frac{\mathcal{L}}{\Xi_{g_{\alpha}}\cdot\Omega_{h_{1/\beta}}}\cdot\frac{\log_{p}(P_{\beta\beta})\cdot
Q-\log_{p}(Q_{\beta\beta})\cdot
P}{\log_{p}(u_{g_{\alpha}})}\pmod{L^{\times}}.$

Proceeding as in [DR2] and [RiRo2] we may also obtain expressions (at least
conjecturally) for the three remaining cohomology classes,
$\kappa^{\prime}(f,g_{\beta},h_{1/\alpha})$,
$\kappa(f,g_{\alpha},h_{1/\alpha})$ and $\kappa(f,g_{\beta},h_{1/\beta})$.

Acknowledgements. It is a pleasure to thank Victor Rotger for his interest on
this project and for a careful reading of the manuscript. I am also indebted
to Daniel Barrera, Henri Darmon, and Giovanni Rosso for several enlightening
conversations around this work. I sincerely thank the anonymous referee for a
careful reading of the text, whose comments notably contributed to improve the
exposition of this article.

The author has been supported by 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\. Preliminaries

This section aims to give an overview of the setting we present, concerned
with triple product $p$-adic $L$-functions, and also recalls some known
results in other related scenarios coming from the theory of elliptic curves
and weight one modular forms.

###  2.1. Hsieh’s triple product p-adic L-function

Fix an algebraic closure $\bar{\mathbb{Q}}$ of $\mathbb{Q}$. For a number
field $K$, let $G_{K}:=\mathrm{Gal\,}(\bar{\mathbb{Q}}/K)$ denote its absolute
Galois group. Fix also an odd prime $p$ and an embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$.

The formal spectrum $\mathcal{W}=\operatorname{Spf}(\Lambda)$ of the Iwasawa
algebra $\Lambda=\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$ is called the
weight space attached to $\Lambda$. The weight space is equipped with a
distinguished class of arithmetic points $\nu_{s,\varepsilon}$ indexed by
integers $s\in\mathbb{Z}$ and Dirichlet characters
$\varepsilon:(\mathbb{Z}/p^{r}\mathbb{Z})^{\times}\rightarrow\bar{\mathbb{Q}}^{\times}$
of $p$-power conductor. The point $\nu_{s,\varepsilon}\in\mathcal{W}$ is
defined by the rule

$\nu_{s,\varepsilon}(n)=\varepsilon(n)n^{s}.$

Let $({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ be a triple of $p$-adic Hida
families of tame levels $N_{{\mathbf{f}}}$, $N_{{\mathbf{g}}}$,
$N_{{\mathbf{h}}}$ and tame characters $\chi_{{\mathbf{f}}}$,
$\chi_{{\mathbf{g}}}$, $\chi_{{\mathbf{h}}}$. Let also
$({\mathbf{f}}^{*},{\mathbf{g}}^{*},{\mathbf{h}}^{*})$ denote the conjugate
triple, and assume that
$\chi_{{\mathbf{f}}}\chi_{{\mathbf{g}}}\chi_{{\mathbf{h}}}=1$ (this is
referred to as the self-duality assumption). Set
$N=\operatorname{lcm}(N_{{\mathbf{f}}},N_{{\mathbf{g}}},N_{{\mathbf{h}}})$,
and suppose that $p\nmid N$.

Let $\Lambda_{{\mathbf{f}}}$, $\Lambda_{{\mathbf{g}}}$ and
$\Lambda_{{\mathbf{h}}}$ be the finite extensions of $\Lambda$ generated by
the coefficients of the Hida families ${\mathbf{f}}$, ${\mathbf{g}}$ and
${\mathbf{h}}$, respectively. The weight space attached to
$\Lambda_{{\mathbf{f}}}$ is
$\mathcal{W}_{{\mathbf{f}}}:=\operatorname{Spf}(\Lambda_{{\mathbf{f}}})$.
Since $\Lambda_{{\mathbf{f}}}$ is a finite flat algebra over $\Lambda$, there
is a natural finite map

$\pi:\mathcal{W}_{{\mathbf{f}}}:=\mathrm{Spf}(\mathcal{W}_{{\mathbf{f}}})\stackrel{{\scriptstyle\mathrm{w}}}{{\longrightarrow}}\mathcal{W},$

and we say that a point $x\in\mathcal{W}_{{\mathbf{f}}}$ is arithmetic of
weight $s$ and character $\varepsilon$ if $\pi(x)=\nu_{s,\varepsilon}$.

A point $x\in\mathcal{W}_{{\mathbf{f}}}$ of weight $k\geq 1$ and character
$\varepsilon$ is said to be crystalline if $\varepsilon=1$ and there exists an
eigenform $f_{x}^{\circ}$ of level $N$ such that $f_{x}$ is the ordinary
$p$-stablization of $f_{x}^{\circ}$. We denote by
$\mathcal{W}_{{\mathbf{f}}}^{\circ}$ the set of crystalline arithmetic points
of $\mathcal{W}_{{\mathbf{f}}}$.

Finally, set $\Lambda_{{\bf
fgh}}=\Lambda_{{\mathbf{f}}}\hat{\otimes}\Lambda_{{\mathbf{g}}}\hat{\otimes}\Lambda_{{\mathbf{h}}}$
and let $\mathcal{W}_{{\bf
fgh}}^{\circ}:=\mathcal{W}_{{\mathbf{f}}}^{\circ}\times\mathcal{W}_{{\mathbf{g}}}^{\circ}\times\mathcal{W}_{{\mathbf{h}}}^{\circ}\subset\mathcal{W}_{{\bf
fgh}}=\operatorname{Spf}(\Lambda_{{\bf fgh}})$ be the set of triples of
crystalline classical points, at which the three Hida families specialize to
modular forms with trivial nebentype at $p$. This set admits the natural
partition

$\mathcal{W}_{{\bf fgh}}^{\circ}=\mathcal{W}_{{\bf
fgh}}^{f}\sqcup\mathcal{W}_{{\bf fgh}}^{g}\sqcup\mathcal{W}_{{\bf
fgh}}^{h}\sqcup\mathcal{W}_{{\bf fgh}}^{\operatorname{bal}},$

where

  * •

$\mathcal{W}_{{\bf fgh}}^{f}$ denotes the set of points
$(x,y,z)\in\mathcal{W}_{{\bf fgh}}^{\circ}$ of weights $(k,\ell,m)$ such that
$k\geq\ell+m$.

  * •

$\mathcal{W}_{{\bf fgh}}^{g}$ and $\mathcal{W}_{{\bf fgh}}^{h}$ are defined
similarly, replacing the role of ${\mathbf{f}}$ by ${\mathbf{g}}$ (resp.
${\mathbf{h}}$).

  * •

$\mathcal{W}_{{\bf fgh}}^{\operatorname{bal}}$ is the set of balanced triples,
consisting of points $(x,y,z)$ of weights $(k,\ell,m)$ such that each of the
weights is strictly smaller than the sum of the other two.

Recall from [DR3, Section 1.4] the notion of test vector. As proved in Section
3.5 of loc. cit. following [Hs], there is a canonical choice of test vectors
for which there exists a square-root $p$-adic $L$-function

${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}):\mathcal{W}_{{\bf
fgh}}\rightarrow\mathbb{C}_{p},$

characterized by an interpolation property relating its values at classical
points $(x,y,z)\in\mathcal{W}_{{\bf fgh}}^{f}$ to the square root of the
central critical value of Garrett’s triple-product complex $L$-function
$L({\mathbf{f}}_{x},{\mathbf{g}}_{y},{\mathbf{h}}_{z},s)$ associated to the
triple of classical eigenforms
$({\mathbf{f}}_{x},{\mathbf{g}}_{y},{\mathbf{h}}_{z})$. For the following
proposition, let $\alpha_{{\mathbf{f}}_{x}}$ and $\beta_{{\mathbf{f}}_{x}}$ be
the roots of the $p$-th Hecke polynomial of ${\mathbf{f}}_{x}$, ordered in
such a way that
${\mathrm{ord}}_{p}(\alpha_{{\mathbf{f}}_{x}})\leq{\mathrm{ord}}_{p}(\beta_{{\mathbf{f}}_{x}})$.
The following result is [DR3, Proposition 5.1].

######  Proposition 2.1.

Fix test vectors
$(\tilde{{\mathbf{f}}},\tilde{{\mathbf{g}}},\tilde{{\mathbf{h}}})$ as in [Hs,
Section 3]. Then
${\mathscr{L}_{p}}^{f}(\tilde{{\mathbf{f}}},\tilde{{\mathbf{g}}},\tilde{{\mathbf{h}}})$
lies in $\Lambda_{{\bf fgh}}$ and for every $(x,y,z)\in\mathcal{W}_{{\bf
fgh}}^{f}$ of weights $(k,\ell,m)$ we have

${\mathscr{L}_{p}}^{f}(\tilde{{\mathbf{f}}},\tilde{{\mathbf{g}}},\tilde{{\mathbf{h}}})^{2}(x,y,z)=\frac{\mathfrak{a}(k,\ell,m)}{\langle{\mathbf{f}}_{x}^{\circ},{\mathbf{f}}_{x}^{\circ}\rangle^{2}}\cdot\mathfrak{e}^{2}(x,y,z)\times
L({\mathbf{f}}_{x}^{\circ},{\mathbf{g}}_{y}^{\circ},{\mathbf{h}}_{z}^{\circ},c),$

where

  1. (1)

$c=\frac{k+\ell+m-2}{2}$.

  2. (2)

$\mathfrak{a}(k,\ell,m)=(2\pi
i)^{-2k}\cdot\Big{(}\frac{k+\ell+m-4}{2}\Big{)}!\cdot\Big{(}\frac{k+\ell-m-2}{2}\Big{)}!\cdot\Big{(}\frac{k-\ell+m-2}{2}\Big{)}!\cdot\Big{(}\frac{k-\ell-m}{2}\Big{)}!$,

  3. (3)

$\mathfrak{e}(x,y,z)=\mathcal{E}(x,y,z)/\mathcal{E}_{0}(x)\mathcal{E}_{1}(x)$
with

$\displaystyle\mathcal{E}_{0}(x)$ $\displaystyle:=$ $\displaystyle
1-\chi_{f}^{-1}(p)\beta_{{\mathbf{f}}_{x}}^{2}p^{1-k},$
$\displaystyle\mathcal{E}_{1}(x)$ $\displaystyle:=$ $\displaystyle
1-\chi_{f}(p)\alpha_{{\mathbf{f}}_{x}}^{-2}p^{k-2},$
$\displaystyle\mathcal{E}(x,y,z)$ $\displaystyle:=$
$\displaystyle\Big{(}1-\chi_{f}(p)\alpha_{{\mathbf{f}}_{x}}^{-1}\alpha_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}p^{\frac{k-\ell-m}{2}}\Big{)}\times\Big{(}1-\chi_{f}(p)\alpha_{{\mathbf{f}}_{x}}^{-1}\alpha_{{\mathbf{g}}_{y}}\beta_{{\mathbf{h}}_{z}}p^{\frac{k-\ell-m}{2}}\Big{)}$
$\displaystyle\times\Big{(}1-\chi_{f}(p)\alpha_{{\mathbf{f}}_{x}}^{-1}\beta_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}p^{\frac{k-\ell-m}{2}}\Big{)}\times\Big{(}1-\chi_{f}(p)\alpha_{{\mathbf{f}}_{x}}^{-1}\beta_{{\mathbf{g}}_{y}}\beta_{{\mathbf{h}}_{z}}p^{\frac{k-\ell-m}{2}}\Big{)}.$

For simplicity, we fix once and for all these test vectors and remove their
dependence in the notation.

There is an ostensible parallelism between this $p$-adic $L$-function and the
so-called Hida–Rankin $p$-adic $L$-function attached to a pair of Hida
families $({\mathbf{g}},{\mathbf{h}})$, but where the cyclotomic variable $s$
is allowed to move freely. It may be instructive to keep in mind this analogy
for the subsequent results.

Some of the easiest cases to understand these triple product $p$-adic
$L$-functions arise when the representation attached to $V_{gh}$ is
irreducible. In particular, assume that $g$ is a weight one theta series
attached to a quadratic field $K$ (either real of imaginary) where $p$ remains
inert. Then, $V_{gh}=V_{\psi_{1}}\oplus V_{\psi_{2}}$, and under the
assumption that at least one between $\psi_{1}$ or $\psi_{2}$ is a genus
(quadratic) character, the works [BSV2] and [DR3] show that

(6) ${\mathscr{L}_{p}}^{f}({\mathbf{f}},g,h)^{2}=\mathfrak{f}(k)\cdot
L_{p}({\mathbf{f}}/K,\psi_{1})\cdot L_{p}({\mathbf{f}}/K,\psi_{2}),$

where $\mathfrak{f}(k)$ is a bounded analytic function on
$\Lambda_{{\mathbf{f}}}$ such that $\mathfrak{f}(x_{0})\in L^{\times}$. Here,
${\mathscr{L}_{p}}({\mathbf{f}}/K,\psi)$ is the two-variable $p$-adic
$L$-function attached to a Hida family ${\mathbf{f}}$ and a character $\psi$
of a quadratic field.

As a word of caution, observe that there are three different $p$-adic
$L$-functions, depending on the region of classical interpolation (associated
to the dominant weight).

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

It is a natural phenomenon in the study of $p$-adic $L$-functions that some of
the Euler factors arising in the interpolation process are analytic along a
subvariety of the weight space. When this happens, one is tempted to define
improved $p$-adic $L$-functions, that is, functions over the corresponding
subvariety characterized by the same interpolation property, but with these
Euler factors removed. This is a quite well-known phenomenon, which dates back
to Greenberg–Stevens [GS] and their study of the Mazur–Kitagawa $p$-adic
$L$-function. This was one of the key ingredients in the proof of our main
results in [RiRo1] and we would like to stress the limitations of the method
in this triple product setting. The interest of this study is that we also
want to discuss later on its applicability from the Euler system side in order
to construct improved cohomology classes.

For the sake of simplicity, assume that $\chi_{f}$ is trivial. In the setting
of triple product $p$-adic $L$-functions we have just discussed, one of the
Euler factors appearing in the interpolation property of
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ is

$1-\frac{\alpha_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}}{\alpha_{{\mathbf{f}}_{x}}}p^{\frac{k-\ell-m}{2}},$

which is an Iwasawa function along the surface $\mathcal{S}_{k=\ell+m}$
defined by

$\mathcal{S}_{k=\ell+m}=\\{(x,y,z)\in\mathcal{W}_{{\mathbf{f}}}^{\circ}\times\mathcal{W}_{{\mathbf{g}}}^{\circ}\times\mathcal{W}_{{\mathbf{h}}}^{\circ}\text{
such that }k=\ell+m\\}.$

The definitions given in [DR1, Def. 4.4] can be adapted to yield an improved
$p$-adic $L$-function
${\mathscr{L}_{p}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}$ on
$\mathcal{S}_{k=\ell+m}$, by replacing the family ${\mathbf{h}}\times
d^{t}{\mathbf{g}}^{[p]}$ with the family ${\mathbf{h}}\times{\mathbf{g}}$,
whose coefficients vary analytically because $t=0$ on
$\mathcal{S}_{k=\ell+m}$.

######  Proposition 2.2.

There exists an analytic $p$-adic $L$-function over the surface
$\mathcal{S}_{k=\ell+m}$, denoted by
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}$, such that
the following equality holds in $\mathcal{S}_{k=\ell+m}$:

(7)
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})=(1-\alpha_{{\mathbf{f}}}^{-1}\alpha_{{\mathbf{g}}}\alpha_{{\mathbf{h}}}){\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}.$

###### Proof.

This follows from the proof of [BSV1, Proposition 8.2] and the discussion
after it. ∎

We point out that the improved $p$-adic $L$-function we have considered,
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}$,
interpolates classical $L$-values, in the same way than
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$, but with the
vanishing Euler factor removed. Therefore, its value at $(x_{0},y_{0},y_{0})$
is given by an explicit non-zero multiple of the square root of the algebraic
part of $L(f,g,h,1)$. In particular, $L(f,g,h,1)\neq 0$ if and only if the
improved $p$-adic $L$-function does not vanish at $(x_{0},y_{0},y_{0})$.

Observe however that we may also consider other Euler factors. Take for
example

$1-\frac{\bar{\chi}_{h}(p)\alpha_{{\mathbf{h}}_{z}}}{\alpha_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{f}}_{x}}}p^{\frac{k+\ell-m-2}{2}},$

which is analytic along $k+\ell=m+2$.

We would expect that one can establish that these factors (each along its
respective region) divide the $p$-adic $L$-function and yield other improved
$p$-adic $L$-functions satisfying mild analytic and interpolation properties.

###  2.3. Exceptional zeros and L-invariants

The situations we study in this article are mostly concerned with the so-
called exceptional zero phenomenon. We now recall several results which appear
in the literature around that, mainly in [GS], [Ven] and [RiRo1]. As
anticipated before, the point is that the $\mathcal{L}$-invariant governing
the arithmetic of $V_{f}\otimes V_{gh}$ is a combination of the
$\mathcal{L}$-invariants attached to $f$ and the adjoint of $g$ (or the
adjoint of $h$, according to the direction we choose). For a brief summary of
the usual definition and the main properties of the adjoint representation in
this scenario, see [DLR1, Section 1.2].

The aim of this section is to give an arithmetic description of the different
$\mathcal{L}$-invariants that later appear in the setting of triple products,
to have a complete description of our picture.

Let us define, to ease notations,

(8)
$\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha})):=\frac{\alpha_{g}^{\prime}}{\alpha_{g}},\qquad\mathcal{L}(E):=\frac{\alpha_{f}^{\prime}}{\alpha_{f}},$

where the derivative is taken along the unique Hida family passing through
$g_{\alpha}$ and $f$, respectively, and then evaluating at the points
corresponding to $g_{\alpha}$ and $f$.

I. The $\mathcal{L}$-invariant of the adjoint of a modular form. One of the
main results of [RiRo1] was the computation of the $\mathcal{L}$-invariant
associated to the adjoint of a modular form.

As shown in [DLR2, Lemma 1.1], we have

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

Fix a generator $u$ of
$(\mathcal{O}_{H}^{\times}\otimes\operatorname{ad}^{0}(g))^{G_{\mathbb{Q}}}$
and also an element $v$ of
$(\mathcal{O}_{H}^{\times}[1/p]^{\times}\otimes\operatorname{ad}^{0}(g))^{G_{\mathbb{Q}}}$
such that $\\{u,v\\}$ is a basis of
$(\mathcal{O}_{H}[1/p]^{\times}\otimes\operatorname{ad}^{0}(g))^{G_{\mathbb{Q}}}$.
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, $\operatorname{ad}^{0}(g)$ decomposes as
$\operatorname{ad}^{0}(g)=L^{1}\oplus L^{\alpha/\beta}\oplus
L^{\beta/\alpha}$, where all the summands are 1-dimensional subspaces
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

$u_{1},\,\,u_{\alpha/\beta},\,\,u_{\beta/\alpha},\,\,v_{1},\,\,v_{\alpha/\beta},\,\,v_{\beta/\alpha}\in
H_{p}^{\times}\otimes_{\mathbb{Q}}L\quad\pmod{L^{\times}}$

denote the projection of the elements $u$ and $v$ in
$(H_{p}^{\times}\otimes\operatorname{ad}^{0}(g))^{G_{\mathbb{Q}_{p}}}$ to the
above lines. By construction we have $u_{1},v_{1}\in\mathbb{Q}_{p}^{\times}$
and

$\mathrm{Fr}_{p}(u_{\alpha/\beta})=\frac{\beta}{\alpha}u_{\alpha/\beta},\quad\mathrm{Fr}_{p}(v_{\alpha/\beta})=\frac{\beta}{\alpha}v_{\alpha/\beta},\quad\mathrm{Fr}_{p}(u_{\beta/\alpha})=\frac{\alpha}{\beta}u_{\beta/\alpha},\quad\mathrm{Fr}_{p}(v_{\beta/\alpha})=\frac{\alpha}{\beta}v_{\beta/\alpha}.$

Let

$\log_{p}:H_{p}^{\times}\otimes L\longrightarrow H_{p}\otimes L$

denote the usual $p$-adic logarithm.

Then, one the main results of [RiRo1] was the computation of
$\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))$, which can be expressed as

(9)
$\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))=-\frac{\log_{p}(v_{1})\log_{p}(u_{\alpha/\beta})-\log_{p}(u_{1})\log_{p}(v_{\alpha/\beta})}{2{\mathrm{ord}}_{p}(v_{1})\cdot\log_{p}(u_{\alpha/\beta})}.$

II. The $\mathcal{L}$-invariant of an elliptic curve (rank 0). In [GS], the
authors prove a conjecture of Mazur, Tate and Teitelbaum [MTT] expressing the
quantity $L_{p}(E,1)$ in terms of the derivative of $L(E,1)$ when the rank is
zero. As a consequence of this, they show that an elliptic curve with split
multiplicative reduction at $p$ satisfies

(10) $\mathcal{L}(E)=-\frac{\log_{p}(q_{E})}{2{\mathrm{ord}}_{p}(q_{E})},$

where $q_{E}$ is Tate’s uniformizer for the elliptic curve $E$. We write
$L_{p}({\mathbf{f}})(x,s)$ for the usual two-variable Mazur–Kitagawa $p$-adic
$L$-function, and $x_{0}$ for the weight two point satisfying
${\mathbf{f}}_{x_{0}}=f$, with $f$ the modular form attached to $E$ by
modularity.

As recalled for instance in the discussion of [BD2, Remark 1.13], there exists
an improved $p$-adic $L$-function along $s=1$, that we denote here as
$L_{p}^{*}({\mathbf{f}})(x)$ and which is characterized by

$L_{p}({\mathbf{f}})(x,1)=(1-a_{p}({\mathbf{f}}_{x})^{-1})\cdot
L_{p}^{*}({\mathbf{f}})(x).$

Observe that in a rank 0 situation $L_{p}^{*}({\mathbf{f}}_{x_{0}})$ is a non-
zero algebraic number which agrees (up to constant) with the algebraic part of
the classical $L$-value.

III. The $\mathcal{L}$-invariant of an elliptic curve (rank 1). In a rank 1
situation, Venerucci relates the second derivatives of the Mazur–Kitagawa
$p$-adic $L$-function with certain heights of Heegner points. Observe that in
this setting, $L_{p}({\mathbf{f}})(x_{0},1)=0$ and the same happens for its
first derivatives. To determine the second order derivatives, he recasts in
[Ven] to the theory of Selmer complexes and Nekovář’s Selmer groups, as
introduced in [Nek].

Following the conventions used in loc. cit., let
$\tilde{H}_{\operatorname{f}}^{1}$ be Nekovar’s extended Selmer group. It is a
$\mathbb{Q}_{p}$-module, equipped with a natural inclusion of the extended
Mordell-Weil group of $E$, that we denote by
$E^{{\dagger}}(\mathbb{Q})\otimes\mathbb{Q}_{p}$. In general,

$\tilde{H}_{\operatorname{f}}^{1}(\mathbb{Q},V_{p}(E))=H_{\operatorname{f}}^{1}(\mathbb{Q},V_{p}(E))\oplus\mathbb{Q}_{p}\cdot
q_{E},$

where $H_{\operatorname{f}}^{1}(\mathbb{Q},V_{p}(E))$ is the Bloch–Kato
$p$-adic Selmer group. Using Nekovar and Venerucci’s results, there is a
canonical $\mathbb{Q}_{p}$-bilinear form

$\langle\cdot,\cdot\rangle:\tilde{H}_{\operatorname{f}}^{1}(\mathbb{Q},V_{p}(E))\otimes_{\mathbb{Q}_{p}}\tilde{H}_{\operatorname{f}}^{1}(\mathbb{Q},V_{p}(E))\rightarrow
I/I^{2},$

where $I$ stands for the augmentation ideal of the cyclotomic Iwasawa algebra,
and which may be thought as the ring of functions vanishing at the point
$(x,s)=(x_{0},1)$, that with a slight abuse of notation we denote by $(2,1)$.
This is the so-called height-weight pairing, which decomposes as

$\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{p}^{\operatorname{cyc}}\cdot\\{s-1\\}+\langle\cdot,\cdot\rangle_{p}^{\operatorname{wt}}\cdot\\{k-2\\},$

where $\langle\cdot,\cdot\rangle_{p}^{\operatorname{cyc}}$ and
$\langle\cdot,\cdot\rangle_{p}^{\operatorname{wt}}$ are canonical
$\mathbb{Q}_{p}$-valued pairings on the extended Selmer group. Finally, the
Schneider height is defined by

$\langle\cdot,\cdot\rangle_{p}^{\operatorname{Sch}}=\langle
x,y\rangle_{p}^{\operatorname{cyc}}-\frac{\log_{p}(\mathrm{res}_{p}(x))\cdot\log_{p}(\mathrm{res}_{p}(y))}{\log_{p}(q_{E})},$

where $\mathrm{res}_{p}(x)$ is the localization-at-$p$ map. The following
result provides expressions for the second derivative of $L_{p}({\mathbf{f}})$
along different directions of the weight space.

######  Proposition 2.3.

The following formulas hold, where $P$ is a generator of the Mordell–Weil
group $E(\mathbb{Q})$.

  1. (a)

$\frac{d^{2}L_{p}({\mathbf{f}})(k,k/2)}{dk^{2}}\Big{|}_{k=2}=\log_{E}(P)^{2}\pmod{L^{\times}};$

  2. (b)

$\frac{d^{2}L_{p}({\mathbf{f}})(k,1)}{dk^{2}}\Big{|}_{k=2}=\mathcal{L}(E)\cdot\langle
P,P\rangle^{\operatorname{cyc}}\pmod{L^{\times}};$

  3. (c)

$\frac{d^{2}L_{p}({\mathbf{f}})(x_{0},s)}{ds^{2}}\Big{|}_{s=1}=\mathcal{L}(E)\cdot\langle
P,P\rangle^{\operatorname{Sch}}\pmod{L^{\times}}.$

###### Proof.

The first part follows from the main result of [BD2], and the other two are
[Ven, Theorems D and E]. We refer the reader to loc. cit. for a definition of
the corresponding pairings. ∎

IV. Results beyond modular forms of weight 2. The main result of Bertolini and
Darmon [BD2] was generalized by Seveso [Se] to modular forms of even weight.
Let us recall here his main result for the sake of completeness and to
illustrate that most of our results generalize to the situation of weights
$(k,1,1)$, by replacing the points over the elliptic curve by the
corresponding Heegner cycles. Let $f_{k}\in S_{k}(N)$, where $N=pN^{+}N^{-}$
and $N^{-}$ is the squarefree product of an odd number of prime factors. The
modular form corresponds, via the Jacquet-Langlands correspondence, to a
modular form on a certain Shimura curve $X=X_{N^{+},pN^{-}}$ uniformized by
the $p$-adic upper half-plane. In this framework, Iovita and Spiess [IS]
constructed a Chow motive $\mathcal{M}_{k-2}$ attached to modular forms on
$X$. Let $m=k/2-1$.

We fix $K/\mathbb{Q}$ a quadratic imaginary field extension, of discriminant
$D_{K}$ prime to $pN$, such that $N^{+}$ is a product of primes that are split
in $K$, while $pN^{-}$ is a product of primes that are inert in $K$; we
further fix an order of $K$ of conductor $c$ prime to $ND_{K}$. Hence, one may
consider a higher weight analogue of Heegner points, the Heegner cycles
$y_{\psi}^{(n)}\in{\mathrm{CH}}^{m+1}(\mathcal{M}_{n})$ attached to a
character $\psi$. The $p$-adic étale Abel-Jacobi map takes the form

${\mathrm{AJ}}_{p}:{\mathrm{CH}}^{m+1}(\mathcal{M}_{n})\rightarrow
M_{k}^{\vee}.$

In the following result, the Mazur–Kitagawa $p$-adic $L$-function is replaced
by the $p$-adic $L$-function attached to the quadratic imaginary field and the
character $\psi$, that we denote by $\mathcal{L}({\mathbf{f}}/K,\psi)(k,s)$
following the notations of [Se].

######  Proposition 2.4 (Seveso).

The first derivative of $\mathcal{L}(f/K,\psi)(k,s)$ in the weight direction
is given by

$2\frac{d}{dx}\Big{(}\mathcal{L}({\mathbf{f}}/K,\psi)(x,x/2)\Big{)}\Big{|}_{x=k}={\mathrm{AJ}}_{p}(y_{\psi}^{(n)})(f)+(-1)^{m}{\mathrm{AJ}}_{p}(y_{\bar{\psi}}^{(n)})(f).$

This suggests that some of our results can be transposed to a higher weight
situation, replacing the points over the elliptic curve by the corresponding
Heegner cycles. More precisely, the results relying on the work of Darmon,
Lauder and Rotger on the elliptic Stark conjecture [DLR1] can be adapted
following the generalizations of Gatti and Guitart to higher weights [GG].
Similarly, the construction of derived cohomology classes, anticipated in the
introduction and developed in Section 3, can be also carried out for general
weights $(k,1,1)$.

##  3\. Derived diagonal cycles and an explicit reciprocity law

###  3.1. Diagonal cycles and an explicit reciprocity law

Darmon and Rotger constructed in [DR3] an element

$\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\bf fgh}}^{{\dagger}})$

arising from the interpolation of diagonal cycles along the balanced region.
An alternative construction has been given by Bertolini, Seveso and Venerucci
[BSV1, Section 3]. This class is symmetric in all three variables. Let

$\mathrm{res}_{p}:H^{1}(\mathbb{Q},\mathbb{V}_{{\bf
fgh}}^{{\dagger}})\rightarrow H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\bf
fgh}}^{{\dagger}})$

denote the restriction map to the local cohomology at $p$, and set

$\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}):=\mathrm{res}_{p}(\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}))\in
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\bf fgh}}^{{\dagger}}).$

One of the main results of both [BSV1] and [DR3] is the proof of an explicit
reciprocity law. As showed in loc. cit., the Galois representation
$\mathbb{V}_{{\bf fgh}}^{{\dagger}}$ is endowed with a four-step filtration

(11) $0\subset\mathbb{V}_{{\bf fgh}}^{++}\subset\mathbb{V}_{{\bf
fgh}}^{+}\subset\mathbb{V}_{{\bf fgh}}^{-}\subset\mathbb{V}_{{\bf
fgh}}^{{\dagger}}$

by $G_{\mathbb{Q}_{p}}$-stable $\Lambda_{{\bf fgh}}$-submodules of ranks 0, 1,
4, 7 and 8, respectively. Moreover,

$\mathbb{V}_{{\bf fgh}}^{+}/\mathbb{V}_{{\bf
fgh}}^{++}=\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}\oplus\mathbb{V}_{{\mathbf{g}}}^{{\mathbf{h}}{\mathbf{f}}}\oplus\mathbb{V}_{{\mathbf{h}}}^{{\mathbf{f}}{\mathbf{g}}}.$

We discuss now the definition of
$\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$. Let
$\Theta_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$ be the $\Lambda_{{\bf
fgh}}$-adic cyclotomic character whose specialization at a point of weight
$(k,\ell,m)$ is $\varepsilon_{\operatorname{cyc}}^{t}$, with
$t:=(-k+\ell+m)/2$, and let $\psi_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$
be the unramified character of $G_{\mathbb{Q}_{p}}$ sending $\mathrm{Fr}_{p}$
to $\chi_{f}^{-1}(p){\bf a}_{p}({\mathbf{f}}){\bf
a}_{p}({\mathbf{g}})^{-1}{\bf a}_{p}({\mathbf{h}})^{-1}$. Define $\mathbb{U}$
as the unramified $\Lambda_{{\bf fgh}}$-adic representation of
$G_{\mathbb{Q}_{p}}$ given by several copies of the character
$\psi_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$, and let

$\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}=\mathbb{U}(\Theta_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}).$

We finally introduce the $\Lambda$-adic Dieudonné module

$\mathbb{D}(\mathbb{U}):=(\mathbb{U}\hat{\otimes}\mathbb{Z}_{p}^{\operatorname{nr}})^{G_{\mathbb{Q}_{p}}}.$

Then, one may construct a Perrin-Riou regulator map whose source is
$H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}})\rightarrow\Lambda_{{\bf
fgh}}$ and which interpolates either the Bloch–Kato logarithm or the dual
exponential map, according to the value of a certain Hodge–Tate weight. In
order to state their main properties, we need to introduce more terminology.
Let $c=\frac{k+\ell+m-2}{2}$, and with the previous notations, define

$\mathcal{E}^{\operatorname{PR}}(x,y,z)=\frac{1-p^{-c}\beta_{{\mathbf{f}}_{x}}\alpha_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}}{1-p^{-c}\alpha_{{\mathbf{f}}_{x}}\beta_{{\mathbf{g}}_{y}}\beta_{{\mathbf{h}}_{z}}}.$

The following result is discussed e.g. in [DR3, Proposition 5.6] and follows
from the general theory of Perrin-Riou maps. For this statement we implicitly
assume that neither
$\beta_{{\mathbf{f}}_{x}}\alpha_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}$
nor
$\alpha_{{\mathbf{f}}_{x}}\beta_{{\mathbf{g}}_{y}}\beta_{{\mathbf{h}}_{z}}$
vanish. If this were the case, and as we will later see, we need to work with
the expression

$\mathcal{E}^{\operatorname{PR}}(x,y,z)=\frac{1-\chi_{f}(p)p^{\frac{k-\ell-m}{2}}\alpha_{{\mathbf{f}}_{x}}^{-1}\alpha_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}}{1-\bar{\chi}_{f}(p)p^{\frac{\ell+m-k-2}{2}}\alpha_{{\mathbf{f}}_{x}}\alpha_{{\mathbf{g}}_{y}}^{-1}\alpha_{{\mathbf{h}}_{z}}^{-1}},$

which agrees with the former in the non-exceptional case.

######  Proposition 3.1.

There is a homomorphism (usually named Perrin-Riou regulator)

$\mathcal{L}_{{\mathbf{f}},{\mathbf{g}}{\mathbf{h}}}:H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}})\rightarrow\mathbb{D}(\mathbb{U})$

such that for all $\kappa_{p}\in
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}})$
the image $\mathcal{L}_{{\mathbf{f}},{\mathbf{g}}{\mathbf{h}}}(\kappa_{p})$
satisfies the following interpolation properties:

  1. (1)

For all points $(x,y,z)\notin\mathcal{W}_{{\bf fgh}}^{f}$,

$\nu_{x,y,z}(\mathcal{L}_{{\mathbf{f}},{\mathbf{g}}{\mathbf{h}}}(\kappa_{p}))=\frac{(-1)^{t}}{t!}\mathcal{E}^{\operatorname{PR}}(x,y,z)\cdot\log_{\operatorname{BK}}(\nu_{x,y,z}(\kappa_{p})),$

  2. (2)

For all points $(x,y,z)\in\mathcal{W}_{{\bf fgh}}^{f}$,

$\nu_{x,y,z}(\mathcal{L}_{{\mathbf{f}},{\mathbf{g}}{\mathbf{h}}}(\kappa_{p}))=(-1)^{t}\cdot(1-t)!\cdot\mathcal{E}^{\operatorname{PR}}(x,y,z)\cdot\exp_{\operatorname{BK}}^{*}(\nu_{x,y,z}(\kappa_{p})).$

Following [DR3], one can define

(12) $\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{f}\in
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}})$

as the projection of the local class
$\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ to
$\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{h}}}$. Let
$\eta_{{\mathbf{f}}^{*}}$, $\omega_{{\mathbf{g}}^{*}}$ and
$\omega_{{\mathbf{h}}^{*}}$ be the canonical differentials attached to Hida
families as introduced for instance in [KLZ, Section 10]. The following result
has been independently established in [BSV1, Thoerem A] and [DR3, Theorem 10].

######  Proposition 3.2.

For any triplet of $\Lambda$-adic test vectors
$(\tilde{{\mathbf{f}}},\tilde{{\mathbf{g}}},\tilde{{\mathbf{h}}})$, the
following equality holds in the ring of fractions of $\Lambda_{{\bf fgh}}$:

$\langle\mathcal{L}_{{\mathbf{f}},{\mathbf{g}}{\mathbf{h}}}(\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{f}),\eta_{\tilde{\mathbf{f}}^{*}}\otimes\omega_{\tilde{\mathbf{g}}^{*}}\otimes\omega_{\tilde{\mathbf{h}}^{*}}\rangle=\mathcal{L}_{p}^{f}(\tilde{\mathbf{f}},\tilde{\mathbf{g}},\tilde{\mathbf{h}}).$

######  Remark 3.3.

There exist analogue reciprocity laws for ${\mathscr{L}_{p}}^{g}$ and
${\mathscr{L}_{p}}^{h}$.

We can also formulate an explicit reciprocity law for the improved $p$-adic
$L$-function. Since along the region $k=\ell+m$ the Perrin-Riou map
interpolates the dual exponential, we have that

(13)
$\frac{1}{1-p^{-k+1}\alpha_{{\mathbf{f}}_{x}}\beta_{{\mathbf{g}}_{y}}\beta_{{\mathbf{h}}_{z}}}\cdot\langle\exp_{\operatorname{BK}}^{*}(\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{f}(x,y,z)),\eta_{\tilde{\mathbf{f}}_{x}^{*}}\otimes\omega_{\tilde{\mathbf{g}}_{y}^{*}}\otimes\omega_{\tilde{\mathbf{h}}_{z}^{*}}\rangle=\mathcal{L}_{p}^{f}(\tilde{\mathbf{f}}_{x},\tilde{\mathbf{g}}_{y},\tilde{\mathbf{h}}_{z})^{*},$

and in particular the dual exponential map vanishes at $(x_{0},y_{0},y_{0})$
(i.e. the class is crystalline) if and only if the improved $p$-adic
$L$-function is zero at that point.

######  Remark 3.4.

In [GGMR], the authors study the cohomology classes in a generic rank zero
situation, where they are non-crystalline. This yields a formula for the
special value ${\mathscr{L}_{p}}^{g}$ in terms of ${\mathscr{L}_{p}}^{f}$ in
absence of exceptional zeros. Again, the key point is that each component of
the cohomology class encodes information about a different $p$-adic
$L$-function.

###  3.2. Vanishing of cohomology classes

In [BSV1, Section 8.2], the authors deal with a situation where the numerator
of the Perrin-Riou map $\mathcal{L}_{{\mathbf{f}},{\mathbf{g}}{\mathbf{h}}}$
vanishes, defining an improved map whose derivatives may be explicitly
computed. We come back to this question later on. Let us analyze, firstly, the
vanishing of the denominator of the Perrin-Riou map, but in the case of the
Perrin-Riou map $\mathcal{L}_{{\mathbf{g}},{\mathbf{h}}{\mathbf{f}}}$, that
is:

(14)
$1-\bar{\chi}_{g}(p)p^{\frac{k-\ell+m-2}{2}}\alpha_{{\mathbf{f}}_{x}}^{-1}\alpha_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}^{-1}=0.$

Since we have placed ourselves in the ordinary setting, a necessary condition
for this to happen is $k+m=\ell+2$, which moreover suffices to guarantee the
analyticity of the Euler factor in the denominator.

Hence, when $f$ is of weight 2 with split multiplicative reduction at $p$, and
$g$ and $h$ are self-dual of the same weight ($h=g\otimes\chi_{g}^{-1}$), the
denominator of the Perrin-Riou map vanishes. This means that we expect

$\log_{\operatorname{BK}}(\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{g}(x,y,z))=0.$

However, the self-duality condition is not necessary for this vanishing, and
following the conventions of the introduction in the case of weights $(2,1,1)$
(again, with $E$ of split multiplicative reduction), it suffices to impose
that $\alpha_{g}\beta_{h}=1$. This encompasses for example the case of theta
series of quadratic fields where the prime $p$ is inert. Nevertheless, there
are certain phenomena which are exclusive from the self-dual case: indeed, the
fact that the Hida families interpolating both $g$ and $h$ keep the self-
duality condition gives us a vanishing along the whole line $(2,\ell,\ell)$.
We treat both the self-dual and the non self-dual case, emphasizing the main
differences between them.

We begin by showing that when $\alpha_{g}\beta_{h}=1$ and $g$ and $h$ are
self-dual, the local class $\kappa_{p}(f,g_{\alpha},h_{\alpha})$ vanishes,
using the techniques of our prior work [RiRo1]. Although this is not strictly
necessary since we will later see that the whole global class is zero, we
believe that it may be instructive for the reader to compare the formalism of
[RiRo1], which relies on the basic properties of the Perrin-Riou maps, with
the more conceptual proof of [BSV1, Section 8], based on the geometric
construction of an improved cohomology class.

######  Proposition 3.5.

With the running assumptions, the specialization of the $\Lambda$-adic
cohomology class $\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{g}}^{*})$ at
$(x_{0},y_{0},y_{0})$ vanishes, that is,
$\kappa_{p}(f,g_{\alpha},g_{1/\beta}^{*})=0$.

###### Proof.

We will follow the same strategy used in [RiRo1, Theorem 3.5]. First of all we
show, invoking [BSV1, Theorem 7.1], that any specialization of the three-
variable $\Lambda$-adic class at a point of weights $(2,\ell,\ell)$, with
$\ell\geq 2$, is zero. In order to achieve this, we just use the comparison
provided by the aforementioned result with the twisted class
$\kappa^{{\dagger}}$, twisting now in the $g$-variable, that is, applying the
operator $\operatorname{Id}\otimes w_{p}^{\prime}\otimes\operatorname{Id}$
according to the definitions given at the beginning of Section 7.2 of loc.
cit., where $w_{p}$ stands for the Atkin–Lehner involution. As we later
discuss, this class may be understood as an improved cohomology class, since
it agrees with the former up to multiplication by the Euler factor

$1-\frac{\bar{\chi}(p)\alpha_{{\mathbf{g}}_{y}}}{\alpha_{{\mathbf{f}}_{x}}\alpha_{{\mathbf{h}}_{z}}}p^{\frac{k-\ell+m-2}{2}}.$

This factor is zero over the line $(2,\ell,\ell)$ when we take Hida families
such that ${\mathbf{h}}={\mathbf{g}}^{*}$, since
$\bar{\chi}(p)\alpha_{{\mathbf{g}}_{y}}=\alpha_{{\mathbf{h}}_{y}}$. Observe
that we are implicitly using Lemma 8.4 of loc. cit., which assert that the
class $\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ is symmetric in all
three variables.

The second part of the proof consists on applying a limit argument
componentwise, via the corresponding Perrin-Riou maps, to conclude that the
limit when $\ell$ goes to one is also zero. For this last step, we look at the
four different components of the local class
$\kappa_{p}(f,g_{\alpha},g_{1/\beta}^{*})$ corresponding to the balanced
subspace $\mathbb{V}_{fgg^{*}}^{+}$. This suffices according to the results
established in [BSV1, Corollary 7.2] and following the notations of Section
6.2 in loc. cit., which asserts that the three-variable cohomology class lies
in the balanced subspace. The components of the balanced subspace are denoted
by $V_{f}^{gg^{*}}$, $V_{g}^{g^{*}f}$, $V_{g^{*}}^{fg}$ and
$V_{fgg^{*}}^{++}$, where $V_{f}^{gg^{*}}$ stands for the specialization of
$\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{g}}^{*}}$ and similarly for
the other factors (recall the filtration of (11)).

  * •

We first prove that the component associated to the rank one subspace
$V_{f}^{gg^{*}}$ is zero. Observe that along the line $(2,\ell,\ell)$, the
specialization of the module
$H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{f}}}^{{\mathbf{g}}{\mathbf{g}}^{*}})$
agrees with $H^{1}(\mathbb{Q}_{p},\mathbb{Z}_{p}(\psi_{g_{y}}^{-2})(\ell-1))$,
where $y$ is a point of weight $\ell$. Then, the Perrin-Riou map is an
application

(15)
$H^{1}(\mathbb{Q}_{p},\Lambda_{{\mathbf{g}}}(\psi_{{\mathbf{g}}}^{-2})\hat{\otimes}\Lambda(\underline{\varepsilon}_{\operatorname{cyc}}))\rightarrow\mathbb{D}(\Lambda_{{\mathbf{g}}}(\psi_{{\mathbf{g}}}^{-2}))\hat{\otimes}\Lambda.$

Since $\psi_{{\mathbf{g}}}^{-2}\neq 1$, we have
$H^{0}(\mathbb{Q}_{p},\Lambda_{{\mathbf{g}}}(\psi_{{\mathbf{g}}}^{-2}))=0$ and
it follows from [KLZ, Theorem 8.2.3] that the above map is an isomorphism.
Moreover, using the same argument of the proof of the last step of [RiRo1,
Theorem 3.5], we conclude that the $\Lambda$-module of (15) is non-canonically
isomorphic to $\Lambda_{{\mathbf{g}}}$. Therefore, and since infinitely many
specializations vanish according to the previously quoted result of [BSV1],
the corresponding $H^{1}$ is zero.

  * •

The components associated to $V_{g}^{g^{*}f}$ and $V_{g^{*}}^{fg}$ are zero;
this is because

$H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{g}}}^{{\mathbf{g}}^{*}{\mathbf{f}}}|_{(2,\ell,\ell)})\simeq
H^{1}(\mathbb{Q}_{p},\Lambda_{{\mathbf{g}}}(1))\simeq\Lambda_{{\mathbf{g}}}\oplus\Lambda_{{\mathbf{g}}},$

and although the Perrin-Riou map only kills one of the above two components,
the restriction of the class is zero since again infinitely many
specializations are zero.

  * •

For the remaining component, the one corresponding to
$\mathbb{V}_{fgg^{*}}^{++}$, the same argument used in the first step works
once we have established that the remaining projections vanish.

∎

Consider now the surface

$\mathcal{S}=\mathcal{S}_{k,k+m-2,m}:=\\{(x,y,z)\in\mathcal{W}_{{\mathbf{f}}}\times\mathcal{W}_{{\mathbf{g}}}\times\mathcal{W}_{{\mathbf{h}}}:\operatorname{wt}(x)+\operatorname{wt}(z)=\operatorname{wt}(y)+2\\},$

and also the line

$\mathcal{C}:=\\{(x,y,z)\in\mathcal{W}_{{\mathbf{f}}}\times\mathcal{W}_{{\mathbf{g}}}\times\mathcal{W}_{{\mathbf{h}}}:\operatorname{wt}(x)=2,\quad\operatorname{wt}(y)=\operatorname{wt}(z)\\}.$

Observe that the surface $\mathcal{S}$ is just a finite cover of the plane in
$\mathcal{W}^{3}$ arising as the Zariski closure of weights $(k,k+m-2,m)$.

Using the results of [BSV1, Section 8.2], we may upgrade Proposition 3.5 to
the vanishing of the global class $\kappa(f,g_{\alpha},h_{\alpha})$ when
$\alpha_{g}\beta_{h}=1$ (and hence we can work beyond the setting of the
adjoint, covering for example the case of theta series of quadratic fields
where the prime $p$ remains inert).

In particular, we have the following result.

######  Proposition 3.6.

The global class $\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ vanishes
along the line $\mathcal{C}$ in the self dual case. Moreover, the class
$\kappa(f,g_{\alpha},h_{1/\beta})$ is zero when $\alpha_{g}\beta_{h}=1$.

###### Proof.

Following again [BSV1, Section 8.2], there is an improved class
$\kappa_{g}^{*}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ along the surface
$\mathcal{S}$ satisfying

(16)
$\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})_{|\mathcal{S}}=\Big{(}1-\frac{\bar{\chi}(p)\alpha_{{\mathbf{g}}_{y}}}{\alpha_{{\mathbf{f}}_{x}}\alpha_{{\mathbf{h}}_{z}}}\Big{)}\kappa_{g}^{*}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}).$

Hence, the vanishing of $\kappa(f,g_{\alpha},h_{1/\beta})$ follows from the
vanishing of the corresponding Euler factor. ∎

###  3.3. Derived classes and reciprocity laws

Following the analogy with [RiRo1], let us focus firstly on the self-dual case
to discuss the notion of derived classes. We shrink the weight space
$\mathcal{W}$ to a rigid-analytic open disk $\mathcal{U}\subset\mathcal{W}$
centered at 2 at which the finite cover
$w:\mathcal{W}_{{\mathbf{f}}}\rightarrow\mathcal{W}$ restricts to an
isomorphism $w:\mathcal{U}_{{\mathbf{f}}}\xrightarrow{\sim}\mathcal{U}$ with
$x_{0}\in\mathcal{U}_{{\mathbf{f}}}$. Let
$\Lambda_{\mathcal{U}_{{\mathbf{f}}}}=\mathcal{O}(\mathcal{U}_{{\mathbf{f}}})$
denote the Iwasawa algebra of analytic functions on
$\mathcal{U}_{{\mathbf{f}}}$ whose supremum norm is bounded by $1$. Shrink
likewise $\mathcal{C}$ and $\mathcal{S}$ so that projection to the weight
space restricts to an isomorphism with $\mathcal{U}$ and
$\mathcal{U}\times\mathcal{U}$ respectively. Having done that, their
associated Iwasawa algebras are respectively
$\mathcal{O}(\mathcal{C})=\Lambda_{\mathcal{U}_{{\mathbf{f}}}}\simeq\mathbb{Z}_{p}[[X]]$
and
$\mathcal{O}(\mathcal{S})=\Lambda_{\mathcal{U}_{{\mathbf{f}}}}\hat{\otimes}\Lambda_{\mathcal{U}_{{\mathbf{h}}}}\simeq\mathbb{Z}_{p}[[X,Z]]$.
The isomorphism
$\Lambda_{\mathcal{U}_{{\mathbf{f}}}}\simeq\mathbb{Z}_{p}[[X]]$ is not
canonical and depends on the choice of an element
$\gamma\in\Lambda_{\mathcal{U}_{{\mathbf{f}}}}^{\times}$ which is sent to
$1+X$.

Then, consider the short exact sequence of $\mathbb{Z}_{p}$-modules

$0\rightarrow\mathbb{Z}_{p}[[X,Z]]\xrightarrow{\cdot
X}\mathbb{\mathbb{missing}}Z_{p}[[X,Z]]\rightarrow\mathbb{Z}_{p}[[Z]]\rightarrow
0.$

Under the usual isomorphisms, $\Lambda_{{\mathbf{f}}}$ may be identified with
$\mathbb{Z}_{p}[[X]]$ after fixing a topological generator $\gamma$ of
$\Lambda_{\mathcal{U}_{{\mathbf{f}}}}^{\times}$ and sending $[\gamma]$ to
$1+X$. Then, $\Lambda_{{\mathbf{f}}}\hat{\otimes}\Lambda_{{\mathbf{h}}}$
becomes isomorphic to $\mathbb{Z}_{p}[[X,Z]]$ and the previous exact sequence
may be recast as

(17)
$0\rightarrow\mathcal{O}_{\mathcal{S}}\xrightarrow{\delta}\mathcal{O}_{\mathcal{S}}\rightarrow\mathcal{O}_{\mathcal{C}}\rightarrow
0$

with $\delta=(\gamma-1)\otimes 1$ in
$\mathcal{O}_{\mathcal{S}}\simeq\Lambda_{{\mathbf{f}}}\hat{\otimes}\Lambda_{{\mathbf{h}}}$.

######  Proposition 3.7.

In the self-dual case, there exists a unique derived class
$\kappa_{\gamma}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{g}}^{*})\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\bf fgg^{*}}|\mathcal{S}})$ such that

$\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{g}}^{*})|_{\mathcal{S}}=\delta\cdot\kappa_{\gamma}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{g}}^{*}).$

###### Proof.

This follows by considering the long exact sequence in cohomology attached to
(17):

$H^{0}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{g}}^{*}|\mathcal{C}})\rightarrow
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{g}}^{*}|\mathcal{S}})\rightarrow
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{g}}^{*}|\mathcal{S}})\rightarrow
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{g}}^{*}|\mathcal{C}}).$

Since the restriction of $\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{g}}^{*})$
to
$H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{g}}^{*}|\mathcal{C}})$
is zero by Proposition 3.6, one may assure the existence of a derived class as
in the statement, which is moreover unique due to the vanishing of the $H^{0}$
for weight reasons (the Hodge–Tate weights corresponding to the balanced part
cannot be zero, as shown in [DR3, Corollary 5.3]). ∎

Normalizing by $\log_{p}(\gamma)$, the specializations of this class over the
line $(2,\ell,\ell)$ can be proved to be independent of the choice of
$\gamma$.

In general, if we are no longer in the self-dual case but the condition
$\alpha_{g}\beta_{h}=1$ still holds, the notion of derived class makes sense
at the point $(x_{0},y_{0},z_{0})$. For that purpose, let $\mathcal{D}$ stand
for the codimension two subvariety

$\mathcal{D}:=\\{(x,y,z)\in\mathcal{W}_{{\mathbf{f}}}\times\mathcal{W}_{{\mathbf{g}}}\times\mathcal{W}_{{\mathbf{h}}}:\operatorname{wt}(x)=\operatorname{wt}(y)+1,\quad
z=z_{0}\\}.$

The following result is the analogue of [RiRo1, Proposition 3.13] and its
proof follows from the same argument of Proposition 3.7.

######  Proposition 3.8.

Assume that $\alpha_{g}\beta_{h}=1$. Then, there exists a unique class
$\kappa_{\gamma}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\bf fgh}|\mathcal{D}})$ such that

$\kappa({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})|_{\mathcal{D}}=\delta\cdot\kappa_{\gamma}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}).$

Let
$\mathcal{L}=\frac{\alpha_{g}^{\prime}}{\alpha_{g}}-\frac{\alpha_{f}^{\prime}}{\alpha_{f}}$,
and consider the normalization of
$\kappa_{\gamma}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ by
$\gamma$, that is,

$\kappa^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})=\frac{\kappa_{\gamma}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})}{\log_{p}(\gamma)}.$

######  Theorem 3.9.

The logarithm of the derived class satisfies the following

$\langle\log_{\operatorname{BK}}(\kappa_{p}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{g}(x_{0},y_{0},z_{0})),\eta_{f}\otimes\omega_{g}\otimes\omega_{h}\rangle=\mathcal{L}\cdot{\mathscr{L}_{p}}^{g_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})(x_{0},y_{0},z_{0})\pmod{L^{\times}},$

where as in (12) the superindex $g$ refers to the projection to
$\mathbb{V}_{{\mathbf{g}}}^{{\mathbf{h}}{\mathbf{f}}}$.

###### Proof.

Consider the reciprocity law of Proposition 3.2, now for
${\mathscr{L}_{p}}^{g_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$,
restricted to $\mathcal{D}$, and multiply both sides by the Euler factor in
the denominator of the Perrin-Riou map. Then, we have an equality of the form

$\Big{(}1-\frac{\bar{\chi}(p)\alpha_{{\mathbf{g}}_{y}}}{p\alpha_{{\mathbf{f}}_{x}}\alpha_{{\mathbf{h}}_{z}}}\Big{)}\cdot\langle\log_{\operatorname{BK}}(\kappa_{p}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{g}),\eta_{{\mathbf{f}}}\otimes\omega_{{\mathbf{g}}}\otimes\omega_{{\mathbf{h}}}\rangle=\Big{(}1-\frac{\alpha_{{\mathbf{f}}_{x}}\alpha_{{\mathbf{h}}_{z}}}{\bar{\chi}(p)\alpha_{{\mathbf{g}}_{y}}}\Big{)}\cdot{\mathscr{L}_{p}}^{g_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}),$

since along $\mathcal{D}$ the Perrin-Riou interpolates the Bloch–Kato
logarithm. At the point $(x_{0},y_{0},z_{0})$ both the cohomology class at the
left hand side and the Euler factor at the right are zero. Taking derivatives
along the direction $(k+1,k,1)$, and evaluating then at the point
$(x_{0},y_{0},z_{0})$, the result follows (see [Ri1, Remark 4.8] for a more
exhaustive discussion on the identifications we are considering). ∎

An analogue formula holds for any point over the line $(2,\ell,\ell)$ in the
self-dual case, but of course the description of the $\mathcal{L}$-invariant
is not so explicit and relies on the results of [Se].

It may be instructive to compare this derived cohomology class with the
improved cohomology class considered by Bertolini, Seveso and Venerucci. We
can prove the following.

######  Proposition 3.10.

Consider the map given by the projection

$\phi_{g}^{hf}:H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{f}}{\mathbf{g}}{\mathbf{h}}|\mathcal{S}})\rightarrow
H^{1}(\mathbb{Q}_{p},\mathbb{V}_{{\mathbf{g}}}^{{\mathbf{h}}{\mathbf{f}}}|_{\mathcal{S}}).$

Then, there is a relation between the improved class
$\phi_{g}^{hf}(\kappa_{g}^{*}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}))$ and
$\phi_{g}^{hf}(\kappa^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}))$,
given by

$\phi_{g}^{hf}(\kappa^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}))=\mathcal{L}\cdot\phi_{g}^{hf}(\kappa_{g}^{*}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}}))\pmod{L^{\times}}.$

###### Proof.

This is proved by applying the map
$\langle\log_{\operatorname{BK}}(\cdot),\eta_{{\mathbf{f}}}\otimes\omega_{{\mathbf{g}}}\otimes\omega_{{\mathbf{h}}}\rangle$
to both sides, and then comparing the results. For that purpose, we use that
the Euler factors involved in the Perrin-Riou map are analytic along
$\mathcal{S}$ and can be cancelled out. That way, we obtain an improved
reciprocity law

${\mathscr{L}_{p}}^{g_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})(x,y,z)=\langle\log_{\operatorname{BK}}(\phi_{g}^{hf}(\kappa_{g}^{*}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{g}(x,y,z))),\eta_{{\mathbf{f}}_{x}}\otimes\omega_{{\mathbf{g}}_{y}}\otimes\omega_{{\mathbf{h}}_{z}}\rangle\pmod{L^{\times}},$

which holds for all the points $(x,y,z)$ of $\mathcal{S}$. ∎

Finally, we point out that we may expect a relation between
$\kappa_{p}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})(x,y,z)$ and the
Gross–Kudla–Schoen cycle of [DR1], that we denote by $\Delta_{k,\ell,m}\in
H^{1}(\mathbb{Q},V_{fgh}((4-k-\ell-m)/2)$. In particular, we expect the
following result to be true (or at least, a slight variant of it). Here,
$\operatorname{loc}_{p}$ stands for the localization at $p$-map.

######  Question 3.11.

Can we establish that, up to multiplication by a non-zero constant in
$L^{\times}$ and for any point $(x,y,z)$ of weights $(2,\ell,\ell)$ with
$\ell\geq 2$, we have the equality

$\kappa_{p}^{\prime}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})(x,y,z)=\mathcal{L}\cdot\operatorname{loc}_{p}(\Delta_{2,\ell,\ell})?$

Of course, this would require the proof of an analogue result to [DR1, Theorem
5.1] in a situation where $f$ has split multiplicative reduction.

##  4\. Derivatives of triple product p-adic L-functions

In this section, we discuss a variant of the elliptic Stark conjecture for the
derivative of the triple product $p$-adic $L$-function ${\mathscr{L}_{p}}^{f}$
in a situation of exceptional zeros. As before, we keep the assumption that
$f$ has split multiplicative reduction at $p$ and that an exceptional zero
condition occurs.

There are two main instances we want to consider: the rank zero situation and
the rank two situation. While the former is quite well understood after the
results developed in [BSV1] and [BSV2], the latter is more subtle and we will
propose a conjectural formula in this scenario. Along this section, by the
word rank, we refer to the rank of the $V_{gh}$-isotypic component of $E(H)$.
According to our general assumptions on the local signs, the rank is always
even. The $V_{gh}$-component of $E(H)$ is endowed with an inclusion in the
Selmer group, that is,

$\operatorname{Hom}_{G_{\mathbb{Q}}}(E(H),V_{gh})\simeq(E(H)\otimes
V_{gh}^{\vee})^{G_{\mathbb{Q}}}\subset
H_{\operatorname{f}}^{1}(\mathbb{Q},V_{fgh}),$

where $H_{\operatorname{f}}^{1}(\mathbb{Q},V_{fgh})$ is the group of
extensions of $\mathbb{Q}_{p}$ by $V_{fgh}$ in the category of
$\mathbb{Q}_{p}$-linear representations of $G_{\mathbb{Q}}$ which are
crystalline at $p$.

Recall that for higher ranks the computations performed in [DLR1] lead us to
expect that the special value ${\mathscr{L}_{p}}^{g_{\alpha}}$ presented in
the introduction is zero, and that the second derivative of
${\mathscr{L}_{p}}^{f}$ along the $f$-direction vanishes, too. The odd rank
situation is equally interesting, and we hope to come back to this question in
a further work. We keep the notations of the previous section.

######  Proposition 4.1.

The value
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})(x_{0},y_{0},z_{0})$
is zero. Moreover, the jacobian matrix of
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ at the point
$(x_{0},y_{0},z_{0})$ is given by

$(0\quad\mathcal{L}_{g_{\alpha}}-\mathcal{L}_{f}\quad\mathcal{L}_{h_{\alpha}}-\mathcal{L}_{f})\cdot{\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}.$

###### Proof.

This directly follows from [BSV1, Proposition 8.2]. ∎

######  Remark 4.2.

Observe that, towards the rationality conjectures we are interested in, the
value ${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}$ is
an algebraic number, and it is non-zero if and only if the cohomology class
$\kappa(f,g_{\alpha},h_{\alpha})$ is non-crystalline.

In particular, the derivative along the direction $(2+k,1,1)$ vanishes, and
along the direction $(2,1+\ell,1+\ell)$ is given by
$\mathcal{L}_{g_{\alpha}}+\mathcal{L}_{h_{\alpha}}-2\mathcal{L}_{f}$, up to an
explicit algebraic number in the number field $L$.

Suppose from now on that
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}$ vanishes
at $(x_{0},y_{0},z_{0})$. Therefore, the cohomology class
$\kappa(f,g_{\alpha},h_{\alpha})$ is crystalline, and following [BSV2, Section
2.1] we may define a new Bloch–Kato logarithm, denoted by $\log_{\beta\beta}$
in loc. cit. Roughly speaking, it can be understood as a projection to the
rank one subspace $V_{fgh}^{++}$ arising in the filtration (11), followed by
the Bloch–Kato logarithm and the pairing with
$\omega_{f}\otimes\omega_{g}\otimes\omega_{h}$. To be coherent with the other
notations we will need later on, write $\log^{++}$ for this map.
Alternatively, we may consider the local class
$\kappa_{p}(f,g_{\alpha},h_{\alpha})$ and take its decomposition according to
the action of the Frobenius element, in such a way that $\kappa_{\beta\beta}$
is the part corresponding to the $(\beta_{g},\beta_{h})$ component.

Assume further that $\alpha_{g}\alpha_{h}=1$ (in particular, this also implies
that $\beta_{g}\beta_{h}=1$). The following result is the content of [BSV2,
Section 2.1].

######  Proposition 4.3.

Under the given conditions, the value
${\mathscr{L}_{p}}^{f}({\mathbf{f}},g_{\alpha},h_{\alpha})$ vanishes and

$\frac{d^{2}}{dx^{2}}{\mathscr{L}_{p}}^{f}({\mathbf{f}},g_{\alpha},h_{\alpha})|_{x=x_{0}}=\frac{1}{2{\mathrm{ord}}_{p}(q_{E})}\cdot(1-p^{-1})^{-1}\cdot\log^{++}(\kappa_{p}(f,g_{\alpha},h_{\alpha})).$

######  Remark 4.4.

In the adjoint case, when we take $h_{1/\alpha}=g_{1/\beta}^{*}$ we do have a
relation between $\mathcal{L}_{g}$ and $\mathcal{L}_{h}$: indeed

$\frac{(1/\alpha_{g})^{\prime}}{1/\alpha_{g}}=-\frac{\alpha_{g}^{\prime}}{\alpha_{g}};$

however when $h_{1/\alpha}=g_{1/\alpha}^{*}$ both quantities are a priori
unrelated.

###  4.1. A conjecture for the second derivative

As we have discussed before, the improved $p$-adic $L$-function
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})^{*}$
interpolates an explicit non-zero multiple of $L(f\otimes g\otimes h,1)$, and
we expect this value to be zero when the rank of the corresponding isotypic
component of the Selmer group is two. In those cases, we would like to compare
the Kato class with a basis of $(E(H)\otimes V_{gh}^{\vee})^{G_{\mathbb{Q}}}$,
that we write as $\\{P,Q\\}$. We also assume that
$H_{\operatorname{f}}^{1}(\mathbb{Q},V_{fgh})$ has dimension 2.

To fix notations, observe that $V_{gh}$ decomposes as a
$G_{\mathbb{Q}_{p}}$-module as the direct sum of four different lines
$V_{gh}^{\alpha\alpha}:=V_{g}^{\alpha_{g}}\otimes
V_{h}^{\alpha_{h}},\ldots,V_{gh}^{\beta\beta}$. After choosing a basis of
$V_{gh}^{\vee}$, we may write this decomposition as

$V_{gh}=L\cdot e_{\alpha\alpha}^{\vee}\oplus L\cdot
e_{\alpha\beta}^{\vee}\oplus L\cdot e_{\beta\alpha}^{\vee}\oplus L\cdot
e_{\beta\beta}^{\vee},$

where

$\mathrm{Fr}_{p}(e_{\lambda\mu}^{\vee})=a_{\lambda\mu}\cdot
e_{\lambda\mu}^{\vee}\quad\text{ for any
}\quad\lambda,\mu\in\\{\alpha,\beta\\}.$

Here, $a_{\lambda\mu}=\beta_{g}\beta_{h}$ if $(\lambda,\mu)=(\alpha,\alpha)$
and similarly in the other three cases.

In particular, restricting the elements $\\{P,Q\\}$ to a decomposition group
at $p$ gives expressions

$P=P_{\beta\beta}\otimes e_{\beta\beta}^{\vee}+P_{\beta\alpha}\otimes
e_{\beta\alpha}^{\vee}+P_{\alpha\beta}\otimes
e_{\alpha\beta}^{\vee}+P_{\alpha\alpha}\otimes e_{\alpha\alpha}^{\vee},$

and similarly for $Q$, where as recalled in the introduction $\mathrm{Fr}_{p}$
acts on $P_{\beta\beta}$ with eigenvalue $\beta_{g}\beta_{h}$ and analogously
for the remaining components.

######  Conjecture 4.5.

Under the running assumptions, the following equality holds:

$\frac{d^{2}}{dx^{2}}{\mathscr{L}_{p}}^{f}({\mathbf{f}},g_{\alpha},h_{\alpha})|_{x=x_{0}}=\log_{p}(P_{\beta\beta})\log_{p}(Q_{\alpha\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\alpha\alpha})\pmod{L^{\times}}.$

This conjecture can be seen as a quite natural analogue for the first part of
Proposition 2.3; that is, we are proposing an expression for the second
derivative of the $p$-adic $L$-function along the line $(k,k/2)$ since the
central critical point corresponding to $(k,1,1)$ is precisely $k/2$. It would
be interesting to understand the derivatives along different directions; we
expect that they would be related with appropriate height pairings. See [CaHs]
for an approximation to that question when $g$ and $h$ are theta series
attached to the same quadratic imaginary field where the prime $p$ splits.

###  4.2. Some reducible cases

We continue by recalling some factorization formulas in special cases where
the representation $V_{gh}$ becomes reducible. See [DLR2, Section 2] for a
complete discussion of the different cases where this may occur.

A first case occurs when $g$ and $h$ are theta series of the same quadratic
field $K$, but the behavior is ostensibly different depending on whether $K$
is real or imaginary, and on whether $p$ is inert or split in $K$. While the
inert case was worked out in [BSV2], the split case was not considered in loc.
cit. However, it turns out that it is not specially interesting, at least when
$K$ is imaginary: the second derivative of ${\mathscr{L}_{p}}^{f}$ along the
$x$-direction is 0 for trivial reasons.

######  Remark 4.6.

It may be tempting to prove a factorization formula for
${\mathscr{L}_{p}}^{f}$ as in [CR], or even when all three variables
$(k,\ell,m)$ are allowed to move along a Hida family. However, the two-
variable Castella’s $p$-adic $L$-functions considered in loc. cit. would have
infinity types

$\Big{(}\frac{k+\ell+m}{2}-1,\frac{k+\ell+m}{2}-\ell-m+1\Big{)},\quad\Big{(}\frac{k+\ell+m}{2}-m,\frac{k+\ell+m}{2}-\ell\Big{)}.$

This precludes the possibility of comparing the different $p$-adic $L$-values
along the region of classical interpolation, since they are disjoint.

Finally, in the case where $h=g^{*}$, the situation is also quite simple and
the right hand of the conjecture is zero when the component corresponding to
the adjoint has rank two. For details on that, see the case by case analysis,
completely analogue to our situation, of [DR2].

######  Proposition 4.7.

Conjecture 4.5 holds whenever (a) $g$ and $h$ are theta series of an imaginary
quadratic field where $p$ splits, $V_{gh}=V_{\psi_{1}}\oplus V_{\psi_{2}}$,
with each component of rank one; (b) $g$ and $h$ are theta series of a
quadratic field where $p$ is inert, $V_{gh}=V_{\psi_{1}}\oplus V_{\psi_{2}}$,
and either $\psi_{1}$ or $\psi_{2}$ is a genus character.

###### Proof.

Consider first the case of imaginary quadratic fields, where we can prove that
both the left and the right hand side are zero for trivial reasons. For that
purpose, recall the notations introduced in the discussion before Proposition
4.3. In order to see that the second derivative vanishes, it is enough to
conclude that the component $\kappa_{\beta\beta}=0$, and this follows after
adapting the results of [DR2, Section 4.3] to the multiplicative situation,
where one may invoke the discussion of [CR]. In particular, if we assume
without loss of generality that $P_{\beta\alpha}\neq 0$, then
$P_{\alpha\alpha}=P_{\beta\beta}=0$, and similarly
$Q_{\alpha\beta}=Q_{\beta\alpha}=0$. See [GGMR, Section 4] for a similar
treatment of an analogue situation.

The case of theta series for quadratic fields where the prime is inert follows
from the main results of Bertolini–Seveso–Venerucci [BSV2, Section 3], taking
into account the identifications among the different eigenspaces for the
Frobenius action of e.g. [DLR1, Section 3.3] and [DR2, Sections 4.3, 4.4]. ∎

###  4.3. The conjecture in other settings

We would like to make some comments regarding the case
$\alpha_{g}\beta_{h}=1$. Observe that the previous Euler factor that gave rise
to the improved $p$-adic $L$-function does not vanish, but the factors

$1-\frac{\chi_{f}(p)\alpha_{{\mathbf{g}}_{y}}\beta_{{\mathbf{h}}_{z}}}{\alpha_{{\mathbf{f}}_{x}}}p^{\frac{k-\ell-m}{2}}\quad\text{
and }\quad
1-\frac{\chi_{f}(p)\beta_{{\mathbf{g}}_{y}}\alpha_{{\mathbf{h}}_{z}}}{\alpha_{{\mathbf{f}}_{x}}}p^{\frac{k-\ell-m}{2}}$

do. The first one is analytic along the region $\mathcal{S}_{k+m=\ell+2}$,
while the second is analytic along the region $\mathcal{S}_{k+\ell=m+2}$. In
this case we cannot assure the existence of an improved $p$-adic $L$-function,
but at least we can guarantee that
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ vanishes.

We get indeed a very similar result.

######  Proposition 4.8.

The value
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})(x_{0},y_{0},z_{0})=0$.
Moreover, the jacobian matrix of
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ at
$(x_{0},y_{0},z_{0})$ is given by an $L$-multiple of

$(0\quad\mathcal{L}_{g_{\alpha}}-\mathcal{L}_{f}\quad\mathcal{L}_{h_{\alpha}}-\mathcal{L}_{f}).$

Observe that Conjecture 4.5 still makes sense in this framework. And again, we
can also take the derivative along the line $(2+k,1+k,1)$ and we would expect
to relate it with an explicit multiple of an appropriate height pairing
$\langle P,P\rangle$.

##  5\. Applications to the elliptic Stark conjecture

###  5.1. Interplay between both settings and a conjecture of Darmon–Rotger

Let $H$ denote the smallest number field cut out by the representation
$V_{gh}$, with coefficients in a finite extension $L/\mathbb{Q}$. By enlarging
it if necessary, assume throughout that $L$ contains both the Fourier
coefficients of $g$ and $h$, and the roots of their $p$-th Hecke polynomials.
Fix a prime ideal $\wp$ of $H$ lying above $p$, thus determining an embedding
$H\subset H_{p}\subset\bar{\mathbb{Q}}_{p}$ of $H$ into its completion $H_{p}$
at $\wp$, and an arithmetic Frobenius
$\mathrm{Fr}_{p}\in\mathrm{Gal\,}(H_{p}/\mathbb{Q}_{p})$. Due to our
regularity assumptions, $V_{g}$ and $V_{h}$ decompose as

$V_{g}=V_{g}^{\alpha}\oplus V_{g}^{\beta},\quad V_{h}=V_{h}^{\alpha}\oplus
V_{h}^{\beta},$

where $\mathrm{Fr}_{p}$ acts on $V_{g}^{\alpha}$ with eigenvalue $\alpha_{g}$,
and similarly for the remaining summands.

Fix eigenbases $\\{e_{g}^{\alpha},e_{g}^{\beta}\\}$ and
$\\{e_{h}^{\alpha},e_{h}^{\beta}\\}$ of $V_{g}$ and $V_{h}$, respectively,
which are compatible with the choice of the basis for $V_{gh}$, i.e.,

$e_{\alpha\alpha}=e_{g}^{\alpha}\otimes e_{h}^{\alpha},\quad
e_{\alpha\beta}=e_{g}^{\alpha}\otimes e_{h}^{\beta},\quad
e_{\beta\alpha}=e_{g}^{\beta}\otimes e_{h}^{\alpha},\quad
e_{\beta\beta}=e_{g}^{\beta}\otimes e_{h}^{\beta}$

(recall that in previous sections we were using the dual basis). Let
$\eta_{g_{\alpha}}\in(H_{p}\otimes V_{g}^{\beta})^{G_{\mathbb{Q}_{p}}}$ and
$\omega_{h_{\alpha}}\in(H_{p}\otimes V_{g}^{\alpha})^{G_{\mathbb{Q}_{p}}}$
denote the canonical periods arising as the weight one specializations of the
$\Lambda$-adic periods $\eta_{{\mathbf{g}}}$ and $\omega_{{\mathbf{h}}}$
introduced in [KLZ, Section 10.1]. Then, we can define $p$-adic periods
$\Xi_{g_{\alpha}}\in H_{p}^{\mathrm{Fr}_{p}=\beta_{g}^{-1}}$ and
$\Omega_{h_{\alpha}}\in H_{p}^{\mathrm{Fr}_{p}=\alpha_{h}^{-1}}$ by setting

$\Xi_{g_{\alpha}}\otimes
e_{g}^{\beta}=\eta_{g_{\alpha}},\quad\Omega_{h_{\alpha}}\otimes
e_{h}^{\alpha}=\omega_{h_{\alpha}},$

and

(18)
$\mathcal{L}_{g_{\alpha}}:=\frac{\Omega_{g_{\alpha}}}{\Xi_{g_{\alpha}}}\in(H_{p})^{\mathrm{Fr}_{p}=\frac{\beta_{g}}{\alpha_{g}}}.$

At the same time, recall that $u_{g_{\alpha}}$ is the Stark unit attached to
the adjoint representation of $g_{\alpha}$, which arises as a normalization
term in the conjectures of [DLR1] and [DLR2] involving a second-order
regulator. Then, it was conjectured by Darmon and Rotger [DR2] that

(19) $\mathcal{L}_{g_{\alpha}}=\log_{p}(u_{g_{\alpha}})\pmod{L^{\times}}.$

This relation gives a relatively easy interpretation of the apparently
mysterious unit $u_{g_{\alpha}}$. This suggests that more natural descriptions
of this object should be available, involving only the arithmetic of the
modular form $g$. However, this conjecture seems to be hard to prove, even in
cases where the elliptic Stark conjecture is known (theta series of quadratic
imaginary fields where the prime $p$ splits). The main difficulty is the lack
of an explicit description of the periods $\Omega_{g_{\alpha}}$ and
$\Xi_{g_{\alpha}}$: in weights greater than one, these periods can be
understood as certain algebraic numbers and be explicitly described, but in
weight one this description is no longer available and $\Omega_{g_{\alpha}}$
and $\Xi_{g_{\alpha}}$ are $p$-adic transcendental numbers.

The main point of this section is that the knowledge of different conjectures
involving these periods can be enough to determine the value of the ratio
$\mathcal{L}_{g_{\alpha}}$. Indeed, the generalized cohomology classes
described in Section 3 can be decomposed as the sum of different components,
each one encoding information about different $p$-adic $L$-functions. When
combining these results, we may relate the different periods which are
involved.

As a first application of this technique, let us prove a result of this kind
using the theory of Beilinson–Flach elements. This corresponds to the limit
case where the modular form $f$ is Eisenstein and the arithmetic governing the
triple product are ostensibly different. For the following discussion, the
notations are the same of [RiRo2]. Let
$U_{gg^{*}}=\mathcal{O}_{H}^{\times}\otimes L$ and
$U_{gg^{*}}[1/p]=\mathcal{O}_{H}[1/p]^{\times}\otimes L$, and assume that the
hypothesis (H1)-(H3) of the introduction of [RiRo2] hold. Fix a basis
$\\{u,v\\}$ of the two dimensional space
$(U_{gg^{*}}[1/p]/p^{\mathbb{Z}}\otimes\operatorname{ad}^{0}(V_{g}))^{G_{\mathbb{Q}}}$
such that
$u\in(\mathcal{O}_{H}^{\times}\otimes\operatorname{ad}^{0}(V_{g}))^{G_{\mathbb{Q}}}$.
As in the case of elliptic curves, these unit groups are endowed with a
Frobenius action, since the restriction to a decomposition group allows us to
decompose $\operatorname{ad}^{0}(V_{g})=L^{1}\oplus L^{\alpha/\beta}\oplus
L^{\beta/\alpha}$ and we may take the projection of $u$ and $v$ to each of
those components. Let

$\begin{array}[]{ccc}R_{g_{\alpha}}&=&\log_{p}(u_{1})\log_{p}(v_{\alpha/\beta})-\log_{p}(v_{1})\log_{p}(u_{\alpha/\beta}),\\\
R_{g_{\beta}}&=&\log_{p}(u_{1})\log_{p}(v_{\beta/\alpha})-\log_{p}(v_{1})\log_{p}(u_{\beta/\alpha})\end{array}$

be the regulators which appear in the formulation of the main conjecture of
[DLR2] and [RiRo2].

######  Proposition 5.1.

Assume that $R_{g_{\alpha}}$ and $R_{g_{\beta}}$ are both non-zero. Then,

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

###### Proof.

Recall the maps $\log^{+-}$ and $\log^{-+}$ introduced in [RiRo2, Section 3.3]
as the composition of the corresponding projection maps from $V_{gh}$, the
Bloch–Kato logarithm, and the pairing with the canonical differentials. Apply
then [RiRo2, Proposition 4.3] twice, first with the map $\log^{-+}$ (and hence
taking the $\beta/\alpha$ component of both $u$ and $v$), and then with the
map $\log^{+-}$ (taking the $\alpha/\beta$ component of both $u$ and $v$).
Then, comparing both expressions we have that

$\Xi_{g_{\alpha}}\cdot\Omega_{g_{1/\alpha}^{*}}\cdot\log_{p}(u_{g_{\alpha}})=\Omega_{g_{\alpha}}\cdot\Xi_{g_{1/\alpha}^{*}}\cdot\log_{p}(u_{g_{\beta}})\pmod{L^{\times}}.$

We now proceed as in [RiRo1, Section 5.2] (see the discussion after display
(56)), observing that

$\Omega_{g_{1/\alpha}^{*}}=\Xi_{g_{\beta}}^{-1},\quad\Xi_{g_{1/\alpha}^{*}}=\Omega_{g_{\beta}}^{-1}\pmod{L^{\times}},$

and we are done. ∎

We would like to go a step beyond and aim for stronger results, so in a
certain way we would like to keep the period attached to $h$ fixed and vary
just the one attached to $g$, which would yield the desired equality.

We do this by analyzing first the prototypical case of the elliptic Stark
conjecture, where the Selmer group is two-dimensional and we may fix a basis
$\\{P,Q\\}$ of the $L$-vector space

$(E(H)\otimes V_{gh}^{\vee})^{G_{\mathbb{Q}}}.$

For the following Proposition we assume the hypothesis discussed in the
introduction of [DLR1], and in particular, that $L(f\otimes g\otimes h,1)=0$.
Recall the decomposition

$P=P_{\beta\beta}\otimes e_{\beta\beta}^{\vee}+P_{\beta\alpha}\otimes
e_{\beta\alpha}^{\vee}+P_{\alpha\beta}\otimes
e_{\alpha\beta}^{\vee}+P_{\alpha\alpha}\otimes e_{\alpha\alpha}^{\vee},$

and similarly for $Q$.

Define the regulators

$\operatorname{Reg}_{g_{\alpha}}(V_{gh})=\log_{p}(P_{\beta\beta})\log_{p}(Q_{\beta\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\beta\alpha})$

and

$\operatorname{Reg}_{f}(V_{gh})=\log_{p}(P_{\beta\beta})\log_{p}(Q_{\alpha\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\alpha\alpha}).$

To shorten notations, write

$\log^{-+}(\kappa)=\langle\log_{\operatorname{BK}}(\kappa_{p}^{g}),\omega_{f}\otimes\eta_{g}\otimes\omega_{h}\rangle,$

and whenever $\kappa$ is crystalline, write $\log^{++}$ for the Bloch–Kato
logarithm of [BSV2, Section 2.1], as recalled before during the proof of
Proposition 4.3.

######  Proposition 5.2.

Assume that
$\operatorname{Reg}_{g_{\alpha}}(V_{gh}),\,\operatorname{Reg}_{f}(V_{gh})\neq
0$. Suppose that two of the following three equalities are true modulo
$L^{\times}$. Then, the third automatically holds.

  1. (a)

${\mathscr{L}_{p}}^{g_{\alpha}}(f,g_{\alpha},h_{\alpha})=\frac{\log_{p}(P_{\beta\beta})\log_{p}(Q_{\beta\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\beta\alpha})}{\log_{p}(u_{g_{\alpha}})}.$

  2. (b)

$\frac{\partial^{2}{\mathscr{L}_{p}}^{f}({\mathbf{f}},g_{\alpha},h_{\alpha})}{\partial
x^{2}}\Big{|}_{x=x_{0}}=\log_{p}(P_{\beta\beta})\log_{p}(Q_{\alpha\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\alpha\alpha}).$

  3. (c)

$\mathcal{L}_{g_{\alpha}}=\log_{p}(u_{g_{\alpha}}).$

###### Proof.

The proof is based on the study of the local cohomology class
$\kappa_{p}(f,g_{\alpha},h_{\alpha})$ introduced in the preceding sections.

Observe that (a) and (b) are equivalent to

$\log^{-+}(\kappa_{p}(f,g_{\alpha},h_{\alpha}))=\frac{\log_{p}(P_{\beta\beta})\log_{p}(Q_{\beta\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\beta\alpha})}{\log_{p}(u_{g_{\alpha}})}\pmod{L^{\times}}$

and

$\log^{++}(\kappa_{p}(f,g_{\alpha},h_{\alpha}))=\log_{p}(P_{\beta\beta})\log_{p}(Q_{\alpha\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\alpha\alpha})\pmod{L^{\times}},$

respectively, by virtue of the explicit reciprocity laws of [BSV1] (both in
the usual version and improved version based on the techniques of Venerucci).

Let us define the local class

(20)
$\kappa_{0}=\frac{1}{\Xi_{g_{\alpha}}\cdot\Omega_{h_{\alpha}}}\cdot\frac{1}{\log_{p}(u_{g_{\alpha}})}\cdot(\log_{p}(P_{\beta\beta})\cdot
Q-\log_{p}(Q_{\beta\beta})\cdot P),$

where we have implicitly identified a point over the elliptic curve with its
image under the Kummer map; take then $\tilde{\kappa}=\kappa-\kappa_{0}$. The
element $\tilde{\kappa}$ clearly belongs to the kernel of the Bloch–Kato
logarithm $\log^{-+}$, that we have defined by

$\log^{-+}:H^{1}(\mathbb{Q}_{p},V_{fgh})\xrightarrow{\operatorname{pr}^{-+}}H^{1}(\mathbb{Q}_{p},V_{f}\otimes
V_{gh}^{\alpha\beta})\rightarrow\mathbb{C}_{p},$

the last map being the composition of the Perrin-Riou map and the pairing with
the differentials
$\omega_{f}\otimes\eta_{g_{\alpha}}\otimes\omega_{h_{\alpha}}$. Then,
$\tilde{\kappa}=\lambda(\log_{p}(P_{\beta\alpha})\cdot
Q-\log_{p}(Q_{\beta\alpha})\cdot P)$. But observe that by [BSV1, Corollary
7.2] we know that the cohomology class $\kappa$ lies in the balanced part for
the filtration attached to $H^{1}(\mathbb{Q}_{p},V_{fgh})$ and hence
$\tilde{\kappa}$ lies in the kernel of the map $\log^{--}$

$\log^{--}:H^{1}(\mathbb{Q}_{p},V_{fgh})\xrightarrow{\operatorname{pr}^{--}}H^{1}(\mathbb{Q}_{p},V_{f}\otimes
V_{gh}^{\alpha\alpha})\rightarrow\mathbb{C}_{p}.$

Hence, the non-vanishing of the regulator
$\operatorname{Reg}_{g_{\alpha}}(V_{gh})$, implies that $\tilde{\kappa}=0$ and
therefore $\kappa=\kappa_{0}$.

From the same argument and under the assumption that
$\operatorname{Reg}_{f}(V_{gh})$, the second equation yields

(21)
$\kappa=\frac{1}{\Omega_{g_{\alpha}}\cdot\Omega_{h_{\alpha}}}\cdot(\log_{p}(P_{\beta\beta})\cdot
Q-\log_{p}(Q_{\beta\beta})\cdot P)\pmod{L^{\times}},$

where again we have identified the points with their image under the Kummer
map.

Now the statement is clear. For instance, if both (a) and (b) are true,
comparing the previous expressions, we get

$\log_{p}(u_{g_{\alpha}})=\frac{\Xi_{g_{\alpha}}}{\Omega_{g_{\alpha}}}\pmod{L^{\times}}.$

The proof of the other implications is equally straightforward. ∎

It would be interesting to prove an analogue result in a more general
situation, beyond the case of split multiplicative reduction. The discussion
around cohomology classes is still valid, but the point is that one needs a
replacement for the results expressing the second derivative of
${\mathscr{L}_{p}}^{f}$ in terms of the Bloch–Kato logarithm of the cohomology
class. While we can assure that the special value ${\mathscr{L}_{p}}^{f}$ is
zero, it is not clear how to proceed with its derivatives.

######  Question 5.3.

Is there a reciprocity law relating the second derivative of
${\mathscr{L}_{p}}^{f}$ (or some variation of it) with the logarithm
$\log^{--}$ of the cohomology class $\kappa(f,g,h)$ in a generic situation
(non exceptional zeros)?

###  5.2. Case (a)

We assume first that $\alpha_{g}\alpha_{h}=1$. The results we have proved
until now showing a deep interaction between the value of the derivatives of
${\mathscr{L}_{p}}^{f}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$ and the value
of ${\mathscr{L}_{p}}^{g_{\alpha}}({\mathbf{f}},{\mathbf{g}},{\mathbf{h}})$
may be applied to study new instances of the elliptic Stark conjecture.

Let us analyze some particular cases describing the exact shape of the
generalized cohomology classes. For example, according to the results of
[BSV1], when $g$ is a theta series attached to a quadratic field where the
prime $p$ is inert and $V_{gh}=V_{\psi_{1}}\oplus V_{\psi_{2}}$ with
$\psi_{1}$ being a genus character, we have

$\frac{d^{2}{\mathscr{L}_{p}}^{f}({\mathbf{f}},g_{\alpha},h_{\alpha})}{dx^{2}}\Big{|}_{x=x_{0}}=\log^{++}(\kappa_{p}(f,g_{\alpha},h_{\alpha}))=\log(P_{\psi_{1}}^{+})\cdot\log(P_{\psi_{2}}^{+})\pmod{L^{\times}},$

where $P_{\psi_{i}}^{+}=P_{\psi_{i}}+\sigma_{p}P_{\psi_{i}}$, being
$\sigma_{p}\in\mathrm{Gal\,}(H/\mathbb{Q})$ a Frobenius element at $p$.

######  Remark 5.4.

This situation occurs in general when at least one of $\psi_{1}$ or $\psi_{2}$
is a genus character. See for example the discussion after [DLR1, Lemma 3.10]
where the authors explain how the regulator of the elliptic Stark conjecture
admits a particularly simple expression in this case.

However, from the results we already know around the elliptic Stark
conjecture, one obtains that

(22)
${\mathscr{L}_{p}}^{g_{\alpha}}(f,g_{\alpha},h_{\alpha})=\frac{\log(P_{\psi_{1}}^{+})\cdot\log(P_{\psi_{2}}^{-})}{\mathcal{L}_{g_{\alpha}}}\pmod{L^{\times}},$

where with the previous notations,
$P_{\psi_{i}}^{-}=P_{\psi_{i}}-\sigma_{p}P_{\psi_{i}}$. This is quite
significant, since it establishes the elliptic Stark conjecture only up to a
conjecture about periods of weight one modular forms.

######  Corollary 5.5.

Let $g$ and $h$ be theta series attached to a quadratic field (either real or
imaginary) where the prime $p$ remains inert, with $V_{gh}=V_{\psi_{1}}\oplus
V_{\psi_{2}}$ and $\psi_{1}$ or $\psi_{2}$ being a genus character. Then, the
elliptic Stark conjecture of [DLR1] is equivalent to the conjecture about
periods of [DR2].

###### Proof.

This follows from the fact that part (b) of Proposition 5.2 holds in this
setting. ∎

Moreover, we expect conjectural expressions for the generalized Kato classes.
In particular, the previous result suggests the following conjecture.

######  Proposition 5.6.

In the setting of Proposition 5.2, if the formulas which appear in that
statement are satisfied, then the equality

$\kappa(f,g_{\alpha},h_{1/\alpha})=\frac{1}{\Omega_{g_{\alpha}}\cdot\Omega_{h_{\alpha}}}\cdot(\log_{p}(P_{\beta\beta})\cdot
Q-\log_{p}(Q_{\beta\beta})\cdot P),$

holds in $H_{\operatorname{f}}^{1}(\mathbb{Q},V_{fgh})$ up to multiplication
by $L^{\times}$.

###### Proof.

This follows verbatim the proof of Proposition 5.2, using the third statement
to simplify the different period relations. ∎

The same result holds for $\kappa(f,g_{\beta},h_{1/\alpha})$.

###  5.3. Case (b)

In the case where $\alpha_{g}\beta_{h}=1$, the explicit reciprocity law gives
a connection between ${\mathscr{L}_{p}}^{g_{\alpha}}$ and the Bloch–Kato
logarithm of $\kappa(f,g_{\alpha},h_{1/\beta})$, but unfortunately both the
latter class and one of the Euler factors involved in the equality vanish.
Therefore, that result is meaningless in this setting.

In previous sections we saw how to overcome that difficulty, proving a derived
reciprocity law after having observed that certain Euler factors are analytic
along the line $k+m=\ell+2$. There are two natural directions for considering
the derivative over that plane (although of course it makes sense to take any
combination of them): the line $(2,\ell,\ell)$ and the line $(k+1,k,1)$; the
former is not quite interesting since both the class
$\kappa(f,g_{\alpha},g_{1/\beta}^{*})$ and the Euler factor in the denominator
of the Perrin-Riou map vanish identically. Hence, we may take derivative along
$(k+1,k,1)$ and we get an equality of the form

$\mathcal{L}\cdot{\mathscr{L}_{p}}^{g_{\alpha}}(f,g_{\alpha},h_{1/\beta})=\log^{-+}(\kappa_{p}^{\prime}(f,g_{\alpha},h_{1/\beta}))\pmod{L^{\times}},$

where $\mathcal{L}$ is the $\mathcal{L}$-invariant which already appeared in
previous sections. Hence, if the elliptic Stark conjecture for
${\mathscr{L}_{p}}^{g_{\alpha}}$ were true, the class
$\kappa_{p}^{\prime}(f,g_{\alpha},h_{\alpha})$ could be expressed as a
combination of points, normalized by appropriate $\mathcal{L}$-invariants. In
particular, this would yield an equality of the form

(23)
$\kappa^{\prime}(f,g_{\alpha},h_{1/\beta})=\frac{\mathcal{L}}{\Xi_{g_{\alpha}}\cdot\Omega_{h_{1/\beta}}}\cdot\frac{\log_{p}(P_{\beta\beta})\cdot
Q-\log_{p}(Q_{\beta\beta})\cdot
P}{\log_{p}(u_{g_{\alpha}})}\pmod{L^{\times}}.$

One may obtain a symmetric expression for
$\kappa^{\prime}(f,g_{\beta},h_{1/\alpha})$. Recall that this is the analogue
of [RiRo1, Theorem B].

######  Conjecture 5.7.

The equality

$\kappa^{\prime}(f,g_{\alpha},h_{1/\beta})=\frac{\mathcal{L}}{\Xi_{g_{\alpha}}\cdot\Omega_{h_{1/\beta}}}\cdot\frac{\log_{p}(P_{\beta\beta})\cdot
Q-\log_{p}(Q_{\beta\beta})\cdot P}{\log_{p}(u_{g_{\alpha}})}\pmod{L^{\times}}$

holds in $H_{\operatorname{f}}^{1}(\mathbb{Q},V_{fgh})$.

As it was pointed out before, in the self dual case the product
$\Xi_{g_{\alpha}}\Omega_{h_{1/\beta}}$ is an element of $L^{\times}$.

We finish our work with the following result.

######  Proposition 5.8.

Assume that Conjecture 5.7 is true. Then, the special value
${\mathscr{L}_{p}}^{g_{\alpha}}$ satisfies

${\mathscr{L}_{p}}^{g_{\alpha}}(f,g_{\alpha},h_{\alpha})=\frac{\log_{p}(P_{\beta\beta})\log_{p}(Q_{\beta\alpha})-\log_{p}(Q_{\beta\beta})\log_{p}(P_{\beta\alpha})}{\log_{p}(u_{g_{\alpha}})}\pmod{L^{\times}}.$

###### Proof.

This follows by applying the Bloch–Kato logarithm $\log^{-+}$ to the
cohomology class $\kappa^{\prime}(f,g_{\alpha},h_{1/\beta})$, and using the
derived reciprocity law of Theorem 3.9. ∎

The converse can also be established with some extra assumptions, including
the conjecture about periods of [DR2].

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