# The structure of Selmer groups and the Iwasawa main conjecture for elliptic
curves

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

We reveal a new and refined application of (a _weaker_ statement than) the
Iwasawa main conjecture for elliptic curves to the _structure_ of Selmer
groups of elliptic curves of _arbitrary_ rank. For a large class of elliptic
curves, we obtain the following arithmetic consequences.

  * •

Kato’s Kolyvagin system is non-trivial. It is the cyclotomic analogue of the
Kolyvagin conjecture.

  * •

The structure of Selmer groups of elliptic curves over the rationals is
completely determined in terms of certain modular symbols. It is a structural
refinement of Birch and Swinnerton-Dyer conjecture.

  * •

The rank zero $p$-converse, the $p$-parity conjecture, and a new upper bound
of the ranks of elliptic curves are obtained.

  * •

The conjecture of Kurihara on the semi-local description of mod $p$ Selmer
groups is confirmed.

  * •

An application of the $p$-adic Birch and Swinnerton-Dyer conjecture to the
structure of Iwasawa modules is discussed.

###### Key words and phrases:

Birch and Swinnerton-Dyer conjecture, elliptic curves, Iwasawa theory, refined
Iwasawa theory, Kato’s Euler systems, Kolyvagin systems, Kurihara numbers,
modular symbols

###### 2010 Mathematics Subject Classification:

11F67, 11G40, 11R23

###### Contents

  1. 1 Introduction
  2. 2 Kolyvagin systems for elliptic curves
  3. 3 A refined explicit reciprocity law and Kurihara numbers
  4. 4 $\Lambda$-adic Kolyvagin systems for elliptic curves: Proof of Theorem 1.4
  5. 5 The structure of Selmer groups: Proof of Theorem 1.9
  6. 6 The conjecture of Kurihara: Proof of Theorem 1.11
  7. 7 Iwasawa modules and $p$-adic BSD conjectures: Proof of Theorem 1.16
  8. 8 Numerical examples

##  1\. Introduction

###  1.1. Overview

#### 1.1.1.

In modern number theory, one of the most important themes is to understand the
arithmetic meaning of special values of $L$-functions by developing the
connection with the _size_ of arithmetically interesting groups. We go beyond
this philosophy by giving a description of the _structure_ of Selmer groups of
elliptic curves in terms of a certain discrete variation of their special
$L$-values.

Let $p$ be a prime and $E$ an elliptic curve over $\mathbb{Q}$. Without a
doubt, the Selmer group $\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ of the
$p$-power torsion points of $E$ plays a central role in studying the
arithmetic of elliptic curves. The Selmer group encodes the information of the
Mordell–Weil group $E(\mathbb{Q})$ and the Tate–Shafarevich group
$\textrm{{SH}}(E/\mathbb{Q})$ via the fundamental exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(\mathbb{Q})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The celebrated Birch and Swinnerton-Dyer (BSD) conjecture predicts that the
rank of an elliptic curve $E$ equals the vanishing order of the complex
$L$-function of $E$ at $s=1$ and the leading term of the $L$-function knows
the size of $\textrm{{SH}}(E/\mathbb{Q})$, which is conjecturally finite, and
other arithmetic invariants.

The $p$-adic BSD conjecture à la Mazur–Tate–Teitelbaum [MTT86] also predicts
that the rank of an elliptic curve equals the vanishing order of its $p$-adic
$L$-function at the trivial character and the sizes of similar arithmetic
invariants are also encoded in the leading term of the $p$-adic $L$-function.

Here, the key common observation is that a certain _complex or $p$-adic
variation of $L$-values_ detects the rank of an elliptic curve and the leading
term of the Taylor expansion of the variation is related to the size of the
Tate–Shafarevich group and other arithmetic invariants.

In this article, we focus on the _discrete variation of $L$-values_ naturally
arising from Kato’s Kolyvagin systems and establish the corresponding refined
Birch and Swinnerton-Dyer type conjecture, which determines the structure of
Selmer groups. We call the discrete variation the _collection of Kurihara
numbers_ , which are explicitly built out from modular symbols (as defined in
$\S$1.4) and are also realized as the image of Kato’s Kolyvagin system under a
refinement of the dual exponential map. The collection of Kurihara numbers
exactly plays the role of $L$-functions in the context of _refined_ Iwasawa
theory for elliptic curves à la Kurihara [Kur14a, Kur14b].

#### 1.1.2.

The most interesting features of this article are the cyclotomic analogue of
the Kolyvagin conjecture and the structural refinement of Birch and
Swinnerton-Dyer conjecture. These are special cases of Corollaries 1.5 and 1.6
with help of Theorem 1.9.

######  Theorem 1.1 (Corollaries 1.5 and 1.6).

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a semi-stable
reduction prime111It means that $E$ has good _or_ multiplicative reduction at
$p$. for $E$ such that the mod $p$ representation
$\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{Aut}_{\mathbb{F}_{p}}(E[p])$
is surjective. If the Iwasawa main conjecture inverting $p$ holds222The
meaning of ‘inverting $p$’ here is ‘up to $\mu$-invariants’. Indeed, the
equality between the vanishing orders of both sides of Conjecture 1.3 at the
trivial character is enough as explained in Theorem 1.4. or
$\mathrm{ord}_{s=1}L(E,s)\leq 1$, then

  1. (1)

Kato’s Kolyvagin system is non-trivial, and

  2. (2)

the structure of Selmer group $\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ as a
co-finitely generated $\mathbb{Z}_{p}$-module is completely determined by the
collection of Kurihara numbers as described in Theorem 1.9.

If we assume the finiteness of $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ as
well, then the explicit formulas for the rank of $E(\mathbb{Q})$ and the exact
size of $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ are also provided in Theorem
1.9.

Conclusion (2) of Theorem 1.1 says that all the non-trivial (moreover,
linearly independent) elements of the Selmer group can be detected in terms of
modular symbols. In particular, it contains more information on Selmer groups
than Birch and Swinnerton-Dyer conjecture.

When $E$ has good ordinary reduction at $p$, the Iwasawa main conjecture
inverting $p$ is confirmed [Kat04, SU14, Wan15]; thus, all the conclusions of
Theorem 1.1 are valid. We refer to [Wan, CLW] for the recent development of
the Iwasawa main conjecture for elliptic curves with supersingular reduction.
It is believed that the Iwasawa main conjecture inverting $p$ does not help to
know the exact sizes of Selmer and Tate–Shafarevich groups due to the
$p$-power subtlety. However, our result shows that it is not true at all.
Furthermore, we extract the arithmetic information of elliptic curves of
_arbitrary_ rank from the Iwasawa main conjecture (inverting $p$) via Theorem
1.1.

When $\mathrm{ord}_{s=1}L(E,s)\leq 1$, the Iwasawa main conjecture is _not_
used to have Theorem 1.1. In particular, we obtain an explicit formula for the
exact size of $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ for semi-stable
elliptic curve $E$ of analytic rank $\leq 1$ and every prime $p\geq 5$ with
the surjectivity condition _but with no use of_ the Iwasawa main conjecture
(cf. [JSW17, Cas18]). When $E$ is semi-stable and $p\geq 11$,
$\overline{\rho}$ is always surjective [Maz78, Theorem 4]. We refer to
$\S$1.3.6 for details.

As far as we know, all the known results on the $p$-part of the Birch and
Swinnerton-Dyer formula depend heavily on the _full_ Iwasawa main conjecture
and only cover elliptic curves of analytic rank $\leq 1$. In this sense,
Theorem 1.1 provides us with a _better_ consequence on Selmer groups from a
_weaker_ input. We refer to $\S$1.9 for a more detailed comparison with
related conjectures and results.

We computed several numerical examples in $\S$8. In particular, the structural
information of the Selmer group in Example (5) cannot be observed from Birch
and Swinnerton-Dyer conjecture. Both the Iwasawa main conjecture and the low
analytic rank assumption in Theorem 1.1 are _not_ essential _at all_ when we
compute numerical examples in practice. We refer to $\S$1.5.3 for the
computational aspect.

#### 1.1.3.

In order to prove Theorem 1.1, we improve the following well-known
applications of the Euler system argument and Iwasawa theory for elliptic
curves.

  1. (1)

If the $L$-value does not vanish, then the Selmer group is finite and bounded
by the valuation of the $L$-value [Rub00, Kat99, PR98].

  2. (2)

If the Iwasawa main conjecture holds, then the $p$-part of the Birch and
Swinnerton-Dyer formula for elliptic curves of analytic rank $\leq 1$ holds.
We refer to $\S$1.9.1 for the list of relevant references on this result.

The improvements can be described as follows.

  1. (1’)

If the collection of Kurihara numbers does not vanish identically, then the
structure of the Selmer group is completely determined by the collection of
Kurihara numbers (Theorem 1.9).

  2. (2’)

If the Iwasawa main conjecture localized at the height one prime ideal
corresponding to the trivial character holds, then the collection of Kurihara
numbers does not vanish identically (Theorem 1.4 and Proposition 3.15).

As mentioned before, these improvements lead us to obtain a better result on
the study of Selmer groups with a weaker input. The following flowchart
explains how Theorem 1.1 follows from the improvements with the comparison
with well-known applications of the full Iwasawa main conjecture

Iwasawa main conjecture$\scriptstyle{{\begin{subarray}{c}\textrm{well-
known}\\\ \textrm{applications}\end{subarray}}}$listed in $\S$1.9.1
$\textstyle{{\begin{subarray}{c}\textrm{The $p$-part of the BSD formula}\\\
\textrm{when the analytic rank $\leq
1$}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{Iwasawa main
conjecture}\\\ \textrm{inverting
$p$}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\begin{subarray}{c}\textrm{elliptic
curves}\\\ \textrm{with good ordinary reduction at
$p$}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{
\cite[cite]{[\@@bibref{}{kato-euler-systems, skinner-urban, wan_hilbert}{}{}]}
}\\\ \textrm{(Corollary \ref{cor:main-
ks})}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{Iwasawa main
conjecture}\\\ \textrm{``localized at
$X\Lambda$"}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Theorem
1.4$\textstyle{{\begin{subarray}{c}\textrm{elliptic curves}\\\ \textrm{of
analytic rank $\leq
1$}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\begin{subarray}{c}\textrm{
\cite[cite]{[\@@bibref{}{bertolini-darmon-venerucci, burungale-skinner-tian,
kazim-pollack-sasaki}{}{}]} }\\\ \textrm{(Corollary \ref{cor:main-
ks})}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{The non-
triviality of}\\\ \textrm{Kato's Kolyvagin
systems}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Proposition
3.15$\scriptstyle{{\begin{subarray}{c}\textrm{ \cite[cite]{[\@@bibref{}{mazur-
rubin-book}{}{}]} }\\\ \textrm{(Theorem \ref{thm:kato-kolyvagin-
main})}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{The structure
of $p$-strict Selmer groups}\\\ \textrm{with no (analytic) rank
restriction}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{The non-
vanishing of}\\\ \textrm{the collection of Kurihara
numbers}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Theorem 1.9 $\textstyle{{\begin{subarray}{c}\textrm{The structure of Selmer
groups}\\\ \textrm{with no (analytic) rank restriction.}\end{subarray}}}$

The main idea of the first improvement (Theorem 1.9) is the extension of
Mazur–Rubin’s structure theorem of $p$-strict Selmer groups (in the above
diagram) to the case of classical Selmer groups. Although this idea looks
simple, the actual proof requires various technical input (studied in $\S$2
and $\S$3) and contains significant computations (given in $\S$5). The second
improvement (Theorem 1.4 and Proposition 3.15) can be viewed as a bridge from
Iwasawa theory to refined Iwasawa theory for elliptic curves following the
above diagram, and it is proved in $\S$4. Since the Iwasawa main conjecture
inverting $p$ for elliptic curves with good ordinary reduction is completely
resolved under the surjectivity assumption with $p\geq 5$ [Kat04, SU14,
Wan15], both Kato’s Kolyvagin system and the collection of Kurihara numbers
are non-trivial. Therefore, the cyclotomic analogue of the Kolyvagin
conjecture is confirmed for elliptic curves with good ordinary reduction, and
we are able to describe the structures of both $p$-strict (fine) Selmer groups
and classical Selmer groups.

We expect that our strategy generalizes to various settings. We refer to
[Kima] for the cases of Heegner point Kolyvagin systems and bipartite Euler
systems. In particular, the Kolyvagin conjecture is a trivial consequence of
the Heegner point main conjecture.

#### 1.1.4.

In the mod $p$ situation, we confirm the conjecture of Kurihara [Kur14b, Kur]
on the semi-local description of mod $p$ Selmer groups
$\mathrm{Sel}(\mathbb{Q},E[p])$ (Theorem 1.11). More precisely, when $p$ does
not divide any Tamagawa factor, we obtain the following implications

Iwasawa main conjecture$\textstyle{{\begin{subarray}{c}\textrm{The non-
vanishing of}\\\ \textrm{the collection of mod $p$ Kurihara
numbers}\end{subarray}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$the
conjecture of Kurihara$\textstyle{{\begin{subarray}{c}\textrm{The non-
triviality of}\\\ \textrm{mod $p$ Kato's Kolyvagin
systems}\end{subarray}}}$$\textstyle{{\begin{subarray}{c}\textrm{The semi-
local decription of}\\\ \textrm{mod $p$ Selmer groups
$\mathrm{Sel}(\mathbb{Q},E[p])$}\\\ \textrm{via one non-zero mod $p$ Kurihara
number. }\end{subarray}}}$

This diagram generalizes [KKS20, KN20, Sak] in various ways.

#### 1.1.5.

Regarding the applications to Birch and Swinnerton-Dyer conjecture, we obtain
the following “rank zero converse” result _without using_ any control theorem
or the interpolation formula of $p$-adic $L$-functions (cf. [SU14, Theorem 2],
[Wan15, Theorem 7]).

######  Theorem 1.2 (Corollary 1.12).

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then the following statements
are equivalent.

  1. (1)

Both $E(\mathbb{Q})$ and $\textrm{{SH}}(E/\mathbb{Q})$ are finite.

  2. (2)

$L(E,1)\neq 0$.

We also give a different proof of the $p$-parity conjecture (Corollary 1.13)
and a new computational upper bound of the ranks of elliptic curves (Corollary
1.14).

#### 1.1.6.

Last, we discuss how the Iwasawa module structure of the fine Selmer groups
over the cyclotomic tower can be understood via a version of the $p$-adic
Birch and Swinnerton-Dyer conjecture (Theorem 1.16). It is remarkable that the
$p$-adic BSD conjecture also contains the structural information of fine
Selmer groups over the Iwasawa algebra as Iwasawa modules.

The rest of this section is devoted to state the precise statements of all the
results mentioned above.

###  1.2. Kato’s Kolyvagin systems and the Iwasawa main conjecture

As a preliminary material, we quickly review the notion of Kato’s Kolyvagin
systems and the Iwasawa main conjecture for elliptic curves.

#### 1.2.1.

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$ and $p\geq 5$
a prime. Let $T$ be the $p$-adic Tate module of $E$ and denote by

$\mathbf{z}^{\mathrm{Kato}}=\left\\{z^{\mathrm{Kato}}_{F}\in\mathrm{H}^{1}(F,T)\right\\}_{F}$

Kato’s Euler system for $E$ where $F$ runs over finite abelian extensions of
$\mathbb{Q}$. See $\S$2.3.1 for the precise convention.

#### 1.2.2.

Let $k\geq 1$ be an integer. Let

$\mathcal{P}_{k}=\left\\{\ell,\textrm{ a prime}:(\ell,Np)=1,\ell\equiv
1\pmod{p^{k}},a_{\ell}(E)\equiv\ell+1\pmod{p^{k}}\right\\}$

and $\mathcal{N}_{k}$ the set of square-free products of primes in
$\mathcal{P}_{k}$. Write
$I_{\ell}=(\ell-1,a_{\ell}-\ell-1)\mathbb{Z}_{p}\subseteq\mathbb{Z}_{p}$ and
$I_{n}=\sum_{\ell|n}I_{\ell}\subseteq\mathbb{Z}_{p}$ where
$\ell\in\mathcal{P}_{k}$ and $n\in\mathcal{N}_{k}$.

#### 1.2.3.

Let $\mathbb{Q}_{\infty}$ be the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$, $\mathbb{Q}_{m}$ the cyclic subextention of $\mathbb{Q}$ of
degree $p^{m}$ in $\mathbb{Q}_{\infty}$, and
$\Lambda=\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\rrbracket$
the Iwasawa algebra. By using the $\Lambda$-adic version of the Euler-to-
Kolyvagin system map (recalled in Theorem 4.3), we obtain the $\Lambda$-adic
Kato’s Kolyvagin system

$\boldsymbol{\kappa}^{\mathrm{Kato},\infty}=\left\\{\kappa^{\mathrm{Kato},\infty}_{n}\in\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T\otimes\Lambda)\right\\}_{n\in\mathcal{N}_{1}}$

from $\mathbf{z}^{\mathrm{Kato}}$. In particular, we have

$\kappa^{\mathrm{Kato},\infty}_{1}=z^{\mathrm{Kato}}_{\mathbb{Q}_{\infty}}=\varprojlim_{m}z^{\mathrm{Kato}}_{\mathbb{Q}_{m}}$

where the projective limit is taken with respect to the corestriction map. Of
course, this element lies in the first Iwasawa cohomology group
$\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)=\varprojlim_{m}\mathrm{H}^{1}(\mathbb{Q}_{m},T)\simeq\mathrm{H}^{1}(\mathbb{Q},T\otimes\Lambda)$.
More details on Kolyvagin systems and $\Lambda$-adic Kolyvagin systems are
reviewed in $\S$2 and $\S$4.

#### 1.2.4.

Denote by $\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])$ the $p$-strict
(“fine”) Selmer group of $E[p^{\infty}]$ over $\mathbb{Q}_{\infty}$. Write
$(-)^{\vee}=\mathrm{Hom}(-,\mathbb{Q}_{p}/\mathbb{Z}_{p})$. We recall the
Iwasawa main conjecture without $p$-adic $L$-functions à la Kato [Kat04,
Conjecture 12.10].

######  Conjecture 1.3 (IMC).

The following equality as (principal) ideals of $\Lambda$ holds

(1.1)
$\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda\kappa^{\mathrm{Kato},\infty}_{1}}\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right).$

#### 1.2.5.

If $\overline{\rho}$ is surjective and $p\geq 5$, then all the required
hypotheses for the Kolyvagin system argument are satisfied [MR04, Lemma
6.2.3]. The surjectivity condition excludes the CM case. The $p\geq 5$
condition is required only for the Chebotarev density type argument in [MR04,
$\S$3.6 and Proposition 4.3.11], and it seems possible to extend to the $p=3$
case following the recent work of Sakamoto [Sak, Appendix].

Also, all the argument works when $\overline{\rho}$ is irreducible and there
exists a prime $\ell$ exactly dividing the conductor of $E$ such that
$\overline{\rho}$ is ramified at $\ell$ [Ski16, $\S$2.5]. The absolute
irreducibility and the irreducibility of $\overline{\rho}$ are equivalent for
the case of elliptic curves over $\mathbb{Q}$ [Rub97, Lemma 5].

###  1.3. The non-triviality of Kato’s Kolyvagin systems

#### 1.3.1.

Let $\mathfrak{P}$ be a height one prime ideal of $\Lambda$. We first recall
the following concepts from the theory of Kolyvagin systems [MR04].

  * •

$\mathfrak{P}$ is a blind spot of $\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$
if $\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$ vanishes modulo
$\mathfrak{P}$.

  * •

$\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$ is $\Lambda$-primitive if any
height one prime ideal of $\Lambda$ is not a blind spot of
$\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$.

These concepts play essential roles in the new and refined applications of
Kolyvagin systems to the structure of Selmer groups and the Iwasawa main
conjecture, which are not observed directly from the theory of Euler systems
[KKS20]. It is known that Kato’s Kolyvagin system
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ is non-trivial if and only if $X\Lambda$
is not a blind spot of $\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$ (recalled
in Proposition 4.11). Here, $\boldsymbol{\kappa}^{\mathrm{Kato}}$ is defined
by the image of $\mathbf{z}^{\mathrm{Kato}}$ under the Euler-to-Kolyvagin
system map (recalled in Theorem 2.1).

#### 1.3.2.

Fix an isomorphism $\Lambda\simeq\mathbb{Z}_{p}\llbracket X\rrbracket$ by
sending a topological generator of
$\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$ to $X+1$. Let $M$ be a finitely
generated torsion $\Lambda$-module. Write
$\mathrm{char}_{\Lambda}\left(M\right)=p^{\mu}\cdot\prod_{i}g_{i}(X)^{m_{i}}\cdot\Lambda$
where $g_{i}(X)$ is an irreducible distinguished polynomial and $g_{i}(X)\neq
g_{j}(X)$ if $i\neq j$. We define
$\mathrm{ord}_{\mathfrak{P}_{i}}\left(\mathrm{char}_{\Lambda}\left(M\right)\right)=m_{i}$
and
$\mathrm{ord}_{p\Lambda}\left(\mathrm{char}_{\Lambda}\left(M\right)\right)=\mu$
where $\mathfrak{P}_{i}$ is the height one prime ideal generated by
$g_{i}(X)$.

#### 1.3.3.

One of the technical innovation of this article is the following statement.

######  Theorem 1.4.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective. Let $\mathfrak{P}$ be a height one prime
ideal of $\Lambda$. The following statements are equivalent.

  1. (1)

$\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$ does not vanish modulo
$\mathfrak{P}$, i.e. $\mathfrak{P}$ is not a blind spot of
$\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$.

  2. (2)

The “Iwasawa main conjecture localized at $\mathfrak{P}$” holds; in other
words,

(1.2)
$\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda\kappa^{\mathrm{Kato},\infty}_{1}}\right)\right)=\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right)\right).$

In particular, $\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$ is
$\Lambda$-primitive if and only if the Iwasawa main conjecture (Conjecture
1.3) holds.

###### Proof.

See $\S$4.3. ∎

The one direction $(1)\Rightarrow(2)$ and $\Lambda$-primitivity $\Rightarrow$
IMC are established in the work of Mazur–Rubin [MR04, Theorem 5.3.10.(2) and
(3)] (recalled in Theorem 4.7). Our contribution is to prove the converse,
which has many interesting arithmetic consequences. In particular, we observe
that the non-triviality of Kato’s Kolyvagin system
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ is _strictly weaker_ than the Iwasawa
main conjecture.

More generally, if the Iwasawa main conjecture (inverting $p$) is valid, then
we are able to control the structures of the specializations of the $p$-strict
Selmer group of $E[p^{\infty}]$ over $\mathbb{Q}_{\infty}$ at all height one
primes except $p\Lambda$ via the $\Lambda$-adic Kato’s Kolyvagin system
$\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$.

Let us focus on the $\mathfrak{P}=X\Lambda$ case. The most striking aspect of
Theorem 1.4 is that we only need _a small piece of_ the full Iwasawa main
conjecture to determine the structure (and so the size) of both $p$-strict and
usual Selmer groups of elliptic curves (Corollaries 1.5 and 1.6). More
precisely, the Iwasawa main conjecture _localized at $X\Lambda$_ is strong
enough to have such applications.

Furthermore, this approach applies to elliptic curves of _arbitrary_ rank
while currently known applications of various Iwasawa main conjectures to BSD
conjecture are restricted to the case of analytic rank $\leq 1$.

One small disadvantage of our structural approach is that the exact bound of
the Tamagawa factors is missing. See Conjecture 1.10 for this aspect.

#### 1.3.4.

After having Theorem 1.4, we immediately obtain the following important
application, which confirms the cyclotomic analogue of the Kolyvagin
conjecture [Kol91, Zha14b], from the establishment of the Iwasawa main
conjecture (inverting $p$) for elliptic curves with good ordinary reduction
and Perrin-Riou’s conjecture on Kato’s zeta elements.

######  Corollary 1.5.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective. If the Iwasawa main conjecture localized at
$\mathfrak{P}=X\Lambda$ (1.2) holds, then Kato’s Kolyvagin system
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ is non-trivial. In particular, if one of
the following holds:

  1. (1)

$E$ has good ordinary reduction at $p$,

  2. (2)

$E$ has analytic rank zero, or

  3. (3)

$E$ has analytic rank one and $E$ has semi-stable reduction at $p$,

then $\boldsymbol{\kappa}^{\mathrm{Kato}}$ is non-trivial. In this case, the
structure of $\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$ is completely
determined by $\boldsymbol{\kappa}^{\mathrm{Kato}}$ as described in Theorem
2.14.

###### Proof.

Since the Iwasawa main conjecture _inverting $p$_ for elliptic curves with
good ordinary reduction at $p$ is established under our setting [Kat04, SU14,
Wan15], Theorem 1.4 immediately implies the non-triviality of
$\boldsymbol{\kappa}^{\mathrm{Kato}}$.

When the analytic rank is zero, the non-triviality of
$\kappa^{\mathrm{Kato}}_{1}=z^{\mathrm{Kato}}_{\mathbb{Q}}$ for _every_ prime
$p$ follows from Kato’s explicit reciprocity law [Kat04, Theorem 12.5].

When the analytic rank is one, the non-triviality of
$\kappa^{\mathrm{Kato}}_{1}=z^{\mathrm{Kato}}_{\mathbb{Q}}$ for _every semi-
stable_ reduction prime $p>2$ follows from the recent settlement of Perrin-
Riou’s conjecture [PR93, BDV22, BST, BPS].

The Kolyvagin system description of the structure of $p$-strict Selmer groups
follows from the work of Mazur–Rubin [MR04] (recalled in Theorem 2.14). ∎

#### 1.3.5.

With help of Theorem 1.9, we also obtain the following application to the
structure of Selmer groups, which is the structural refinement of Birch and
Swinnerton-Dyer conjecture.

######  Corollary 1.6.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that

  * •

$\overline{\rho}$ is surjective, and

  * •

the Manin constant is prime to $p$.

If the Iwasawa main conjecture localized at $\mathfrak{P}=X\Lambda$ (1.2)
holds, then the collection of Kurihara numbers, denoted by
$\widetilde{\boldsymbol{\delta}}$, does not vanish identically. In particular,
if one of the following holds:

  1. (1)

$E$ has good ordinary reduction at $p$,

  2. (2)

$E$ has analytic rank zero, or

  3. (3)

$E$ has analytic rank one and $E$ has semi-stable reduction at $p$,

then $\widetilde{\boldsymbol{\delta}}$ does not vanish identically. In this
case, the structure of $\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ is completely
determined by $\widetilde{\boldsymbol{\delta}}$ as described in Theorem 1.9.

###### Proof.

By Proposition 3.15, the non-triviality of
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ is equivalent to the non-vanishing of
$\widetilde{\boldsymbol{\delta}}$ under our running hypotheses. The modular
symbol description of the structure of Selmer groups follows from Theorem 1.9.
∎

It is conjectured that the Manin constant is 1. The Manin constant is not
divisible by a prime $p\geq 3$ if $E$ has semi-stable reduction at $p$ [Maz78,
Corollary 4.1]. Thus, the Manin constant assumption is needed only when $E$
has additive reduction at $p$.

#### 1.3.6.

It is remarkable that Corollary 1.6 is even stronger than the main results of
[JSW17, Cas18] on the $p$-part of the BSD formula for semi-stable elliptic
curves of analytic rank $\leq 1$ since the structure of Selmer groups is
completely determined. When $\mathrm{ord}_{s=1}L(E,s)=1$, we use Perrin-Riou’s
conjecture [PR93] recently settled in [BDV22, BST, BPS] instead of the Iwasawa
main conjecture. All the known results on the $p$-part of the BSD formula for
elliptic curves of analytic rank $\leq 1$ depend heavily on the full Iwasawa
main conjecture (as listed in $\S$1.9.1).

#### 1.3.7.

The following corollary is immediate.

######  Corollary 1.7.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$. If there exists a prime
$p\geq 5$ such that $\overline{\rho}$ is surjective, $E$ has good ordinary
reduction at $p$, and $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ is finite,
then there exists a deterministic algorithm to compute the exact rank of
$E(\mathbb{Q})$.

We refer to [Man71, $\S$11] and [SW13, Proposition 2.2] for Manin’s approach
to compute the ranks under the BSD conjecture.

###  1.4. Modular symbols and Kurihara numbers

In order to give a precise statement of Theorem 1.9, we quickly review the
notion of modular symbols and Kurihara numbers.

#### 1.4.1.

Let $p\geq 5$ be a prime and $E$ an elliptic curve over $\mathbb{Q}$ such that
the residual representation $\overline{\rho}$ is irreducible and the Manin
constant is prime to $p$.

Let $f=\sum_{n\geq 1}a_{n}(E)q^{n}\in S_{2}(\Gamma_{0}(N))$ be the newform
corresponding to $E$ [BCDT01]. For each $\dfrac{a}{b}\in\mathbb{Q}$, the
modular symbol $\left[\dfrac{a}{b}\right]^{+}$ is defined by equality

$2\pi\cdot\int^{\infty}_{0}f(\dfrac{a}{b}+iy)dy=\left[\dfrac{a}{b}\right]^{+}\cdot\Omega^{+}_{E}+\left[\dfrac{a}{b}\right]^{-}\cdot\sqrt{-1}\cdot\Omega^{-}_{E}$

where $\left[\dfrac{a}{b}\right]^{+}$ and $\left[\dfrac{a}{b}\right]^{-}$ are
rational numbers [MT87, (1.1)]. Especially, the real Néron period
$\Omega^{+}_{E}$ of $E$ is taken as the absolute value of the integral of an
invariant differential of a global minimal Weierstrass model of $E$ over
$E(\mathbb{R})$. (cf. [MT87, (1.1)].) Under our assumptions, we have
$\left[\dfrac{a}{b}\right]^{+}\in\mathbb{Z}_{(p)}$, i.e. the modular symbols
are $p$-integral.

#### 1.4.2.

We follow the convention in $\S$1.2.2. For each prime
$\ell\in\mathcal{P}_{k}$, we fix a primitive root $\eta_{\ell}$ mod $\ell$ and
define $\mathrm{log}_{\eta_{\ell}}(a)\in\mathbb{Z}/(\ell-1)$ by
$\eta^{\mathrm{log}_{\eta_{\ell}}(a)}_{\ell}\equiv a\pmod{\ell}$. Let
$\mathcal{N}_{k}$ be the set of square-free products of the primes in
$\mathcal{P}_{k}$. Note that $1\in\mathcal{N}_{k}$ for every $k$ by
convention. Obviously, $\mathcal{P}_{k+1}\subseteq\mathcal{P}_{k}$ and
$\mathcal{N}_{k+1}\subseteq\mathcal{N}_{k}$.

#### 1.4.3.

For each $n\in\mathcal{N}_{1}$, the (mod $I_{n}$) Kurihara number at $n$ is
defined333In [KKS20] and [Kimc], the mod $p$ Kurihara numbers are only
considered. by

$\widetilde{\delta}_{n}=\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\left(\overline{\left[\dfrac{a}{n}\right]^{+}}\cdot\prod_{\ell\mid
n}\overline{\mathrm{log}_{\eta_{\ell}}(a)}\right)\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$

where $\overline{\left[\dfrac{a}{n}\right]^{+}}$ is the mod $I_{n}$ reduction
of $\left[\dfrac{a}{n}\right]^{+}$ and
$\overline{\mathrm{log}_{\eta_{\ell}}(a)}$ is also the mod $I_{n}$ reduction
of $\mathrm{log}_{\eta_{\ell}}(a)$. Note that $\widetilde{\delta}_{n}$ is
well-defined up to $(\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p})^{\times}$. When
$n\in\mathcal{N}_{k}$, we write
$\widetilde{\delta}^{(k)}_{n}=\widetilde{\delta}_{n}\pmod{p^{k}}\in\mathbb{Z}/p^{k}\mathbb{Z}$.
When $n=1$, we have

$\widetilde{\delta}_{1}=[0]^{+}=\dfrac{L(E,1)}{\Omega^{+}_{E}}\in\mathbb{Z}_{(p)}.$

#### 1.4.4.

The collection of Kurihara numbers is defined by

$\widetilde{\boldsymbol{\delta}}=\left\\{\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}:n\in\mathcal{N}_{1}\right\\}.$

For a square-free integer $n$, denote by $\nu(n)$ the number of prime factors
of $n$ with convention $\nu(1)=0$. The vanishing order of
$\widetilde{\boldsymbol{\delta}}$ is defined by

$\mathrm{ord}(\widetilde{\boldsymbol{\delta}})=\mathrm{min}\left\\{\nu(n):n\in\mathcal{N}_{1},\widetilde{\delta}_{n}\neq
0\right\\}.$

We write $\mathrm{ord}(\widetilde{\boldsymbol{\delta}})=\infty$ if
$\widetilde{\delta}_{n}=0$ for all $n\in\mathcal{N}_{1}$, i.e. the collection
of Kurihara numbers vanishes identically. The following conjecture on the non-
vanishing of $\widetilde{\boldsymbol{\delta}}$ follows from the Iwasawa main
conjecture (Conjecture 1.3) thanks to Theorem 1.4 and Proposition 3.15.

######  Conjecture 1.8.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective and the Manin constant is prime to $p$. Then
$\mathrm{ord}(\widetilde{\boldsymbol{\delta}})<\infty$.

###  1.5. The structure of Selmer groups and Kurihara numbers

#### 1.5.1.

Denote by $\partial^{(0)}(\widetilde{\boldsymbol{\delta}})$ by the $p$-adic
valuation of $\widetilde{\delta}_{1}$, and write
$\widetilde{\delta}^{(k)}_{n}=\widetilde{\delta}_{n}\pmod{p^{k}}\in\mathbb{Z}/p^{k}\mathbb{Z}$
where $n\in\mathcal{N}_{k}$. For $i\in\mathbb{Z}_{>0}$, we also define

$\partial^{(i)}(\widetilde{\boldsymbol{\delta}}^{(k)})=\mathrm{min}\left\\{j:\widetilde{\delta}^{(k)}_{n}\in
p^{j}\mathbb{Z}/p^{k}\mathbb{Z}\textrm{ for every }n\in\mathcal{N}_{k}\textrm{
with }\nu(n)=i\right\\}$

and

$\displaystyle\partial^{(i)}(\widetilde{\boldsymbol{\delta}})$
$\displaystyle=\mathrm{min}\left\\{j:\widetilde{\delta}_{n}\in
p^{j}\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}\textrm{ for every
}n\in\mathcal{N}_{1}\textrm{ with }\nu(n)=i\right\\}$
$\displaystyle=\lim_{k\to\infty}\partial^{(i)}(\widetilde{\boldsymbol{\delta}}^{(k)}).$

Note that
$\lim_{k\to\infty}\partial^{(i)}(\widetilde{\boldsymbol{\delta}}^{(k)})$ is
well-defined under our running hypotheses. We also write

$\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})=\mathrm{min}\\{\partial^{(i)}(\widetilde{\boldsymbol{\delta}}):0\leq
i\\}.$

#### 1.5.2.

Denote by $M_{/\mathrm{div}}$ the quotient of $M$ by its maximal divisible
submodule.

######  Theorem 1.9.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective and the Manin constant is prime to $p$. If
$\mathrm{ord}(\widetilde{\boldsymbol{\delta}})<\infty$, then

  1. (1)

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])=\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$,
i.e.
$\mathrm{Fitt}_{i,\mathbb{Z}_{p}}\left(\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])^{\vee}\right)=0$
for all $0\leq i\leq\mathrm{ord}(\widetilde{\boldsymbol{\delta}})-1$,

  2. (2)

$\mathrm{Fitt}_{i,\mathbb{Z}_{p}}\left(\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])^{\vee}\right)=p^{\partial^{(i)}(\widetilde{\boldsymbol{\delta}})-\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})}\mathbb{Z}_{p}$
for all $i\geq\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$ with
$i\equiv\mathrm{ord}(\widetilde{\boldsymbol{\delta}})\pmod{2}$, and

  3. (3)

$\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])_{/\mathrm{div}}\right)=\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}))}(\widetilde{\boldsymbol{\delta}})-\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})$.

In other words, we have

$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\simeq\left(\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\oplus\mathrm{ord}(\widetilde{\boldsymbol{\delta}})}\oplus\bigoplus_{i\geq
1}\left(\mathbb{Z}/p^{(\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}})+2(i-1))}(\widetilde{\boldsymbol{\delta}})-\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}})+2i)}(\widetilde{\boldsymbol{\delta}}))/2}\mathbb{Z}\right)^{\oplus
2}.$

If we further assume the finiteness of
$\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$, then we have

  1. (4)

$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$,

  2. (5)

$\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]\simeq\bigoplus_{i\geq
1}\left(\mathbb{Z}/p^{(\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}})+2(i-1))}(\widetilde{\boldsymbol{\delta}})-\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}})+2i)}(\widetilde{\boldsymbol{\delta}}))/2}\mathbb{Z}\right)^{\oplus
2}$, and so

  3. (6)

$\mathrm{length}_{\mathbb{Z}_{p}}\left(\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]\right)=\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}))}(\widetilde{\boldsymbol{\delta}})-\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})$.

###### Proof.

See $\S$5. The first two statements imply all the other statements. ∎

When $\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})=0$, the ideal
$p^{\partial^{(i)}(\widetilde{\boldsymbol{\delta}})}\mathbb{Z}_{p}$ is closely
related to the $i$-th higher Stickelberger ideal $\Theta_{i}=\sum_{0\leq
i_{0}\leq
i}p^{\partial^{(i_{0})}(\widetilde{\boldsymbol{\delta}})}\mathbb{Z}_{p}$
[Kur14a, Kur14b]. The conclusion of Theorem 1.9 should be viewed as the
structural refinement of Birch and Swinnerton-Dyer conjecture, and it was
first observed by Kurihara from a completely different perspective under
various technical assumptions. See $\S$1.9.4 for details.

We are able to obtain an approximate structural upper bound of
$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ in practice since each $i$-th higher
Fitting ideal can be approximated by computing the valuations of
$\widetilde{\delta}_{n}$’s with fixed $\nu(n)=i$.

#### 1.5.3.

Considering the compatibility with the conjectural classical BSD formula, we
expect the following equality, which is a quantitative refinement of
Conjecture 1.8. See [Zha14a, Conjecture 4.5] for the Heegner point version.

######  Conjecture 1.10.

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$ and $p\geq 5$
a prime such that $\overline{\rho}$ is surjective and the Manin constant is
prime to $p$. Then

$\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})=\sum_{\ell\mid
N}\mathrm{ord}_{p}(c_{\ell})$

where $c_{\ell}$ is the Tamagawa factor of $E$ at $\ell$.

We expect that the only obstructions to observe the non-vanishing of
$\widetilde{\boldsymbol{\delta}}$ are the Selmer corank (Theorem 1.9),
Tamagawa factors (Conjecture 1.10), and the functional equation (Proposition
3.16). In this sense, the numerical verification of Conjecture 1.8 is not very
difficult. It is easy to compute each Kurihara number numerically, at least in
the mod $p$ situation [Kur14b, KKS20, Kim21, Kur]444An efficient algorithm to
compute $\widetilde{\delta}_{n}$ is available at
https://github.com/aghitza/kurihara_numbers thanks to Alexandru Ghitza..

###  1.6. The conjecture of Kurihara

#### 1.6.1.

In the mod $p$ situation, we confirm the conjecture of Kurihara [Kur14b,
Conjecture 2], which says that a single $n$ with
$\widetilde{\delta}^{(1)}_{n}\neq 0$ entirely determines the structure of mod
$p$ Selmer groups in terms of purely local data when all the Tamagawa factors
are prime to $p$. In this case, the non-vanishing of
$\widetilde{\boldsymbol{\delta}}^{(1)}$ is equivalent to the Iwasawa main
conjecture [Kat04, Conjecture 12.10].

######  Theorem 1.11.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that

  * •

$\overline{\rho}$ is surjective,

  * •

the Manin constant is prime to $p$,

  * •

$E(\mathbb{Q}_{p})[p]=0$, and

  * •

all the Tamagawa factors are prime to $p$.

Then the following statements are equivalent.

  1. (1)

$\widetilde{\delta}^{(1)}_{n}\neq 0$ in $\mathbb{F}_{p}$ for some
$n\in\mathcal{N}_{1}$ with
$\nu(n)=\mathrm{ord}(\widetilde{\boldsymbol{\delta}}^{(1)})$.

  2. (2)

The mod $p$ Kato’s Kolyvagin system $\boldsymbol{\kappa}^{\mathrm{Kato},(1)}$
is non-trivial.

  3. (3)

The Iwasawa main conjecture holds.

In this case, the canonical homomorphism

$\mathrm{Sel}(\mathbb{Q},E[p])\to\bigoplus_{\ell\mid
n}(E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}/p\mathbb{Z})\simeq\bigoplus_{\ell\mid
n}(E(\mathbb{F}_{\ell})\otimes\mathbb{Z}/p\mathbb{Z})$

is an isomorphism, and Conjecture 1.10 follows with value zero. If we further
assume that $\textrm{{SH}}(E/\mathbb{Q})[p]$ is trivial, then we have a mod
$p$ exact rank formula

$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=\mathrm{ord}(\widetilde{\boldsymbol{\delta}}^{(1)})=\nu(n).$

###### Proof.

See $\S$6. ∎

The implication (1) $\Rightarrow$ (2) $\Rightarrow$ (3) is the main result of
[KKS20]. The implication (3) $\Rightarrow$ (1) is proved by Kurihara [Kur14a]
(under the assumption on the non-degeneracy of $p$-adic height pairings) and
Sakamoto [Sak] when $E$ has good ordinary reduction at $p$ via Kolyvagin
systems of Gauss sum type and Kolyvagin systems of rank zero, respectively. In
particular, Theorem 1.11 recovers the main result of [Sak]. See also [Kur14b,
Kur]. The non-anomalous good reduction case is studied in [Kimc].

#### 1.6.2.

The reader can observe that our notation related to
$\widetilde{\boldsymbol{\delta}}$ is similar to that related to Kolyvagin
systems recalled in $\S$2 (see also [MR04, $\S$4.5 and $\S$5.2]). This seems
reasonable since we obtain $\widetilde{\boldsymbol{\delta}}$ directly from
Kato’s Kolyvagin system $\boldsymbol{\kappa}^{\mathrm{Kato}}$ via an
improvement of the dual exponential map discussed in $\S$3 (see also [KKS20]).
In this sense, $\widetilde{\boldsymbol{\delta}}$ can be understood as an
analytic counterpart of Kato’s Kolyvagin system
$\boldsymbol{\kappa}^{\mathrm{Kato}}$. This idea has played a fundamental role
in [KKS20] in the mod $p$ situation. However, it also turns out that the
behaviors of $\boldsymbol{\kappa}^{\mathrm{Kato}}$ and
$\widetilde{\boldsymbol{\delta}}$ are _fundamentally different_ (Proposition
2.2 versus Proposition 3.16). The discrepancy is closely related to that
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ controls fine Selmer groups but
$\widetilde{\boldsymbol{\delta}}$ controls classical Selmer groups.

###  1.7. Applications to Birch and Swinnerton-Dyer conjecture

We discuss some consequences of Theorem 1.9 towards Birch and Swinnerton-Dyer
conjecture.

#### 1.7.1.

If $\widetilde{\boldsymbol{\delta}}$ does not vanish, then the rank zero
$p$-converse to a theorem of Gross–Zagier–Kolyvagin follows.

######  Corollary 1.12.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime. We
assume that $\overline{\rho}$ is surjective, the Manin constant is prime to
$p$, and $\mathrm{ord}(\widetilde{\boldsymbol{\delta}})<\infty$. Then
$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ is finite if and only if $L(E,1)\neq
0$. In particular, Theorem 1.2 follows.

###### Proof.

It immediately follows from Theorem 1.9 except the ‘In particular’ part. For
CM elliptic curves, it it a result of Rubin [Rub91b, Theorem 11.1]. See also
[BT22, BT]. For non-CM elliptic curves, we are always possible to choose a
good ordinary prime $p$ such that $\overline{\rho}$ is surjective. By Theorem
1.4 and Proposition 3.15, the Iwasawa main conjecture inverting $p$ [Kat04,
SU14, Wan15] implies $\mathrm{ord}(\widetilde{\boldsymbol{\delta}})<\infty$.
By Theorem 1.9, $\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ is finite if and only
if $\widetilde{\delta}_{1}=\dfrac{L(E,1)}{\Omega^{+}_{E}}\neq 0$. It is known
that $L(E,1)\neq 0$ implies the finiteness of both $E(\mathbb{Q})$ and
$\textrm{{SH}}(E/\mathbb{Q})$ thanks to the work of Kato [Rub00, Theorem
3.5.4], [Kat04, Corollary 14.3]. Thus, Theorem 1.2 follows. ∎

The reader is invited to compare Corollary 1.12 with [SU14, Theorem 2],
[Wan15, Theorem 7], and [Ski20, Theorem E in $\S$4], which use the control
theorem and the interpolation formula of $p$-adic $L$-functions instead of
Theorem 1.9.

#### 1.7.2.

By using the functional equation for $\widetilde{\delta}_{n}$ discussed in
(3.6), we immediately obtain the $p$-parity conjecture from Theorem 1.9.

######  Corollary 1.13.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective and the Manin constant is prime to $p$. If
$\mathrm{ord}(\widetilde{\boldsymbol{\delta}})<\infty$, then we have

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\equiv\mathrm{ord}_{s=1}L(E,s)\pmod{2}.$

In particular, the $p$-parity conjecture holds for elliptic curves having good
ordinary reduction at $p$.

Although the $p$-parity conjecture is established in [Nek01, Kim07, Nek09,
DD10], our approach is completely independent of theirs.

#### 1.7.3.

We obtain the following “easy and practical” upper bound of the ranks of
elliptic curves.

######  Corollary 1.14.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective and the Manin constant is prime to $p$. If
$\widetilde{\delta}_{n}\neq 0$ for some $n\in\mathcal{N}_{1}$, then

$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})\leq\nu(n).$

###  1.8. Iwasawa modules and a $p$-adic Birch and Swinnerton-Dyer conjecture

#### 1.8.1.

Compared with the $p$-adic Birch and Swinnerton-Dyer conjecture for elliptic
curves [MTT86], we consider the following variant, namely the $p$-adic Birch
and Swinnerton-Dyer conjecture for Kato’s zeta elements.

######  Conjecture 1.15 ($p$-adic BSD for Kato’s zeta elements).

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])=\mathrm{ord}_{X\Lambda}\left(\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\kappa^{\mathrm{Kato},\infty}_{1}}\right)\right).$

This type of $p$-adic BSD conjecture can be found in [BKSb, Conjecture 4.16].
Indeed, it turns out that Conjecture 1.15 implies the usual $p$-adic BSD
conjecture [BKSb, Corollary 6.6].

#### 1.8.2.

The interesting feature we observed is that Conjecture 1.15 has an application
to the Iwasawa module structure of fine Selmer groups. In particular,
Conjecture 1.15 is slightly more refined than IMC localized at
$\mathfrak{P}=X\Lambda$ (1.2) from the viewpoint of Iwasawa modules. Fix a
pseudo-isomorphism

(1.3)
$\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\to\bigoplus_{i}\Lambda/X^{m_{i}}\Lambda\oplus\bigoplus_{j}\Lambda/f_{j}\Lambda$

where each $f_{j}$ is prime to $X\Lambda$.

######  Theorem 1.16.

Conjecture 1.15 implies $m_{i}=1$ for every $i$ in (1.3).

###### Proof.

See $\S$7. ∎

###  1.9. Comparison with related conjectures and results

####  1.9.1. The classical and $p$-adic BSD conjectures

######  Conjecture 1.17 (Birch–Swinnerton-Dyer).

Let $E$ be an elliptic curve over $\mathbb{Q}$.

  1. (1)

$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=\mathrm{ord}_{s=1}L(E,s)=r$.

  2. (2)

If the rank part holds and Tate–Shafarevich group
$\textrm{{SH}}(E/\mathbb{Q})$ is finite, then

(1.4)
$\dfrac{L^{(r)}(E,1)}{r!\cdot\Omega^{+}_{E}\cdot\mathrm{Reg}(E/\mathbb{Q})}=\dfrac{\\#\textrm{{SH}}(E/\mathbb{Q})}{(\\#E(\mathbb{Q})_{\mathrm{tor}})^{2}}\cdot\prod_{\ell\mid
N}c_{\ell}$

where $\Omega^{+}_{E}$ is the real Néron period and
$\mathrm{Reg}(E/\mathbb{Q})$ is the regulator of $E(\mathbb{Q})$.

The rank part is completely open when $\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})$
or $\mathrm{ord}_{s=1}L(E,s)$ is larger than 1. Thanks to the work of
Coates–Wiles [CW77], Rubin [Rub87], Gross–Zagier [GZ86], Kolyvagin [Kol90],
and Kato [Kat04] based on the method of Euler systems, it is known that if
$\mathrm{ord}_{s=1}L(E,s)\leq 1$, then the rank part is true and the
Tate–Shafarevich group is finite. Also, its $p$-converse is recently developed
enormously; see [SU14, Ski20, Zha14b, BT20, Kimb] for example. Furthermore,
when $\mathrm{ord}_{s=1}L(E,s)\leq 1$, a large amount of the $p$-part of the
BSD formula (1.4) is resolved through the establishment of various Iwasawa
main conjectures and complex and $p$-adic Gross–Zagier formulas thanks to the
work of Skinner–Urban [SU14], Kobayashi [Kob13], Zhang [Zha14b], Wan [Wan15,
Wan20], Berti–Bertolini–Venerucci [BBV16], Jetchev–Skinner–Wan [JSW17],
Castella [Cas18], Castella–Grossi–Lee–Skinner [CGLS22], and
Büyükboduk–Pollack–Sasaki [BPS].

######  Conjecture 1.18 (Mazur–Tate–Teitelbaum).

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at
$p$. Let $L_{p}(E)$ be the $p$-adic $L$-function of $E$. We regard $L_{p}(E)$
as an element of $\mathbb{Z}_{p}\llbracket X\rrbracket$ by choosing a
topological generator $\gamma$ of the Galois group of the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$. Then the following statements
hold.

  1. (1)

$\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})=\mathrm{ord}_{X=0}L_{p}(E)=r$.

  2. (2)

If the first part holds and $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$ is
finite, then

$L^{(r)}_{p}(E)=\left(\dfrac{1}{1-\alpha}\right)^{2}\cdot\dfrac{\\#\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]}{(\\#E(\mathbb{Q})_{\mathrm{tor}})^{2}}\cdot\prod_{\ell\mid
N}c_{\ell}\cdot\mathrm{Reg}_{\gamma}(E/\mathbb{Q})$

where $\alpha$ is the unit root of $x^{2}-a_{p}(E)\cdot x+p$ and
$\mathrm{Reg}_{\gamma}(E/\mathbb{Q})$ is a certain $p$-adic regulator with
normalization to nullify the choice of $\gamma$ (see [SW13, $\S$4.4] for
details).

Sine we only consider the good ordinary reduction case here, we do not see the
exceptional zero in the above statement. There is also a version for the
supersingular reduction case due to Bernardi–Perrin-Riou [BPR93], but we omit
it here. We refer to [MTT86, SW13, BMS16] for details.

Regarding the rank part, Kato proved
$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\leq\mathrm{ord}_{X=0}L_{p}(E)$
based on the one-sided divisibility of the Iwasawa main conjecture [Rub91b,
Kat04]. In order to prove the equality of the rank part, we need the Iwasawa
main conjecture, the non-degeneracy of the $p$-adic height pairing on $E$, and
the finiteness of $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$. The leading term
part also follows from the combination of these conjectures. See [Sch85,
Theorem 2’], [PR93, Proposition 3.4.6], and [BMS16, Theorem 1.7].

Compared with progresses towards both conjectures, our structural refinement
(Theorem 1.9) works for elliptic curves of arbitrary rank and depends only on
Conjecture 1.8, which is strictly weaker than the Iwasawa main conjecture.
Also, no regulator term is involved in our results. Furthermore, we may ask
the following naïve questions.

######  Question 1.19.

Let $r=\mathrm{rk}_{\mathbb{Z}}E(\mathbb{Q})$ and assume that
$\textrm{{SH}}(E/\mathbb{Q})$ is finite. Denote by $s$ the number of
generators of $\textrm{{SH}}(E/\mathbb{Q})$, and by $s_{p}$ the number of
generators of $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$.

  1. (1)

Can we determine the structure of $\textrm{{SH}}(E/\mathbb{Q})$ from the
values $L^{(r)}(E,1),\cdots,L^{(r+s)}(E,1)$?

  2. (2)

Can we determine the structure of $\textrm{{SH}}(E/\mathbb{Q})[p^{\infty}]$
from the values $L^{(r)}_{p}(E),\cdots,L^{(r+s_{p})}_{p}(E)$?

####  1.9.2. Other (classical) Kolyvagin systems and the structure of Selmer
groups

In [Kol91], Kolyvagin illustrated how one can obtain the structure of Selmer
group $\mathrm{Sel}(K,E[p^{\infty}])$ where $K$ is an imaginary quadratic
field satisfying the Heegner hypothesis from a _non-trivial_ Heegner point
Kolyvagin system. See also [Zha14b].

For the structural results on the classical Iwasawa theory based on cyclotomic
unit Kolyvagin systems and Kolyvagin systems of Gauss sums, we refer to
[Kol90, Rub91a, Kur03b, Aok05].

####  1.9.3. Refined BSD type conjectures

In [MT87], Mazur and Tate proposed several conjectures refining the BSD
conjectures equivariantly. Since the initial Fitting ideals are used in the
formulation of their weak main conjecture, it would be regarded as the first
approach towards the structure of Selmer groups. Theorem 1.9 refines their
weak vanishing conjecture [MT87, Conjecture 1] by making their inequality into
the equality when the character is trivial and also refines their weak main
conjecture [MT87, Conjecture 3] and a “Birch–Swinnerton-Dyer type” conjecture
[MT87, Conjecture 4] by considering all the higher Fitting ideals but with no
equivariant variation. We refer to [Ota18, KK21, Kat21, Kat22, BSS, BMC22,
BKSb, BKSa] for the recent developments.

####  1.9.4. Refined Iwasawa theory

In a series of his papers [Kur03a, Kur03b, Kur12, Kur14b, Kur14a], Kurihara
developed refined Iwasawa theory and the theory of Kolyvagin systems of Gauss
sum type to study the structure of Selmer groups. It is remarkable that
Iwasawa himself was also interested in Kurihara’s refinement of the Iwasawa
main conjecture [Kat17].

In [Kur14a, Theorem B], Kurihara obtained Theorem 1.9 via the theory of
Kolyvagin systems of Gauss sum type under the following assumptions:

  * •

$E$ has non-anomalous good ordinary reduction at $p$,

  * •

$p$ does not divide all the Tamagawa factors of $E$,

  * •

$\overline{\rho}$ is surjective,

  * •

the $\mu$-invariant of of the $p$-adic $L$-function of $E$ is zero,

  * •

the Iwasawa main conjecture for $E$ holds, and

  * •

the $p$-adic height pairing is non-degenerate.

His argument is completely different from ours, and this difference can be
compared with the difference between cyclotomic unit Kolyvagin systems and
Kolyvagin systems of Gauss sums in classical Iwasawa theory. In [Kur14a],
Kurihara explicitly constructed a certain skew-Hermitian relation matrix to
present the structure of Selmer groups and the matrix is essentially
equivalent to the organizing matrix constructed by Mazur–Rubin [MR05]. In our
approach, we completely bypass the construction of such matrices, and our work
does not involve any $p$-adic height pairing. Recently, Kurihara also obtained
the same result for the supersingular reduction case with $a_{p}(E)=0$ under
similar assumptions [Kur]. In this sense, Theorem 1.9 generalizes Kurihara’s
results by removing all the serious Iwasawa-theoretic assumptions.

##  2\. Kolyvagin systems for elliptic curves

We summarize some materials in [MR04] with an emphasis on the structure of
fine Selmer groups of elliptic curves.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective. Let $T$ be the $p$-adic Tate module of $E$.

###  2.1. Selmer structures and Selmer groups

#### 2.1.1.

Following [MR04, $\S$1], we recall the standard local conditions for Selmer
groups of $T/p^{k}T$ (with $k\geq 1$) and review their properties. Since $p$
is odd, we ignore the infinite place.

Let $K$ be a local field. We recall some examples of local conditions, which
are submodules of $\mathrm{H}^{1}(K,T/p^{k}T)$.

  * •

the finite condition:
$\mathrm{H}^{1}_{f}(K,T/p^{k}T)=E(K)\otimes\mathbb{Z}/p^{k}\mathbb{Z}$ via the
Kummer map.

  * •

the relaxed condition:
$\mathrm{H}^{1}_{\mathrm{rel}}(K,T/p^{k}T)=\mathrm{H}^{1}(K,T/p^{k}T)$.

  * •

the strict condition: $\mathrm{H}^{1}_{\mathrm{str}}(K,T/p^{k}T)=0$.

  * •

the unramified condition:
$\mathrm{H}^{1}_{\mathrm{ur}}(K,T/p^{k}T)=\mathrm{H}^{1}(K^{\mathrm{ur}}/K,\mathrm{H}^{0}(K^{\mathrm{ur}},T/p^{k}T))$.

  * •

the transverse condition (when $K=\mathbb{Q}_{\ell}$ and $\ell\equiv
1\pmod{p^{k}}$):

$\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},T/p^{k}T)=\mathrm{H}^{1}(\mathbb{Q}_{\ell}(\zeta_{\ell})/\mathbb{Q}_{\ell},\mathrm{H}^{0}(\mathbb{Q}_{\ell}(\zeta_{\ell}),T/p^{k}T)).$

The first four local conditions are also defined on $T$ and $E[p^{\infty}]$.
We write
$\mathrm{H}^{1}_{/f}(K,T/p^{k}T)=\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{\ell},T/p^{k}T)}{\mathrm{H}^{1}_{f}(K,T/p^{k}T)}$.
If $T/p^{k}T$ is unramified as a representation of $G_{K}$, then we have
$\mathrm{H}^{1}_{f}(K,T/p^{k}T)=\mathrm{H}^{1}_{\mathrm{ur}}(K,T/p^{k}T)$.

#### 2.1.2.

The Selmer structure $\mathcal{F}$ on $T/p^{k}T$ consists of

  * •

a finite set $\Sigma$ of places of $\mathbb{Q}$ containing the primes where
$E$ has bad reduction, $p$, and the infinite place, and

  * •

the choices of local conditions
$\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{\ell},T/p^{k}T)$ at primes in
$\Sigma$.

For a prime $q\not\in\Sigma$, we fix
$\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{q},T/p^{k}T)=\mathrm{H}^{1}_{f}(\mathbb{Q}_{q},T/p^{k}T)$.

If $\ell\in\mathcal{N}_{k}$, we identify
$\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},T/p^{k}T)=\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},T/p^{k}T)$
[MR04, Lemma 1.2.4]. For $\ell\in\mathcal{N}_{k}$,
$Q(X)=\frac{\mathrm{det}(1-\mathrm{Fr}_{\ell}\cdot
X|T/p^{k}T)}{X-1}\in\mathbb{Z}/p^{k}\mathbb{Z}[X]$ is well-defined where
$\mathrm{Fr}_{\ell}$ is the arithmetic Frobenius at $\ell$, and we have
isomorphism
$\phi^{\mathrm{fs}}_{\ell}:\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},T/p^{k}T)\simeq\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},T/p^{k}T)$
defined by the composition of isomorphisms

$\textstyle{\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},T/p^{k}T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{(T/p^{k}T)/(\mathrm{Fr}_{\ell}-1)(T/p^{k}T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q(\mathrm{Fr}^{-1}_{\ell})}$$\scriptstyle{\simeq}$$\textstyle{(T/p^{k}T)^{\mathrm{Fr}_{\ell}=1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},T/p^{k}T).}$

See [MR04, $\S$1.2] for details.

#### 2.1.3.

For a given Selmer structure $\mathcal{F}$ on $T/p^{k}T$ and
$n\in\mathcal{N}_{k}$, the Selmer structure $\mathcal{F}(n)$ is defined by

  * •

$\mathrm{H}^{1}_{\mathcal{F}(n)}(\mathbb{Q}_{\ell},T/p^{k}T)=\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{\ell},T/p^{k}T)$
for $\ell$ not dividing $n$, and

  * •

$\mathrm{H}^{1}_{\mathcal{F}(n)}(\mathbb{Q}_{\ell},T/p^{k}T)=\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},T/p^{k}T)$
for $\ell$ dividing $n$.

#### 2.1.4.

The Selmer group $\mathrm{Sel}_{\mathcal{F}}(\mathbb{Q},T/p^{k}T)$ of
$T/p^{k}T$ with respect to $\mathcal{F}$ is defined by the exact sequence

$0\to\mathrm{Sel}_{\mathcal{F}}(\mathbb{Q},T/p^{k}T)\to\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},T/p^{k}T)\to\bigoplus_{\ell\in\Sigma}\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{\ell},T/p^{k}T)}{\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{\ell},T/p^{k}T)}$

where $\mathbb{Q}_{\Sigma}$ is the maximal extension of $\mathbb{Q}$
unramified outside $\Sigma$. This definition is independent of the choice of
$\Sigma$.

#### 2.1.5.

Let $(-)^{*}=\mathrm{Hom}(-,\mu_{p^{\infty}})$ be the Cartier dual and then we
have $(T/p^{k}T)^{*}\simeq E[p^{k}]\simeq T/p^{k}T$ via the Weil pairing. The
corresponding dual Selmer structure $\mathcal{F}^{*}$ on $(T/p^{k}T)^{*}$ is
defined by the choices of local conditions
$\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q}_{\ell},(T/p^{k}T)^{*})=\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{\ell},T/p^{k}T)^{\perp}$
under the local Tate pairing with the same $\Sigma$. The Selmer group
$\mathrm{Sel}_{\mathcal{F}^{*}}(\mathbb{Q},E[p^{k}])$ with respect to
$\mathcal{F}^{*}$ is defined in a similar way.

#### 2.1.6.

We recall two natural Selmer structures on $T/p^{k}T$.

The classical Selmer structure $\mathcal{F}_{\mathrm{cl}}$ is defined by
$\mathrm{H}^{1}_{\mathcal{F}_{\mathrm{cl}}}(\mathbb{Q}_{\ell},T/p^{k}T)=\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},T/p^{k}T)$,
the image of of $E(\mathbb{Q}_{\ell})/p^{k}E(\mathbb{Q}_{\ell})$ under the
Kummer map for every prime $\ell$ so that we have

$\textstyle{\mathrm{Sel}(\mathbb{Q},T/p^{k}T)=\mathrm{Sel}_{\mathcal{F}_{\mathrm{cl}}}(\mathbb{Q},T/p^{k}T),}$$\textstyle{\mathrm{Sel}(\mathbb{Q},E[p^{k}])=\mathrm{Sel}_{\mathcal{F}^{*}_{\mathrm{cl}}}(\mathbb{Q},E[p^{k}]).}$

Thus, the classical Selmer structure recovers classical Selmer groups.

The canonical Selmer structure $\mathcal{F}_{\mathrm{can}}$ is defined by
$\mathrm{H}^{1}_{\mathcal{F}_{\mathrm{can}}}(\mathbb{Q}_{\ell},T/p^{k}T)=\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},T/p^{k}T)$
for every prime $\ell\neq p$ and
$\mathrm{H}^{1}_{\mathcal{F}_{\mathrm{can}}}(\mathbb{Q}_{p},T/p^{k}T)=\mathrm{H}^{1}(\mathbb{Q}_{p},T/p^{k}T)$.
We write

$\textstyle{\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},T/p^{k}T)=\mathrm{Sel}_{\mathcal{F}_{\mathrm{can}}}(\mathbb{Q},T/p^{k}T),}$$\textstyle{\mathrm{Sel}_{0}(\mathbb{Q},E[p^{k}])=\mathrm{Sel}_{\mathcal{F}^{*}_{\mathrm{can}}}(\mathbb{Q},E[p^{k}]).}$

In other words, the canonical Selmer structure defines the $p$-relaxed Selmer
group and the dual canonical Selmer structure defines the $p$-strict Selmer
group.

#### 2.1.7.

Let $n\in\mathcal{N}_{k}$ and $\Sigma_{n}=\Sigma\cup\\{\ell:\ell\mid n\\}$.
The $n$-transverse $p$-relaxed Selmer group, the $n$-transverse Selmer group,
and the $n$-transverse fine Selmer group of $T/I_{n}T$ are defined by

$\displaystyle\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)$
$\displaystyle=\mathrm{ker}\left(\mathrm{H}^{1}(\mathbb{Q}_{\Sigma_{n}}/\mathbb{Q},T/I_{n}T)\to\bigoplus_{\ell\in\Sigma\setminus\\{p\\}}\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{\ell},T/I_{n}T)}{\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},T/I_{n}T)}\oplus\bigoplus_{\ell\mid
n}\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{\ell},T/I_{n}T)}{\mathrm{H}^{1}_{\mathrm{tr}}(\mathbb{Q}_{\ell},T/I_{n}T)}\right),$
$\displaystyle\mathrm{Sel}_{n}(\mathbb{Q},T/I_{n}T)$
$\displaystyle=\mathrm{ker}\left(\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)\to\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)}{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)}\right),$
$\displaystyle\mathrm{Sel}_{0,n}(\mathbb{Q},T/I_{n}T)$
$\displaystyle=\mathrm{ker}\left(\mathrm{Sel}_{n}(\mathbb{Q},T/I_{n}T)\to\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)\right),$

respectively. We focus on $n=1$. Then all the above Selmer groups can be
defined for $T/p^{k}T$ for every $k\geq 1$. The compact Selmer groups
$\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},T)$, $\mathrm{Sel}(\mathbb{Q},T)$, and
$\mathrm{Sel}_{0}(\mathbb{Q},T)$ are defined by taking the projective limit
$\varprojlim_{k}$, respectively. By using $T/p^{k}T\simeq E[p^{k}]$ as Galois
modules, the discrete Selmer groups
$\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},E[p^{\infty}])$,
$\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$, and
$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$ are defined by taking the direct
limit $\varinjlim_{k}$, respectively.

###  2.2. Euler and Kolyvagin systems

#### 2.2.1.

We recall the notion of Euler systems following [MR04, Definition 3.2.2]. Let
$\mathcal{P}=\mathcal{P}_{1}$ be the set of primes defined in $\S$1.2.2 and
$\mathcal{K}$ a possibly infinite abelian extension of $\mathbb{Q}$. An Euler
system $\mathbf{z}$ for $(T,\mathcal{K},\mathcal{P})$ is a collection of
cohomology classes

$\mathbf{z}=\left\\{z_{F}\in\mathrm{H}^{1}(F,T):F/\mathbb{Q}\textrm{
finite},F\subseteq\mathcal{K}\right\\}$

such that whenever $F^{\prime}/F$ are finite subextenions of $\mathbb{Q}$ in
$\mathcal{K}$, satisfying the norm relation

$\mathrm{Nm}_{F^{\prime}/F}(z_{F^{\prime}})=\left(\prod_{\ell}P_{\ell}(\mathrm{Fr}^{-1}_{\ell})\right)\cdot
z_{F}$

where $\mathrm{Nm}_{F^{\prime}/F}$ is the corestriction map from $F^{\prime}$
to $F$, the product runs over primes in $\mathcal{P}$ which are ramified in
$F^{\prime}/\mathbb{Q}$ but not in $F/\mathbb{Q}$,
$P_{\ell}(X)=\mathrm{det}(1-\mathrm{Fr}_{\ell}\cdot X|T)$, and
$\mathrm{Fr}_{\ell}$ is the arithmetic Froenius at $\ell$. Let
$\mathbf{ES}(T)=\mathbf{ES}(T,\mathcal{K},\mathcal{P})$ denote the module of
Euler systems over
$\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rrbracket$.

#### 2.2.2.

A Kolyvagin system $\boldsymbol{\kappa}$ for $(T,\mathcal{F},\mathcal{P})$ is
a collection of cohomology classes

$\boldsymbol{\kappa}=\left\\{\kappa_{n}\in\mathrm{Sel}_{\mathcal{F}(n)}(\mathbb{Q},T/I_{n}T):n\in\mathcal{N}_{1}\right\\}$

such that
$\mathrm{loc}^{s}_{\ell}(\kappa_{n\ell})=\phi^{\mathrm{fs}}_{\ell}\circ\mathrm{loc}_{\ell}(\kappa_{n})\in\mathrm{H}^{1}_{/f}(\mathbb{Q}_{\ell},T/I_{n\ell}T)$
for $\ell\in\mathcal{N}_{1}$ with $(n,\ell)=1$ where
$\mathrm{loc}_{\ell}:\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)\to\mathrm{H}^{1}(\mathbb{Q}_{\ell},T/I_{n}T)$,
and
$\mathrm{loc}^{s}_{\ell}:\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)\to\mathrm{H}^{1}_{/f}(\mathbb{Q},T/I_{n}T)$.
Let $\mathbf{KS}(T)=\mathbf{KS}(T,\mathcal{F},\mathcal{P})$ denote the module
of Kolyvagin systems and
$\overline{\mathbf{KS}}(T)=\overline{\mathbf{KS}}(T,\mathcal{F},\mathcal{P})$
denote the generalized module of Kolyvagin systems defined by the completion.
See [MR04, $\S$3.1] for details.

Our convention of Kolyvagin systems depends on the choice of generators of
$\mathrm{Gal}(\mathbb{Q}(\zeta_{\ell})/\mathbb{Q})$ for each prime $\ell$
dividing $n$, and it corresponds to the choice of the primitive roots in the
definition of Kurihara numbers.

The natural map $\mathbf{KS}(T)\to\overline{\mathbf{KS}}(T)$ is an isomorphism
when the core rank is one ($\S$2.4.2 and Theorem 2.4). See [MR04, Corollary
4.5.3, Proposition 5.2.9, and Proposition 6.2.2] for details.

#### 2.2.3.

Under our working hypotheses, all the conditions in Theorem 2.1 below are
satisfied.

######  Theorem 2.1 (Mazur–Rubin).

Suppose that $\mathcal{K}$ contains the maximal abelian $p$-extension of
$\mathbb{Q}$ which is unramified outside $p$ and $\mathcal{P}$, and

  1. (1)

$T/(\mathrm{Fr}_{\ell}-1)T$ is a cyclic $\mathbb{Z}_{p}$-module for every
$\ell\in\mathcal{P}$,

  2. (2)

$\mathrm{Fr}^{p^{k}}_{\ell}-1$ is injective on $T$ for every
$\ell\in\mathcal{P}$ and every $k\geq 0$.

Then there exists a canonical homomorphism

$\textstyle{\mathbf{ES}(T,\mathcal{K},\mathcal{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathbf{KS}}(T,\mathcal{F}_{\mathrm{can}},\mathcal{P})}$$\textstyle{\mathbf{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\boldsymbol{\kappa}}$

such that $\kappa_{1}=z_{\mathbb{Q}}$. If we further assume that
$\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])$ is a divisible
$\mathbb{Z}_{p}$-module, then
$\overline{\mathbf{KS}}(T,\mathcal{F}_{\mathrm{can}},\mathcal{P})$ can be
replaced by $\mathbf{KS}(T,\mathcal{F}_{\mathrm{can}},\mathcal{P})$.

###### Proof.

See [MR04, Theorem 3.2.4 and $\S$6.2]. ∎

###  2.3. Kato’s Euler systems and Kato’s Kolyvagin systems

#### 2.3.1.

Let $\mathbf{z}^{\mathrm{Kato}}=(z^{\mathrm{Kato}}_{F})_{F}$ be Kato’s Euler
system for $E$ associated to the real Néron period $\Omega^{+}_{E}$ where
$z^{\mathrm{Kato}}_{F}\in\mathrm{H}^{1}(F,T)$ and $F$ runs over abelian
extensions of $\mathbb{Q}$ following the convention of [Kat21, Theorem 6.1].
More explicitly, each $z^{\mathrm{Kato}}_{F}\in\mathrm{H}^{1}(F,T)$ is
characterized by the interpolation formula

$\sum_{\sigma\in\mathrm{Gal}(F/\mathbb{Q})}\sigma\left(\mathrm{exp}^{*}\circ\mathrm{loc}^{s}_{p}(z^{\mathrm{Kato}}_{F})\right)\cdot\chi(\sigma)=\dfrac{L_{(Sp)}(E,\chi,1)}{\Omega^{+}_{E}}\cdot\omega_{E}$

where $\chi$ is any even character of $\mathrm{Gal}(F/\mathbb{Q})$, $S$ is the
product of the ramified primes of $F/\mathbb{Q}$, $L_{(Sp)}(E,\chi,1)$ is the
$Sp$-imprimitive $\chi$-twisted $L$-value of $E$ at $s=1$, and $\omega_{E}$ is
the Néron differential. For odd characters of $\mathrm{Gal}(F/\mathbb{Q})$,
the interpolation formula is zero.

#### 2.3.2.

Kato’s Kolyvagin system $\boldsymbol{\kappa}^{\mathrm{Kato}}$ is defined by
the image of $\mathbf{z}^{\mathrm{Kato}}$ under the map in Theorem 2.1. The
explicit construction of $\boldsymbol{\kappa}^{\mathrm{Kato}}$ is as follows.

For each $n\in\mathcal{N}_{1}$, the Kolyvagin derivative operator at $n$ is
defined by
$D_{\mathbb{Q}(\zeta_{n})}=\prod_{\ell|n}\sum_{i=1}^{\ell-2}i\cdot\sigma^{i}_{\eta_{\ell}}$
where $\eta_{\ell}$ is a primitive root mod $\ell\in\mathcal{P}_{1}$. Then
$\kappa^{\mathrm{Kato}}_{n}$ is defined by the image of
$D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})}\in\mathrm{H}^{1}(\mathbb{Q}(\zeta_{n}),T)$
in $\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)$ under the composition of the
following maps

$\textstyle{\mathrm{H}^{1}(\mathbb{Q}(\zeta_{n}),T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bmod{I_{n}}}$$\textstyle{\left(\mathrm{H}^{1}(\mathbb{Q}(\zeta_{n}),T)/I_{n}\right)^{G_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}(\zeta_{n}),T/I_{n}T)^{G_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{res}^{-1}}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)}$

where $G_{n}=\mathrm{Gal}(\mathbb{Q}(\zeta_{n})/\mathbb{Q})$ and
$\mathrm{res}^{-1}$ is the inverse of the restriction map [Rub00, $\S$4.4].
Note that we do not need any modification of the image of
$D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})}$ to obtain
$\kappa^{\mathrm{Kato}}_{n}$ (cf. [MR04, (33) in Page 80]).

###  2.4. Properties of Kolyvagin systems

#### 2.4.1.

Recall that $\mathcal{P}_{k}$ is the set of primes $\ell$ such that
$(\ell,Np)=1$, $\ell\equiv 1\pmod{p^{k}}$, and
$a_{\ell}(E)\equiv\ell+1\pmod{p^{k}}$, which we call the set of Kolyvagin
primes for $T/p^{k}T$. Also, $\mathcal{N}_{k}$ is the set of square-free
products of primes in $\mathcal{P}_{k}$.

Following [MR04, $\S$3.6 and Theorem 4.4.1], a Kolyvagin prime
$\ell\in\mathcal{P}_{k}$ is said to be useful for (non-zero) $\kappa_{n}$ with
$n\in\mathcal{N}_{k}$ if

  * •

$(\ell,n)=1$,

  * •

$\mathrm{loc}_{\ell}(\kappa_{n})\neq 0$,

  * •

$\mathrm{loc}_{\ell}:\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)[p]\to\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},T/I_{n}T)$
is non-zero, and

  * •

$\mathrm{loc}_{\ell}:\mathrm{Sel}_{0,n}(\mathbb{Q},E[I_{n}])[p]\to\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},E[I_{n}])$
is non-zero.

######  Proposition 2.2.

If $\kappa_{n}\neq 0$ and $\ell$ is a useful prime for $\kappa_{n}$, then
$\kappa_{n\ell}\neq 0$.

###### Proof.

This immediately follows from the axiom of Kolyvagin systems in $\S$2.2.2. ∎

The following proposition shows that there are infinitely many useful primes
for a given non-zero $\kappa_{n}$.

######  Proposition 2.3.

Let $c_{1},c_{2}\in\mathrm{H}^{1}(\mathbb{Q},T/p^{k}T)$ and
$c_{3},c_{4}\in\mathrm{H}^{1}(\mathbb{Q},E[p^{k}])$ be non-zero elements. For
every $k\in\mathbb{Z}_{>0}$, there exists a set $S\subseteq\mathcal{P}_{k}$ of
positive density such that for every $\ell\in S$, the localizations
$\mathrm{loc}_{\ell}(c_{i})$ are all non-zero.

###### Proof.

See [MR04, Proposition 3.6.1]. ∎

#### 2.4.2.

Following [MR04, $\S$4], we write

$\displaystyle\lambda(n,E[I_{n}])$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}}\mathrm{Sel}_{0,n}(\mathbb{Q},E[I_{n}]),$
$\displaystyle\lambda(n,T/I_{n}T)$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}}\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)$

where $n\in\mathcal{N}_{1}$. We say $n\in\mathcal{N}_{1}$ is a core vertex if
$\lambda(n,E[I_{n}])$ or $\lambda(n,T/I_{n}T)$ is zero [MR04, Definition
4.1.8] and the core rank $\chi(T)$ (with the $p$-relaxed Selmer structure) is
defined by
$\mathrm{cork}_{\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}}\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)$
for any core vertex $n$ [MR04, Definition 4.1.11 and Definition 5.2.4].

#### 2.4.3.

The following theorem plays an important role to have the equivalence between
the non-triviality of $\boldsymbol{\kappa}^{\mathrm{Kato}}$ and the non-
vanishing of $\widetilde{\boldsymbol{\delta}}$ (Proposition 3.15).

######  Theorem 2.4.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime such that
$\overline{\rho}$ is surjective. Let $T$ be the $p$-adic Tate module of $E$.

  1. (1)

$\mathbf{KS}(T,\mathcal{F}_{\mathrm{cl}},\mathcal{P})=0$.

  2. (2)

$\mathbf{KS}(T,\mathcal{F}_{\mathrm{can}},\mathcal{P})$ is free of rank one
over $\mathbb{Z}_{p}$.

###### Proof.

See [MR04, Theorem 4.2.2, Theorem 5.2.10, and Proposition 6.2.2]. The first
case lies in the core rank zero case and the second case lies in the core rank
one case. ∎

Theorem 2.4 explains why Kato’s Kolyvagin systems cannot control classical
Selmer groups directly.

#### 2.4.4.

We discuss the precise location of Kolyvagin system classes in $n$-transverse
$p$-relaxed Selmer groups.

######  Theorem 2.5.

For every $k\geq 1$ and $n\in\mathcal{N}_{k}$, there exists a non-canonical
isomorphism

(2.1)
$\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)\simeq\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}\oplus\mathrm{Sel}_{0,n}(\mathbb{Q},E[I_{n}]).$

###### Proof.

See [MR04, Theorem 4.1.13.(i)] and [MR04, Theorem 5.2.5]. ∎

For each $n\in\mathcal{N}_{1}$, write

$\textstyle{\mathcal{H}(n)=\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T),}$$\textstyle{\mathcal{H}^{\prime}(n)=p^{\lambda(n,E[I_{n}])}\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T),}$

and the latter is said to be the stub Selmer submodule at $n$.

######  Theorem 2.6.

For every $n\in\mathcal{N}_{1}$, $\kappa_{n}\in\mathcal{H}^{\prime}(n)$.

###### Proof.

See [MR04, Theorem 4.4.1] with Theorem 2.4. ∎

######  Remark 2.7.

Theorem 2.6 says that
$\kappa_{n}\in\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)$ indeed lies
in
$p^{\lambda(n,E[I_{n}])}\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}\subseteq\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$,
the first factor in the decomposition (2.1).

The following proposition illustrates the precise location of $\kappa_{n}$ in
$\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)$.

######  Proposition 2.8.

Suppose that $\kappa_{n}\neq 0$ for some $n\in\mathcal{N}_{1}$. Let $j\geq 0$
such that $\kappa_{n}$ generates $p^{j}\mathcal{H}^{\prime}(n)$. Then
$\kappa_{n^{\prime}}$ generates
$p^{j}\mathcal{H}^{\prime}(n^{\prime})=p^{j+\lambda(n^{\prime},E[I_{n^{\prime}}])}\mathcal{H}(n^{\prime})\simeq
p^{j+\lambda(n^{\prime},E[I_{n^{\prime}}])}\mathbb{Z}_{p}/I_{n^{\prime}}\mathbb{Z}_{p}$
for all $n^{\prime}\in\mathcal{N}_{1}$.

###### Proof.

See [MR04, Corollary 4.5.2.(ii)]. ∎

######  Corollary 2.9.

If $\kappa_{n}\in\mathrm{Sel}_{0,n}(\mathbb{Q},T/I_{n}T)$, then
$\kappa_{n}=0$.

###### Proof.

Consider the localization map
$\mathrm{loc}_{p}:\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)\to\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)$.
Then we have
$\mathrm{Sel}_{0,n}(\mathbb{Q},T/I_{n}T)=\mathrm{ker}(\mathrm{loc}_{p})$.
Also, the restriction of $\mathrm{loc}_{p}$ to $\mathcal{H}^{\prime}(n)$ is
injective since $p^{\lambda(n,E[I_{n}])}$ annihilates
$\mathrm{Sel}_{0,n}(\mathbb{Q},E[I_{n}])\simeq\mathrm{Sel}_{0,n}(\mathbb{Q},T/I_{n}T)$.
Since $\kappa_{n}\in
p^{j}\mathcal{H}^{\prime}(n)\subseteq\mathcal{H}^{\prime}(n)$, we are done. ∎

We say Kato’s Kolyvagin system $\boldsymbol{\kappa}$ is primitive if
$\boldsymbol{\kappa}^{(1)}=\left(\kappa_{n}\pmod{p}\right)_{n\in\mathcal{N}_{1}}$
is non-zero. This is equivalent to $j=0$ in Proposition 2.8. See [MR04,
Corollary 4.5.4 and Definition 4.5.5].

###  2.5. Kolyvagin systems over $\mathbb{Z}/p^{k}\mathbb{Z}$

#### 2.5.1.

Let $\boldsymbol{\kappa}^{(k)}$ be a Kolyvagin system over $T/p^{k}T$ and
$r\geq 0$ an integer. Define

$\partial^{(r)}(\boldsymbol{\kappa}^{(k)})=\mathrm{min}\left\\{k-\mathrm{length}(\mathbb{Z}/p^{k}\mathbb{Z}\cdot\kappa^{(k)}_{n}):n\in\mathcal{N}_{k},\nu(n)=r\right\\}$

and the elementary divisors of $\boldsymbol{\kappa}$ are defined by

$e_{i}(\boldsymbol{\kappa}^{(k)})=\partial^{(i)}(\boldsymbol{\kappa}^{(k)})-\partial^{(i+1)}(\boldsymbol{\kappa}^{(k)}),i\geq
0.$

#### 2.5.2.

Suppose that $\kappa^{(k)}_{n}\neq 0$ for some $n\in\mathcal{N}_{k}$.
Following [MR04, Example 3.1.12], we have another Kolyvagin system
$\boldsymbol{\kappa}^{n,(k)}$ defined by

$\kappa^{n,(k)}_{m}=\kappa^{(k)}_{n\cdot m}$

where $m\in\mathcal{N}_{k}$ with $(m,n)=1$.

#### 2.5.3.

The following lemma is important when we proceed the Kolyvagin system argument
in Proposition 2.11.

######  Lemma 2.10.

Let $n\in\mathcal{N}_{k}$ and $\ell\in\mathcal{P}_{k}$ with $(n,\ell)=1$, and
‘$\ell\textrm{-str}$’ denotes the strict local condition at $\ell$.

  1. (1)

If the localization map $\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/p^{k}T)\to
E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}_{p}/p^{k}\mathbb{Z}_{p}$ is surjective,
then we have
$\mathrm{Sel}_{0,n\ell}(\mathbb{Q},E[p^{k}])=\mathrm{Sel}_{0,n,\ell\textrm{-}\mathrm{str}}(\mathbb{Q},E[p^{k}])$.

  2. (2)

If the localization map $\mathrm{Sel}_{n}(\mathbb{Q},T/p^{k}T)\to
E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}_{p}/p^{k}\mathbb{Z}_{p}$ is surjective,
then we have
$\mathrm{Sel}_{n\ell}(\mathbb{Q},E[p^{k}])=\mathrm{Sel}_{n,\ell\textrm{-}\mathrm{str}}(\mathbb{Q},E[p^{k}])$.

###### Proof.

See [MR04, Lemma 4.1.7]. ∎

#### 2.5.4.

The following proposition is [MR04, Proposition 4.5.8]. Since the idea of its
proof is used repeatedly in $\S$5, we recall the detail here.

######  Proposition 2.11.

Suppose that $\mathrm{ord}(\boldsymbol{\kappa}^{(k)})<\infty$, i.e
$\kappa^{(k)}_{n}\neq 0$ for some $n\in\mathcal{N}_{1}$. Write

$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{k}])\simeq\bigoplus_{i\geq
1}\mathbb{Z}/p^{d_{i}}\mathbb{Z}$

with non-negative integers $d_{1}\geq d_{2}\geq\cdots$, and fix $j\geq 0$ such
that $\kappa^{(k)}_{n}$ generates $p^{j}\mathcal{H}^{\prime}(n)$ (Proposition
2.8). Then for every $r\geq 0$,

$\partial^{(r)}(\boldsymbol{\kappa}^{(k)})=\mathrm{min}\\{k,j+\sum_{i>r}d_{i}\\}.$

###### Proof.

We first prove
$\partial^{(r)}(\boldsymbol{\kappa}^{(k)})\geq\mathrm{min}\\{k,j+\sum_{i>r}d_{i}\\}$.
By Proposition 2.8, $\kappa^{(k)}_{n}$ generates
$p^{j}\mathcal{H}^{\prime}(n)$ for every $n\in\mathcal{N}_{k}$. By using
Theorem 2.5, we have

$\partial^{(r)}(\boldsymbol{\kappa}^{(k)})=\mathrm{min}\\{k,j+\lambda(n,E[p^{k}]):n\in\mathcal{N}_{k},\nu(n)=r\\}.$

In particular, when $r=0$, we have equality
$\lambda(1,E[p^{k}])=\sum_{i}d_{i}$.

Suppose that $n\in\mathcal{N}_{k}$ with $\nu(n)=r$. Consider the natural map

$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{k}])\to\bigoplus_{\ell\mid
n}E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}/p^{k}\mathbb{Z}.$

Since $\bigoplus_{\ell\mid
n}E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}/p^{k}\mathbb{Z}\simeq(\mathbb{Z}/p^{k}\mathbb{Z})^{\oplus
r}$, the image is a quotient of $\mathrm{Sel}_{0}(\mathbb{Q},E[p^{k}])$
generated by at most $r$ elements. Thus, the length of the image is at most
$\sum_{i\leq r}d_{i}$ and the length of the kernel is at least
$\sum_{i>r}d_{i}$. Since the kernel is contained in
$\mathrm{Sel}_{0,n}(\mathbb{Q},E[p^{k}])$, we have
$\lambda(n,E[p^{k}])\geq\sum_{i>r}d_{i}$, so the desired inequality follows.

We prove the opposite inequality
$\partial^{(r)}(\boldsymbol{\kappa}^{(k)})\leq\mathrm{min}\\{k,j+\sum_{i>r}d_{i}\\}$
by induction on $r$. The $r=0$ case is already proved. By Theorem 2.5,
$p^{k-1}\cdot\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},T/p^{k}T)\neq 0$ and
choose a non-zero element $c\in
p^{k-1}\cdot\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},T/p^{k}T)\subseteq\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},T/p^{k}T)[p]$
If $d_{1}>0$, then choose a non-zero element $c^{\prime}\in
p^{d_{1}-1}\cdot\mathrm{Sel}_{0}(\mathbb{Q},E[p^{k}])\subseteq\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},E[p^{k}])[p]$.
By using Proposition 2.3, we choose a prime $\ell\in\mathcal{P}_{k}$ such that
$\mathrm{loc}_{\ell}(c)\neq 0\in\mathrm{H}^{1}(\mathbb{Q}_{\ell},T/p^{k}T)$
and $\mathrm{loc}_{\ell}(c^{\prime})\neq
0\in\mathrm{H}^{1}(\mathbb{Q}_{\ell},E[p^{k}])$ if $d_{1}>0$. It follows that

  * •

$\mathrm{loc}_{\ell}:\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},T/p^{k}T)\to
E(\mathbb{Q}_{\ell})\otimes\mathbb{Z}/p^{k}\mathbb{Z}$ is surjective, and

  * •

$\mathrm{Sel}_{0,\ell\textrm{-}\mathrm{str}}(\mathbb{Q},E[p^{k}])\simeq\bigoplus_{i\geq
1}\mathbb{Z}/p^{d_{i+1}}\mathbb{Z}$.

By Lemma 2.10, we have
$\mathrm{Sel}_{0,\ell\textrm{-}\mathrm{str}}(\mathbb{Q},E[p^{k}])\simeq\mathrm{Sel}_{0,\ell}(\mathbb{Q},E[p^{k}])$.
Let $\boldsymbol{\kappa}^{\ell,(k)}$ be the Kolyvagin system constructed from
$\boldsymbol{\kappa}^{(k)}$ as in $\S$2.5.2. Then we have
$\partial^{(r)}(\boldsymbol{\kappa}^{(k)})\geq\partial^{(r-1)}(\boldsymbol{\kappa}^{\ell,(k)})=\mathrm{min}\\{k,j+\sum_{i>r-1}d_{i+1}\\}$
from the induction hypothesis applied to $\boldsymbol{\kappa}^{\ell,(k)}$.
This completes the proof. ∎

###  2.6. Kolyvagin systems over $\mathbb{Z}_{p}$

#### 2.6.1.

Let $\boldsymbol{\kappa}$ be a Kolyvagin system for $T$. Define

$\partial^{(0)}(\boldsymbol{\kappa})=\mathrm{max}\left\\{j:\kappa_{1}\in
p^{j}\mathrm{Sel}_{\mathrm{rel}}(\mathbb{Q},T)\right\\}$

and we allow $\partial^{(0)}(\boldsymbol{\kappa})=\infty$ (when
$\kappa_{1}=0$).

######  Theorem 2.12.

Let $\boldsymbol{\kappa}$ be a Kolyvagin system for $T$. Then

$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])\leq\partial^{(0)}(\boldsymbol{\kappa}).$

In particular, if $\kappa_{1}\neq 0$, then
$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$ is finite.

###### Proof.

See [MR04, Theorem 5.2.2]. ∎

######  Definition 2.13.

  1. (1)

The vanishing order of a non-zero Kolyvagin system
$\boldsymbol{\kappa}=(\kappa_{n})_{n\in\mathcal{N}_{1}}$ is defined by

$\mathrm{ord}(\boldsymbol{\kappa})=\mathrm{min}\left\\{\nu(n):n\in\mathcal{N}_{1},\kappa_{n}\neq
0\right\\}$

  2. (2)

For $r\in\mathbb{Z}_{>0}$, define

$\displaystyle\partial^{(r)}(\boldsymbol{\kappa})$ $\displaystyle=\ $
$\displaystyle\mathrm{max}\left\\{j:\kappa_{n}\in
p^{j}\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)\textrm{ for every
}n\in\mathcal{N}_{1}\textrm{ with }\nu(n)=r\right\\}$ $\displaystyle=\ $
$\displaystyle\lim_{k\to\infty}\mathrm{max}\left\\{j:\kappa^{(k)}_{n}\in
p^{j}\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/p^{k}T)\textrm{ for every
}n\in\mathcal{N}_{k}\textrm{ with }\nu(n)=r\right\\}$

and the second equality follows from Theorem 2.14.(1) below.

  3. (3)

The sequence of elementary divisors is defined by

$e_{i}(\boldsymbol{\kappa})=\partial^{(i)}(\boldsymbol{\kappa})-\partial^{(i+1)}(\boldsymbol{\kappa})$

where $i\geq\mathrm{ord}(\boldsymbol{\kappa})$.

#### 2.6.2.

The following theorem illustrates how a non-trivial Kolyvagin system
determines the structure of $p$-strict Selmer groups.

######  Theorem 2.14.

Suppose that $\boldsymbol{\kappa}$ is a non-trivial Kolyvagin system for $T$.
Then the following statements hold.

  1. (1)

For every $s\geq 0$,

$\partial^{(s)}(\boldsymbol{\kappa})=\lim_{k\to\infty}\partial^{(s)}(\boldsymbol{\kappa}^{(k)}).$

  2. (2)

The sequence $\partial^{(s)}(\boldsymbol{\kappa})$ is non-increasing, and
finite for $s\geq\mathrm{ord}(\boldsymbol{\kappa})$.

  3. (3)

The sequence $e_{i}(\boldsymbol{\kappa})$ is non-increasing, non-negative, and
finite for $i\geq\mathrm{ord}(\boldsymbol{\kappa})$.

  4. (4)

$\mathrm{ord}(\boldsymbol{\kappa})$ and the $e_{i}(\boldsymbol{\kappa})$ are
independent of the choice of non-zero $\boldsymbol{\kappa}\in\mathbf{KS}(T)$.

  5. (5)

$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])=\mathrm{ord}(\boldsymbol{\kappa})$.

  6. (6)

$\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])_{/\mathrm{div}}\simeq\bigoplus_{i\geq\mathrm{ord}(\boldsymbol{\kappa})}\dfrac{\mathbb{Z}_{p}}{p^{e_{i}(\boldsymbol{\kappa})}\mathbb{Z}_{p}}$

  7. (7)

$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])_{/\mathrm{div}}=\partial^{(\mathrm{ord}(\boldsymbol{\kappa}))}(\boldsymbol{\kappa})-\partial^{(\infty)}(\boldsymbol{\kappa})$

  8. (8)

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

###### Proof.

See [MR04, Theorem 5.2.12]. ∎

######  Corollary 2.15.

Suppose that $\boldsymbol{\kappa}$ is a non-trivial Kolyvagin system for $T$.

  1. (1)

$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])$
is finite if and only if $\kappa_{1}\neq 0$.

  2. (2)

$\mathrm{length}_{\mathbb{Z}_{p}}\mathrm{Sel}_{0}(\mathbb{Q},E[p^{\infty}])\leq\partial^{(0)}(\boldsymbol{\kappa})$,
with equality if and only if $\boldsymbol{\kappa}$ is primitive.

###### Proof.

See [MR04, Corollary 5.2.13]. ∎

##  3\. A refined explicit reciprocity law and Kurihara numbers

The goal of this section is to describe the precise connection between
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ and $\widetilde{\boldsymbol{\delta}}$.
We first investigate when the local torsion $E(\mathbb{Q}_{p})[p]$ is trivial.
Then we compute the integral image of the Bloch–Kato dual exponential map.
Using this computation, we extend the the Bloch–Kato dual exponential map to
torsion coefficients. By using the compatibility between the classical
Bloch–Kato dual exponential map and the torsion Bloch–Kato dual exponential
map, we obtain the derivative of Kato’s explicit reciprocity law. It
significantly refines the computation in [KKS20].

###  3.1. Local torsions of elliptic curves

#### 3.1.1.

Consider an exact sequence

(3.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{E}(p\mathbb{Z}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(\mathbb{Q}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{E}(\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\widehat{E}$ is the formal group associated to $E$ and $\widetilde{E}$
is the mod $p$ reduction of $E$. Since $\widehat{E}(p\mathbb{Z}_{p})$ is
isomorphic to $\mathbb{Z}_{p}$ as additive groups, $E(\mathbb{Q}_{p})[p]$ is
non-trivial if and only if $\widetilde{E}(\mathbb{F}_{p})[p]$ is non-trivial
and (3.1) splits. Thus, if $E$ has non-anomalous good reduction at $p$, then
$E(\mathbb{Q}_{p})[p]$ is trivial.

Now we consider the case that $E$ has anomalous good reduction at $p$,
equivalently, $\widetilde{E}(\mathbb{F}_{p})[p]$ is non-trivial. Indeed,
$a_{p}(E)\equiv 1\pmod{p}$ implies $a_{p}(E)=1$ if $p\geq 7$, and it is known
that the set $\\{p:a_{p}(E)=1\\}$ has zero density [Gre99, Proposition 5.1].

Suppose that $E(\mathbb{Q}_{p})[p]$ is non-trivial, i.e. (3.1) splits. By
applying Gross’ tameness criterion [Gro90, Proposition 13.2]555The unchecked
compatibility in [Gro90] is confirmed in the Ph.D. thesis of Bryden Cais
[Cai07]., this is equivalent to that the image of the restriction of the
residual representation to
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ is diagonalizable.

Recall that $f=\sum_{n\geq 1}a_{n}(E)q^{n}$ admits a mod $p$ companion form if
there exists a mod $p$ eigenform $g=\sum_{n\geq 1}b_{n}q^{n}$ of weight $p-1$
such that $n^{2}\cdot b_{n}\equiv n\cdot a_{n}(E)\pmod{p}$ for all $n\geq 1$.
By [Gro90], the above diagonalizability is also equivalent to the existence of
a mod $p$ companion form of $f$.

#### 3.1.2.

By using the parametrization by the Tate curve, it is not difficult to observe
the following statement for the split multiplicative reduction case. The Tate
parameter $q$ can be taken from the $q$-expansion of the $j$-invariant.

######  Proposition 3.1.

Suppose that $E$ has split multiplicative reduction at $p$. Then the following
statements are equivalent.

  1. (1)

$E(\mathbb{Q}_{p})[p]$ is non-trivial.

  2. (2)

The Tate parameter $q\in\mathbb{Q}^{\times}_{p}$ satisfies
$p\mid\mathrm{ord}_{p}(q)$.

  3. (3)

The $j$-invariant $j\in\mathbb{Q}^{\times}_{p}$ satisfies
$p\mid\mathrm{ord}_{p}(j)$.

When $E$ admits non-split multiplicative reduction at a prime $p>2$, the
corresponding Galois representation is the Galois representation coming from
the Tate curve but twisted by a quadratic unramified character. Thus,
$E(\mathbb{Q}_{p})[p]$ is trivial.

#### 3.1.3.

To sum up, we have the following statement.

######  Proposition 3.2.

Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$ with a prime $p$. Then
$E(\mathbb{Q}_{p})[p]\neq 0$ if and only if

  1. (1)

$E$ has anomalous good ordinary reduction at $p$ and admits a mod $p$
companion form,

  2. (2)

$E$ has split multiplicative reduction at $p$ and the $j$-invariant satisfies
$p\mid\mathrm{ord}_{p}(j)$, or

  3. (3)

$E$ has additive reduction at $p$ with minimal Weierstrass equation

$y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$

where $a_{i}\in p\mathbb{Z}_{p}$ for each $i$ satisfying

    1. (a)

$p=2$ and $a_{1}+a_{3}\equiv 2\pmod{4}$,

    2. (b)

$p=3$ and $a_{2}\equiv 6\pmod{9}$,

    3. (c)

$p=5$ and $a_{4}\equiv 10\pmod{25}$, or

    4. (d)

$p=7$ and $a_{6}\equiv 14\pmod{49}$.

###### Proof.

Thanks to the above discussion, it suffices to consider the additive reduction
case, which is covered in [KN20, $\S$2] and [KP]. ∎

#### 3.1.4.

For convenience, we also record some related results.

######  Proposition 3.3.

  1. (1)

If $E(\mathbb{Q})[2]$ is non-trivial and $p>5$ is a good reduction prime for
$E$, then $p$ is a non-anomalous prime for $E$.

  2. (2)

If $E$ is $\mathbb{Q}$-isogenous to an elliptic curve $E^{\prime}$ such that
$E^{\prime}(\mathbb{Q})[q]$ is non-trivial with $q>2$, and $p\nmid q$, then
$p$ is a non-anomalous prime for $E$.

  3. (3)

If $E$ has CM and $p>3$ is a good reduction prime for $E$, then
$E(\mathbb{Q}_{p})[p]\neq 0$ if and only if $a_{p}(E)=1$.

  4. (4)

If the corresponding modular form $f$ does not admit a mod $p$ companion form,
then $f$ is not congruent to a CM modular form mod $p$.

###### Proof.

  1. (1)

See [Gre99, Lemma 5.2].

  2. (2)

See [Gre99, Lemma 5.2].

  3. (3)

See [DW08, Proposition 2.1.(3)].

  4. (4)

It immediately follows from [Gha05, Proposition 6].

∎

#### 3.1.5.

It is conjectured that there are only finitely many primes $p$ such that
$E(\mathbb{Q}_{p})[p]\neq 0$ when $E$ is a non-CM elliptic curve [DW08,
Conjecture 1.1], and its average version is proved in [DW08, Theorem 1.2].

###  3.2. The computation of the integral image of Bloch–Kato’s dual
exponential map

We recall the computation of the integral image of Bloch–Kato’s dual
exponential map following [Rub00, Proposition 3.5.1], [Kat04, Lemma 14.18] and
[KKS20, $\S$6 and $\S$7] with some refinement in order to handle the
$E(\mathbb{Q}_{p})[p^{\infty}]\neq 0$ case.

#### 3.2.1.

Following [Rub00, $\S$3.5], fix a minimal Weierstrass model of $E$ over
$\mathbb{Q}_{p}$ and $\omega_{E}$ is the corresponding holomorphic
differential. Let $\omega^{*}_{E}$ be the basis of the tangent space of the
minimal Weierstrass model such that the natural pairing
$\langle\omega^{*}_{E},\omega_{E}\rangle=1$. Write
$\mathrm{exp}^{*}_{\omega_{E}}(-)=\langle\omega^{*}_{E},\mathrm{exp}^{*}(-)\rangle$
where $\mathrm{exp}^{*}$ is the Bloch–Kato dual exponential map. We write

$t=\left\\{\begin{array}[]{ll}0&\textrm{ if $E$ has split multiplicative
reduction at $p$,}\\\
\mathrm{length}_{\mathbb{Z}_{p}}\left(E(\mathbb{Q}_{p})[p^{\infty}]\right)&\textrm{
otherwise.}\end{array}\right.$

#### 3.2.2.

We first consider the good reduction case.

######  Lemma 3.4.

If $p>2$, we have

$\mathrm{exp}^{*}_{\omega_{E}}:\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{E(\mathbb{Q}_{p})\otimes\mathbb{Z}_{p}}\simeq\dfrac{\\#\widetilde{E}(\mathbb{F}_{p})}{\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])}\cdot\dfrac{1}{p}\cdot\mathbb{Z}_{p}.$

If $E$ has good reduction at $p>2$, then we have

$\mathrm{exp}^{*}_{\omega_{E}}:\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{E(\mathbb{Q}_{p})\otimes\mathbb{Z}_{p}}\simeq\dfrac{\\#\widetilde{E}(\mathbb{F}_{p})}{p^{1+t}}\cdot\mathbb{Z}_{p}=\dfrac{p-a_{p}(E)+1}{p^{1+t}}\cdot\mathbb{Z}_{p}.$

###### Proof.

See [Rub00, Proposition 3.5.1] and [Kat04, Lemma 14.18]. ∎

When $E$ has good reduction at $p$, we have
$\\#\widetilde{E}(\mathbb{F}_{p})=p-a_{p}(E)+1$, and
$\dfrac{p-a_{p}(E)+1}{p}=1-a_{p}(E)p^{-1}+p^{-1}$ is the Euler factor at $p$.
Due to the Hasse bound, $1-a_{p}(E)+p$ is non-zero.

######  Corollary 3.5.

Suppose that $E$ has good reduction at $p\geq 5$. Let $K$ be a finite
unramified extension of $\mathbb{Q}_{p}$ with residue field $k$. We assume
either

  1. (1)

$p$ does not divide $\\#\widetilde{E}(k)$, or

  2. (2)

the corresponding modular form does not admit a mod $p$ companion form.

Then we have $E(K)[p]=0$ and

$\mathrm{exp}^{*}_{\omega_{E}}:\dfrac{\mathrm{H}^{1}(K,T)}{E(K)\otimes\mathbb{Z}_{p}}\simeq\dfrac{\\#\widetilde{E}(k)}{p}\cdot\mathcal{O}_{K}.$

###### Proof.

The first isomorphism in Lemma 3.4 works for any finite unramified extension
$K$. The first case is immediate. For the second case, we know the restriction
of $\overline{\rho}$ to
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ is wildly ramified
thanks to [Gro90, Proposition 13.7]. Since $K/\mathbb{Q}_{p}$ is unramified,
the restriction of $\overline{\rho}$ to $\mathrm{Gal}(\overline{K}/K)$ is
still wildly ramified. Thus, it cannot be a direct sum of two characters.
Thus, we have $E(K)[p]=0$ even when $\widetilde{E}(k)[p]$ is non-trivial. ∎

######  Remark 3.6.

See [Ots09, Theorem 4.1], [Ota18, Lemma 5.5], and [Kat21, Assumption 3.7] for
the application of a $E(K)[p]=0$ type assumption in the construction of
integral Mazur–Tate elements over $K$.

#### 3.2.3.

We move to the bad reduction cases. Consider the following exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}(\mathbb{Q}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(\mathbb{Q}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{E}(\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}(\mathbb{Q}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}(\mathbb{Q}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\widetilde{E}(\mathbb{F}_{p})$ is the mod $p$ reduction of
$E(\mathbb{Q}_{p})$ and $E_{1}(\mathbb{Q}_{p})$ is the kernel of the mod $p$
reduction map. Also, $\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})$ is the non-
singular locus of $\widetilde{E}(\mathbb{F}_{p})$ and $E_{0}(\mathbb{Q}_{p})$
is the preimage of $\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})$ in
$E(\mathbb{Q}_{p})$. As in Lemma 3.4, if $p>2$, we have

$\mathrm{exp}^{*}_{\omega_{E}}:\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{E(\mathbb{Q}_{p})\otimes\mathbb{Z}_{p}}\simeq\dfrac{\widetilde{E}(\mathbb{F}_{p})}{\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])}\cdot\dfrac{1}{p}\cdot\mathbb{Z}_{p}\simeq\dfrac{\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])}\cdot[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]\cdot\dfrac{1}{p}\cdot\mathbb{Z}_{p}.$

######  Theorem 3.7.

Let $E$ be an elliptic curve over $\mathbb{Q}_{p}$ (with minimal Weierstrass
equation), and $\Delta$ the discriminant of $E$.

  1. (1)

If $E$ admits split multiplicative reduction at $p$, then
$[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]=\mathrm{ord}_{p}(\Delta)=-\mathrm{ord}_{p}(j)$.

  2. (2)

If $E$ admits non-split multiplicative reduction at $p$, then
$[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]\leq 2$.

  3. (3)

Otherwise, $[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]\leq 4$.

###### Proof.

It follows from Tate’s algorithm [Tat75]. See [Sil09, Theorem VII.6.1] and
[Lor11, $\S$2.2]. ∎

######  Remark 3.8.

When $E$ has additive reduction at $p$ with $p\leq 3$, then
$[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]$ is not divisible by $p$. See
[Sil99, Table 4.1, Page 365]

Thus, when $E$ admits non-split multiplicative reduction at $p\geq 3$, we have

$\mathrm{exp}^{*}_{\omega_{E}}(\mathrm{H}^{1}(\mathbb{Q}_{p},T))=\dfrac{\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])}\cdot[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]\cdot\dfrac{1}{p}\cdot\mathbb{Z}_{p}=\dfrac{1}{p}\cdot\mathbb{Z}_{p}.$

Now we suppose that $E$ admits split multiplicative reduction at a prime
$p>2$. Then the $p$-adic uniformization by the Tate curve yields the following
exact sequence of representations of
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $T_{1}\simeq\mathbb{Z}_{p}(1)$ and $T_{2}\simeq\mathbb{Z}_{p}$. By
tensoring $\mathbb{Q}_{p}$, we also have

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $V_{\bullet}=T_{\bullet}\otimes\mathbb{Q}_{p}$ and
$\bullet\in\\{\emptyset,1,2\\}$. Following [Kob06, $\S$4], we have the
commutative diagram

$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\mathrm{exp}^{*}_{E}}$$\textstyle{\mathbb{Q}_{p}\omega_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{*}_{E}}$$\textstyle{\mathbb{Q}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},V_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{exp}^{*}_{\mathbb{G}_{m}}}$$\textstyle{\mathbb{Q}_{p}\omega_{\mathbb{G}_{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega^{*}_{\mathbb{G}_{m}}}$$\textstyle{\mathbb{Q}_{p}}$

where $\pi$ is induced from the quotient map $V\to V_{2}$,
$\mathrm{exp}^{*}_{E}$ is the dual exponential map for $E$,
$\mathrm{exp}^{*}_{\mathbb{G}_{m}}$ is the dual exponential map for
$\mathbb{G}_{m}$, $\omega_{\mathbb{G}_{m}}$ is the invariant differential of
$\mathbb{G}_{m}$, which is $\dfrac{dX}{1+X}$ on the formal multiplicative
group $\widehat{\mathbb{G}}_{m}$, and $\omega^{*}_{\mathbb{G}_{m}}$ is the
dual basis for $\omega_{\mathbb{G}_{m}}$ with
$\omega^{*}_{\mathbb{G}_{m}}(\omega_{\mathbb{G}_{m}})=1$. Write
$\mathrm{exp}^{*}_{\omega_{E}}=\omega^{*}_{E}\circ\mathrm{exp}^{*}_{E}$ and
$\mathrm{exp}^{*}_{\omega_{\mathbb{G}_{m}}}=\omega^{*}_{\mathbb{G}_{m}}\circ\mathrm{exp}^{*}_{\mathbb{G}_{m}}$.
In order to compute the lattice
$\mathrm{exp}^{*}_{\omega_{E}}(\mathrm{H}^{1}(\mathbb{Q}_{p},T))\subseteq\mathbb{Q}_{p}$,
it suffices to compute the lattice
$\mathrm{exp}^{*}_{\omega_{\mathbb{G}_{m}}}(\mathrm{H}^{1}(\mathbb{Q}_{p},\mathbb{Z}_{p}))\subseteq\mathbb{Q}_{p}$.
Considering the local Tate duality, we compute the lattice
$\mathrm{log}(\mathbb{G}_{m}(\mathbb{Q}_{p}))\subseteq\mathbb{Q}_{p}$ where
$\mathrm{log}:\mathbb{G}_{m}(\mathbb{Q}_{p})\otimes\mathbb{Q}_{p}\to\mathbb{Q}_{p}$
is the linear extension of the formal logarithm map on
$\widehat{\mathbb{G}}_{m}(p\mathbb{Z}_{p})$. Note that $\mathbb{Z}_{p}(1)$ is
the Tate module of the multiplicative group $\mathbb{G}_{m}$. We have an exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathbb{G}}_{m}(p\mathbb{Z}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{G}_{m}(\mathbb{Q}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathbb{G}}_{m}(\mathbb{F}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By using the formal logarithm, $\widehat{\mathbb{G}}_{m}(p\mathbb{Z}_{p})$
maps to $p\mathbb{Z}_{p}\subseteq\mathbb{Q}_{p}$. Also,
$\widetilde{\mathbb{G}}_{m}(\mathbb{F}_{p})=\mathbb{F}^{\times}_{p}$ has size
prime to $p$. Thus, we also have

$\mathrm{exp}^{*}_{\omega_{E}}(\mathrm{H}^{1}(\mathbb{Q}_{p},T))=\mathrm{exp}^{*}_{\omega_{\mathbb{G}_{m}}}(\mathrm{H}^{1}(\mathbb{Q}_{p},\mathbb{Z}_{p}))=\dfrac{1}{p}\mathbb{Z}_{p}.$

To sum up, we have the following statement.

######  Lemma 3.9.

If $E$ has multiplicative reduction at $p\geq 3$, then

$\mathrm{exp}^{*}_{\omega_{E}}(\mathrm{H}^{1}(\mathbb{Q}_{p},T))=\dfrac{1}{p}\mathbb{Z}_{p}.$

It is remarkable that Lemma 3.9 does not involve $t$. When $E$ has non-split
multiplicative reduction at $p$, we have $t=0$. When $E$ has split
multiplicative reduction at $p$,
$\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])$ and
$[E(\mathbb{Q}_{p}):E_{0}(\mathbb{Q}_{p})]$ cancel each other. See [Sil99,
Table 4.1, Page 365].

######  Lemma 3.10.

If $E$ has additive reduction at $p\geq 3$, then

$\mathrm{exp}^{*}_{\omega_{E}}(\mathrm{H}^{1}(\mathbb{Q}_{p},T))=\dfrac{1}{p^{t}}\mathbb{Z}_{p}.$

###### Proof.

It follows from Theorem 3.7 and Remark 3.8. ∎

###  3.3. The extension of the dual exponential map to torsion coefficients

#### 3.3.1.

Let $n\in\mathcal{N}_{1}$. Recall the following exact sequence

(3.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{2}(\mathbb{Q}_{p},T)[I_{n}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

If $E(\mathbb{Q}_{p})[p^{\infty}]=0$, then
$\mathrm{H}^{2}(\mathbb{Q}_{p},T)[I_{n}]=0$ by the local Tate duality. In this
case, we easily define the dual exponential map on
$\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)$ by

$\mathrm{exp}^{*}_{\omega_{E}}:\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)\simeq\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)}\to\dfrac{\mathrm{exp}^{*}_{\omega_{E}}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},T)\right)}{I_{n}\mathrm{exp}^{*}_{\omega_{E}}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},T)\right)}$

where the second map is the naïve mod $I_{n}$ reduction of the integral dual
exponential map discussed in $\S$3.2.

#### 3.3.2.

Now we assume that $E(\mathbb{Q}_{p})[p^{\infty}]\neq 0$. From (3.2), we
obtain the following exact sequence

(3.3)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)}{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{2}(\mathbb{Q}_{p},T)[I_{n}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

thanks to the following lemma.

######  Lemma 3.11.

The map

$\textstyle{\phi:\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)}{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)}.}$

is injective.

###### Proof.

Since $\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)$ is the Kummer image of
$E(\mathbb{Q}_{p})\otimes\mathbb{Z}_{p}$ and
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)$ is the Kummer image of
$E(\mathbb{Q}_{p})\otimes\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$, we have
$\dfrac{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}=\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)$.

Let
$[c]\in\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}$
such that $\phi([c])=0$. Choose $c\in\mathrm{H}^{1}(\mathbb{Q}_{p},T)$ which
lifts $[c]$. Since $\phi([c])=0$, the image of $c$ in
$\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)$ lies in
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)$. Thanks to the following
commutative diagram

$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)}$$\textstyle{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dfrac{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

the image of $c$ in
$\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)}$
also lies in
$\dfrac{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}$.
Thus, $c$ lies in
$I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)$,
so $[c]=0$. This is independent of the choice of the lift of $[c]$. ∎

######  Proposition 3.12.

The sequence (3.3) splits as $\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$-modules.

###### Proof.

We first recall the isomorphism
$\mathrm{exp}^{*}_{\omega_{E}}:\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}\simeq\mathrm{exp}^{*}_{\omega_{E}}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},T)\right)(\simeq\mathbb{Z}_{p})$.
Then its naïve mod $I_{n}$ reduction also induces an isomorphism

$\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}\simeq\dfrac{\mathrm{exp}^{*}_{\omega_{E}}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},T)\right)}{I_{n}\mathrm{exp}^{*}_{\omega_{E}}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},T)\right)}(\simeq\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}).$

In particular,
$\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}$
is free of rank one over $\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$. In the
Pontryagin dual of (3.3)

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\mathrm{H}^{2}(\mathbb{Q}_{p},T)[I_{n}]\right)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)}{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)}\right)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\vee}}$$\textstyle{\left(\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}\right)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

the last term is also free of rank one over
$\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$. Thus, the dual sequence splits, and
(3.3) also splits. ∎

We define the dual exponential map on
$\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)$ by the composition

$\mathrm{exp}^{*}_{\omega_{E}}:\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)\to\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)}{\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T/I_{n}T)}\to\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T)}{I_{n}\mathrm{H}^{1}(\mathbb{Q}_{p},T)+\mathrm{H}^{1}_{f}(\mathbb{Q}_{p},T)}\to\dfrac{\mathrm{exp}^{*}_{\omega_{E}}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},T)\right)}{I_{n}\mathrm{exp}^{*}_{\omega_{E}}\left(\mathrm{H}^{1}(\mathbb{Q}_{p},T)\right)}$

where the first map is the natural quotient map, the second map comes from the
splitting of (3.3) (as proved in Proposition 3.12), and the third map is the
naïve mod $I_{n}$ reduction of the integral dual exponential map again.

###  3.4. “Derivatives” of Kato’s explicit reciprocity law

#### 3.4.1.

Let $n\in\mathcal{N}_{1}$ and fix an isomorphism
$\xi:\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}\simeq\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$.
Then we have the map

(3.4) $\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
41.05309pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-16.36555pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.04167pt\hbox{$\scriptstyle{\mathrm{loc}^{s}_{p}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-26.69778pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.97258pt\raise 6.83833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.8839pt\hbox{$\scriptstyle{\xi\circ\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 100.27405pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
100.27405pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}}$}}}}}}}{\hbox{\kern-35.75417pt\raise-45.83554pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dfrac{\mathrm{H}^{1}(\mathbb{Q}_{p},T/I_{n}T)}{E(\mathbb{Q}_{p})\otimes\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
41.0564pt\raise-39.8861pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.99501pt\hbox{$\scriptstyle{\mathrm{exp}^{*}_{\omega_{E}}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.6814pt\raise-50.45865pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.62312pt\hbox{$\scriptstyle{\simeq}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
65.05309pt\raise-45.83554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
65.05309pt\raise-45.83554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
108.64023pt\raise-17.15417pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.62312pt\hbox{$\scriptstyle{\simeq}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
120.08469pt\raise-17.15417pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\xi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 120.08469pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{split}$

thanks to Lemma 3.4, Lemma 3.9, Lemma 3.10, and the extension of the dual
exponential map to torsion coefficients. Note that
$\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})$ is prime to $p$ if $E$ has
multiplicative reduction and
$\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})=p$ if $E$ has additive
reduction.

######  Theorem 3.13.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ is a prime such
that $\overline{\rho}$ is surjective and the Manin constant is prime to $p$.
Then we have formula

$\xi\circ\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}(\kappa^{\mathrm{Kato}}_{n})=u\cdot
p^{t}\cdot\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}.$

where $u\in(\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p})^{\times}$.

###### Proof.

The following computation is essentially done in [KKS20, $\S$6 and $\S$7]

$\displaystyle\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}(D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})})$
$\displaystyle=E_{p}(\sigma_{p})\cdot
D_{\mathbb{Q}(\zeta_{n})}\left(\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\zeta^{a}_{n}\cdot\left[\dfrac{a}{n}\right]^{+}\right)\in\mathbb{Q}_{p}\otimes\mathbb{Q}(\zeta_{n})$

where

$E_{p}(\sigma_{p})=\left\\{\begin{array}[]{ll}1-a_{p}(E)\cdot
p^{-1}\cdot\sigma_{p}+p^{-1}\cdot\sigma^{2}_{p}&\textrm{if }p\nmid N\\\
1-a_{p}(E)\cdot p^{-1}\cdot\sigma_{p}&\textrm{if }p\|N\\\ 1&\textrm{if
}p^{2}|N\end{array}\right.$

and $\sigma_{p}\in\mathrm{Gal}(\mathbb{Q}(\zeta_{n})/\mathbb{Q})$ is the
arithmetic Frobenius at $p$. We also have

$D_{\mathbb{Q}(\zeta_{n})}\left(\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\zeta^{a}_{n}\cdot\left[\dfrac{a}{n}\right]^{+}\right)\equiv\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$

as in [KKS20, Theorem 7.5] and [Kur14a, Page 190].

By the extension of the dual exponential map to torsion coefficients discussed
in $\S$3.3, we have

(3.5)
$\mathrm{exp}^{*}_{\omega_{E}}:\mathrm{H}^{1}_{/f}(\mathbb{Q}_{p},T/I_{n}T)\simeq\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}.$

Because
$\kappa^{\mathrm{Kato}}_{n}\in\mathrm{Sel}_{\mathrm{rel},n}(\mathbb{Q},T/I_{n}T)\subseteq\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)$
comes from the mod $I_{n}$ reduction of
$D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})}$, we
regard $E_{p}(\sigma_{p})\cdot
D_{\mathbb{Q}(\zeta_{n})}\left(\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\zeta^{a}_{n}\cdot\left[\dfrac{a}{n}\right]^{+}\right)$
as an element in
$\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}\otimes\mathbb{Z}[\zeta_{n}]\subseteq\mathbb{Q}_{p}\otimes\mathbb{Q}(\zeta_{n})$
via (3.5) in order to compute
$\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}(\kappa^{\mathrm{Kato}}_{n})$.

We record all the integral lattices in each step of the following computation
to avoid confusion.

$\displaystyle\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}(\kappa^{\mathrm{Kato}}_{n})\in\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}$
$\displaystyle=\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}(D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})}\pmod{I_{n}})\in\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}$
$\displaystyle=\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}(D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})})\pmod{I_{n}}\in\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}\otimes\mathbb{Z}[\zeta_{n}]/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}\otimes\mathbb{Z}[\zeta_{n}]$
$\displaystyle=E_{p}(\sigma_{p})\cdot
D_{\mathbb{Q}(\zeta_{n})}\left(\sum_{a\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\zeta^{a}_{n}\cdot\left[\dfrac{a}{n}\right]^{+}\right)\in\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}\otimes\mathbb{Z}[\zeta_{n}]/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}\otimes\mathbb{Z}[\zeta_{n}]$
$\displaystyle=E_{p}(\sigma_{p})\cdot\widetilde{\delta}_{n}\in\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}$
$\displaystyle=E_{p}\cdot\widetilde{\delta}_{n}\in\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}\mathbb{Z}_{p}/\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p^{1+t}}I_{n}\mathbb{Z}_{p}$

where

$E_{p}=\left\\{\begin{array}[]{ll}1-a_{p}(E)p^{-1}+p^{-1}&\textrm{if }p\nmid
N\\\ 1-a_{p}(E)p^{-1}&\textrm{if }p\|N\\\ 1&\textrm{if
}p^{2}|N\end{array}\right..$

This computation is done in the following commutative diagram

}$\textstyle{\left\\{D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})}\right\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
---

where the dotted arrow means that the image of
$D_{\mathbb{Q}(\zeta_{n})}z^{\mathrm{Kato}}_{\mathbb{Q}(\zeta_{n})}$ under
$\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}$ actually lies in
$\dfrac{\\#\widetilde{E}_{\mathrm{ns}}(\mathbb{F}_{p})}{p}\mathbb{Z}_{p}\otimes\mathbb{Z}[\zeta_{n}]$.
By using the fixed isomorphism $\xi$, the conclusion immediately follows. ∎

Theorem 3.13 can be understood as Kolyvagin derivatives of (an equivariant
refinement of) Kato’s explicit reciprocity law [Kat04, Theorem 12.5], [Kat21,
Theorem 6.1].

#### 3.4.2.

######  Lemma 3.14.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ is a prime such
that $\overline{\rho}$ is surjective and the Manin constant is prime to $p$.
Then the non-vanishing of $p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ and the
non-vanishing of $\widetilde{\boldsymbol{\delta}}$ are equivalent. In
particular,
$\mathrm{ord}(p^{t}\cdot\widetilde{\boldsymbol{\delta}})=\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$.

###### Proof.

One direction $p^{t}\cdot\widetilde{\boldsymbol{\delta}}\neq
0\Rightarrow\widetilde{\boldsymbol{\delta}}\neq 0$ and inequality
$\mathrm{ord}(p^{t}\cdot\widetilde{\boldsymbol{\delta}})\geq\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$
are obvious.

Suppose that $p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ is zero. Then $p^{t}$
annihilates $\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$ for
every $n\in\mathcal{N}_{1}$, so
$k-t\leq\partial^{(i)}(\widetilde{\boldsymbol{\delta}}^{(k)})$ for every
$i\geq 0$ and $k\geq 1$. As $k$ goes to the infinity,
$\partial^{(i)}(\widetilde{\boldsymbol{\delta}})$ diverges for every $i\geq
0$, so we have $\widetilde{\boldsymbol{\delta}}=0$.

Suppose that $\widetilde{\delta}_{n}\neq 0$ with
$\nu(n)=i=\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$. This implies that
$\partial^{(i)}(\widetilde{\boldsymbol{\delta}})<\infty$, and we also have
$\partial^{(i)}(p^{t}\cdot\widetilde{\boldsymbol{\delta}})=\partial^{(i)}(\widetilde{\boldsymbol{\delta}})+t<\infty$.
Thus,
$\mathrm{ord}(p^{t}\cdot\widetilde{\boldsymbol{\delta}})\leq\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$,
so the equality holds. ∎

######  Proposition 3.15.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ is a prime such
that $\overline{\rho}$ is surjective and the Manin constant is prime to $p$.
Then the non-triviality of $\boldsymbol{\kappa}^{\mathrm{Kato}}$ and the non-
vanishing of $\widetilde{\boldsymbol{\delta}}$ are equivalent.

###### Proof.

Recall the formula

$\xi\circ\mathrm{exp}^{*}_{\omega_{E}}\circ\mathrm{loc}^{s}_{p}(\kappa^{\mathrm{Kato}}_{n})=u\cdot
p^{t}\cdot\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$

where $u\in(\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p})^{\times}$ in Theorem 3.13. The
formula immediately implies one direction
$p^{t}\cdot\widetilde{\boldsymbol{\delta}}\neq
0\Rightarrow\boldsymbol{\kappa}^{\mathrm{Kato}}\neq 0$.

Suppose that $p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ vanishes identically.
Then every $\kappa^{\mathrm{Kato}}_{n}$ lies in
$\mathrm{Sel}_{n}(\mathbb{Q},T/I_{n}T)$ by the formula, so we have

$\boldsymbol{\kappa}^{\mathrm{Kato}}\in\mathbf{KS}(T,\mathcal{F}_{\mathrm{cl}},\mathcal{P})\subseteq\mathbf{KS}(T,\mathcal{F}_{\mathrm{can}},\mathcal{P}),$

but $\mathbf{KS}(T,\mathcal{F}_{\mathrm{cl}},\mathcal{P})=0$ (Theorem 2.4).
The conclusion follows from Lemma 3.14. ∎

Proposition 3.15 does _not_ mean that $\kappa^{\mathrm{Kato}}_{n}\neq 0$ if
and only if $\widetilde{\delta}_{n}\neq 0$ for each $n\in\mathcal{N}_{1}$.

###  3.5. Mazur–Tate elements and functional equations

Following [Kur14b, (31)], the Kurihara number $\widetilde{\delta}_{n}$ also
arises from the mod $I_{n}$ Taylor expansion of Mazur–Tate element

$\theta_{\mathbb{Q}(\zeta_{n})}(E)\equiv\widetilde{\delta}_{n}\cdot\prod_{i=1}^{\nu(n)}(\sigma_{\eta_{\ell_{i}}}-1)\pmod{I_{n},(\sigma_{\eta_{\ell_{1}}}-1)^{2},\cdots,(\sigma_{\eta_{\ell_{\nu(n)}}}-1)^{2}}$

where $\theta_{\mathbb{Q}(\zeta_{n})}(E)$ is the Mazur–Tate element,
$n=\ell_{1}\cdot\cdots\cdot\ell_{\nu(n)}$ and $\sigma_{\eta_{\ell_{i}}}$ is a
generator of $\mathrm{Gal}(\mathbb{Q}(\zeta_{\ell_{i}})/\mathbb{Q})$. Thus,
the functional equation of Mazur–Tate elements [MT87, (1.6.2)] affects the
non-vanishing problem of $\widetilde{\delta}_{n}$. Following [Kur14b, Lemma 4
(Page 347)] and [Kur14a, Page 220], we have

(3.6)
$w(E)\cdot(-1)^{\nu(n)}\cdot\widetilde{\delta}_{n}=\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I_{n}\mathbb{Z}_{p}$

where $w(E)$ is the root number of $E$. Comparing with Proposition 2.2, the
following statement shows the fundamental difference between
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ and $\widetilde{\boldsymbol{\delta}}$.

######  Proposition 3.16.

If $(-1)^{\nu(n)}\neq w(E)$, then $\widetilde{\delta}_{n}=0$. In particular,
if $\widetilde{\delta}_{n}\neq 0$, then
$\widetilde{\delta}_{n\ell}=0\in\mathbb{Z}_{p}/I_{n\ell}\mathbb{Z}_{p}$ for
all $\ell\in\mathcal{N}_{1}$ with $(n,\ell)=1$.

##  4\. $\Lambda$-adic Kolyvagin systems for elliptic curves: Proof of Theorem
1.4

Throughout this section, we assume that $\overline{\rho}$ is surjective.

###  4.1. The Iwasawa-theoretic set up

#### 4.1.1.

Recall the notation in $\S$1.2.3. Let $\mathbb{Q}_{\infty}$ be the cyclotomic
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ and
$\mathbb{Q}_{m}\subseteq\mathbb{Q}_{\infty}$ the cyclic subextension of
$\mathbb{Q}$ of degree $p^{m}$ in $\mathbb{Q}_{\infty}$. Let

$\Lambda=\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\rrbracket=\varprojlim_{m}\mathbb{Z}_{p}[\mathrm{Gal}(\mathbb{Q}_{m}/\mathbb{Q})]$

be the Iwasawa algebra.

#### 4.1.2.

Fix a finite set $\Sigma$ of the places of $\mathbb{Q}$ containing $p$,
$\infty$, and the primes where $T$ is ramified. Denote by
$\mathbb{Q}_{\Sigma}$ the maximal extension of $\mathbb{Q}$ unramified outside
$\Sigma$.

######  Lemma 4.1.

  1. (1)

$\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},T\otimes\Lambda)\simeq\varprojlim_{m}\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q}_{m},T)$.

  2. (2)

If $\ell\neq p$, then
$\mathrm{H}^{1}(\mathbb{Q}_{\ell},T\otimes\Lambda)=\mathrm{H}^{1}_{\mathrm{ur}}(\mathbb{Q}_{\ell},T\otimes\Lambda)$.

  3. (3)

$\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},T\otimes\Lambda)$ is
independent of the choice of $\Sigma$.

###### Proof.

See [MR04, Lemma 5.3.1]. ∎

######  Remark 4.2.

Lemma 4.1.(2) indicates that Tamagawa factors cannot be observed over the
cyclotomic tower.

The Selmer structure $\mathcal{F}_{\Lambda}$ on $T\otimes\Lambda$ is defined
by

$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(\mathbb{Q}_{v},T\otimes\Lambda)=\mathrm{H}^{1}(\mathbb{Q}_{v},T\otimes\Lambda)$

for all $v\in\Sigma$. Also, by Lemma 4.1.(2), we have
$\mathrm{H}^{1}_{f}(\mathbb{Q}_{v},T\otimes\Lambda)=\mathrm{H}^{1}(\mathbb{Q}_{v},T\otimes\Lambda)$
for all $v\not\in\Sigma$. Thus, this Selmer structure is independent of the
choice of $\Sigma$, and

$\mathrm{Sel}_{\mathcal{F}_{\Lambda}}(\mathbb{Q},T\otimes\Lambda)=\mathrm{H}^{1}(\mathbb{Q},T\otimes\Lambda).$

######  Theorem 4.3 (Mazur–Rubin).

Suppose that $\mathcal{K}$ contains the maximal abelian $p$-extension of
$\mathbb{Q}$ which is unramified outside $p$ and $\mathcal{P}$, and

  1. (1)

$T/(\mathrm{Fr}_{\ell}-1)T$ is a cyclic $\mathbb{Z}_{p}$-module for every
$\ell\in\mathcal{P}$,

  2. (2)

$\mathrm{Fr}^{p^{k}}_{\ell}-1$ is injective on $T$ for every
$\ell\in\mathcal{P}$ and every $k\geq 0$.

Then there exists a canonical homomorphism

$\textstyle{\mathbf{ES}(T,\mathcal{K},\mathcal{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathbf{KS}}(T\otimes\Lambda,\mathcal{F}_{\Lambda},\mathcal{P})}$$\textstyle{\mathbf{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\boldsymbol{\kappa}^{\infty}}$

such that

$\kappa^{\infty}_{1}=z_{\mathbb{Q}_{\infty}}=\varprojlim_{m}z_{\mathbb{Q}_{m}}\in\mathrm{H}^{1}(\mathbb{Q},T\otimes\Lambda).$

###### Proof.

See [MR04, Theorem 5.3.3]. ∎

Under our working hypotheses, all the assumptions in Theorem 4.3 are
satisfied.

#### 4.1.3.

######  Lemma 4.4.

For every $i\geq 0$,
$\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},T\otimes\Lambda)$ and
$\mathrm{H}^{i}(\mathbb{Q}_{p},T\otimes\Lambda)$ are finitely generated, and
$\mathrm{H}^{1}(\mathbb{Q}_{\Sigma}/\mathbb{Q},(T\otimes\Lambda)^{*})$ is co-
finitely generated. Furthermore,
$\mathrm{H}^{2}(\mathbb{Q}_{p},T\otimes\Lambda)$ is a torsion
$\Lambda$-module.

###### Proof.

See [MR04, Lemma 5.3.4]. ∎

######  Theorem 4.5 (Kato–Rohrlich).

Kato’s zeta element
$\kappa^{\mathrm{Kato},\infty}_{1}=z^{\mathrm{Kato}}_{\mathbb{Q}_{\infty}}$
over $\mathbb{Q}_{\infty}$ is non-trivial.

###### Proof.

By using Kato’s explicit reciprocity law [Kat04, Theorem 12.5], the conclusion
follows from the generic non-vanishing of twisted $L$-values [Roh84, Roh88]. ∎

######  Theorem 4.6 (Kato).

  1. (1)

If $\overline{\rho}$ is irreducible, then
$\mathrm{Sel}_{\mathcal{F}_{\Lambda}}(\mathbb{Q},T\otimes\Lambda)=\mathrm{H}^{1}(\mathbb{Q},T\otimes\Lambda)$
is free of rank one over $\Lambda$.

  2. (2)

$\mathrm{Sel}_{\mathcal{F}^{*}_{\Lambda}}(\mathbb{Q},(T\otimes\Lambda)^{*})=\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])$
is a co-finitely generated co-torsion $\Lambda$-module.

###### Proof.

See [Kat04, Theorem 12.4]. See also [MR04, Lemma 5.3.5 and Theorem 5.3.6]. ∎

We recall the notion in $\S$1.3.1. We say that $\boldsymbol{\kappa}^{\infty}$
is $\Lambda$-primitive if $\boldsymbol{\kappa}^{\infty}\pmod{\mathfrak{P}}$
does not vanish for every height one prime $\mathfrak{P}$ of $\Lambda$ [MR04,
Definition 5.3.9].

######  Theorem 4.7 (Mazur–Rubin).

Let $\boldsymbol{\kappa}^{\infty}\in\mathbf{KS}(T\otimes\Lambda)$ be a
$\Lambda$-adic Kolyvagin system.

  1. (1)

$\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda\kappa^{\infty}_{1}}\right)\subseteq\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right).$

  2. (2)

Suppose that $\kappa^{\infty}_{1}\neq 0$. If $\mathfrak{P}$ is not a blind
spot of $\boldsymbol{\kappa}^{\infty}$, then

$\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda\kappa^{\infty}_{1}}\right)\right)=\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right)\right).$

  3. (3)

Suppose that $\kappa^{\infty}_{1}\neq 0$. If $\boldsymbol{\kappa}$ is
$\Lambda$-primitive, then

$\mathrm{char}_{\Lambda}\left(\dfrac{\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)}{\Lambda\kappa^{\infty}_{1}}\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])^{\vee}\right).$

###### Proof.

See [MR04, Theorem 5.3.10]. ∎

Theorem 1.4 proves the converse to the second and the third statements of
Theorem 4.7. Thanks to [MR04, Remark 5.3.11], both
$\mathbf{KS}(T\otimes\Lambda)$ and $\overline{\mathbf{KS}}(T\otimes\Lambda)$
work for all the argument below.

###  4.2. Specializations at height one primes

Let $\mathfrak{P}$ be a height one prime ideal of $\Lambda$. Denote by
$S_{\mathfrak{P}}$ the integral closure of $\Lambda/\mathfrak{P}$, which is a
discrete valuation ring. Then $[S_{\mathfrak{P}}:\mathbb{Z}_{p}]$ is finite
and
$T\otimes\Lambda\otimes_{\Lambda}S_{\mathfrak{P}}=T\otimes_{\mathbb{Z}_{p}}S_{\mathfrak{P}}$.
The canonical Selmer structure on $T\otimes_{\mathbb{Z}_{p}}S_{\mathfrak{P}}$
is induced from the $\Lambda$-adic Selmer structure on $T\otimes\Lambda$. Let

$\Sigma_{\Lambda}=\left\\{\mathfrak{P}:\mathrm{H}^{2}(\mathbb{Q}_{\Sigma}/\mathbb{Q},T\otimes\Lambda)[\mathfrak{P}]\textrm{
is
infinite}\right\\}\cup\left\\{\mathfrak{P}:\mathrm{H}^{2}(\mathbb{Q}_{p},T\otimes\Lambda)[\mathfrak{P}]\textrm{
is infinite}\right\\}\cup\left\\{p\Lambda\right\\}$

be the exceptional set of height one primes of $\Sigma_{\Lambda}$ and it is
finite by Lemma 4.4.

######  Proposition 4.8.

For every height one prime $\mathfrak{P}$ of $\Lambda$, the composition map
$T\otimes\Lambda\twoheadrightarrow T\otimes\Lambda/\mathfrak{P}\hookrightarrow
T\otimes S_{\mathfrak{P}}$ induces maps