# Eisenstein congruences among Euler systems

Óscar Rivero and Victor Rotger  O. R.: Simons Laufer Mathematical Sciences
Institute, 17 Gauss Way, Berkeley, CA 94720, United States of America
<EMAIL_ADDRESS>V. R.: IMTech, UPC and Centre de Recerca Matemàtiques,
C. Jordi Girona 1-3, 08034 Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract.

We investigate Eisenstein congruences between the so-called Euler systems of
Garrett–Rankin–Selberg type. This includes the cohomology classes of
Beilinson–Kato, Beilinson–Flach and diagonal cycles. The proofs crucially rely
on different known versions of the Bloch–Kato conjecture, and are based on the
study of the Perrin-Riou formalism and the comparison between the different
$p$-adic $L$-functions.

###### 2010 Mathematics Subject Classification:

11F33 (primary); 11F67, 11F80 (secondary).

###### Contents

  1. 1 Introduction
  2. 2 Background: Euler systems and reciprociy laws
  3. 3 Congruences between Beilinson–Flach and Beilinson–Kato classes
  4. 4 Congruences between diagonal cycles and Beilinson–Flach elements

##  1\. Introduction

The theory of Euler systems is a powerful machinery that provides deep insight
on the arithmetic of Galois representations, Selmer groups and Iwasawa main
conjectures. There are relatively few instances of such Euler systems in the
literature, and their construction is far from being a systematic issue. This
work focuses on the interaction between different kinds of Euler systems, and
can be seen as a natural continuation of our previous article [RiRo], where we
had studied congruence relations between the system of circular units and
Beilinson–Kato elements.

This note is concerned with the theory of Eisenstein congruences and the
development of a $p$-adic Artin formalism at the level of Euler systems. The
easiest instance of an Euler system is that of circular units, obtained as a
weighted combination of cyclotomic units after applying the Kummer map. Its
image under the Perrin-Riou regulator (a. k. a. Coleman map) allows us to
recover the Kubota–Leopoldt $p$-adic $L$-function.

The three Euler systems we want to discuss here are slightly harder to
describe. They are sometimes referred as Euler systems of
Garrett–Rankin–Selberg type, and the most well-known example comes from Kato’s
work [Ka], which allowed the proof of the Bloch–Kato conjecture for elliptic
curves defined over $\mathbb{Q}$ in rank 0 and of one of the divisibilities of
the Iwawawa main conjecture. The other two instances come from generalizations
of that previous construction, when the two modular units involved in Kato’s
construction are replaced by cuspidal forms. The case of Beilinson–Flach
classes has been explored in different works of Bertolini, Darmon and Rotger
[BDR2], and Kings, Lei, Loeffler and Zerbes [LLZ], [KLZ2], [KLZ]. The case of
diagonal cycles, where a proper Euler system is not available (but a family of
cohomology classes varying over Hida or Coleman families) has been worked out
by Darmon and Rotger [DR2], [DR4] and by Bertolini, Seveso and Venerucci
[BSV1], [BSV2].

###  1.1. The set-up

Let $N,k>1$ be positive integers,
$\chi_{f}:(\mathbb{Z}/N\mathbb{Z})^{\times}\rightarrow\bar{\mathbb{Q}}^{\times}$
a Dirichlet character and $f\in S_{k}(N,\chi_{f})$ a normalized cuspidal
eigenform of level $N$, weight $k$ and nebentype $\chi_{f}$. Fix a prime
$p\nmid 6N\varphi(N)$. Similarly, we fix eigenforms $g\in
S_{\ell}(N,\chi_{g})$ and $h\in S_{m}(N,\chi_{h})$, with $\ell,m\geq 2$ and
$\chi_{g},\chi_{h}$ Dirichlet characters. Let $F$ be the finite extension of
$\mathbb{Q}$ generated by the field of coefficients of $f$, $g$, $h$ and the
values of all Dirichlet characters of conductor $N$; let $\mathcal{O}$ be its
ring of integers and $\mathfrak{p}\subset{\mathcal{O}}$ a prime ideal above
$p$. Before continuing, we need to introduce the following objects:

  * -

$\mathscr{H}_{\mathbb{Z}_{p}}$ is the étale $\mathbb{Z}_{p}$-sheaf on the open
modular curve $Y_{1}(N)$ given by the Tate module of the universal elliptic
curve $\mathcal{E}/Y_{1}(N)$, as introduced in [KLZ, Def. 3.1.1] or [KLZ2,
§2.3].

  * -

$\operatorname{TSym}^{k}\mathscr{H}_{\mathbb{Z}_{p}}$ is the sheaf of
symmetric tensors of degree $k-2$ over $\mathscr{H}_{\mathbb{Z}_{p}}$ defined
in [KLZ, Def. 3.1.2] or [KLZ2, §2.2]; after inverting $(k-2)!$ it is
isomorphic to the $(k-2)$-th symmetric power.

  * -

$T_{f,Y}$ is the integral $p$-adic Galois representations given as the
$f$-isotypical quotient of
$H^{1}_{\operatorname{et}}(\bar{Y},\operatorname{TSym}^{k-2}(\mathscr{H}_{\mathbb{Z}_{p}})(1))\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{\mathfrak{p}}$
of the closed modular curve of level $\Gamma_{1}(N)$, as defined in [RiRo, eq.
(11)] in the case of trivial coefficients; see also (1).

  * -

$T_{f,X}$ is defined in the same way as $T_{f,Y}$ (see [RiRo, eq. (11)]), but
using compactly supported cohomology instead. We set, as usual,
$V_{f,Y}=T_{f,Y}\otimes\mathbb{Q}$ and $V_{f,X}=T_{f,Y}\otimes\mathbb{Q}$, and
note that it holds $V_{f,Y}=V_{f,X}$.

We also attach analogous lattices to $g$ and $h$. All along the article, we
impose that $k+\ell+m$ is even and that $\chi_{f}\chi_{g}\chi_{h}=1$. We
assume that the triple $(k,\ell,m)$ is balanced, that is, $\ell+m\geq k$,
$m+k\geq\ell$ and $k+\ell\geq m$. This means that we are in the so-called
geometric region, where a construction of cohomology classes in an appropriate
$H^{1}$ is meaningful and the corresponding $L$-value vanishes. For
simplicity, we also assume $\chi_{h}\not\equiv 1\pmod{\mathfrak{p}}$.

Along this short note we deal with three different objects, whose
characterizations and main properties are recalled in Section §2. According to
the discussions of [BSV2, Rk. 3.3], the global classes we introduce may have
some bounded denominators, but along the introduction we assume that they are
integral at $p$, which is always the case provided that $p$ is large enough
(in particular, we need that $p>\max\\{k,\ell,m\\}$ to invert the factorials
appearing in the formulas of loc. cit.); we come back to this issue later on
in the text.

Set $c=(k+\ell+m-2)/2$ and consider:

  1. (a)

The Beilinson–Kato class $\kappa_{f}\in
H^{1}(\mathbb{Q},T_{f,Y}(1)(\ell+m-c-1)$, whose characterization is recalled
in Section 2.3 following [Ka] and [BD].

  2. (b)

The Beilinson–Flach class $\kappa_{f,g}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}(m-c))$; we recall its main properties in Section 2.4 following [BDR1],
[BDR2] and [KLZ].

  3. (c)

The diagonal cycle class $\kappa_{f,g,h}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes T_{h,Y}(1-c))$, introduced in Section 2.5 following [DR1],
[DR2] and [BSV1].

Moreover, and as a piece of notation, we denote $\bar{\kappa}_{f}$,
$\bar{\kappa}_{f,g}$, $\bar{\kappa}_{f,g,h}$ the reduction of the above
integral classes modulo a suitable power of the prime ideal $\mathfrak{p}$
(usually denoted by $\mathfrak{p}^{t}$).

###  1.2. Main results

We use the notations about Eisenstein series introduced in [RiRo]; in
particular, they are indexed by the weight and by a pair of Dirichlet
characters whose products of nebentypes is $N$. When $g$ is congruent modulo
$\mathfrak{p}^{t}$ with the Eisenstein series $E_{\ell}(\chi_{g},1)$, as we
discussed in loc. cit., there is a projection map

$T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}\longrightarrow\mathcal{O}/\mathfrak{p}^{t}(\chi_{g}).$

Hence, the class $\bar{\kappa}_{f,g}$ projects to an element

$\bar{\kappa}_{f,g,1}\in
H^{1}(\mathbb{Q},T_{f,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(\chi_{g})(m-c)).$

When this projection vanishes modulo $\mathfrak{p}^{t}$, and proceeding as in
[RiRo, §4], we may lift $\kappa_{f,g}$ to a class in
$H^{1}(\mathbb{Q},T_{f,Y}\otimes T_{g,X})$ and consider instead the projection

$T_{g,X}\otimes\mathcal{O}/\mathfrak{p}^{t}\longrightarrow\mathcal{O}/\mathfrak{p}^{t}(\ell-1).$

Thus, we obtain a class

$\bar{\kappa}_{f,g,2}\in
H^{1}(\mathbb{Q},T_{f,Y}(\ell+m-c-1)\otimes\mathcal{O}/\mathfrak{p}^{t}).$

To state our first theorem, let $L_{p}(f,\chi_{g},s)$ stand for the
Mazur–Swinnerton-Dyer $p$-adic $L$-function attached to the pair
$(f,\chi_{g})$, with the conventions of [RiRo, §3]. The result depends on a
weak Gorenstein assumption (properly introduced as Assumption 2.1) and on the
conditions introduced as Assumption 3.1 and Assumption 3.2. The latter makes
reference to the conditions of big image and non-exceptional zeros which give
a Bloch–Kato type result, which allows us to conclude that the corresponding
spaces on which the classes live are one-dimensional. As discussed in the
corresponding section, these results are known to hold in many different
instances, so the assumptions are quite mild in nature.

######  Theorem 1.1.

Suppose that Assumption 2.1 holds for $g$, that the Galois representation
$T_{f,Y}$ is absolutely irreducible, and that Assumptions 3.1 and 3.2 also
hold.

If $L_{p}(f,\chi_{g},m-c+1)\neq 0\pmod{\mathfrak{p}^{t}}$, then
$\bar{\kappa}_{f,g,1}=0$ and the following congruence holds in
$H^{1}(\mathbb{Q},T_{f,Y}(\ell+m-c-1)\otimes\mathcal{O}/\mathfrak{p}^{t})$:

$\bar{\kappa}_{f,g,2}=\bar{\kappa}_{f}.$

The proof is inspired by the ideas introduced in our previous work, using the
Perrin-Riou formalism to perform a comparison at the level of $p$-adic
$L$-functions. However, here, we have to proceed in a slightly different way:
the first natural projection is 0 because of the local condition satisfied by
the Beilinson–Flach class, and it is at second order where we can do a proper
comparison.

We now mimic that approach in the framework of diagonal cycles. When $h$ is
congruent modulo $\mathfrak{p}^{t}$ with $E_{m}(\chi_{h},1)$ we may define a
class

$\bar{\kappa}_{f,g,h,1}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(\chi_{h})(1-c)).$

When it vanishes, and proceeding as in the previous case, we have the
refinement

$\bar{\kappa}_{f,g,h,2}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}(m-c)\otimes\mathcal{O}/\mathfrak{p}^{t}).$

Let $L_{p}(f,g,\chi_{h},c)$ (resp. $L_{p}(g,f,\chi_{h},c)$) stand for the
value at $s=c$ of the Rankin–Selberg $p$-adic $L$-function attached to
$(f,g,\chi_{h})$ with $f$ dominant (resp. $g$ dominant), as recalled in
Section 2.4.

######  Theorem 1.2.

Suppose that Assumption 2.1 holds for $h$, that the Galois representations
$T_{f,Y}$ and $T_{g,Y}$ are absolutely irreducible, and that Assumptions 4.1
and 4.2 hold.

If $L_{p}(f,g,\chi_{h},c)\neq 0\pmod{\mathfrak{p}^{t}}$ or
$L_{p}(g,f,\chi_{h},c)\neq 0\pmod{\mathfrak{p}^{t}}$, then
$\bar{\kappa}_{f,g,h,1}=0$ and the following congruence holds in
$H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}(m-c)\otimes\mathcal{O}/\mathfrak{p}^{t})$:

$\bar{\kappa}_{f,g,h,2}=\bar{\kappa}_{f,g}.$

We believe that most of the results we discuss are amenable to generalizations
to Hida or Coleman families, but in that case one must be more circumspect
about integrality issues. Deformations to weight one of the above congruence
formulae may be particularly interesting, given their potential connection to
the equivariant Birch and Swinnerton-Dyer conjecture. Furthermore, it would be
interesting to explore whether our results could have applications to non-
vanishing results, as in [Va, §3], or to the study of classical problems like
Goldfeld’s conjecture, as in [KrLi], where they establish congruences between
elliptic units and Heegner points using a comparison between the $p$-adic
$L$-functions of Katz and Bertolini–Darmon–Prasanna.

We finally point out that some of the results we state contain several
assumptions which are likely to be relaxed, but we have not pursued this in an
attempt to keep this work short and to avoid technical complications in that
regard.

###  1.3. Relation to other works

This note grew up as an attempt to better understand the interactions among
Euler systems coming from Eisenstein congruences with those discussed by
Loeffler and the first author [LR], where Coleman families passing through the
critical $p$-stabilization of an Eisenstein series are studied. In that
setting, we obtain analogous relations (in characteristic zero) between
Beilinson–Flach and Beilinson–Kato classes, and between Beilinson–Kato and
Beilinson–Flach. Compare [LR, Thms. C1.13, C2.11] in loc. cit. with the main
result of this paper. The reason behind this parallelism is that in both
scenarios we have two natural filtrations (one corresponding to the usual
étale cohomology of the modular curve and the other to étale cohomology with
compact support) whose interplay is used in an analogous way. To be more
precise, in this article we consider the lattices attached to the open and
closed modular curve, that we denote by $T_{f,Y}$ and $T_{f,X}$, respectively.
In this case, there is an inclusion $T_{f,Y}\supset T_{f,X}\supset I\cdot
T_{f,Y}\supset I\cdot T_{f,X}\supset\ldots$, where $I$ is the Eisenstein
ideal. This is a parallel situation to what we had described in the discussion
after [LR, Cor. 6.1.3], where the maximal ideal $\mathfrak{m}$ considered in
loc. cit. plays the role of the Eisenstein ideal. We hope that the
interactions between these two settings can be pushed forward in subsequent
works.

Another recent work dealing with the connections between different kinds of
Euler systems is the article of Bertolini–Darmon–Venerucci [BDV], where they
prove a conjecture of Perrin-Riou relating Heegner points and Beilinson–Kato
elements via the use of Beilinson–Flach elements. We believe that this kind of
techniques can have multiple applications to arithmetic problems and to the
construction of new Euler systems, and we expect to come back to these issues
in forthcoming work.

###  1.4. Acknowledgments and funding

We thank Kazim Büyükboduk and David Loeffler for stimulating comments and
discussions around the topics studied in this note.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 682152). O.R. was further supported by the Royal Society
Newton International Fellowship NIF\R1\202208. V. R. is supported by Icrea
through an Icrea Academia Grant. This material is based upon work supported by
the National Science Foundation under Grant No. DMS-1928930 while the first
author was in residence at the Mathematical Sciences Research Institute in
Berkeley, California, during the Spring 2023 semester.

##  2\. Background: Euler systems and reciprociy laws

###  2.1. Modular curves and lattices

We begin by recalling some notations about modular curves and lattices. For
that purpose, fix algebraic closures $\bar{\mathbb{Q}}$,
$\bar{\mathbb{Q}}_{p}$ of $\mathbb{Q}$ and $\mathbb{Q}_{p}$ respectively, and
embeddings of $\bar{\mathbb{Q}}$ into $\bar{\mathbb{Q}}_{p}$ and $\mathbb{C}$.
The former singles out a prime ideal $\mathfrak{p}$ of ${\mathcal{O}}$ lying
above $p$ and we let $\mathcal{O}_{\mathfrak{p}}$ denote the completion of
$\mathcal{O}$ at $\mathfrak{p}$. We also fix throughout an uniformizer
$\varpi$ of $\mathcal{O}_{\mathfrak{p}}$ and an isomorphism
$\mathbb{C}_{p}\simeq\mathbb{C}$.

Given a variety $Y/\mathbb{Q}$ and a field extension $F/\mathbb{Q}$, let
$Y_{F}=Y\times F$ denote the base change of $Y$ to $F$ and set
$\bar{Y}=Y_{\bar{\mathbb{Q}}}$. Fix an integer $N\geq 3$ and let
$Y_{1}(N)\subset X_{1}(N)$ denote the canonical models over $\mathbb{Q}$ of
the (affine and projective, respectively) modular curves classifying pairs
$(A,i)$ where $A$ is a (generalized) elliptic curve and $i:\mu_{N}\rightarrow
A$ is an embedding of group schemes.

Let $k\geq 2$ be an integer and $\theta$ a Dirichlet character. For an
ordinary newform $f\in S_{k}(N,\theta)$ satisfying the congruence $f\equiv
E_{k}(\theta,1)\,\,\mathrm{mod}\,\mathfrak{p}^{t}$, let $f(q)=\sum
a_{n}(f)q^{n}$ denote its $q$-expansion at the cusp $\infty$. Enlarge $F$ so
that it also contains the eigenvalues $\\{a_{n}(f)\\}_{n\geq 1}$, and still
write ${\mathcal{O}}$ for its ring of integers and
${\mathcal{O}}_{{\mathfrak{p}}}$ for its completion at the prime ideal
$\mathfrak{p}$ over $p$. For our further use, we denote by
$\alpha_{f}\in{\mathcal{O}}_{\mathfrak{p}}^{\times}$ (resp.
$\beta_{f}\in{\mathcal{O}}_{\mathfrak{p}}$) the unit root (resp. non-unit
root) of the $p$-th Hecke polynomial of $f$. Let
$\psi_{f}:G_{\mathbb{Q}_{p}}\longrightarrow{\mathcal{O}}_{\mathfrak{p}}^{\times}$
denote the unramified character characterized by
$\psi_{f}(\mathrm{Fr}_{p})=\alpha_{f}$.

Let $I_{f}^{*}=(T^{*}_{\ell}-a_{\ell}(f))\subset\mathbb{T}^{*}$ denote the
ideal associated to the system of eigenvalues of $f$ with respect to the dual
Hecke operators. As in the introduction, consider the
${\mathcal{O}}_{{\mathfrak{p}}}$-module

(1)
$T_{f,Y}=H^{1}_{\operatorname{et}}(\bar{Y},\operatorname{TSym}^{k-2}(\mathscr{H}_{\mathbb{Z}_{p}})(1))\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{\mathfrak{p}}/I^{*}_{f},$

and define similarly $T_{f,X}$, using compact support cohomology instead.

Note that the two lattices $T_{f,X}$ and $T_{f,Y}$ may give rise to different
${\mathcal{O}}_{{\mathfrak{p}}}[G_{\mathbb{Q}}]$-modules in spite of the fact
that the associated rational Galois representations

$V_{f}:=T_{f,X}\otimes F_{{\mathfrak{p}}}\simeq T_{f,Y}\otimes
F_{{\mathfrak{p}}}$

are isomorphic.

Finally, let $\Sigma_{X}$ and $\Sigma_{Y}$ denote the torsion submodules of
$T_{f,X}$ and $T_{f,Y}$ respectively, and let

(2)
$T_{f,Y,\circ}^{\operatorname{quo}}:=T_{f,Y}^{\operatorname{quo}}/\Sigma_{Y}\simeq{\mathcal{O}}_{\mathfrak{p}}(\psi_{f})$

denote the free quotient.

Since $f\equiv E_{k}(\theta,1)\,\mbox{mod}\,\mathfrak{p}^{t}$, we have

(3)
$T_{f,Y,\circ}^{\operatorname{quo}}\otimes\mathcal{O}/\mathfrak{p}^{t}\simeq\mathcal{O}/\mathfrak{p}^{t}(\theta),$

which amounts to the congruence
$\psi_{f}\equiv\theta\,(\mbox{mod}\,\mathfrak{p}^{t})$ as unramified
characters of $G_{\mathbb{Q}_{p}}$.

In the cases where we deal with several modular forms of common level $N$, we
slightly abuse notations and still write $F$ for the finite extension of
$\mathbb{Q}$ generated by the field of coefficients of all of the them and the
values of all Dirichlet characters of conductor $N$. Similarly, $\mathcal{O}$
stands for the ring of integers and $\mathfrak{p}\subset\mathcal{O}$ for a
prime ideal above $p$.

As in our previous work, we need the following assumption to establish our
main results.

######  Assumption 2.1.

The $G_{\mathbb{Q}_{p}}$-module $\mathcal{O}/\mathfrak{p}^{t}(\theta)$ does
not show up as a quotient of $\Sigma_{Y}/\Sigma_{X}$.

In this case, we will say that $T_{f,Y}$ (or simply $f$, for short) satisfies
the weak-Gorenstein condition. The reason is that this assumption
automatically follows if the localization of the Hecke algebra acting on
$M_{k}(\Gamma_{1}(N))$ at the Eisenstein ideal is Gorenstein.

Several of our results are suitably formulated in the language of families.
Let $\mathcal{W}$ denote the weight space defined as the formal spectrum of
the Iwasawa algebra
$\Lambda=\mathcal{O}_{\mathfrak{p}}[[\mathbb{Z}_{p}^{\times}]]$. The set of
classical points in $\mathcal{W}$ is given by characters $\nu_{s,\xi}$ of the
form $\xi\varepsilon_{\operatorname{cyc}}^{s-1}$ where $\xi$ is a Dirichlet
character of $p$-power conductor, $\varepsilon_{\operatorname{cyc}}$ is the
cyclotomic character and $s$ is an integer; this forms a dense subset in
$\mathcal{W}$ for the Zariski topology. Let $\mathcal{W}^{\circ}$ further
denote the set of those points with $\xi=1$; we shall often write $s$ in place
of $\nu_{s}=\nu_{s,1}$. Let $\mathcal{W}^{\pm}\subset\mathcal{W}$ denote the
topological closure of the set of points
$\xi\varepsilon_{\operatorname{cyc}}^{s-1}$ with $(-1)^{s-1}\xi(-1)=\pm 1$. We
have $\mathcal{W}=\mathcal{W}^{+}\sqcup\mathcal{W}^{-}$ and we write
$\Lambda=\Lambda^{+}\oplus\Lambda^{-}$ for the corresponding decomposition of
the Iwasawa algebra.

###  2.2. Congruences between canonical periods

Recall that the de Rham Dieudonné module
$D_{\operatorname{\mathrm{dR}}}(V_{f})$ associated to the eigenform $f$ is a
$F_{\mathfrak{p}}$-filtered vector space of dimension $2$. Poincaré duality
yields a perfect pairing

$\langle\,,\,\rangle:D_{\operatorname{\mathrm{dR}}}(V_{f}(-1))\times
D_{\operatorname{\mathrm{dR}}}(V_{f^{*}})\rightarrow F_{\mathfrak{p}}$

and there is an exact sequence of Dieudonné modules

(4) $0\rightarrow
D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{sub}})\rightarrow
D_{\operatorname{\mathrm{dR}}}(V_{f})\rightarrow
D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{quo}})\rightarrow 0,$

where $D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{sub}})$ and
$D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{quo}})$ have both
dimension $1$.

Falting’s theorem associates to $f$ a regular differential form
$\omega_{f}\in\mathrm{Fil}(D_{\operatorname{\mathrm{dR}}}(V_{f}))$, which
gives rise to an element in
$D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{quo}})$ via the right-
most map in (4) and in turn induces a linear form

(5)
$\omega_{f}\,:\,D_{\operatorname{\mathrm{dR}}}(V_{f^{*}}^{\operatorname{sub}}(-1))\rightarrow
F_{\mathfrak{p}},\quad\eta\mapsto\langle\omega_{f},\eta\rangle$

that we continue to denote with same symbol by a slight abuse of notation.

There is also a differential $\eta_{f}$, which is characterized by the
property that it spans the line
$D_{\operatorname{\mathrm{dR}}}(V_{f}^{\operatorname{sub}}(-1))$ and is
normalized so that

(6) $\langle\eta_{f},\omega_{f^{*}}\rangle=1.$

Again, it induces a linear form

(7)
$\eta_{f}\,:\,D_{\operatorname{\mathrm{dR}}}(V_{f^{*}}^{\operatorname{quo}})\rightarrow
F_{\mathfrak{p}},\quad\omega\mapsto\langle\eta_{f},\omega\rangle$

Now, let $T$ be an unramified
${\mathcal{O}}_{\mathfrak{p}}[\mathrm{Gal\,}(\bar{\mathbb{Q}}_{p}/\mathbb{Q}_{p})]$-module
and set $V=T\otimes F_{\mathfrak{p}}$. Let
$\hat{\mathbb{Z}}_{p}^{\mathrm{ur}}$ denote the completion of the ring of
integers of the maximal unramified extension of $\mathbb{Q}_{p}$, and define
the integral Dieudonné module

$D(T):=(T\,\hat{\otimes}_{\mathbb{Z}_{p}}\,\hat{\mathbb{Z}}_{p}^{\mathrm{ur}})^{\mathrm{Fr}_{p}=1}.$

As shown in loc. cit. we have $D_{\operatorname{\mathrm{dR}}}(V)=D(T)\otimes
F_{\mathfrak{p}}$.

As explained e.g. in [FK, Prop. 1.7.6], there is a functorial isomorphism of
${\mathcal{O}}_{\mathfrak{p}}$-modules (forgetting the Galois structure) given
by

(8) $T\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}D(T).$

This map is not canonical as it depends on a choice of root of unity; for
$T={\mathcal{O}}_{\mathfrak{p}}(\chi)$ we take it to be given by the rule
$1\mapsto\,\mathfrak{g}(\chi)$.

Recall the free ${\mathcal{O}}_{\mathfrak{p}}$-quotient

$T_{f,Y,\circ}^{\operatorname{quo}}\simeq{\mathcal{O}}_{\mathfrak{p}}(\psi_{f});$

note in particular that $T_{f,Y,\circ}^{\operatorname{quo}}$ is unramified.

Recall the uniformizer $\varpi$ fixed at the outset in §2.1. Let
$C_{f^{*}}=(\varpi^{r})\subseteq O_{\mathfrak{p}}$ denote the congruence ideal
attached to $f^{*}$ as defined e.g. in [Oh3]. In Hida’s terminology,
$\varpi^{r}$, is sometimes called a congruence divisor. According to [KLZ2,
§10.1.2], the image of $\eta_{f^{*}|D(T_{f,Y}^{\operatorname{quo}})}$ is
precisely the inverse $C_{f^{*}}^{-1}$ of the congruence ideal, and hence
there is an isomorphism

$\eta_{f^{*}}\,:\,D(T_{f,Y,\circ}^{\operatorname{quo}})\longrightarrow
C_{f^{*}}^{-1},\quad\omega\mapsto\langle\eta_{f^{*}},\omega\rangle.$

Setting $\tilde{\eta}_{f^{*}}:=\varpi^{r}\cdot\eta_{f^{*}}$, the above map
gives rise to an isomorphism of $O_{\mathfrak{p}}$-modules

(9)
$\tilde{\eta}_{f^{*}}\,:\,D(T_{f,Y,\circ}^{\operatorname{quo}})\longrightarrow\mathcal{O}_{\mathfrak{p}},\quad\omega\mapsto\langle\tilde{\eta}_{f^{*}},\omega\rangle.$

Choose an isomorphism of $G_{\mathbb{Q}_{p}}$-modules.

(10)
$\bar{\iota}_{1}:\bar{T}_{f,Y,\circ}^{\operatorname{quo}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}/\mathfrak{p}^{t}(\theta).$

Fixing such a map amounts to choosing the class
$(\mathrm{mod}\,{\mathfrak{p}^{t}})$ of an isomorphism of local modules
$\iota_{1}:T_{f,Y,\circ}^{\operatorname{quo}}\simeq\mathcal{O}_{\mathfrak{p}}(\psi_{f})$.
In light of the functoriality provided by (8) this determines and is
determined by the class $(\mathrm{mod}\,{\mathfrak{p}^{t}})$ of an isomorphism
$D(\iota_{1}):D(T_{f,Y,\circ}^{\operatorname{quo}})\simeq
D({\mathcal{O}}_{\mathfrak{p}}(\psi_{f}))$.

We choose $\iota_{1}$ as the single isomorphism making the following diagram
commutative:

(11)
$\textstyle{D(T_{f,Y,\circ}^{\operatorname{quo}})\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\langle\,,\tilde{\eta}_{f^{*}}\rangle}$$\scriptstyle{D(\iota_{1})}$$\textstyle{\quad\mathcal{O}_{\mathfrak{p}}}$$\textstyle{D({\mathcal{O}}_{\mathfrak{p}}(\psi_{f}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
1/\mathfrak{g}(\theta)}$

Indeed, since both $\langle\,,\tilde{\eta}_{f^{*}}\rangle$ and $\cdot
1/\mathfrak{g}(\theta)$ are isomorphisms, it follows that such a map
$D(\iota_{1})$ exists and is unique, and this in turn pins down $\iota_{1}$
and $\bar{\iota}_{1}$ in light of (8).

There are similar results for the differential $\omega_{f}$. Using again
[KLZ2, §10.1.2], there is an isomorphism

$\omega_{f^{*}}\,:\,D(T_{f,Y}^{\operatorname{sub}})\longrightarrow\mathcal{O}_{\mathfrak{p}},\quad\eta\mapsto\langle\omega_{f^{*}},\eta\rangle.$

Hence, we may choose an isomorphism of $G_{\mathbb{Q}_{p}}$-modules

(12)
$\bar{\iota}_{2}:\bar{T}_{f,Y}^{\operatorname{sub}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{O}}/\mathfrak{p}^{t}(1).$

Fixing such a map amounts to choosing the class
$(\mathrm{mod}\,{\mathfrak{p}^{t}})$ of an isomorphism of local modules
$\iota_{2}:T_{f,Y}^{\operatorname{sub}}\simeq\mathcal{O}_{\mathfrak{p}}(k-1)$.
This determines and is determined by the class
$(\mathrm{mod}\,{\mathfrak{p}^{t}})$ of an isomorphism
$D(\iota_{2}):D(T_{f,Y}^{\operatorname{sub}})\simeq
D({\mathcal{O}}_{\mathfrak{p}})$.

We choose $\iota_{2}$ as the single isomorphism making the following diagram
commutative:

(13)
$\textstyle{D(T_{f,Y}^{\operatorname{sub}})\quad\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\langle\,,\omega_{f^{*}}\rangle}$$\scriptstyle{D(\iota_{2})}$$\textstyle{\quad\mathcal{O}_{\mathfrak{p}}}$$\textstyle{D({\mathcal{O}}_{\mathfrak{p}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
1}$

Indeed, since both $\langle\,,\omega_{f^{*}}\rangle$ and $\cdot 1$ are
isomorphisms, it follows that such a map $D(\iota_{2})$ exists and is unique,
and this in turn pins down $\iota_{2}$ and $\bar{\iota}_{2}$.

###  2.3. The Beilinson–Kato Euler system

In this section we recall the reciprocity law for Beilinson–Kato elements of
arbitrary weight. For details on the constructions, see e.g. [Ka] or [BD]. As
in the introduction, let $f\in S_{k}(N,\chi_{f})$, and consider two auxiliary
positive integers $(\ell,m)$, with $\ell,m\geq 2$ and the triple $(k,\ell,m)$
balanced. The Beilinson–Kato class is an element

(14) $\kappa_{f}\in H^{1}(\mathbb{Q},T_{f,Y}(\psi)(1))$

obtained as the cup product of two Eisenstein classes $\delta_{g}$ and
$\delta_{h}$ of weights $\ell-2$ and $m-2$, respectively. In particular,

$\delta_{g}\in
H_{\operatorname{et}}^{1}(\bar{Y},\operatorname{TSym}^{\ell-2}(\mathscr{H}_{\mathbb{Z}_{p}})(1))\quad\text{and}\quad\delta_{h}\in
H_{\operatorname{et}}^{1}(\bar{Y},\operatorname{TSym}^{m-2}(\mathscr{H}_{\mathbb{Z}_{p}})(1)).$

As a piece of notation, we write $c=(k+\ell+m-2)/2$ and $q=(\ell+m-k)/2>0$.
Note that the classes may also depend on an auxiliary Dirichlet character
$\chi$. However, when the choices of $\ell$, $m$ and $\chi$ are clear from the
setting, we drop them from the notation as in (16). Note that the previous
class is integral at $p$ provided that $p\nmid N$, as it follows from the
discussions of e.g. [Kings, §5.4] and [KLZ, §4,7].

It may be instructive to keep in mind the parallelism with the weight two
scenario considered in [RiRo, §2]. In that case, where $k=\ell=m=2$, the
Beilinson–Kato class is obtained as the cup product of the modular units

$u_{1,\chi}\cup u_{1,\xi}\in
H_{\operatorname{et}}^{2}(\bar{Y},\mathbb{Z}_{p}(2)),$

where $u_{1,\chi}\in H^{1}(\mathbb{Q},\mathbb{Z}_{p}(1))$, $u_{1,\xi}\in
H^{1}(\mathbb{Q},\mathbb{Z}_{p}(1))$. Applying now the Hochschild–Serre map
and projecting to the $f$-isotypical component, we get the desired class.

As a piece of notation, let

$\underline{\varepsilon}_{\operatorname{cyc}}:G_{\mathbb{Q}}\rightarrow\Lambda^{\times}$

denote the $\Lambda$-adic cyclotomic character which sends a Galois element
$\sigma$ to the group-like element
$[\varepsilon_{\operatorname{cyc}}(\sigma)]$. It interpolates the powers of
the $\mathbb{Z}_{p}$-cyclotomic character, in the sense that for any
arithmetic point of the form $\nu_{s,\xi}\in\mathcal{W}$,

(15)
$\nu_{s,\xi}\circ\underline{\varepsilon}_{\operatorname{cyc}}=\xi\cdot\varepsilon_{\operatorname{cyc}}^{s-1}.$

######  Proposition 2.2.

Assume that $\alpha_{f}\psi(p)\not\equiv 1\pmod{\mathfrak{p}}$. Then there
exists a homomorphism of $\Lambda$-modules

$\mathcal{L}_{f}^{-}:H^{1}(\mathbb{Q}_{p},T_{f,Y,\circ}^{\operatorname{quo}}\otimes\Lambda(\psi)(\varepsilon_{\operatorname{cyc}}^{q}\underline{\varepsilon}_{\operatorname{cyc}}))\longrightarrow\Lambda$

satisfying the following interpolation property: for
$s\in\mathcal{W}^{\circ}$, the specialization of $\mathcal{L}_{f}^{-}$ at $s$
is the homomorphism

$\mathcal{L}_{f,s}^{-}:H^{1}(\mathbb{Q}_{p},T_{f,Y,\circ}^{\operatorname{quo}}(\psi)(s+q-1))\longrightarrow\mathcal{O}_{\mathfrak{p}}$

given by

$\mathcal{L}_{f,s}^{-}=\frac{1-\bar{\psi}(p)\alpha_{f}^{-1}p^{-s}}{1-\psi(p)\alpha_{f}p^{s-1}}\times\begin{cases}\frac{(-1)^{s}}{(s-1)!}\times\langle\log_{\operatorname{BK}},t^{s}\tilde{\eta}_{f^{*}}\rangle&\text{
if }s\geq 2-q\\\
(-s)!\times\langle\exp_{\operatorname{BK}}^{*},t^{s}\tilde{\eta}_{f^{*}}\rangle&\text{
if }s<2-q,\end{cases}$

where $\log_{\operatorname{BK}}$ is the Bloch-Kato logarithm and
$\exp_{\operatorname{BK}}^{*}$, the dual exponential map.

###### Proof.

This follows from Coleman and Perrin-Riou’s theory of $\Lambda$-adic logarithm
maps as extended by Loeffler and Zerbes in [LZ]. In particular, [KLZ2, Theorem
8.2.3] yields an injective map

$H^{1}(\mathbb{Q}_{p},T_{f,Y,\circ}^{\operatorname{quo}}\otimes\Lambda(\psi)(\varepsilon_{\operatorname{cyc}}^{q}\underline{\varepsilon}_{\operatorname{cyc}}))\longrightarrow
D(T_{f,Y,\circ}^{\operatorname{quo}})\otimes\Lambda,$

since $H^{0}(\mathbb{Q}_{p},T_{f,\circ}^{\operatorname{quo}}(\psi))=0$ because
of the assumption that $\alpha_{f}\psi(p)\not\equiv 1$ modulo $\mathfrak{p}$.
This map is characterized by the interpolation property formulated in [LZ,
Appendix B]. Next we apply the pairing of (9) and the result follows. ∎

In particular, for $q=0$, we recover the homomorphism introduced in [RiRo,
§3.4]. The assumption $\alpha_{f}\psi(p)\not\equiv 1$ modulo $\mathfrak{p}$ is
necessary to have image in $\Lambda$, but it may be still possible that for a
fixed value of $s$ the image is integral even though the condition is not
satisfied (see [KLZ2, §8.2] for a precise description of the kernel and
cokernel of that map).

The following result, essentially due to Kato (and reformulated in our
language by Bertolini and Darmon), shows that there exists a $\Lambda$-adic
cohomology class whose bottom layer is precisely $\kappa_{f}$ and whose image
under a suitable regulator map recovers the $p$-adic $L$-function attached to
$f^{*}$. To fix notations, let $L_{p}(f^{*},\psi,s)$ stand for the
Mazur–Swinnerton-Dyer $p$-adic $L$-function attached to a modular form $f^{*}$
and a Dirichlet character $\psi$, with the conventions used in Section 3 of
loc. cit.. Let $L_{p}(f^{*},E_{k}(\chi_{1},\chi_{2}),s)$ be the Hida–Rankin
$p$-adic $L$-function attached to the two modular forms $f^{*}$ and
$E_{k}(\chi_{1},\chi_{2})$.

######  Proposition 2.3.

Under the assumptions of Proposition 2.2, there exists a $\Lambda$-adic
cohomology class

$\kappa_{f,\infty}\in
H^{1}(\mathbb{Q},T_{f,Y}\otimes\Lambda^{-}(\varepsilon_{\operatorname{cyc}}^{q}\underline{\varepsilon}_{\operatorname{cyc}}))$

such that:

  1. (a)

There is an explicit reciprocity law

$\mathcal{L}_{f}^{-}(\mathrm{res}_{p}(\kappa_{f,\infty})^{-})=L_{p}(f^{*},E_{\ell}(1,\chi),q+s),$

and there is a factorization

$L_{p}(f^{*},E_{\ell}(1,\chi),q+s)=\frac{-L_{p}(f^{*},\chi,\frac{k+\ell-m}{2})}{\mathfrak{g}(\bar{\chi})}\times
L_{p}(f^{*},q+s).$

Here, $\mathrm{res}_{p}$ stands for the map corresponding to localization at
$p$ and $\mathrm{res}_{p}(\kappa_{f,\infty})^{-}$ is the map induced in
cohomology from the projection map $T_{f,Y}\rightarrow
T_{f,Y,\circ}^{\operatorname{quo}}$.

  2. (b)

The bottom layer $\kappa_{f}(q)$ lies in
$H_{\operatorname{f}}^{1}(\mathbb{Q},T_{f}(q))$ and satisfies

$\kappa_{f}(q)=\mathcal{E}_{f}\cdot\kappa_{f},$

where $\mathcal{E}_{f}$ is the Euler factor

$\mathcal{E}_{f}=(1-\bar{\chi}(p)\beta_{f}p^{m-c-1})(1-\bar{\chi}_{f}\chi(p)\beta_{f}p^{\ell-c-1})(1-\alpha_{f}p^{-1})(1-\beta_{f}p^{-1}).$

###### Proof.

This is due to Kato [Ka]; with the current formulation, see more specifically
the work of Bertolini and Darmon [BD, Theorems 4.4 and 5.1]. ∎

###  2.4. The Beilinson–Flach Euler system

We now introduce the Beilinson–Flach classes and a reciprocity law for them.
See [KLZ] and [KLZ2] for details on the construction. Let $f\in
S_{f}(N,\chi_{f})$ and $g\in S_{\ell}(N,\chi_{g})$ be two cuspidal eigenforms
with $\chi_{f}\chi_{g}\not\equiv 1\pmod{\mathfrak{p}}$, and let $s$ be an
integer such that $1\leq s<\min(k,\ell)$. The Beilinson–Flach element
constructed in [KLZ] is a class

(16) $\kappa_{f,g}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}(1-s)),$

obtained via the pushforward of a class

$\delta_{h}\in
H_{\operatorname{et}}^{1}(\bar{Y},\operatorname{TSym}^{m-2}(\mathscr{H}_{\mathbb{Z}_{p}})(1))$

along the diagonal $Y\hookrightarrow Y^{2}$. Here, we can see the cyclotomic
twist $s$ as a shortcut for $c-m+1$. Observe once more that these classes have
no denominators at $p$; see [KLZ, Rk. 7.1.2] and [KLZ2, §3.2].

In the case where $k=\ell=2$ and $s=1$, the construction can be summarized as
follows: take the Eisenstein element $u_{1,\chi_{f}\chi_{g}}\in
H_{\operatorname{et}}^{1}(Y,\mathbb{Z}_{p}(1))$ and consider the Gysin map
(push-forward of appropriate sheaves) along the diagonal $Y\hookrightarrow
Y\times Y$ to get a class in
$H_{\operatorname{et}}^{3}(Y^{2},\mathbb{Z}_{p}(2))$. Applying the
Hochschild–Serre map and projecting to the $(f,g)$-istoypic component we get
the desired class. See e.g. [KLZ, §4]. (It must be pointed out that we are
allowed to use the Hochschild–Serre identification since the $H^{3}$ of a
surface over $\bar{\mathbb{Q}}$ vanishes.)

######  Remark 2.4.

The classes we are working with are the so-called Eisenstein classes in the
terminology of [KLZ2, §8]. Observe that the Euler factors that appear at
Proposition 8.1.3 of loc. cit. are precisely the same as those of the previous
section (except for the $c$-factor that arises in the reciprocity law for
Beilinson–Flach elements when we vary $f$ and $g$ in families and which plays
no role when $\chi_{f}\chi_{g}\neq 1$).

######  Proposition 2.5.

Assume that $\alpha_{f}\alpha_{g}^{-1}\chi_{g}(p)\not\equiv
1\pmod{\mathfrak{p}}$. Then there exists a homomorphism of $\Lambda$-modules

$\mathcal{L}_{f,g}^{-+}:H^{1}(\mathbb{Q}_{p},T_{f,Y,\circ}^{\operatorname{quo}}\otimes
T_{g,Y}^{\operatorname{sub}}\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))\longrightarrow\Lambda$

satisfying the following interpolation property: for
$r\in\mathcal{W}^{\circ}$, the specialization of $\mathcal{L}_{f,g}^{-}$ at
$s$ is the homomorphism

$\mathcal{L}_{f,g,s}^{-+}:H^{1}(\mathbb{Q}_{p},T_{f,Y,\circ}^{\operatorname{quo}}\otimes
T_{g,Y}^{\operatorname{sub}}(s))\longrightarrow\mathcal{O}_{\mathfrak{p}}$

given by

$\mathcal{L}_{f,g,s}^{-+}=\frac{1-\alpha_{f}^{-1}\beta_{g}^{-1}p^{-s}}{1-\alpha_{f}\beta_{g}p^{s-1}}\times\begin{cases}\frac{(-1)^{s+1}}{s!}\times\langle\log_{\operatorname{BK}},t^{s+1}\tilde{\eta}_{f^{*}}\otimes\omega_{g^{*}}\rangle&\text{
if }s\geq 0\\\
(-s-1)!\times\langle\exp_{\operatorname{BK}}^{*},t^{s+1}\tilde{\eta}_{f^{*}}\otimes\omega_{g^{*}}\rangle&\text{
if }s<0.\end{cases}$

###### Proof.

As in Proposition 2.2, this follows from the general theory of Perrin-Riou
maps (see [KLZ2, Thm. 8.2.8] for this formulation). ∎

For the following result, let $L_{p}(f^{*},g^{*},s)$ stand for the Hida–Rankin
$p$-adic $L$-function attached to the convolution of two modular forms. Note
that this function is not symmetric on the two modular forms. Let
$\chi=\bar{\chi}_{f}\bar{\chi}_{g}$. The following result, established by
Kings, Loeffler and Zerbes, shows that there exists a $\Lambda$-adic class
whose bottom layer is $\kappa_{f,g}$ (up to a suitable Euler factor) and which
recovers the Hida–Rankin $p$-adic $L$-functions.

######  Proposition 2.6.

Under the assumptions of Proposition 2.5, there exists a $\Lambda$-adic
cohomology class

$\kappa_{f,g,\infty}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes\Lambda(\varepsilon_{\operatorname{cyc}}\underline{\varepsilon}_{\operatorname{cyc}}))$

such that:

  1. (a)

There is an explicit reciprocity law

$\mathcal{L}_{f}^{-}(\mathrm{res}_{p}(\kappa_{f,g,\infty})^{-+})=L_{p}(f^{*},g^{*},1+s),$

where $\mathrm{res}_{p}$ stands for the map corresponding to localization at
$p$ and $\mathrm{res}_{p}(\kappa_{f,g,\infty})^{-+}$ is the map induced in
cohomology from the projection map $T_{f,Y}\otimes T_{g,Y}\rightarrow
T_{f,Y,\circ}^{\operatorname{quo}}\otimes T_{g,Y}^{\operatorname{sub}}$.

  2. (b)

The bottom layer $\kappa_{f,g}(s)\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}(s))$ satisfies

$\kappa_{f,g}(s)=\mathcal{E}_{f,g}\cdot\kappa_{f,g},$

where $\mathcal{E}_{f,g}$ is the Euler factor

$\mathcal{E}_{f,g}=(1-\beta_{f}\alpha_{g}p^{-s})(1-\beta_{f}\beta_{g}\chi(p)p^{s+1-k-\ell})(1-\beta_{g}\alpha_{f}p^{-s})(1-\beta_{g}\beta_{f}p^{-s}).$

###### Proof.

This follows from considering the result of [KLZ2, Thm. 10.2.2] when the two
modular forms are fixed. ∎

In particular, for our applications we do not need to consider the whole
cyclotomic variation, and it is enough to consider our attention to geometric
twists, that is, values of $s$ for which certain inequalities are satisfied
(and in particular, the results of [KLZ] suffice).

###  2.5. Diagonal cycles

We finally introduce the cohomology classes obtained via diagonal cycles and
their reciprocity laws, following [DR4], [BSV1] and [BSV2, §3]. We also assume
that the classes we are going to deal with have no denominators at $p$, and
following the discussion of [BSV2, Rk. 3.3] this is always the case provided
that $p$ is greater than the weights. Consider a triple of cuspidal eigenforms
$f\in S_{k}(N,\chi_{f})$, $g\in S_{\ell}(N,\chi_{g})$ and $h\in
S_{m}(N,\chi_{h})$, with $k+\ell+m$ even and $\chi_{f}\chi_{g}\chi_{h}=1$; we
also assume that the Galois representation attached to $f$ is absolutely
irreducible. Let $c=(k+\ell+m-2)/2$. The diagonal cycle class is an element

(17) $\kappa_{f,g,h}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}\otimes
T_{h,Y}(1-c)).$

When the weights are all two, the class is constructed by pushing forward the
class $1\in H_{\operatorname{et}}^{0}(Y,\mathbb{Z}_{p})$ along the diagonal
inclusion $Y\hookrightarrow Y^{3}$; this gives an element in
$H^{4}(Y^{3},\mathbb{Z}_{p}(2))$, which gives the desired class after applying
the Hochschild–Serre map following by projection to the $(f,g,h)$-isotypic
component. Following [DR1], one can also view the diagonal class directly over
the closed modular curve, which in our case is a consequence of the congruence
relations we will develop (at least modulo $p$).

Contrary to the previous settings, the absence of a cyclotomic variable makes
the situation slightly different. Fortunately, we only need the reciprocity
law pointwise, at a point of weight $(k,\ell,m)$. If a greater level of
generality were needed, we are allowed to move all three modular forms
$(f,g,h)$ along a Hida (or Coleman) family, but restricting to the central
twist. Although the Perrin-Riou map makes sense for the whole family of
twists, since the classes are only available for the central ones, we restrict
to those ones from now on.

######  Definition 2.7.

Assume that
$\alpha_{f}\chi_{f}^{-1}(p)\alpha_{g}^{-1}\alpha_{h}^{-1}\not\equiv 1$ modulo
$\mathfrak{p}$. Then we define $\mathcal{L}_{f,g,h}^{-++}$ as the morphism of
$\mathbb{Z}_{p}$-modules

$\mathcal{L}_{f,g,h}^{-++}:H^{1}(\mathbb{Q}_{p},T_{f,Y,\circ}^{\operatorname{quo}}\otimes
T_{g,Y}^{\operatorname{sub}}\otimes
T_{h,Y}^{\operatorname{sub}}(1-c))\longrightarrow\mathcal{O}_{\mathfrak{p}}$

given by

$\mathcal{L}_{f,g,h}^{-++}=\frac{1-\alpha_{f}^{-1}\beta_{g}^{-1}\beta_{h}^{-1}p^{-1}}{1-\alpha_{f}\beta_{g}\beta_{h}}\times\langle\log_{\operatorname{BK}},\tilde{\eta}_{f^{*}}\otimes\omega_{g^{*}}\otimes\omega_{h^{*}}\rangle$

For the following result, let $L_{p}(f^{*},g^{*},h^{*},c)$ stand for the
special value at $s=c$ of the triple product $p$-adic $L$-function associated
to the Hida families passing through $(f,g,h)$, as introduced in [DR1, §4]. As
recalled in loc. cit., we can also interpret these values in terms of a
suitable pairing, from which it is clear the parallelism with the Hida–Rankin
one used in previous sections.

When dealing with diagonal cycles, we implicitly fix a choice of test vectors,
for instance as in [Hs] (note that we can do it since the dominant $p$-adic
$L$-function will never be the Eisenstein one). To keep the parallelism with
the previous settings, let

$\kappa_{f,g,h,\operatorname{fam}}=\mathcal{E}_{f,g,h}\cdot\kappa_{f,g,h},$

where $\mathcal{E}_{f,g,h}$ is the Euler factor

$\mathcal{E}_{f,g,h}=(1-\beta_{f}\alpha_{g}\beta_{h}p^{-c})(1-\beta_{f}\beta_{g}\alpha_{h}p^{-c})(1-\alpha_{f}\beta_{g}\beta_{h}p^{-c})(1-\beta_{f}\beta_{g}\beta_{h}p^{-c}).$

We can understand this class as the result of the specialization of a class
varying in Hida families.

######  Proposition 2.8.

Under the conditions of Definition 2.7, there is an equality

$\mathcal{L}_{f,g,h}^{-++}(\mathrm{res}_{p}(\kappa_{f,g,h,\operatorname{fam}})^{-++})=L_{p}(f^{*},g^{*},h^{*},c),$

where $\mathrm{res}_{p}$ stands for the map corresponding to localization at
$p$ and $\mathrm{res}_{p}(\kappa_{f,g,h,\operatorname{fam}})^{-++}$ is the map
induced in cohomology from the projection map $T_{f,Y}\otimes T_{g,Y}\otimes
T_{h,Y}(1-c)\rightarrow T_{f,Y,\circ}^{\operatorname{quo}}\otimes
T_{g,Y}^{\operatorname{sub}}\otimes T_{h,Y}^{\operatorname{sub}}(1-c)$.

###### Proof.

This follows from the reciprocity law for diagonal cycles as in e.g. [DR1,
Thm. 1.3]; see also [BSV2, §3]. ∎

##  3\. Congruences between Beilinson–Flach and Beilinson–Kato classes

Let $f\in S_{k}(N,\chi_{f})$ and $g\in S_{\ell}(N,\chi_{g})$ be two cuspidal
eigenforms with $\chi_{f}\chi_{g}\not\equiv 1\pmod{\mathfrak{p}}$. We also fix
an integer $s$ with $1\leq s<\min\\{k,\ell\\}$, and introduce the auxiliary
variable $m=k+\ell-2s$. Recall that $F$ is a finite extension of $\mathbb{Q}$
containing the field of coefficients of $f$ and $g$, and $\mathcal{O}$ is its
ring of integers. Fix algebraic closures $\bar{\mathbb{Q}}$,
$\bar{\mathbb{Q}}_{p}$ of $\mathbb{Q}$ and $\mathbb{Q}_{p}$ and an embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$. This singles out a
prime ideal $\mathfrak{p}$ of ${\mathcal{O}}$ lying above $p$ and we let
$\mathcal{O}_{\mathfrak{p}}\subset\bar{\mathbb{Q}}_{p}$ denote the completion
of $\mathcal{O}$ at $\mathfrak{p}$. The Beilinson–Flach class we consider here
is the element

$\kappa_{f,g}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}(1-s))$

introduced in Section 2.4, where we omit the dependence on $s$.

In this section, we carry out a comparison with the Beilinson–Kato element

$\kappa_{f}\in
H^{1}(\mathbb{Q},T_{f,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(\ell-s))$

of Section 2.3.

Throughout all the section, we keep the following simplifying assumptions,
which are implicitly used in the different results we develop. In case of
needing some extra condition for a particular claim, we will state it
explicitly.

  * •

The weak-Gorenstein condition for $g$, as presented in Assumption 2.1.

  * •

The residual Galois representation attached to $f$ is absolutely irreducible.

Consider now the Eisenstein congruence setting, where

$g\equiv E_{\ell}(\chi_{g},1)\pmod{\mathfrak{p}^{t}},$

or alternatively $g^{*}\equiv E_{\ell}(1,\bar{\chi}_{g})$, which implies that
$\mathfrak{p}^{t}$ divides the Bernoulli number $B_{2}(\bar{\chi}_{g})$.

Since $g\equiv E_{\ell}(\chi_{g},1)$, we may consider the projection
$T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}\longrightarrow\mathcal{O}/\mathfrak{p}^{t}(\chi_{g})$,
that gives rise to a class

$\bar{\kappa}_{f,g,1}\in
H^{1}(\mathbb{Q},T_{f,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(\chi_{g})(1-s)).$

When $\bar{\kappa}_{f,g,1}$ vanishes, and following the arguments of [RiRo,
Prop. 4.1], we may consider a putative refinement of it, namely

$\bar{\kappa}_{f,g,2}\in
H^{1}(\mathbb{Q},T_{f,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(\ell-s)).$

The objective of this section is studying these cohomology classes and relate
them with the arithmetic of Beilinson–Kato classes. Our study rests on the
Perrin-Riou theory, which require suitable bounds on the corresponding Selmer
groups coming from Iwasawa theory.

We state the assumptions that are needed to establish our results. We write
$H_{\operatorname{Gr}}^{1}$ for the $H^{1}$ with Greenberg local conditions,
resulting from imposing the unramified condition at the primes $v\neq p$ and
the Bloch–Kato condition at $p$ (see e.g. [KLZ, §11]).

We introduce now two different sets of assumptions, that are needed for the
two different results that will be established.

######  Assumption 3.1.

  1. (a)

The space $H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}(\chi_{g})(1-s))$ is
one-dimensional.

  2. (b)

It holds that

$H^{0}(\mathbb{Q},T_{f,Y}(\chi_{g})(1-s)\otimes\mathcal{O}/\mathfrak{p}^{t})=0,\quad
H^{2}(\mathbb{Q},T_{f,Y}(\chi_{g})(1-s))=0$

and $\alpha_{f}\chi_{g}(p)\not\equiv 1\pmod{\mathfrak{p}}$.

######  Assumption 3.2.

  1. (a)

The space $H_{\operatorname{f}}^{1}(\mathbb{Q},T_{f,Y}(\ell-s))$ is one-
dimensional.

  2. (b)

It holds that $H^{2}(\mathbb{Q},T_{f,Y}(\ell-s))=0$ and $\alpha_{f}\not\equiv
1\pmod{\mathfrak{p}}$.

Kato [Ka] proved that under big image assumptions for the representation
$T_{f,Y}$, condition (a) is true, so this can be seen as a mild condition (but
in particular we want to exclude the case where $f$ is also congruent to an
Eisenstein series). Assumption (b) may be regarded as a condition regarding
non-exceptional zeros which also assures that the denominator of the Perrin-
Riou map does not vanish modulo $\mathfrak{p}$.

######  Remark 3.3.

The triviality of both the local $H^{0}$ or the local $H^{2}$ may be
formulated in terms on a condition over the Hecke eigenvalues. If
$p>k+\ell-2$, these are the following ones:

  * •

The group
$H^{0}(\mathbb{Q}_{p},T_{f,Y}(\chi_{g})(1-s)\otimes\mathcal{O}/\mathfrak{p}^{t})\neq
0$ if and only if $s=1$ and $\alpha_{f}\chi_{g}(p)\equiv
1\pmod{\mathfrak{p}}$.

  * •

The group $H^{2}(\mathbb{Q}_{p},T_{f,Y}(\chi_{g})(1-s))\neq 0$ if and only if
$s=1$ and $\chi_{g}(p)\equiv 1\pmod{\mathfrak{p}}$.

  * •

The group $H^{2}(\mathbb{Q},T_{f,Y}(\ell-s))\neq 0$ if and only if $s=\ell-1$
and $\alpha_{f}(p)\equiv 1\pmod{\mathfrak{p}}$.

If $p>k+\ell-2$ and under Assumption 2.1,
$H^{0}(\mathbb{Q}_{p},T_{f,Y}(\ell-s)\otimes\mathcal{O}/\mathfrak{p}^{t})=0$.
(Observe however that the global $H^{0}$ may be zero even though the
corresponding local group is non-trivial.)

Finally, note that if $p\leq k+\ell-2$ we must be more cautious:
$H^{0}(\mathbb{Q}_{p},\mathcal{O}/\mathfrak{p}(r))$ is non-zero when $r\equiv
0$ modulo $p-1$ (the cyclotomic character pulls back to the Teichmüller lift).
See [Man, §3.1] for a more elaborated discussion on that issue.

######  Proposition 3.4.

Under the conditions of Assumptions 3.1 (resp. 3.2), the Selmer group with
Greenberg condition
$H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}(\chi_{g})(1-s)\otimes\mathcal{O}/\mathfrak{p}^{t})$
(resp.
$H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}(\ell-s)\otimes\mathcal{O}/\mathfrak{p}^{t})$)
is one-dimensional.

###### Proof.

We do the proof for the case of $T_{f,Y}(\ell-s)$, and the other case is
completely analogous. Beginning with

$0\longrightarrow T_{f,Y}(\ell-s)\xrightarrow{\times
p}T_{f,Y}(\ell-s)\longrightarrow
T_{f,Y}(\ell-1)/pT_{f,Y}(\ell-s)\longrightarrow 0,$

our assumptions guarantee that there is a short exact sequence

$0\rightarrow H^{1}(\mathbb{Q},T_{f,Y}(\ell-s))\xrightarrow{\times
p}H^{1}(\mathbb{Q},T_{f,Y}(\ell-s))\rightarrow
H^{1}(\mathbb{Q},T_{f,Y}(\ell-s)\otimes\mathcal{O}/\mathfrak{p}^{t})\rightarrow
0,$

and therefore

$\frac{H^{1}(\mathbb{Q},T_{f,Y}(\ell-s))}{pH^{1}(\mathbb{Q},T_{f,Y}(\ell-s))}\simeq
H^{1}(\mathbb{Q},T_{f,Y}(\ell-s)\otimes\mathcal{O}/\mathfrak{p}^{t}(\ell-s)).$

The conclusion now follows. ∎

######  Lemma 3.5.

The Bloch–Kato logarithm of Proposition 2.2 is injective.

###### Proof.

Note that $H^{0}(\mathbb{Q}_{p},T_{f,Y}(\ell-s))=0$ by an immediate inspection
of the Hodge–Tate weights. Then the result follows from the construction of
the Bloch–Kato logarithm discussed in [KLZ, Section 8.2]. ∎

The first result of this note establishes the vanishing of
$\operatorname{loc}_{p}(\bar{\kappa}_{f,g,1})$ under the previous assumptions.
Note however that there may be interesting cases where the hypotheses fail,
leading to a tantalizing connection with the theory of circular units that we
expect to explore in forthcoming work.

######  Proposition 3.6.

Suppose the conditions of Assumption 3.1 are true. If $L_{p}(f,\chi_{g},s)\neq
0$ modulo $p$, then the class $\bar{\kappa}_{f,g,1}$ vanishes.

###### Proof.

Using the running assumptions, the space
$H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}(\chi_{g})(1-s)\otimes\mathcal{O}/\mathfrak{p}^{t})$
is one-dimensional and contains a canonical element, the Beilinson–Kato class
$\kappa_{f}$ resulting from the specialization of the element
$\kappa_{f,\infty}$ in Proposition 2.3.

The projection of this class to
$H^{1}(\mathbb{Q}_{p},T_{f,Y}^{-}(\chi_{g})(1-s)\otimes\mathcal{O}/\mathfrak{p}^{t})$
is non-zero, since its image under the Perrin-Riou regulator is precisely
$L_{p}(f,\chi_{g},s)$, which does not vanish due to the assumptions. However,
the projection to that quotient of $\bar{\kappa}_{f,g,1}$ is zero because of
the local properties satisfied by the Beilinson–Flach class (see [KLZ, Lemma
8.1.5, Prop. 8.1.7]). ∎

When $\bar{\kappa}_{f,g,1}$ vanishes modulo $\mathfrak{p}$, we may consider
the class $\bar{\kappa}_{f,g,2}$, and compare it with the previous Kato class.

The comparison we want to understand is summarized in the following diagram in
the case of weight two (and therefore trivial coefficients), where the upper
row gives the Beilinson–Flach class and the bottom one, the Belinson–Kato
element.

$\textstyle{H_{\operatorname{et}}^{1}(Y,\mathbb{Z}_{p}(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\operatorname{et}}^{3}(Y^{2},\mathbb{Z}_{p}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},H_{\operatorname{et}}^{2}(\bar{Y}^{2},\mathbb{Z}_{p}(2)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y})}$$\textstyle{H_{\operatorname{et}}^{2}(Y,\mathbb{Z}_{p}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},H_{\operatorname{et}}^{1}(\bar{Y},\mathbb{Z}_{p}(2)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},T_{f,Y}(1)).}$

When the cohomology class $\bar{\kappa}_{f,g,1}$ vanishes we may lift the
class $\kappa_{f,g}$ to an element in the cohomology with compact support and
use the opposite projection.

######  Theorem 3.7.

Suppose that $\bar{\kappa}_{f,g,1}=0$ and that the conditions of Assumption
3.2 hold. Then the following congruence holds in
$H^{1}(\mathbb{Q},T_{f,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(\ell-s))$:

$\bar{\kappa}_{f,g,2}=\bar{\kappa}_{f}.$

###### Proof.

From the reciprocity laws stated as Proposition 2.2 and Proposition 2.5, we
have that

$\mathcal{L}_{f}^{-}(\mathrm{res}_{p}(\kappa_{f,\infty})^{-})\equiv\mathcal{L}_{f,g}^{-+}(\mathrm{res}_{p}(\kappa_{f,g,\infty})^{-+})\pmod{\mathfrak{p}^{t}}$

because both $p$-adic $L$-functions agree. Using Lemma 3.5 we can upgrade this
to the desired equality using the period relations discussed in [RiRo, Section
3.5], and the result follows from the assumptions of the statement. Since the
Greenberg subspace of $H^{1}(\mathbb{Q},T_{f,Y}(\ell-s))$ is torsion-free of
rank 1, we conclude that the Greenberg module in the right has rank 1 over
$\mathcal{O}/\mathfrak{p}^{t}\mathcal{O}$. ∎

######  Remark 3.8.

A similar approach works for extending our results to the Iwasawa cohomology
classes, that is, incorporating the cyclotomic variation.

##  4\. Congruences between diagonal cycles and Beilinson–Flach elements

We continue the development of the theory of Eisenstein congruences between
Euler systems with the case of diagonal cycles and Beilinson–Flach elements.
Consider now a triple of three cuspidal eigenforms $(f,g,h)$ and nebentype
$(\chi_{f},\chi_{g},\chi_{h})$, with common level $N$ and with $p\nmid N$. We
assume that they satisfy the self-duality condition
$\chi_{f}\chi_{g}\chi_{h}=1$, and for simplicity we assume that
$\chi_{h}\not\equiv 1\pmod{\mathfrak{p}}$.

The diagonal cycles class we consider is a global element

$\kappa_{f,g,h}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}\otimes
T_{h,Y}(1-c)).$

In this section, we carry out a comparison with the Beilinson–Flach element

$\kappa_{f,g}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}(m-c)).$

Throughout all the section, we keep the following simplifying assumptions:

  * •

The weak-Gorenstein condition for $h$, as presented in Assumption 2.1.

  * •

The residual Galois representations attached to $f$ and $g$ are absolutely
irreducible, and the class $\kappa_{f,g,h}$ is integral at $p$.

We consider the Eisenstein congruence setting $h\equiv E_{m}(\chi_{h},1)$.
Then there is a class

$\bar{\kappa}_{f,g,h,1}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(\chi_{h})(1-c)).$

When $\bar{\kappa}_{f,g,h,1}$ vanishes, and as discussed before, we may
consider a putative refinement of it, namely

$\bar{\kappa}_{f,g,h,2}\in H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(m-c)).$

We begin with the following assumptions.

######  Assumption 4.1.

  1. (a)

The space $H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}(\chi_{h})(1-c))$ is one-dimensional.

  2. (b)

It holds that

$H^{0}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}(\chi_{h})(1-c)\otimes\mathcal{O}/\mathfrak{p}^{t})=0,\quad
H^{2}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}(\chi_{h})(1-c))=0$

and
$\alpha_{f}\alpha_{g}^{-1}\chi_{h}(p),\alpha_{f}^{-1}\alpha_{g}\chi_{h}(p)\not\equiv
1\pmod{\mathfrak{p}}$.

######  Assumption 4.2.

  1. (a)

The space $H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}(m-c))$
is one-dimensional.

  2. (b)

It holds that

$H^{0}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(m-c))=0,\quad
H^{2}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}(m-c))=0$

and $\alpha_{f}\alpha_{g}^{-1}\not\equiv 1\pmod{\mathfrak{p}}$.

Kings–Loeffler–Zerbes [KLZ] proved that under big image assumptions for the
representation $T_{f,Y}\otimes T_{g,Y}$, condition (a) is true, so this is a
relatively mild condition. Assumption (b) may be regarded as a condition for
non-exceptional zeros. As before, the previous conditions may be rephrased as
a condition over the Frobenius eigenvalues modulo $p$ whenever $p$ is large
enough (in particular, $p>k+\ell+m$ is enough):

  * •

The group $H^{0}(\mathbb{Q}_{p},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(m-c))\neq 0$ if and only if
$\alpha_{f}(p)\equiv 1\pmod{\mathfrak{p}}$ and $\alpha_{g}(p)\equiv
1\pmod{\mathfrak{p}^{t}}$.

  * •

The group $H^{2}(\mathbb{Q}_{p},T_{f,Y}\otimes T_{g,Y}(\chi_{h})(1-c))=0$ if
and only if $\chi_{h}(p)\equiv 1\pmod{\mathfrak{p}}$.

  * •

The group $H^{0}(\mathbb{Q}_{p},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(m-c))=0$ if and only if
$\chi_{h}(p)\equiv 1\pmod{\mathfrak{p}^{t}}$.

  * •

The group $H^{2}(\mathbb{Q},T_{f,Y}\otimes T_{g,Y}(m-c))=0$ if and only if
$\chi_{f}(p)\equiv 1$ or $\chi_{g}(p)\equiv 1\pmod{\mathfrak{p}}$ .

######  Proposition 4.3.

Under the Assumptions 4.2, the space
$H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t})$ is one-dimensional.

The following result is proved in the same way as in the previous section
using the general theory of Perrin-Riou maps.

######  Lemma 4.4.

Assume that $H^{0}(\mathbb{Q}_{p},T_{f,Y}\otimes T_{g,Y}(m-c))=0$. Then the
Bloch–Kato logarithm is injective.

We now state the first congruence concerning diagonal cycles.

######  Proposition 4.5.

Suppose the conditions of Assumption 4.1 hold. Then if
$L_{p}(f,g,\chi_{h},c)\neq 0$ or $L_{p}(g,f,\chi_{h},c)\neq 0$ modulo $p$, the
class $\bar{\kappa}_{f,g,h,1}$ vanishes.

###### Proof.

The space $H_{\operatorname{Gr}}^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}(\chi_{h})(1-c)\otimes\mathcal{O}/\mathfrak{p}^{t})$ is one-dimensional
under the running assumptions. It is spanned by the Beilinson–Flach class when
this is non-zero; this is the class resulting from the specialization of the
element $\kappa_{f,g,\infty}$ in Theorem 2.6.

If $L_{p}(f,g,\chi_{h},c)\neq 0$, the projection of this class to
$H^{1}(\mathbb{Q}_{p},T_{f,Y}\otimes
T_{g,Y}^{-}(\chi_{h})(1-c)\otimes\mathcal{O}/\mathfrak{p}^{t})$ is non-zero,
since its image under the Perrin-Riou regulator is precisely
$L_{p}(f,g,\chi_{h},c)$, which does not vanish due to the assumptions.
However, the projection to that quotient of $\bar{\kappa}_{f,g,h,1}$ is zero
because of the local properties satisfied by the diagonal cycle class (see
[DR4, Prop. 5.8]). ∎

When $\bar{\kappa}_{f,g,h,1}$ vanishes, we may lift the class to the
cohomology of the open modular curve and consider a diagram like the following
(in weight 2), where the upper row gives the diagonal cycle and the bottom
one, the Belinson–Flach element. (We have shortened the notation and written
$T_{f,g,h}:=T_{f,Y}\otimes T_{g,Y}\otimes T_{h,Y}$ and
$T_{f,g}:=T_{f,Y}\otimes T_{g,Y}$.)

$\textstyle{H_{\operatorname{et}}^{0}(Y,\mathbb{Z}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\operatorname{et}}^{4}(Y^{3},\mathbb{Z}_{p}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},H_{\operatorname{et}}^{3}(\bar{Y}^{3},\mathbb{Z}_{p}(2)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},T_{f,g,h}(-1))}$$\textstyle{H_{\operatorname{et}}^{3}(Y^{2},\mathbb{Z}_{p}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},H_{\operatorname{et}}^{2}(\bar{Y}^{2},\mathbb{Z}_{p}(2)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\mathbb{Q},T_{f,g}).}$

######  Theorem 4.6.

Suppose that $\bar{\kappa}_{f,g,h,1}=0$ and that the conditions of Assumption
4.2 hold. The following congruence holds in
$H^{1}(\mathbb{Q},T_{f,g}\otimes\mathcal{O}/\mathfrak{p}^{t}(m-c))$:

$\bar{\kappa}_{f,g,h,2}=\bar{\kappa}_{f,g}.$

###### Proof.

We study the congruence via the theory of Perrin-Riou maps. Consider the
projection

$\pi_{f,g}^{-+}\,:\,H^{1}(\mathbb{Q},T_{f,g}(m-c)\otimes\mathcal{O}/\mathfrak{p}^{t})\longrightarrow
H^{1}(\mathbb{Q}_{p},T_{f,Y,\circ}^{\operatorname{quo}}\otimes
T_{g,Y}^{\operatorname{sub}}(m-c)\otimes\mathcal{O}/\mathfrak{p}^{t}).$

From the previous proposition and the reciprocity laws of [KLZ] and [DR4], we
have that the images of $\bar{\kappa}_{f,g,h,2}$ and $\kappa_{f,g}$ under the
appropriate Bloch–Kato logarithms agree, since the corresponding $p$-adic
$L$-functions take the same value; this can be easily seen, for instance,
using the expressions in terms of Petersson products, as in [DR1, Cor. 4.13].
We can upgrade this to the equality of the statement using the period
relations discussed in [RiRo, Section 3.5], and the statement follows.

Using now Lemma 4.4, we may upgrade that result to an equality in
$H^{1}(\mathbb{Q},T_{f,Y}\otimes
T_{g,Y}\otimes\mathcal{O}/\mathfrak{p}^{t}(m-c))$. ∎

######  Remark 4.7.

It would be interesting to understand these congruences in terms of the
explicit geometric description of the cycles, following the seminal work of
Bloch [Bl] and the approach of [MVW]. This approach, as well as others
connecting different Euler systems, is based on a comparison at the level of
$p$-adic $L$-functions, and it would be nice to have a more geometric
understanding of this phenomenon.

######  Remark 4.8.

A case which is specially intriguing occurs when more than one of the modular
forms are congruent with Eisenstein series. However, the assumptions needed to
conclude that the corresponding Greenberg cohomology spaces are one-
dimensional do not hold (the big image condition does not hold since the
modulo $\mathfrak{p}$ representations are not irreducible). We expect to end
up with a single class in $H^{1}(\mathbb{Q},\mathcal{O}/\mathfrak{p}^{t}(2))$,
which could potentially be understood using similar techniques to those
developed by Sharifi and Fukaya–Kato.

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###### Abstract

This contribution explores the ability to probe BSM physics by using the
experimental prospects for measuring the forward-backward asymmetry ($A_{FB}$)
in $e^{+}e^{-}\rightarrow b\bar{b}$ and $e^{+}e^{-}\rightarrow c\bar{c}$
processes at the baseline energy points of ILC: 250 and 500 GeV. The studies
are based on the full simulation samples and reconstruction chains from the
ILD concept group. The BSM models studied are two different types of gauge-
Higgs unification (GHU) models that predict BSM Z′ resonances at the TeV
scale.

##  1 Introduction

In the Standard Model (SM) of particle physics, the Higgs boson does not
follow the gauge principle as the rest of the SM bosons do. As a result, the
values of the Higgs couplings with quarks and leptons, as well as its self-
coupling, are fixed parameters. Additionally, the Higgs mass adquires large
quantum corrections that have to be canceled by fine-tuning. A possible
solution for the aforementioned inconsistencies is achieving stabilization of
the Higgs mass against quantum corrections by extending the gauge group of the
SM. These models are referred to as Gauge-Higgs Unification (GHU) models.

In this document, a first approach to the phenomenology of GHU models will be
addressed in the context of a future Higgs factory: The International Linear
Collider (ILC) [1, 2, 3, 4, 5]. This study has been done featuring the
International Large Detector (ILD) [5, 6] at ILC with full simulation studies
at 250 and 500 GeV collision energies and using the forward-backward asymmetry
($A_{FB}$) in $e^{+}e^{-}\rightarrow b\bar{b}$ and $e^{+}e^{-}\rightarrow
c\bar{c}$ processes as the observable to discriminate these models. The
$A_{FB}$ is defined as:

$A^{q\bar{q}}_{FB}=\frac{\sigma^{F}_{e{-}e^{+}\rightarrow
q\bar{q}}-\sigma^{B}_{e{-}e^{+}\rightarrow
q\bar{q}}}{\sigma^{F}_{e{-}e^{+}\rightarrow
q\bar{q}}+\sigma^{B}_{e{-}e^{+}\rightarrow q\bar{q}}}$ (1)

where $\sigma^{F/B}_{e{-}e^{+}\rightarrow q\bar{q}}$ is the cross-section in
the forward/backward hemisphere as defined by the polar angle $\theta_{q}$.

##  2 Gauge-Higgs Unification (GHU) models

Gauge-Higgs unification (GHU) introduces the Higgs boson as an Aharonov-Bohm
(AB) phase $\theta_{H}$ in an extra dimension. The GHU models extend the gauge
theory of the SM to $SU(3)_{c}\times SO(5)_{W}\times U(1)_{x}$ in a Randall-
Sundrum warped space[7, 8]. In this context, gauge symmetry stabilizes the
Higgs-boson mass against quantum corrections. At low energies the
phenomenology of GHU matches the SM but the prediction of existence of
$Z^{\prime}$ bosons produces deviations at high energies. The $Z^{\prime}$
bosons predicted by GHU are Kaluza-Klein (KK) resonances of $\gamma$, $Z$, and
$Z_{R}$. The angle $\theta_{H}$ and KK mass $m_{KK}$ and are set in a way that
reproduces the masses and decay width of the SM $W$ and $Z$ bosons, which are
the zeroth mode of the 5 dimensional fields. In this work, two different types
of GHU models are considered: $A$ models [7] and $B$ models [8].

In ref. [7], three variations of the $A$ models ($A_{1}$, $A_{2}$, $A_{3}$)
are proposed (see Tab. 1). In these models, quark-lepton multiplets are
introduced in the vector representation of $SO(5)$. These models use
$\theta_{H}\simeq 0.11-0.07$ and KK resonances masses $m_{KK}\simeq 8-11$ TeV.
The non-observation in LHC for direct measurements of $Z^{\prime}$ set a limit
of $\theta_{H}\lesssim 0.11$. They also predict large couplings of the right-
handed (down-type) fermions to the $Z^{\prime}$ resonances. In all $A$ models,
the parameters are chosen such that $Z$ couplings to fermions (except top-
quark) agree with the SM within one part in $10^{4}$.

Model |  | $\theta_{H}$
---
[rad.]
| $m_{KK}$
---
[TeV]
| $m_{Z^{(1)}}$
---
[TeV]
| $\Gamma_{Z^{(1)}}$
---
[TeV]
| $m_{\gamma^{(1)}}$
---
[TeV]
| $\Gamma_{\gamma^{(1)}}$
---
[TeV]
| $m_{Z_{R}^{(1)}}$
---
[TeV]
| $\Gamma_{Z_{R}^{(1)}}$
---
[TeV]
$A_{1}$ | 0.115 | 7.41 | 6.00 | 0.406 | 6.01 | 0.909 | 5.67 | 0.729
$A_{2}$ | 0.0917 | 8.81 | 7.19 | 0.467 | 7.20 | 0.992 | 6.74 | 0.853
$A_{3}$ | 0.0737 | 10.3 | 8.52 | 0.564 | 8.52 | 1.068 | 7.92 | 1.058
Table 1: Parameter setting, masses and decay widths of $Z^{\prime}$ bosons
($Z^{(1)}$, $\gamma^{(1)}$ and $Z_{R}^{(1)}$) in the $A$ models

In ref. [8], five variations of the $B$ models ($B$, $B_{L}$, $B_{H}$,
$B_{-}$, $B_{+}$) are proposed (see Tab. 2). The $B$ models are inspired by
Grand Unification Theories, leading to quark-lepton multiples introduced in
the spinor, vector and singlet representations of $SO(5)$. The parameters are
fixed in a way that predicts values of
$A_{FB}(e^{-}e^{+}\rightarrow\mu^{-}\mu^{+})$ compatible with current
measurements. For all $B$ models, the parameters are chosen such that $Z$
couplings to fermions (except top-quark) agree with the SM within one part in
$10^{3}$. The $B$ models, in contrast with the $A$ models, predict a large
deviation for the couplings of the left-handed (up-type) fermions to the new
bosons.

Model |  | $\theta_{H}$
---
[rad.]
| $m_{KK}$
---
[TeV]
| $m_{Z^{(1)}}$
---
[TeV]
| $\Gamma_{Z^{(1)}}$
---
[TeV]
| $m_{\gamma^{(1)}}$
---
[TeV]
| $\Gamma_{\gamma^{(1)}}$
---
[TeV]
| $m_{Z_{R}^{(1)}}$
---
[TeV]
| $\Gamma_{Z_{R}^{(1)}}$
---
[TeV]
$B$ | 0.10 | 13.00 | 10.20 | 8.713 | 10.20 | 3.252 | 9.951 | 0.816
$B_{L}$ | 0.10 | 11.00 | 8.713 | 4.773 | 8.715 | 2.080 | 8.420 | 0.603
$B_{H}$ | 0.10 | 15.00 | 11.69 | 11.82 | 11.69 | 4.885 | 11.48 | 1.253
$B_{-}$ | 0.09 | 13.00 | 10.26 | 6.413 | 10.26 | 2.723 | 9.951 | 0.732
$B_{+}$ | 0.11 | 13.00 | 10.15 | 9.374 | 10.15 | 3.836 | 9.951 | 0.924
Table 2: Parameter setting, masses and decay widths of $Z^{\prime}$ bosons
($Z^{(1)}$, $\gamma^{(1)}$ and $Z_{R}^{(1)}$) in the $B$ models

##  3 Experimental prospects for $A_{FB}$ at ILC250 and ILC500

The ILC run plan considered in this study is the Horizon-2020 (H20) as
detailed in [9] which foresees $e^{+}e^{-}$ collisions at different center of
mass energies ranging from the Z-pole (Giga-Z) up to 1 TeV. The nominal
program has an initial stage at 250 GeV (ILC250), including a luminosity
upgrade, and a following stage at 500 GeV (ILC500), after an energy upgrade.
ILC also features longitudinal polarization for both the electron and positron
beams. For H20, the ILC250 physics program foresees a total integrated
luminosity of 2000 fb-1 distributed between four different beam polarization
schemes: $45\%$ in $P_{\mathrm{e^{-}e^{+}}}=(-0.8,+0.3)$, $45\%$ in
$P_{\mathrm{e^{-}e^{+}}}=(+0.8,+0.3)$, $5\%$ in
$P_{\mathrm{e^{-}e^{+}}}=(-0.8,-0.3)$ and $5\%$ in
$P_{\mathrm{e^{-}e^{+}}}=(+0.8,-0.3)$. The ILC500 program foresees a total
integrated luminosity of 4000 fb-1 distributed among the different
polarization as $40\%$, $40\%$, $10\%$ and $10\%$, with the same polarization
configurations as the ILC250 case.

The ILD is one of the two proposed detector concepts for the ILC. It is a
highly hermetic multi-purpose detector designed for the maximal exploitation
of particle flow techniques in event reconstruction. The ILD layout, from the
IP to the outside consists of: a high-precision vertex detector (VTX), silicon
tracking systems, a time projection chamber (TPC), a highly granular
calorimeter system (ECAL and HCAL) and a muon catcher. All the aforementioned
subdetectors are placed inside a solenoid providing a magnetic field of $3.5$
T, surrounded by an iron yoke instrumented for muon detection. The ILD TPC
[10, 11] allows continuous 3D tracking and particle identification (PID).

The full simulations for this study have been performed via ILCSOFT
v02-02-03111Link: https://github.com/iLCSoft, which merge different software
packages and algorithms that operate in a modular way: from MC events to final
reconstruction. All simulations use the ILD-L[6] model, whose geometry,
material and readout systems are implemented in the DD4HEP framework[12],
interfaced with Geant4 toolkit. Both signal and background events from QED ISR
are generated with the WHIZARD[13] event generator at leading order. The beam
energy spectrum and beam-beam interaction is generated via Guinea-Pig[14]. The
non-perturbative effects of hadronization are provided by the Pythia event
generator[15], as well as the final state QCD and QED radiations. The signal
and background cross sections are listed in the references: Tables 1 and 2
from [16] for ILC250 and in Tables 1 and 2 from [17] for ILC500. Only
backgrounds leading to fully hadronic final states are considered. Backgrounds
involving leptons in the final states are ignored since those are expected to
be easily identified. The simulations feature fully polarized beams.

The reconstruction of the events begins with the track reconstruction
performed by the MarlinTrk framework. Later, Pandora[18] runs the particle
flow algorithm (PFA) that matches the tracking with the high-granular
calorimetry information building the particle flow objects (PFO), which are
then treated as single particles. By using LCFI+ software tool the PFOs are
merged into vertices and jets. Then, also in LCFI+, the flavor tagging of the
jets is performed.

All the methods (Preselection, Double tag, Double charge and fitting) for
reconstructing the $q$$\overline{q}$ system have been developed for the
analysis at 250 GeV and are explained in detail in [16] together with a
comprehensive study of the most dominant systematic uncertainties. The same
methods are used for the study at 500 GeV shown in [17]. Additionally, in this
later document, the improvement gain by using cluster counting techniques
($dN/\/dx$) instead of the traditional energy loss ($dE/\/dx$) approach was
reviewed.

##  4 Discrimination power for GHU at ILC250/500

Through the detailed studies described in the previous section, we obtain a
realistic estimation of the uncertainties on $A_{FB}$ for the $b$ and
$c$-quark cases at ILC250 and ILC500, with existing detector models and
reconstruction tools. Systematic uncertainties are being ignored for the
analysis discussed in this section since $A_{FB}$ measurements above the
$Z$-pole are expected to be statistically dominated at ILC, as shown in [16]
and [17]. The uncertainties are considered as normally distributed and with no
correlations between measurements. This second approximation is motivated by
the nature of the analysis, given that the Double Tag and Double Charge
methods lead to a selection of fully independent samples for the different
flavors and polarization. Also, statistically independent MC simulations have
been used to analyze the various polarization scenarios.

The statistical significance ($\sigma$-level) when comparing two models, $i$
and $j$, is defined as

$d_{ij}=\frac{|A_{FB,i}-A_{FB,j}|}{\Delta A_{FB,j}}$ (2)

with $A_{FB,i/j}$ being the $A_{FB}$ predicted at leading order by the model
$i$ or $j$, introduced in Sec. 2. The $\Delta A_{FB,i/j}$ corresponds to the
expected statistical uncertainty of the forward-backward measurement at ILC,
obtained as explained in Sec. 3.

|
---|---
|
Figure 1: Statistical discrimination power between SM and GHU models using $A_{FB}$ at ILC250 under the H20 scenario. Plots assuming only statistical uncertainties and the use of $dN/\/dx$ for PID. |
---|---
|
Figure 2: Statistical discrimination power between SM and GHU models using
$A_{FB}$ at ILC500 under the H20 scenario. Plots assuming only statistical
uncertainties and the use of $dN/\/dx$ for PID.

In Fig. 1 the four $A_{FB}$ observables at 250 GeV $c.m.e.$ are shown. In this
plot is clear how the $A^{b}_{FB}$ with right-handed electrons and left-handed
positrons ($P_{\mathrm{e^{-}e^{+}}}=(+0.8,-0.3)$) collisions provide the best
resolution for the $A$ models, so that they could be discriminate from the SM
by only this measurement. Given the expected discrimination power, in case of
a measurement compatible with any of the $A$ models all of the $B$ models
could be ruled out.

Alternatively, in Fig. 2 the $A_{FB}$ observables at ILC500 are plotted. This
plot made manifest the aforementioned sensibility of the $A$ models to the
$P_{\mathrm{e^{-}e^{+}}}=(+0.8,-0.3)$ case, and now also for $A^{c}_{FB}$.
Furthermore, at this energy, the $B$ models could also be distinguished from
the SM by its sensibility to $A^{c}_{FB}$ for
$P_{\mathrm{e^{-}e^{+}}}=(-0.8,+0.3)$.

##  5 Conclusions

This document shows the first insights into the potential of ILC for measuring
GHU models by using heavy quark production in its two main stages: ILC250 and
ILC500. All the prospects have been done assuming only statistical
uncertainties. The $A$ models could be ruled out by measuring $A^{b}_{FB}$
with $P_{\mathrm{e^{-}e^{+}}}=(+0.8,-0.3)$ at ILC250 whereas accessing to
higher energies at ILC500 in mandatory to rule out the $B$ models, that could
be performed by using the $A^{c}_{FB}$ in
$P_{\mathrm{e^{-}e^{+}}}=(-0.8,+0.3)$ beam operation. However, full between-
model separation is not possible under these assumptions only using these
channels.

The statistical combination of these measurements to obtain a greater
discrimination power as well as using other running scenarios such as a 1 TeV
ILC run is work yet to be explored. Additionally, other models predicting
heavy Z′ resonances could also be studied in the future, including other GHU
models [19] or different ones such as the minimal $U(1)_{X}$ models [20].

##  6 Acknowledgments

We would like to thank the LCC generator working group and the ILD software
working group for providing the simulation and reconstruction tools and
producing the Monte Carlo samples used in this study. This work has benefited
from computing services provided by the ILC Virtual Organization, supported by
the national resource providers of the EGI Federation and the Open Science
GRID. This work was supported by the PlanGenT program from the Generalitat
Valenciana (Spain) with the grant number CIDEGENT/2020/21 and by CSIC via the
mobility grant iMOVE23245.

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