# On the adjoint Selmer groups of semi-stable elliptic curves and Flach’s zeta
elements

Chan-Ho Kim  June E Huh Center for Mathematical Challenges, Korea Institute
for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

We explicitly construct the rank one primitive Stark (equivalently, Kolyvagin)
system extending a constant multiple of Flach’s zeta elements for semi-stable
elliptic curves. As its arithmetic applications, we obtain the equivalence
between a specific behavior of the Stark system and the minimal modularity
lifting theorem, and we also discuss the cyclicity of the adjoint Selmer
groups. Our Stark system construction yields a more refined interpretation of
the collection of Flach’s zeta elements than the “geometric Euler system”
approach due to Flach, Wiles, Mazur, and Weston.

###### Key words and phrases:

Stark systems, symmetric squares of elliptic curves, Flach’s zeta elements,
modularity lifting theorem, adjoint Selmer groups, refined Iwasawa theory

###### 2020 Mathematics Subject Classification:

11R23, 11R39 (Primary); 11F33, 11F67,11G40 (Secondary)

This research was partially supported by a KIAS Individual Grant (SP054103)
via the Center for Mathematical Challenges at Korea Institute for Advanced
Study and by the National Research Foundation of Korea(NRF) grant funded by
the Korea government(MSIT) (No. 2018R1C1B6007009).

## Introduction

### Flach’s zeta elements

As explained in Introduction of his cerebrated work [Wil95], A. Wiles first
tried to prove the semi-stable Shimura–Taniyama conjecture and Fermat’s last
theorem via the construction of an Euler system for symmetric squares of semi-
stable elliptic curves extending the cohomology classes constructed by M.
Flach [Fla92], which we call _Flach’s zeta elements_ in this article. However,
the construction of the Euler system was incomplete, and it was regarded as a
gap.

After that, Wiles, together with R. Taylor, found a completely different
approach towards the semi-stable Shimura–Taniyama conjecture, which is now
known as the Taylor–Wiles system argument [TW95], to fill the gap. See [Kur93,
Kol94, RS94, Dar95] for the survey articles written before the gap was filled,
and see [Cal23] for the far-reaching development of the Taylor–Wiles system
argument.

This Euler system approach is often known as the “geometric Euler system”
strategy, and it was further developed mainly by B. Mazur and T. Weston [Maz,
MW, Wes01, Wes02]. However, it was still incomplete to obtain the exact bound
of Selmer groups.

The main goals of this article are to investigate the behavior of the
collection of Flach’s zeta elements beyond the scope of this geometric Euler
system strategy and to deduce certain structural results for adjoint Selmer
groups. However, it is well-known that the construction question of an _Euler_
system extending Flach’s zeta element is extremely difficult (see [Rub00,
$\S$3.6], [Wes01, pp. 359–360], [MR04, Rem. 6.3.4], and [LZ19, Intro.] for
example).

What is the new idea to overcome this notorious difficulty? We are greatly
benefited from various remarkable and significant developments of the theory
of Euler systems. In particular, it turns out that Kolyvagin systems are more
essential than Euler systems to obtain the exact bound of the corresponding
dual Selmer groups [MR04]. Also, the notion of Stark systems is developed by
generalizing the units predicted by Stark-type conjectures [MR16]. The key
observation is that the collection of Flach’s zeta elements forms a “Stark
system of Gauss sum type”111No Gauss sums actually appear in this article.
Since we initially have only _finitely many_ cohomology classes forming a
Stark system, we call them “Gauss sum type”.. Since the module of Stark
systems (equivalently, Kolyvagin systems) is free of rank one, the Gauss sum
type Stark system naturally extends to the _bona fide_ one. In other words,
_we construct derivative (so ramified) classes defined over $\mathbb{Q}$
directly from Flach’s zeta elements by using the framework of Stark systems_.

Since there is a certain constant multiple subtlety in our Stark system
approach, our result does not give a new proof of the semi-stable
Shimura–Taniyama conjecture or Fermat’s last theorem yet. Also, we do not know
whether our Stark system actually comes from an Euler system222We are not sure
whether it is possible to recover an Euler system from the Kolyvagin system
associated to the Euler system.. Indeed, we prove that the minimal modularity
lifting theorem is equivalent to showing that the constant multiple is a
$p$-adic unit.

### The cyclicity of adjoint Selmer groups

The cyclicity question has long history in Iwasawa theory. This is just a
quick summary of the preface of Hida’s book [Hid22a] and we strongly recommend
the reader to read it.

K. Iwasawa himself was interested in the structure of various classical
Iwasawa modules including the cyclicity over the Iwasawa algebra. In [Iwa69],
he obtained the cyclicity of certain classical Iwasawa modules over the
Iwasawa algebra (as well as the main conjecture) under the Kummer–Vandiver
conjecture. Even without assuming the Kummer–Vandiver conjecture, he proposed
several interesting questions on the cyclicity around 1980’s [Iwa88, Iwa14].

H. Hida explored the non-abelian analogue of this nature, and [Hid86] was
written with this purpose in mind. In particular, he studied the cyclicity
question of the adjoint Selmer groups over the universal deformation ring in a
series of his papers[Hid, Hid20, Hid22b, Hid22a]. In [Hid22a, Chap. 7], he
proved many cyclicity results for the adjoint Selmer groups of Artin
representations. As far as we understand, the elliptic curve case seems quite
open [Hid22a, pp. 287–288 and 339] unless the length of adjoint Selmer groups
is $\leq 1$. As an application of the above Stark system construction, we
obtain a cyclicity result for the adjoint Selmer groups of elliptic curves
(Corollary 1.5).

### Comparison with other recent work

Recent years, some groups of mathematicians have developed new approaches
towards the construction of symmetric square and adjoint Euler systems from
different sources [LZ19, Urb21, LZ, SS]. We provide brief remarks on their
constructions and highlight the novelty of our approach in contrast to theirs,
as per the request to address this aspect.

In [LZ19], an Euler system is constructed from Beilinson–Flach’s elements and
bounds Selmer groups of the adjoint representations of an ordinary modular
form _twisted by a non-trivial Dirichlet character_. This twist is inevitable
to have the correct Euler system relation. In [Urb21], an Euler system for
adjoint representations of ordinary Hilbert modular forms is constructed
directly from congruence modules, and the construction _uses_ the
$R=\mathbb{T}$ theorem as an essential ingredient. Also, the corresponding
explicit reciprocity law depends on a conjecture on the Fitting ideals of some
equivariant congruence modules for abelian base change. In [LZ], an Euler
system for adjoint representations of ordinary modular forms is constructed
from Asai–Flach classes and is used to prove one-sided divisibility of the
main conjecture for the symmetric square Selmer group over the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ _except two height one prime
ideals, namely, $(p)$ and the augmentation ideal_. Due to the exception of
these height-one prime ideals, further work is needed to get an upper bound of
symmetric square Selmer groups over $\mathbb{Q}$. In the ongoing work [SS], an
Euler system is constructed from $\mathrm{GSp}_{4}$ Eisenstein classes (as
indicated in [LZ]). According to Skinner’s IAS talk333An Euler System for the
Symmetric Square of a Modular Form - Chris Skinner (Feb. 16, 2023)., this
Euler system lies in the Selmer group of the symmetric square representation
of a modular form _twisted by an odd Dirichlet character_. In summary, none of
the work mentioned here actually bounds what Wiles wanted to bound in their
current form. Also, the $p$-ordinary condition is assumed in [LZ19, Urb21,
LZ].

Our Stark system detects the precise upper bound of the adjoint Selmer groups
which Wiles concerned, so the direct connection with the modularity lifting
theorem is obtained. It appears that our approach is the most akin to Wiles’
original method since we work with Flach’s zeta elements as Wiles did, so it
might have some historical meaning. Also, our argument works for any good
reduction prime. In addition, the structural refinement like the cyclicity is
not observed in other approaches.

### The organization

In §1, we state our main result on the existence of the rank one primitive
Stark systems for symmetric square representations and its arithmetic
applications. In §2, we review Flach’s zeta elements and state the explicit
reciprocity law for Flach’s zeta elements. In §3, we verify the working
hypotheses for rank one Stark systems. In §4, we recall Weston’s computation
on the explicit reciprocity law. In §5, we prove our main result by
constructing the Stark system explicitly. This is the heart of this article.
In §6, we prove corollaries to the main result.

##  1\. Statement of the main result

Let $E$ be a semi-stable _modular_ 444We do not use the semi-stable modularity
theorem of Wiles and Taylor–Wiles [Wil95, TW95] and its generalizations in
this article. elliptic curve over $\mathbb{Q}$ of conductor $N$ and $p\geq 5$
a good reduction prime for $E$. Denote by $T_{p}E$ the $p$-adic Tate module
for $E$. Assume that
$\overline{\rho}:G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{Aut}_{\mathbb{F}_{p}}(E[p])$
is irreducible and $N=N(\overline{\rho})$ where $N(\overline{\rho})$ is the
conductor of $\overline{\rho}$.

The main goal of this article is to construct the (rank one) Stark system from
Flach’s zeta elements. It partially resolves a question of Mazur–Rubin in
[MR04, Rem. 6.3.4]. Indeed, we completely settle their question under the
semi-stable modularity lifting theorem (see Corollary 1.4 below).

######  Theorem 1.1 (Flach–Stark systems).

For a prime $\ell$ with $(Np,\ell)=1$, $\ell\equiv 1\pmod{p}$, and
$a_{\ell}(E)\not\equiv\pm 2\pmod{p}$, there exists the non-trivial primitive
rank one Stark system

$\boldsymbol{\epsilon}^{\mathrm{Flach},\ell}$

for $\mathrm{Sym}^{2}(T_{p}E)$ extending a constant multiple of Flach’s zeta
element $c(\ell)$ (recalled in Theorem 2.2) which determines the structure of
the $\ell$-strict (trace zero) adjoint Bloch–Kato Selmer group

$\mathrm{Sel}_{\ell\textrm{-}\mathrm{str}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}])).$

In particular, the constant multiple is independent of $\ell$. We call
$\boldsymbol{\epsilon}^{\mathrm{Flach},\ell}$ the Flach–Stark system.

###### Proof.

The explicit construction of $\boldsymbol{\epsilon}^{\mathrm{Flach},\ell}$ is
given in §5. ∎

In the rest of this section, we fix a prime $\ell$ satisfying the conditions
in Theorem 1.1. Let $\boldsymbol{\kappa}^{\mathrm{Flach},\ell}$ be the (non-
trivial and primitive) Flach–Kolyvagin system corresponding to
$\boldsymbol{\epsilon}^{\mathrm{Flach},\ell}$ under the isomorphism between
the modules of Stark systems and Kolyvagin systems under our setting [MR16,
Rem. 11.5 and Thm. 12.4]. Indeed, the constant multiple in Theorem 1.1 is
designed to deduce the following formula for the exact bound of the genuine
Bloch–Kato Selmer group in terms of
$\boldsymbol{\kappa}^{\mathrm{Flach},\ell}$.

######  Corollary 1.2 (Exact bound).

Let

$\mathrm{loc}^{s}_{\ell}(\boldsymbol{\kappa}^{\mathrm{Flach},\ell})=\left\\{\mathrm{loc}^{s}_{\ell}(\kappa^{\mathrm{Flach},\ell}_{n}):n\in\mathcal{N}^{\mathrm{Flach},(\ell)}_{1}\right\\}$

be the singular quotient of the localization of
$\boldsymbol{\kappa}^{\mathrm{Flach},\ell}$ at $\ell$ where
$\mathcal{N}^{\mathrm{Flach},(\ell)}_{1}$ is defined in Definition 2.3. Then

$\mathrm{ord}_{p}(\mathrm{loc}^{s}_{\ell}(\kappa^{\mathrm{Flach},\ell}_{1}))=\mathrm{length}_{\mathbb{Z}_{p}}(\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}])))$

where $\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}]))$ is the adjoint
Bloch–Kato Selmer group and
$\mathrm{ord}_{p}(\mathrm{loc}^{s}_{\ell}(\kappa^{\mathrm{Flach},\ell}_{1}))$
is the $p$-divisibility index of
$\mathrm{loc}^{s}_{\ell}(\kappa^{\mathrm{Flach},\ell}_{1})$ in
$\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(T_{p}E))$.

###### Proof.

See $\S$6.1. ∎

######  Remark 1.3.

It is known that the primitivity of a (rank one) Kolyvagin system is closely
related to showing the corresponding main conjecture [MR04, KKS20, Sak22,
Kim]. This principle would apply to our setting as follows if such a
formulation is valid. Let $\mathbb{Q}_{\infty}$ be the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ and $\Lambda$ the cyclotomic
Iwasawa algebra. _If_ the $\Lambda$-adic deformation
$\boldsymbol{\kappa}^{\mathrm{Flach},\ell,\infty}$ of
$\boldsymbol{\kappa}^{\mathrm{Flach},\ell}$ exists and
$\kappa^{\mathrm{Flach},\ell,\infty}_{1}$ is non-zero in
$\widehat{\mathrm{Sel}_{\ell\textrm{-}\mathrm{rel}}}(\mathbb{Q}_{\infty},\mathrm{Sym}^{2}(T_{p}E))$,
then the _hypothetical_ main conjecture

$``\mathrm{char}_{\Lambda}\left(\widehat{\mathrm{Sel}_{\ell\textrm{-}\mathrm{rel}}}(\mathbb{Q}_{\infty},\mathrm{Sym}^{2}(T_{p}E))/\Lambda\kappa^{\mathrm{Flach},\ell,\infty}_{1}\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{\ell\textrm{-}\mathrm{str}}(\mathbb{Q}_{\infty},\mathrm{ad}^{0}(E[p^{\infty}]))^{\vee}\right)"$

follows from the primitivity of $\boldsymbol{\kappa}^{\mathrm{Flach},\ell}$
[MR04, $\S$5.3] where
$\widehat{\mathrm{Sel}_{\ell\textrm{-}\mathrm{rel}}}(\mathbb{Q}_{\infty},-)=\varprojlim_{n}\mathrm{Sel}_{\ell\textrm{-}\mathrm{rel}}(\mathbb{Q}_{n},-)$
with respect to the corestriction maps and $(-)^{\vee}$ is the Pontryagin
dual. However, we do not have any clue of the construction of
$\boldsymbol{\kappa}^{\mathrm{Flach},\ell,\infty}$ yet. We may need to put the
$p$-ordinary assumption for $E$ in order to have the correct formulation.

Denote by $\mathrm{deg}(\phi)$ the minimal modular degree of $E$. We obtain
the following connection between Flach’s zeta elements and the minimal
modularity lifting theorem from Theorem 1.1.

######  Corollary 1.4 (Minimal modularity lifting).

The following three statements are equivalent:

  1. (1)

The constant multiple in Theorem 1.1 is a $p$-adic unit so that

$\mathrm{ord}_{p}(\mathrm{loc}^{s}_{\ell}c(\ell))=\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}])).$

  2. (2)

The Bloch–Kato conjecture for the adjoint representation of $E$ holds, i.e.

$\mathrm{ord}_{p}(\mathrm{deg}(\phi))=\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}])).$

  3. (3)

The minimal deformation ring of $\overline{\rho}$ and the minimal Hecke
algebra localized at the maximal ideal $\mathfrak{m}_{\overline{\rho}}$
associated to $\overline{\rho}$ are isomorphic as complete intersections, i.e.

$R^{\mathrm{min}}_{\overline{\rho}}\simeq\mathbb{T}^{\mathrm{min}}_{\mathfrak{m}_{\overline{\rho}}}.$

###### Proof.

See $\S$6.2. ∎

Of course, all the statements in Corollary 1.4 are true thanks to the work of
Wiles and Taylor–Wiles [Wil95, TW95]. One remarkable feature of Corollary 1.4
is that the modularity lifting theorem can be deduced from a certain property
regarding Flach’s zeta elements (finally!).

On the other hand, the modularity lifting theorem (Corollary 1.4.(3)) yields
the explicit construction of the rank one primitive Kolyvagin system for
$\mathrm{Sym}^{2}(T_{p}E)$ extending Flach’s zeta elements (without
introducing any constant multiple) for every elliptic curve $E$ which occurs
in the minimal deformation space of $\overline{\rho}$555We thank H. Hida for
bringing this deformation-theoretic observation to our attention. .

From the viewpoint of Corollary 1.4, we anticipate that our construction of
the Stark system will contribute directly to the revival of Wiles’ abandoned
“geometric Euler system” approach to the semi-stable Shimura–Taniyama
conjecture in a future.

Although we do not know whether or how
$\mathrm{loc}^{s}_{\ell}(\boldsymbol{\kappa}^{\mathrm{Flach},\ell})$
determines the structure of
$\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}]))$ in general yet666In
[Kim], the self-duality of Galois representations is used in an essential way.
We are informed that R. Sakamoto is working on the general structure theorem
with rank _zero_ Stark/Kolyvagin systems, and it seems to be closely related
to our approach., we still have the following structural application under the
$p$-indivisibility assumption on $c(\ell)$.

######  Corollary 1.5 (Cyclicity).

Suppose that any (equivalent) statement of Corollary 1.4 holds. If we further
assume that
$c(\ell)\in\mathrm{Sel}_{\ell\textrm{-}\mathrm{rel}}(\mathbb{Q},\mathrm{Sym}^{2}(T_{p}E))$
is not divisible by $p$, then the adjoint Selmer group is cyclic, i.e.

$\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}]))\simeq\mathbb{Z}_{p}/\mathrm{deg}(\phi)\mathbb{Z}_{p}$

where
$\mathrm{Sel}_{\ell\textrm{-}\mathrm{rel}}(\mathbb{Q},\mathrm{Sym}^{2}(T_{p}E))$
is the $\ell$-relaxed compact symmetric square Bloch–Kato Selmer group.

###### Proof.

See $\S$6.3. ∎

This cyclicity can be understood as the structural refinement of the
Bloch–Kato conjecture (cf. [Kim, Kim24]). We make the following question (not
conjecture).

######  Question 1.6.

Does the $p$-indivisibility assumption in Corollary 1.5 hold in general? It is
equivalent to proving that
$\mathrm{Sel}_{\ell\textrm{-}\mathrm{str}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}]))$
is trivial.

The higher weight generalization of Theorem 1.1 and its interpretation in
terms of rank _zero_ Stark/Kolyvagin systems will be investigated in a joint
work with Ryotaro Sakamoto.

##  2\. Flach’s zeta elements and the explicit reciprocity law

Let $G_{k}\subseteq\mathrm{GL}_{2}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the image
of $G_{\mathbb{Q}}$ under the mod $p^{k}$ reduction of
$\rho:G_{\mathbb{Q}}\to\mathrm{Aut}_{\mathbb{Z}_{p}}(T_{p}E)\simeq\mathrm{GL}_{2}(\mathbb{Z}_{p})$.
The following annihilation result is the very starting point of this story.

######  Theorem 2.1 (Flach).

Assume that $p$ is odd, $\overline{\rho}$ is absolutely irreducible, $p$ does
not divide $N$, and $N=N(\overline{\rho})$. If we further assume that $N$ is
square-free and $\mathrm{H}^{1}(G_{k},\mathrm{ad}^{0}(E[p^{k}]))=0$ for all
$k\geq 1$, then

$\mathrm{deg}(\phi)\cdot\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}]))=0.$

In particular, $\mathrm{Sel}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}]))$ is
finite, and it is trivial for all but finitely many primes $p$.

###### Proof.

See [Fla95, Thm. 1] and [Fla92, Thm. 1]. ∎

In order to obtain Theorem 2.1, Flach constructed the following important
cohomology classes, which we call Flach’s zeta elements.

######  Theorem 2.2 (Flach’s zeta elements).

Assume that $p$ is odd, $\overline{\rho}$ is absolutely irreducible, $p$ does
not divide $N$, and $N=N(\overline{\rho})$. For a prime $\ell$ not dividing
$2Np$, there exists a cohomology class

$c(\ell)\in\mathrm{Sel}_{\ell\textrm{-}\mathrm{rel}}(\mathbb{Q},\mathrm{Sym}^{2}(T_{p}E))$

such that

  * •

$\mathrm{loc}_{q}(c(\ell))\in\mathrm{H}^{1}_{f}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(T_{p}E))$
for any prime $q\neq\ell$, and

  * •

$\mathrm{ord}_{p}(\mathrm{loc}^{s}_{\ell}(c(\ell)))\leq\mathrm{ord}_{p}(\mathrm{deg}(\phi))+\mathrm{ord}_{p}(\alpha_{\ell}-\beta_{\ell})$

where $\alpha_{\ell}$ and $\beta_{\ell}$ are the roots of
$X^{2}-a_{\ell}(E)X+\ell$.

###### Proof.

See [Fla95, Prop. 1 and $\S$5]. ∎

######  Definition 2.3.

Let $k\geq 1$ be an integer. A prime $\ell$ is said to be a $k$-Flach prime if
$(\ell,Np)=1$, $\ell\equiv 1\pmod{p^{k}}$, and $a_{\ell}(E)\not\equiv\pm
2\pmod{p}$. This condition is equivalent to that $(\ell,Np)=1$ and the
arithmetic Frobenius at $\ell$ acts on $E[p^{k}]$ as $\tau$ chosen in Choice
3.4 below. We call 1-Flach primes by Flach primes for convenience. Denote by
$\mathcal{P}^{\mathrm{Flach},(\ell)}_{k}$ the set of all $k$-Flach primes
except $\ell$ for fixed $(E,p)$ and by
$\mathcal{N}^{\mathrm{Flach},(\ell)}_{k}$ the set of square-free products of
primes in $\mathcal{P}^{\mathrm{Flach},(\ell)}_{k}$.

It is easy to check that $\alpha_{\ell}-\beta_{\ell}$ is a $p$-adic unit for a
Flach prime $\ell$. From Theorem 2.2, we have inequality

$\mathrm{ord}_{p}(\mathrm{loc}^{s}_{\ell}(c(\ell)))\leq\mathrm{ord}_{p}(\mathrm{deg}(\phi))$

for a Flach prime $\ell$. Here, $\mathrm{deg}(\phi)$ plays the role of the
_depth of a partial geometric Euler system_ in the sense of Mazur and Weston
[Maz, Wes01, Wes02].

This inequality can be upgraded to equality thanks to Weston’s computation
[Wes01, Wes02]. The equality (2.1) below should be viewed as the _explicit
reciprocity law for Flach’s zeta elements_ since $\mathrm{deg}(\phi)$ has a
precise connection with the adjoint $L$-value via the formula of Shimura and
Hida.

######  Proposition 2.4 (Weston’s explicit reciprocity law).

If $\ell$ is a Flach prime, the inequality in Theorem 2.2 becomes equality

(2.1)
$\mathrm{ord}_{p}(\mathrm{loc}^{s}_{\ell}c(\ell))=\mathrm{ord}_{p}(\mathrm{deg}(\phi)).$

###### Proof.

See $\S$4. ∎

The collection of $c(\ell)$ varying Flach primes $\ell$ forms a (cohesive)
Flach system of depth $\mathrm{deg}(\phi)$ in the sense of Mazur and Weston
[Maz, Wes02].

##  3\. Verifying the hypotheses for Stark systems

We review the running hypotheses of Stark systems (of core rank one) in [MR16,
$\S$4] and verify them for our setting.

###  3.1. Running hypotheses

Let $\Sigma=\\{p,\infty,\textrm{ramified primes for }T_{p}E\\}$ and fix a
Flach prime $\ell$.

######  Definition 3.1.

By Selmer data, we mean the triple

$(\mathrm{Sym}^{2}(T_{p}E),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{\mathrm{Flach},(\ell)})$

where

  * •

$\mathrm{Sym}^{2}(T_{p}E)$ as a $G_{\mathbb{Q}}$-module,

  * •

$\mathcal{F}^{\ell}_{\mathrm{BK}}$ is the Bloch–Kato Selmer structure except
the relaxed local condition at $\ell$, and

  * •

$\mathcal{P}^{\mathrm{Flach},(\ell)}$ is the set of Flach primes excluding
$\ell$.

For a $G_{\mathbb{Q}}$-module $A$, we write $\mathbb{Q}(A)$ for the fixed
field in $\overline{\mathbb{Q}}$ of the kernel of the map
$G_{\mathbb{Q}}\to\mathrm{Aut}(A)$.

######  Assumption 3.2.

We list the assumptions to proceed the Stark system argument [MR16, $\S$4]:

  1. (H.1)

$\mathrm{H}^{0}(\mathbb{Q},\mathrm{Sym}^{2}(E[p]))=\mathrm{H}^{0}(\mathbb{Q},\mathrm{ad}^{0}(E[p]))=0$
and $\mathrm{Sym}^{2}(E[p])$ is absolutely irreducible.

  2. (H.2)

There exists an element $\tau\in
G_{\mathbb{Q}(\zeta_{p^{\infty}})}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\zeta_{p^{\infty}}))$
such that $\mathrm{Sym}^{2}(T_{p}E)/(\tau-1)\mathrm{Sym}^{2}(T_{p}E)$ is free
of rank one over $\mathbb{Z}_{p}$.

  3. (H.3)

$\mathrm{H}^{1}(\mathbb{Q}_{T}/\mathbb{Q},\mathrm{Sym}^{2}(E[p]))=\mathrm{H}^{1}(\mathbb{Q}_{T}/\mathbb{Q},\mathrm{ad}^{0}(E[p]))=0$
where
$\mathbb{Q}_{T}=\mathbb{Q}(\mathrm{Sym}^{2}(T_{p}E),\zeta_{p^{\infty}})$.

  4. (H.4)

$\mathrm{Sym}^{2}(E[p])\not\simeq\mathrm{ad}^{0}(E[p])$ as Galois modules or
$p>3$.

  5. (H.5)

The Selmer structure $\mathcal{F}^{\ell}_{\mathrm{BK}}$ is cartesian.

  6. (H.6)

$\chi(\mathrm{Sym}^{2}(T_{p}E),\mathcal{F}^{\ell}_{\mathrm{BK}})=1$, where
$\chi(\mathrm{Sym}^{2}(T_{p}E),\mathcal{F}^{\ell}_{\mathrm{BK}})$ is the core
rank of $(\mathrm{Sym}^{2}(T_{p}E),\mathcal{F}^{\ell}_{\mathrm{BK}})$.

  7. (H.7)

$I_{q}=0$ for $q\in\mathcal{P}^{\mathrm{Flach},(\ell)}_{k}$ when the
coefficient ring is $\mathbb{Z}/p^{k}\mathbb{Z}$.

The following lemma is useful.

######  Lemma 3.3.

Let $E$ be a semi-stable elliptic curve over $\mathbb{Q}$, $p$ a prime, and
$\overline{\rho}:G_{\mathbb{Q}}\to\mathrm{Aut}_{\mathbb{F}_{p}}(E[p])$. Then
$\overline{\rho}$ is surjective if and only if $\overline{\rho}$ is
irreducible.

###### Proof.

See [Edi97, Prop. 2.1]. ∎

###  3.2. Verifying (H.1)

(H.1) follows from Lemma 3.3 and [Rub00, $\S$3.6].

###  3.3. Verifying (H.2)

Thanks to Lemma 3.3, we are able to make the following choice of a Galois
element.

######  Choice 3.4.

We choose an element $\tau\in G_{\mathbb{Q}(\zeta_{p^{\infty}})}$ such that

$\rho(\tau)\sim\begin{pmatrix}\alpha&0\\\ 0&\alpha^{-1}\end{pmatrix}$

where $\alpha\in\mathbb{Z}_{p}$ with $\alpha^{2}\not\equiv 1\pmod{p}$ when
$p>3$, or $\alpha\in\mathbb{Z}_{p^{2}}$ with
$\alpha\equiv\sqrt{-1}\in\mathbb{F}_{3^{2}}$ when $p=3$.

This choice of $\tau$ is possible and satisfies (H.2) for
$\mathrm{Sym}^{2}(T_{p}E)$ as explained in [Rub00, $\S$3.6]. The choice of
$\tau$ determines the type of auxiliary primes when we apply the Chebotarev
density theorem as in Definition 2.3. In the case of Heegner points, $\tau$ is
chosen to be the complex conjugation. In [Fla92], $\tau$ is chosen to be the
complex conjugation again. For Kato’s Euler systems, $\tau$ is chosen to be a
unipotent element as explained in [Rub00, Prop. 3.5.8]. We also recommend the
reader to read [Wil95, Intro.] for his change of auxiliary primes.

###  3.4. Verifying (H.3)

(H.3) follows from [Rub00, $\S$3.6]. See also [Fla92, Lem. 1.2] with [Fla95,
Rem. 1 in $\S$4], and [DDT97, Lem. 2.48] when $p>5$.

###  3.5. Verifying (H.4)

This is immediate.

###  3.6. Verifying (H.5)

(H.5) follows from [MR04, Lem. 3.7.1 and Rem. 3.7.2].

###  3.7. Verifying (H.6)

As explained in [MR04, Rem. 6.3.4], the core rank of Kolyvagin systems for
$\left(\mathrm{Sym}^{2}(T_{p}E),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{\mathrm{Flach},(\ell)}\right)$
is one since $\ell$ is a Flach prime. The core rank of Stark systems is
defined in the exactly same way. See also [MR04, Thm. 4.1.13] and [MR16, Prop.
3.3].

###  3.8. Verifying (H.7)

Suppose that we are working over the
$\mathbb{Z}/p^{k}\mathbb{Z}$-coefficients. For
$q\in\mathcal{P}^{\mathrm{Flach},(\ell)}_{k}$, $I_{q}$ is defined by the ideal
of $\mathbb{Z}/p^{k}\mathbb{Z}$ generated by
$(1-\alpha^{2}_{q})\cdot(1-q)\cdot(1-\beta^{2}_{q})$ and $(1-q)$. This means
that $I_{q}=(1-q)=(0)$, so we are done.

##  4\. A proof of the explicit reciprocity law

We recall the computation done by Weston in [Wes01] but with a very slight
modification. Proposition 2.4 follows from this computation.

###  4.1. A lemma

######  Lemma 4.1.

Suppose that $q$ is prime to $Np$. Then
$\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(T_{p}E))\simeq\mathrm{H}^{0}(\mathbb{F}_{q},\mathrm{ad}^{0}(T_{p}E))$.
If $q$ is a $k$-Flach prime, then we have isomorphism

$\psi_{q}:\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(E[p^{k}]))\simeq\mathbb{Z}/p^{k}\mathbb{Z}.$

###### Proof.

See [Wes01, Lem. 10.9] for the first statement and the mod $p^{k}$ version
also holds by the same reasoning. Since [Wes01, Lem. 10.9] concerns the
complex conjugation, we give the detail for the latter. Since $q$ is a
$k$-Flach prime, the arithmetic Frobenius $\mathrm{Fr}_{q}$ at $q$ acts on
$E[p^{k}]$ by $\tau$ in Choice 3.4. Choose a basis $x$, $y$ of $E[p^{k}]$ such
that $\mathrm{Fr}_{q}(x)=\alpha\cdot x$ and
$\mathrm{Fr}_{q}(y)=\alpha^{-1}\cdot y$. Then $x\otimes x$, $x\otimes
y+y\otimes x$, $y\otimes y$ forms a basis of $\mathrm{Sym}^{2}(E[p^{k}])$.
Since $\mathrm{Sym}^{2}(E[p^{k}])(-1)=\mathrm{ad}^{0}(E[p^{k}])$,
$\mathrm{Fr}_{q}$ acts on the induced basis of $\mathrm{ad}^{0}(E[p^{k}])$ by
multiplication by $\alpha^{2}q^{-1}$, $q^{-1}$, and $\alpha^{-2}q^{-1}$,
respectively. Since $q\equiv 1\pmod{p^{k}}$ and $\alpha^{2}\not\equiv
1\pmod{p}$,
$\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(E[p^{k}]))\simeq\mathrm{H}^{0}(\mathbb{F}_{q},\mathrm{ad}^{0}(E[p^{k}]))$
is free of rank one over $\mathbb{Z}/p^{k}\mathbb{Z}$, so we are done. ∎

###  4.2. The computation of the image

We recap Weston’s computation of the image of $c(\ell)$ under
$\mathrm{loc}^{s}_{\ell}$ but with Flach prime $\ell$. This computation
slightly refines [Fla95, Lem. 2.5], and see [Wes02, Thm. 3.1.1] for a more
theoretical background of this computation.

As in Lemma 4.1, we choose a basis $x$, $y$ of $T_{p}E$ with respect to which
$\mathrm{Fr}_{\ell}$ has matrix

$\begin{pmatrix}\alpha&0\\\ 0&\alpha^{-1}\cdot\ell\end{pmatrix}$

where $\alpha\in\mathbb{Z}_{p}$ with $\alpha^{2}\not\equiv 1\pmod{p}$. By
using the same idea of Lemma 4.1 again, we have isomorphism

$\displaystyle\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(T_{p}E))$
$\displaystyle\simeq\mathrm{H}^{0}(\mathbb{F}_{\ell},\mathrm{ad}^{0}(T_{p}E))$
$\displaystyle=\mathbb{Z}_{p}\cdot\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

Following Weston’s computation in [Wes01, p. 369], the image of
$\mathrm{loc}^{s}_{\ell}c(\ell)$ in
$\mathrm{H}^{0}(\mathbb{F}_{\ell},\mathrm{ad}^{0}(T_{p}E))$ is

$6\cdot\mathrm{deg}(\phi)\cdot(\alpha_{\ell}-\beta_{\ell})\cdot\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$

Since $p\geq 5$ and $\ell$ is a Flach prime, we have an isomorphism of cyclic
modules

$\dfrac{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(T_{p}E))}{\mathbb{Z}_{p}\cdot\mathrm{loc}^{s}_{\ell}c(\ell)}\simeq\mathbb{Z}_{p}/\mathrm{deg}(\phi)\mathbb{Z}_{p},$

which implies the explicit reciprocity law (2.1).

##  5\. The construction of Flach–Stark systems

We freely use the language in [MR04, MR16] in order to keep the argument
concise.

###  5.1. Basic setup for Stark systems

Due to Theorem 2.1 and [MR04, Lem. 3.5.3], we are able to and do fix an
integer $k\gg 0$ such that

$\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))=\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{\infty}]))$

where $\mathcal{F}_{\mathrm{BK}}$ is the Bloch–Kato Selmer structure.

Fix a Flach prime $\ell$ and let

(5.1) $n=q_{1}\cdot q_{2}\cdot\cdots\cdot
q_{s}\in\mathcal{N}^{\mathrm{Flach},(\ell)}_{k}$

throughout this section.

######  Remark 5.1.

In the case of Kolyvagin systems of Gauss sums [MR04, Rem. 6.3.3], the choice
of $n$ depends seriously on $\ell$. Namely, every prime divisor of $n$ should
divide $\ell-1$. In our case, there are no such restrictions, so each $q_{i}$
is just a $k$-Flach prime not equal to $\ell$.

Following [MR16, $\S$6], recall that

$\displaystyle W_{n}$
$\displaystyle=\bigoplus^{s}_{i=1}\mathrm{Hom}\left(\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q_{i}},\mathrm{Sym}^{2}(E[p^{k}])),\mathbb{Z}/p^{k}\mathbb{Z}\right),$
$\displaystyle Y_{n}$
$\displaystyle=\bigwedge^{1+\nu(n)}\mathrm{Sel}_{\mathcal{F}^{\ell
n}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\otimes\bigwedge^{\nu(n)}W_{n}$

where $\nu(n)=s$ is the number of prime divisors of $n$, and
$\mathcal{F}^{\ell n}_{\mathrm{BK}}$ is the $\ell n$-relaxed Bloch–Kato Selmer
structure. For $m$ dividing $n$, we have the cartesian square

$\textstyle{\mathrm{Sel}_{\mathcal{F}^{\ell
m}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\oplus_{q|m}\mathrm{loc}^{s}_{q}}$$\textstyle{\mathrm{Sel}_{\mathcal{F}^{\ell
n}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\oplus_{q|n}\mathrm{loc}^{s}_{q}}$$\textstyle{\bigoplus_{q|m}\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{q|n}\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(E[p^{k}]))}$

and the canonical map $\Psi_{n,m}:Y_{n}\to Y_{m}$ attached to the above square
as in [MR16, Prop. A.2]. For a $k$-Flach prime $q$, we have an isomorphism

$\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(E[p^{k}]))\simeq\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{q},\mathrm{Sym}^{2}(E[p^{k}]))$

by [MR04, Lem. 1.2.4] where $\mathrm{H}^{1}_{\mathrm{tr}}$ means the
transverse local condition.

###  5.2. Definition and properties of Stark systems

The canonical map $\Psi_{n,m}$ given above has the following compatibility.

######  Proposition 5.2.

Suppose $n_{0}\in\mathcal{N}^{\mathrm{Flach},(\ell)}_{k}$, $n_{1}$ divides
$n_{0}$, and $n_{2}$ divides $n_{1}$. Then

$\Psi_{n_{0},n_{2}}=\Psi_{n_{1},n_{2}}\circ\Psi_{n_{0},n_{1}}.$

###### Proof.

See [MR16, Prop. 6.4]. ∎

We briefly recall the definition of Stark systems [MR16, Def. 6.5] and their
important properties.

######  Definition 5.3.

The $\mathbb{Z}/p^{k}\mathbb{Z}$-module of the (rank one) Stark system
$\mathbf{SS}_{1}(\mathrm{Sym}^{2}(E[p^{k}]))=\mathbf{SS}_{1}(\mathrm{Sym}^{2}(E[p^{k}]),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{\mathrm{Flach},(\ell)}_{k})$
for Selmer data
$(\mathrm{Sym}^{2}(E[p^{k}]),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{\mathrm{Flach},(\ell)}_{k})$
is defined to be the inverse limit

$\mathbf{SS}_{1}(\mathrm{Sym}^{2}(E[p^{k}]))=\varprojlim_{n_{0}\in\mathcal{N}^{\mathrm{Flach},(\ell)}_{k}}Y_{n_{0}}$

with respect to the maps $\Psi_{n_{0},n_{1}}$ with $n_{1}|n_{0}$. An element
$\boldsymbol{\epsilon}^{\ell}=\varprojlim_{n_{0}\in\mathcal{N}^{\mathrm{Flach},(\ell)}_{k}}\epsilon^{\ell}_{n_{0}}$
in $\mathbf{SS}_{1}(\mathrm{Sym}^{2}(E[p^{k}]))$ is called a Stark system.

######  Theorem 5.4.

Under our working hypotheses, we have
$\mathbf{SS}_{1}(\mathrm{Sym}^{2}(E[p^{k}]))\simeq\mathbb{Z}/p^{k}\mathbb{Z}$.

###### Proof.

See [MR16, Thm. 6.7]. ∎

A Stark system $\boldsymbol{\epsilon}^{\ell}$ is primitive if the image of
$\boldsymbol{\epsilon}^{\ell}$ in $\mathbf{SS}_{1}(\mathrm{Sym}^{2}(E[p]))$ is
non-zero [MR16, Prop. 7.3 and Thm. 7.4]. The theory of Stark systems yields
the following result on the exact size of the dual Selmer groups.

######  Theorem 5.5.

We keep our working hypotheses, and let $\boldsymbol{\epsilon}^{\ell}$ be a
Stark system.

  1. (1)

If $\boldsymbol{\epsilon}^{\ell}$ is non-trivial, then

$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{\ell\textrm{-}\mathrm{str}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))=\mathrm{ord}(\epsilon^{\ell}_{1})-\partial\varphi_{\boldsymbol{\epsilon}^{\ell}}(\infty)$

where $\partial\varphi_{\boldsymbol{\epsilon}^{\ell}}(\infty)$ is the minimal
valuation of the whole Stark system $\boldsymbol{\epsilon}^{\ell}$.

  2. (2)

If $\boldsymbol{\epsilon}^{\ell}$ is primitive, then

$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{\ell\textrm{-}\mathrm{str}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))=\mathrm{ord}(\epsilon^{\ell}_{1}).$

  3. (3)

$\boldsymbol{\epsilon}^{\ell}$ is primitive if and only if
$\partial\varphi_{\boldsymbol{\epsilon}^{\ell}}(\infty)=0$.

###### Proof.

See [MR16, Thm. 8.7]. In fact, the module structure of
$\mathrm{Sel}_{\ell\textrm{-}\mathrm{str}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))$
is also determined by $\boldsymbol{\epsilon}^{\ell}$. ∎

###  5.3. The key computation

We keep the choices of $\ell$ and $n$ in (5.1) from now on. We explicitly
compute

$\Psi_{n,n/q_{s}}\left(c(\ell)\wedge\bigwedge^{s}_{i=1}c(q_{i})\otimes\bigwedge^{s}_{i=1}\psi_{q_{i}}\right)$

where $c(\ell)$ and $c(q_{i})$’s are Flach’s zeta elements in Theorem 2.2 and
$\psi_{q_{i}}:\mathrm{H}^{1}_{/f}(\mathbb{Q}_{q_{i}},\mathrm{Sym}^{2}(E[p^{k}]))\simeq\mathbb{Z}/p^{k}\mathbb{Z}$
is the map in Lemma 4.1. Here we use the same notation $c(\ell)$ and
$c(q_{i})$ for their image in
$\mathrm{H}^{1}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))$. Note that $c(\ell)$
and all $c(q_{i})$’s are linearly independent due to their local conditions in
Theorem 2.2. We first apply [MR16, Prop. A.1] to our setting. Consider exact
sequence

$\textstyle{\mathrm{Sel}_{\mathcal{F}^{\ell
n/q_{s}}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{\mathcal{F}^{\ell
n}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}}$$\textstyle{\mathbb{Z}/p^{k}\mathbb{Z}.}$

Then there exists a unique map

$\widehat{\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}}:\bigwedge^{s+1}\mathrm{Sel}_{\mathcal{F}^{\ell
n}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\to\mathbb{Z}/p^{k}\mathbb{Z}\otimes\bigwedge^{s}\mathrm{Sel}_{\mathcal{F}^{\ell
n/q_{s}}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))$

such that the composition

$\textstyle{\bigwedge^{s+1}\mathrm{Sel}_{\mathcal{F}^{\ell
n}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}}}$$\textstyle{\mathbb{Z}/p^{k}\mathbb{Z}\otimes\bigwedge^{s}\mathrm{Sel}_{\mathcal{F}^{\ell
n/q_{s}}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/p^{k}\mathbb{Z}\otimes\bigwedge^{s}\mathrm{Sel}_{\mathcal{F}^{\ell
n}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))}$

is given by

$c(\ell)\wedge\bigwedge_{i=1}^{s}c(q_{i})\mapsto\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}(c(\ell))\cdot\bigwedge_{i=1}^{s}c(q_{i})+\sum_{i=1}^{s}\left((-1)^{i}\cdot\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}(c(q_{i}))\cdot
c(\ell)\wedge\bigwedge^{s}_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}c(q_{j})\right),$

and the image of $\widehat{\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}}$ is the
image of
$\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}\left(\mathrm{Sel}_{\mathcal{F}^{\ell
n}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\right)\otimes\bigwedge^{s}\mathrm{Sel}_{\mathcal{F}^{\ell
n/q_{s}}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))$ in
$\mathbb{Z}/p^{k}\mathbb{Z}\otimes\bigwedge^{s}\mathrm{Sel}_{\mathcal{F}^{\ell
n/q_{s}}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))$.

By the local conditions of Flach’s zeta elements in Theorem 2.2, we have

$\displaystyle\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}(c(\ell))\cdot\bigwedge_{i=1}^{s}c(q_{i})+\sum_{i=1}^{s}\left((-1)^{i}\cdot\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}(c(q_{i}))\cdot
c(\ell)\wedge\bigwedge^{s}_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}c(q_{j})\right)$
$\displaystyle=(-1)^{s}\cdot\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}(c(q_{s}))\cdot
c(\ell)\wedge\bigwedge^{s-1}_{i=1}c(q_{i}).$

Following the concrete description of $\Psi_{n,m}$ in [MR16, pp. 161], we have

(5.2)
$\Psi_{n,n/q_{s}}\left(c(\ell)\wedge\bigwedge^{s}_{i=1}c(q_{i})\otimes\bigwedge^{s}_{i=1}\psi_{q_{i}}\right)=(-1)^{s}\cdot\psi_{q_{s}}\circ\mathrm{loc}^{s}_{q_{s}}(c(q_{s}))\cdot
c(\ell)\wedge\bigwedge^{s-1}_{i=1}c(q_{i})\otimes\bigwedge^{s-1}_{i=1}\psi_{q_{i}}.$

The computation of the image under $\Psi_{n,n/q_{i}}$ is identical for every
$i$ possibly except the sign. By Proposition 5.2, we also have

(5.3) $\Psi_{n,1}=\Psi_{q_{1},1}\circ\Psi_{q_{1}\cdot
q_{2},q_{1}}\circ\cdots\circ\Psi_{n/q_{s},n/(q_{s-1}\cdot
q_{s})}\circ\Psi_{n,n/q_{s}}.$

###  5.4. A Stark system of Gauss sum type and its extension

We keep the choices of $\ell$ and $n$ in (5.1) as before. Define

$\epsilon^{\mathrm{Flach}?,\ell}_{n}:=(-1)^{\frac{s(s+1)}{2}}\cdot
c(\ell)\wedge\bigwedge^{s}_{i=1}c(q_{i})\otimes\bigwedge^{s}_{i=1}\psi_{q_{i}}\in
Y_{n}$

so that

$\Psi_{n,1}(\epsilon^{\mathrm{Flach}?,\ell}_{n})=\left(\prod_{i=1}^{s}\psi_{q_{i}}\circ\mathrm{loc}^{s}_{q_{i}}(c(q_{i}))\right)\cdot
c(\ell).$

This formula follows from the iteration of (5.2) via (5.3).

For $m$ dividing $n$, we define

$\epsilon^{\mathrm{Flach}?,\ell}_{m}:=\Psi_{n,m}(\epsilon^{\mathrm{Flach}?,\ell}_{n}).$

By Definition 5.3, $\left\\{\epsilon^{\mathrm{Flach}?,\ell}_{m}\in
Y_{m}:m|n\right\\}$ forms a _finite_ Stark system of core rank one, i.e. a
Stark system for Selmer data
$(\mathrm{Sym}^{2}(E[p^{k}]),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{n})$
where $\mathcal{P}^{n}$ is the (finite!) set of the primes dividing $n$.

Let $m$ is a divisor of $n$ and
$G_{m}=\otimes_{q|m}\mathrm{Gal}(\mathbb{Q}(\zeta^{(p)}_{q})/\mathbb{Q})$
where $\mathbb{Q}(\zeta^{(p)}_{q})$ is the maximal $p$-subextension of
$\mathbb{Q}$ in $\mathbb{Q}(\zeta_{q})$. Recall the map in [MR16, $\S$12]

$\Pi_{m}:Y_{m}\to\mathrm{Sel}_{\mathcal{F}^{\ell}_{\mathrm{BK}}(m)}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\otimes
G_{m}$

where $\mathcal{F}^{\ell}_{\mathrm{BK}}(m)$ is the $\ell$-relaxed and
$m$-transverse Bloch–Kato Selmer structure. Then [MR16, Prop. 12.3] implies
that

$\left\\{(-1)^{\nu(m)}\cdot\Pi_{m}(\epsilon^{\mathrm{Flach}?,\ell}_{m}):m|n\right\\}$

forms a _finite_ Kolyvagin system for
$\left(\mathrm{Sym}^{2}(E[p^{k}]),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{n}\right)$.
Thanks to the core rank one property recalled in $\S$3.7, the finite Kolyvagin
system extends to the rank one Kolyvagin system
$\boldsymbol{\kappa}^{\mathrm{Flach}?,\ell}$ for
$\left(\mathrm{Sym}^{2}(E[p^{k}]),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{\mathrm{Flach},(\ell)}_{k}\right)$
as in the case of Gauss sum Kolyvagin systems [MR04, Rem. 6.3.3].

######  Remark 5.6.

If $n\in\mathcal{P}^{\mathrm{Flach},(\ell)}_{k}$ with
$\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK},\ell
n}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))=0$, then
$\kappa^{\mathrm{Flach}?,\ell}_{n}$ is uniquely determined by
$\kappa^{\mathrm{Flach}?,\ell}_{m}$ for $m$ properly dividing $n$ [MR04, Rem.
3.1.9] where $\mathcal{F}_{\mathrm{BK},\ell n}$ is the $\ell n$-strict
Bloch–Kato Selmer structure. From this point of view, the extension of a
finite Kolyvagin system is not a miracle and it shows the strength of the
rigidity of Kolyvagin systems.

The modules of Stark and Kolyvagin systems over $\mathbb{Z}/p^{k}\mathbb{Z}$
are isomorphic as free $\mathbb{Z}/p^{k}\mathbb{Z}$-modules of rank one
(Theorem 5.4 and [MR16, Rem. 11.5 and Thm. 12.4]). By using this isomorphism,
the finite Stark system also extends to the rank one Stark system
$\boldsymbol{\epsilon}^{\mathrm{Flach}?,\ell}$ for
$\left(\mathrm{Sym}^{2}(E[p^{k}]),\mathcal{F}^{\ell}_{\mathrm{BK}},\mathcal{P}^{\mathrm{Flach},(\ell)}_{k}\right)$.

###  5.5. The primitive normalization

We construct the _primitive_ Stark system
$\boldsymbol{\epsilon}^{\mathrm{Flach},\ell}$ from
$\boldsymbol{\epsilon}^{\mathrm{Flach}?,\ell}$ and compute the constant
multiple in Theorem 1.1.

By global Poitou–Tate duality [Rub00, Thm. 1.7.3], we have exact sequence

(5.4)
$\textstyle{\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK},\ell}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\dfrac{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(E[p^{k}]))}{\mathrm{loc}^{s}_{\ell}\left(\mathrm{Sel}_{\mathcal{F}^{\ell}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\right)}\right)^{\vee}}$

where $(-)^{\vee}$ is the Pontryagin dual. We also have

$\displaystyle\kappa^{\mathrm{Flach}?,\ell}_{1}$
$\displaystyle=\epsilon^{\mathrm{Flach}?,\ell}_{1}$
$\displaystyle=\left(\prod_{i=1}^{s}\psi_{q_{i}}\circ\mathrm{loc}^{s}_{q_{i}}(c(q_{i}))\right)\cdot
c(\ell)$
$\displaystyle\in\mathrm{Sel}_{\mathcal{F}^{\ell}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))$
$\displaystyle\simeq\mathbb{Z}/p^{k}\mathbb{Z}\oplus\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK},\ell}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))$

where the last isomorphism is non-canonical and follows from [MR16, Cor.
3.5.(i)]. By [MR04, Thm. 4.4.1], we know
$\langle\kappa^{\mathrm{Flach}?,\ell}_{1}\rangle\subseteq\mathbb{Z}/p^{k}\mathbb{Z}$
in the above decomposition. Thus, we have equality

(5.5)
$\mathrm{ord}_{p}(c(\ell))+\mathrm{length}_{\mathbb{Z}_{p}}\left(\dfrac{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(E[p^{k}]))}{\mathrm{loc}^{s}_{p}\left(\mathrm{Sel}_{\mathcal{F}^{\ell}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\right)}\right)^{\vee}=\mathrm{ord}_{p}(\mathrm{loc}^{s}_{p}(c(\ell)))$

where $\mathrm{ord}_{p}(a)$ means the $p$-divisibility index of $a$ in the
natural module containing $a$.

Write
$\lambda=\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))$
for convenience. Consider the _tautological_ bound

$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))=\lambda=\mathrm{ord}_{p}\left(\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}\cdot\mathrm{loc}^{s}_{\ell}(c(\ell))\right)$

where
$\psi_{\ell}:\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(E[p^{k}]))\simeq\mathbb{Z}/p^{k}\mathbb{Z}$
is the map in Lemma 4.1. By (5.4) and (5.5), we also obtain the exact bound
the $\ell$-strict one

(5.6)
$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK},\ell}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))=\mathrm{ord}_{p}\left(\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}\cdot
c(\ell)\right).$

We apply Theorem 5.5 to $\boldsymbol{\epsilon}^{\mathrm{Flach}?,\ell}$ and
compare it with (5.6). Then we obtain

(5.7)
$\displaystyle\begin{split}&\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK},\ell}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))\\\
&=\mathrm{ord}_{p}\left(\left(\prod_{i=1}^{s}\psi_{q_{i}}\circ\mathrm{loc}^{s}_{q_{i}}(c(q_{i}))\right)\cdot
c(\ell)\right)+\mathrm{ord}_{p}\left(\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}\right)-\mathrm{ord}_{p}\left(\prod_{i=1}^{s}\psi_{q_{i}}\circ\mathrm{loc}^{s}_{q_{i}}(c(q_{i}))\right),\end{split}$

so the minimal valuation of the whole Stark system
$\boldsymbol{\epsilon}^{\mathrm{Flach}?,\ell}$ is

$\mathrm{ord}_{p}\left(\dfrac{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}{p^{\lambda}}\cdot\prod_{i=1}^{s}\psi_{q_{i}}\circ\mathrm{loc}^{s}_{q_{i}}(c(q_{i}))\right).$

Thus, we define the primitive Flach–Stark system by

$\boldsymbol{\epsilon}^{\mathrm{Flach},\ell}=\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}\cdot\left(\prod_{i=1}^{s}\dfrac{1}{\psi_{q_{i}}\circ\mathrm{loc}^{s}_{q_{i}}(c(q_{i}))}\right)\cdot\boldsymbol{\epsilon}^{\mathrm{Flach}?,\ell}$

in the module of Stark systems, and we have explicit formulas

$\displaystyle\epsilon^{\mathrm{Flach},\ell}_{1}$
$\displaystyle=\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}\cdot
c(\ell),$ $\displaystyle\epsilon^{\mathrm{Flach},\ell}_{n^{\prime}}$
$\displaystyle=\dfrac{(-1)^{\frac{s^{\prime}(s^{\prime}+1)}{2}}\cdot
p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}\cdot\left(\prod_{i=1}^{s^{\prime}}\dfrac{1}{\psi_{q^{\prime}_{i}}\circ\mathrm{loc}^{s}_{q^{\prime}_{i}}(c(q^{\prime}_{i}))}\right)\cdot
c(\ell)\wedge\bigwedge^{s^{\prime}}_{i=1}c(q^{\prime}_{i})\otimes\bigwedge^{s^{\prime}}_{i=1}\psi_{q^{\prime}_{i}}$

for _every_ $n^{\prime}=q^{\prime}_{1}\cdot q^{\prime}_{2}\cdot\cdots\cdot
q^{\prime}_{s^{\prime}}\in\mathcal{N}^{\mathrm{Flach},(\ell)}_{k}$ with
$\nu(n^{\prime})=s^{\prime}\geq 1$. Note that our choice of
$n\in\mathcal{N}^{\mathrm{Flach},(\ell)}_{k}$ in (5.1) was arbitrary (Remark
5.1).

In particular, the constant multiple in Theorem 1.1 is
$\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}=\dfrac{p^{\lambda}}{\mathrm{deg}(\phi)}$
by construction and by the explicit reciprocity law (2.1). Therefore, the
Bloch–Kato conjecture (Corollary 1.4.(2)) is equivalent to showing that this
constant multiple is a $p$-adic unit.

######  Remark 5.7.

Although each
$\dfrac{c(q^{\prime}_{i})}{\psi_{q^{\prime}_{i}}\circ\mathrm{loc}^{s}_{q^{\prime}_{i}}(c(q^{\prime}_{i}))}$
makes sense if and only if $\mathrm{loc}^{s}_{q^{\prime}_{i}}$ is surjective,
$\epsilon^{\mathrm{Flach},\ell}_{n^{\prime}}$ makes sense without assuming the
surjectivity of each $\mathrm{loc}^{s}_{q^{\prime}_{i}}$. In fact, the
possible failure of the surjectivity of these localization maps makes the
Stark system argument non-vacuous.

##  6\. Proofs of corollaries

###  6.1. Proof of Corollary 1.2

This follows from the tautological bound in §5.5 and the constant multiple
$\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}=\dfrac{p^{\lambda}}{\mathrm{deg}(\phi)}$.

###  6.2. Proof of Corollary 1.4

The equivalence between (1) and (2) follows from the last argument in $\S$5.5.
The equivalence between the Bloch–Kato conjecture (2) and the modularity
lifting theorem (3) follows from Wiles’ numerical criterion [DDT97, Thm. 5.3].
See also [DFG04].

###  6.3. Proof of Corollary 1.5

The assumption says that

$\mathrm{ord}_{p}\left(\dfrac{p^{\lambda}}{\psi_{\ell}\circ\mathrm{loc}^{s}_{\ell}(c(\ell))}\cdot
c(\ell)\right)=\mathrm{ord}_{p}\left(c(\ell)\right)=0.$

Thus, (5.6) implies
$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK},\ell}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))=0$.
By global duality (5.4), we have

$\mathrm{Sel}_{\mathcal{F}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{ad}^{0}(E[p^{k}]))\simeq\left(\dfrac{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(E[p^{k}]))}{\mathrm{loc}^{s}_{\ell}\left(\mathrm{Sel}_{\mathcal{F}^{\ell}_{\mathrm{BK}}}(\mathbb{Q},\mathrm{Sym}^{2}(E[p^{k}]))\right)}\right)^{\vee}$

and
$\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},\mathrm{Sym}^{2}(E[p^{k}]))\simeq\mathbb{Z}/p^{k}\mathbb{Z}$
by Lemma 4.1, so the proof is complete.

## Acknowledgement

We thank Francesc Castella, Henri Darmon, Matthias Flach, Minhyong Kim, Masato
Kurihara, Tom Weston, and Christopher Skinner for their interests in our work.
We are benefited from the discussion with Ashay Burungale, Haruzo Hida,
Antonio Lei, Gyujin Oh, and Ryotaro Sakamoto.

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