# Non-vanishing of Kolyvagin systems and Iwasawa theory

Ashay Burungale  University of Texas at Austin, USA<EMAIL_ADDRESS>,
Francesc Castella  University of California Santa Barbara, South Hall, Santa
Barbara, CA 93106, USA<EMAIL_ADDRESS>,  Giada Grossi  CNRS, Institut
Galilée, Université Sorbonne Paris Nord, 93430 Villetaneuse, FRANCE
<EMAIL_ADDRESS>and  Christopher Skinner  Princeton University,
Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991
Kolyvagin conjectured that the system of cohomology classes for the $p$-adic
Tate module of $E$ derived from Heegner points on $E$ over ring class fields
of an imaginary quadratic field $K$ is non-trivial. In this paper we prove
Kolyvagin’s conjecture in the cases where $p$ is a prime of good ordinary
reduction for $E$ that splits in $K$ and a $p$-adic anticyclotomic Iwasawa
Main Conjecture for $E/K$ holds. In particular, our results cover many cases
where $p$ is an Eisenstein prime, complementing W. Zhang’s progress towards
the conjecture.

Our methods also yield a proof of a refinement of Kolyvagin’s conjecture
expressing the divisibility index of the $p$-adic Heegner point Kolyvagin
system in terms of the Tamagawa numbers of $E$, as conjectured by W. Zhang in
2014, as well as proofs of analogous results for the Kolyvagin systems
obtained from the cyclotomic Euler system of Beilinson–Kato elements.

###### Contents

  1. 0.1 Main results
  2. 0.2 Strategy
  3. 0.3 Relation to previous works
  4. 1 Heegner points and anticyclotomic Iwasawa theory
    1. 1.1 The Kolyvagin system of Heegner points
    2. 1.2 Anticyclotomic Iwasawa Main Conjecture
    3. 1.3 The Kolyvagin system bound with error term
  5. 2 Proofs of Theorem A and Theorem B
    1. 2.1 Kolyvagin’s conjecture
    2. 2.2 The refined Kolyvagin’s conjecture
  6. 3 The cyclotomic analogue: the non-triviality of Kato’s Kolyvagin system
    1. 3.1 The Kolyvagin system of Kato classes
    2. 3.2 Cyclotomic Main Conjecture and results
    3. 3.3 Nonvanishing of Kato’s Kolyvagin system
    4. 3.4 Extension of Mazur–Rubin’s Selmer group bound

## Introduction

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, and let $p$ be an
odd prime of good ordinary reduction for $E$. In this paper we prove
Kolyvagin’s conjecture on the non-vanishing of the $p$-adic Heegner point
Kolyvagin system under mild conditions (cf. Theorem A). When $p$ is non-
Eisenstein for $E$, the conjecture was first proved by W. Zhang in many cases.
The approach introduced in this paper covers the general non-Eisenstein case,
as well as the first general cases where $p$ is Eisenstein for $E$. Moreover,
in the former case we prove the _refined_ Kolyvagin conjecture due to W.
Zhang, expressing the divisibility index of the Heegner point Kolyvagin system
in terms of the Tamagawa numbers of $E$ (cf. Theorem B). Following a similar
strategy, we also prove analogous results for the Kolyvagin system derived
from Beilinson–Kato elements (cf. Theorem C).

###  0.1. Main results

#### Kolyvagin’s conjecture

Let $K$ be an imaginary quadratic field of discriminant $-D_{K}<0$ such that

(Heeg) every prime $\ell|N$ splits in $K$,

and fix an integral ideal $\mathfrak{N}\subset\mathcal{O}_{K}$ with
$\mathcal{O}_{K}/\mathfrak{N}=\mathbb{Z}/N\mathbb{Z}$. Assume also that

(disc) $D_{K}$ is odd and $D_{K}\neq-3$.

For each positive integer $m$ prime to $N$, let
$\mathcal{O}_{m}=\mathbb{Z}+m\mathcal{O}_{K}$ be the order of $K$ of conductor
$m$ and denote by $K[m]$ the corresponding ring class field extension of $K$.

By the theory of complex multiplication, the cyclic $N$-isogeny between
complex CM elliptic curves

$\mathbb{C}/\mathcal{O}_{m}\rightarrow\mathbb{C}/(\mathfrak{N}\cap\mathcal{O}_{m})^{-1}$

defines a point $x_{m}\in X_{0}(N)(K[m])$. Fix a modular parameterisation

(0.1) $\pi:X_{0}(N)\rightarrow E.$

The _Heegner point_ of conductor $m$ is defined by

(0.2) $P[m]:=\pi(x_{m})\in E(K[m]).$

We call $\ell$ a _Kolyvagin prime_ if $\ell$ is inert in $K$, coprime to $Np$,
and

$M(\ell):=\min\\{\operatorname{ord}_{p}(\ell+1),\operatorname{ord}_{p}(a_{\ell})\\}>0,$

where $a_{\ell}:=\ell+1-|\tilde{E}(\mathbb{F}_{\ell})|$ and
$\operatorname{ord}_{p}(x)$ denotes the $p$-adic valuation of an integer $x$.
Let $\mathcal{N}_{\rm Heeg}$ be the set of squarefree products of Kolyvagin
primes, and for $n\in\mathcal{N}_{\rm Heeg}$ put

$M(n):=\min\\{M(\ell)\;\colon\;\ell\mid n\\}$

if $n>1$ and $M(1):=\infty$. Let $T=T_{p}E$ be the $p$-adic Tate module of
$E$, and suppose that

(tor) $E(K)[p]=0.$

From the Kummer images of the Heegner points $P[n]$, Kolyvagin constructed a
system of classes

$\bigl{\\{}\kappa_{n}^{\rm
Heeg}\in\mathrm{H}^{1}(K,T/I_{n}T)\;\colon\;n\in\mathcal{N}_{\rm
Heeg}\bigr{\\}},\quad\textrm{where $I_{n}=p^{M(n)}\mathbb{Z}_{p}$.}$

In particular, $\kappa_{1}^{\rm Heeg}$ is the image of the Heegner point
$P_{K}:=\operatorname{Tr}_{K[1]/K}(P[1])\in E(K)$ under the Kummer map

$E(K)\otimes\mathbb{Z}_{p}\rightarrow\mathrm{H}^{1}(K,T),$

and so by the Gross–Zagier formula [GZ86], $\kappa_{1}^{\rm Heeg}\neq 0$ if
and only if $L^{\prime}(E/K,1)\neq 0$.

In [Kol91] Kolyvagin conjectured that even when the analytic rank of $E/K$ is
greater than one, the _system_ $\\{\kappa_{n}^{\rm Heeg}\\}_{n}$ is non-
trivial, i.e. there exists $n\in\mathcal{N}_{\rm Heeg}$ such that

$\kappa_{n}^{\rm Heeg}\neq 0.$

The first major progress towards this conjecture was due to W. Zhang [Zha14b]:
about a decade ago, he proved Kolyvagin’s conjecture when $p\nmid 6N$ is a
prime of good ordinary reduction for $E$ under the assumption that

(sur)
$\bar{\rho}_{E}:G_{\mathbb{Q}}=\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to{\rm
Aut}_{\mathbb{F}_{p}}(E[p])$ is surjective

and $\bar{\rho}_{E}$ satisfies certain ramification hypotheses. More recently,
some of the hypotheses have been relaxed by N. Sweeting [Swe20] using an
ultrapatching method for bipartite Euler systems.

In the first part of this paper we prove Kolyvagin’s conjecture in cases where
(sur) is not necessarily satisfied, and we also relax the ramification
hypotheses. The paper considers Eisenstein primes $p$ as well as cases where

(irr) $E[p]$ is an irreducible $G_{\mathbb{Q}}$-module.

Our method relies on anticyclotomic Iwasawa theory, recent developments
towards anticyclotomic main conjectures, and ideas introduced in [CGS23]. A
key ingredient is a Kolyvagin system bound with “error terms” obtained in _op.
cit._ allowing us to control the size of the Tate–Shafarevich group of certain
anticyclotomic twists of $T$ by characters $\alpha$ with $\alpha\equiv
1\pmod{p^{m}}$ for $m\gg 0$. When (sur) is satisfied, the error terms are zero
and when a certain Iwasawa Main Conjecture is known, our methods also yield a
proof of the _refined Kolvyagin conjecture_ due to W. Zhang [Zha14a, Conj.
4.5].

We now describe the main result precisely. Let $G_{\mathbb{Q}}$ be the
absolute Galois group of $\mathbb{Q}$ and $G_{p}\subset G_{\mathbb{Q}}$ a
decomposition group at $p$. Denote by
$\omega:G_{\mathbb{Q}}\rightarrow\mathbb{F}_{p}^{\times}$ the Teichmüller
character.

######  Theorem A (Kolyvagin’s conjecture).

Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of good
ordinary reduction for $E$. Let $K$ be a quadratic imaginary field satisfying
(Heeg), (disc), (tor), and such that $p$ splits in $K$. Assume that the
rational anticyclotomic Main Conjecture 1.2.10 holds. Then

there exists $n\in\mathcal{N}_{\rm Heeg}$ such that $\kappa_{n}^{\rm Heeg}\neq
0$.

In particular, $\\{\kappa_{n}^{\rm Heeg}\\}\neq 0$ in both of the following
cases:

  * $\circ$

$E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_{p}(\phi)\subset
E[p]$, where $\phi:G_{\mathbb{Q}}\to\mathbb{F}_{p}^{\times}$ is a character
such that $\phi|_{G_{p}}\neq\mathds{1},\omega$.

  * $\circ$

$p>3$ satisfies (irr).

The ‘In particular’ part of Theorem A relies on the results of [CGS23, BCS23],
respectively, on the anticyclotomic Main Conjecture.

The non-vanishing of the Kolyvagin system in combination with [Kol91, Theorem
4] leads to a link between the _order of vanishing_ of the Kolyvagin system
and the rank of the $p^{\infty}$-Selmer group of $E/K$. More precisely, for
any $n$, let $\nu(n)$ denote the number of primes factors of $n$, and for
$n\in\mathcal{N}_{\rm Heeg}$ let

$\operatorname{ord}(\kappa^{\rm Heeg}):=\min\\{r:\text{ there exists
}n\in\mathcal{N}_{\rm Heeg}\text{ with }\nu(n)=r\text{ such that
}\kappa_{n}^{\rm Heeg}\neq 0\\}.$

The $p^{\infty}$-Selmer group ${\rm Sel}_{p^{\infty}}(E/K)$ has an action of
complex conjugation. Denote by ${\rm Sel}_{p^{\infty}}(E/K)^{\pm}$ the
$\pm$-eigenspaces with respect to this action and let

$r(E/K)^{\pm}={\rm corank}_{\mathbb{Z}_{p}}{\rm Sel}_{p^{\infty}}(E/K)^{\pm}.$

######  Corollary A.

For $E$, $p$ and $K$ as in Theorem A, we have

$\operatorname{ord}(\kappa^{\rm Heeg})=\max\\{r(E/K)^{+},r(E/K)^{-}\\}-1.$

######  Remark.

The corollary applies for arbitrary values of $r(E/K)^{\pm}$. In the rank one
case it yields a $p$-converse to the Gross–Zagier and Kolyvagin theorem:

${\rm corank}_{\mathbb{Z}_{p}}{\rm
Sel}_{p^{\infty}}(E/K)=1\implies\operatorname{ord}_{s=1}L(E/K,s)=1$

as the Birch and Swinnerton-Dyer conjecture predicts (cf. [Zha14b, Theorem
1.3]). One can also deduce the $p$-parity conjecture for $E/\mathbb{Q}$ (cf.
[Zha14b, Theorem 1.2, Remark 1]).

#### Refined Kolyvagin’s conjecture

As formulated by W. Zhang [Zha14b], a natural refinement of Kolyvagin’s
conjecture predicts a formula for the divisibility index of the Heegner point
Kolyvagin system $\\{\kappa_{n}^{\rm Heeg}\\}$ in terms of the Tamagawa
numbers

$c_{\ell}=[E(\mathbb{Q}_{\ell}):E^{0}(\mathbb{Q}_{\ell})]$

at the primes $\ell|N$.

For each $n\in\mathcal{N}_{\rm Heeg}$, define
$\mathscr{M}(n)\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$ by
$\mathscr{M}(n)=\infty$ if $\kappa_{n}^{\rm Heeg}=0$, and by

$\mathscr{M}(n)=\max\\{\mathscr{M}\colon\kappa_{n}^{\rm Heeg}\in
p^{\mathscr{M}}\mathrm{H}^{1}(K,T/I_{n}T)\\}$

otherwise. Put $\mathscr{M}_{r}=\min\\{\mathscr{M}(n)\colon\nu(n)=r\\}$. We
have $\mathscr{M}_{r}\geq\mathscr{M}_{r+1}\geq 0$ for all $r\geq 0$ (cf.
[Kol91]). Put

$\mathscr{M}_{\infty}=\lim_{r\to\infty}\mathscr{M}_{r}.$

Note that Kolyvagin’s conjecture is equivalent to the finiteness of
$\mathscr{M}_{\infty}$. W. Zhang’s refinement [Zha14b, Conj. 4.5] provides a
conjectural formula for $\mathscr{M}_{\infty}$, which we establish:

######  Theorem B (Refined Kolyvagin’s conjecture).

Let $(E,p,K)$ be as in Theorem A, and assume that $p>3$ and (sur) holds.
Assume also that the fixed modular parametrisation $\pi:X_{0}(N)\to E$ as in
(0.1) is $p$-optimal111That is $\operatorname{ord}_{p}(\deg(\pi))$ is minimal
among all modular parametrisations of all curves in the $\mathbb{Q}$-isogeny
class of $E$. . Then we have

$\mathscr{M}_{\infty}=\sum_{\ell\mid N}{\rm ord}_{p}(c_{\ell}).$

######  Remark.

In the rank one case Theorem B implies the $p$-part of the conjectural Birch
and Swinnerton-Dyer formula for $E/K$ (cf. [Zha14b, Theorem 1.6]).

Our proof relies on the cases of the anticyclotomic Main Conjecture 1.2.10
established in [BCS23].

In [Jet08] Jetchev showed that $\mathscr{M}_{\infty}\geq{\rm
ord}_{p}(c_{\ell})$ for any prime $\ell|N$. The first cases of the refined
Kolyvagin’s conjecture were proved by W. Zhang: the ramification hypotheses in
[Zha14b] imply that $p\nmid c_{\ell}$ for all primes $\ell|N$, and the main
result of _op. cit._ shows that $\mathscr{M}_{\infty}=0$.

#### Non-vanishing of Kato’s Kolyvagin system

The strategy introduced in this paper for the proofs of Theorem A and Theorem
B can be adapted to establish analogous results for the Kolyvagin system
derived from Beilinson–Kato elements. The second part of the paper is devoted
to the proof of these results.

In [Kat04] Kato constructed an Euler system for $T$ (viewed as a
$G_{\mathbb{Q}}$-representation) building on Siegel units on the modular curve
of level $N$. Let $\mathcal{L}_{\rm Kato}$ be the set of primes $\ell\nmid Np$
such that

$I_{\ell}:=(\ell-1,a_{\ell}-\ell-1)\subset p^{k}\mathbb{Z}_{p}$

for some $k>0$ (in other words, $\ell\equiv 1\;({\rm mod}\,p^{k})$ and
$a_{\ell}\equiv\ell+1\;({\rm mod}\,p^{k})$), and let $\mathcal{N}_{\rm Kato}$
be the set of squarefree products of primes in $\mathcal{L}_{\rm Kato}$. For
$n\in\mathcal{N}_{\rm Kato}$, let

$I_{n}:=\sum_{\ell\mid n}I_{\ell}$

for $n>1$, and put $I_{1}:=\\{0\\}$. By the process of Kolyvagin derivatives,
from Kato’s Euler system one obtains a system of cohomology classes

$\bigl{\\{}\kappa^{\rm
Kato}_{n}\in\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)\;\colon\;n\in\mathcal{N}_{\rm
Kato}\bigr{\\}}$

forming a Kolyvagin system for $T$ in the sense of Mazur–Rubin (see [MR04,
Thm. 3.2.4]).

In this setting our main result is the following.

######  Theorem C (Non-vanishing of Kato’s Kolyvagin system).

Let $E/\mathbb{Q}$ be an elliptic curve without CM, and let $p$ be an odd
prime of good ordinary reduction for $E$ such that $E(\mathbb{Q})[p]=0$.
Assume that the rational cyclotomic Main Conjecture 3.2.4 holds. Then

there exists $n\in\mathcal{N}_{\rm Kato}$ such that $\kappa_{n}^{\rm Kato}\neq
0$.

In particular, $\\{\kappa_{n}^{\rm Kato}\\}\neq 0$ in both of the following
cases:

  * $\circ$

$E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_{p}(\phi)\subset
E[p]$, where $\phi:G_{\mathbb{Q}}\to\mathbb{F}_{p}^{\times}$ is a character
such that $\phi|_{G_{p}}\neq\mathds{1},\omega$.

  * $\circ$

$p>3$ satisfies (irr).

The ‘In particular’ part relies on the cases of Conjecture 3.2.4 established
in [Kat04, CGS23] for Eisenstein $p$ and in [Kat04, SU14, Wan15] for non-
Eisenstein $p$.

We now describe an application of Theorem C. Similarly as in the Heegner point
case, for any $n\in\mathcal{N}_{\rm Kato}$, define the order of vanishing of
the Kolyvagin system $\kappa^{\rm Kato}$ by

$\operatorname{ord}(\kappa^{\rm Kato}):=\min\\{r:\text{ there exists
}n\in\mathcal{N}_{\rm Kato}\text{ with }\nu(n)=r\text{ such that
}\kappa_{n}^{\rm Kato}\neq 0\\}.$

Let $\mathrm{H}^{1}_{\rm str}(\mathbb{Q},E[p^{\infty}])$ denote the Selmer
group for $E[p^{\infty}]$ obtained by imposing the strict local condition at
$p$ (see $\S$3.1). Let

$r_{\rm str}(E/\mathbb{Q})={\rm
corank}_{\mathbb{Z}_{p}}\left(\mathrm{H}^{1}_{\rm
str}(\mathbb{Q},E[p^{\infty}])\right).$

Combining the theorem above with [MR04, Theorem 5.2.12(v)], we obtain an
analogue of Corollary A for Kato’s Kolyvagin system.

######  Corollary C.

For $E$ and $p$ as in Theorem C, we have

$\operatorname{ord}(\kappa^{\rm Kato})=r_{\rm str}(E/\mathbb{Q}).$

The divisibility index $\mathscr{M}_{\infty}^{\rm Kato}$ of
$\\{\kappa_{n}^{\rm Kato}\\}$ was studied by Mazur and Rubin [MR04, §6.2], who
showed that if ${\rm ord}_{p}(c_{\ell})>0$ for some prime $\ell|N$, then
$\mathscr{M}_{\infty}^{\rm Kato}>0$. This was refined by Büyükboduk [Buy09],
who showed that $\mathscr{M}_{\infty}^{\rm Kato}\geq{\rm ord}_{p}(c_{\ell})$
for any prime $\ell|N$. Our approach to Theorem C also leads to a proof of the
conjecture [Kim22b, Conj. 1.10] of Kim:

$\mathscr{M}_{\infty}^{\rm Kato}=\sum_{\ell\mid N}{\rm ord}_{p}(c_{\ell})$

under some hypotheses (see Remark 3.3.4; these also include that $\pi$ is
$p$-optimal).

###  0.2. Strategy

Let $\Gamma={\rm Gal}(K_{\infty}/K)$ be the Galois group of the anticyclotomic
$\mathbb{Z}_{p}$-extension of $K$, and denote by
$\Lambda=\mathbb{Z}_{p}[[\Gamma]]$ the anticyclotomic Iwasawa algebra. The key
ingredients in the proof of Theorem A are:

  1. (a)

The non-vanishing of the base class $\bm{\kappa}_{1}^{\rm Heeg}$ of the
_$\Lambda$ -adic_ Heegner point Kolyvagin system (i.e., the proof by Cornut
and Vastal of Mazur’s conjecture).

  2. (b)

Letting $\bm{\kappa}_{1}^{\rm Heeg}(\alpha)$ be the specialisation of
$\bm{\kappa}_{1}^{\rm Heeg}$ at a character $\alpha$ of $\Gamma$ with
$\alpha\equiv 1\;({\rm mod}\,p^{m})$ for suitable $m\gg 0$ such that

(0.3) $\bm{\kappa}_{1}^{\rm Heeg}(\alpha)\neq 0,$

an estimate on the divisibility index of $\bm{\kappa}_{1}^{\rm Heeg}(\alpha)$
in terms of Tamagawa numbers and the Shafarevich–Tate group of the twist of
$E/K$ by $\alpha$.

  3. (c)

The Kolyvagin system bound with controlled error terms obtained in [CGS23].

Most of the work in the first part of the paper goes into the proof of (b),
which we deduce from the explicit reciprocity law222This is a $\Lambda$-adic
avatar of the Bertolini–Darmon–Prasanna formula [BDP13]. for
$\bm{\kappa}_{1}^{\rm Heeg}$ in [CH18] and an extension of the anticyclotomic
control theorem in [JSW17] allowing character twists. Together with a
congruence relation between $\bm{\kappa}_{n}^{\rm Heeg}(\alpha)$ and
$\kappa_{n}^{\rm Heeg}$, the results (a)–(c) lead to a proof of Theorem A.

To approach the refined form of Kolyvagin’s conjecture as in Theorem B, we
replace the Kolyvagin system bound in (c) with an _exact formula_ for the size
of the above Tate–Shafarevich group in terms of the divisibility index of
$\bm{\kappa}_{1}^{\rm Heeg}(\alpha)$ and the divisibility of the system
$\\{\bm{\kappa}_{n}^{\rm Heeg}(\alpha)\\}_{n}$, generalising a consequence of
the structure theorem [Kol91, Thm. 1] in the case $\alpha=1$. We emphasize
that our result applies to the twists of $E$ by any anticyclotomic character
$\alpha$ not necessarily $p$-adically close to the trivial one. Relating the
index of divisibility of $\\{\bm{\kappa}_{n}^{\rm Heeg}(\alpha)\\}_{n}$ to
that of $\\{\kappa_{n}^{\rm Heeg}\\}_{n}$ gives Theorem B.

The proof of Theorem C proceeds along similar lines. The analogue of (a)
follows from results of Rohrlich and Kato, and we deduce the analogue of (b)
from Kato’s explicit reciprocity law and a twisted variant of the Euler
characteristic computation originally due to Schneider and Perrin-Riou. The
analogue of (c) in the setting of Kato’s Euler system requires more work; we
deduce it from the results developed in the last section of the paper that
extend the Kolyvagin system bound of Mazur and Rubin [MR04] to a more general
setting, allowing in particular for Eisenstein primes, a result that may be of
independent interest.

Finally, we note that the links established in this paper between non-
triviality of a Kolyvagin system (horizontal variation), and Iwasawa theory
(vertical variation) of elliptic curves seem amenable to generalisations,
which we plan to explore in the near future.

###  0.3. Relation to previous works

#### Comparison with [Zha14b, Swe20].

The approach used by W. Zhang in his breakthrough work on Kolyvagin’s
conjecture is based on the principle of level-raising and rank-lowering, where
rank refers to that of an associated mod $p$ Selmer group; this was extended
by N. Sweeting to mod $p^{m}$ Selmer groups for large $m$. The approach
proceeds by induction on the rank, the base case being: the triviality of the
mod $p$ Selmer group associated to a weight two elliptic newform implies the
$p$-indivisibility of the algebraic part of the associated central $L$-value.
This rank zero implication is a consequence of the (integral) cyclotomic Main
Conjecture for the newform, as established by Skinner and Urban [SU14] under
the hypotheses of [Zha14b] (by working mod $p^{m}$, some of the ramification
hypotheses were removed in [Swe20]). It is precisely at this stage that
Iwasawa theory enters into the approach of W. Zhang, albeit only implicitly.
In contrast, the strategy in this paper is inherently Iwasawa-theoretic, based
on the variation of Heegner points along the anticyclotomic
$\mathbb{Z}_{p}$-extension. Moreover, a rational Main Conjecture suffices for
its implementation. The approach of [Zha14b, Swe20] essentially excludes the
Eisenstein case. However, while the results of this paper on Kolyvagin’s
conjecture are for $p$ split in $K$, the results of [Zha14b, Swe20] apply also
when $p$ is inert in $K$.

#### Comparison with [Kim22a, Kim22b]

In the cases where (sur) holds, results similar to Theorems A and C are
obtained by C.-H. Kim [Kim22a, Kim22b]: he proves that if the anticyclotomic
(resp. cyclotomic) Main Conjecture holds at the trivial character, then the
system $\\{\kappa_{n}^{\rm Heeg}\\}$ (resp. $\\{\kappa_{n}^{\rm Kato}\\}$) is
non-trivial. To draw a parallel with our approach, while the ingredient (a) is
common, his proof interestingly replaces our ingredients (b) and (c) with a
structure theorem for Selmer groups in terms of divisibility indices of
certain specialisations of $\Lambda$-adic classes. In contrast, under (sur)
our approach also leads to a proof of the refined Kolyvagin conjecture and its
cyclotomic analogue.

### Acknowledgements

We thank Kâzim Büyükboduk, Masataka Chida, Christophe Cornut, Shinichi
Kobayashi and Wei Zhang for helpful discussions. We also thank Chan-Ho Kim for
his interest in our work, and his comments on an earlier draft. During the
preparation of this paper, A.B. was partially supported by the NSF grants
DMS-2303864 and DMS-2302064; F.C. was partially supported by the NSF grant
DMS-2101458; C.S. was partially supported by the Simons Investigator Grant
#376203 from the Simons Foundation and by the NSF grant DMS-1901985. This work
was partially supported by the National Science Foundation under Grant No.
DMS-1928930 while the authors were in residence at the Simons Laufer
Mathematical Sciences Institute in Berkeley, California during the Spring 2023
semester.

##  1\. Heegner points and anticyclotomic Iwasawa theory

###  1.1. The Kolyvagin system of Heegner points

In this section we recall the construction of the classes
$\kappa_{n}=\kappa_{n}^{\rm Heeg}\in\mathrm{H}^{1}(K,T/I_{n}T)$ and of their
Iwasawa-theoretic analogues. See [How04a, $\S$1.7,$\S$2.3] for more details.

####  1.1.1. Selmer structures

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, let $p\nmid 2N$ be a
prime of good ordinary reduction for $E$, and let $K$ be an imaginary
quadratic field of discriminant $D_{K}$ prime to $Np$. We assume that the
triple $(E,p,K)$ satisfies hypotheses (Heeg), (disc), and (tor) from the
Introduction.

Let $\Gamma={\rm Gal}(K_{\infty}/K)$ be the Galois group of the anticyclotomic
$\mathbb{Z}_{p}$-extension of $K$ and
$\alpha:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ a character. Let
$\rho_{E}:G_{\mathbb{Q}}\rightarrow{\rm Aut}_{\mathbb{Z}_{p}}(T_{p}E)$ give
the action of $G_{\mathbb{Q}}$ on the $p$-adic Tate module of $E$ and consider
the $G_{K}$-modules

$T_{\alpha}:=T_{p}E\otimes\mathbb{Z}_{p}(\alpha),\quad
V_{\alpha}:=T_{\alpha}\otimes\mathbb{Q}_{p},\quad
W_{\alpha}:=T_{\alpha}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\simeq
V_{\alpha}/T_{\alpha},$

where $\mathbb{Z}_{p}(\alpha)$ is the free $\mathbb{Z}_{p}$-module of rank one
on which $G_{K}$ acts via the projection $G_{K}\twoheadrightarrow\Gamma$
composed with $\alpha$, and the $G_{K}$-action on $T_{\alpha}$ is via
$\rho_{\alpha}=\rho_{E}\otimes\alpha$. For $k>0$ let
$T_{\alpha}^{(k)}=T_{\alpha}/p^{k}T_{\alpha}$; this is identified with
$W_{\alpha}[p^{k}]=E[p^{k}]\otimes\mathbb{Z}_{p}(\alpha)$ via $T_{\alpha}\ni
a\mapsto a\otimes\frac{1}{p^{k}}\in W_{\alpha}$.

A _Selmer structure_ $\mathcal{F}$ on any of the $G_{K}$-modules
$M=T_{\alpha},V_{\alpha},W_{\alpha},T_{\alpha}^{(k)}$ is a finite set
$\Sigma=\Sigma(\mathcal{F})$ of places of $K$ containing $\infty$, the primes
above $p$, and the primes where $T_{\alpha}$ is ramified, together with a
choice of $\mathbb{Z}_{p}$-submodules (“local conditions”)
$\mathrm{H}^{1}_{\mathcal{F}}(K_{w},M)\subset\mathrm{H}^{1}(K_{w},M)$ for
every $w\in\Sigma$. The associated _Selmer group_ is then defined by

$\mathrm{H}^{1}_{\mathcal{F}}(K,M):={\rm
ker}\biggl{\\{}\mathrm{H}^{1}(K^{\Sigma}/K,M)\rightarrow\prod_{w\in\Sigma}\frac{\mathrm{H}^{1}(K_{w},M)}{\mathrm{H}^{1}_{\mathcal{F}}(K_{w},M)}\biggr{\\}},$

where $K^{\Sigma}$ is the maximal extension of $K$ unramified outside
$\Sigma$.

We recall the definition of the local conditions of interest in this section:

  * •

For a finite prime $w\nmid p$, the _finite_ (or _unramified_) local condition
for $V_{\alpha}$ is

$\mathrm{H}^{1}_{f}(K_{w},V_{\alpha}):={\rm
ker}\bigl{\\{}\mathrm{H}^{1}(K_{w},V_{\alpha})\rightarrow\mathrm{H}^{1}(K_{w}^{\rm
ur},V_{\alpha})\bigr{\\}};$

the corresponding local conditions $\mathrm{H}^{1}_{f}(K_{w},T_{\alpha})$ and
$\mathrm{H}^{1}_{f}(K_{w},W_{\alpha})$ are defined to be the inverse image and
the image, respectively, of $\mathrm{H}^{1}_{f}(K_{w},V_{\alpha})$ under the
natural maps

(1.1)
$\mathrm{H}^{1}(K_{w},T_{\alpha})\rightarrow\mathrm{H}^{1}(K_{w},V_{\alpha})\rightarrow\mathrm{H}^{1}(K_{w},W_{\alpha}).$

Similarly, the local condition $\mathrm{H}^{1}_{f}(K_{w},T_{\alpha}^{(k)})$ is
the preimage of $\mathrm{H}^{1}_{f}(K_{w},W_{\alpha})$ under the natural map
$\mathrm{H}^{1}(K_{w},T_{\alpha}^{(k)})\twoheadrightarrow\mathrm{H}^{1}(K_{w},W_{\alpha})[p^{k}]$.
If $T_{\alpha}$ is not ramified at $w$, then all these are just the submodules
of unramified classes (see also Remark 1.1.1 below).

  * •

Denote by $\mathcal{L}_{0}$ the set of rational primes $\ell\neq p$ such that

    * –

$\ell$ is inert in $K$,

    * –

$T_{p}E$ is unramified at $\ell$,

    * –

$p\mid\ell+1$.

Letting $K[\ell]$ be the ring class field of $K$ of conductor $\ell$, define
the _transverse_ local condition at $\lambda|\ell\in\mathcal{L}_{0}$ by

$\mathrm{H}^{1}_{\rm tr}(K_{\lambda},T_{\alpha}^{(k)}):={\rm
ker}\bigl{\\{}\mathrm{H}^{1}(K_{\lambda},T_{\alpha}^{(k)})\rightarrow\mathrm{H}^{1}(K[\ell]_{\lambda^{\prime}},T_{\alpha}^{(k)})\bigr{\\}},$

where $K[\ell]_{\lambda^{\prime}}$ is the completion of $K[\ell]$ at any prime
$\lambda^{\prime}$ above $\lambda$ and $k>0$ is any integer such that
$p^{k}\mid\ell+1$.

  * •

For $w$ a prime of $K$ above $p$, set

${\rm Fil}_{w}^{+}(T_{p}E):={\rm ker}\bigl{\\{}T_{p}E\rightarrow
T_{p}\tilde{E}\bigr{\\}},$

where $\tilde{E}$ is the reduction of $E$ at $w$, and put

${\rm Fil}_{w}^{+}(T_{\alpha}):={\rm
Fil}_{w}^{+}(T_{p}E)\otimes\mathbb{Z}_{p}(\alpha),\quad{\rm
Fil}_{w}^{+}(V_{\alpha}):={\rm Fil}_{w}^{+}(T_{\alpha})\otimes\mathbb{Q}_{p}.$

The _ordinary_ local condition for $V_{\alpha}$ is defined by

$\mathrm{H}^{1}_{\rm ord}(K_{w},V_{\alpha}):={\rm
im}\bigl{\\{}\mathrm{H}^{1}(K_{w},{\rm
Fil}_{w}^{+}(V_{\alpha}))\rightarrow\mathrm{H}^{1}(K_{w},V_{\alpha})\bigr{\\}}.$

Similarly as before, the corresponding local conditions $\mathrm{H}^{1}_{\rm
ord}(K_{w},T_{\alpha})$ and $\mathrm{H}^{1}_{\rm ord}(K_{w},W_{\alpha})$ are
defined to be the inverse image and the image, respectively, of
$\mathrm{H}^{1}_{\rm ord}(K_{w},V_{\alpha})$ under the natural maps (1.1), and
$\mathrm{H}^{1}_{\rm ord}(K_{w},T_{\alpha}^{(k)})$ is the preimage of
$\mathrm{H}^{1}_{\rm ord}(K_{w},W_{\alpha})$.

######  Remark 1.1.1.

For $w\nmid p$, it follows immediately from the definitions that
$\mathrm{H}^{1}_{f}(K_{w},W_{\alpha})$ is contained in

$\mathrm{H}^{1}_{\rm ur}(K_{w},W_{\alpha}):={\rm
ker}\bigl{\\{}\mathrm{H}^{1}(K_{w},W_{\alpha})\rightarrow\mathrm{H}^{1}(K_{w}^{\rm
ur},W_{\alpha})\bigr{\\}}.$

In fact, for $V_{\alpha}$ one even has
$\mathrm{H}^{1}_{f}(K_{w},V_{\alpha})=0$, and therefore
$\mathrm{H}^{1}_{f}(K_{w},W_{\alpha})$ is also trivial. Thus the $p$-part of
the Tamagawa number of $W_{\alpha}$ is given by

$c_{w}^{(p)}(W_{\alpha}):=[\mathrm{H}^{1}_{\rm
ur}(K_{w},W_{\alpha})\colon\mathrm{H}^{1}_{f}(K_{w},W_{\alpha})]=\\#\mathrm{H}^{1}_{\rm
ur}(K_{w},W_{\alpha}).$

######  Definition 1.1.2.

The _ordinary_ Selmer structure $\mathcal{F}_{\rm ord}$ on $V_{\alpha}$ is
defined by taking $\Sigma(\mathcal{F}_{\rm\operatorname{ord}})=\\{w\mid pN\\}$
and

$\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{w},V_{\alpha}):=\begin{cases}\mathrm{H}^{1}_{\rm
ord}(K_{w},V_{\alpha})&\textrm{if $w\mid p$,}\\\\[1.99997pt]
\mathrm{H}^{1}_{f}(K_{w},V_{\alpha})&\textrm{else.}\end{cases}$

Let $\mathcal{F}_{\rm ord}$ also denote the resulting Selmer structure on
$T_{\alpha}$, $W_{\alpha}$, and $T_{\alpha}^{(k)}$.

######  Remark 1.1.3.

For $\alpha=\mathds{1}$, one knows that $\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,W_{\alpha})={\rm Sel}_{p^{\infty}}(E/K)$ (see e.g. [Rub00, Prop.
1.6.8]).

Given a subset $\mathcal{L}\subset\mathcal{L}_{0}$, we let
$\mathcal{N}(\mathcal{L})$ be the set of squarefree products of primes
$\ell\in\mathcal{L}$; when the choice of $\mathcal{L}$ is irrelevant or clear
from the context, we shall simply denote this by $\mathcal{N}$. Given a Selmer
structure $\mathcal{F}$ on $T_{\alpha}^{(k)}$, and $n\in\mathcal{N}$, we
define the modified Selmer structure $\mathcal{F}(n)$ by

$\mathrm{H}^{1}_{\mathcal{F}(n)}(K_{\lambda},T_{\alpha}^{(k)})=\begin{cases}\mathrm{H}^{1}_{\rm
tr}(K_{\lambda},T_{\alpha}^{(k)})&\text{if }\lambda\mid n,\\\\[1.99997pt]
\mathrm{H}^{1}_{\mathcal{F}}(K_{w},T_{\alpha}^{(k)})&\text{if }\lambda\nmid
n.\end{cases}$

####  1.1.2. The Kolyvagin system of Heegner points

Let $T=T_{\mathds{1}}=T_{p}E$. We start by defining the derived Heegner
classes $\kappa_{n,k}^{\rm
Heeg}\in\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)})$, where
$T^{(k)}=T/p^{k}T\simeq E[p^{k}]$ and $k\leq M(n)$.

We recall the following result.

######  Lemma 1.1.4 ([CGLS22, Lem. 3.3.1]).

For every $n\in\mathcal{N}$ and $0<i\leq k\leq M(n)$ there are natural
isomorphisms

$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(i)})=\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)}/p^{i}T^{(k)})\xrightarrow{\sim}\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)}[p^{i}])\xrightarrow{\sim}{\rm
H}_{\mathcal{F}(n)}^{1}(K,T^{(k)})[p^{i}]$

induced by the maps
$T^{(i)}=T^{(k)}/p^{i}T^{(k)}\xrightarrow{p^{k-i}}T^{(k)}[p^{i}]\rightarrow
T^{(k)}$.

We first define $\kappa_{n}^{\rm Heeg}=\kappa_{n,M(n)}^{\rm
Heeg}\in\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(M(n))})$; the classes
$\kappa_{n,k}^{\rm Heeg}$ for $k<M(n)$ are then defined to be the pre-image of
$p^{M(n)-k}\kappa_{n}^{\rm Heeg}$ under the isomorphism
$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)})\xrightarrow{\sim}\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(M(n))})[p^{k}]$
of Lemma 1.1.4 above.

Put

$\mathcal{L}_{\rm
Heeg}:=\\{\ell\in\mathcal{L}_{0}\,\colon\,a_{\ell}\equiv\ell+1\equiv 0\;({\rm
mod}\,p)\\},$

where $a_{\ell}=\ell+1-\\#\tilde{E}(\mathbb{F}_{\ell})$. For
$\ell\in\mathcal{L}_{\rm Heeg}$ we let
$G_{\ell}=\operatorname{Gal}(K[\ell]/K[1])$, and for $n\in\mathcal{N}_{\rm
Heeg}:=\mathcal{N}(\mathcal{L}_{\rm Heeg})$ we set
$\mathcal{G}(n)=\operatorname{Gal}(K[n]/K)$ and $G(n)=\prod_{\ell\mid
n}G_{\ell}$. Then for $m$ dividing $n$ we have the natural identification

$\operatorname{Gal}(K[n]/K[m])\simeq\prod_{\ell\mid(n/m)}G_{\ell}=G(n/m)$

induced by the projections $\operatorname{Gal}(K[n]/K[m])\twoheadrightarrow
G_{\ell}$. The Kolyvagin derivative operator
$D_{\ell}\in\mathbb{Z}_{p}[G_{\ell})]$ is defined by

(1.2) $D_{\ell}=\sum_{i=1}^{\ell}i\sigma_{\ell}^{i},$

where $\sigma_{\ell}$ is a fixed generator of $G_{\ell}$. Let
$D_{n}=\prod_{\ell\mid n}D_{\ell}\in\mathbb{Z}_{p}[G(n)]$.

Recall the Heegner points $P[n]\in E(K[n])$ in (0.2). Choosing a set $S$ of
representatives for the cosets $\mathcal{G}(n)/G(n)$, one can show the
inclusion

$\tilde{\kappa}_{n}:=\sum_{\sigma\in S}\sigma
D_{n}(P[n])\in\left({\raisebox{1.99997pt}{$E(K[n])$}\left/\raisebox{-1.99997pt}{$p^{M(n)}E(K[n])$}\right.}\right)^{\mathcal{G}(n)}$

(see [How04a, Lem. 1.7.1]). Hence, applying the Kummer map we may consider
$\tilde{\kappa}_{n}\in\mathrm{H}^{1}(K[n],T^{(M(n))})^{\mathcal{G}(n)}$.
Hypothesis (tor) ensures that the restriction map

$\mathrm{H}^{1}(K,T^{(M(n))})\to\mathrm{H}^{1}(K[n],T^{(M(n))})^{\mathcal{G}(n)}$

is an isomorphism, and the derived Heegner class $\kappa_{n}^{\rm Heeg}$ is
defined to be the unique class in $\mathrm{H}^{1}(K,T^{(M(n))})$ which
restricts to the image of $\tilde{\kappa}_{n}$ in
$\mathrm{H}^{1}(K[n],T^{(M(n))})^{\mathcal{G}(n)}$.

The classes $\kappa_{n}^{\rm Heeg}$ land in the $n$-transverse Selmer group
$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(M(n))})$ (see [How04a, Lem. 1.7.3]),
and after a slight modification – which we still denote by $\kappa_{n}^{\rm
Heeg}$ – the resulting collection of classes

$\\{\kappa_{n}^{\rm Heeg}\\}_{n\in\mathcal{N}_{\rm Heeg}}$

forms a _Kolyvagin system_ for $(T,\mathcal{F}_{\rm ord},\mathcal{L}_{E})$ in
the sense of [_op. cit._ , Def. 1.2.3].

####  1.1.3. The $\Lambda$-adic Kolyvagin system

Let $K_{\infty}/K$ be the anticyclotomic $\mathbb{Z}_{p}$-extension, let
$\Gamma=\operatorname{Gal}(K_{\infty}/K)$ and
$\Lambda=\mathbb{Z}_{p}[[\Gamma]]$ the anticyclotomic Iwasawa algebra. Put

$\mathbf{T}=T_{p}E\otimes_{\mathbb{Z}_{p}}\Lambda,$

and consider the _ordinary_ Selmer structure $\mathcal{F}_{\Lambda}$ on
$\mathbf{T}$ given by

$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K_{w},\mathbf{T})=\begin{cases}{\rm
im}\\{\mathrm{H}^{1}(K_{w},{\rm
Fil}_{w}^{+}(T_{p}E)\otimes\Lambda)\rightarrow\mathrm{H}^{1}(K_{w},\mathbf{T})\\}&\textrm{if
$w\mid p$,}\\\\[1.99997pt] \mathrm{H}^{1}_{\rm
ur}(K_{w},\mathbf{T})&\textrm{else.}\end{cases}$

We now briefly recall the construction of the $\Lambda$-adic Heegner point
Kolyvagin system, which is a collection of classes $\bm{\kappa}_{n}^{\rm
Heeg}\in\mathrm{H}^{1}(K,\mathbf{T}/p^{M(n)}\mathbf{T})=\mathrm{H}^{1}(K,E[p^{M(n)}]\otimes\Lambda)$.

Let $K_{k}$ be the subfield of $K_{\infty}$ of degree $p^{k}$ over $K$. For
each $n\in\mathcal{N}_{\rm Heeg}$ as above set

$P_{k}[n]:={\rm Norm}_{K[np^{d(k)}]/K_{k}[n]}(P[np^{d(k)}])\in E(K_{k}[n]),$

where $d(k)=\min\\{d\in\mathbb{Z}_{\geq 0}\colon K_{k}\subset K[p^{d(k)}]\\}$,
and $K_{k}[n]$ is the compositum of $K_{k}$ and $K[n]$. Let $H_{k}[n]$ be the
$\mathbb{Z}_{p}[{\rm Gal}(K_{k}[n]/K)]$-submodule of
$E(K_{k}[n])\otimes\mathbb{Z}_{p}$ generated by $P_{j}[n]$ for $j\leq k$, and
consider the $\Lambda[\mathcal{G}(n)]$-module

$\mathbf{H}[n]=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{k}H_{k}[n],$

where the limit is taken with respect to the norm maps. By [How04a, Lem.
2.3.3], there is a family

$\\{Q[n]=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{k}Q_{k}[n]\in\mathbf{H}[n]\\}_{n\in\mathcal{N}}$

such that

(1.3) $Q_{0}[n]=\Phi
P[n],\quad\textrm{where}\;\Phi=\begin{cases}(p-a_{p}\sigma_{p}+\sigma_{p}^{2})(p-a_{p}\sigma_{p}^{*}+\sigma_{p}^{*2})&\text{$p$
splits in $K$},\\\ (p+1)^{2}-a_{p}&\text{$p$ inert in $K$}.\end{cases}$

Here, when $p$ splits in $K$, $\sigma_{p}$ and $\sigma_{p}^{*}$ are the
Frobenius elements in $\mathcal{G}(n)$ of the primes above $p$ in $K$.

Letting $D_{n}\in\mathbb{Z}_{p}[G(n)]$ be the Kolyvagin’s derivative operators
defined in (1.2), and choosing a set $S$ of representatives for
$\mathcal{G}(n)/G(n)$, the class $\bm{\kappa}_{n}^{\rm
Heeg}\in\mathrm{H}^{1}(K,\mathbf{T}/I_{n}\mathbf{T})$ is defined as the
natural image of

$\tilde{\bm{\kappa}}_{n}=\sum_{\sigma\in S}\sigma D_{n}(Q[n])\in\mathbf{H}[n]$

under the composite map

$\bigl{(}\mathbf{H}[n]/p^{M(n)}\mathbf{H}[n]\bigr{)}^{\mathcal{G}(n)}\xrightarrow{\delta(n)}\mathrm{H}^{1}(K[n],\mathbf{T}/p^{M(n)}\mathbf{T})^{\mathcal{G}(n)}\overset{\simeq}{\longleftarrow}\mathrm{H}^{1}(K,\mathbf{T}/p^{M(n)}\mathbf{T}),$

where $\delta(n)$ is induced by the limit of Kummer maps
$\delta_{k}(n):E(K_{k}[n])\otimes\mathbb{Z}_{p}\rightarrow{\rm
H}^{1}(K_{k}[n],T)$, and the second arrow is given by restriction. The latter
is an isomorphism since the extensions $K[n]$ and $\mathbb{Q}(E[p])$ are
linearly disjoint, and $E(K_{\infty})[p]=0$ by hypothesis (tor).

Finally, [How04a, Lem. 2.3.4] _et seq._ show that the classes
$\bm{\kappa}_{n}^{\rm Heeg}$ land in
$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}(n)}(K,\mathbf{T}/p^{M(n)}\mathbf{T})$
and they can be slightly modified - and still denoted $\bm{\kappa}_{n}^{\rm
Heeg}$ \- so that the result is a system $\bm{\kappa}^{\rm
Hg}=\\{\bm{\kappa}_{n}^{\rm Heeg}\\}_{n\in\mathcal{N}_{\rm Heeg}}$ satisfying
the Kolyvagin system relations. Here $\mathcal{F}_{\Lambda}(n)$ denotes the
modification of the Selmer structure on $\mathbf{T}/p^{M(n)}\mathbf{T})$
induced from $\mathcal{F}_{\Lambda}$ that includes the obvious analog of the
transverse local conditions at the primes $\ell\mid n$.

####  1.1.4. Kolyvagin system for anticyclotomic twists

We now consider a height one prime $\mathfrak{P}\subset\Lambda$ with
$\mathfrak{P}\neq p\Lambda$, and denote by $R_{\mathfrak{P}}$ the integral
closure of $\Lambda/\mathfrak{P}$. We let $G_{K}$ act on $R_{\mathfrak{P}}$ by
the character
$\alpha_{\mathfrak{P}}:G_{K}\twoheadrightarrow\Gamma\to\Lambda\to
R_{\mathfrak{P}}$, and consider

$T_{\mathfrak{P}}=\mathbf{T}\otimes_{\Lambda}R_{\mathfrak{P}}=T_{\alpha_{\mathfrak{P}}}$

with diagonal $G_{K}$-action. By Remark 1.2.4 and Lemma 2.2.7 in [How04a], the
natural map induced by $\mathbf{T}\to T_{\mathfrak{P}}$ sends
$\bm{\kappa}^{\rm Hg}$ to a Kolyvagin system for $T_{\mathfrak{P}}$ (for the
Selmer structure $\mathcal{F}_{\rm ord}$ defined in §1.1.1), which we denote
by $\\{\bm{\kappa}_{n}^{\rm
Heeg}(\alpha_{\mathfrak{P}})\\}_{n\in\mathcal{N}_{\rm Heeg}}$. In general,
$R_{\mathfrak{P}}$ may not be $\mathbb{Z}_{p}$ and the definitions in §1.1.1
have to be suitably modified. However, for the purposes of this paper we will
only be interested in the case $\mathfrak{P}=(\gamma-\alpha(\gamma))$ where
$\gamma$ is a topological generator of $\Gamma$ and
$\alpha:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ is a character. In this case
$\alpha_{\mathfrak{P}}$ is just the composition of $\alpha$ and the projection
$G_{K}\twoheadrightarrow\Gamma$, and we write

$\\{\bm{\kappa}_{n}^{\rm Heeg}(\alpha)\\}_{n\in\mathcal{N}_{\rm Heeg}}$

for the corresponding Kolyvagin system.

######  Lemma 1.1.5.

Suppose $\alpha\equiv 1\pmod{p^{m}}$. For all $n\in\mathcal{N}_{\rm Heeg}$
with $M(n)\geq m$ we have the congruence

$\bm{\kappa}_{n}^{\rm
Heeg}(\alpha)\equiv\begin{cases}(\alpha_{p}-1)^{2}(\beta_{p}-1)^{2}\kappa_{n}^{\rm
Heeg}&\text{$p$ splits in $K$}\\\ ((p+1)^{2}-a_{p}^{2})\kappa_{n}^{\rm
Heeg}&\text{$p$ inert in $K$}\end{cases}\;({\rm mod}\,p^{m}),$

where $\alpha_{p},\beta_{p}$ are the roots of the characteristic polynomial of
the Frobenius at $p$, with $\alpha_{p}$ the $p$-adic unit root.

###### Proof.

Let $\mathfrak{P}=(\gamma-\alpha(\gamma))$ and $\mathfrak{P}_{0}=(\gamma-1)$.
The image of $Q[n]$ under the map on $\mathbf{H}[n]$ induced by the
composition
$\Lambda\to\Lambda/\mathfrak{P}\to(\Lambda/\mathfrak{P})/p^{m}\simeq\mathbb{Z}_{p}/p^{m}\mathbb{Z}_{p}$
is the same as the image induced by the composition
$\Lambda\to\Lambda/\mathfrak{P}_{0}\to(\Lambda/\mathfrak{P}_{0})/p^{m}\simeq\mathbb{Z}_{p}/p^{m}\mathbb{Z}_{p}$
(since both are given by quotienting by the ideal $(\gamma-1,p^{m})$). By
(1.3), the latter image is $\Phi P[n]\;({\rm mod}\,p^{m})$ from which the
result follows immediately by the constructions of $\bm{\kappa}_{n}^{\rm
Heeg}$ and $\kappa_{n}^{\rm Heeg}$ and the definition of $\Phi$. ∎

In particular, noting that $\mathrm{H}^{1}(K,T)$ is torsion-free as a
consequence of (tor), Lemma 1.1.5 implies, combined with the Gross–Zagier
formula [GZ86], that

$\bm{\kappa}_{1}^{\rm Heeg}(\mathds{1})\neq 0\quad\Longleftrightarrow\quad{\rm
ord}_{s=1}L(E/K,s)=1.$

More generally, without any conditions on the analytic rank of $E/K$, we have
the following.

######  Theorem 1.1.6 (Cornut–Vatsal).

For $m\gg 0$ and any $\alpha\equiv 1\pmod{p^{m}}$, the class
$\bm{\kappa}_{1}^{\rm Heeg}(\alpha)$ is non-zero.

###### Proof.

This is immediate from the main result of [Cor02] and [Vat03], showing that
$\bm{\kappa}_{1}^{\rm Heeg}$ is not $\Lambda$-torsion. ∎

######  Remark 1.1.7.

Based on an extension of the Bertolini–Darmon–Prasanna formula and the
nonvanishing of the $p$-adic $L$-function $\mathcal{L}_{p}^{\rm BDP}(f/K)$
recalled in Theorem 1.2.1 below, one can give alternate proof of Theorem 1.1.6
when $p$ splits in $K$ (see [Bur20] and [CH18]).

###  1.2. Anticyclotomic Iwasawa Main Conjecture

We keep the hypotheses on $(E,p,K)$ from the preceding section, and assume in
addition that

(spl) $p=v\bar{v}$ splits in $K$,

with $v$ the prime of $K$ above $p$ induced by a fixed embedding
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$. Fix an
integral ideal $\mathfrak{N}\subset\mathcal{O}_{K}$ with

$\mathcal{O}_{K}/\mathfrak{N}\simeq\mathbb{Z}/N\mathbb{Z}.$

In this section we show how a certain anticyclotomic Iwasawa Main Conjecture
(see Conjecture 1.2.10) yields an expression for the divisibility index of the
Kolyvagin systems $\\{\bm{\kappa}_{n}^{\rm
Heeg}(\alpha)\\}_{n\in\mathcal{N}_{\rm Heeg}}$, for $\alpha\equiv
1\pmod{p^{m}}$ as above with $m\gg 0$, in terms of the Tamagawa numbers and
the Shafarevich–Tate group of (a twist of) $E$.

Below we shall write $a\sim_{p}b$ to denote that $a$ and $b$ have the same
$p$-adic valuation.

####  1.2.1. $p$-adic $L$-function

Let $f\in S_{2}(\Gamma_{0}(N))$ be the newform attached to $E$. Recall that
$\Lambda=\mathbb{Z}_{p}[[\Gamma]]$ denotes the anticyclotomic Iwasawa algebra
of $K$, and put $\Lambda^{\rm
ur}=\Lambda\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{Z}_{p}^{\rm ur}$, where
$\mathbb{Z}_{p}^{\rm ur}$ is the completion of the ring of integers of the
maximal unramified extension of $\mathbb{Q}_{p}$.

The next result refines the construction of a continuous anticyclototomic
$p$-adic $L$-function due to Bertolini–Darmon–Prasanna [BDP13]. The $p$-adic
measure underlying their construction was explicitly given in [CH18], whose
formulation we follow.

######  Theorem 1.2.1.

There exists an element $\mathcal{L}_{p}^{\rm BDP}(f/K)\in\Lambda^{\rm ur}$
characterised by the following interpolation property: For every character
$\xi$ of $\Gamma$ crystalline at both $v$ and $\bar{v}$ and corresponding to a
Hecke character of $K$ of infinity type $(n,-n)$ with $n\in\mathbb{Z}_{>0}$
and $n\equiv 0\pmod{p-1}$, we have

$\mathcal{L}_{p}^{\rm
BDP}(f/K)^{2}(\xi)=\frac{\Omega_{p}^{4n}}{\Omega_{\infty}^{4n}}\cdot\frac{\Gamma(n)\Gamma(n+1)\xi(\mathfrak{N}^{-1})}{4(2\pi)^{2n+1}\sqrt{D_{K}}^{2n-1}}\cdot\bigl{(}1-a_{p}\xi(\bar{v})p^{-1}+\xi(\bar{v})^{2}p^{-1}\bigr{)}^{2}\cdot
L(f/K,\xi,1),$

where $\Omega_{p}$ and $\Omega_{\infty}$ are CM periods attached to $K$ as in
[CH18, §2.5]. Moreover, $\mathcal{L}_{p}^{\rm BDP}(f/K)$ is nonzero.

###### Proof.

This follows from results contained in [CH18, §3], as explained in the proof
[CGLS22, Thm. 2.1.1]. ∎

######  Definition 1.2.2.

A collection of characters as in Theorem 1.2.1 with $n$ essentially arbitrary
can be defined as follows. Let $u\in
1+p\mathbb{Z}_{p}=1+p\mathcal{O}_{K_{v}}\subset\mathcal{O}_{K_{v}}^{\times}$
be a $\mathbb{Z}_{p}$-generator and let $\gamma\in\Gamma$ be a topological
generator such that there exists $h>0$ a power of $p$ for which $\gamma^{h}$
is the image of $u$ under the reciprocity map of class field theory. For
$n\in\mathbb{Z}$ such that $n\equiv 0\pmod{p-1}$, let
$\xi_{n}:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ be the unique continuous
character such $\xi_{n}(\gamma)=u^{n}$. Then the composition of $\xi_{n}$ with
the projection $G_{K}\twoheadrightarrow\Gamma$ is crystalline at $v$ (and
hence also $\bar{v}$) and the corresponding Hecke character of $K$ has
infinity type $(hn,-hn)$. For $m>0$ we let

$\alpha_{m}=\xi_{(p-1)p^{m-1}}.$

Then $\alpha_{m}\equiv 1\pmod{p^{m}}$.

Assume now that $E/\mathbb{Q}$ is an elliptic curve in the isogeny class
attached to $f$ having a modular parameterisation $\pi:X_{0}(N)\rightarrow E$
that identifies $T_{p}E$ with $T_{f}$, where $T_{f}$ is the quotient of
$\mathrm{H}^{1}(Y_{1}(N),\mathbb{Z}_{p}(1))$ associated with $f$ (cf. [Wut14,
Prop. 8]); we use such a choice of $\pi$ in the following. If (irr) holds,
then $\pi$ is $p$-optimal and $\pi$ (and $E$) is uniquely determined up to a
prime-to-$p$ isogeny, and the exact choice of $\pi$ has no effect on the
formulas and relations below.

Let $\kappa_{\infty}\in\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$
be the $\Lambda$-adic class constructed in [CH18, §5.2]. As explained in
[CGLS22, Remark 4.1.3], the class $\kappa_{\infty}$ generates the same
$\Lambda$-submodule of $\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$
as the base class $\bm{\kappa}_{1}^{\rm Heeg}$ of the $\Lambda$-adic Kolyvagin
system $\bm{\kappa}^{\rm Heeg}$ associated to $E$. For any character
$\alpha:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$, put
$T_{\alpha}=T_{p}E\otimes\mathbb{Z}_{p}(\alpha)$ and denote by

$\bm{\kappa}_{1}^{\rm Heeg}(\alpha)\in\mathrm{H}^{1}(K,T_{\alpha})$

the image of $\bm{\kappa}_{1}^{\rm Heeg}$ under the specialisation map
$\mathrm{H}^{1}(K,\mathbf{T})\rightarrow\mathrm{H}^{1}(K,T_{\alpha})$.

As recalled in [CH18], there exists a $\Lambda$-module map
$\mathscr{L}_{T^{+}}:\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T\otimes_{\mathbb{Z}_{p}}\Lambda)\rightarrow\Lambda^{\rm
ur}$ (a ‘big logarithm’ map) such that

$\mathscr{L}_{T^{+}}(\operatorname{loc}_{v}(\bm{\kappa}_{1}^{\rm
Heeg}))=\mathcal{L}_{p}^{\rm BDP}(f/K).$

In the setting of this paper, $\mathscr{L}_{T^{+}}$ induces an isomorphism
$\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T\otimes_{\mathbb{Z}_{p}}\Lambda)\otimes_{\Lambda}\Lambda^{\rm
ur}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Lambda^{\rm ur}$. From this we
deduce the following.

######  Lemma 1.2.3.

Assume $E/\mathbb{Q}$ is as above. Let
$\alpha:\Gamma\to\mathbb{Z}_{p}^{\times}$ be an anticyclotomic character as in
Theorem 1.2.1 with $n\geq 0$ and such that $\mathcal{L}_{p}^{\rm
BDP}(f/K)(\alpha^{-1})\neq 0$. If $\alpha\equiv 1\;({\rm mod}\,p^{m})$ with
$m\gg 0$, then

$p^{t_{\alpha}}\cdot\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])=\\#\bigl{(}{\rm
H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})/\mathbb{Z}_{p}\cdot\bm{\kappa}^{\rm
Heeg}_{1}(\alpha)\bigr{)}\cdot\\#{\rm coker}(\operatorname{loc}_{v}),$

where

$t_{\alpha}=\mathrm{length}_{\mathbb{Z}_{p}^{\rm
ur}}\bigl{(}\mathbb{Z}_{p}^{\rm ur}/\mathcal{L}_{p}^{\rm
BDP}(f/K)(\alpha^{-1})\bigr{)}$

and $\operatorname{loc}_{v}:\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})\to\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{\alpha})/\mathrm{H}^{1}(K_{v},T_{\alpha})_{\rm
tors}=:\mathrm{H}^{1}(K_{v},{\rm Fil}_{v}^{+}T_{\alpha})_{/\rm tors}$.

###### Proof.

The nonvanishing of $\mathcal{L}_{p}^{\rm BDP}(f/K)(\alpha^{-1})$ implies that
${\rm loc}_{v}(\bm{\kappa}^{\rm Heeg}_{1}(\alpha))$ has nonzero image in
$\mathrm{H}^{1}(K_{v},{\rm Fil}_{v}^{+}T_{\alpha})$ (note that the latter is a
torsion-free $\mathbb{Z}_{p}$-module). Furthermore the cokernel of the
specialization map
$\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T\otimes_{\mathbb{Z}_{p}}\Lambda)\rightarrow\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T_{\alpha})$
is naturally isomorphic to
$\mathrm{H}^{2}(K_{v},\mathrm{Fil}_{v}^{+}T\otimes_{\mathbb{Z}_{p}}\Lambda)[\gamma-\alpha(\gamma)]$,
whose order is easily computed via local duality to equal
$\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])$ if $m\gg 0$. As
$\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T_{\alpha})$ is a torsion-free
$\mathbb{Z}_{p}$-module of rank one, it follows that

$\\#\bigl{(}\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T_{\alpha})/\mathbb{Z}_{p}\operatorname{loc}_{v}(\bm{\kappa}^{\rm
Heeg}_{1}(\alpha))\bigr{)}=p^{t_{\alpha}}\cdot\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]).$

On the other hand, since $\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})$ is also a torsion-free $\mathbb{Z}_{p}$-module of rank
one, $\operatorname{loc}_{v}$ induces an injection
$\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})/\mathbb{Z}_{p}\bm{\kappa}^{\rm
Heeg}_{1}(\alpha)\hookrightarrow\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T_{\alpha})/\mathbb{Z}_{p}\operatorname{loc}_{v}(\bm{\kappa}^{\rm
Heeg}_{1}(\alpha))$, from which it follows that

$\\#\bigl{(}\mathrm{H}^{1}(K_{v},\mathrm{Fil}_{v}^{+}T_{\alpha})/\mathbb{Z}_{p}\operatorname{loc}_{v}(\bm{\kappa}^{\rm
Heeg}_{1}(\alpha))\bigr{)}=\\#\bigl{(}\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})/\mathbb{Z}_{p}\bm{\kappa}^{\rm
Heeg}_{1}(\alpha)\bigr{)}\cdot\\#{\rm coker}(\operatorname{loc}_{v}).$

Combining the last two displayed equations yields the lemma. ∎

######  Remark 1.2.4.

In the case that $\alpha=\mathds{1}$, then $\kappa_{1}^{\rm
Heeg}(\mathds{\alpha})=(1-\alpha_{p})^{2}(1-\beta_{p})^{2}\kappa_{1}^{\rm
Heeg}$ and — noting that
$\\#(\mathbb{Z}_{p}/(1-\alpha_{p})(1-\beta_{p}))=\\#(\mathbb{Z}_{p}/(1-\alpha_{p}^{-1}))=\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])$
— the formula in the Lemma can be rewritten as

$\\#\left(\mathbb{Z}_{p}/\mathcal{L}_{p}^{\rm
BDP}(f/K)(\mathds{1})\right)=\\#\bigl{(}{\rm H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})/\mathbb{Z}_{p}\cdot\bm{\kappa}^{\rm
Heeg}_{1}\bigr{)}\cdot\\#{\rm
coker}(\operatorname{loc}_{v})\cdot\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]).$

Lemma 1.2.3 links the specialisation of the $p$-adic $L$-function at $\alpha$
with the based class in the Heegner point Kolyvagin system for a specific
elliptic curve (up to prime to $p$ isogeny) associated with a specific choice
of lattice, namely the curve $E_{\bullet}$ associated with the lattice
$T_{f}$. The following lemma tells us that we can rephrase it in terms of the
corresponding class attached to any elliptic curve in the isogeny class
associated with $f$. Let $E/\mathbb{Q}$ be a such a curve. We write
$\bm{\kappa}_{1,f}^{\rm Heeg}(\alpha)$ for the classes $\bm{\kappa}_{1}^{\rm
Heeg}(\alpha)$ associate as above with the lattice $T_{f}$ and
$\bm{\kappa}^{\rm Heeg}_{1,E}(\alpha)$ for the ones for $T=T_{p}E$.

######  Lemma 1.2.5.

Let $E/\mathbb{Q}$ be any elliptic curve in the isogeny class associated with
$f$, and let $T=T_{p}E$. Assume that $\bm{\kappa}^{\rm Heeg}_{1,f}(\alpha)\neq
0$. Consider the map

$\operatorname{loc}_{v,E}:\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})\to\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{\alpha})/\mathrm{H}^{1}(K_{v},T_{\alpha})_{\rm
tors}=:\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(K_{v},T_{\alpha})_{/\rm tors},$

and let ${\rm loc}_{v,f}$ be the analogous map with $T_{f}$ in place of $T$.
Then

$\\#\bigl{(}{\rm H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{f,\alpha})/\mathbb{Z}_{p}\cdot\bm{\kappa}^{\rm
Heeg}_{1,f}(\alpha)\bigr{)}\cdot\\#{\rm
coker}(\operatorname{loc}_{v,f})=\\#\bigl{(}{\rm H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})/\mathbb{Z}_{p}\cdot\bm{\kappa}^{\rm
Heeg}_{1,E}(\alpha)\bigr{)}\cdot\\#{\rm coker}(\operatorname{loc}_{v,E}).$

###### Proof.

Let $E_{\bullet}$ be the elliptic curve as before such that $T_{p}E$ is
identifified with $T_{f}$. We may assume without loss of generality that $E$
is _closer_ than $E_{\bullet}$ to the optimal curve, i.e. the degree of the
modular parametrisation for $E$ is smaller than the one for $E_{\bullet}$.
Then there is an étale isogeny $\psi:E\to E_{\bullet}$ such that the induced
injective map

$\psi:\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})\to\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(K,T_{f,\alpha})$

satisfies $\psi(\bm{\kappa}_{1,E}^{\rm Heeg}(\alpha))=\bm{\kappa}^{\rm
Heeg}_{1,f}(\alpha)$. Let $d\in\mathbb{Z}_{\geq 0}$ be such that
$[\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{f,\alpha}):\psi(\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha}))]=p^{d}$. Since we are assuming that $\bm{\kappa}^{\rm
Heeg}_{1,f}(\alpha)\neq 0$, we also have $\bm{\kappa}^{\rm
Heeg}_{1,E}(\alpha)\neq 0$, and so by [CGS23, Thm. 5.5.1] that
$\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(K,T_{f,\alpha})$ and ${\rm
H}^{1}_{\mathcal{F}_{\rm ord}}(K,T_{\alpha})$ are both free of rank one. Hence

(1.4) $\\#\bigl{(}\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{f,\alpha}))/\mathbb{Z}_{p}\cdot\bm{\kappa}_{1,f}(\alpha)\bigr{)}=p^{d}\cdot\\#\bigl{(}\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{E,\alpha}))/\mathbb{Z}_{p}\cdot\bm{\kappa}_{1,E}(\alpha)\bigr{)}.$

Furthermore, since ${\rm Fil}_{w}^{+}(T_{\alpha})={\rm
Fil}_{w}^{+}(T_{f,\alpha})$, we have the following commutative diagram:

${\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{f,\alpha})}$${\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{f,\alpha})_{/\rm tors}}$${\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})}$${\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{\alpha})_{/\rm
tors}}$$\scriptstyle{\operatorname{loc}_{v,f}}$$\scriptstyle{\psi}$$\scriptstyle{\operatorname{loc}_{v,E}}$

We thus obtain ${\rm
coker}(\operatorname{loc}_{v,E})=\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{f,\alpha})_{/\rm tors}/\psi(\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha}))=\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{f,\alpha})_{/\rm tors}/p^{d}\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{f,\alpha})$, and hence the equality

(1.5) $\\#{\rm coker}(\operatorname{loc}_{v,E})=p^{d}\\#{\rm
coker}(\operatorname{loc}_{v,f}).$

Combining (1.4) and (1.5) yields the desired result. ∎

####  1.2.2. Anticyclotomic Selmer group

Let $\Sigma$ be a finite set of places of $K$ containing $\infty$ and the
primes dividing $Np$. We assume that all finite primes in $\Sigma$ split in
$K$. For a discrete $\mathbb{Z}_{p}$-module $A$, we let $A^{\vee}={\rm
Hom}_{\rm cts}(A,\mathbb{Q}/\mathbb{Z}_{p})$ be the Pontryagin dual. Put

$M:=T_{p}E\otimes\Lambda^{\vee},$

where $\Lambda^{\vee}$ is equipped with a $G_{K}$-action via the inverse of
the character $G_{K}\rightarrow\Lambda^{\times}$ arising from the projection
$G_{K}\twoheadrightarrow\Gamma$.

######  Definition 1.2.6.

Let $G_{K,\Sigma}$ denote the Galois group of the maximal extension of $K$
unramified outside $\Sigma$. We define the _Greenberg Selmer group_ of $M$ as
follows:

$\mathrm{H}^{1}_{\mathcal{F}_{\rm
Gr}}(K,M)=\ker\biggl{\\{}\mathrm{H}^{1}(G_{K,\Sigma},M)\to{\mathrm{H}^{1}(K_{v},M)}\times\prod_{w\in\Sigma,w\nmid
p}\mathrm{H}^{1}(K_{w},M)\biggr{\\}}.$

We also put

$\displaystyle\mathfrak{X}_{\rm
Gr}(E/K_{\infty}):=\mathrm{H}^{1}_{\mathcal{F}_{\rm Gr}}(K,M)^{\vee}.$

It is a standard fact that $\mathfrak{X}_{\rm Gr}(E/K_{\infty})$ is a finitely
generated $\Lambda$-module. We next recall a twisted anticyclotomic variant of
the Euler characteristic calculation for Selmer groups [Gre99, §4]. Here
$E/\mathbb{Q}$ is any elliptic curve in the isogeny class associated with $f$,
and we let $T=T_{p}E$.

######  Theorem 1.2.7 (Anticyclotomic control theorem at $\alpha$).

Let $\alpha:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ be an anticyclotomic
character as in Theorem 1.2.1 such that $\mathcal{L}_{p}^{\rm
BDP}(f/K)(\alpha^{-1})\neq 0$ and $\alpha\equiv 1\;({\rm mod}\,p^{m})$ with
$m\gg 0$. Then $\mathfrak{X}_{\rm Gr}(E/K_{\infty})$ is $\Lambda$-torsion, and
if $\mathcal{F}_{E}\in\Lambda$ generates ${\rm
char}_{\Lambda}(\mathfrak{X}_{\rm Gr}(E/K_{\infty}))$, we have

$\\#\bigl{(}\mathbb{Z}_{p}/\mathcal{F}_{E}(\alpha^{-1})\bigr{)}=\\#\text{\wncyr{W}}(W_{\alpha^{-1}}/K)\cdot\\#{\rm
coker}(\operatorname{loc}_{v})^{2}\cdot\prod_{w\mid
N}c_{w}^{(p)}(\alpha^{-1})\cdot\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])^{2},$

where:

  * •

$c_{w}^{(p)}(\alpha^{-1})=\\#\mathrm{H}^{1}_{\rm ur}(K_{w},W_{\alpha^{-1}})$
is the $p$-part of the Tamagawa number,

  * •

$\text{\wncyr{W}}(W_{\alpha^{-1}}/K)=\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,W_{\alpha^{-1}})/\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,W_{\alpha^{-1}})_{\rm div}$ is the Bloch–Kato Shafarevich–Tate group,

  * •

$\operatorname{loc}_{v}:\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})\to\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{\alpha})_{/\rm tors}$ is the restriction map at $v$ composed
with the projection.

###### Proof.

This follows from $\alpha$-twisted analogues of [JSW17, Prop. 3.2.1] and
[JSW17, Thm. 3.3.1]. More precisely, we begin by verifying the following
properties (i)–(iii):

  * (i)

$\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(K,W_{\alpha^{-1}})_{\rm
div}\simeq\mathbb{Q}_{p}/\mathbb{Z}_{p}$,

  * (ii)

$\mathrm{H}^{1}_{{\rm
ord}}(K_{v},W_{\alpha^{-1}})\simeq\mathbb{Q}_{p}/\mathbb{Z}_{p}$,

  * (iii)

$\operatorname{loc}_{v}:\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,W_{\alpha^{-1}})_{\rm div}\to\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},W_{\alpha^{-1}})$ is surjective.

By Lemma 1.2.3, the nonvanishing of $\mathcal{L}_{p}^{\rm
BDP}(f/K)(\alpha^{-1})$ implies the nonvanishing of $\bm{\kappa}^{\rm
Heeg}_{1}(\alpha)$; the proof of (i) thus follows from [CGS23, Thm. 5.1.1]
(Theorem 1.3.1 below). For (ii), since $\mathrm{H}^{1}_{\rm
ord}(K_{w},W_{\alpha^{-1}})$ is divisible by definition, it suffices to show
that it has $\mathbb{Z}_{p}$-corank one. But this is immediate from local Tate
duality and Tate’s local Euler characteristic formula, noting that because $p$
is a prime of good reduction and $\chi$ is pure of weight $0$, the invariant
subspaces $\mathrm{H}^{0}(K_{v},{\rm Fil}_{w}^{+}W_{\alpha^{-1}})$ and
$\mathrm{H}^{0}(K_{v},({\rm Fil}_{w}^{+}W_{\alpha^{-1}})^{\vee}(1))$ are both
finite. Finally, as in the proof of Lemma 1.2.3, the nonvanishing of
$\mathcal{L}_{p}^{\rm BDP}(f/K)(\alpha^{-1})$ implies that the map
$\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})\rightarrow\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(K_{v},T_{\alpha})_{/{\rm tors}}$ is nonzero, from which (iii) follows.

The same proof as [JSW17, Prop. 3.2.1] thus yields that
$\mathrm{H}^{1}_{\mathcal{F}_{\rm Gr}}(K,W_{\alpha^{-1}})$ is finite, with

(1.6) $\\#\mathrm{H}^{1}_{\mathcal{F}_{\rm
Gr}}(K,W_{\alpha^{-1}})=\\#\text{\wncyr{W}}(W_{\alpha^{-1}}/K)\cdot\\#{\rm
coker}({\rm loc}_{v})^{2}.$

Now put $M_{\alpha^{-1}}=M\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}(\alpha^{-1})$
and $\mathfrak{X}_{\rm
Gr}(E(\alpha^{-1})/K_{\infty})=\mathrm{H}^{1}_{\mathcal{F}_{\rm
Gr}}(K,M_{\alpha^{-1}})^{\vee}$, and denote by ${\rm
Tw}_{\alpha^{-1}}:\Lambda\rightarrow\Lambda$ the $\mathbb{Z}_{p}$-linear
isomorphism given by $\gamma\mapsto\alpha^{-1}(\gamma)\gamma$ for
$\gamma\in\Gamma$. In view of (i)–(iii), the hypotheses in Theorem 3.3.1 of
[JSW17] are satisfied by $V_{\alpha^{-1}}$, from which we conclude (taking
$\Sigma=\emptyset$ in _loc. cit._) that $\mathfrak{X}_{\rm
Gr}(E(\alpha^{-1})/K_{\infty})$ is $\Lambda$-torsion, and letting
$\mathcal{F}_{E(\alpha^{-1})}\in\Lambda$ be a generator of ${\rm
char}_{\Lambda}(\mathfrak{X}_{\rm Gr}(E(\alpha^{-1})/K_{\infty}))$ we have

(1.7)
$\\#\bigl{(}\mathbb{Z}_{p}/\mathcal{F}_{E(\alpha^{-1})}(0)\bigr{)}=\\#\mathrm{H}^{1}_{\mathcal{F}_{\rm
Gr}}(K,W_{\alpha^{-1}})\cdot\prod_{w\mid
N}c_{w}^{(p)}(\alpha^{-1})\cdot\\#\mathrm{H}^{0}(K_{v},W_{\alpha^{-1}})\cdot\\#\mathrm{H}^{0}(K_{\bar{v}},W_{\alpha^{-1}}).$

Since clearly ${\rm Tw}_{\alpha^{-1}}(\mathcal{F}_{E})$ gives a generator of
${\rm char}_{\Lambda}(\mathfrak{X}_{\rm Gr}(E(\alpha^{-1})/K_{\infty})$ and
$\mathrm{H}^{0}(\mathbb{Q}_{p},W_{\alpha^{-1}})=\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}])$
for $\alpha$ sufficiently close to $1$, the combination of (1.6) and (1.7)
yields the equality in the theorem. ∎

As in Theorem 1.2.7, denote by $c_{w}^{(p)}(\alpha)$ the $p$-part of the
Tagamawa number of $W_{\alpha}$ at $w$, and similarly put
$c_{w}^{(p)}=\\#\mathrm{H}^{1}_{\rm ur}(K_{w},W)$ (see also Remark 1.1.1).

######  Lemma 1.2.8.

Assume that $\alpha:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ is an
anticyclotomic character, and $w\nmid p$ is a finite prime of $K$. If
$\alpha\equiv 1\pmod{p^{m}}$, then $c_{w}^{(p)}(\alpha)\equiv
c_{w}^{(p)}\pmod{p^{m}}$.

###### Proof.

By definition, we have

$\mathrm{H}^{1}_{\rm
ur}(K_{w},W_{\alpha})=\mathrm{H}^{1}(\mathbb{F}_{w},(W_{\alpha})^{I_{w}})\simeq(W_{\alpha})^{I_{w}}/({\rm
Fr}_{w}-1)(W_{\alpha})^{I_{w}},$

where $\mathbb{F}_{w}$ is the residue field of $K_{w}$ and $I_{w}\subset
G_{K_{w}}$ is the inertia subgroup at $w$. Since $\alpha$ is unramified at
$w$, we have $(W_{\alpha})^{I_{w}}=W^{I_{w}}$, and using that $\alpha\equiv
1\pmod{p^{m}}$, we obtain

$\mathrm{H}^{1}_{\rm ur}(K_{w},W_{\alpha})/p^{m}\mathrm{H}^{1}_{\rm
ur}(K_{w},W_{\alpha})\simeq\mathrm{H}^{1}_{\rm
ur}(K_{w},W)/p^{m}\mathrm{H}^{1}_{\rm ur}(K_{w},W),$

from which the desired result follows. ∎

######  Remark 1.2.9.

If $\alpha:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ satisfies $\alpha\equiv
1\pmod{p^{m}}$ with $m>\sum_{w\mid N}{\rm ord}_{p}(c^{(p)}_{w})$, Lemma 1.2.8
implies the equality

$\prod_{w\mid N}c_{w}^{(p)}(\alpha)=\prod_{w\mid N}c^{(p)}_{w}.$

####  1.2.3. The Main Conjecture

We recall the statement of the anticyclotomic Main Conjecture for
$\mathcal{L}_{p}^{\rm BDP}(f/K)^{2}$. Note that this can be seen as a special
case of the Iwasawa Main Conjecture for $p$-adic deformations of motives
formulated by Greenberg [Gre94].

######  Conjecture 1.2.10.

Suppose $K$ satisfies (Heeg), (disc), and (spl). Then $\mathfrak{X}_{\rm
Gr}(E/K_{\infty})$ is $\Lambda$-torsion, and

${\rm char}_{\Lambda}(\mathfrak{X}_{\rm Gr}(E/K_{\infty}))\Lambda^{\rm
ur}=(\mathcal{L}_{p}^{\rm BDP}(f/K)^{2})$

as ideals in $\Lambda^{\rm ur}$.

In the following, we shall refer to the statement of Conjecture 1.2.10 with
$\Lambda^{\rm ur}$ replaced by $\Lambda^{\rm ur}\otimes\mathbb{Q}_{p}$ as the
_rational anticyclotomic Main Conjecture_.

Combining the results of Lemma 1.2.3 and Theorem 1.2.7 we arrive at the main
result of this section.

######  Corollary 1.2.11.

Let $\alpha:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ be an anticyclotomic
character as in Definition 1.2.2 with $\mathcal{L}_{p}^{\rm
BDP}(f/K)(\alpha^{-1})\neq 0$ and $\alpha\equiv 1\;({\rm mod}\,p^{m})$ with
$m\gg 0$. If Conjecture 1.2.10 holds, then

$\\#({\rm H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})/\mathbb{Z}_{p}\cdot\bm{\kappa}^{\rm
Heeg}_{1}(\alpha))^{2}=\\#\text{\wncyr{W}}(W_{\alpha^{-1}}/K)\cdot\prod_{w\mid
N}c_{w}^{(p)}(\alpha^{-1})\cdot\\#(\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]))^{4}.$

We conclude this section by recording some results on Conjecture 1.2.10. The
first cases of the conjecture were proved by X. Wan [Wan21], assuming that the
$G_{K}$-action on $T_{p}E$ is surjective and some mild ramification hypotheses
on $E[p]$. More recently, we have the following.

######  Theorem 1.2.12.

Suppose $K$ satisfies (Heeg), (disc), and (spl).

  * (i)

If $E[p]^{ss}=\mathbb{F}_{p}(\phi)\oplus\mathbb{F}_{p}(\psi)$ as
$G_{\mathbb{Q}}$-modules with $\phi|_{G_{p}}\neq\mathds{1},\omega$, then
Conjecture 1.2.10 holds.

  * (ii)

If $p>3$ satisfies (irr), then the rational anticyclotomic Main Conjecture
holds.

  * (iii)

If $p>3$ satisfies (sur), then Conjecture 1.2.10 holds.

###### Proof.

This is shown in [CGS23] in situation (i), and in [BCS23] in situations (ii)
and (iii). ∎

###  1.3. The Kolyvagin system bound with error term

For a positive integer $e$ put

$\mathcal{L}_{e}=\\{\ell\in\mathcal{L}_{0}\;\colon\;a_{\ell}\equiv\ell+1\equiv
0\;({\rm mod}\,p^{e})\\}.$

We shall need the following mild extension of [CGS23, Thm. 5.1.1].

######  Theorem 1.3.1.

Suppose $\mathcal{L}\subset\mathcal{L}^{\rm Heeg}$ satisfies
$\mathcal{L}_{e}\subset\mathcal{L}$ for $e\gg 0$. Let
$\alpha:\Gamma\to\mathbb{Z}_{p}^{\times}$ be an anticyclotomic character such
that $\alpha\equiv 1\;({\rm mod}\,p^{m})$. Suppose that there is a collection
of cohomology classes

$\\{\tilde{\kappa}_{n}\in\mathrm{H}^{1}(K,T_{\alpha}/I_{n}T_{\alpha})\;\colon\;n\in\mathcal{N}(\mathcal{L})\\}$

with $\tilde{\kappa}_{1}\neq 0$ and that there is an integer $t\geq 0$,
independent of $n$, such that
$\\{p^{t}\tilde{\kappa}_{n}\\}_{n\in\mathcal{N}(\mathcal{L})}$ is a Kolyvagin
system for $T_{\alpha}$ and the Selmer structure $\mathcal{F}_{\rm ord}$. Then
there exist non-negative integers $\mathcal{M}$ and $\mathcal{E}$, depending
only on $T_{p}E$, such that if $m\geq\mathcal{M}$ then
$\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(K,T_{\alpha})$ has
$\mathbb{Z}_{p}$-rank one, and there is a finite $\mathbb{Z}_{p}$-module $M$
such that

${\rm H}^{1}_{\mathcal{F}_{\rm
ord}}(K,W_{\alpha^{-1}})\simeq\mathbb{Q}_{p}/\mathbb{Z}_{p}\oplus M\oplus M$

with

${\rm
length}_{\mathbb{Z}_{p}}(M)\leq\operatorname{ind}(\tilde{\kappa}_{1})+\mathcal{E},$

where $\operatorname{ind}(\tilde{\kappa}_{1})={\rm
length}_{\mathbb{Z}_{p}}\bigl{(}{\rm H}^{1}_{\mathcal{F}_{\rm
ord}}(K,T_{\alpha})/\mathbb{Z}_{p}\cdot\tilde{\kappa}_{1}\bigr{)}$. Moreover,
if (sur) holds, then $\mathcal{E}=0$.

###### Proof.

Take $e\gg 0$ so that $\tilde{\kappa}_{1}\not\equiv 0\;({\rm mod}\,p^{e})$ and
such that $\tilde{\mathcal{L}}:=\mathcal{L}_{e+t}$ is contained in
$\mathcal{L}$, and put $\tilde{\mathcal{N}}=\mathcal{N}(\tilde{\mathcal{L}})$.
As in the proof of [How04b, Thm. 2.2.2], the collection of classes
$\\{\tilde{\kappa}_{n}\\}_{n\in\tilde{\mathcal{N}}}$ gives a (non-trivial)
Kolyvagin system for $(T/p^{e}T,\mathcal{F}_{\rm ord},\tilde{\mathcal{L}})$,
and a slight modification of the argument yielding [CGS23, Thm. 5.1.1] applied
to this system yields the result. The claim that $\mathcal{E}=0$ when (sur)
holds follows from [CGS23, Rem. 3.3.5]. ∎

##  2\. Proofs of Theorem A and Theorem B

###  2.1. Kolyvagin’s conjecture

We can now prove our main result towards Kolyvagin’s original conjecture.

###### Proof of Theorem A.

Assume that the rational anticyclotomic Main Conjecture holds. In particular,
letting $\mathcal{F}_{p}(E/K)\in\Lambda$ be any generator of ${\rm
char}_{\Lambda}(\mathfrak{X}_{\rm Gr}(E/K_{\infty}))$ we have the divisibility

(2.1) $\mathcal{F}_{p}(E/K)\cdot p^{R}=\mathcal{L}_{p}^{\rm BDP}(f/K)^{2}\cdot
h$

for some $R\geq 0$ and $h\in\Lambda^{\rm ur}$. Let also

$x:=v_{p}(\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]))=v_{p}(1-\alpha_{p}).$

Assume for contradiction that $\kappa_{n}^{\rm Heeg}=0$ for all
$n\in\mathcal{N}_{\rm Heeg}$. Take $t\in\mathbb{Z}_{>0}$ such that

(2.2) $t>\tfrac{1}{2}\sum_{w\mid
N}\operatorname{ord}_{p}(c_{w}^{(p)})+R+\mathcal{E},$

where $\mathcal{E}$ is the error term from Theorem 1.3.1. Consider a character
$\alpha=\alpha_{m}:\Gamma\to\mathbb{Z}_{p}^{\times}$ as in Definition 1.2.2
such that the following conditions hold:

  * (a)

$\mathcal{L}_{p}^{\rm BDP}(f/K)(\alpha^{-1})\neq 0$,

  * (b)

$m>\mathcal{M}+2t+2x$,

  * (c)

The conclusion of Corollary 1.2.11 holds.

By the last claim in Theorem 1.2.1, we can always find such $\alpha$. In
particular, by Corollary 1.2.3 we then have $\bm{\kappa}_{1}^{\rm
Heeg}(\alpha)\neq 0$, and moreover

$\prod_{w\mid N}c_{w}^{(p)}(\alpha)=\prod_{w\mid N}c_{w}^{(p)}$

by Lemma 1.2.8 and Remark 1.2.9.

Let $\mathcal{N}=\mathcal{N}(\mathcal{L}_{m})$, where
$\mathcal{L}_{m}=\\{\ell\in\mathcal{L}_{0}\;\colon\;a_{\ell}\equiv\ell+1\equiv
0\;({\rm mod}\,p^{m})\\}$, so $\mathscr{M}(n)\geq m$ for all
$n\in\mathcal{N}$. Since we assume that all $\kappa_{n}^{\rm Heeg}$ vanish, by
Lemma 1.1.5 and the assumption $m>t+2x$ we find that $\bm{\kappa}_{n}^{\rm
Heeg}(\alpha)\equiv 0\;({\rm mod}\,p^{t+2x})$ for every $n\in\mathcal{N}$. We
can therefore consider a collection of cohomology classes
$\\{\tilde{\kappa}_{n,\alpha}\in\mathrm{H}^{1}(K,T_{\alpha}/I_{n}T_{\alpha})\\}_{n\in\mathcal{N}}$
such that

$p^{t+2x}\cdot\tilde{\kappa}_{n,\alpha}=\bm{\kappa}_{n}^{\rm Heeg}(\alpha).$

Note that these classes are not canonically defined, but their order (and the
order of their localisations) is. Since $\mathrm{H}^{1}(K,T_{\alpha})$ is
torsion-free by hypothesis (tor), the nonvanishing of $\bm{\kappa}_{1}^{\rm
Heeg}(\alpha)$ implies that the bottom class $\tilde{\kappa}_{1,\alpha}$ is
non-zero and its index is equal to $\operatorname{ind}(\bm{\kappa}_{1}^{\rm
Heeg}(\alpha))-t-2x$. Hence from Theorem 1.3.1 we get

(2.3) $\tfrac{1}{2}\,{\rm
length}_{\mathbb{Z}_{p}}(\text{\wncyr{W}}(W_{\alpha^{-1}}/K))\leq\operatorname{ind}(\bm{\kappa}_{1}^{\rm
Heeg}(\alpha))-t-2x+\mathcal{E}.$

On the other hand, from the combination of Lemma 1.2.3, Lemma 1.2.5, Theorem
1.2.7, and (2.1), we obtain

(2.4) $\operatorname{ind}(\bm{\kappa}_{1}^{\rm
Heeg}(\alpha))\leq\tfrac{1}{2}{\rm
length}_{\mathbb{Z}_{p}}(\text{\wncyr{W}}(W_{\alpha^{-1}}/K))+\tfrac{1}{2}\sum_{w\mid
N}\operatorname{ord}_{p}(c_{w}^{(p)})+R+2x,$

using $m>\sum_{w\mid N}{\rm ord}_{p}(c_{w}^{(p)})$ to apply the observation in
Remark 1.2.9. Combining (2.3) and (2.4), we get

$t\leq\tfrac{1}{2}\sum_{w\mid N}{\rm ord}_{p}(c_{w}^{(p)})+R+\mathcal{E},$

which contradicts (2.2), thereby proving the result. ∎

######  Remark 2.1.1.

In particular, if $E[p]$ satisfies (sur), the above argument shows that
$\mathscr{M}_{\infty,\mathcal{N}}\leq\tfrac{1}{2}\sum_{w\mid N}{\rm
ord}_{p}(c_{w}^{(p)})$, where
$\mathscr{M}_{\infty,\mathcal{N}}=\mathscr{M}_{\infty}(\\{\kappa_{n}^{\rm
Heeg}\\}_{n})$ is the divisibility index of $\\{\kappa_{n}^{\rm
Heeg}\\}_{n\in\mathcal{N}}$ for any
$\mathcal{N}\supset\mathcal{N}(\mathcal{L}_{m})$.

###  2.2. The refined Kolyvagin’s conjecture

Let $R$ be the ring of integers of a finite extension of $\mathbb{Q}_{p}$ with
maximal ideal $\mathfrak{m}$ and uniformiser $\varpi\in\mathfrak{m}$. Let
$\alpha:\Gamma_{K}\to R^{\times}$ be a character such that $\alpha\equiv
1\;({\rm mod}\,\mathfrak{m}^{m})$, and put
$T_{\alpha}=T_{p}E\otimes_{\mathbb{Z}_{p}}R(\alpha)$, as above. In this
section, we do not assume $m\gg 0$, in particular we could also have $m=1$.

Similarly as in the introduction, if
$\bm{\kappa}=\\{\kappa_{n}\\}_{n\in\mathcal{N}}$ is a Kolyvagin system for
$(T_{\alpha},\mathcal{F}_{\rm ord},\mathcal{L})$, for each
$n\in\mathcal{N}=\mathcal{N}(\mathcal{L})$, define
$\mathscr{M}(n)\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$ by
$\mathscr{M}(n)=\infty$ if $\kappa_{n}=0$, and by

$\mathscr{M}(n)=\max\\{\mathscr{M}\colon\kappa_{n}\in\mathfrak{m}^{\mathscr{M}}\mathrm{H}^{1}(K,T/I_{n}T)\\}=\operatorname{ind}(\kappa_{n})$

otherwise. For any $\mathcal{N}^{\prime}\subset\mathcal{N}$, put
$\mathscr{M}_{r,\mathcal{N}^{\prime}}=\min\\{\mathscr{M}(n)\colon\nu(n)=r,n\in\mathcal{N}^{\prime}\\}$,
where $\nu(n)$ denotes the number of prime factors of $n$. Firstly, one shows,
similarly as in [Kol91], that
$\mathscr{M}_{r,\mathcal{N}^{\prime}}\geq\mathscr{M}_{r+1,\mathcal{N}^{\prime}}\geq
0$ for all $r\geq 0$. Put

$\mathscr{M}_{\infty,\mathcal{N}^{\prime}}(\bm{\kappa})=\lim_{r\to\infty}\mathscr{M}_{r,\mathcal{N}^{\prime}}.$

We let
$\mathscr{M}_{\infty}(\bm{\kappa})=\mathscr{M}_{\infty,\mathcal{N}}(\bm{\kappa})$.
Note that
$\mathscr{M}_{\infty,\mathcal{N}^{\prime}}(\bm{\kappa})\geq\mathscr{M}_{\infty}(\bm{\kappa})$
for every $\mathcal{N}^{\prime}\subset\mathcal{N}$.

In addition to the ingredients that went into the proof of Theorem A, the
proof of the refinement in Theorem B relies on the independence of the integer
$\mathscr{M}_{\infty,\mathcal{N}^{\prime}}$ from
$\mathcal{N}^{\prime}\subset\mathcal{N}$ and an exact formula for the length
of $\text{\wncyr{W}}(W_{\alpha^{-1}}/K)$ in the spirit of Kolyvagin’s
structure theorem [Kol91, Thm. 1].

######  Proposition 2.2.1.

Assume that $p\geq 3$ and that (sur) holds. Let
$\bm{\kappa}=\\{\kappa_{n}\\}_{n\in\mathcal{N}}$ be a Kolyvagin system for
$(T,\mathcal{F},\mathcal{L})$. Let $e>0$ be any integer such that
$\mathcal{L}_{e}\subset\mathcal{L}$ and let
$\mathcal{N}^{(e)}=\mathcal{N}(\mathcal{L}_{e})$. Then

$\mathscr{M}_{\infty,\mathcal{N}^{(e)}}(\bm{\kappa})=\mathscr{M}_{\infty,\mathcal{N}}(\bm{\kappa}).$

######  Theorem 2.2.2.

Assume $p>3$. Suppose $\mathcal{L}\subset\mathcal{L}_{0}$ and let
$\bm{\kappa}=\\{\kappa_{n}\\}_{n\in\mathcal{N}}$ be a Kolyvagin system for
$(T,\mathcal{F},\mathcal{L})$ with $\kappa_{1}\neq 0$. Assume further that
(sur) holds. Then

${\rm
length}_{\mathbb{Z}_{p}}\bigl{(}\text{\wncyr{W}}(W_{\alpha^{-1}}/K)\bigr{)}=2\bigl{(}\mathscr{M}_{0}(\bm{\kappa})-\mathscr{M}_{\infty}(\bm{\kappa})\bigr{)},$

where $\mathscr{M}_{0}(\bm{\kappa})=\operatorname{ind}(\kappa_{1})$.

######  Remark 2.2.3.

Note that this theorem applies to any anticylotomic character $\alpha$ of
$\Gamma$, as will be clear from the proof below: the need to distinguish
trivial characters sufficiently close to the trivial character in [CGS23,
5.1.1] (close with respect to [CGLS22, Thm. 3.3.8], where essentially the
maximum $m$ such that $\alpha\equiv 1\mod\mathfrak{m}^{m}$ was part of the
error term) disappears when (sur) holds. In particular, Theorem 2.2.2 applies
for characters $\alpha:\Gamma\to\Lambda/(\mathfrak{P}+p^{m})^{\times}$ for any
height one prime ideal $\mathfrak{P}$ of $\Lambda$.

Let $\mathcal{F}=\mathcal{F}_{\rm ord}$ and $T=T_{\alpha}$. For any $k\geq 1$,
let $R^{(k)}=R/\mathfrak{m}^{k}$ and $T^{(k)}=T/\mathfrak{m}^{k}$. Recall
that, by [CGLS22, Prop. 3.3.2], there is an integer $\epsilon\in\\{0,1\\}$
such that for all $k$ and every every $n\in\mathcal{N}^{(k)}$ there is an
$R^{(k)}$-module $M^{(k)}(n)$ such that

(2.5)
$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)})\simeq(R/\mathfrak{m}^{k})^{\epsilon}\oplus
M^{(k)}(n)\oplus M^{(k)}(n).$

Moreover, by [CGLS22, Thm. 3.3.8], if
$\bm{\kappa}=\\{\kappa_{n}\\}_{n\in\mathcal{N}}$ is a Kolyvagin system for
$(T,\mathcal{F},\mathcal{L})$ with $\kappa_{1}\neq 0$, then $\epsilon=1$ and
for $k\gg 0$, ${\rm exp}(M^{(k)}(1))<k$. We also denote by
$\bm{\kappa}^{(k)}=\\{\kappa_{n}^{(k)}\\}_{n\in\mathcal{N}^{(k)}}$ the
Kolyvagin system for $(T^{(k)},\mathcal{F},\mathcal{L}^{(k)})$ obtained from
$\bm{\kappa}$, where
$\mathcal{L}^{(k)}=\\{\ell\in\mathcal{L}:I_{\ell}\subset\mathfrak{m}^{k}\\}$.
In particular, if $\mathcal{L}=\mathcal{L}_{\rm Heeg}$ and $R=\mathbb{Z}_{p}$,
then $\mathcal{L}^{(k)}=\mathcal{L}_{k}$.

We adopt the following notation: a finite torsion $R$-module $X$ is isomorphic
to a sum of cyclic $R$-modules:
$X\simeq\oplus_{i=1}^{s(X)}R/\mathfrak{m}^{d_{i}(X)}$ for some uniquely-
determined integers $d_{i}(X)\geq 0$, with $d_{1}(X)\geq
d_{2}(X)\geq\cdots\geq d_{s(X)}(X)$. For any $x\in X$, let the order and index
of $x$ in $X$ be

$\operatorname{ord}(x):=\min\\{n\geq 0:\varpi^{n}\cdot x=0\\},\ \ \
\operatorname{ind}(x):=\max\\{t\geq 0:x\in\mathfrak{m}^{t}X\\}.$

For an integer $t\geq 0$ we let $\rho_{t}(X)=\\#\\{i:d_{i}(X)>t\\}$. Assume
$\alpha\equiv 1\mod\mathfrak{m}^{m}$ for some $m\geq 0$. For
$n\in\mathcal{N}^{(k)}$ with $k\geq m$, we let

$\rho(n)^{\pm}:=\rho_{0}(\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(m)})^{\pm}),\ \
\ \rho(n):=\rho(n)^{+}+\rho(n)^{-},$

that is, we are counting the summands of the $\pm$-components of
$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(m)})$, on which, since $\alpha\equiv
1\mod\mathfrak{m}^{m}$, there is a well-defined action of the complex
conjugation. Note that, by [CGLS22, Lem. 3.3.1], we are identifying

$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(m)})\simeq\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)})[\mathfrak{m}^{m}]$

and, in particular, $\rho(n)$ does not depend on $k$ or $\alpha$.

Note also that, by (2.5), if $\rho(n)=1$, we must have
$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)})\simeq R/\mathfrak{m}^{k}$, i.e.
$M^{(k)}(n)=\\{0\\}$. If this is the case, we say that $n$ is a _core vertex_
, as in [MR04, Def. 4.1.8] (and [Zan19] in the Heegner point case).

###### Proof of Proposition 2.2.1.

The essential idea of this proof can be found in the work of Kolyvagin.
Indeed, the case $\alpha=1$ can be easily extracted from the proof of [McC91,
Prop. 5.2].

Let $r\geq 0$ be an integers such that
$\mathscr{M}_{r,\mathcal{N}}^{(k)}=\mathscr{M}_{\infty,\mathcal{N}}^{(k)}$.
Let $M:=\mathscr{M}_{r,\mathcal{N}^{(k)}}$ and
$\mathcal{N}_{r}:=\\{n\in\mathcal{N}:\nu(n)=r\\}$. By definition, there exists
some $n\in\mathcal{N}_{r}$ such that $\mathscr{M}(n)=M$. Necessarily
$n\in\mathcal{N}^{(M+1)}$, so we may assume that $e>M+1$. Suppose there exists
a prime $\ell_{0}\mid n$ such that $\ell_{0}\not\in\mathcal{L}_{e}$. We will
show that there exists a prime $\ell\in\mathcal{L}_{e}$ such that
$n^{\prime}=\ell n/\ell_{0}\in\mathcal{N}_{r}$ is such that
$\mathscr{M}(n^{\prime})=M$. Replacing $n$ with $n^{\prime}$ and repeating as
necessary, we arrive at an $n\in\mathcal{N}_{r}^{(e)}$ such that
$\mathscr{M}(n)=M$, whence the proposition.

We have
$\kappa^{(M+1)}_{n}\in\mathfrak{m}^{M}\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(M+1)})$
but
$\kappa^{(M+1)}_{n}\not\in\mathfrak{m}^{M+1}\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(M+1)})$.
From the identification
$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(1)})=\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(M+1)})[\mathfrak{m}]$,
we see that $\kappa^{(M+1)}_{n}$ is identified with a non-zero class
$0\neq\bar{\kappa}\in\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(1)})$. Complex
conjugation acts on $T^{(1)}$ and hence on
$\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(1)})$ and we may fix a sign $s=\pm$
such that $\bar{\kappa}^{s}\neq 0$.

Let $0\neq
c\in\mathrm{H}^{1}_{\mathcal{F}(n/\ell_{0})^{\ell_{0}}}(K,T^{(1)})^{-s}$. Here
the superscript $\ell_{0}$ on the $\mathcal{F}(n/\ell_{0})$ means that we
impose no restrictions on the classes at the prime $\ell_{0}$. The existence
of such a $c$ follows easily from global duality and the fact that
$\ell_{0}\in\mathcal{L}$.

Now let $\ell\in\mathcal{L}_{e}$, $\ell\nmid n$, be a prime such that
$\operatorname{loc}_{\ell}(\bar{\kappa}^{s})\neq 0$ and
$\operatorname{loc}_{\ell}(c)\neq 0$. The existence of such a prime $\ell$
follows from [CGS23, Prop. 5.3.1] (applied with $m=1$ and $k=e$ in the
notation of _op. cit._) noting that the error term is zero since we are
assuming (sur) holds. Then $n\ell\in\mathcal{N}_{r+1}^{(M+1)}$ and so
$\kappa_{n\ell}^{(M+1)}$ is defined. By the Kolyvagin relations,
$\operatorname{loc}_{\ell}(\kappa_{n\ell}^{(M+1)})$ has the same order as
$\operatorname{loc}_{\ell}(\kappa_{n}^{(M+1)})$, and the latter equals $1$ as
$\operatorname{loc}_{\ell}(\bar{\kappa})\neq 0$. In particular
$\kappa_{n\ell}^{(M+1)}\not\in
p^{M+1}\mathrm{H}^{1}_{\mathcal{F}(n\ell)}(K,T^{(M+1)})$. On the other hand,
$n\ell\in\mathcal{N}_{r+1}$, so
$\mathscr{M}(n\ell)\geq\mathscr{M}_{r+1,\mathcal{N}}=\mathscr{M}_{r,\mathcal{N}}=M$
by the choice of $r$. It follows that
$\kappa_{n\ell}^{(M+1)}\in\mathfrak{m}^{M}\mathrm{H}^{1}_{\mathcal{F}(n\ell)}(K,T^{(M+1)})$
and so defines a non-zero class
$\bar{\kappa}_{\ell}\in\mathrm{H}^{1}_{\mathcal{F}(n\ell)}(K,T^{(1)})$. The
Kolyvagin relations relate $\operatorname{loc}_{\ell}(\kappa_{n})$ with
$\operatorname{loc}_{\ell}(\kappa_{n\ell})$ via the usual finite-singular
(iso)morphism. Since this morphism intertwines the eigenspaces for complex
conjugation (see for example [Nek07, 5.15.1], recalling that
$\ell\in\mathcal{L}_{e}$ and $e>M+1$ imply that $\operatorname{Frob}_{\ell}$
acts as complex conjugation on $T^{(M+1)}$), this shows that
$\operatorname{loc}_{\ell}(\bar{\kappa}_{\ell}^{\pm})\neq 0$ if and only if
$\operatorname{loc}_{\ell}(\bar{\kappa}^{\mp})\neq 0$, so
$\operatorname{loc}_{\ell}(\bar{\kappa}_{\ell}^{-s})\neq 0$.

By global duality we have

$0=\sum_{v}\langle\operatorname{loc}_{v}(c),\operatorname{loc}_{v}(\bar{\kappa}_{\ell}^{-s})\rangle_{v}=\langle\operatorname{loc}_{\ell}(c),\operatorname{loc}_{\ell}(\bar{\kappa}_{\ell}^{-s})\rangle_{\ell}+\langle\operatorname{loc}_{\ell_{0}}(c),\operatorname{loc}_{\ell_{0}}(\bar{\kappa}_{\ell}^{-s})\rangle_{\ell_{0}}.$

As
$0\neq\operatorname{loc}_{\ell}(c)\in\mathrm{H}^{1}_{\mathrm{ur}}(K,T^{(1)})^{-s}$
and
$0\neq\operatorname{loc}_{\ell}(\bar{\kappa}_{\ell}^{-s})\in\mathrm{H}^{1}_{\mathrm{tr}}(K,T^{(1)})^{-s}$,
it follows that
$\langle\operatorname{loc}_{\ell}(c),\operatorname{loc}_{\ell}(\bar{\kappa}_{\ell}^{-s})\rangle_{\ell}\neq
0$. This in turn implies that
$\langle\operatorname{loc}_{\ell_{0}}(c),\operatorname{loc}_{\ell_{0}}(\bar{\kappa}_{\ell}^{-s})\rangle_{\ell_{0}}\neq
0$, which means that
$\operatorname{loc}_{\ell_{0}}(\bar{\kappa}_{\ell}^{-s})\neq 0$. So the order
of $\operatorname{loc}_{\ell_{0}}(\kappa_{n\ell}^{(M+1)})$ is $1$. But then
the Kolyvagin relations imply that
$\operatorname{loc}_{\ell}(\kappa_{n\ell/\ell_{0}}^{(M+1)})$ has order $1$,
and so $\mathscr{M}(n\ell/\ell_{0})=M$ by the same reasoning that showed that
$\mathscr{M}(n\ell)=M$. ∎

For the proof of Theorem 2.2.2, we will also need the following result.

######  Lemma 2.2.4.

Assume $p>3$. If $k\gg 0$ and
$\langle\kappa_{h}^{(k)}\rangle=\mathfrak{m}^{j+\operatorname{length}_{R}(M^{(k)}(h))}\mathrm{H}^{1}_{\mathcal{F}(h)}(K,T^{(k)})$
for some $h\in\mathcal{N}$, then

$\langle\kappa_{n}^{(k)}\rangle=\mathfrak{m}^{j+\operatorname{length}_{R}(M^{(k)}(n))}\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)})$
for every $n\in\mathcal{N}^{(2k-1)}$.

###### Proof.

The proof runs along the lines of [Zan19, $\S$ 2.2]. First, one shows that for
every two core vertexes $n,m\in\mathcal{N}^{(k)}$, the $R^{(k)}$-module
spanned by $\kappa_{n}^{(k)},\kappa_{m}^{(k)}$ are isomorphic, i.e.,

$\rho(n)=\rho(m)=1\ \ \Rightarrow\ \
R/\mathfrak{m}^{k-\operatorname{ind}(\kappa_{n}^{(k)})}\simeq\langle\kappa_{n}^{(k)}\rangle\simeq\langle\kappa_{m}^{(k)}\rangle\simeq
R/\mathfrak{m}^{k-\operatorname{ind}(\kappa_{m}^{(k)})}.$

This follows as in [Zan19, Cor. 2.2.13]. From this one deduces that for any
$k$ and any $n\in\mathcal{N}^{(k)}$ with $\rho(n)=1$,

$\langle\kappa_{n}^{(k)}\rangle\simeq R/\mathfrak{m}^{k-d}$

for some $d$ independent on $k$ and $n$. One then shows that

(2.6)
$\langle\kappa_{n}^{(k)}\rangle=\mathfrak{m}^{d+\operatorname{length}_{R}(M^{(k)}(n))}\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(k)})\
\ \text{if $n\in\mathcal{N}^{(2k-1)}$}.$

Following the proof of [Zan19, Lemma 2.3.1], one proves this statement without
the assumption that $\rho(n)=1$ using [How04a, Lemma 1.6.4] and induction on
$r(k,n)=k+\max\\{\rho(n)^{+},\rho(n)^{-}\\}$. ∎

###### Proof of Theorem 2.2.2.

As in the proof of [CGS23, Thm. 5.1.1], let
$s(n)=\dim_{\mathbb{F}}\mathrm{H}^{1}_{\mathcal{F}(n)}(K,T^{(1)})-1=2\cdot\dim_{\mathbb{F}}M(n)[\mathfrak{m}]$,
where $\mathbb{F}=R/\mathfrak{m}$. Let $\rho=\rho_{0}(1)$. Note that
$\rho\geq\dim_{\mathbb{F}}\mathrm{H}^{1}_{\mathcal{F}}(K,T^{(1)})=s(1)+1$. By
[CGLS22, Thm 3.3.8], we can write
$\mathrm{H}^{1}_{\mathcal{F}}(K,W_{\alpha^{-1}})\simeq{\rm Frac}(R)/R\oplus
M(1)\oplus M(1)$, where $M(1)$ is a finite torsion $R$-module. Let $k$ be a
fixed integer such that

(2.7) $k>\mathrm{length}_{R}(M(1))+\operatorname{ind}(\kappa_{1})+m$

(recall that $\alpha\equiv 1\pmod{\mathfrak{m}^{m}}$). In particular,
$\mathrm{H}^{1}_{\mathcal{F}}(K,T^{(k)})\simeq R^{(k)}\oplus M(1)\oplus M(1)$.
In _op. cit._ we found a sequence of integers
$1=n_{0},n_{1},...,n_{\rho}\in\mathcal{N}^{(k)}$ (which we can actually assume
to be in $\mathcal{N}^{(2k-1)}$ applying [CGS23, Prop. 5.3.1] with a different
choice of $k$) satisfying certain conditions (denoted by (a)-(g) in _op.
cit._), in particular, if (sur) holds, the error term $e$ is zero and we
found:

  * (e)

$\operatorname{ind}(\kappa_{n_{i-1}}^{(k)})\geq\operatorname{ind}(\kappa_{n_{i}}^{(k)})$;

  * (f)

$\rho(n_{i})\leq\rho(n_{i-1})$, and $\rho(n_{i})=\rho(n_{i-1})>1$ only if
$\rho_{0}(\mathrm{H}^{1}_{\mathcal{F}(n_{i})}(K,T^{(m)})^{\pm})\geq 1$;

  * (g)

if $\rho(n_{i-2})>1$ then $\rho(n_{i})<\rho(n_{i-2})$.

From (g), we get that $\rho(n_{\rho})=1$, implying in particular that

$\mathrm{H}^{1}_{\mathcal{F}(n_{\rho})}(K,T^{(k)})\simeq R/\mathfrak{m}^{k},\
\ \text{ i.e.}\ \ M^{(k)}(n_{\rho})=0.$

Note that we do not need to assume $m$ to be large in this case, as the proof
of (B) in [CGS23] only requires $m>\mathcal{M}$ and $\mathcal{M}=0$ when (sur)
holds. In particular, we can take $m=1$ and work with the
$\mathfrak{m}$-torsion in the case where the character $\alpha$ is attached to
a height one prime ideal of $\Lambda$ different from $(p)$.

Moreover, by Lemma 2.2.4 and the choice (2.7), we have

(2.8)
$j+\mathrm{length}_{R}(M(1))=\operatorname{ind}(\kappa_{1})<k-\operatorname{length}_{R}(M(1))<k,\
\ \ j=\operatorname{ind}(\kappa_{n_{\rho}}^{(k)}).$

In order to prove the theorem, we need to show that
$\operatorname{ind}(\kappa_{n_{\rho}}^{(k)})=\mathscr{M}_{\infty}(\bm{\kappa})$.
By (2.8), $\operatorname{ind}(\kappa_{n_{\rho}}^{(k)})<k$, so

$\mathscr{M}(n_{\rho})=\operatorname{ind}(\kappa_{n_{\rho}})=\operatorname{ind}(\kappa_{n_{\rho}}^{(k)})=j.$

Let $\mathcal{N}^{\prime}=\mathcal{N}^{2k-1}$. Let $i\geq 0$ and let
$n\in\mathcal{N}^{\prime}_{i}$ such that
$\mathscr{M}(n)=\mathscr{M}_{\rho+i,\mathcal{N}^{\prime}}$. As
$\mathscr{M}(n)\geq\operatorname{ind}(\kappa^{(k)}_{n})$ by the definition of
$\mathscr{M}(n)$ and since
$\operatorname{ind}(\kappa^{(k)}_{n})=\min\\{k,j+\operatorname{length}_{R}(M(n))\\}$
by Lemma 2.2.4, we have

$\mathscr{M}_{\rho+i,\mathcal{N}^{\prime}}=\mathscr{M}(n)\geq\operatorname{ind}(\kappa^{(k)}_{n})=\min\\{k,j+\operatorname{length}_{R}(M(n))\\}\geq
j=\mathscr{M}(n_{\rho})\geq\mathscr{M}_{\rho,\mathcal{N}^{\prime}}.$

Taking $i=0$, it follows that
$\mathscr{M}_{\rho,\mathcal{N}^{\prime}}=\mathscr{M}(n_{\rho})$. Furthermore,
since
$\mathscr{M}_{\rho+i,\mathcal{N}^{\prime}}\leq\mathscr{M}_{\rho,\mathcal{N}^{\prime}}$,
it follows that
$\mathscr{M}_{\rho+i,\mathcal{N}^{\prime}}=\mathscr{M}_{\rho,\mathcal{N}^{\prime}}$
for all $i\geq 0$. Hence
$\mathscr{M}_{\infty,\mathcal{N}^{\prime}}=\mathscr{M}_{\rho,\mathcal{N}^{\prime}}=\mathscr{M}(n_{\rho})$.
It then follows from Proposition 2.2.1 that
$\mathscr{M}_{\infty}(\bm{\kappa})=\mathscr{M}_{\infty,\mathcal{N}^{\prime}}=\mathscr{M}(n_{\rho})$.
∎

Now are ready to prove the refined Kolyvagin’s conjecture.

###### Proof of Theorem B.

Assume (sur), $p>3$ and Conjecture 1.2.10. By the nonvanishing of
$\mathcal{L}_{p}^{\rm BDP}(f/K)$, we can choose a character
$\alpha=\alpha_{m}:\Gamma\rightarrow\mathbb{Z}_{p}^{\times}$ as in
$\S\ref{subsec:tw}$ with $m>\sum_{w\mid N}{\rm
ord}_{p}(c_{w}^{(p)})+\mathscr{M}_{\infty}(\kappa^{\rm Heeg})$, and such that
$\mathcal{L}_{p}^{\rm BDP}(f/K)(\alpha^{-1})\neq 0$. By Corollary 1.2.11 and
Remark 1.2.9 we then have

(2.9) ${\rm
length}_{\mathbb{Z}_{p}}(\text{\wncyr{W}}(W_{\alpha^{-1}}/K))=2\cdot\operatorname{ind}(\bm{\kappa}_{1}^{\rm
Heeg}(\alpha))-4x-\sum_{w\mid N}{\rm ord}_{p}(c_{w}^{(p)}),$

where
$x=v_{p}(\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]))=v_{p}(1-\alpha_{p})$.
Together with Theorem 2.2.2 for the Kolyvagin system
$\bm{\kappa}(\alpha):=\\{\bm{\kappa}^{\rm
Heeg}_{n}(\alpha)\\}_{n\in\mathcal{N}}$, where
$\mathcal{N}=\mathcal{N}(\mathcal{L}_{m})$,

$\mathscr{M}_{\infty,\mathcal{N}}(\bm{\kappa}(\alpha))=\tfrac{1}{2}\sum_{w\mid
N}{\rm ord}_{p}(c_{w}^{(p)})+2x.$

By Remark 2.1.1 we have $\mathscr{M}_{\infty,\mathcal{N}}(\kappa^{\rm
Heeg})\leq\tfrac{1}{2}\sum_{w\mid N}{\rm ord}_{p}(c_{w}^{(p)})$. If the
inequality was strict, there would exist $n\in\mathcal{N}$ such that
$\operatorname{ind}(\kappa_{n}^{\rm Heeg})\lneq\tfrac{1}{2}\sum_{w\mid N}{\rm
ord}_{p}(c_{w}^{(p)})$. By the congruence of Lemma 1.1.5,
$\operatorname{ind}(\kappa_{n}^{\rm
Heeg})+2x=\operatorname{ind}(\bm{\kappa}_{n}^{\rm Heeg}(\alpha))$, and hence
by the choice of $m$ and the definition of
$\mathscr{M}_{\infty,\mathcal{N}}(\bm{\kappa}(\alpha))$, this would imply
$\mathscr{M}_{\infty,\mathcal{N}}(\bm{\kappa}(\alpha))\lneq\tfrac{1}{2}\sum_{w\mid
N}{\rm ord}_{p}(c_{w}^{(p)})+2x$, giving a contradiction. Therefore we have
$\mathscr{M}_{\infty,\mathcal{N}}(\kappa^{\rm Heeg})=\tfrac{1}{2}\sum_{w\mid
N}{\rm ord}_{p}(c_{w}^{(p)})$ and, by Proposition 2.2.1, we find

$\mathscr{M}_{\infty,\mathcal{N}_{\rm Heeg}}(\kappa^{\rm
Heeg})=\tfrac{1}{2}\sum_{w\mid N}{\rm ord}_{p}(c_{w}^{(p)}).$

∎

######  Remark 2.2.5.

Note that, conversely, the refined Kolyvagin conjecture together with Theorem
2.2.2 implies the “anticyclotomic Iwasawa main conjecture at $\alpha_{m}$” for
$m\gg 0$, namely:

$\mathcal{F}_{p}(E/K)(\alpha_{m})\sim_{p}\mathcal{L}_{p}^{\rm
BDP}(f/K)^{2}(\alpha_{m}),$

where $\mathcal{F}_{p}(E/K)\in\Lambda$ is any generator of ${\rm
char}_{\Lambda}(\mathfrak{X}_{\rm Gr}(E/K_{\infty}))$. This follows repeating
the proof of Theorem B backwards and applying Lemma 1.2.3 and Theorem 1.2.7.

##  3\. The cyclotomic analogue: the non-triviality of Kato’s Kolyvagin system

In this section we work with the cyclotomic Iwasawa algebra over $\mathbb{Q}$.
More precisely, let $\mathbb{Q}_{\infty}$ be the unique
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$, let
$\Gamma_{\mathbb{Q}}=\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\simeq\mathbb{Z}_{p}$,
and

$\Lambda=\Lambda_{\mathbb{Q}}=\mathbb{Z}_{p}[[\Gamma_{\mathbb{Q}}]].$

###  3.1. The Kolyvagin system of Kato classes

We recall the construction of the Kolyvagin system of classes
$\kappa_{n}=\kappa_{n}^{\rm Kato}\in\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)$
derived from Kato’s Euler system and their Iwasawa-theoretic analogues.

####  3.1.1. Selmer structures

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, without complex
multiplication, and let $p\nmid 2N$ be a prime of good ordinary reduction for
$E$. We assume

(${\rm tor}_{\mathbb{Q}}$)
$E[p]^{ss}=\mathbb{F}_{p}(\phi)\oplus\mathbb{F}_{p}(\phi^{-1}\omega)\implies\phi\neq
1,\omega,$

where, of course, $\omega$ is the mod $p$ cyclotomic character. This is
equivalent to the assertion that $E^{\prime}(\mathbb{Q})[p]=0$ for all curves
$E^{\prime}$ in the isogeny class of $E$.

For a character $\alpha:\Gamma_{\mathbb{Q}}\to R^{\times}$ with values in the
ring of integers $R$ of a finite extension $\Phi/\mathbb{Q}_{p}$, we consider
the $G_{\mathbb{Q}}$-modules

(3.1) $T_{\alpha}:=T_{p}E\otimes_{\mathbb{Z}_{p}}R(\alpha),\quad
V_{\alpha}:=T_{\alpha}\otimes_{R}\Phi,\quad
W_{\alpha}:=T_{\alpha}\otimes_{R}\Phi/R\simeq V_{\alpha}/T_{\alpha},$

where $R(\alpha)$ is the free $R$-module of rank one on which $G_{\mathbb{Q}}$
acts via the projection $G_{\mathbb{Q}}\twoheadrightarrow\Gamma_{\mathbb{Q}}$
composed with $\alpha$, and the $G_{\mathbb{Q}}$-action on $T_{\alpha}$ is via
$\rho_{\alpha}=\rho_{E}\otimes\alpha$. Recall that as $E$ has ordinary
reduction at $p$, there is a unique $G_{\mathbb{Q}_{p}}$-stable
$\mathbb{Z}_{p}$-summand $\mathrm{Fil}^{+}(T_{p}E)\subset T_{p}E$ of rank one
such that $T_{p}E/\mathrm{Fil}^{+}(T_{p}E)$ is unramified. We put
$\mathrm{Fil}^{+}(T_{\alpha}):=\mathrm{Fil}^{+}(T_{p}E)\otimes_{\mathbb{Z}_{p}}R(\alpha)$,
$\mathrm{Fil}^{+}(V_{\alpha}):=\mathrm{Fil}^{+}(T_{\alpha})\otimes_{R}\Phi$,
and
$\mathrm{Fil}^{+}(W_{\alpha}):=\mathrm{Fil}^{+}(T_{\alpha})\otimes_{R}\Phi/R$.

Let $M$ denote any of the modules in (3.1) or a quotient
$T_{\alpha}/IT_{\alpha}$ for a non-zero ideal $I\subset R$. We define a
_Selmer structure_ $\mathcal{F}$ on $M$ to be a finite set
$\Sigma=\Sigma(\mathcal{F})$ of places of $\mathbb{Q}$ containing $\infty$,
the prime $p$, and the primes where $M$ is ramified, together with a choice of
a submodule of _local conditions_
$\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{w},M)\subset\mathrm{H}^{1}(\mathbb{Q}_{w},M)$
for every $w\in\Sigma$, similarly as in $\S$1.1.1. The associated _Selmer
group_ is then defined by

$\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},M):={\rm
ker}\biggl{\\{}\mathrm{H}^{1}(\mathbb{Q}^{\Sigma}/\mathbb{Q},M)\rightarrow\prod_{w\in\Sigma}\frac{\mathrm{H}^{1}(\mathbb{Q}_{w},M)}{\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{w},M)}\biggr{\\}},$

where $\mathbb{Q}^{\Sigma}$ is the maximal extension of $\mathbb{Q}$
unramified outside $\Sigma$. The local conditions of interest in this setting
are the following:

  * •

For a finite prime $\ell\neq p$, the _finite_ (or _unramified_) local
condition:

$\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},V_{\alpha}):={\rm
ker}\bigl{\\{}\mathrm{H}^{1}(\mathbb{Q}_{\ell},V_{\alpha})\rightarrow\mathrm{H}^{1}(\mathbb{Q}_{\ell}^{\rm
ur},V_{\alpha})\bigr{\\}}.$

  * •

The _ordinary_ condition at $p$:

$\mathrm{H}^{1}_{\rm ord}(\mathbb{Q}_{p},V_{\alpha}):={\rm
im}\bigl{\\{}\mathrm{H}^{1}(\mathbb{Q}_{p},{\rm
Fil}_{p}^{+}(V_{\alpha}))\rightarrow\mathrm{H}^{1}(\mathbb{Q}_{p},V_{\alpha})\bigr{\\}}.$

  * •

The _relaxed condition_ at $\ell$: $\mathrm{H}^{1}_{\rm
rel}(\mathbb{Q}_{\ell},V_{\alpha})=\mathrm{H}^{1}(\mathbb{Q}_{\ell},V_{\alpha})$.

  * •

The _strict condition_ at $\ell$: $\mathrm{H}^{1}_{\rm
str}(\mathbb{Q}_{\ell},V_{\alpha})=\mathrm{H}^{1}(\mathbb{Q}_{\ell},V_{\alpha})$.

The corresponding local conditions for
$M\in\\{T_{\alpha},W_{\alpha},T_{\alpha}/IT_{\alpha}\\}$ are defined by
propagation. (Thus, for example, $\mathrm{H}^{1}_{\rm
str}(\mathbb{Q}_{\ell},T_{\alpha})=\mathrm{H}^{1}(\mathbb{Q}_{\ell},T_{\alpha})_{\rm
tor}$ and $\mathrm{H}^{1}_{\rm
rel}(\mathbb{Q}_{\ell},W_{\alpha})=\mathrm{H}^{1}(\mathbb{Q}_{\ell},W_{\alpha})_{\rm
div}$.) For $M=T_{\alpha}/IT_{\alpha}$, with $I\subset R$ a non-zero ideal, we
also consider:

  * •

The _transverse_ local condition at a prime $\ell\neq p$ is

$\mathrm{H}^{1}_{\rm tr}(\mathbb{Q}_{\ell},M):={\rm
ker}\bigl{\\{}\mathrm{H}^{1}(\mathbb{Q}_{\ell},M)\rightarrow\mathrm{H}^{1}(\mathbb{Q}_{\ell}(\mu_{\ell}),M)\bigr{\\}}.$

For $\circ\in\\{{\rm rel},{\rm ord},{\rm str}\\}$ we put

$\mathrm{H}^{1}_{\mathcal{F}_{\circ}}(\mathbb{Q}_{\ell},V_{\alpha}):=\begin{cases}\mathrm{H}^{1}_{\circ}(\mathbb{Q}_{p},V_{\alpha})&\textrm{if
$\ell=p$,}\\\\[1.99997pt]
\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{\ell},V_{\alpha})&\textrm{else,}\end{cases}$

and let $\mathcal{F}_{\circ}$ also denote the Selmer structures similarly
defined on $T_{\alpha}$, $W_{\alpha}$, and $T_{\alpha}/IT_{\alpha}$.

Given a Selmer structure $\mathcal{F}$, and $n\in\mathbb{Z}$ square-free and
coprime to $p$, we define the Selmer structure $\mathcal{F}(n)$ as follows:

$\mathrm{H}^{1}_{\mathcal{F}(n)}(\mathbb{Q}_{\ell},T_{\alpha})=\begin{cases}\mathrm{H}^{1}_{\rm
tr}(\mathbb{Q}_{\ell},T_{\alpha})&\text{if }\ell\mid n,\\\\[1.99997pt]
\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{\ell},T_{\alpha})&\text{if }\ell\nmid
n.\end{cases}$

Also, for $M_{\alpha}^{*}=\operatorname{Hom}(M_{\alpha},\mu_{p^{\infty}})$ (so
$T_{\alpha}^{*}\simeq W_{\alpha^{-1}}$), we let $\mathcal{F}^{*}$ be the
Selmer structure on $M^{*}$ determined by the orthogonal complements of
$\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q}_{\ell},M)$ under local Tate duality
$\mathrm{H}^{1}(\mathbb{Q}_{\ell},M)\times\mathrm{H}^{1}(\mathbb{Q}_{\ell},M^{*})\rightarrow\mathrm{H}^{2}(\mathbb{Q}_{\ell},\mu_{p^{\infty}})=\mathbb{Q}_{p}/\mathbb{Z}_{p}$.

We define the Bloch–Kato Selmer structure $\mathcal{F}_{\rm{BK}}$ to be the
Selmer structure given by
$\mathrm{H}^{1}_{\mathcal{F}_{\rm{BK}}}(\mathbb{Q}_{\ell},M)=\mathrm{H}^{1}_{f}(\mathbb{Q}_{\ell},M)$
for all primes $\ell$ and
$M\in\\{T_{\alpha},V_{\alpha},W_{\alpha},T_{\alpha}/IT_{\alpha}\\}$. Note that
$\mathcal{F}_{\rm{BK}}^{*}$ is just the Bloch–Kato Selmer structure for
$M^{*}=\operatorname{Hom}(M,\mu_{p^{\infty}})$. From here on, we put

$\mathcal{F}=\mathcal{F}_{\rm{BK},\operatorname{rel}},\quad\mathcal{F}^{*}=\mathcal{F}_{\rm{BK},\operatorname{str}}.$

Later on, in the proof of Proposition 3.3.1, we shall also need to consider
the Selmer structure $\mathcal{F}_{\rm ord}=\mathcal{F}_{\rm{BK},\rm{ord}}$.

####  3.1.2. Kato’s Kolyvagin system for $T_{f}$

Let $f=\sum_{n=1}^{\infty}a_{n}q^{n}\in S_{2}(\Gamma_{0}(N))$ be the newform
attached to $E$, with associated $p$-adic Galois representation $V_{f}$
constructed as the maximal quotient of $\mathrm{H}^{1}_{\rm
et}(\overline{Y_{1}(N)},\mathbb{Q}_{p}(1))$ on which the Hecke operators
$T_{n}$ act as multiplication by $a_{n}$. After possibly replacing $E$ by an
isogenous elliptic curve, we may suppose that the $p$-adic Tate module
$T_{p}E$ is isomorphic to the $\mathbb{Z}_{p}$-lattice $T_{f}\subset V_{f}$
generated by the image of $\mathrm{H}^{1}_{\rm
et}(\overline{Y_{1}(N)},\mathbb{Z}_{p}(1))$ in $V_{f}$ (in other words, $E$ is
the elliptic curve denoted $E_{\bullet}$ in [Wut14]).

Put $T=T_{p}E$, and let

$\\{z_{n}\in\mathrm{H}^{1}(\mathbb{Q}(\mu_{n}),T)\\}_{n}$

be the Euler system construction by Kato [Kat04] (see esp. [_op. cit._ , Ex.
1.3.3]). Here $n$ runs over the positive integers coprime to $cdN$, for a
fixed pair of integers $c,d\equiv 1\;({\rm mod}\,N)$ used in the construction
of the classes $z_{n}$, and $\mathbb{Q}(\mu_{n})$ denotes the cyclotomic
extension of $\mathbb{Q}$ obtained by adjoining the $n$-th roots of unity. For
varying $n$, the classes $z_{n}$ satisfy the following _norm relations_ :

(3.2)
$\text{cores}^{\mathbb{Q}(\mu_{n\ell})}_{\mathbb{Q}(\mu_{n})}(z_{n\ell})=\begin{cases}z_{n}&\ell\mid
n,\\\\[1.99997pt]
P_{\ell}(\operatorname{Frob}_{\ell}^{-1})z_{n}&\text{otherwise},\end{cases}$

where $P_{\ell}(x)=\det(1-{\rm Frob}_{\ell}^{-1}x\,|\,T)$ is the
characteristic polynomial of a geometric Frobenius at $\ell$.

We outline the construction of the Kolyvagin system derived from Kato’s Euler
system classes. Denote by $\mathcal{L}_{0}$ the set of primes $\ell$ with
$(\ell,cdpN)=1$. Consider

$\mathcal{L}_{E}:=\\{\ell\in\mathcal{L}_{0}\;\colon\;\textrm{$\ell\equiv
1\;({\rm mod}\,p)$ and $a_{\ell}\equiv\ell+1\;({\rm mod}\,p)$}\\}.$

Fix a subset $\mathcal{\mathcal{L}}\subset\mathcal{L}_{E}$, and let
$\mathcal{N}=\mathcal{N}(\mathcal{L})$ be the set of squarefree products of
primes $\ell\in\mathcal{L}$. For $\ell\in\mathcal{L}$ define the ideal

(3.3) $I_{\ell}=(\ell-1,a_{\ell}-\ell-1)\subset\mathbb{Z}_{p}.$

and for $n\in\mathcal{N}$, let $I_{n}=\sum_{\ell\mid n}I_{\ell}$. Let
$G_{n}={\rm Gal}(\mathbb{Q}(\mu_{n})/\mathbb{Q})$. The Kolyvagin derivative
operator $D_{n}\in\mathbb{Z}[G_{n}]$ at $n\in\mathcal{N}$ is defined by

$D_{n}=\prod_{\ell\mid n}\sum_{i=1}^{\ell-2}i\sigma_{\ell}^{i},$

where $\sigma_{\ell}$ is a generator of $G_{\ell}$. Then one easily checks
that the natural image of $D_{n}z_{n}$ in
$\mathrm{H}^{1}(\mathbb{Q}(\mu_{n}),T/I_{n}T)$ is fixed by $G_{n}$, and
$\kappa_{n}^{\rm Kato}\in\mathrm{H}^{1}(\mathbb{Q},T/I_{n}T)$ is defined to be
its preimage under the restriction map

(3.4) $\mathrm{H}^{1}\left(\mathbb{Q},T/I_{n}T\right)\xrightarrow{\rm
res}\mathrm{H}^{1}(\mathbb{Q}(\mu_{n}),T/I_{n}T)^{G_{n}}$

By (${\rm tor}_{\mathbb{Q}}$) and our assumptions on $n$, the restriction map
(3.4) is an isomorphism, and so $\kappa_{n}^{\rm Kato}$ is well-defined. One
can show that after a slight modification (see [MR04, $\S\S{3.2},6.2$] and
[Rub11, Thm. 4.3.1]) the resulting classes – still denoted $\kappa_{n}^{\rm
Kato}$ – satisfy

$\kappa_{n}^{\rm Kato}\in\mathrm{H}^{1}_{\mathcal{F}(n)}(\mathbb{Q},T/I_{n}T)$

and form a Kolyvagin system for $(T,\mathcal{F},\mathcal{L})$ in the sense of
[MR04], i.e., they satisfy the _finite-singular relations_ , which in
particular imply that

(3.5) $\operatorname{ord}(\operatorname{loc}_{\ell}(\kappa_{n}^{\rm
Kato}))=\operatorname{ord}(\operatorname{loc}_{\ell}(\kappa^{\rm
Kato}_{n\ell}))\ \ \text{ for every $n\in\mathcal{N},\ell\in\mathcal{L}$ such
that $\ell\nmid n$}.$

Moreover, similarly as in the setting of Heegner points, one can show that
reducing such a Kolyvagin system modulo $p^{k}$ gives a Kolyvagin system for
$E[p^{k}]$ (see [MR04, Prop. 5.2.9 and Prop. 6.2.2]).

####  3.1.3. The $\Lambda$-adic Kolyvagin system

The Iwasawa algebra $\Lambda=\mathbb{Z}_{p}[[\Gamma_{\mathbb{Q}}]]$ is
equipped with a $G_{\mathbb{Q}}$-action given by the character
$\Psi:G_{\mathbb{Q}}\rightarrow\Lambda^{\times}$ arising from the projection
$G_{\mathbb{Q}}\twoheadrightarrow\Gamma_{\mathbb{Q}}$.

Let $\mathbb{Q}(\mu_{p^{\infty}})$ be the extension of $\mathbb{Q}_{\infty}$
obtained by adjoining to $\mathbb{Q}$ all $p$-power roots of unity. In light
of the norm relations (3.2), for every $n\in\mathcal{N}$ the classes
$z_{np^{k}}$ define an element

$z_{n,\infty}:=\\{z_{np^{k}}\\}_{k}\in\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{k}\mathrm{H}^{1}(\mathbb{Q}(\mu_{np^{k}}),T)\simeq\mathrm{H}^{1}(\mathbb{Q}(\mu_{n}),T\otimes\mathbb{Z}_{p}[[\operatorname{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})]]),$

where the inverse limits are with respect to the corestriction maps and the
isomorphism follows from Shapiro’s lemma. Writing
$\operatorname{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})=\Delta\times\Gamma_{\mathbb{Q}}$
and letting $\mathbf{T}=T\otimes_{\mathbb{Z}_{p}}\Lambda$, we can take the
projection to the component where the action of $\Delta$ is trivial, obtaining
classes

$\mathbf{z}_{n}\in\mathrm{H}^{1}(\mathbb{Q}(\mu_{n}),\mathbf{T}).$

Since $\Gamma_{\mathbb{Q}}$ is a pro-$p$ group, the assumption (${\rm
tor}_{\mathbb{Q}}$) implies that $E(\mathbb{Q}_{\infty})[p]=0$. Therefore the
same construction as above gives classes

$\bm{\kappa}_{n}^{\rm
Kato}\in\mathrm{H}^{1}(\mathbb{Q},\mathbf{T}/I_{n}\mathbf{T})$

that form a Kolyvagin system for
$(\mathbf{T},\mathcal{F}_{\Lambda},\mathcal{L})$, where
$\mathcal{F}_{\Lambda}$ is the Selmer structure given by
$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(\mathbb{Q}_{\ell},\mathbf{T})=\mathrm{H}^{1}(\mathbb{Q}_{\ell},\mathbf{T})$
for all $\ell\in\Sigma$ (see [MR04, $\S\S{5.3},6.2$]).

####  3.1.4. Kolyvagin system for cyclotomic twists

We are interested in the image of this system in the cohomology of
$T_{\alpha}$ for characters
$\alpha:\Gamma_{\mathbb{Q}}\to\mathbb{Z}_{p}^{\times}$ with $\alpha\equiv
1\mod p^{m}$ and $m\gg 0$.

For any $\alpha:\Gamma_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$,
$\mathfrak{P}=(\gamma-\alpha(\gamma))$ is a height one prime ideal of
$\Lambda$ such that the completion of $\Lambda/\mathfrak{P}$ is
$\mathbb{Z}_{p}$. Composing $\Psi:G_{\mathbb{Q}}\rightarrow\Lambda^{\times}$
with the natural map $\Lambda\rightarrow\Lambda/\mathfrak{P}$ recovers the
character $\alpha$. By [MR04, $\S$5.3] (see in particular Corollary 5.3.15 in
_op.cit._), the image of the Kolyvagin system $\\{\bm{\kappa}_{n}^{\rm
Kato}\\}_{n\in\mathcal{N}}$ via the natural map $\mathbf{T}\to T_{\alpha}$
specializes to a Kolyagin system

$\\{\bm{\kappa}_{n}^{\rm Kato}(\alpha)\\}_{n\in\mathcal{N}}$

for $(T_{\mathfrak{P}},\mathcal{F},\mathcal{L})$. The following is the
analogue of Lemma 1.1.5, and is proved in the same manner.

######  Lemma 3.1.1.

Suppose $\alpha\equiv 1\;({\rm mod}\,p^{m})$. For all $n\in\mathcal{N}^{(m)}$
we have $\bm{\kappa}_{n}^{\rm Kato}(\alpha)\equiv\kappa_{n}^{\rm Kato}\;({\rm
mod}\,p^{m})$.

Finally, to conclude the parallel with $\S\ref{seckolylambda}$, we recall the
result on the nonvanishing of $\bm{\kappa}_{1}^{\rm Kato}$ and, therefore, by
the previous lemma, of $\bm{\kappa}_{1}^{\rm Kato}(\alpha_{m})$ for $m\gg 0$.

######  Theorem 3.1.2 (Kato–Rohrlich).

For $m\gg 0$ and any $\alpha\equiv 1\;({\rm mod}\,p^{m})$, the class
$\bm{\kappa}_{1}^{\rm Kato}(\alpha)$ is nonzero.

###### Proof.

This follows from a combination of Rohrlich’s results [Roh84], implying the
nonvanishing of the Mazur–Swinnerton-Dyer $p$-adic $L$-function
$\mathcal{L}_{p}^{\rm MSD}(E/\mathbb{Q})$, and Kato’s explicit reciprocity law
[Kat04, Thm. 16.6] (which we recall in the next section) relating
$\bm{\kappa}_{1}^{\rm Kato}$ with $\mathcal{L}_{p}^{\rm MSD}(E/\mathbb{Q})$ .
∎

###  3.2. Cyclotomic Main Conjecture and results

####  3.2.1. The $p$-adic $L$-function

We recall the existence of the cyclotomic $p$-adic $L$-function attached to
$E$. Given a modular parametrisation $\pi_{E}:X_{0}(N)\rightarrow E$, we
denote by $c_{E}\in\mathbb{Z}$ the corresponding Manin constant. Let
$\omega_{E}$ be a minimal differential on $E$. Pick generators $\delta^{\pm}$
of $H_{1}(E,\mathbb{Z})^{\pm}$, and define the Néron periods
$\Omega_{E}^{\pm}$ by

$\Omega_{E}^{\pm}=\int_{\delta^{\pm}}\omega_{E}.$

We normalize the $\delta^{\pm}$ so that $\Omega_{E}^{+}\in\mathbb{R}_{>0}$ and
$\Omega_{E}^{-}\in i\mathbb{R}_{\geq 0}$. We also let $\alpha_{p}$ be the
$p$-adic unit root of $x^{2}-a_{p}(f)x+p$, where $a_{p}(f)$ is the $p$-th
Fourier coefficient of the modular form $f$ attached to $E$.

######  Theorem 3.2.1 (Mazur–Swinnerton-Dyer [MSD74], Wüthrich [Wut14]).

There exists an element $\mathcal{L}^{\rm
MSD}_{p}(E/\mathbb{Q})\in\Lambda_{\mathbb{Q}}$ such that for any finite order
character $\chi$ of $\Gamma_{\mathbb{Q}}$ of conductor $p^{r}$, we have

$\mathcal{L}^{\rm
MSD}_{p}(E/\mathbb{Q})(\chi)=\begin{cases}\frac{p^{r}}{\tau(\overline{\chi})\alpha_{p}^{r}}\cdot\frac{L(E,\overline{\chi},1)}{\Omega_{E}^{+}}&\text{
if }r>0,\\\ (1-\alpha_{p}^{-1})^{2}\cdot\frac{L(E,1)}{\Omega_{E}^{+}}&\text{
if }r=0,\end{cases}$

where $\tau(\overline{\chi})=\sum_{a\;{\rm
mod}\;p^{r}}\overline{\chi}(a)e^{2\pi ia/p^{r}}$ is the usual Gauss sum.

####  3.2.2. Kato’s explicit reciprocity law

Kato defines in [Kat04, $\S{13.9}$] a modification
$\mathbf{z}_{\gamma,0}^{(p)}$ of the class $\bm{\kappa}_{1}^{\rm Kato}$, with
$\gamma=\delta(f,1,\xi)$ in the notations of _op. cit._ , where $\xi\in{\rm
SL}_{2}(\mathbb{Z})$ is the auxiliary matrix used in the construction of
Kato’s Euler system. More precisely, he first defines

$\mathbf{z}_{\gamma}^{(p)}\in\mathrm{H}^{1}(\mathbb{Q},T\otimes\mathbb{Z}_{p}[[{\rm
Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})]])\otimes\mathbb{Q}_{p}$

by multiplying $\\{z_{p^{k}}\\}_{k}$ by some factors depending on the
auxiliary integers $c,d$ and by the Euler factors at the bad primes, and
$\mathbf{z}_{\gamma,0}^{(p)}\in\mathrm{H}^{1}(\mathbb{Q},\mathbf{T})\otimes\mathbb{Q}_{p}$
is defined to be the projection to the trivial component for the action of
$\Delta$. Kato’s explicit reciprocity law [Kat04, Thm. 16.6] then relates
$\mathbf{z}_{\gamma,0}^{(p)}$ to $\mathcal{L}^{\rm MSD}_{p}(E/\mathbb{Q})$,
possibly up to some power of $p$ accounting for the different periods (which
depend on the choice of $\xi$). In particular, Kato’s result implies (see
[Rub98, Cor. 7.2]) that the natural image of $\bm{\kappa}_{1}^{\rm Kato}$
under the Coleman map

${\rm
Col}_{\infty}:\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},\mathbf{T})\rightarrow\Lambda,$

where
$\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},\mathbf{T}):=\mathrm{H}^{1}(\mathbb{Q}_{p},\mathbf{T})/\mathrm{H}^{1}(\mathbb{Q}_{p},{\rm
Fil}_{p}^{+}T\otimes_{\mathbb{Z}_{p}}\Lambda)$, is related to an imprimitive
variant of $\mathcal{L}_{p}^{\rm MSD}(E/\mathbb{Q})$. More precisely, taking
$c,d$ to be $\not\equiv 1\;({\rm mod}\,p)$, there exists $t_{1}\in\mathbb{Z}$
depending only on $E$ such that

(3.6) ${\rm Col}_{\infty}(\operatorname{loc}_{s}(\bm{\kappa}^{\rm
Kato}_{1}))\sim_{p}p^{t_{1}}\prod_{\ell\mid
N}(1-a_{\ell}\ell^{-1}\gamma_{\ell}^{-1})\mathcal{L}^{\rm
MSD}_{p}(E/\mathbb{Q}),$

where $\gamma_{\ell}$ denotes the image of the arithmetic Frobenius at $\ell$
in $\Gamma_{\mathbb{Q}}$.

In particular, specialising at cyclotomic characters, we can obtain the
following result.

######  Lemma 3.2.2.

Let $\alpha:\Gamma_{\mathbb{Q}}\to\mathbb{Z}_{p}^{\times}$ be a crystalline
character with Hodge–Tate weight $n\leq 0$ with $n\equiv 0\pmod{p-1}$ and such
that $\alpha\equiv 1\;({\rm mod}\,p^{m})$. Then there exist integers
$\mathcal{M}\in\mathbb{Z}_{>0}$ and $h\in\mathbb{Z}$, both independent of
$\alpha$, such that if $m\geq\mathcal{M}$ then $\mathcal{L}_{p}^{\rm
MSD}(E/\mathbb{Q})(\alpha^{-1})\neq 0$ and

$\\#\left(\mathbb{Z}_{p}/\mathcal{L}^{\rm
MSD}_{p}(E/\mathbb{Q})(\alpha^{-1})\right)\cdot
p^{h}\,\sim_{p}\,\\#\bigl{(}\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},T_{\alpha})/\mathbb{Z}_{p}\cdot\operatorname{loc}_{s}(\bm{\kappa}^{\rm
Kato}_{1}({\alpha}))\bigr{)},$

where
$\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},T_{\alpha}):=\frac{\mathrm{H}^{1}(\mathbb{Q}_{p},T_{\alpha})}{\mathrm{H}^{1}_{\rm
ord}(\mathbb{Q},T_{\alpha})}$ and $\operatorname{loc}_{s}$ is the composite of
the localisation map
$\operatorname{loc}_{p}:\mathrm{H}^{1}(\mathbb{Q},T_{\alpha})\to\mathrm{H}^{1}(\mathbb{Q}_{p},T_{\alpha})$
and the natural quotient map.

###### Proof.

This follows from a similar argument as in Lemma 1.2.3. Indeed, the map ${\rm
Col}_{\infty}$ is an isomorphism in our setting (see [Kat04, Prop. 17.11]),
and we claim that the specialization map
$\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},\mathbf{T})\rightarrow\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},T_{\alpha})$
is surjective. From the definitions we have a commutative diagram with exact
rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},\mathbf{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},\mathrm{Fil}^{-}\mathbf{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{e}$$\textstyle{\mathrm{H}^{2}(\mathbb{Q}_{p},\mathrm{Fil}^{+}\mathbf{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},T_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(\mathbb{Q}_{p},\mathrm{Fil}^{-}T_{\alpha})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{H}^{2}(\mathbb{Q}_{p},\mathrm{Fil}^{+}T_{\alpha}),}$

where we have written $\mathrm{Fil}^{\pm}\mathbf{T}$ for
$\mathrm{Fil}^{\pm}T\otimes_{\mathbb{Z}_{p}}\Lambda$. The vertical arrows are
the natural specialization maps. By Tate’s local duality,
$\mathrm{H}^{2}(\mathbb{Q}_{p},F^{+}\mathbf{T})$ is dual to

$\mathrm{H}^{0}(\mathbb{Q}_{p},\mathrm{Fil}^{-}T\otimes_{\mathbb{Z}_{p}}\Lambda^{*})=\mathrm{H}^{0}(\mathbb{Q}_{\infty,p},\tilde{E}[p^{\infty}])=\tilde{E}(\mathbb{F}_{p})[p^{\infty}],$

using that $\mathbb{Q}_{\infty,p}/\mathbb{Q}_{p}$ is totally ramified for the
last equality. The same argument shows that
$\mathrm{H}^{2}(\mathbb{Q}_{p},\mathrm{Fil}^{+}T_{\alpha})$ is dual to
$\tilde{E}(\mathbb{F}_{p})[p^{\infty}]$ and it follows that $c$ is an
isomorphism. So replacing the right-most column with the images of the maps
$d$ and $e$ yields a commutative diagram with short exact rows and columns and
with the right-most arrow an injection. Applying the snake lemma to this
diagram then yields $\mathrm{coker}(a)\simeq\mathrm{coker}(b)$. The latter
cokernel is given by
$\mathrm{H}^{2}(\mathbb{Q}_{p},\mathrm{Fil}^{-}\mathbf{T})[\gamma-\alpha(\gamma)]$,
which is dual to
$\mathrm{H}^{0}(\mathbb{Q}_{\infty,p},F^{+}E[p^{\infty}])/(\gamma-\alpha(\gamma))$.
Since the $\mathbb{Q}_{p,n}$-rational $p$-torsion in the formal group of $E$
is trivial for all $n$, this shows that $b$ is surjective and hence that $a$
is surjective, as claimed.

Now setting $h:=t_{1}+t_{2}$ where $t_{2}:=\sum_{\ell\mid N}{\rm
ord}_{p}(1-a_{\ell}\ell^{-1})$, and noting that $\sum_{\ell\mid N}{\rm
ord}_{p}(1-a_{\ell}\ell^{-1}\alpha(\gamma_{\ell}^{-1}))$ is equal to $t_{2}$
for $m$ sufficiently large, (3.6) yields the result. ∎

######  Remark 3.2.3.

Note that crystalline characters satisfying the assumptions of Lemma 3.2.2 are
just the characters $\epsilon^{n}$, $n\geq 0$ an integer such that $n\equiv
0\mod(p-1)$, for $\epsilon$ the $p$-adic cyclotomic character.

####  3.2.3. Rational cyclotomic Main Conjecture

The Pontryagin dual $\Lambda^{\vee}$ is equipped with a
$G_{\mathbb{Q}}$-action via the inverse of the character
$\Psi:G_{\mathbb{Q}}\rightarrow\Lambda^{\times}$ arising from the projection
$G_{\mathbb{Q}}\twoheadrightarrow\Gamma_{\mathbb{Q}}$.

Let $\mathbf{M}=T_{p}E\otimes\Lambda^{\vee}$, and put

$\mathrm{H}^{1}_{\operatorname{ord}}(\mathbb{Q}_{p},\mathbf{M})={\rm
im}\bigl{\\{}\mathrm{H}^{1}(\mathbb{Q}_{p},{\rm
Fil}^{+}(T_{p}E)\otimes\Lambda^{\vee})\rightarrow\mathrm{H}^{1}(\mathbb{Q}_{p},\mathbf{M})\bigr{\\}},$

where ${\rm Fil}^{+}(T_{p}E)={\rm ker}\\{T_{p}E\rightarrow T_{p}\tilde{E}\\}$
is the kernel of reduction. The _ordinary Selmer group_
$\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(\mathbb{Q},\mathbf{M})$ is

$\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(\mathbb{Q},\mathbf{M}):=\ker\biggl{\\{}\mathrm{H}^{1}(\mathbb{Q}^{\Sigma}/\mathbb{Q},\mathbf{M})\to\prod_{w\in\Sigma,w\nmid
p}\mathrm{H}^{1}(\mathbb{Q}_{w},\mathbf{M})\times\frac{\mathrm{H}^{1}(\mathbb{Q}_{p},\mathbf{M})}{\mathrm{H}^{1}_{\rm
ord}(\mathbb{Q}_{p},\mathbf{M})}\biggr{\\}},$

and we write $\mathfrak{X}_{\rm ord}(E/\mathbb{Q}_{\infty})$ for its
Pontryagin dual.

We recall the statement of the Iwasawa Main Conjecture formulated by Mazur in
[Maz72].

######  Conjecture 3.2.4.

The module $\mathfrak{X}_{\rm ord}(E/\mathbb{Q}_{\infty})$ is
$\Lambda$-torsion, with

${\rm char}_{\Lambda}\left(\mathfrak{X}_{\rm
ord}(E/\mathbb{Q}_{\infty})\right)=(\mathcal{L}^{\rm MSD}_{p}(E/\mathbb{Q}))$

as ideals in $\Lambda$.

In the following, we shall refer to the statement of Conjecture 3.2.4 with
$\Lambda$ replaced by $\Lambda\otimes\mathbb{Q}_{p}$ as the _rational
cyclotomic Main Conjecture_. It follows from the results of [Sch87, PR89] that
this is invariant under isogenies.

The first cases of Conjecture 3.2.4 were proved by Rubin [Rub91] when
$E/\mathbb{Q}$ has complex multiplication. In the non-CM case, the conjecture
was proved in [Kat04, SU14] (residually irreducible case) and [Kat04, GV00]
(residually reducible case) under mild hypotheses. More recently, we have the
following.

######  Theorem 3.2.5.

Suppose $p>2$ is a prime of good ordinary reduction for $E$.

  * (i)

If $E[p]^{ss}=\mathbb{F}_{p}(\phi)\oplus\mathbb{F}_{p}(\psi)$ as
$G_{\mathbb{Q}}$-modules with $\phi|_{G_{p}}\neq\mathds{1},\omega$, then
Conjecture 3.2.4 holds.

  * (ii)

If $E[p]$ satisfies (irr), then the rational cyclotomic Main Conjecture holds.

###### Proof.

This is shown in [CGS23] in case (i), and in [Wan15] in case (ii). ∎

###  3.3. Nonvanishing of Kato’s Kolyvagin system

In this section we prove Theorem C in the Introduction. The key ingredients
are:

  1. (1)

The nonvanishing of $\bm{\kappa}^{\rm Kato}_{1}$ (Theorem 3.1.2)

  2. (2)

For a character $\alpha:\Gamma_{\mathbb{Q}}\rightarrow R^{\times}$
sufficiently close to $1$, an estimate on the divisibility index of
$\bm{\kappa}^{\rm Kato}_{1}(\alpha)$ in terms the length of the dual Selmer
group and the Tamagawa factors of $E$ (Proposition 3.3.1 below).

  3. (3)

A Kolyvagin system bound with controlled error terms (Theorem 3.3.3 below).

The cyclotomic Main Conjecture333In fact, as will be clear from the proof, the
“lower bound” on the size the Selmer group predicted by Conjecture 3.2.4 (or
its rational version) suffices for the application to nonvanishing. enters
into the proof of (2). With these ingredients in hand, the proof of Theorem C
proceeds along similar lines as in $\S\ref{sec:Kolyproof}$.

####  3.3.1. Cyclotomic control theorem

As above, we consider the $G_{\mathbb{Q}}$-modules

$T_{\alpha}:=T_{p}E\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}(\alpha),\quad
V_{\alpha}:=T_{\alpha}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p},\quad
W_{\alpha}:=T_{\alpha}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}/Z_{p}$

for a character
$\alpha:\Gamma_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$. Recall that we
set $\mathcal{F}=\mathcal{F}_{\rm{BK},\rm rel}$ and
$\mathcal{F}^{*}=\mathcal{F}_{\rm{BK},\rm str}$.

######  Proposition 3.3.1.

Let $\mathcal{M}$ be as in Lemma 3.2.2, and suppose
$\alpha:\Gamma_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$ is such that
$\alpha\equiv 1\;({\rm mod}\,p^{m})$ for $m\geq\mathcal{M}$ (so in particular
$\mathcal{L}_{p}^{\rm MSD}(E/\mathbb{Q})(\alpha^{-1})\neq 0$ and
$\mathcal{L}^{\rm MSD}_{p}(E/\mathbb{Q})(\alpha)\neq 0$). If Conjecture 3.2.4
holds then

${\rm
length}_{R}(\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}}))+\sum_{\ell\mid
N}\operatorname{ord}_{p}(c_{\ell}(\alpha^{-1}))+2t_{3}+h=[\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha}):\mathbb{Z}_{p}\cdot\bm{\kappa}^{\rm
Kato}_{1}(\alpha)],$

where $c_{\ell}(\alpha^{-1})=\\#\mathrm{H}^{1}_{\rm
ur}(\mathbb{Q}_{\ell},W_{\alpha^{-1}})$, $t_{3}={\rm
ord}_{p}(\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]))$, and
$h=t_{1}+t_{2}$ is as in Lemma 3.2.2.

###### Proof.

Poitou–Tate duality gives rise to the exact sequence

(3.7)
$0\rightarrow\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}})\rightarrow\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(\mathbb{Q},W_{\alpha^{-1}})\xrightarrow{{\rm
loc}_{p}}\mathrm{H}^{1}_{\rm
ord}(\mathbb{Q}_{p},W_{\alpha^{-1}})\xrightarrow{\beta}\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})^{\vee}\rightarrow\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(\mathbb{Q},T_{\alpha})^{\vee}\rightarrow 0.$

By the cyclotomic Main Conjecture and Mazur’s control theorem, the
nonvanishing of $\mathcal{L}_{p}^{\rm MSD}(E/\mathbb{Q})(\alpha)$ and
$\mathcal{L}_{p}^{\rm MSD}(E/\mathbb{Q})(\alpha^{-1})$ implies that
$\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(\mathbb{Q},W_{\alpha^{-1}})$ and
$\mathrm{H}^{1}_{\mathcal{F}_{\rm ord}}(\mathbb{Q},T_{\alpha})$ are both
finite. From (3.7), it follows that
$\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}})$ is also finite,
and that $\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})$ has
$\mathbb{Z}_{p}$-rank one; since the latter is torsion-free by (${\rm
tor}_{\mathbb{Q}}$), we have in fact
$\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})\simeq\mathbb{Z}_{p}$.
Moreover, by local Tate duality the map $\beta$ is identified with the
Pontryagin dual of the natural map

${\rm
loc}_{s}:\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})\rightarrow\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},T_{\alpha}):=\frac{\mathrm{H}^{1}(\mathbb{Q}_{p},T_{\alpha})}{\mathrm{H}^{1}_{\rm
ord}(\mathbb{Q},T_{\alpha})},$

and from the above we see that ${\rm loc}_{s}$ has finite cokernel, with

(3.8) $\\#{\rm im}({\rm loc}_{p})=\\#{\rm coker}({\rm loc}_{s}).$

Since by Lemma 3.2.2 the class $\bm{\kappa}^{\rm
Kato}_{1}(\alpha)\in\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})$ is
nonzero and has non-torsion image in
$\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},T_{\alpha})$, we find

(3.9) $\\#{\rm coker}({\rm
loc}_{s})=\frac{[\mathrm{H}^{1}_{s}(\mathbb{Q}_{p},T_{\alpha}):\mathbb{Z}_{p}\cdot{\rm
loc}_{s}(\bm{\kappa}_{1}^{\rm
Kato}(\alpha))]}{[\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha}):\mathbb{Z}_{p}\cdot\bm{\kappa}_{1}^{\rm
Kato}(\alpha)]}=\frac{\\#(\mathbb{Z}_{p}/\mathcal{L}_{p}^{\rm
MSD}(E/\mathbb{Q})(\alpha))\cdot
p^{h}}{[\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha}):\mathbb{Z}_{p}\cdot\bm{\kappa}^{\rm
Kato}_{1}(\alpha)]}.$

On the other hand, letting $\mathcal{F}_{E}\in\Lambda$ be a generator of ${\rm
char}_{\Lambda}(\mathfrak{X}_{\rm ord}(E/\mathbb{Q}_{\infty}))$, by a variant
of [Gre99, Thm. 4.1] incorporating the twist by $\alpha$ with $m\gg 0$, we
have

(3.10) $\displaystyle\\#\mathbb{Z}_{p}/(\mathcal{F}_{E}(\alpha^{-1}))$
$\displaystyle\sim_{p}\\#\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(\mathbb{Q},W_{\alpha^{-1}})\cdot\prod_{\ell|N}c_{\ell}(\alpha^{-1})\cdot(\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]))^{2}$
$\displaystyle\sim_{p}\\#\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}})\cdot\\#{\rm
coker}({\rm
loc}_{s})\cdot\prod_{\ell|N}c_{\ell}(\alpha^{-1})\cdot(\\#\mathrm{H}^{0}(\mathbb{Q}_{p},E[p^{\infty}]))^{2},$

using (3.7) and (3.8) for the second equality. Since Conjecture 3.2.4 implies

$\\#\mathbb{Z}_{p}/(\mathcal{F}_{E}(\alpha^{-1}))=\\#\mathbb{Z}_{p}/(\mathcal{L}_{p}^{\rm
MSD}(E/\mathbb{Q})(\alpha^{-1}))$

the result now follows from (3.9) and (3.10). ∎

We have the following analogue of Lemma 1.2.8.

######  Lemma 3.3.2.

Assume that $\alpha:\Gamma_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$ is
such that $\alpha\equiv 1\;({\rm mod}\,p^{m})$, and $\ell\nmid p$ is a finite
prime. Then $c_{\ell}(\alpha)\equiv c_{\ell}\;({\rm mod}\,\varpi^{m})$, where
$c_{\ell}$ is the $p$-part of the Tamagawa factor $E$ at $\ell$.

###### Proof.

The proof is exactly the same as the one of Lemma 1.2.8. ∎

Thus taking $\alpha:\Gamma_{\mathbb{Q}}\to\mathbb{Z}_{p}^{\times}$ as in
Proposition 3.3.1 sufficiently close to $1$, from Lemma 3.3.2 we arrive at

(3.11) ${\rm
length}_{R}(\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}}))+\sum_{\ell\mid
N}\operatorname{ord}_{p}(c_{\ell})+h^{\prime}=\operatorname{ind}(\bm{\kappa}^{\rm
Kato}_{1}(\alpha),\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})),$

where $h^{\prime}:=t_{1}+t_{2}+2t_{3}$.

####  3.3.2. Kolyvagin system bound

The last ingredient we need is the following extension of [MR04, Thm. 5.2.2].
For a positive integer $e$ we put

$\mathcal{L}_{E,e}=\\{\ell\in\mathcal{L}_{E}\;:\;I_{\ell}\subset
p^{e}\mathbb{Z}_{p}\\},$

where $I_{\ell}\subset\mathbb{Z}_{p}$ is as in (3.3).

######  Theorem 3.3.3.

Suppose $\mathcal{L}\subset\mathcal{L}_{E}$ satisfies
$\mathcal{L}_{E,e}\subset\mathcal{L}$ for $e\gg 0$. Let
$\alpha:\Gamma_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$ be a cyclotomic
character such that $\alpha\equiv 1\;({\rm mod}\,p^{m})$. Suppose that there
is a collection of cohomology classes

$\\{\tilde{\kappa}_{n}\in\mathrm{H}^{1}(\mathbb{Q},T_{\alpha}/I_{n}T_{\alpha})\;:\;n\in\mathcal{N}\\}$

with $\tilde{\kappa}_{1}\neq 0$ and that there is an integer $t\geq 0$,
independent of $n$, such that
$\\{p^{t}\tilde{\kappa}_{n}\\}_{n\in\mathcal{N}}\in{\mathbf{KS}}(T_{\alpha},\mathcal{F},\mathcal{L})$.
Then $\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})$ has
$\mathbb{Z}_{p}$-rank $1$,
$\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}})$ is finite, and
there is a non-negative integer $\mathcal{E}$ depending only on $T_{p}E$ such
that

${\rm
length}_{R}(\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}}))\leq\operatorname{ind}(\tilde{\kappa}_{1})+\mathcal{E},$

where $\operatorname{ind}(\tilde{\kappa}_{1})={\rm
length}_{\mathbb{Z}_{p}}(\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha})/\mathbb{Z}_{p}\cdot\tilde{\kappa}_{1})$.

###### Proof.

By the same argument as in the proof Theorem 1.3.1, the result for $t=0$
easily implies the result for any $t\geq 0$, so it suffices to prove the
former. Under hypothesis (sur), the result is shown in [MR04, Thm. 5.2.2]; the
proof in the general case is given in $\S\ref{subsec:MR-refined}$ (see Theorem
3.4.1). ∎

Granted the results in $\S\ref{subsec:MR-refined}$, we are now ready to
conclude the proof of Theorem C.

###### Proof of Theorem C.

Arguing by contradiction, we assume that $\kappa_{n}=0$ for every
$n\in\mathcal{N}$, and take $t$ such that

$t>\sum_{\ell\mid N}{\rm ord}_{p}(c_{\ell})+\mathcal{E}+h^{\prime},$

where $\mathcal{E}$ and $h^{\prime}$ are as in Theorem 3.3.3 and (3.11),
respectively. Take
$\alpha:\Gamma_{\mathbb{Q}}\rightarrow\mathbb{Z}_{p}^{\times}$, $\alpha\equiv
1\;({\rm mod}\,p^{m})$, as in Proposition 3.3.1, with $m\geq t$ and such that
(3.11) holds. Then, letting $\mathcal{N}=\mathcal{N}(\mathcal{L}_{m})$, from
Lemma 3.1.1 we deduce the existence of a collection of cohomology classes
$\\{\tilde{\kappa}_{n,\alpha}\in\mathrm{H}^{1}(\mathbb{Q},T_{\alpha}/I_{n}T_{\alpha})\\}_{n\in\mathcal{N}}$
defined by the relation
$p^{t}\cdot\tilde{\kappa}_{n,\alpha}=\bm{\kappa}_{n}^{\rm Kato}(\alpha)$. By
Theorem 3.1.2 and Theorem 3.3.3 we obtain

${\rm
length}_{\mathbb{Z}_{p}}(\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}}))\leq{\rm
ind}(\bm{\kappa}^{\rm
Kato}_{1}(\alpha),\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T_{\alpha}))-t+\mathcal{E},$

which by our choice of $t$ contradicts (3.11). This concludes the proof of
Theorem C assuming Conjecture 3.2.4. A straightforward modification of the
above argument leads to the same conclusion just assuming the rational
anticyclotomic Main Conjecture. ∎

######  Remark 3.3.4.

Denote by $\mathscr{M}_{\infty}^{\rm Kato}$ the divisibility index of
$\\{\kappa_{n}^{\rm Kato}\\}$. Similarly as in the proof of Theorem B, the
above argument combined with Mazur–Rubin’s structure theorem for
$\mathrm{H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},W_{\alpha^{-1}})$ in terms of
$\bm{\kappa}^{\rm Kato}(\alpha)$ (see [MR04, Thm. 5.2.12]) shows that under
the following hypotheses:

  * (i)

(sur) holds (so that $\mathcal{E}=0$ in Theorem 3.3.3),

  * (ii)

$h=0$ in the notations of Lemma 3.2.2,

  * (iii)

$a_{p}\not\equiv 1\pmod{p}$ (so that $E(\mathbb{Q}_{p})[p^{\infty}]=0$ and
$t_{3}=0$ in the notation of Proposition 3.3.1),

if Conjecture 3.2.4 holds, then

(3.12) $\mathscr{M}_{\infty}^{\rm Kato}=\sum_{\ell\mid N}{\rm
ord}_{p}(c_{\ell}),$

confirming [Kim22b, Conj. 1.10]. In particular, by the cases of Conjecture
3.2.4 established in [BCS23], (3.12) holds under the following hypotheses in
addition to (i)-(iii): $p>3$ and there exists an element $\sigma\in
G_{\mathbb{Q}}$ fixing $\mathbb{Q}(\mu_{p^{\infty}})$ such that
$T/(\sigma-1)T$ is a free $\mathbb{Z}_{p}$-module of rank one.

###  3.4. Extension of Mazur–Rubin’s Selmer group bound

In this section we prove Theorem 3.4.1 below. The result extends [MR04, Thm.
5.2.2] and might be of independent interest.

Let $R$ be the ring of integers of a finite extension of $\mathbb{Z}_{p}$ with
maximal ideal $\mathfrak{m}$ and uniformiser $\varpi\in\mathfrak{m}$. Let
$\alpha:\Gamma_{\mathbb{Q}}\to R^{\times}$ be a cyclotomic character such that
$\alpha\equiv 1\;({\rm mod}\,\mathfrak{m})$, and put

$T=T_{p}E\otimes_{\mathbb{Z}_{p}}R(\alpha).$

As above, we put $\mathcal{F}=\mathcal{F}_{\rm{BK},\rm rel}$,
$\mathcal{F}^{*}=\mathcal{F}_{\rm{BK},\rm str}$, and let
$\mathcal{L}\subset\mathcal{L}_{E}$ be a set of primes with
$\mathcal{L}_{E,e}\subset\mathcal{L}$ for $e\gg 0$. We also put
$\mathcal{N}=\mathcal{N}(\mathcal{L})$.

######  Theorem 3.4.1.

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication, and
let $p$ be an odd prime of good reduction for $E$ such that
$E(\mathbb{Q})[p]=0$. Suppose that there is a Kolyvagin system
$\kappa=\\{\kappa_{n}\\}_{n\in\mathcal{N}}\in\mathbf{KS}(T,\mathcal{F},\mathcal{L})$
with $\kappa_{1}\neq 0$. Then $\mathrm{H}^{1}_{\mathcal{F}}(\mathbb{Q},T)$ has
$R$-rank one, ${\rm H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},T^{*})$ is finite, and
there exists a non-negative integer $\mathcal{E}$ depending only on $T_{p}E$
and ${\rm rank}_{\mathbb{Z}_{p}}(R)$ such that

${\rm length}_{R}({\rm
H}^{1}_{\mathcal{F}^{*}}(\mathbb{Q},T^{*}))\leq\operatorname{ind}(\kappa_{1})+\mathcal{E},$

where $\operatorname{ind}(\kappa_{1})={\rm length}_{R}\bigl{(}{\rm
H}^{1}_{\mathcal{F}}(K,T)/R\cdot\kappa_{1}\bigr{)}$. Moreover, $\mathcal{E}=0$
if (sur) holds.

As preparation for the proof of Theorem 3.4.1, we collect some preliminary
results from [MR04, $\S{4.1}$], whose proof applies verbatim under the
assumption $E(\mathbb{Q})[p]=0$.

######  Lemma 3.4.2.

Let $k>0$ and $T^{(k)}=T/\mathfrak{m}^{k}$. For every $n\in\mathcal{N}$ and
$0<i\leq k$ there are natural isomorphisms

$\mathrm{H}^{1}_{\mathcal{F}(n)}(\mathbb{Q},T^{(k)}/\mathfrak{m}^{i}T^{(k)})\xrightarrow{\sim}\mathrm{H}^{1}_{\mathcal{F}(n)}(\mathbb{Q},T^{(k)}[\mathfrak{m}^{i}])\xrightarrow{\sim}{\rm
H}_{\mathcal{F}(n)}^{1}(\mathbb{Q},T^{(k)})[\mathfrak{m}^{i}]$

and

$\mathrm{H}^{1}_{\mathcal{F}(n)^{*}}(\mathbb{Q},T^{*}[\mathfrak{m}^{i}])\xrightarrow{\sim}\mathrm{H}^{1}_{\mathcal{F}(n)^{*}}(\mathbb{Q},T^{*})[\mathfrak{m}^{i}]$

induced by the maps
$T^{(k)}/\mathfrak{m}^{i}T^{(k)}\xrightarrow{\varpi^{k-i}}T^{(k)}[\mathfrak{m}^{i}]\hookrightarrow
T^{(k)}$ and the inclusion $T^{*}[\mathfrak{m}^{i}]\hookrightarrow T^{*}$,
respectively.

###### Proof.

This is [MR04, Lem. 4.1.1]. ∎

######  Theorem 3.4.3.

For every $k>0$ and $n\in\mathcal{N}$, we have

$\mathrm{H}^{1}_{\mathcal{F}(n)}(\mathbb{Q},T/\mathfrak{m}^{k})\simeq
R/\mathfrak{m}^{k}\oplus\mathrm{H}^{1}_{\mathcal{F}(n)^{*}}(\mathbb{Q},T^{*}[\mathfrak{m}^{k}]).$

###### Proof.

Since our Selmer structure $\mathcal{F}$ has _core rank_
$\chi(T,\mathcal{F})=1$ in the sense of [MR04] (see [_op. cit._ , Prop.
6.2.2]), the result follows from [MR04, Thm. 4.1.13]. ∎

By assumption the elliptic curve does not have CM, so there exists $\tau\in
G_{\mathbb{Q}}$ such that $V_{p}E/(\tau-1)V_{p}E\simeq\mathbb{Q}_{p}$ since it
follows by Serre’s open image theorem [Ser72] that
${\tiny\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}}$ is in the image of $\rho_{E}$
for some $0\neq x\in\mathbb{Z}_{p}$. We fix $\tau$ such that
$\rho_{E}(\tau)={\tiny\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}}$ with
$t^{\prime}:=v_{p}(x)=\min_{m}\\{m:{\tiny\begin{pmatrix}1&y\\\
0&1\end{pmatrix}}\in{\rm Im}(\rho_{E}),v_{p}(y)=m\\}$. In particular, we have

($\tau$) $\tau_{|\mu_{p^{\infty}}}=1\ \ \ \text{and}\ \ \
T_{p}E/(\tau-1)T_{p}E\simeq\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}/p^{t^{\prime}}.$

Note that, since the first condition implies $\tau_{|\mathbb{Q}_{\infty}}=1$
and $\alpha$ is a character of $\Gamma_{\mathbb{Q}}$, we also have
$T/(\tau-1)T\simeq R\oplus R/\mathfrak{m}^{t}$, where