# The exceptional zero phenomenon for elliptic units

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

The exceptional zero phenomenon has been widely studied in the realm of
$p$-adic $L$-functions, where the starting point lies in the foundational
works of Ferrero–Greenberg, Katz, Gross, or Mazur–Tate–Teitelbaum, among
others. This phenomenon also appears in the study of Euler systems, and in
this case one is led to study higher order derivatives of cohomology classes
in order to extract the arithmetic information which is usually encoded in the
explicit reciprocity laws which make the connection with $p$-adic
$L$-funtions. In this work, we focus on the elliptic units of an imaginary
quadratic field and study this exceptional zero phenomenon, proving an
explicit formula relating the logarithm of a derived elliptic unit either to
special values of Katz’s two variable $p$-adic $L$-function or to its
derivatives. Further, we interpret this fact in terms of an
$\mathcal{L}$-invariant and relate this result to other approaches to the
exceptional zero phenomenon, most notably to the work of Bley, and we also
compare this setting with other scenarios concerning Heegner points and
Beilinson–Flach elements.

###### 2010 Mathematics Subject Classification:

11G16; 11F67

###### Contents

  1. 1 Introduction
  2. 2 Circular units
    1. 2.1 Leopoldt’s formula
    2. 2.2 Exceptional zeros and circular units
  3. 3 Elliptic units
    1. 3.1 Elliptic units
    2. 3.2 Katz’s two-variable p-adic L-function of an imaginary quadratic field
    3. 3.3 A reciprocity law for elliptic units
  4. 4 Exceptional zeros and elliptic units
    1. 4.1 Specialization at the character $\psi$
    2. 4.2 Specialization at the character $\psi\mathcal{N}$
    3. 4.3 Interactions with the theory of circular units
  5. 5 The exceptional zero phenomenon: Beilinson–Flach elements and elliptic units
    1. 5.1 Elliptic units and Beilinson–Flach elements
    2. 5.2 Beilinson–Flach elements and exceptional zeros

##  1\. Introduction

Since the introduction of the exceptional zero phenomenon for the
Kubota–Leopoldt $p$-adic $L$-function by Ferrero and Greenberg [FG] and for
the $p$-adic $L$-function attached to an elliptic curve by Mazur, Tate and
Teitelbaum [MTT], a lot of progress has been made in the study of this topic.
The main goal of this article is to study exceptional zero phenomena for
Katz’s two-variable $p$-adic $L$-function at points lying outside the region
of classical interpolation, where the Euler system of elliptic units vanishes.
Hence, our setting departs notably from loc. cit. and is closer in spirit to
the study of e. g. [Cas] and [RiRo1], sharing some points in common with
earlier work of Solomon [Sol] and Bley [Bl], and also with the recent preprint
of Büyükboduk and Sakamoto [BS].

Fix once and for all a prime $p$ and a quadratic imaginary field $K$ in which
$p$ splits, and fix embeddings
$\mathbb{C}\hookleftarrow\bar{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$. Let
$\mathfrak{p}$ and $\bar{\mathfrak{p}}$ be the two prime ideals lying over
$p$, with $\mathfrak{p}$ the prime above $p$ induced by the previously fixed
embedding $\bar{\mathbb{Q}}\hookrightarrow\mathbb{C}_{p}$ (assume in the
introduction for notational simplicity that both ideals are principal, and
with a slight abuse of notation we denote by $\mathfrak{p}$ and
$\bar{\mathfrak{p}}$ their generators). Let $K_{\infty}^{\operatorname{cyc}}$
and $K_{\infty}^{\operatorname{ac}}$ be the cyclotomic and anticyclotomic
$\mathbb{Z}_{p}$-extensions of $K$, respectively, and set
$K_{\infty}=K_{\infty}^{\operatorname{ac}}K_{\infty}^{\operatorname{cyc}}$.
Denote $\Gamma_{K}=\mathrm{Gal\,}(K_{\infty}/K)$ and
$\Lambda_{K}=\mathbb{Z}_{p}[[\Gamma_{K}]]$. The latter is a Galois module with
an appropriate tautological action that we later recall. The weight space is
the formal spectrum $\operatorname{Spf}(\Lambda_{K})$ of the two-variable
Iwasawa algebra $\Lambda_{K}$. Let $\psi$ stand for a Hecke character of
finite order, conductor $\mathfrak{n}$ and taking values in a number field
$F$, and let $\mathcal{N}$ denote the norm character of $K$. We denote by $N$
the norm of $\mathfrak{n}$, and assume that $(N,p)=1$. Finally, let $F_{p}$
denote a completion of $F$ at the prime $p$. The Hecke character $\psi$ may be
also understood as a Galois character $\psi:G_{K}\rightarrow F^{\times}$; the
notation $\psi^{\prime}$ will be used to designate the composition of $\psi$
with conjugation by the non-trivial element in $\mathrm{Gal\,}(K/\mathbb{Q})$:
$\psi^{\prime}(\sigma)=\psi(\tau\sigma\tau^{-1})$, where $\tau$ is any element
of $G_{\mathbb{Q}}$ which acts non-trivially on $K$.

As an additional piece of notation, let $H_{\infty}$ denote the unique
$\mathbb{Z}_{p}$-extension of $K$ in which $\bar{\mathfrak{p}}$ is unramified
(therefore, the prime $\mathfrak{p}$ is ramified in $H_{\infty}/K$). We choose
a Galois character $\lambda$ of $\Gamma_{K}$, so that it factors through
$\mathrm{Gal\,}(H_{\infty}/K)$, defining an isomorphism

$\mathrm{Gal\,}(H_{\infty}/K)\longrightarrow 1+p\mathbb{Z}_{p}.$

The choice of $\lambda$ is unique once we require that it is the Galois
representation corresponding to a Grössencharacter for $K$ of infinity type
$(1,0)$. We define in the same way an extension $H_{\infty}^{\prime}$ and a
character $\lambda^{\prime}$, exchanging the roles of $\mathfrak{p}$ and
$\bar{\mathfrak{p}}$. Although characters of $\Gamma_{K}$ are elements of
$\operatorname{Hom}_{\operatorname{cont}}(G_{K},\bar{\mathbb{Q}}_{p}^{\times})$,
we are interested in the restriction to the subspace of
$\operatorname{Hom}_{\operatorname{cont}}(G_{K},\bar{\mathbb{Q}}_{p}^{\times})$
given by

$\Sigma_{\psi}:=\\{\psi\xi\mathcal{N}^{s}\lambda^{t}\text{ such that
}(s,t)\in\mathbb{Z}_{p}^{2}\\},$

where $\xi$ is a finite order character of $p$-power conductor.

One can naturally associate, via the theory of elliptic units of Robert [Rob]
(see also de Shalit and Yager’s approaches [deS], [Yag]) and Kummer maps, a
global cohomology class to $\psi$

(1) $\kappa_{\psi,\infty}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})).$

We denote by $\kappa_{\psi}\in H^{1}(K,F_{p}(\psi^{-1}))$ and
$\kappa_{\psi\mathcal{N}}\in H^{1}(K,F_{p}(\psi^{-1})(\mathcal{N}))$ the
specializations of $\kappa_{\psi,\infty}$ at $\psi$ and $\psi\mathcal{N}$,
respectively. In section 4, we prove that $\kappa_{\psi}$ never vanishes,
while $\kappa_{\psi\mathcal{N}}$ vanishes if and only if
$\psi(\mathfrak{p})=1$ or $\psi(\bar{\mathfrak{p}})=1$. In these cases, there
exists a notion of derived cohomology class along the subvariety
$\mathcal{C}^{\prime}$ of weight space, that we define as the Zariski closure
of the points $\mathcal{N}\bar{\lambda}^{t}$ with $t\in\mathbb{Z}^{\geq 0}$,
in an appropriate sense we make precise later on,

$\kappa_{\psi,\infty}^{\prime}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}}).$

The specialization of this cohomology class at $\psi\mathcal{N}$ encodes the
arithmetic information which is given by $\kappa_{\psi\mathcal{N}}$ in a non-
exceptional situation. To be more explicit, denote by $u_{\mathfrak{n}}$ a
fixed choice of an elliptic unit of conductor $\mathfrak{n}$, and to lighten
notations, define

(2) $u_{\psi}=\prod_{\sigma\in\mathrm{Gal\,}(K_{\mathfrak{n}}/K)}(\sigma
u_{\mathfrak{n}})^{\psi^{-1}(\sigma)}\in(\mathcal{O}_{K_{\mathfrak{n}}}^{\times}\otimes
F)^{\psi},$

where $K_{\mathfrak{n}}$ is the field cut out by $\psi$. We expect a relation
between $\kappa_{\psi\mathcal{N}}^{\prime}$ and $u_{\psi}$, since they both
lie in the same space (after applying a Kummer map), and this is the content
of the main result of this note, which we now state.

Assume that $\psi(\mathfrak{p})=1$, and let us define the following
$\mathcal{L}$-invariant

(3)
$\mathcal{L}(\psi)=(1-\psi^{-1}(\bar{\mathfrak{p}}))\cdot\log_{p}(\mathfrak{p}),$

where $\log_{p}$ stands for the usual $p$-adic Iwasawa logarithm.

Then, we have the following result, proved in Section 4.2.

######  Theorem 1.1.

Suppose that $\psi(\mathfrak{p})=1$. Then,

(4) $\kappa_{\psi\mathcal{N}}^{\prime}=\mathcal{L}(\psi)\cdot u_{\psi}.$

Although the previous result does not require any explicit mention to the
theory of $p$-adic $L$-functions, it is fair to say that Katz’s two variable
$p$-adic $L$-function plays a prominent role in our results. The interplay
between the Euler system of elliptic units and Katz’s two variable $p$-adic
$L$-function can be set as a very particular case of a wider theory. One may
distinguish two main approaches to construct a $p$-adic $L$-function.

  1. (a)

Firstly, interpolating the algebraic parts of the special values of the
classical $L$-function along the so-called critical region. This requires, as
a starting point, the proof of certain algebraicity results.

  2. (b)

Secondly, as the image under a certain Perrin-Riou map of a family of
cohomology classes, constructed along the so-called geometric region. These
classes are typically obtained as the image under certain regulators of
distinguished elements arising from the geometry of algebraic varieties.

In both approaches, the $p$-adic $L$-function is completely characterized by
the value at the points lying either at the critical or at the geometric
region. Moreover, some Euler factors arise, measuring the discrepancy between
the interpolation of $L$-values in the critical region and the interpolation
of cohomology classes in the geometric region. The vanishing of these factors
lead us to study exceptional zero phenomena. In the case of the Perrin-Riou
map, the shape of these factors is $\frac{1-p^{j}\phi}{1-p^{-1-j}\phi^{-1}}$,
where $j$ is related to the Hodge–Tate type of the character at which we are
specializing, and $\phi$ refers to a Frobenius eigenvalue. As it is suggested
for instance in [KLZ, Section 8] or [LZ], there are two kinds of Euler factors
in the usual Perrin-Riou maps: those appearing in the numerator (which
typically lead to an exceptional vanishing of the $p$-adic $L$-function via
explicit reciprocity laws) and those appearing in the denominator (which lead
to an exceptional vanishing of the cohomology class). While the former
phenomenon has been widely studied, as far as we know the latter has only been
discussed with the tools from Perrin-Riou theory in the setting of Heegner
points in [Cas] and for Beilinson–Flach elements in [RiRo1]. Nevertheless,
similar results have been obtained by Bley [Bl], although there are some
differences we later discuss.

Katz’s two-variable $p$-adic $L$-function $L_{\mathfrak{p}}(K)(\cdot)$ is
defined on the domain
$\operatorname{Hom}_{\operatorname{cont}}(G_{K},\bar{\mathbb{Q}}_{p}^{\times})$,
but we can consider its restriction to

(5) $\Sigma_{\psi}=\\{\psi\xi\mathcal{N}^{s}\lambda^{t}\text{ such that
}(s,t)\in\mathbb{Z}_{p}^{2}\\},$

where $\xi$ is a finite order character of $p$-power conductor. This allows us
to make use of the techniques relative to $p$-adic variation, sharing some
points in common with Hida theory. We write $L_{\mathfrak{p}}(K,\psi)(\cdot)$
for the restriction of $L_{\mathfrak{p}}(K)(\cdot)$ to the subspace of
characters given in (5), and denote
$L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}}):=L_{\mathfrak{p}}(K)(\psi)$
and
$L_{\mathfrak{p}}(K,\psi)(\mathcal{N}):=L_{\mathfrak{p}}(K)(\psi\mathcal{N})$.
In our case, and because of the dualities involved in the Perrin-Riou
formalism, we are also interested in the function
$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})$.

We can now describe the main ingredients involved in the proof of Theorem 1.1.

  1. (a)

An explicit reciprocity law for Katz’s two-variable $p$-adic $L$-function due
to Yager. This expresses the special value
$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N})$ in terms of the image
under a Perrin-Riou map of the cohomology class $\kappa_{\psi\mathcal{N}}$,
and directly gives us that
$\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\mathcal{N}})=0$ when
$\psi(\mathfrak{p})=1$, due to the vanishing of an Euler factor. Here,
$\operatorname{loc}_{\mathfrak{p}}$ stands for the localization at
$\mathfrak{p}$. The explicit description of the localization-at-$\mathfrak{p}$
map shows that we can conclude that $\kappa_{\psi\mathcal{N}}=0$ and consider
the derived cohomology class. We refer the reader to Sections 3.3 and 4.2 for
a proper definition of derived class and for more details on that.

  2. (b)

A derived reciprocity law, expressing the Bloch–Kato logarithm of the derived
class in terms of $L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N})$.
This requires an explicit description of the Perrin-Riou map, which at the
norm character interpolates the Bloch–Kato logarithm and gives a map

$\log_{\operatorname{BK}}:H^{1}(K_{\mathfrak{p}},F_{p}(\psi^{-1})(\mathcal{N}))\longrightarrow\mathbb{D}_{\operatorname{\mathrm{dR}}}(F_{p}(\psi^{-1}))\simeq
F_{p}.$

Under the identification induced by the Kummer morphism, this map corresponds,
in a sense that we later make precise, to the usual $p$-adic logarithm. Then,
we have the following result, whose proof is given in Section 4.2.

######  Proposition 1.2.

Assume that $\psi(\mathfrak{p})=1$. Then,

$\log_{p}(\mathfrak{p})\cdot
L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N})=-(1-p^{-1})\cdot\log_{p}(\operatorname{loc}_{\mathfrak{p}}(\kappa^{\prime}_{\psi\mathcal{N}})).$

  3. (c)

The functional equation for Katz’s two-variable $p$-adic $L$-function (see
[Gro, p.90–91]), which asserts that

$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N})=L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}}).$

  4. (d)

Katz’s $p$-adic version of Kronecker limit formula, expressing the special
value $L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}})$ in terms of the
elliptic unit $u_{\psi}$

$L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}})=-(1-\psi^{-1}(\bar{\mathfrak{p}}))(1-\psi(\mathfrak{p})p^{-1})\cdot\log_{p}(u_{\psi}).$

In Section 3.2 we properly discuss the main features of Katz’s two-variable
$p$-adic $L$-function.

As a by-product of the previous discussion, along the text we also deal with
other instances of the exceptional zero phenomenon. The results of Section 4
encompass two main situations: the exceptional vanishing of
$\kappa_{\psi\mathcal{N}}$; and the exceptional vanishing of the $p$-adic
$L$-function $L_{\mathfrak{p}}(K,\psi)$, which is a more well-established
phenomenon that has been widely studied in the literature and already appears
in Katz’s original work.

Once these results have been developed, the last section of the article serves
to analyze how our results fit with similar statements concerning exceptional
zero phenomena. In particular, we emphasize the parallelism, but also the
differences, with the theory of Heegner points, as well as the fact that these
elliptic units may be seen as a particular case inside the theory of
Beilinson–Flach elements, where different instances of the exceptional zero
phenomena also appear. When $g$ is a theta series of an imaginary quadratic
field where $p$ splits and we take the pair of modular forms $(g,g^{*})$,
[RiRo1] describes a connection between a derived Beilinson–Flach element, an
elliptic unit and an special value of the Hida–Rankin $p$-adic $L$-function
attached to $(g,g^{*})$. The assumptions considered in loc. cit. (we had
imposed that the Galois representation attached to $g$ was $p$-distinguished)
excluded the possibility of elliptic units presenting an exceptional zero, so
in a certain way the results of this article regarding exceptional zeros of
elliptic units can be thought as a degenerate case inside the theory of
Beilinson–Flach elements. While our main theorem can be seen as the
counterpart of [RiRo1, Theorem B] in the framework of elliptic units, we point
out that there is another exceptional zero phenomenon related to the vanishing
of the numerator of the Perrin-Riou map, which in this case leads to a trivial
zero of the Katz’s two variable $p$-adic $L$-function (see Section 4.1) and
which in the setting of Beilinson–Flach elements has been studied in [LZ2].

Acknowledgements. It is a pleasure to sincerely thank Victor Rotger, who
suggested me to work on this question and carefully read an earlier version of
this manuscript. I also thank Kazim Büyükboduk for useful comments and
correspondence regarding his work, and Francesc Castellà for many enlightening
conversations and helpful remarks during the writing of this note. Finally, it
is a must to thank the anonymous referees for a very careful reading of the
manuscript, whose comments notably contributed to improve the exposition and
corrected several inaccuracies.

The author has been supported by Grant MTM2015-63829-P, as well as from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 682152). The author has
also received financial support through “la Caixa” Fellowship Grant for
Doctoral Studies (grant LCF/BQ/ES17/11600010).

##  2\. Circular units

Circular units constitute one of the first examples of Euler systems, and they
play a key role in the proof of the classical Iwasawa main conjecture. We
recall here some of their most relevant features because of the parallelism
they keep with the theory of elliptic units. We discuss what the exceptional
zero phenomenon represents in this case, and then we will compare this setting
with that of elliptic units.

###  2.1. Leopoldt’s formula

In this section, we denote by
$\Lambda_{\operatorname{cyc}}:=\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$, and
let $\mathcal{W}:=\operatorname{Spf}(\Lambda_{\operatorname{cyc}})$. We fix a
primitive, non-trivial even Dirichlet character of conductor $N$,

$\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow F^{\times}$

where $F$ is a number field and $(p,N)=1$. We write $F_{p}$ for its completion
at a prime lying above $p$. For our applications to exceptional zero
phenomena, we are interested in the case in which $\chi(p)=1$.

The Kubota–Leopoldt $p$-adic $L$-function attached to $\chi$, $L_{p}(\chi,s)$,
can be defined as the $p$-adic analytic function satisfying the interpolation
property

$L_{p}(\chi,n)=(1-\chi(p)p^{-n})L(\chi,n),\quad\text{ for all }n\leq 0.$

Alternatively, we may understand it as a function defined over an appropriate
rigid analytic space, sometimes called the weight space.

######  Definition 2.1.

A classical point of $\mathcal{W}$ is a pair $(k,\xi)$, where $k$ is an
integer and $\xi$ is a Dirichlet character of $p$-power conductor,
corresponding to the homomorphism

$z\mapsto z^{k-1}\xi(z).$

Then, the Kubota–Leopoldt $p$-adic $L$-function can be seen as an application

$L_{p}(\chi,\cdot):\quad\mathcal{W}\longrightarrow\mathbb{C}_{p}$

defined in terms of an interpolation property for a subset of classical
points. We warn the reader that there are several possible conventions
regarding this function. Here, we closely follow the approach of [PR, Section
3], and the $p$-adic $L$-function we have considered satisfies the
interpolation property of Proposition 3.1.4 of loc. cit. See also [Tale] for a
reformulation of those ideas in our language. Another standard way of
presenting this $p$-adic $L$-function is discussed in [Das, Section 3], and we
will come back to this issue later on; there, the interpolation property
involves appropriate twists by powers of the Teichmüller character, but both
approaches are closely connected as shown in [PR, Section 3.1.5]: in
particular, the $p$-adic $L$-values at integers $n$ with $n\equiv 1$ modulo
$p-1$ agree.

Let $H$ denote the field cut out by $\chi$, and for a choice of a primitive
$p^{n}$-th root of unity $\zeta_{p^{n}}$, let $H_{n}=H(\zeta_{p^{n}})$. Define
the units

(6)
$c_{\chi,n}:=\prod_{a=1}^{N-1}(1-\zeta_{Np^{n}}^{a})^{\chi^{-1}(a)}\in(\mathcal{O}_{H_{n}}^{\times}\otimes
F)^{\chi},$

that behave under the norm maps as dictated by the theory of Euler systems:

$\mathcal{N}_{H_{n}}^{H_{n+1}}(c_{\chi,n+1})=\begin{cases}c_{\chi,n}&\text{ if
}n\geq 1,\\\ c_{\chi}\otimes(\chi(p)-1)&\text{ if }n=0.\end{cases},$

where $c_{\chi}=c_{\chi,0}$. As a word of caution, note that we have used the
standard multiplicative notation, where the exponentiation
$(1-\zeta_{Np^{n}}^{a})^{\chi^{-1}(a)}$ means
$(1-\zeta_{Np^{n}}^{a})\otimes\chi^{-1}(a)$.

Hence, one can construct a norm compatible family of cohomology classes taking
the image under the Kummer map $\delta$. More precisely, we consider

(7) $\kappa_{\chi,n}:=\delta(c_{\chi,n})\in
H^{1}(H_{n},F_{p}(1))^{\chi}=H^{1}(H_{n},F_{p}(\chi^{-1})(1)).$

These classes can be patched all together taking the projective limit for
$n\geq 1$, resulting in an element $\kappa_{\chi,\infty}$

(8)
$\kappa_{\chi,\infty}\in\lim_{\leftarrow}H^{1}(H_{n},F_{p}(\chi^{-1})(1))=H^{1}(\mathbb{Q},\Lambda_{\operatorname{cyc}}\otimes
F_{p}(\chi^{-1})(1)).$

As usual, $\Lambda_{\operatorname{cyc}}$ can be understood as a $p$-adic
interpolation of the Tate twists, parameterized by pairs $(k,\xi)$, where
$k\in\mathbb{Z}$ and $\xi$ is a Dirichlet character of $p$-power conductor.
Hence, we are endowing $\Lambda_{\operatorname{cyc}}$ with the tautological
action of $G_{\mathbb{Q}}$ (this is sometimes written in the literature as
$\Lambda_{\operatorname{cyc}}(\underline{\varepsilon}_{\operatorname{cyc}})$
to emphasize the Galois action, where
$\underline{\varepsilon}_{\operatorname{cyc}}$ is the so-called
$\Lambda_{\operatorname{cyc}}$-adic cyclotomic character). Let $F_{\xi,p}$
stand for the compositum of $F_{p}$ with the field of values of $\xi$. The
specialization maps $\nu_{k,\xi}:\Lambda_{\operatorname{cyc}}\rightarrow
F_{\xi,p}$ are ring homomorphisms sending the group-like element
$a\in\mathbb{Z}_{p}^{\times}$ to $a^{k-1}(\chi\xi)^{-1}(a)$, and induce
$G_{\mathbb{Q}}$-equivariant specialization maps

$\nu_{k,\xi}:\Lambda_{\operatorname{cyc}}\rightarrow
F_{\xi,p}(\xi^{-1})(k-1).$

This gives rise to a collection of global cohomology classes

(9) $\kappa_{k,\chi\xi}:=\nu_{k,\xi}(\kappa_{\chi,\infty})\in
H^{1}(\mathbb{Q},F_{\xi,p}(\chi\xi)^{-1}(k)).$

In order to state the following result, recall that the Gauss sum associated
to a Dirichlet character $\eta$ of conductor $m$ and with values in a number
field $L$ is defined by

(10)
$\mathfrak{g}(\eta)=\sum_{a=1}^{m-1}\zeta_{m}^{a}\otimes\eta(a)\in\mathcal{O}_{\mathbb{Q}(\zeta_{m})}^{\times}\otimes
F.$

From now on, $\exp_{\operatorname{BK}}^{*}$ stands for the Bloch–Kato dual
exponential map and $\log_{\operatorname{BK}}$ for the Bloch–Kato logarithm.
The following proposition is a reformulation of a classical result by Coleman
[Co], using the formalism of Perrin-Riou regulators developed in [PR].

######  Proposition 2.2 (Coleman, Perrin-Riou).

There exists a morphism of $\Lambda$-modules (referred to as the Coleman or
the Perrin-Riou map)

$\mathcal{L}_{p}:H^{1}(\mathbb{Q}_{p},\Lambda_{\operatorname{cyc}}\otimes_{\mathbb{Z}_{p}}F_{p}(\chi^{-1})(1))\longrightarrow
I^{-1}\Lambda_{\operatorname{cyc}}$

satisfying that for all classical points $(k,\xi)$, the specialization map
$\nu_{k,\chi\xi}(\mathcal{L}_{p})$ is the homomorphism

$\nu_{k,\chi\xi}(\mathcal{L}_{p}):H^{1}(\mathbb{Q}_{p},F_{\xi,p}(\chi\xi)^{-1}(k))\longrightarrow\mathbb{D}_{\operatorname{\mathrm{dR}}}(F_{\xi,p}((\chi\xi)^{-1})(k))\simeq
F_{\xi,p}$

given by

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

where $t$ is Fontaine’s $p$-adic analogue of $2\pi i$, and the target of both
the Bloch–Kato logarithm and the dual exponential map is identified with
$F_{\xi,p}$. Here, $I$ is the kernel of the specialization at $\nu_{1,\chi}$.

We finally relate the image of the previously introduced class
$\kappa_{\chi,\infty}$ under the Perrin-Riou regulator with the
Kubota–Leopoldt $p$-adic $L$-function. See for instance [PR] for a more
detailed treatment of this result. Here, $\operatorname{loc}_{p}$ stands for
the localization at $p$ of a global cohomology class.

######  Theorem 2.3.

Let $\chi$ stand for a non-trivial and even Dirichlet character. Then, the
cohomology class $\kappa_{\chi,\infty}\in
H^{1}(\mathbb{Q},\Lambda_{\operatorname{cyc}}\otimes_{\mathbb{Z}_{p}}F_{p}(\chi^{-1})(1))$
introduced in (8) satisfies

$L_{p}(\chi,\cdot)=\mathcal{L}_{p}(\operatorname{loc}_{p}(\kappa_{\chi,\infty})).$

The previous theorem can be seen as an equality in
$\Lambda_{\operatorname{cyc}}$, and we may apply the specialization maps to
both sides at any $\mathbb{C}_{p}$-valued point. From the previous results,
and using Kummer’s identifications again, it turns out that one has the
equality

(11)
$L_{p}(\chi\xi,1)=-\frac{(1-\chi\xi(p)p^{-1})}{(1-(\chi\xi)^{-1}(p))}\cdot\frac{\log_{p}(\operatorname{loc}_{p}(\kappa_{1,\chi\xi}))}{\mathfrak{g}((\chi\xi)^{-1})},$

whenever $(\chi\xi)(p)\neq 1$; if $(\chi\xi)(p)=1$, both the Euler factor in
the denominator and the cohomology class in the numerator vanish. Recall that
here we have identified $t\cdot\log_{\operatorname{BK}}$ with the Iwasawa
$p$-adic logarithm.

Since $\chi$ is non-trivial, one has
$L_{p}(\chi,1)\in\bar{\mathbb{Q}}_{p}^{\times}$. This suggests the existence
of a derived cohomology class $\kappa_{1,\chi}^{\prime}$ related with
$L_{p}(\chi,1)$, which is the content of the following section. We recover
this idea along the article, but anyway it is good to keep in mind that in
this setting one also has a $p$-adic Kronecker’s limit formula expressing the
value of $L_{p}(\chi,1)$ in terms of a unit in the number field cut out by the
character

(12)
$L_{p}(\chi,1)=-\frac{(1-\chi(p)p^{-1})}{\mathfrak{g}(\chi^{-1})}\cdot\log_{p}\Big{(}\prod_{a=1}^{N-1}(1-\zeta^{a})^{\chi^{-1}(a)}\Big{)}.$

This result is generally due to Leopoldt (see also [PR]), and is often called
in the literature Leopoldt’s formula. The quantity
$\prod_{a=1}^{N-1}(1-\zeta^{a})^{\chi^{-1}(a)}$ is typically referred as the
circular unit attached to $\chi$ and we have denoted it by $c_{\chi}$.

As we have pointed out, we may instead consider a slightly different $p$-adic
$L$-function, that we denote $L_{p,1}(\chi,n)$, and defined in terms of the
interpolation property

$L_{p,1}(\chi,n)=(1-\chi\omega^{n-1}(p)p^{-n})\cdot
L(\chi\omega^{n-1},n)\quad\text{ for all }n\leq 0,$

where $\omega$ stands for the modulo $p$ cyclotomic character (Teichmüller). A
very interesting object of study in the theory of $\mathcal{L}$-invariants is
$L_{p,1}^{\prime}(\chi,0)$, in the case where $\chi\omega^{-1}(p)=1$ and
therefore $L_{p,1}(\chi,0)=0$ due to an exceptional vanishing. Washington
[Was] provides a formula for the value of the derivative in terms of Morita’s
$p$-adic Gamma function, $\Gamma_{p}(x)$:

(13)
$L_{p,1}^{\prime}(\chi,0)=\log_{p}\Big{(}\prod_{a=1}^{N}\Gamma_{p}(a/N)^{\chi\omega^{-1}(a)}\Big{)}+L_{p,1}(\chi,0)\log_{p}(N).$

Hence, in the situation of exceptional vanishing $\chi\omega^{-1}(p)=1$, there
is an exceptional zero for $L_{p,1}(\chi,s)$ at $s=0$ and one has that

(14) $L_{p,1}^{\prime}(\chi,0)=\log_{p}(v_{\chi}),$

where

$v_{\chi}=\prod_{a=1}^{N}\Gamma_{p}(a/N)^{\chi\omega^{-1}(a)}.$

In the case where one considers instead an odd Dirichlet character $\eta$ with
$\eta(p)=1$, the determination of $L_{p,1}^{\prime}(\eta\omega,0)$ is a
particular case of Gross’ conjectures, as studied first by Ferrero–Greenberg
[FG], and then by Darmon–Dasgupta–Pollack (among others!) for arbitrary
totally real fields. Here, this derivative is expressed in terms of the
logarithm of a $p$-unit in the field cut out by the character.

###  2.2. Exceptional zeros and circular units

Suppose from now on that $\chi$ is a non-trivial, even, Dirichlet character of
conductor $N$ with $\chi(p)=1$ and $(p,N)=1$. Then, the arguments of the
previous section show that the specialization of the
$\Lambda_{\operatorname{cyc}}$-adic class $\kappa_{\chi,\infty}$ at $\chi$
vanishes, that is, $\kappa_{1,\chi}=0$. Of course, this can be interpreted in
terms of the vanishing of the denominator of the Perrin-Riou map. This
section, where no claim of originality is made, explains how to obtain a
formula for $L_{p}(\chi,1)$ involving $\kappa_{\chi,\infty}$, following for
that the work of Solomon [Sol] and Büyükboduk [Buy1], and also discusses how
the vanishing of the numerator of the Perrin-Riou regulator at $s=0$ can be
studied inside the framework developed in [Ven].

With the previous notations, let $T=F_{p}(\chi^{-1})(1)$ and
$T^{*}=F_{p}(\chi)$, viewed as representations of $G_{\mathbb{Q}}$. We single
out one of the $(p-1)$ connected components of $\mathcal{W}$, which
corresponds to the choice of the residue class of 1 modulo $p-1$ and of the
Iwasawa algebra
$\Lambda=\mathbb{Z}_{p}[[1+p\mathbb{Z}_{p}]]\subset\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$.
After fixing a topological generator $\gamma$ of $1+p\mathbb{Z}_{p}$, one may
consider the short exact sequence of $\mathbb{Z}_{p}$-modules

$0\rightarrow\Lambda\otimes T\xrightarrow{\gamma-1}\Lambda\otimes T\rightarrow
T\rightarrow 0$

which induces a long exact sequence in cohomology. Since
$H^{0}(\mathbb{Q},T)=0$,

$0\rightarrow H^{1}(\mathbb{Q},\Lambda\otimes
T)\xrightarrow{\gamma-1}H^{1}(\mathbb{Q},\Lambda\otimes
T)\xrightarrow{\mathcal{N}}H^{1}(\mathbb{Q},T).$

The image of $\kappa_{\chi,\infty}$ under the map $\mathcal{N}$ vanishes since
$\chi(p)=1$, and hence there exists a unique

$\kappa_{\chi,\infty}^{\prime}\in H^{1}(\mathbb{Q},\Lambda\otimes T)$

such that

$\frac{\gamma-1}{\log_{p}(\gamma)}\cdot\kappa_{\chi,\infty}^{\prime}=\kappa_{\chi,\infty}.$

The reason of normalizing by $\log_{p}(\gamma)$ is, as discussed in [Buy1,
Section 3], that the derived class does not longer depend on the choice of the
topological generator $\gamma$.

Summing all up, we have the following result. We refer the reader to [Buy1,
Proposition 3.4] for a more detailed discussion.

######  Proposition 2.4.

If $\chi(p)=1$, the class $\kappa_{\chi,\infty}$ vanishes at the character
$\chi$ and there exists a derived cohomology class
$\kappa_{\chi,\infty}^{\prime}\in H^{1}(\mathbb{Q},\Lambda\otimes T)$ such
that

$\kappa_{\chi,\infty}=\frac{\gamma-1}{\log_{p}(\gamma)}\cdot\kappa_{\chi,\infty}^{\prime}.$

######  Remark 2.5.

It may be tempting to look for a relation between $\kappa_{1,\chi}^{\prime}$
and the special value $L_{p}(\chi,1)$. However, the fact that the Euler factor
$1-p^{k-1}$ is not analytic in the variable $k$ precludes the possibility of
directly taking derivatives in the reciprocity law of Proposition 2.2.
However, this can be remedied invoking Solomon’s results, as we later see,
connecting the order of the derived class with the special value at $s=1$. In
the following sections we discuss how in a bigger weight space certain
derivatives are related with the $p$-adic logarithm and others with the
$p$-adic valuation.

Let us provide a more explicit description of the previous result. In [Buy1]
the author makes a connection between the value of $L_{p}(\chi,1)$ and
Nekovar’s pairings. In [Buy1, Corollary 2.11] it is shown that

$H_{\operatorname{f},p}^{1}(\mathbb{Q},T)=(\mathcal{O}_{H}^{\times}[1/p])^{\chi}\otimes
F_{p},$

is a two-dimensional space where we may explicitly construct a basis. Here,
$\tilde{H}_{\operatorname{f}}^{1}(\mathbb{Q},T)$ stands for the Bloch–Kato
Selmer group of classes which are unramified outside $p$ and de Rham at $p$.
As before, we have written $H$ for the field cut out by $\chi$. The fact that
this space is two-dimensional reflects the exceptional zero coming from the
condition $\chi(p)=1$, which gives rise to an extra $p$-unit in the field cut
out by the character.

Define

$c_{n}=\mathcal{N}_{\mathbb{Q}(\zeta_{Nn})/H(\zeta_{n})}(1-\zeta_{Nn})\in(\mathcal{O}_{H(\zeta_{n})}^{\times}\otimes
F),$

and consider its $\chi$-part, $c_{\chi,n}$. The element $c_{\chi}:=c_{\chi,1}$
is called the tame cyclotomic unit, and agrees with the definition given in
the previous section. For a finite abelian extension $H^{\prime}$ of
$\mathbb{Q}$ of conductor $m$ we also define

$\xi_{H^{\prime}}=\mathcal{N}_{\mathbb{Q}(\zeta_{mp})/H^{\prime}}(1-\zeta_{mp}).$

With a slight abuse of notation, we may identify the units with the cohomology
classes obtained via the Kummer map. Then, it turns out that the collection

$\xi=\xi_{\chi,\infty}:=\\{e_{\chi}\xi_{H_{n}}\text{ for }n\geq
1\\}\in\lim_{\leftarrow}H^{1}(H_{n},T),$

where for the sake of simplicity we have written $e_{\chi}$ for the
$\chi$-projector, satisfies the Euler system distribution relation, and
moreover $\xi_{H}=1$. Proceeding as before, we obtain an element
$z_{\chi,\infty}$ satisfying

$\frac{\gamma-1}{\log_{p}(\gamma)}\times z_{\chi,\infty}=\xi.$

We call its bottom layer $z_{\chi}:=z_{0,\chi}\in H^{1}(\mathbb{Q},T)$ the
cyclotomic $p$-unit, and $\\{c_{\chi},z_{\chi}\\}$ is a basis of
$H_{\operatorname{f},p}^{1}(\mathbb{Q},T)$. In [Sol], it is proved that
$\log_{p}(c_{\chi})={\mathrm{ord}}_{p}(z_{\chi})\in F_{p}$, where
${\mathrm{ord}}_{p}$ is the usual $p$-adic valuation. Of course, this depends
on the choice of a prime of $F$ lying above $p$.

The interesting fact appears when the denominator of the Perrin-Riou map
vanishes. To circumvent that problem, [Buy1, Section 6.1] recasts to the
principle of improved Perrin-Riou map, which allows to introduce a primitive
$p$-adic $L$-function $\tilde{L}_{p}(\chi,s)$ vanishing at $s=1$. The main
result of [Buy1] is the computation, via the theory of Nekovar’s pairings, of
a formula for $L_{p}(\chi,1)$, which asserts that

(15) $\tilde{L}_{p}^{\prime}(\chi,1)=p\cdot
L_{p}(\chi,1)=\frac{1-p}{\mathfrak{g}(\chi^{-1})}\times\log_{p}(c_{\chi})=\frac{1-p}{\mathfrak{g}(\chi^{-1})}\times{\mathrm{ord}}_{p}(z_{\chi}).$

This also works for the case of an imaginary quadratic field when one only
consider the $\mathbb{Z}_{p}$-extension which is ramified just over a fixed
prime $\mathfrak{p}$ above $p$.

For the sake of completeness, we finish the section by analyzing what happens
for $L_{p,1}(\chi,0)$, where we may follow the approach of [Ven, Section 3] to
analyze the vanishing of the numerator in the Perrin-Riou map. To ease
notations, let $\psi=\chi\omega^{-1}$ and write again $F$ for its field of
values. In particular, we know that when $\psi(p)=1$, $L_{p,1}(\chi,0)=0$.

Since the numerator of the Perrin-Riou regulator vanishes, we can consider its
derivative. Let $I$ stand for the augmentation ideal of $\Lambda$. Then, we
define the derivative of the Perrin-Riou map $\mathcal{L}_{p}$ of Proposition
2.2 as the application

$\mathcal{L}_{p}^{\prime}:H^{1}(\mathbb{Q}_{p},\Lambda\otimes_{\mathbb{Z}_{p}}F_{p}(\psi^{-1})(1))\rightarrow
I/I^{2},$

i. e. , the composition of $\mathcal{L}_{p}$ with the projection
$\\{\cdot\\}:I\rightarrow I/I^{2}$.

Let $\kappa_{\psi}=(\kappa_{n,\psi})$ be the cohomology class we have
previously introduced in (9). Following the same strategy as in [Ven, Prop.
3.6], and identifying $\kappa_{0,\psi}$ with an element in
$\operatorname{Hom}(\mathbb{Q}_{p}^{\times},\mathbb{Q}_{p})\otimes
F_{p}(\psi^{-1})$, one has that

$\mathcal{L}_{p}^{\prime}(\kappa_{\psi})=-\mathfrak{g}(\psi^{-1})^{-1}(1-p^{-1})^{-1}\cdot\frac{\exp_{\operatorname{BK}}^{*}(\operatorname{loc}_{p}(\kappa_{0,\psi}))}{\log_{p}(\gamma)}\cdot\\{\gamma\\},$

where we have identified $I/I^{2}$ with the multiplicative group
$1+p\mathbb{Z}_{p}$. As in [Ven, Section 5], we can relate the derivative of
$\mathcal{L}_{p}$ with the derivative of $L_{p,1}(\chi,s)$ and obtain this way
a formula for $L_{p,1}^{\prime}(\chi,0)$ in terms of
$\exp_{\operatorname{BK}}^{*}(\operatorname{loc}_{p}(\kappa_{0,\psi}))$.

##  3\. Elliptic units

In this section we introduce Katz’s two-variable $p$-adic $L$-function and the
theory of elliptic units, following mainly [deS] and [Yag], but adapting their
results to the framework discussed before. We also recall the Perrin-Riou big
logarithm and recast Yager’s theorem, which gives an explicit reciprocity law
analogue to Theorem 2.3 in this setting. We recover the notations of the
introduction, where $K$ is an imaginary quadratic field and we fix a prime $p$
which splits on $K$, i.e. $p\mathcal{O}_{K}=\mathfrak{p}\bar{\mathfrak{p}}$.
We also fix an identification of $\mathbb{C}_{p}$ with $\mathbb{C}$ and
embeddings of $\bar{\mathbb{Q}}$ to either of these fields, which are
compatible with these identifications. Let $h$ denote the class number of $K$.
Then, let $\pi_{\mathfrak{p}}\in\mathcal{O}_{K}$ be such that
$\mathfrak{p}^{h}=\pi_{\mathfrak{p}}\mathcal{O}_{K}$, and define
$\varpi_{\mathfrak{p}}=\pi_{\mathfrak{p}}/\pi_{\bar{\mathfrak{p}}}$. For
simplicity, we assume that $\mathcal{O}_{K}^{\times}=\pm 1$ and that the
discriminant of $K$ is an odd number $D<0$.

Consider also a non-trivial Hecke character of finite order $\psi$, of
conductor $\mathfrak{n}$, where $(\mathfrak{n},p)=1$. In the particular case
that $\chi$ is a Dirichlet character of conductor
$N:=\mathcal{N}_{K/\mathbb{Q}}(\mathfrak{n})$, the Dirichlet character may be
seen by restriction as an example of the kind of Hecke characters we are
interested in, provided that $K$ is a quadratic field where all primes
dividing $N$ split. As before, let $F$ stand for the field cut out by the
character and $F_{p}$ for its completion.

###  3.1. Elliptic units

Elliptic units are the result of evaluating modular units at CM points. They
give rise to units in abelian extensions of the imaginary quadratic field $K$
and are the counterpart of circular units for cyclotomic fields. They also
constitute one of the key ingredients for the proof of the Iwasawa main
conjecture for imaginary quadratic fields [Rub].

For the general construction of elliptic units, we refer the reader to the
seminal work of Coates and Wiles [CW], or alternatively to Robert’s original
paper [Rob]. Let us give an explicit description in the special setting where
the conductor $\mathfrak{n}$ of $\psi$ satisfies that there exists a rational
integer $N$ such that
$\mathcal{O}_{K}/\mathfrak{n}\simeq\mathbb{Z}/N\mathbb{Z}$, and $\psi$ can be
interpreted as a Dirichlet character of conductor $N$. We closely follow the
survey [Tale] for that purpose.

Special values of $L$-series are encoded in terms of the so-called Siegel
units $g_{a}\in\mathcal{O}_{Y_{1}(N)}^{\times}\otimes\mathbb{Q}$ attached to a
fixed choice of primitive $N$-th root of unity $\zeta_{N}$ and a parameter
$1\leq a\leq N-1$. Its $q$-expansion is given by

(16)
$g_{a}(q)=q^{1/12}(1-\zeta_{N}^{a})\prod_{n>0}(1-q^{n}\zeta_{N}^{a})(1-q^{n}\zeta_{N}^{-a}).$

Let $\tau_{\mathfrak{n}}=\frac{b+\sqrt{D}}{2N}$, where
$\mathfrak{n}=\mathbb{Z}N+\mathbb{Z}\frac{b+\sqrt{D}}{2}$. The classical and
$p$-adic elliptic units are defined by

(17) $u_{a,\mathfrak{n}}:=g_{a}(\tau_{\mathfrak{n}}),\qquad
u_{a,\mathfrak{n}}^{(p)}:=g_{a}^{(p)}(\tau_{\mathfrak{p}\mathfrak{n}}),$

being $g_{a}$ the infinite product of (16) and
$g_{a}^{(p)}:=g_{pa}(q^{p})g_{a}(q)^{-p}$. As we did with circular units, we
may define

(18) $u_{\psi}:=\prod_{\sigma\in\mathrm{Gal\,}(K_{\mathfrak{n}}/K)}(\sigma
u_{1,\mathfrak{n}})^{\psi^{-1}(\sigma)},$

where $K_{\mathfrak{n}}$ is the ray class field of $K$ of conductor
$\mathfrak{n}$ and
$u_{1,\mathfrak{n}}\in\mathcal{O}_{K_{\mathfrak{n}}}^{\times}$. In additive
notation, this corresponds to $(\sigma
u_{1,\mathfrak{n}})\otimes\psi^{-1}(\sigma)$. This construction works in
greater generality and one can always define the element $u_{\psi}$ (see
[Rob]). These units are the bottom layer of a norm compatible family of
elliptic units over the two-variable $\mathbb{Z}_{p}$-extension $K_{\infty}$
of $K$.

Performing a similar construction to that of (7) and (8), the work of Katz
[Ka] and de Shalit [deS] gives a cohomology class

(19) $\kappa_{\psi,\infty}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})),$

where $\Lambda_{K}$ is the two-variable Iwasawa algebra of the introduction
endowed with the tautological Galois action. In particular, if $\eta$ is a
Hecke character of infinity type $(\kappa_{1},\kappa_{2})$, the global class
obtained by specializing $\kappa_{\psi,\infty}$ at $\eta$, although it arises
from elliptic units, encodes information about a Galois representation of $K$
attached to a Hecke character.

###  3.2. Katz’s two-variable p-adic L-function of an imaginary quadratic
field

The classical two-variable $L$-function attached to $K$ and a Hecke $\psi$ is
defined by

$L(K,\psi,\kappa_{1},\kappa_{2}):=\sideset{}{{}^{\prime}}{\sum}_{\alpha\in\mathcal{O}_{K}}\psi(\alpha)\alpha^{-\kappa_{1}}\bar{\alpha}^{-\kappa_{2}},$

where the sum is over the set of non-zero ideals of $\mathcal{O}_{K}$. This
$L$-series allows us to recover the more familiar $L$-function attached to a
character $\psi$ of an imaginary quadratic field, via the relation

(20) $L(K,\psi,s)=\frac{1}{2}L(K,\psi,s,s).$

We follow [DLR, Section 3] for the construction of Katz’s two variable
$p$-adic $L$-function. Let $\mathfrak{c}\subset\mathcal{O}_{K}$ be an integral
ideal of $K$, and let $\Sigma$ be the set of Hecke characters of $K$ of
conductor dividing $\mathfrak{c}$. Define
$\Sigma_{K}=\Sigma_{K}^{(1)}\cup\Sigma_{K}^{(2)}\subset\Sigma$ to be the
disjoint union of the sets

$\Sigma_{K}^{(1)}=\\{\psi\in\Sigma\text{ of infinity type
}(\kappa_{1},\kappa_{2}),\text{ with }\kappa_{1}\leq 0,\kappa_{2}\geq 1\\},$
$\Sigma_{K}^{(2)}=\\{\psi\in\Sigma\text{ of infinity type
}(\kappa_{1},\kappa_{2}),\text{ with }\kappa_{1}\geq 1,\kappa_{2}\leq 0\\}.$

For all $\psi\in\Sigma_{K}$, the complex argument $s=0$ is a critical point
for $L(\psi^{-1},s)$, and Katz’s $p$-adic $L$-function is constructed
interpolating the algebraic part of $L(\psi^{-1},0)$, as $\psi$ ranges over
$\Sigma_{K}^{(2)}$.

Let $\hat{\Sigma}_{K}$ be the completion of $\Sigma_{K}^{(2)}$ with respect to
the compact open topology on the space of $\mathcal{O}_{L_{p}}$-valued
functions on a subset of $\mathbb{A}_{K}^{\times}$. By the work of Katz, there
exists a $p$-adic analytic function

$L_{\mathfrak{p}}(K):\hat{\Sigma}_{K}\longrightarrow\mathbb{C}_{p},$

uniquely determined by the interpolation property that for all
$\xi\in\Sigma_{K}^{(2)}$ of infinity type $(\kappa_{1},\kappa_{2})$,

(21)
$L_{\mathfrak{p}}(K)(\xi)=\mathfrak{a}(\xi)\times\mathfrak{e}(\xi)\times\mathfrak{f}(\xi)\times\frac{\Omega_{p}^{\kappa_{1}-\kappa_{2}}}{\Omega^{\kappa_{1}-\kappa_{2}}}\times
L_{\mathfrak{c}}(\xi^{-1},0),$

where

  1. (1)

$\mathfrak{a}(\xi)=(\kappa_{1}-1)!\pi^{-\kappa_{2}}$,

  2. (2)

$\mathfrak{e}(\xi)=(1-\xi(\mathfrak{p})p^{-1})(1-\xi^{-1}(\bar{\mathfrak{p}}))$,

  3. (3)

$\mathfrak{f}(\xi)=D_{K}^{\kappa_{2}/2}2^{-\kappa_{2}}$,

  4. (4)

$\Omega_{p}\in\mathbb{C}_{p}^{\times}$ is a $p$-adic period attached to $K$,

  5. (5)

$\Omega\in\mathbb{C}^{\times}$ is the complex period associated to $K$,

  6. (6)

$L_{\mathfrak{c}}(\xi^{-1},s)$ is Hecke’s $L$-function associated to
$\xi^{-1}$ with the Euler factors at primes dividing $\mathfrak{c}$ removed.

We have followed the conventions of [BDP2, Proposition 3.1], which in turn
follows from [Ka, Section 5.3.0]. Observe that the definition is not symmetric
with respect to the primes $\mathfrak{p}$ and $\bar{\mathfrak{p}}$ above $p$,
and hence we can also consider the function
$L_{\bar{\mathfrak{p}}}(K)(\cdot)$.

The $p$-adic $L$-function $L_{\mathfrak{p}}(K)(\cdot)$ satisfies a functional
equation

(22)
$L_{\mathfrak{p}}(K)(\xi)=L_{\mathfrak{p}}(K)((\xi^{\prime})^{-1}\mathcal{N}),$

where again $\xi^{\prime}$ is the composition of $\xi$ with the complex
conjugation (see [Gro, pages 90–91]). We remark that since our characters are
unramified at $p$, the Gauss sum that sometimes appears in the interpolation
formula is equal to 1. Finally, and according to this definition, the
interpolation is over the special values of the form
$L_{\mathfrak{c}}(\xi^{-1},0)$; this explains some discrepancies regarding
certain conventions with the case of circular units.

It is possible to obtain an expression for the value of
$L_{\mathfrak{p}}(K)(\psi)$ at finite order characters. This is usually
referred to as the $p$-adic Kronecker’s limit formula, and is due to Katz:

(23)
$L_{\mathfrak{p}}(K)(\psi)=\begin{cases}\frac{1}{2}\Big{(}\frac{1}{p}-1\Big{)}\cdot\log_{p}(\pi_{\mathfrak{p}}^{1/h})&\text{
if }\psi=1;\\\
-(1-\psi^{-1}(\bar{\mathfrak{p}}))(1-\psi(\mathfrak{p})p^{-1})\cdot\log_{p}(u_{\psi})&\text{
if }\psi\neq 1.\end{cases}$

Here, $h$ stands for the class number of $K$ and $\pi_{\mathfrak{p}}$ for a
generator of the $\mathcal{O}_{K}$-ideal $\mathfrak{p}^{h}$. Via the
functional equation, we also have an expression for the value at the
$\psi\mathcal{N}$

(24)
$L_{\mathfrak{p}}(K)(\psi\mathcal{N})=\begin{cases}\frac{1}{2}\Big{(}\frac{1}{p}-1\Big{)}\cdot\log_{p}(\pi_{\mathfrak{p}}^{1/h})&\text{
if }\psi=1;\\\
-(1-\psi(\mathfrak{p}))(1-\psi^{-1}(\bar{\mathfrak{p}})p^{-1})\cdot\log_{p}(u_{(\psi^{\prime})^{-1}})&\text{
if }\psi\neq 1.\end{cases}$

######  Definition 3.1.

Let $L_{\mathfrak{p}}(K,\psi)(\cdot)$ stand for the restriction of the
$p$-adic $L$-function of (21) to characters of the form
$\psi\xi\lambda^{\kappa_{1}}(\lambda^{\prime})^{\kappa_{2}}$, where $\xi$ is a
character of $p$-power conductor and $\lambda$ is the character of infinity
type $(1,0)$ presented in the introduction.

In particular, write
$L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}}):=L_{\mathfrak{p}}(K)(\psi)$
and
$L_{\mathfrak{p}}(K,\psi)(\mathcal{N}):=L_{\mathfrak{p}}(K)(\psi\mathcal{N})$.

######  Remark 3.2.

Depending on the normalization we choose for the two variable $p$-adic
$L$-function, the value $L_{\mathfrak{p}}(K)(\psi)$ may be affected by
multiplication by a non-zero explicit rational number. Further, this number
can depend on the conductor of $\psi$; however, since we are restricting the
function to characters of the form $\psi\xi$, where $\xi$ has $p$-power
conductor, we can adopt a suitable normalization such that our special value
formulas always work.

Further, observe that the $p$-adic $L$-function of [Buy1, Theorem 6.3] also
differs from this one in the factor $(1-\xi^{-1}(\bar{\mathfrak{p}}))$.

We finish this description of Katz’s two-variable $p$-adic $L$-function by
discussing its relation with the theory of improved $p$-adic $L$-functions. We
say that a character is analytic if it is of the form $\psi\lambda^{t}$, with
$t\in\mathbb{Z}^{\geq 0}$. The reason for this terminology is that they
correspond to the subvariety of the weight space along which the Euler factors
appearing in the Perrin-Riou map are analytic as functions in the variable
$t$. Katz constructed in [Ka, Section 7.2] a one-variable $p$-adic
$L$-function $L_{\mathfrak{p}}^{*}(K,\psi,k)$, such that the restriction of
$L_{\mathfrak{p}}(K,\psi)$ to analytic characters yields the equation

(25)
$L_{\mathfrak{p}}(K,\psi)(\lambda^{k})=(1-\psi^{-1}(\bar{\mathfrak{p}})\pi_{\bar{\mathfrak{p}}}^{-k/h})L_{\mathfrak{p}}^{*}(K,\psi,k).$

The ratio of the two $p$-adic $L$-series is a $p$-adic analytic function of
$k$, since $\pi_{\bar{\mathfrak{p}}}$ belongs to
$\mathcal{O}_{K_{\mathfrak{p}}}^{\times}$. This ratio measures the difference
between working with the ordinary $p$-stabilization of the Eisenstein series,
$E_{k,\psi}^{(p)}$, and the $p$-depletion, $E_{k,\psi}^{[p]}$. Further, by a
result of Katz [Ka, Section 7.2], one has a relation between the $p$-adic
$L$-function of a quadratic imaginary field and elliptic units, given by

(26)
$L_{\mathfrak{p}}^{*}(K,\psi,0)=-(1-\psi(\mathfrak{p})p^{-1})\cdot\log_{p}(u_{\psi}).$

We refer the reader to [BD, Section 4.3] for a more detailed exposition of
this material.

###  3.3. A reciprocity law for elliptic units

In this section, we recall the existence of a Perrin-Riou map interpolating
both the dual exponential map and the Bloch–Kato logarithm, as we did with
circular units. We closely follow the treatment of [deS], recalling the main
properties of this regulator map, whose source is the Iwasawa cohomology of
the representation induced by a Hecke character, and which interpolates the
Bloch–Kato logarithm and the dual exponential map, depending on the Hodge–Tate
type of the character at which we specialize.

Although this is part of a rather general theory, we are interested in a more
down-to-earth version of these results, which have been recovered by Loeffler
and Zerbes in [LZ] in the setting of two-variable Perrin-Riou regulators. In
particular, Theorem 4.15 of loc. cit. gives an analogue to Proposition 2.2 in
the setting of elliptic units. We restrict to characters of $K$ of the form
$\psi\lambda^{\kappa_{1}}(\lambda^{\prime})^{\kappa_{2}}$, where $\lambda$ is
the character of infinity type $(1,0)$ of the introduction and
$\lambda^{\prime}$ is its complex conjugate. Of course we may also consider
twists by characters $\xi$ of $p$-power order, but we neglect this possibility
so as to ease the exposition.

Recall that $\Lambda_{K}$ is the two-variable Iwasawa algebra attached to $K$
with the tautological Galois action. As with circular units, for any character
$\eta=\lambda^{\kappa_{1}}(\lambda^{\prime})^{\kappa_{2}}$ as above there
exists a specialization map that we normalize, to ease the exposition, as the
one inducing specializations of the form

$\nu_{\kappa_{1},\kappa_{2}}:\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})\rightarrow
F_{p}(\psi^{-1})(\lambda^{\kappa_{2}}(\lambda^{\prime})^{\kappa_{1}}).$

We identify again the target of both the dual exponential map and the
Bloch–Kato logarithm with $F_{p}$. The following result has been established
in [LZ, Section 6.4, Appendix B], and here is presented in the language of
[KLZ, Section 8].

######  Proposition 3.3.

There exists a morphism

$\mathcal{L}_{\mathfrak{p}}:H^{1}(K_{\mathfrak{p}},\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N}))\longrightarrow\Lambda_{K}$

interpolating both the dual exponential map and the Bloch–Kato logarithm, and
such that for any point $\eta$ of infinity type $(\kappa_{1},\kappa_{2})$ with
$\xi=1$, the specialization of $\mathcal{L}$ at $\eta$ is the homomorphism

$\nu_{\kappa_{1},\kappa_{2}}(\mathcal{L}_{\mathfrak{p}}):H^{1}(K_{\mathfrak{p}},F_{p}(\psi^{-1})(\lambda^{\kappa_{2}}(\lambda^{\prime})^{\kappa_{1}}))\longrightarrow\mathbb{D}_{\operatorname{\mathrm{dR}}}(F_{p}(\psi^{-1})(\lambda^{\kappa_{2}}(\lambda^{\prime})^{\kappa_{1}}))\simeq
F_{p}$

given by

$\nu_{\kappa_{1},\kappa_{2}}(\mathcal{L}_{\mathfrak{p}})=\frac{1-\psi(\mathfrak{p})\pi_{\mathfrak{p}}^{-\kappa_{2}/h}\pi_{\bar{\mathfrak{p}}}^{-\kappa_{1}/h}}{1-\frac{\psi^{-1}(\mathfrak{p})\pi_{\mathfrak{p}}^{\kappa_{2}/h}\pi_{\bar{\mathfrak{p}}}^{\kappa_{1}/h}}{p}}\begin{cases}\frac{(-t)^{\kappa_{2}}}{\kappa_{2}!}\log_{\operatorname{BK}}&\text{
if }\kappa_{2}>0\\\
(-\kappa_{2})!t^{\kappa_{2}}\exp_{\operatorname{BK}}^{*}&\text{ if
}\kappa_{2}\leq 0.\end{cases}$

######  Remark 3.4.

It is interesting to analyze the shape of the Euler factors and compare it
with those of [Cas, Theorem 3.5]. To follow this parallelism, let
$\kappa=\kappa_{1}+\kappa_{2}$ and $r=-\kappa_{2}$. Then,

(27)
$\frac{1-\psi(\mathfrak{p})\pi_{\mathfrak{p}}^{-\kappa_{2}/h}\pi_{\bar{\mathfrak{p}}}^{-\kappa_{1}/h}}{1-\frac{\psi^{-1}(\mathfrak{p})\pi_{\mathfrak{p}}^{\kappa_{2}/h}\pi_{\bar{\mathfrak{p}}}^{\kappa_{1}/h}}{p}}=\frac{1-\psi(\mathfrak{p})\pi_{\mathfrak{p}}^{r/h}\pi_{\bar{\mathfrak{p}}}^{(-\kappa-r)/h}}{1-\frac{\psi^{-1}(\mathfrak{p})\pi_{\mathfrak{p}}^{-r/h}\pi_{\bar{\mathfrak{p}}}^{(\kappa+r)/h}}{p}}=\frac{1-{\psi(\mathfrak{p})}\varpi_{\mathfrak{p}}^{r/h}\pi_{\bar{\mathfrak{p}}}^{-\kappa/h}}{1-\frac{\psi^{-1}(\mathfrak{p})\varpi_{\mathfrak{p}}^{-r/h}\pi_{\bar{\mathfrak{p}}}^{\kappa/h}}{p}}.$

Our results concerning elliptic units can be seen as a counterpart of those
for Heegner points when the cuspidal Hida family is replaced by an Eisentein
series.

######  Remark 3.5.

As we will discuss in the last section, this also fits well with [RiRo1,
Proposition 3.2], which is a reformulation of [KLZ, Theorem 8.1.7]; with the
notations of loc. cit., if we fix $s=0$ and identify the modular forms $g$ and
$h$ with two weight one theta series attached to the imaginary quadratic field
$K$, we recover the map of Proposition 3.3. Further, observe that the
numerology is coherent, and the condition $m>s$ defining the region of
interpolation of the Bloch–Kato logarithm becomes in this case $\kappa_{2}>0$.

The following result is the main theorem of [Yag].

######  Proposition 3.6.

The cohomology class $\kappa_{\psi,\infty}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1}))$ satisfies

$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})=\mathcal{L}_{\mathfrak{p}}(\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi,\infty})).$

Again, the denominator of the Perrin-Riou regulator may vanish. Assume that
$\psi(\mathfrak{p})=1$. Then, we have the following:

  1. (i)

If $\kappa_{1}=\kappa_{2}=0$, then the numerator vanishes and the denominator
equals $1-p^{-1}$.

  2. (ii)

If $\kappa_{1}=\kappa_{2}=1$, the numerator equals $1-p^{-1}$ and the
denominator vanishes.

######  Remark 3.7.

For a fixed $\kappa_{2}$, both the numerator and the denominator are analytic
functions on the variable $\kappa_{1}$.

##  4\. Exceptional zeros and elliptic units

We analyze different instances of exceptional zero phenomena and discuss the
existence of derived cohomology classes and some of their properties. Our main
result, stated as Theorem 1.1 in the introduction, is about the exceptional
vanishing of $\kappa_{\psi\mathcal{N}}$, but for the sake of convenience we
also study the exceptional vanishing of Katz’s two variable $p$-adic
$L$-functon at $\psi$ in Section 4.1. Then, in Section 4.2 we discuss the
different cases of exceptional zeros at $\psi\mathcal{N}$ and prove the main
result of the note. Finally, Section 4.3 discusses the special case where
$\psi$ is the restriction of a Dirichlet character, suggesting a tantalizing
connection with the theory of circular units.

###  4.1. Specialization at the character $\psi$

We assume that the condition $\psi(\bar{\mathfrak{p}})=1$ is satisfied. We
begin this section by discussing the vanishing of Katz’s two-variable $p$-adic
$L$-function at the character $\psi$ under this hypothesis. Indeed, from (25),
it is straightforward that $\psi(\bar{\mathfrak{p}})=1$ is a necessary and
sufficient condition for the vanishing of
$L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}})$ when the classical
value is non-zero.

Until otherwise stated, and so as to shorten the notation, derivatives of
$p$-adic $L$-functions are considered along the character $\lambda$ (that is,
along the $H_{\infty}$ direction).

######  Proposition 4.1.

Assume that $\psi(\bar{\mathfrak{p}})=1$. Then,
$L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}})=0$, and

(28)
$L_{\mathfrak{p}}^{\prime}(K,\psi)(\chi_{\operatorname{triv}})=\log_{p}(\pi_{\mathfrak{p}}^{1/h})\cdot(1-\psi(\mathfrak{p})p^{-1})\cdot\log_{p}(u_{\psi}).$

###### Proof.

This follows by taking derivatives in (25) and applying the special value
formula of (26). Observe that we have also used that
$\log_{p}(\pi_{\mathfrak{p}}^{1/h})=-\log_{p}(\pi_{\bar{\mathfrak{p}}}^{1/h})$.
∎

The Euler factors arising in the Perrin-Riou map are analytic in the variable
$\kappa_{1}$ once the value of $\kappa_{2}$ is fixed. In particular, we may
consider the Perrin-Riou map with $\kappa_{2}=0$ fixed. Writing
$\kappa_{\kappa_{1},\psi}$ for the corresponding specialization of the
$\Lambda$-adic class, one has

$(1-\pi_{\bar{\mathfrak{p}}}^{\kappa_{1}/h}/p)\cdot
L_{\mathfrak{p}}(K,\psi)(\chi_{\operatorname{triv}})=(1-\pi_{\bar{\mathfrak{p}}}^{-\kappa_{1}/h})\cdot\exp_{\operatorname{BK}}^{*}(\operatorname{loc}_{\mathfrak{p}}(\kappa_{\kappa_{1},(\psi^{\prime})^{-1}})),$

for all $\kappa_{1}\geq 0$. Taking derivatives with respect to $\kappa_{1}$ at
both sides and evaluating at $\kappa_{1}=0$, we get that

(29)
$(1-p^{-1})L_{\mathfrak{p}}^{\prime}(K,\psi)(\chi_{\operatorname{triv}})=-\log_{p}(\pi_{\mathfrak{p}}^{1/h})\cdot\exp_{\operatorname{BK}}^{*}(\operatorname{loc}_{\mathfrak{p}}(\kappa_{(\psi^{\prime})^{-1}})),$

where $\kappa_{(\psi^{\prime})^{-1}}$ stands for the specialization at the
trivial character.

Here, by class field theory,

$H^{1}(K_{\mathfrak{p}},F_{p}(\psi^{-1}))\simeq\operatorname{Hom}_{\operatorname{cont}}(K_{\mathfrak{p}}^{\times},K_{\mathfrak{p}})\otimes
F_{p}(\psi^{-1}),$

and $\operatorname{loc}_{\mathfrak{p}}(\kappa_{(\psi^{\prime})^{-1}})$
corresponds to the evaluation at the inverse of a local uniformizer,
$\pi_{\mathfrak{p}}^{-1/h}$. In particular, there is a non-canonical
isomorphism with $F_{p}$. Under this isomorphism, the dual exponential map
corresponds to the ${\mathrm{ord}}$ map. Then, we may identify an element of
the space

$\operatorname{Hom}_{\operatorname{cont}}(K_{\mathfrak{p}}^{\times},K_{\mathfrak{p}})\otimes
F_{p}(\psi^{-1})$

with its image under evaluation at $\pi_{\mathfrak{p}}^{-1/h}$.

Therefore, combining Proposition 4.1 with equation (29), we get the following.

######  Proposition 4.2.

Assume that $\psi(\bar{\mathfrak{p}})=1$. Then, with the previous
identifications,

(30)
$\operatorname{loc}_{\mathfrak{p}}(\kappa_{(\psi^{\prime})^{-1}})=-(1-p^{-1})(1-\psi(\mathfrak{p})p^{-1})\times\log_{p}(u_{\psi}).$

###  4.2. Specialization at the character $\psi\mathcal{N}$

The motivation for this section comes from the following lemma.

######  Lemma 4.3.

The following relation between the unit $u_{\psi}$ and the specialization of
$\kappa_{\psi,\infty}$ at the character $\psi\mathcal{N}$ holds:

(31)
$\log_{p}(\kappa_{\psi\mathcal{N}})=(1-\psi^{-1}(\mathfrak{p}))\cdot(1-\psi^{-1}(\bar{\mathfrak{p}}))\cdot\log_{p}(u_{\psi}).$

In particular, $\log_{p}(\kappa_{\psi\mathcal{N}})$ vanishes if and only if
$\psi(\mathfrak{p})=1$ or $\psi(\bar{\mathfrak{p}})=1$.

###### Proof.

This follows after combining the $p$-adic Kronecker limit formula given in
(24), now applied to $(\psi^{\prime})^{-1}$, with Proposition 3.6. Observe
that the factor $1-\psi(\mathfrak{p})/p$ cancels out, and the same occurs for
the two minus signs (the one coming from the Perrin-Riou map and that of the
$p$-adic Kronecker limit formula). ∎

Further, since both $\kappa_{\psi\mathcal{N}}$ and $u_{\psi}$ are
$\psi$-units, the $p$-adic logarithm defines an isomorphism and we may upgrade
the previous lemma to an equality in
$(\mathcal{O}_{K_{\mathfrak{n}}}^{\times}\otimes F)^{\psi}$. Observe that we
are implicitly using that the localization map corresponds to the injection of
global units inside local units, and this allows us to conclude that the
global class $\kappa_{\psi\mathcal{N}}$ also vanishes when
$\psi(\mathfrak{p})=1$ or $\psi(\bar{\mathfrak{p}})=1$.

We distinguish now three different situations.

  1. (a)

If $\psi(\mathfrak{p})=1$, the cohomology class $\kappa_{\psi\mathcal{N}}=0$.
One can construct a derived class whose derivative along the direction
$\lambda$ is computed and expressed in terms of the unit $u_{\psi}$. If
$\psi(\bar{\mathfrak{p}})\neq 1$, the special value
$L_{\mathfrak{p}}(K,\psi)(\mathcal{N})$ does not vanish and it is related with
the Bloch–Kato logarithm of the derived class.

  2. (b)

If $\psi(\bar{\mathfrak{p}})=1$, both $\kappa_{\psi\mathcal{N}}=0$ and
$L_{\mathfrak{p}}(K,\psi)(\mathcal{N})=0$. The logarithm of the derived local
class $\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\mathcal{N}}^{\prime})$,
once the value of $\kappa_{2}$ is fixed, can be expressed in terms of
$L_{\mathfrak{p}}^{\prime}(K,\psi)(\mathcal{N})$.

  3. (c)

When $\psi(\mathfrak{p})=\psi(\bar{\mathfrak{p}})=1$, both
$\kappa_{\psi\mathcal{N}}=0$ and $L_{\mathfrak{p}}(K,\psi)(\mathcal{N})=0$.
Observe that this can be understood as an extension of the subcases (a) and
(b). Then, $\kappa_{\psi\mathcal{N}}^{\prime}=0$ too, and there is a notion of
second derivative of the cohomology class, whose logarithm is related with
$L_{\mathfrak{p}}^{\prime}(K,\psi)(\mathcal{N})$.

Case (a). Suppose that $\psi(\mathfrak{p})=1$. Let $\mathcal{C}$ be the line
of weight space obtained by taking the Zariski closure of all the points of
the form $\mathcal{N}\lambda^{t}$ (that is, fixing $\kappa_{2}=1$). We are
considering the $\mathbb{C}_{p}$-points as a rigid analytic space, and
realizing all these characters as elements of this space. Analogously, define
$\mathcal{C}^{\prime}$ as the closure of the points of the form
$\mathcal{N}\bar{\lambda}^{t}$. At the level of $p$-adic $L$-functions, and
under the map of Proposition 3.6, the point $\mathcal{N}\lambda^{t}$ in
$\mathcal{C}$ corresponds to
$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N}\bar{\lambda}^{t})$, or
alternatively and via the functional equation, to
$L_{\mathfrak{p}}(K,\psi)(\lambda^{t})$.

In this situation of exceptional zeros, the Euler factor in the denominator of
the Perrin-Riou map of Proposition 3.3 vanishes, and hence one must carry out
some new constructions to obtain a formula relating the special value of the
$p$-adic $L$-function with an appropriate cohomology class. For that purpose,
we can argue the existence of a derived cohomology class arising from elliptic
units, which is directly related with the special value of the derivative of
Katz’s two variable $p$-adic $L$-function. Observe that derivatives of
cohomology classes must be taken along $\mathcal{C}^{\prime}$, since elsewhere
the Euler factors are not analytic (and not even continuous!).

######  Proposition 4.4.

It holds that $\kappa_{\psi\mathcal{N}}=0$, and there exists a derived
cohomology class along $\mathcal{C}^{\prime}$

$\kappa_{\gamma,\psi,\infty}^{\prime}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}})$

satisfying that

$\kappa_{\psi|\mathcal{C}^{\prime}}=(\gamma-1)\kappa^{\prime}_{\gamma,\psi,\infty},$

where $\gamma$ is a fixed topological generator of $1+p\mathbb{Z}_{p}$.

###### Proof.

The vanishing of the local class
$\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\mathcal{N}})$ directly follows
from Lemma 4.3 and the discussion after it.

The construction of the derived class follows the same argument than in the
case of circular units explained in Section 2, and which is already present in
the work of Bley [Bl]. After fixing a topological generator $\gamma$ of
$1+p\mathbb{Z}_{p}$, one may consider the short exact sequence of
$\mathbb{Z}_{p}$-modules

$0\rightarrow\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}}\xrightarrow{\gamma-1}\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}}\rightarrow
F_{p}(\psi^{-1})(\mathcal{N})\rightarrow 0$

which induces a long exact sequence in cohomology. Since
$H^{0}(K,F_{p}(\psi^{-1})(\mathcal{N}))=0$,

$0\rightarrow H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}})\xrightarrow{\gamma-1}H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}})\xrightarrow{\mathcal{N}}H^{1}(K,F_{p}(\psi^{-1})(\mathcal{N})).$

As we have just seen, the image of $\kappa_{\psi,\infty}$ under the map
$\mathcal{N}$ vanishes, and hence there exists a unique

$\kappa_{\gamma,\psi,\infty}^{\prime}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}})$

such that

$\frac{\gamma-1}{\log_{p}(\gamma)}\times\kappa_{\gamma,\psi,\infty}^{\prime}=\kappa_{\gamma,\psi,\infty}.$

∎

######  Remark 4.5.

If we normalize dividing by $\log_{p}(\gamma)$, the specialization of the
resulting class at the character $\psi\mathcal{N}$ does not depend on $\gamma$
(see [Buy1, Section 3]). We define
$\kappa_{\psi,\infty}^{\prime}:=\frac{\kappa_{\gamma,\psi,\infty}^{\prime}}{\log_{p}(\gamma)}$.

We can relate the cohomology class with $L_{\mathfrak{p}}(K)(\cdot)$ at
$\psi\mathcal{N}$.

######  Definition 4.6.

Let $\mathcal{E}_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\cdot)$ stand for the
function defined over the set of characters of the form
$\lambda^{\kappa_{1}}\bar{\lambda}$ (and then extended to their Zariski
closure over weight space), and given by

(32)
$\mathcal{E}_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\lambda^{\kappa_{1}}\bar{\lambda})=\Big{(}1-\frac{\pi_{\mathfrak{p}}^{1/h}\pi_{\bar{\mathfrak{p}}}^{\kappa_{1}/h}}{p}\Big{)}L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\lambda^{\kappa_{1}}\bar{\lambda}).$

######  Lemma 4.7.

The function $\mathcal{E}_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})$ satisfies
that for a character $\eta$ of infinity type $(\kappa_{1},1)$

(33)
$\mathcal{E}_{\mathfrak{p}}(K)(\psi\eta)=-\Big{(}1-\pi_{\mathfrak{p}}^{-1/h}\pi_{\bar{\mathfrak{p}}}^{-\kappa_{1}/h}\Big{)}\cdot
t\log_{\operatorname{BK}}(\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\eta})).$

In particular, the function
$\mathcal{E}_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})$ is zero at the norm
character.

###### Proof.

The first part follows from Proposition 3.6. The second statement is due to
the vanishing of the specialization of the cohomology class at
$\psi\mathcal{N}$ (recall that for characters of infinity type
$(\kappa_{1},1)$ the Bloch–Kato logarithm interpolates the usual logarithm
map). ∎

This is sufficient to prove Theorem 1.1. Recall again that we may identify the
Bloch–Kato logarithm with the usual $p$-adic logarithm under Kummer’s
isomorphism.

######  Remark 4.8.

For the following result, we need to use that
$(\log_{\operatorname{BK}}(\kappa))^{\prime}=\log_{\operatorname{BK}}(\kappa^{\prime})$.
This can be easily seen by considering the natural isomorphism between the
weight space and $\mathbb{Z}_{p}[[X]]$. Then, the class $\kappa$ corresponds
to a function $f$ vanishing at $0$, and hence there is another function $g$
such that $f=X\cdot g$. The Bloch–Kato logarithm is a linear morphism between
a $\mathbb{Z}_{p}[[X]]$-module and a field embedded in $\mathbb{C}_{p}$, that
we may denote with the letter $\Phi$. Then,

$\Phi(f)^{\prime}|_{X=0}=\Phi(g)|_{X=0},$

as desired.

######  Theorem 4.9.

It holds that

$\log_{p}(\pi_{\mathfrak{p}}^{1/h})\cdot
L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N})=-(1-p^{-1})\cdot\log_{p}(\operatorname{loc}_{\mathfrak{p}}(\kappa^{\prime}_{\psi\mathcal{N}})).$

Moreover,

$\kappa_{\psi\mathcal{N}}^{\prime}=\log_{p}(\pi_{\mathfrak{p}}^{1/h})\cdot(1-\psi^{-1}(\bar{\mathfrak{p}}))\cdot
u_{\psi},$

where $\operatorname{loc}_{\mathfrak{p}}$ stands here for the composition of
the Kummer map with localization at $\mathfrak{p}$.

###### Proof.

The first part follows by considering the derivative of the function
$\mathcal{E}_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})$ using both (32) and (33).
That is, the left hand side is the result of deriving (32) and evaluating at
$\kappa_{1}=1$, where the Euler factor vanishes; similarly, the right hand
side is the result of deriving (33) and evaluating again at $\kappa_{1}=1$,
where the class vanishes. Observe that we have identified
$t\log_{\operatorname{BK}}$ with the $p$-adic logarithm, as usual.

The second part directly follows after comparing the first statement with the
special value formula given in (24). Further, since we are taking a derivative
along the $\bar{\mathfrak{p}}$-ramified extension $H_{\infty}^{\prime}/K$, the
resulting derived class could only have non-zero valuation at the prime
$\bar{\mathfrak{p}}$. This cannot be the case when
$\psi(\bar{\mathfrak{p}})\neq 1$ since there are no extra
$\bar{\mathfrak{p}}$-units, and the $p$-adic logarithm is therefore an
isomorphism for the $\psi$-component of the unit group. If
$\psi(\mathfrak{p})=\psi(\bar{\mathfrak{p}})=1$ the same conclusion follows by
recasting to the arguments of [Bl]. We discuss this situation in more detail
in the following sections. ∎

This result slightly differs from the work of Bley [Bl]. There, the author
takes the derivative along the direction $\mathcal{C}$, which corresponds to
the $\mathbb{Z}_{p}$-extension which is ramified at the prime $\mathfrak{p}$.
In that case, the derivative of the Perrin-Riou map agrees with the order (and
not with the logarithm), which is coherent with the fact that when
$\psi(\mathfrak{p})=1$ there is a $\mathfrak{p}$-unit in the $\psi$-component.
Hence, our results are coherent with the computations of [Buy1, Section 6.4]
relating the valuation of the extra $\mathfrak{p}$-unit with the logarithm of
the elliptic unit $u_{\psi}$.

Case (b). Assume now that $\psi(\bar{\mathfrak{p}})=1$. Observe that via the
functional equation for Katz’s two-variable $L$-function, this leads to

$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N})=0,$

and $L_{\mathfrak{p}}^{\prime}(K,(\psi^{\prime})^{-1})(\mathcal{N})$ can be
related with the derived cohomology class constructed in the previous section.
We recall that the derivative of the $p$-adic $L$-function always means
derivative along the $H_{\infty}$ direction. As we did in Proposition 4.4, we
may take the class

$\kappa_{\psi,\infty}^{\prime}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}}).$

######  Proposition 4.10.

Assume that $\psi(\bar{\mathfrak{p}})=1$. Then,

(34) $(1-\psi^{-1}(\mathfrak{p}))\cdot
L_{\mathfrak{p}}^{\prime}(K,(\psi^{\prime})^{-1})(\mathcal{N})=-(1-\psi(\mathfrak{p})p^{-1})\cdot\log_{p}(\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\mathcal{N}}^{\prime})).$

###### Proof.

This directly follows by considering the derivative with respect to the
variable $\kappa_{1}$ in the reciprocity law of Proposition 3.6 when
$\kappa_{2}$ is fixed. ∎

The derivative $L_{\mathfrak{p}}^{\prime}(K,\psi)(\mathcal{N})$ is related,
via the funcional equation, with the derivative at the trivial character along
the direction $\lambda^{\prime}$, and a priori we do not know any expression
for that value in terms of $u_{\psi}$, which would allow us to prove an
analogue of Theorem 1.1 in this setting. Further, in this case the derived
class $\kappa_{\psi\mathcal{N}}^{\prime}$ is no longer a unit, but a
$\bar{\mathfrak{p}}$-unit, and the $p$-adic logarithm is not sufficient to
characterize $\kappa_{\psi\mathcal{N}}^{\prime}$ (one needs to use the
information about its $p$-adic valuation). It would be nice to understand
these results in the framework provided by [BD, Section 4], and we hope to
come back to this issue in forthcoming work.

In the framework of circular units, Gross’ factorization formula [Gro],
combined with the results of the previous section, allows us to express, after
considering the identifications provided by Kummer’s isomorphisms, the class
$\kappa_{\psi\mathcal{N}}^{\prime}$ as a linear combination of a circular unit
and an elliptic unit. We discuss this in Section 4.3

######  Remark 4.11.

Following Proposition 4.10, we can extend our computations for the derivative
of the cohomology class to arbitrary directions. Indeed, we may consider the
function $L_{\bar{\mathfrak{p}}}(K)(\cdot)$, thus complementing our picture.

  * •

When $\psi(\mathfrak{p})=1$, $L_{\bar{\mathfrak{p}}}(K)(\cdot)$ vanishes at
$\psi\mathcal{N}$ and in this case we can relate the derivative of
$\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\mathcal{N}})$ along the
direction $\mathcal{C}$ with the derivative of $L_{\bar{\mathfrak{p}}}(K)$ at
$\psi$ along the $(-1,0)$ direction.

  * •

When $\psi(\bar{\mathfrak{p}})=1$, the derivative of
$\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\mathcal{N}})$ along the
direction $\mathcal{C}$ can be explicitly expressed in terms of
$L_{\bar{\mathfrak{p}}}(\psi\mathcal{N})$.

Case (c). When $\psi(\bar{\mathfrak{p}})=\psi(\mathfrak{p})=1$, we expect that
the cohomology class $\kappa_{\psi\mathcal{N}}^{\prime}$ vanishes since
$L_{\mathfrak{p}}(K,\psi)(\mathcal{N})=0$. More precisely, and due to the
symmetry between both directions $\lambda$ and $\lambda^{\prime}$, we may
combine the results of previous sections with [Buy1, Theorem 6.13] to conclude
that the class $\kappa_{\psi\mathcal{N}}^{\prime}$ is zero.

When this happens, we may take a second derivative of the cohomology class

$\kappa_{\gamma,\psi,\infty}^{\prime\prime}\in H^{1}(K,\Lambda_{K}\otimes
F_{p}(\psi^{-1})(\mathcal{N})_{|\mathcal{C}^{\prime}}).$

Now, we normalize the class dividing by $\log_{p}(\gamma)^{2}$ in such a way
that its value at the character $\psi\mathcal{N}$,
$\kappa_{\psi\mathcal{N}}^{\prime\prime}$, does not depend on $\gamma$.
Proceeding in the same way as before, we may obtain the following result.

######  Proposition 4.12.

Assume that $\psi(\bar{\mathfrak{p}})=\psi(\mathfrak{p})=1$. Then,

(35) $\log_{p}(\pi_{\mathfrak{p}}^{1/h})\cdot
L_{\mathfrak{p}}^{\prime}(K,(\psi^{\prime})^{-1})(\mathcal{N})=-2(1-p^{-1})\cdot\log_{p}(\operatorname{loc}_{\mathfrak{p}}(\kappa_{\psi\mathcal{N}}^{\prime\prime})).$

###### Proof.

Consider again the reciprocity law of Proposition 3.3 and fix again
$\kappa_{2}=1$. When evaluating at the norm character, it happens that both
the Euler factor of the denominator and the special value
$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})(\mathcal{N})$ are zero. Hence,
multiplying both sides by
$1-\pi_{\mathfrak{p}}^{-1/h}\pi_{\bar{\mathfrak{p}}}^{-\kappa_{1}/h}$ and
taking twice the derivative with respect to $\kappa_{1}$, we obtain the
result. ∎

######  Remark 4.13.

We point out, just for the sake of completeness, that another interesting
instance of the exceptional zero phenomenon can be observed in [BDP2,
Proposition 3.5], which asserts that a self-dual character of infinity type
$(1+j,-j)$ with $j\geq 0$ satisfies that the evaluation of Katz’s two-variable
$p$-adic $L$-function at $\nu$ agrees, up to multiplication by some periods
and gamma factors, with a classical $L$-value times
$(1-\nu^{-1}(\bar{\mathfrak{p}}))^{2}$. Again, if $\nu(\bar{\mathfrak{p}})=0$
we observe the presence of an exceptional zero.

In the special case where we consider characters of infinity type $(1,0)$,
Agboola [Agb] studies a variant of the $p$-adic BSD conjecture for CM elliptic
curves concerning special values of Katz’s two-variable $p$-adic $L$-function.
Here, we are again in a situation where our same condition leads to an
exceptional vanishing.

###  4.3. Interactions with the theory of circular units

Observe that in the case where $\psi(\mathfrak{p})=1$ we have described the
derived cohomology class $\kappa_{\psi\mathcal{N}}^{\prime}$ as an explicit
multiple of the elliptic unit $u_{\psi}$. However, when
$\psi(\bar{\mathfrak{p}})=1$ this is no longer possible, since we cannot
express in terms of $u_{\psi}$ the derivative of the Katz’s two variable
$p$-adic $L$-function at $\psi$ along the $\lambda^{\prime}$-direction.

In general, one may wish to determine the derivatives of the $p$-adic
$L$-function along the different directions of the weight space. We know the
derivative along the $\lambda$-direction, and it turns out that in some
particular cases we can further determine the derivative along the norm
direction.

This is the case when $\psi$ is a finite order Hecke character which comes
from the restriction to $G_{K}$ of a Dirichlet character (that is, a
$G_{\mathbb{Q}}$ character), and hence one can invoke Gross’ factorization
formula, which is the main result of [Gro]. To begin with, consider that
$\psi$ comes from a Dirichlet character and that $\psi(\bar{\mathfrak{p}})=1$.
In this case, we have determined the derivative of $L_{\mathfrak{p}}(K,\psi)$
at $\chi_{\operatorname{triv}}$ along the direction $\lambda$, and this is

$\frac{\partial
L_{\mathfrak{p}}(K,\psi)}{\partial\lambda}=\log_{p}(\pi_{\mathfrak{p}}^{1/h})\cdot(1-\psi(\mathfrak{p})p^{-1})\times\log_{p}(u_{\psi}).$

Similarly, using Gross’ factorization with the conventions about Gauss sums
followed in the article, we have that

$\frac{\partial
L_{\mathfrak{p}}(K,\psi)}{\partial\mathcal{N}}=-(1-\psi(\mathfrak{p})p^{-1})\cdot
L_{p,1}^{\prime}(\psi^{-1}\chi_{K}\omega,0)\times\log_{p}(c_{\psi}),$

where $\chi_{K}$ stands for the quadratic character attached to $K$ and
$\omega$ is the Teichmüller character. Observe that here
$L_{p}(\psi^{-1}\chi_{K}\omega,0)=0$ due to the running assumptions.

Then, one can determine the derivative along any direction: for
$\eta=\psi\lambda^{a}\mathcal{N}^{b}$, the derivative of
$L_{\mathfrak{p}}(K,\psi)$ at the trivial character along the direction $\eta$
is given by

(36)
$a\cdot\log_{p}(\pi_{\mathfrak{p}}^{1/h})\cdot(1-\psi(\mathfrak{p})p^{-1})\cdot\log_{p}(u_{\psi})-b\cdot
L_{p,1}^{\prime}(\psi^{-1}\chi_{K}\omega,0)\cdot(1-\psi(\mathfrak{p})p^{-1})\cdot\log_{p}(c_{\psi}).$

######  Remark 4.14.

There is only one direction along which this value is zero; as discussed in
the inspiring presentation [Gr], this has significant applications towards
Iwasawa theory.

Now, the functional equation gives the derivative of
$L_{\mathfrak{p}}(K,(\psi^{\prime})^{-1})$ at the norm character along any
direction, and in particular, for the direction $\lambda$, we have to set
$a=1$ and $b=-1$ (this is the direction $-(\lambda^{\prime})$ at the trivial
character). Then, Proposition 4.10 yields that, up to an element in the kernel
of the logarithm,

$\kappa_{\psi\mathcal{N}}^{\prime}=\alpha u_{\psi}+\beta c_{\psi}$

where $c_{\psi}$ is the circular unit introduced in Section 2, and $\alpha$
and $\beta$ can be determined combining (36) with (12):

$\alpha=-\log_{p}(\pi_{\mathfrak{p}}^{1/h})(1-\psi^{-1}(\mathfrak{p})),$
$\beta=-L_{p,1}^{\prime}(\psi^{-1}\chi_{K}\omega,0)(1-\psi^{-1}(\mathfrak{p})).$

If we further assume that $\psi(\mathfrak{p})=1$, Proposition 4.12 gives a new
expression for the second derivative of the cohomology classm again up to an
element in the kernel of the logarithm,

$\kappa_{\psi\mathcal{N}}^{\prime\prime}=\alpha^{\prime}u_{\psi}+\beta^{\prime}c_{\phi},$

where now

$\alpha^{\prime}=-\log_{p}(\pi_{\mathfrak{p}}^{1/h})^{2}/2,$
$\beta^{\prime}=-\log_{p}(\pi_{\mathfrak{p}}^{1/h})L_{p,1}^{\prime}(\psi^{-1}\chi_{K}\omega,0)/2.$

##  5\. The exceptional zero phenomenon: Beilinson–Flach elements and elliptic
units

In this last section, we emphasize the interplay between the results we have
presented until now and other related works in this direction. In particular,
we study how the phenomena we have discussed arise in some of the other Euler
systems presented in the survey [Tale], focusing on two main aspects:

  1. (a)

Elliptic units can be seen as the natural substitute of Heegner points when
instead of considering a cusp form, one takes an Eisenstein series.

  2. (b)

Elliptic units can be recast in terms of Beilinson–Flach elements, when we
take two weight one modular forms corresponding to theta series of the same
imaginary quadratic field where the prime $p$ splits.

As it has been extensively discussed in the literature, there is a striking
parallelism between the theory of Heegner points and that of elliptic units.
Following this analogy, this note may be read as the counterpart of [Cas] when
the cusp form $f$ is replaced by an Eisenstein series. With the notations
introduced in loc. cit., where the weight space is modeled by a weight
variable (that we denote with the letter $k$) and an anticyclotomic variable
(denoted with the letter $t$), an exceptional zero arises at the point
$(k,t)=(2,0)$ when ${\bf a}_{p}({\mathbf{f}})=1$ (the associated elliptic
curve has split multiplicative reduction at $p$) and we specialize at the
character $\psi\mathcal{N}$. There is a clear interplay between that setting
and ours, but we would like to point out some of the differences:

  * •

In [Cas] the author extends the $p$-adic Gross–Zagier formula of [BDP1] and
finds an explicit expression for its value at the norm character, which is
different from zero. However, in our setting it may occur that the $p$-adic
$L$-function vanishes both at $\psi\mathcal{N}$ and at $\psi$. This
simultaneous vanishing of the Euler factor and the $p$-adic $L$-function gives
rise to a higher order vanishing of the derived class $\kappa_{\psi,\infty}$
at the character $\psi\mathcal{N}$. This can be understood via [Cas, Eq. 0.2],
where the specialization of the higher dimensional Heegner cycle (whose role
is now played by the cohomology class coming from the elliptic unit) and the
Heegner class (whose role is now played by the unit $u_{\psi}$) are related by
the factor

$\Big{(}1-\frac{p^{k/2-1}}{\nu_{k}({\bf a}_{p})}\Big{)}^{2}.$

In our case, however, the link is via the factor

$(1-\psi^{-1}(\bar{\mathfrak{p}})\mathfrak{p}^{(\kappa_{1}-1)/h}\bar{\mathfrak{p}}^{(\kappa_{2}-1)/h})\cdot(1-\psi^{-1}(\mathfrak{p})\mathfrak{p}^{(\kappa_{2}-1)/h}\bar{\mathfrak{p}}^{(\kappa_{1}-1)/h}),$

and hence there are two possible (and independent) sources of vanishing.

  * •

In our setting there are two points where the exceptional zero phenomenon
emerges: the character $\psi$ and the character $\psi\mathcal{N}$. In [Cas]
the vanishing of the numerator in the Perrin-Riou map would occur at
$(k,t)=(0,-1)$, where there is not a clear geometric meaning of this
phenomenon.

In any case, the similitude between his main result and ours is evident,
expressing a derived cohomology class as a certain $\mathcal{L}$-invariant
times a more classical avatar. Obviously, his $\mathcal{L}$-invariant encodes
information both about the elliptic curve and the imaginary quadratic field
$K$.

###  5.1. Elliptic units and Beilinson–Flach elements

Elliptic units can be understood as a degenerate case of the theory of
Beilinson–Flach elements. To make this statement more precise, let

$g=\sum_{n\geq 1}a_{n}q^{n}\in S_{1}(N_{g},\chi_{g}),\quad h=\sum_{n\geq
1}b_{n}q^{n}\in S_{1}(N_{h},\chi_{h})$

be two normalized newforms, and let $V_{g}$ and $V_{h}$ denote the Artin
representations attached to them. Consider also $V_{gh}:=V_{g}\otimes V_{h}$,
and let $H$ be the smallest number field cut out by this representation. We
fix a prime number $p$ which does not divide $N_{g}N_{h}$. Label the roots of
the $p$-th Hecke polynomial of $g$ and $h$ as

$X^{2}-a_{p}(g)X+\chi_{g}(p)=(X-\alpha_{g})(X-\beta_{g})\quad
X^{2}-a_{p}(h)X+\chi_{h}(p)=(X-\alpha_{h})(X-\beta_{h}).$

Let

$g_{\alpha}(q)=g(q)-\beta_{g}q(q^{p}),\qquad
h_{\alpha}(q)=h(q)-\beta_{h}h(q^{p})$

denote the $p$-stabilization of $g$ (resp. $h$) on which the Hecke operator
$U_{p}$ acts with eigenvalue $\alpha_{g}$ (resp. $\alpha_{h}$). Let $F$ be a
number field containing both the Fourier coefficients of $g$ and $h$ and the
eigenvalues for the $p$-th Hecke polynomials. We can attach in a natural way
two canonical differentials $\omega_{g_{\alpha}}$ and $\eta_{g_{\alpha}}$ to
the weight one modular form $g$, as it is recalled in [RiRo1, Sections 2,3].
The reinterpretation of the main results of [KLZ] in [RiRo1] and [RiRo2]
establishes the existence of cohomology classes

$\kappa(g_{\alpha},h_{\alpha}),\kappa(g_{\alpha},h_{\beta}),\kappa(g_{\beta},h_{\alpha}),\kappa(g_{\beta},h_{\beta})\in
H^{1}(\mathbb{Q},V_{gh}\otimes F_{p}(1)),$

and also gives an explicit reciprocity law which in slightly rough terms
asserts that

$\Big{(}1-\frac{1}{p\alpha_{g}\beta_{h}}\Big{)}\cdot\log^{-+}(\kappa_{p}(g_{\alpha},h_{\alpha}))=(1-\alpha_{g}\beta_{h})\cdot
L_{p}(g,h,1)\pmod{F^{\times}}.$

Here, $L_{p}(g,h,s)$ stands for the Hida–Rankin $p$-adic $L$-function attached
to the convolution $g\otimes h$, $\kappa_{p}(g_{\alpha},h_{\alpha})$ is the
restriction of the cohomology class to a decomposition group at $p$, and
$\log^{-+}$ is the result of applying the Bloch–Kato logarithm to a certain
projection of the local class followed by the pairing with the canonical
differentials. The proof of this result is based on considering Hida families
${\mathbf{g}}$, ${\mathbf{h}}$ interpolating $g_{\alpha}$ and $h_{\alpha}$,
and on proving the corresponding equality over a dense set of points of the
weight space. We refer the reader to [RiRo1] for a complete discussion of the
results.

Now, let $g^{*}$ stand for the twist of $g$ by the inverse of its nebentype.
When $h_{\alpha}=g_{1/\beta}^{*}$, the Euler factor $1-\alpha_{g}\beta_{h}$ is
zero, and Proposition 3.12 of loc. cit. establishes that both
$\kappa(g_{\alpha},g_{1/\beta}^{*})$ and $\kappa(g_{\beta},g_{1/\alpha}^{*})$
vanish and moreover the authors prove the existence of a derived cohomology
class $\kappa^{\prime}(g_{\alpha},g_{1/\beta}^{*})\in
H^{1}(\mathbb{Q},V_{gg^{*}}\otimes F_{p}(1))$ satisfying

(37)
$\log^{-+}(\kappa_{p}^{\prime}(g_{\alpha},g_{1/\beta}^{*}))=\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))\cdot
L_{p}(g,g^{*},1)\pmod{F^{\times}},$

being $\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))$ the
$\mathcal{L}$-invariant of the adjoint of the weight one modular form
$g_{\alpha}$.

Other interesting cases arise when both $g$ and $h$ are theta series attached
to the same quadratic imaginary field where the prime $p$ splits. In [RiRo1,
Section 6], the authors prove a formula establishing an explicit connection
between the Hida–Rankin $p$-adic $L$-function attached to the pair of modular
forms $(g,g^{*})$ and Katz’s two variable $p$-adic $L$-function. Indeed, let
$g=\theta(\psi)$, the theta series attached to the character $\psi$. Then,
[RiRo1, Theorem 6.2] asserts that for any $s\in\mathbb{Z}_{p}$ the following
equality holds up to multiplication by $F^{\times}$:

(38)
$L_{p}(g,g^{*},s)=\frac{1}{\log_{p}(u_{\psi_{\operatorname{ad}}})}\cdot\zeta_{p}(s)\cdot
L_{p}(\chi_{K}\omega,s)\cdot
L_{\mathfrak{p}}(K,\psi_{\operatorname{ad}})(\mathcal{N}^{s}),$

being $\psi_{\operatorname{ad}}=\psi/\psi^{\prime}$. Note that
$\psi_{\mathrm{ad}}$ is a ring class character, regardless of whether $\psi$
is so or not. Then, according to [RiRo1],

(39) $L_{p}(g,g^{*},0)=\log_{p}(v_{1})\pmod{F^{\times}},$

where $v_{1}$ is the norm of a generator $v$ of the one-dimensional space
$(\mathcal{O}_{H}^{\times}[1/p]\otimes F)^{G_{\mathbb{Q}}}$.

Observe that according to [RiRo1, Section 5.3], this gives a link between
Beilinson–Flach elements and the cohomology classes coming from elliptic units
via the Kummer map, expressed as

(40) $\kappa^{\prime}(g_{\alpha},g_{1/\beta}^{*})=\log_{p}(v_{1})\cdot
v\pmod{F^{\times}}.$

Additionally, the derived Beilinson–Flach element is also related with the
cohomology class $\kappa_{\psi\mathcal{N}}$ via the factorization formula (38)
and the results of the preceding sections.

As sketched in [RiRo2, Section 5.2], the factorization formula (38) admits a
counterpart in the case where $g$ and $h$ are no longer self-dual. In this
case,

(41)
$L_{p}(g,h,0)=\log^{-+}(\kappa_{p}(g_{\alpha},h_{\alpha}))=\frac{\log_{p}(u_{\psi_{1}})\cdot\log_{p}(u_{\psi_{2}})}{\log_{p}(u_{g_{\alpha}})}\pmod{F^{\times}},$

where $\psi_{1}=\psi_{g}\psi_{h}$ and $\psi_{2}=\psi_{g}\psi_{h}^{\prime}$,
$u_{\psi_{i}}$ is the elliptic unit attached to $\psi_{i}$ and
$u_{g_{\alpha}}$ is the Stark unit attached to the adjoint representation of
$g_{\alpha}$.

Then, and following [RiRo2],

(42) $\kappa(g_{\alpha},h_{\alpha})=\mathfrak{C}\cdot u_{2},$

with $\mathfrak{C}$ an explicit constant involving $u_{\psi_{1}}$,
$u_{g_{\alpha}}$ and certain periods explicitly described in loc. cit., and
$u_{2}=u_{\psi_{2}}u_{\psi_{2}^{\prime}}$, where as usual $\psi_{2}^{\prime}$
is the composition of $\psi_{2}$ with the complex conjugation.

An interesting observation is that the case where the Euler system of elliptic
units presents an exceptional zero never arises in the setting of [RiRo1], due
to the regularity assumptions which are assumed in loc. cit (the fact of $g$
being $p$-distinguished). Hence, our results may be seen as a degenerate case
of the theory of Beilinson–Flach elements for weight one modular forms.

Let us be more precise in this last sentence. Let ${\mathbf{g}}$ stand for the
Hida family of CM theta series whose weight $\kappa_{1}$ specialization has
characteristic Hecke polynomial at $p$ given by

$(x-\mathfrak{p}^{(\kappa_{1}-1)/h})(x-\bar{\mathfrak{p}}^{(\kappa_{1}-1)/h}).$

Similarly, let ${\mathbf{h}}$ be the canonical Hida family of CM forms, such
that its weight $\kappa_{2}$ specialization has characteristic Hecke
polynomial at $p$

$(x-\psi(\mathfrak{p})\mathfrak{p}^{(\kappa_{2}-1)/h})(x-\psi(\bar{\mathfrak{p}})\bar{\mathfrak{p}}^{(\kappa_{2}-1)/h}).$

Then, as we had already anticipated, Proposition 3.3 may be seen as a
degenerate case of [RiRo1, Proposition 3.2], where we do not consider the
twist by the cyclotomic character (we fix $s=0$). Observe that there, the role
played by Katz’s two-variable $p$-adic $L$-function is done not exactly by the
Hida–Rankin $p$-adic $L$-function, but by its product with the $c$-factor

(43) $c^{2}(1-\chi_{g}(c)^{-1}\chi_{h}(c)^{-1}),$

where $c$ is a fixed integer number relatively prime to $6pN_{g}N_{h}$.

######  Remark 5.1.

The connection between Beilinson–Flach elements and units (in this case
circular units) is also exploited in [Das], where the proof of the main
result, a factorization formula for the Rankin–Selberg $p$-adic $L$-function,
rests on a explicit comparison between a certain unit constructed via the
theory of Beilinson–Flach elements and a circular unit. However, the approach
used in loc. cit. is quite different, since the unit is constructed via the
specialization of the Beilinson–Flach class at a point of weight $(2,2,1)$.

###  5.2. Beilinson–Flach elements and exceptional zeros

As we have pointed out, elliptic units may be understood as a special case
inside the theory of Beilinson–Flach elements, where the two modular forms are
theta series attached to the same imaginary quadratic field. Hence, it is
reasonable to expect that the two phenomena we have described concerning
exceptional zeros also arise in this setting. This section serves to recall
the main characteristics of the exceptional zero phenomenon for
Beilinson–Flach elements, following closely the discussions of [RiRo1] and
[LZ2].

With the notations of the previous section, let ${\mathbf{g}}$ and
${\mathbf{h}}={\mathbf{g}}^{*}$ Hida families interpolating two self-dual
modular forms $g_{\alpha}$ and $h_{\alpha}=g_{1/\beta}^{*}$, respectively. We
assume that ${\mathbf{g}}\in\Lambda_{{\mathbf{g}}}[[q]]$, where
$\Lambda_{{\mathbf{g}}}$ is a finite flat extensions of the Iwasawa algebra
$\Lambda_{\operatorname{cyc}}=\mathbb{Z}_{p}[[\mathbb{Z}_{p}^{\times}]]$.
Write $\mathcal{W}=\operatorname{Spf}(\Lambda_{\operatorname{cyc}})$ and
$\mathcal{W}_{{\mathbf{g}}}=\operatorname{Spf}(\Lambda_{{\mathbf{g}}})$. Let
$y_{0}$ be a weight one point of $\Lambda_{{\mathbf{g}}}$ such that
${\mathbf{g}}_{y_{0}}=g_{\alpha}$ and
${\mathbf{g}}^{*}_{y_{0}}=g_{1/\beta}^{*}$.

The work of [KLZ] attaches to $({\mathbf{g}},{\mathbf{g}}^{*})$ a three-
variable family of cohomology classes $\kappa({\mathbf{g}},{\mathbf{g}}^{*})$
parameterized by points
$(y,z,s)\in\mathcal{W}_{{\mathbf{g}}}\times\mathcal{W}_{{\mathbf{g}}}\times\mathcal{W}$
of weights $(\ell,m,s)$. More precisely, if $\mathbb{V}_{{\mathbf{g}}}$ and
$\mathbb{V}_{{\mathbf{g}}^{*}}$ stand for Hida’s $\Lambda$-adic Galois
representations afforded by ${\mathbf{g}}$ and ${\mathbf{g}}^{*}$,
respectively, and $\underline{\varepsilon}_{\operatorname{cyc}}$ is the
$\Lambda$-adic cyclotomic character,

$\kappa({\mathbf{g}},{\mathbf{g}}^{*})\in
H^{1}(\mathbb{Q},\mathbb{V}_{{\mathbf{g}}}\hat{\otimes}\mathbb{V}_{{\mathbf{g}}^{*}}\hat{\otimes}\Lambda_{\operatorname{cyc}}(\underline{\varepsilon}_{\operatorname{cyc}}^{-1})(1)).$

Observe that here, and as a matter of convention, we have used the inverse of
the tautological action over $\Lambda_{\operatorname{cyc}}$, and this is why
we have written
$\Lambda_{\operatorname{cyc}}(\underline{\varepsilon}_{\operatorname{cyc}}^{-1})$
(we may avoid this by invoking the appropriate functional equation).

As it follows from the discussion of [KLZ, Sections 8,10], the three-variable
Hida–Rankin $p$-adic $L$-function $L_{p}({\mathbf{g}},{\mathbf{g}}^{*})$ is
the image of the class $\kappa({\mathbf{g}},{\mathbf{g}}^{*})$ under a Perrin-
Riou map. Again, the numerator or the denominator may vanish in some
exceptional cases.

For the precise statements concerning Beilinson–Flach classes, we refer the
reader to the notations of [RiRo1]. As a first observation, we have that in
this self-dual case, and according to [Das, Theorem 9.4] and the computations
of [RiRo1], one has that $L_{p}(g,g^{*},0)=L_{p}(g,g^{*},1)$. The main results
we want to discuss here are the following ones:

  1. (i)

When we specialize both ${\mathbf{g}}$ and ${\mathbf{g}}^{*}$ at a fixed
weight one point $y_{0}$, and the cyclotomic variable $s$ is set as $s=0$, the
denominator of the Perrin-Riou map is zero and the cohomology class
$\kappa({\mathbf{g}},{\mathbf{g}}^{*})(y_{0},y_{0},0)$ vanishes. Then, the
explicit reciprocity law of [KLZ] is substituted by a derived reciprocity law
relating the derived cohomology class with $L_{p}(g,g^{*},0)$, up to
multiplication by an $\mathcal{L}$-invariant. This is precisely equation (37).

  2. (ii)

When we specialize both ${\mathbf{g}}$ and ${\mathbf{g}}^{*}$ at weight one,
and the cyclotomic variable $s$ is set as $s=1$ the numerator of the Perrin-
Riou map is zero but $L_{p}(g,g^{*},1)$ does not vanish (at least
generically). This is because [KLZ, Thm. B] contains the correction factor of
(43) at the $L$-function side, which vanishes in this case.

Exceptional vanishing of the denominator of the Perrin-Riou big logarithm. The
denominator of the Perrin-Riou regulator introduced in [RiRo1, Prop.3.2]
vanishes at all points $(y,y,\ell-1)$ of weight $(\ell,\ell,\ell-1)$. In
particular, specializing ${\mathbf{g}}$ and ${\mathbf{g}}^{*}$ at the weight
one modular forms $g$ and $g^{*}$ respectively, it turns out that

$\log^{-+}(\kappa_{p}^{\prime}({\mathbf{g}},{\mathbf{g}}^{*})(y_{0},y_{0},0))=\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))\cdot
L_{p}(g,g^{*},0)=\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))\cdot
L_{p}(g,g^{*},1)\pmod{F^{\times}}.$

To shorten our notations, we have written
$\kappa_{p}^{\prime}({\mathbf{g}},{\mathbf{g}}^{*})$ for the localization at
$p$ of the class.

We would like to emphasize the similitude with our main results for elliptic
units, which also relate the logarithm of the derived cohomology class with a
special value of Katz’s two-variable $p$-adic $L$-function, up to
multiplication by a certain $\mathcal{L}$-invariant.

It turns out that the special value
$L_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,0)$ is related via the functional
equation with $L_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,1)$ and here, to
determine its value, we can follow the same approach than in this note: over
the line corresponding to those points of weight $(\ell,\ell,\ell)$, the
$p$-adic $L$-function factors due to the analyticity of an Euler factor (this
is properly developed in [Hi]), and this allows us to obtain an explicit
expression of the special value $L_{p}(g,g^{*},1)$ via Galois deformation
techniques. This expression involves units and $p$-units in the field cut out
by the Galois representation $V_{gg^{*}}$.

Exceptional vanishing of the numerator of the Perrin-Riou big logarithm. The
results we present now closely follow [LZ2] and are the counterpart of those
developed in [RiRo1]. We include it here for the sake of completeness, and to
illustrate how this exceptional phenomenon arises in a setting which is
germane to ours.

Consider specializations of $\kappa({\mathbf{g}},{\mathbf{g}}^{*})$ at weights
$(y,z,m)$, where $\operatorname{wt}(z)=m$. Then, if $\alpha_{g_{y}}$ and
$\beta_{g_{y}}$ stand for the eigenvalues of the $p$-th Hecke polynomial of
$g_{y}$, with
${\mathrm{ord}}(\alpha_{g_{y}})\leq{\mathrm{ord}}(\beta_{g_{y}})$, the Euler
factor in the numerator of the Perrin-Riou map is

$1-\frac{\alpha_{g_{z}}}{\alpha_{g_{y}}}.$

This factor is zero for all points of weight $(\ell,\ell,\ell)$. But this does
not mean that the $p$-adic $L$-function vanishes at those points, since the
explicit reciprocity law of [KLZ, Thm. B] contains the factor

$c^{2}-c^{2s+2-\ell-m}=c^{2}(1-c^{m-\ell})$

multiplying the value of $L_{p}({\mathbf{g}},{\mathbf{g}}^{*})$, where $c$ is
a fixed positive integer coprime with both $6p$ and the level of $g$. At the
points where $\operatorname{wt}(z)=m$, the Perrin-Riou map interpolates the
Bloch–Kato dual exponential map, and

$c^{2}(1-c^{m-\ell})\Big{(}1-\frac{\alpha_{g_{y}}}{p\alpha_{g_{z}}}\Big{)}L_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,z,m)=\Big{(}1-\frac{\alpha_{g_{z}}}{\alpha_{g_{y}}}\Big{)}\cdot\exp_{\operatorname{BK}}^{*-+}(\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,z,s)),$

where $\exp_{\operatorname{BK}}^{*-+}$ stands for the composition of the
projection to a certain subspace of $V_{gg^{*}}\otimes F_{p}(1)$ followed by
the dual exponential map and the pairing with the canonical differentials.

Since both sides of the previous equation vanish along the line $y=z$,
$\operatorname{wt}(z)=m$, we may consider the derivative at a point
$(y,y,\ell)$, obtaining the expression

$c^{2}(1-p^{-1})\dot{\log}_{p}(c)L_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell)=\Big{(}\frac{-\alpha_{{\mathbf{g}}_{y}}^{\prime}}{\alpha_{{\mathbf{g}}_{y}}}\Big{)}\cdot\exp_{\operatorname{BK}}^{*-+}(\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell)),$

up to multiplication by $F^{\times}$. Here, $\alpha_{{\mathbf{g}}}^{\prime}$
stands for the derivative of the Iwasawa function $\alpha_{{\mathbf{g}}}$
along the weight direction.

Additionally, invoking Hida’s result on the existence of an improved $p$-adic
$L$-function [Hi], we get that, whenever $\operatorname{wt}(y)=1$,

$\log_{p}(c)=\exp_{\operatorname{BK}}^{*-+}(\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell))\pmod{F^{\times}}.$

Observe now that

$L_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell)=L_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell-1)\pmod{F^{\times}},$

and hence

$L_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell-1)=\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))^{-1}\cdot\log^{-+}(\kappa_{p}^{\prime}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell-1))\pmod{F^{\times}}.$

Consequently, up to multiplication by $F^{\times}$, we have the equality

(44)
$\frac{\log_{p}(c)}{\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))^{2}}\cdot\log^{-+}(\kappa^{\prime}_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell-1))=\exp_{\operatorname{BK}}^{*-+}(\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell))$

In particular, modulo $F^{\times}$, one has

(45)
$\frac{\log_{p}(c)}{\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))^{2}}\cdot\log^{-+}(\kappa^{\prime}_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y_{0},y_{0},0))=\exp_{\operatorname{BK}}^{*-+}(\kappa_{p}({\mathbf{g}},{\mathbf{g}}^{*})(y_{0},y_{0},1)).$

Observe that this is coherent with the computations of [RiRo1], from where it
follows that, up to multiplication by a scalar in $F^{\times}$,

$\log^{-+}(\kappa_{p}^{\prime}({\mathbf{g}},{\mathbf{g}}^{*})(y,y,\ell-1))=\mathcal{L}(\operatorname{ad}^{0}(g_{\alpha}))^{2}.$

This is a consequence of Hida’s improved factorization and the derived version
of the explicit reciprocity law of [KLZ].

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