# A higher Gross–Zagier formula and the structure of Selmer groups

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

We establish a structural refinement of Gross–Zagier formula by comparing
Heegner point Kolyvagin systems with Kurihara numbers when the root number of
a rational elliptic curve $E$ over an imaginary quadratic field $K$ is $-1$.
When the root number of $E$ over $K$ is 1, we first prove the structure
theorem of the Selmer group of $E$ over $K$ and also obtain a structural
refinement of Waldspurger formula by comparing bipartite Euler systems with
Kurihara numbers. No low rank assumption is imposed in both refinements.

We also discuss how the non-triviality of various “Kolyvagin systems” can be
directly obtained from the corresponding main conjectures.

###### Key words and phrases:

refined Iwasawa theory, Kato’s Kolyvagin systems, Kurihara numbers, modular
symbols, Heegner point Kolyvagin systems, bipartite Euler systems

###### 2010 Mathematics Subject Classification:

11F67, 11G40, 11R23

###### Contents

  1. 1 Introduction
  2. 2 Statement of the main results
  3. 3 Modular symbols and Kurihara numbers
  4. 4 Heegner point Kolyvagin systems
  5. 5 Bipartite Euler systems for elliptic curves
  6. 6 Structural comparisons
  7. 7 The non-triviality of Kolyvagin systems and the main conjectures
  8. A Construction of the Gross point of conductor 1

##  1\. Introduction

###  1.1. A higher Gross–Zagier formula

The cerebrated Gross–Zagier formula [GZ86] is a remarkable formula relating
the first central derivatives of Rankin–Selberg $L$-functions with the heights
of Heegner points, and it has played an important role to study the Birch and
Swinnerton-Dyer conjecture. The Waldspurger formula [Wal85] also presents a
beautiful relation between the Rankin–Selberg $L$-values and certain toric
period integrals, which covers the complement to the Gross–Zagier formula in
terms of the sign of the functional equation. These formulas have been
enormously extended to Shimura curves over totally real fields [YZZ13] and
various higher dimensional Shimura varieties in the context of the arithmetic
Gan–Gross–Prasad conjecture [Zha12, Zha21] when the rank is at most one (in an
appropriate sense).

When the rank is larger than one, the situation becomes extremely puzzled. The
only known work is essentially Yun–Zhang’s ground-breaking work on the
Gross–Zagier formula for the function field case [YZ17, YZ19] and its
subsequent work. It seems that no similar result for the number field case is
addressed yet.

The main goal of this article is to present certain _structural refinements_
of Gross–Zagier formula and Waldspurger formula for elliptic curves over the
rationals from the viewpoint of Kolyvagin systems and the structure of Selmer
groups ($\S$6.2 and $\S$6.3). In particular, no low rank assumption is
imposed. A simple consequence of these structural refinements is stated in
Theorems 2.3 and 2.5 as _higher_ Gross–Zagier and Waldspurger formulas.

When the Heegner hypothesis is satisfied, the structural Gross–Zagier formula
compares certain modular symbols of an elliptic curve and its quadratic twist
with the corresponding Heegner point Kolyvagin system.

When the Heegner hypothesis breaks down, the structural Waldspurger formula
compares the same modular symbols with the values of families of quaternionic
automorphic forms at CM points.

These comparisons are purely algebraic and strongly inspired by refined
Iwasawa theory [Kur14a, Kur14b], especially, the structure theorems of Selmer
groups via the Kolyvagin system method [Kol91, MR04]. This is completely
different from the relative trace formula approach to the arithmetic
Gan–Gross–Prasad conjecture. It is remarkable that our approach captures the
structure of Selmer groups and Kolyvagin _derivatives_ of certain Mazur–Tate
elements are involved instead of _derivatives_ of Rankin–Selberg
$L$-functions.

###  1.2. From Euler systems to Kolyvagin systems

In the arithmetic of elliptic curves, the importance of the theory of Euler
systems cannot be overestimated. In particular, two completely different Euler
systems are extensively studied to understand Selmer groups of elliptic
curves: Kato’s Euler systems [Kat04] and Heegner points [Kol90]. The following
table summarizes their differences.

|  Kato’s Euler systems |  Heegner points
---|---|---
field variation |  abelian extensions of the rationals (“cyclotomic”) |  ring class extensions of imaginary quadratic fields (“anticyclotomic”)
local condition at $p$ |  the $p$-relaxed condition |  the classical condition
bounded Selmer groups |  the $p$-strict Selmer groups over the rationals |  the Selmer groups over imaginary quadratic fields
additional hypothesis |  |  the Heegner hypothesis

Although the natures of these two Euler systems are very different, B. Perrin-
Riou [PR93] formulated a conjecture making the connection between the bottom
class of Kato’s Euler system and the bottom class of the Heegner point Euler
system when the analytic rank is one, and it is recently settled [BDV22, BST,
BPS]. Except these bottom classes, we do _not_ expect the existence of a more
general comparison between Kato’s Euler systems and Heegner point Euler
systems since their field variations are _disjoint_ except the base imaginary
quadratic field. However, we still hope that Kato’s _Kolyvagin systems_ and
Heegner point _Kolyvagin systems_ can be compared, and we establish an
explicit relation between their divisibilities by using the modular symbol
interpretation of Kato’s Kolyvagin systems, namely Kurihara numbers. This is
possible because Kurihara numbers determine the structure of Selmer groups
over the rationals [Kur14a, Kur14b, Kim] and the Heegner point Kolyvagin
system also determines the structure of Selmer groups over imaginary quadratic
fields under certain assumptions [Kol91].

On the way to prove the structural Gross–Zagier formula, we also reveal a
_deeper_ relation between Kato’s zeta elements and Heegner points and it can
be viewed as a structural refinement of Perrin-Riou’s conjecture (Remark 6.1).
The following picture summarizes the context.

Kato’s Euler systems for $E$ and $E^{K}$ cyclotomic Kolyvagin derivatives
Heegner point Euler systems for $E$ over $K$ anticyclotomic Kolyvagin
derivatives Kato’s Kolyvagin systems for $E$ and $E^{K}$ dual exponential
structural Perrin-Riou Heegner point Kolyvagin systems determine the structure
structural Gross–Zagier Kurihara numbers for $E$ and $E^{K}$ determine the
structure
$\textstyle{{\mathrm{Sel}(K,E[p^{\infty}])\simeq\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\oplus\mathrm{Sel}(\mathbb{Q},E^{K}[p^{\infty}])}}$

where $p\geq 5$ is a prime, $E$ is a rational elliptic curve, $K$ is an
imaginary quadratic field, and $E^{K}$ is the quadratic twist of $E$ by $K$
satisfying the (generalized) Heegner hypothesis.

###  1.3. Bipartite Euler systems and the structure of Selmer groups

When the Heegner hypothesis is not satisfied, the Heegner point Kolyvagin
system method is certainly not available. In the extremely inspiring paper
[BD05], Bertolini–Darmon developed a completely different Euler system
argument involving the systematic use of rank lowering congruences and proved
one-sided divisibility of the anticyclotomic Iwasawa main conjecture for
elliptic curves. The theory of bipartite Euler systems [How06] captures the
essence of this Euler system argument axiomatically. We push the bipartite
Euler system method further to obtain the description of the structure of
Selmer groups over imaginary quadratic fields in terms of the values of level-
raising families of quaternionic automorphic forms (Theorem 5.23). This is the
counterpart of Kolyvagin’s work on the structure of Selmer groups mentioned
above. As a consequence, we obtain the structural Waldspurger formula as in
the following picture again

Kato’s Kolyvagin systems for $E$ and $E^{K}$ dual exponential bipartite Euler
systems (values of quaternionic forms) determine the structure structural
Waldspurger Kurihara numbers for $E$ and $E^{K}$ determine the structure
$\textstyle{{\mathrm{Sel}(K,E[p^{\infty}])\simeq\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])\oplus\mathrm{Sel}(\mathbb{Q},E^{K}[p^{\infty}])}}$

with the same notation above but with the opposite root number. In this sense,
bipartite Euler systems are very much like Kolyvagin systems rather than Euler
systems, as pointed out in [BD05, Introduction] and [How06, $\S$1].

###  1.4. The non-triviality questions on various “Kolyvagin systems”

Although various structure theorems are discussed above, the non-triviality of
the corresponding Kolyvagin system is _required_ for each structure theorem.
When Kolyvagin proved his structure theorem [Kol91], he assumed the non-
triviality of Heegner point Kolyvagin systems, and formulated it as a
conjecture. Later, Wei Zhang proved this conjecture for a large class of
elliptic curves [Zha14b] by using the Iwasawa main conjecture resolved by
Skinner–Urban [SU14].

The non-triviality of Kato’s Kolyvagin systems and the collection of Kurihara
numbers are also required for Mazur–Rubin’s structure theorem [MR04] and
Kurihara’s structure theorem [Kur14a, Kur14b, Kim], respectively. In [Kim], we
recently found a method to establish the non-triviality of Kato’s Kolyvagin
systems and the collection of Kurihara numbers by making connection with the
Iwasawa main conjecture.

By extending this idea, we show that the non-triviality of Heegner point
Kolyvagin systems and bipartite Euler systems follows from a _small piece of_
the corresponding main conjectures, respectively (Theorem 2.2). As a result,
we have a plenty of examples satisfying all the required non-triviality
statements. In particular, a more direct and simplified proof of the Heegner
point main conjecture [BCK21] is obtained (Corollary 7.8).

### Acknowledgement

The author thanks to Kâzim Büyükboduk and Naomi Sweeting for helpful
discussions. This research was partially supported by a KIAS Individual Grant
(SP054103) via the Center for Mathematical Challenges at Korea Institute for
Advanced Study and by the National Research Foundation of Korea(NRF) grant
funded by the Korea government(MSIT) (No. 2018R1C1B6007009).

##  2\. Statement of the main results

###  2.1. Working hypotheses

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ of conductor $N$ and
$p\geq 5$ a prime such that

  * (a)

the associated mod $p$ Galois representation
$\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{Aut}_{\mathbb{F}_{p}}(E[p])$
is surjective, and

  * (b)

the Manin constant is prime to $p$.

It is expected that (b) is always true and it actually holds if $E$ has semi-
stable reduction at $p$ [Maz78, Corollary 4.1].

Let $K$ be an imaginary quadratic field of discriminant $D_{K}$ such that
$(D_{K},Np)=1$. Write

$N=N^{+}\cdot N^{-}$

where a prime divisor of $N^{+}$ splits in $K/\mathbb{Q}$ and a prime divisor
of $N^{-}$ is inert in $K/\mathbb{Q}$. For a square-free integer $M$, denote
by $\nu(M)$ the number of prime divisors of $M$. We always assume that

  * (c)

$N^{-}$ is square-free,

  * (d)

if $\nu(N^{-})$ is odd, then $E$ has good reduction at $p$, and

  * (e)

if $\nu(N^{-})$ is odd and a prime $q$ satisfies $q|N^{-}$ and $q\equiv\pm
1\pmod{p}$, then $\overline{\rho}$ is ramified at $q$. (_Condition CR_)

Hypotheses (a) and (b) also hold for the quadratic twist $E^{K}$of $E$ by $K$.

Let $T$ be the $p$-adic Tate module of $E$ and $\mathcal{F}_{\mathrm{cl}}$ the
classical Selmer structure for $T$.

###  2.2. Kurihara numbers

Let $k\geq 1$ be an integer,

$\mathcal{P}^{\mathrm{cyc}}_{k}=\left\\{q,\textrm{ a prime}:q\nmid Np,q\equiv
1\ (\text{mod}\ p^{k}),a_{q}(E)\equiv q+1\ (\text{mod}\ p^{k})\right\\},$

and write $I^{\mathrm{cyc}}_{q}=(q-1,a_{q}(E)-q-1)\mathbb{Z}_{p}$ for
$q\in\mathcal{P}^{\mathrm{cyc}}_{k}$. Let $\mathcal{N}^{\mathrm{cyc}}_{k}$ be
the set of square-free products of the primes in
$\mathcal{P}^{\mathrm{cyc}}_{k}$. For $n\in\mathcal{N}^{\mathrm{cyc}}_{k}$,
write $I^{\mathrm{cyc}}_{n}=\sum_{q|n}I^{\mathrm{cyc}}_{q}$.

The collection of Kurihara numbers for $E$ is denoted by

$\widetilde{\boldsymbol{\delta}}(E)=\widetilde{\boldsymbol{\delta}}=\left\\{\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I^{\mathrm{cyc}}_{n}\mathbb{Z}_{p}\right\\}_{n\in\mathcal{N}^{\mathrm{cyc}}_{1}}$

and each $\delta_{n}$ is explicitly built out from modular symbols. See $\S$3
for details. We write

$\displaystyle\mathrm{ord}(\widetilde{\boldsymbol{\delta}})$
$\displaystyle=\mathrm{min}\left\\{\nu(n):n\in\mathcal{N}^{\mathrm{cyc}}_{1},\widetilde{\delta}_{n}\neq
0\right\\},$ $\displaystyle\partial^{(i)}(\widetilde{\boldsymbol{\delta}})$
$\displaystyle=\mathrm{min}\left\\{j:\widetilde{\delta}_{n}\in
p^{j}\mathbb{Z}_{p}/I^{\mathrm{cyc}}_{n}\mathbb{Z}_{p}\textrm{ for every
}n\in\mathcal{N}^{\mathrm{cyc}}_{1}\textrm{ with }\nu(n)=i\right\\},$
$\displaystyle\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})$
$\displaystyle=\mathrm{min}\left\\{\partial^{(i)}(\widetilde{\boldsymbol{\delta}}):i\geq
0\right\\}.$

###  2.3. Heegner point Kolyvagin systems

Suppose that $\nu(N^{-})$ is even. Let $k\geq 1$ be an integer,

$\mathcal{P}^{\mathrm{ac}}_{k}=\left\\{q,\textrm{ a prime}:q\nmid Np,q\textrm{
inert in }K/\mathbb{Q},a_{q}(E)\equiv q+1\equiv 0\ (\text{mod}\
p^{k})\right\\},$

and write $I^{\mathrm{ac}}_{q}=(q+1,a_{q}(E))\mathbb{Z}_{p}$ for
$q\in\mathcal{P}^{\mathrm{ac}}_{k}$. Let $\mathcal{N}^{\mathrm{ac}}_{k}$ be
the set of square-free products of the primes in
$\mathcal{P}^{\mathrm{ac}}_{k}$. For $n\in\mathcal{N}^{\mathrm{ac}}_{k}$,
write $I^{\mathrm{ac}}_{n}=\sum_{q|n}I^{\mathrm{ac}}_{q}$.

The Heegner point Kolyvagin system for
$(T,\mathcal{F}_{\mathrm{cl}},\mathcal{N}^{\mathrm{ac}}_{1})$ is the family of
cohomology classes

$\boldsymbol{\kappa}^{\mathrm{Heeg}}=\left\\{\kappa^{\mathrm{Heeg}}_{n}\in\mathrm{Sel}_{\mathcal{F}_{\mathrm{cl}}(n)}(K,T/I^{\mathrm{ac}}_{n}T)\right\\}_{n\in\mathcal{N}^{\mathrm{ac}}_{1}}$

satisfying certain relations [How04, Definition 1.2.3] where
$\mathcal{F}_{\mathrm{cl}}(n)$ means the classical Selmer structure except
ordinary local conditions at primes dividing $n$ (cf. $\S$5.1.2). See $\S$4
for details. We write

$\displaystyle\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})$
$\displaystyle=\mathrm{min}\left\\{\nu(n):n\in\mathcal{N}^{\mathrm{ac}}_{1},\kappa^{\mathrm{Heeg}}_{n}\neq
0\right\\},$
$\displaystyle\partial^{(i)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})$
$\displaystyle=\mathrm{min}\left\\{j:\kappa^{\mathrm{Heeg}}_{n}\in
p^{j}\mathrm{Sel}_{\mathcal{F}_{\mathrm{cl}}(n)}(K,T/I^{\mathrm{ac}}_{n}T)\textrm{
for every }n\in\mathcal{N}^{\mathrm{ac}}_{1}\textrm{ with }\nu(n)=i\right\\},$
$\displaystyle\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})$
$\displaystyle=\mathrm{min}\left\\{\partial^{(i)}(\boldsymbol{\kappa}^{\mathrm{Heeg}}):i\geq
0\right\\}.$

###  2.4. Bipartite Euler systems

Suppose that $\nu(N^{-})$ is odd. Let $k\geq 1$ be an integer,

$\displaystyle\mathcal{P}^{\mathrm{adm}}_{k}=\left\\{q,\textrm{ a
prime}:\right.$ $\displaystyle q\nmid Np,q\textrm{ inert in
}K/\mathbb{Q},q\not\equiv\pm 1\ (\text{mod}\ p),$
$\displaystyle\left.a_{q}(E)\equiv\epsilon_{q}\cdot(q+1)\ (\text{mod}\
p^{k})\textrm{ with }\epsilon_{q}=1\textrm{ or }-1\right\\},$

and $I^{\mathrm{adm}}_{\ell}=(a_{q}(E)-\epsilon_{q}\cdot(q+1))\mathbb{Z}_{p}$.
Let $\mathcal{N}^{\mathrm{adm}}_{k}$ be the set of square-free products of the
primes in $\mathcal{P}^{\mathrm{adm}}_{k}$. For
$n\in\mathcal{N}^{\mathrm{adm}}_{k}$, write
$I^{\mathrm{adm}}_{n}=\sum_{\ell|n}I^{\mathrm{adm}}_{\ell}$.

Let $\mathcal{N}^{\mathrm{def}}_{k}\subseteq\mathcal{N}^{\mathrm{adm}}_{k}$ be
the subset defined by putting the additional condition: every
$n\in\mathcal{N}^{\mathrm{def}}_{k}$ satisfies that $\nu(nN^{-})$ is odd. Then
$\mathcal{N}^{\mathrm{ind}}_{k}$ is defined by
$\mathcal{N}^{\mathrm{adm}}_{k}\setminus\mathcal{N}^{\mathrm{def}}_{k}$ so
that every $n\in\mathcal{N}^{\mathrm{ind}}_{k}$ satisfies that $\nu(nN^{-})$
is even. A bipartite Euler system for
$(T,\mathcal{F}_{\mathrm{cl}},\mathcal{N}^{\mathrm{adm}}_{1})$ consists of two
families

$\textstyle{\boldsymbol{\lambda}^{\mathrm{bip}}=\left\\{\lambda^{\mathrm{bip}}_{n}\in\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}\right\\}_{n\in\mathcal{N}^{\mathrm{def}}_{1}},}$$\textstyle{\boldsymbol{\kappa}^{\mathrm{bip}}=\left\\{\kappa^{\mathrm{bip}}_{n}\in\mathrm{Sel}_{\mathcal{F}_{\mathrm{cl}}(n)}(K,T/I^{\mathrm{adm}}_{n}T)\right\\}_{n\in\mathcal{N}^{\mathrm{ind}}_{1}}}$

with the connection via the first and second explicit reciprocity laws
(Definition 5.12) where $\boldsymbol{\lambda}^{\mathrm{bip}}$ is constructed
from the values of congruent families of quaternionic automorphic forms
varying definite quaternion algebras and $\boldsymbol{\kappa}^{\mathrm{bip}}$
is constructed from Heegner points on _different_ Shimura curves varying
indefinite quaternion algebras. See $\S$5 for details. We write

$\displaystyle\mathrm{ord}(\boldsymbol{\lambda}^{\mathrm{bip}})$
$\displaystyle=\mathrm{min}\left\\{\nu(n):n\in\mathcal{N}^{\mathrm{def}}_{1},\lambda^{\mathrm{bip}}_{n}\neq
0\right\\},$
$\displaystyle\partial^{(i)}(\boldsymbol{\lambda}^{\mathrm{bip}})$
$\displaystyle=\mathrm{min}\left\\{j:\lambda^{\mathrm{bip}}_{n}\in
p^{j}\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}\textrm{ for every
}n\in\mathcal{N}^{\mathrm{def}}_{1}\textrm{ with (even) }\nu(n)=i\right\\},$
$\displaystyle\partial^{(\infty)}(\boldsymbol{\lambda}^{\mathrm{bip}})$
$\displaystyle=\mathrm{min}\left\\{\partial^{(i)}(\boldsymbol{\lambda}^{\mathrm{bip}}):i\geq
0\right\\}.$

######  Remark 2.1.

Our formulation of bipartite Euler systems works even when an elliptic curve
has supersingular reduction at $p$ _and_ $p$ is inert in $K/\mathbb{Q}$. In
particular, we do not use anticyclotomic Iwasawa theory at all except $\S$7.

###  2.5. The non-triviality questions

The following non-triviality statements play important roles in this article.

  * •

$\widetilde{\boldsymbol{\delta}}$ does not vanish.

  * •

$\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial.

  * •

$\boldsymbol{\lambda}^{\mathrm{bip}}$ does not vanish.

All these non-triviality statements hold for elliptic curves of analytic rank
$\leq 1$ with good reduction at $p$ [BDV22, BST, BPS]. In [Kim], we proved the
equivalence between the the localization of the Iwasawa main conjecture at the
height one prime ideal corresponding to the trivial character and the non-
vanishing of $\widetilde{\boldsymbol{\delta}}$. Extending this idea to the
cases of $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ and
$\boldsymbol{\lambda}^{\mathrm{bip}}$, we obtain the following result in
$\S$7.

######  Theorem 2.2.

We keep all the working hypotheses in $\S$2.1 as well as additional hypotheses
given in Assumption 7.4 and Assumption 7.11 below.

  1. (1)

If the localization of the Iwasawa main conjecture at the height one prime
ideal corresponding to the trivial character holds, then
$\widetilde{\boldsymbol{\delta}}$ does not vanish (Corollary 7.2).

  2. (2)

Suppose that $\nu(N^{-})$ is even. If the localization of the Heegner point
main conjecture at the height one prime ideal corresponding to the trivial
character holds, then $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial
(Corollary 7.8).

  3. (3)

Suppose that $\nu(N^{-})$ is odd. If the localization of the anticyclotomic
main conjecture for elliptic curves at the height one prime ideal
corresponding to the trivial character holds, then
$\boldsymbol{\lambda}^{\mathrm{bip}}$ does not vanish (Corollary 7.16).

It is remarkable that Theorem 2.2 provides us with numerous examples
satisfying the non-triviality statements we need. In particular, when an
elliptic curve has good ordinary reduction at $p$ and $p$ splits in $K$, these
non-triviality questions are also addressed in [Sak, Kim] for
$\widetilde{\boldsymbol{\delta}}$, [Zha14b] for
$\boldsymbol{\kappa}^{\mathrm{Heeg}}$, and [BCK21] for
$\boldsymbol{\lambda}^{\mathrm{bip}}$ under mild additional hypotheses, which
allows us to invoke the Iwasawa main conjecture resolved by Skinner–Urban
[SU14].

###  2.6. Higher Gross–Zagier and Waldspurger formulas

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

Our higher Gross–Zagier formula relates $\boldsymbol{\kappa}^{\mathrm{Heeg}}$
with $\widetilde{\boldsymbol{\delta}}(E)$ and
$\widetilde{\boldsymbol{\delta}}(E^{K})$ and reveals the exact size of the
Selmer group as follows.

######  Theorem 2.3 (Higher Gross–Zagier).

We keep all the working hypotheses in $\S$2.1 with even $\nu(N^{-})$. If
$\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial, and
$\widetilde{\boldsymbol{\delta}}(E)$ and
$\widetilde{\boldsymbol{\delta}}(E^{K})$ are both non-zero, then

$\displaystyle\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1$
$\displaystyle=\mathrm{max}\left\\{\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E)),\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))\right\\}$
$\displaystyle=\mathrm{max}\left\\{\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}]),\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E^{K}[p^{\infty}])\right\\}.$

If we further assume that
$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])$ and
$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E^{K}[p^{\infty}])$
differ by 1, then

$\displaystyle\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})$
$\displaystyle=\mathrm{min}\left\\{\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E)),\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))\right\\}$
$\displaystyle=\mathrm{min}\left\\{\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}]),\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E^{K}[p^{\infty}])\right\\},$

and

(2.1)
$\displaystyle\begin{split}2\cdot\left(\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}}))}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})\right)=\
&\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E)))}(\widetilde{\boldsymbol{\delta}}(E))-\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}}(E))\\\
&+\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K})))}(\widetilde{\boldsymbol{\delta}}(E^{K}))-\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}}(E^{K}))\\\
=\
&\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,E[p^{\infty}])_{/\mathrm{div}}\right).\end{split}$

###### Proof.

It is immediate from $\S$6.2. ∎

######  Remark 2.4.

  1. (1)

The additional assumption on Selmer coranks comes from Kolyvagin’s structure
theorem [Kol91], [Zha14b, Remark 18]. See also [RS07] for the variation of
ranks in quadratic twist families of elliptic curves.

  2. (2)

Conceptually, the RHS of (2.1) corresponds to the derivative of the
Rankin–Selberg product $L$-function, and the LHS of (2.1) corresponds to the
height of Heegner points. See $\S$2.7 for a more detailed comparison with
classical Gross–Zagier formula. In some sense,
$\partial^{(i)}(\boldsymbol{\alpha})$ plays the role of $i$-th “Taylor
coefficient” of $\boldsymbol{\alpha}$.

  3. (3)

Theorem 2.3 is a piece of the comparison between Kolyvagin’s structure theorem
[Kol91] and Kurihara’s structure theorem [Kur14a, Kur14b, Kim]. The full
comparison is given in $\S$6.2.

  4. (4)

Let $\boldsymbol{\kappa}^{\mathrm{Kato}}(E)$ be Kato’s Kolyvagin system for
$E$. Since $\widetilde{\boldsymbol{\delta}}(E)$ is the image of
$\boldsymbol{\kappa}^{\mathrm{Kato}}(E)$ under the dual exponential map (cf.
Corollary 7.2), Theorem 2.3 also explicitly relates
$\boldsymbol{\kappa}^{\mathrm{Heeg}}$ to
$\boldsymbol{\kappa}^{\mathrm{Kato}}(E)$ and
$\boldsymbol{\kappa}^{\mathrm{Kato}}(E^{K})$ from the viewpoint of Perrin-
Riou’s conjecture [PR93, BDV22, BST, BPS].

Our higher Waldspurger formula relates $\boldsymbol{\lambda}^{\mathrm{bip}}$
with $\widetilde{\boldsymbol{\delta}}(E)$ and
$\widetilde{\boldsymbol{\delta}}(E^{K})$ and reveals the exact size of the
Selmer group again.

######  Theorem 2.5 (Higher Waldspurger).

We keep all the working hypotheses in $\S$2.1 with odd $\nu(N^{-})$. If
$\boldsymbol{\lambda}^{\mathrm{bip}}$ is non-zero, and
$\widetilde{\boldsymbol{\delta}}(E)$ and
$\widetilde{\boldsymbol{\delta}}(E^{K})$ are non-zero, then

$\mathrm{ord}(\boldsymbol{\lambda}^{\mathrm{bip}})=\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E))+\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))=\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(K,E[p^{\infty}])$

and

(2.2)
$\displaystyle\begin{split}2\cdot\left(\partial^{(\mathrm{ord}(\boldsymbol{\lambda}^{\mathrm{bip}}))}(\boldsymbol{\lambda}^{\mathrm{bip}})-\partial^{(\infty)}(\boldsymbol{\lambda}^{\mathrm{bip}})\right)=\
&\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E)))}(\widetilde{\boldsymbol{\delta}}(E))-\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}}(E))\\\
&+\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K})))}(\widetilde{\boldsymbol{\delta}}(E^{K}))-\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}}(E^{K}))\\\
=\
&\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,E[p^{\infty}])_{/\mathrm{div}}\right).\end{split}$

###### Proof.

It is immediate from $\S$6.3. ∎

######  Remark 2.6.

  1. (1)

The LHS of (2.2) corresponds to the toric period integral.

  2. (2)

In order to obtain Theorem 2.5, we first develop the structure theorem of
Selmer groups via bipartite Euler systems (Theorem 5.23) and compare it with
Kurihara’s structure theorem. The full comparison is given in $\S$6.3.

###  2.7. The connection with the classical Gross–Zagier formula

We first recall the classical Gross–Zagier formula following [GZ86].

Let $K$ be an imaginary quadratic field of discriminant $D_{K}$. We assume
that $D_{K}$ is odd, hence square-free and congruent to 1 (mod 4), for
convenience.

Let $\pi:X_{0}(N)\to E$ be the modular parametrization sending the infinity
cusp to the point at infinity. Let $f\in S_{2}(\Gamma_{0}(N))$ be the newform
corresponding to $E$ and write $\omega_{f}=2\pi\sqrt{-1}\cdot
f(z)dz=\sum_{n\geq 1}a_{n}q^{n}\cdot\dfrac{dq}{q}$. Denote by $\omega_{E}$ a
Néron differential on $E$, which is unique up to sign. Then
$\pi^{*}(\omega_{E})=c\cdot\omega_{f}$ for some non-zero integer $c$. We
normalize $\omega_{E}$ so that $c>0$. Following [GZ86, Page 310], we define

$\|\omega_{E}\|^{2}:=\int\int_{E(\mathbb{C})}\omega_{E}\wedge\overline{\omega_{E}}=c^{2}\cdot\dfrac{\|\omega_{f}\|^{2}}{\mathrm{deg}(\pi)}.$

where
$\|\omega_{f}\|^{2}=\int\int_{X_{0}(N)(\mathbb{C})}\omega_{f}\wedge\overline{i\cdot\omega_{f}}=8\pi^{2}\cdot(f,f)$
as described in [GZ86, Page 230]

Let

$P_{K}=\sum_{\sigma\in\mathrm{Gal}(H/K)}\pi(\sigma x)\in E(K)$

where $x\in X_{0}(N)(H)$ is any Heegner point of discriminant $D_{K}$ and $H$
is the Hilbert class field of $K$. In particular, $P_{K}$ is independent of
the choice of $x$.

######  Theorem 2.7 (Gross–Zagier).

Suppose that $\mathrm{ord}_{s=1}L(E/K,s)=1$. Then

$\dfrac{L^{\prime}(E/K,1)}{\|\omega_{E}\|^{2}}=\dfrac{\widehat{h}(P_{K})}{c^{2}\cdot
u^{2}_{K}\cdot|D_{K}|^{1/2}}$

where $\widehat{h}(-)$ is the canonical height on $E(K)$.

###### Proof.

See [GZ86, (2.1) Theorem, Page 311]. ∎

Denote by $\langle-,-\rangle_{\mathrm{NT}}$ the Néron–Tate height pairing.
Since $\mathrm{Reg}(E/K)=\dfrac{\langle
P_{K},P_{K}\rangle_{\mathrm{NT}}}{[E(K):\mathbb{Z}\cdot P_{K}]^{2}}$ and
$\langle P_{K},P_{K}\rangle_{\mathrm{NT}}=\widehat{h}(P_{K})$ (e.g. [Zha14a,
Page 175]), Theorem 2.7 is equivalent to

$\dfrac{L^{\prime}(E/K,1)}{\|\omega\|^{2}\cdot\mathrm{Reg}(E/K)}=\dfrac{[E(K):\mathbb{Z}\cdot
P_{K}]^{2}}{c^{2}\cdot u^{2}_{K}\cdot|D_{K}|^{1/2}}.$

Let $E^{K}$ be the quadratic twist of $E$ by $K$. Following [GZ86, Page 312],
we have

$\dfrac{\|\omega_{E}\|^{2}}{|D_{K}|^{1/2}}=[E(\mathbb{R}):E(\mathbb{R})^{0}]\cdot\Omega^{+}_{E}\cdot\Omega^{+}_{E^{K}}.$

Now we assume that the analytic rank of $E$ is one and the analytic rank of
$E^{K}$ is zero. Then we have

$L^{\prime}(E/K,1)=L^{\prime}(E,1)\cdot L(E^{K},1)$

since $L(E/K,s)=L(E,s)\cdot L(E^{K},s)$. Combining the above three equations,
we obtain

(2.3) $\dfrac{c^{2}\cdot
u^{2}_{K}}{[E(\mathbb{R}):E(\mathbb{R})^{0}]}\cdot\dfrac{L^{\prime}(E,1)\cdot
L(E^{K},1)}{\Omega^{+}_{E}\cdot\Omega^{+}_{E^{K}}\cdot\mathrm{Reg}(E/K)}=[E(K):\mathbb{Z}\cdot
P_{K}]^{2}.$

If $E(K)[p]=0$, then
$\mathrm{ord}_{p}\left(\dfrac{\mathrm{Reg}(E/K)}{\mathrm{Reg}(E/\mathbb{Q})}\right)=0$
since $P_{K}-\overline{P}_{K}\in E^{K}(\mathbb{Q})_{\mathrm{tors}}\subseteq
E^{K}(K)_{\mathrm{tors}}=E(K)_{\mathrm{tors}}$. Thus, for a good reduction
prime $p>3$ with $E(K)[p]=0$, the $p$-part of (2.3) is equivalent to

(2.4)
$\mathrm{ord}_{p}\left(\dfrac{L^{\prime}(E,1)}{\Omega^{+}_{E}\cdot\mathrm{Reg}(E/\mathbb{Q})}\right)+\mathrm{ord}_{p}\left(\dfrac{L(E^{K},1)}{\Omega^{+}_{E^{K}}}\right)=2\cdot\mathrm{ord}_{p}\left([E(K):\mathbb{Z}\cdot
P_{K}]\right).$

For simplicity, we assume that both $E(\mathbb{Q}_{p})[p^{\infty}]$ and
$E^{K}(\mathbb{Q}_{p})[p^{\infty}]$ are trivial. Then one of our formulas
(2.1) can be regarded as a conceptual generalization of (2.4) with equalities

(2.5)
$\textstyle{\partial^{(1)}(\widetilde{\boldsymbol{\delta}}(E))=\mathrm{ord}_{p}\left(\dfrac{L^{\prime}(E,1)}{\Omega^{+}_{E}\cdot\mathrm{Reg}(E/\mathbb{Q})}\right),}$$\textstyle{\partial^{(0)}(\widetilde{\boldsymbol{\delta}}(E^{K}))=\mathrm{ord}_{p}\left(\dfrac{L(E^{K},1)}{\Omega^{+}_{E^{K}}}\right)}$

“up to Tamagawa factors” since we also expect that

$\displaystyle\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})$
$\displaystyle=\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}}(E))+\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}}(E^{K}))$
$\displaystyle=\textrm{the sum of the $p$-valuations of Tamagawa factors of
$E$ over $K$}.$

See [Jet08, Büy09, Zha14a, Kim] for the details of the Tamagawa factor issue.
It is remarkable that the first equality of (2.5) shows how the derivative of
the $L$-function and the regulator are absorbed in modular symbols explicitly.

##  3\. Modular symbols and Kurihara numbers

We review the structural refinement of Birch and Swinnerton-Dyer conjecture
given in [Kim].

###  3.1. Modular symbols and Kurihara numbers

Let $p\geq 5$ be a prime and $E$ an elliptic curve over $\mathbb{Q}$
satisfying (a) and (b) of the working hypotheses in $\S$2.1. Let
$f=\sum_{n\geq 1}a_{n}(E)q^{n}\in S_{2}(\Gamma_{0}(N))$ be the newform
corresponding to $E$ [BCDT01]. For each $\dfrac{a}{b}\in\mathbb{Q}$, the
modular symbol $\left[\dfrac{a}{b}\right]^{+}$ is defined by equality

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

where $\left[\dfrac{a}{b}\right]^{+}$ and $\left[\dfrac{a}{b}\right]^{-}$ are
rational numbers where $\Omega^{\pm}_{E}$ is the real and imaginary Néron
period of $E$, respectively [MT87, (1.1)]. Under our working hypotheses,
$\left[\dfrac{a}{b}\right]^{+}\in\mathbb{Z}_{(p)}$.

For each prime $\ell\in\mathcal{P}^{\mathrm{cyc}}_{k}$, we fix a primitive
root $\eta_{\ell}$ mod $\ell$ and define
$\mathrm{log}_{\eta_{\ell}}(a)\in\mathbb{Z}/(\ell-1)$ by
$\eta^{\mathrm{log}_{\eta_{\ell}}(a)}_{\ell}\equiv a\pmod{\ell}$. For each
$n\in\mathcal{N}^{\mathrm{cyc}}_{1}$, the (mod $I^{\mathrm{cyc}}_{n}$)
Kurihara number at $n$ is defined by

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

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

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

The collection of Kurihara numbers is defined by

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

Recall that the vanishing order of $\widetilde{\boldsymbol{\delta}}$ is
defined by

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

We assume that $\mathrm{ord}(\widetilde{\boldsymbol{\delta}})<\infty$, i.e.
the collection of Kurihara numbers does not vanish identically.

###  3.2. The modular symbol description of Selmer groups

Denote by $\partial^{(0)}(\widetilde{\boldsymbol{\delta}})$ by the $p$-adic
valuation of $\widetilde{\delta}_{1}$. For $i\in\mathbb{Z}_{>0}$, we define

$\partial^{(i)}(\widetilde{\boldsymbol{\delta}})=\mathrm{min}\left\\{j:\widetilde{\delta}_{n}\in
p^{j}\mathbb{Z}_{p}/I^{\mathrm{cyc}}_{n}\mathbb{Z}_{p}\textrm{ for every
}n\in\mathcal{N}^{\mathrm{cyc}}_{1}\textrm{ with }\nu(n)=i\right\\}$

and
$\partial^{(\infty)}(\widetilde{\boldsymbol{\delta}})=\mathrm{min}\left\\{\partial^{(i)}(\widetilde{\boldsymbol{\delta}}):i\geq
0\right\\}$.

######  Theorem 3.1.

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p\geq 5$ a prime
satisfying (a) and (b) of the working hypotheses in $\S$2.1. If
$\mathrm{ord}(\widetilde{\boldsymbol{\delta}})<\infty$, then

  1. (1)

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

  2. (2)

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

  3. (3)

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

In other words, we have

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

###### Proof.

See [Kim]. ∎

##  4\. Heegner point Kolyvagin systems

We review the main result of [Kol91]. We keep all the working hypotheses in
$\S$2.1 and assume that $\nu(N^{-})$ is even.

###  4.1. Convention

Let $X^{N^{-}}(N^{+})$ be the (compactified) Shimura curve associated to the
indefinite quaternion algebra of discriminant $N^{-}$ and an Eichler order of
level $N^{+}$. Let $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ be the Heegner point
Kolyvagin system coming from Heegner points on $X^{N^{-}}(N^{+})$. See [How04,
Nek07, BCK21] for the explicit construction of
$\boldsymbol{\kappa}^{\mathrm{Heeg}}$. Let
$\mathrm{Sel}(K,E[p^{\infty}])^{\pm}$ be the eigenspace with eigenvalue $\pm
1$ of $\mathrm{Sel}(K,E[p^{\infty}])$ with respect to the complex conjugation,
respectively. Since $p>2$, we have decomposition

$\mathrm{Sel}(K,E[p^{\infty}])=\mathrm{Sel}(K,E[p^{\infty}])^{+}\oplus\mathrm{Sel}(K,E[p^{\infty}])^{-}.$

Write

$\textstyle{r(E/K)=\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(K,E[p^{\infty}]),}$$\textstyle{r^{\pm}(E/K)=\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(K,E[p^{\infty}])^{\pm}.}$

Denote by $W(E/\mathbb{Q})$ the root number of $E$ over $\mathbb{Q}$.

###  4.2. Structure theorem

######  Theorem 4.1.

Suppose that $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial. Then the
following statements are valid.

  1. (1)

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

  2. (2)

$r^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1}}(E/K)=\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1$.

  3. (3)

$\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-r^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}(E/K)\geq
0$ and is even.

###### Proof.

See [Kol91, Theorem 4] and also [Zha14b, Theorems 1.2 and 11.2]. ∎

The following corollary is immediate.

######  Corollary 4.2.

If
$\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E[p^{\infty}])-\mathrm{cork}_{\mathbb{Z}_{p}}\mathrm{Sel}(\mathbb{Q},E^{K}[p^{\infty}])=\pm
1$, then we have

$\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-r^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}(E/K)=0.$

Write

(4.1)
$\mathrm{Sel}(K,E[p^{\infty}])^{\pm}_{/\mathrm{div}}\simeq\bigoplus_{i\geq
1}\left(\mathbb{Z}/p^{a^{\pm}_{i}}\mathbb{Z}\right)^{\oplus 2}$

where $a^{\pm}_{1}\geq a^{\pm}_{2}\geq\cdots$.

######  Theorem 4.3.

Suppose that $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial. Then
$a^{\pm}_{i}$’s in (4.1) are determined by
$\boldsymbol{\kappa}^{\mathrm{Heeg}}$ as follows:

$\displaystyle
a^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1}}_{i}$
$\displaystyle=\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i)}(\boldsymbol{\kappa}^{\mathrm{Heeg}}),$
$\displaystyle
a^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}_{i+(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-r^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}(E/K))}$
$\displaystyle=\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-2)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})$

for $i\geq 1$.

###### Proof.

See [Kol91, Theorem 1]. See also [Zha14b, Remark 18]. ∎

######  Remark 4.4.

We do not know whether
$a^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}_{i}$’s
are determined by $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ or not for $1\leq
i\leq\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-r^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}(E/K)$.
Write

$\mathrm{Err}=\bigoplus_{i}\mathbb{Z}/p^{a^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}_{i}}\mathbb{Z}$

where $i$ runs over $1\leq
i\leq\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-r^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})}}(E/K)$.
Then we have

$\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,E[p^{\infty}])_{/\mathrm{div}}\right)=2\cdot\left(\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}}))}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+\mathrm{length}_{\mathbb{Z}_{p}}(\mathrm{Err})\right).$

##  5\. Bipartite Euler systems for elliptic curves

In this section, we deal with the counterpart of Kolyvagin’s work on the
structure of Selmer groups reviewed in §4. Let $E$ be an elliptic curve over
the rationals, $p\geq 5$ a _good_ reduction prime for $E$, and $K$ an
imaginary quadratic field. When the (generalized) Heegner hypothesis for
$(E,K)$ is satisfied, Theorem 4.3 describes the module structure of
$\mathrm{Sel}(K,E[p^{\infty}])$ in terms of the divisibilities of the (non-
trivial) Heegner point Kolyvagin system under mild assumptions as we have seen
[Kol91, How04, Zha14b].

The opposite case is studied in this section. In other words, the root number
of $E/K$ is 1. The main result is the description of the structure of
$\mathrm{Sel}(K,E[p^{\infty}])$ in terms of the valuations of the values of
certain families of quaternionic automorphic forms (Theorem 5.23). One
remarkable feature is that our argument works even when $E$ has supersingular
reduction at a prime $p$ and $p$ is inert in $K$ since we do not use Iwasawa
theory in this section.

###  5.1. Formalism of the theory of bipartite Euler systems

We first review the theory of bipartite Euler systems _over the imaginary
quadratic field_ closely following [How06]. Then we prove the structure
theorem of Selmer groups (Theorem 5.18). We fix one principal Artinian local
ring $R$ as the coefficient ring in this subsection.

####  5.1.1. Preliminaries

Let $K$ be an imaginary quadratic field and $G_{K}$ the absolute Galois group
of $K$. Let $R$ be a principal Artinian local ring with maximal ideal
$\mathfrak{m}$ and residue characteristic $p>3$ (e.g.
$R=\mathbb{Z}/p^{k}\mathbb{Z}$). Let $T$ be a free $R$-module of rank two
endowed with continuous action of $G_{K}$ such that there exists a perfect,
$G_{K}$-equivariant, and alternating pairing

$T\times T\to R(1).$

######  Remark 5.1.

The notation $T$ in this subsection is _not_ a Galois representation with a
coefficient ring of characteristic zero.

If $B$ is any $R$-module and $b\in B$ we define $\mathrm{ind}(b,B)$, the index
of divisibility of $b$ in $B$, to be the largest $k\leq\infty$ such that
$b\in\mathfrak{m}^{k}B$.

####  5.1.2. Selmer groups and variants

A Selmer structure $(\mathcal{F},\Sigma)$ on $T$ consists of

  * •

a finite set $\Sigma$ of places of $K$ containing the archimedean places, the
primes where $T$ is ramified, and the prime $p$.

  * •

a choice of submodule
$\mathrm{H}^{1}_{\mathcal{F}}(K_{v},T)\subseteq\mathrm{H}^{1}(K_{v},T)$ for
every place $v$ of $K$ such that
$\mathrm{H}^{1}_{\mathcal{F}}(K_{v},T)=\mathrm{H}^{1}_{\mathrm{ur}}(K_{v},T)$
for all $v\not\in\Sigma$.

The Selmer group $\mathrm{Sel}_{\mathcal{F}}(K,T)$ associated to $\mathcal{F}$
is defined by the exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{\mathcal{F}}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{v}\dfrac{\mathrm{H}^{1}(K_{v},T)}{\mathrm{H}^{1}_{\mathcal{F}}(K_{v},T)}}$

where the direct sum runs over all places $v$ of $K$. A Selmer structure
$\mathcal{F}$ is self-dual if $\mathrm{H}^{1}_{\mathcal{F}}(K_{v},T)$ is
maximal isotropic under the (symmetric) local Tate pairing for every finite
place $v$ of $K$. Since $p>2$, $\mathrm{H}^{1}(K_{v},T)=0$ for $v$
archimedean.

We assume that we are given a fixed self-dual Selmer structure throughout this
section.

For a submodule or a quotient $S$ of $T$ and the Selmer structure
$\mathcal{F}$ on $T$, there exists an induced Selmer structure on $S$ defined
by using the natural map $\mathrm{H}^{1}(K_{v},S)\to\mathrm{H}^{1}(K_{v},T)$
or $\mathrm{H}^{1}(K_{v},T)\to\mathrm{H}^{1}(K_{v},S)$, respectively. See
[How06, Remark 2.1.3] for details.

A rational prime $\ell$ is an admissible prime for $(T,K)$ if

  1. (1)

$\ell$ is coprime to the residue characteristic $p$,

  2. (2)

$T$ is unramified at $\ell$,

  3. (3)

$\ell$ is inert in $K/\mathbb{Q}$,

  4. (4)

$\ell^{2}\not\equiv 1\pmod{p}$,

  5. (5)

$a_{\ell}=\pm(\ell+1)$ in $R$ where $a_{\ell}$ is the trace of the arithmetic
Frobenius at $\ell$ acting on $T$.

When $R=\mathbb{Z}/p^{k}\mathbb{Z}$, $\ell$ is called a $k$-admissible prime,
and write $\ell\in\mathcal{P}^{\mathrm{adm}}_{k}$. Denote by
$\mathcal{P}^{\mathrm{adm}}$ the set of admissible primes for $(T,K)$ and by
$\mathcal{N}^{\mathrm{adm}}$ the set of square-free products of primes in
$\mathcal{P}^{\mathrm{adm}}$. By the definition of admissible primes, we have
unique decomposition $T\simeq R\oplus R(1)$ as $G_{K_{\ell}}$-modules where
$\ell\in\mathcal{P}^{\mathrm{adm}}$. By using the decomposition, we define the
ordinary cohomology $\mathrm{H}^{1}_{\mathrm{ord}}(K_{\ell},T)$ by the image
of $\mathrm{H}^{1}(K_{\ell},R(1))$ in $\mathrm{H}^{1}(K_{\ell},T)$ for each
$\ell\in\mathcal{P}^{\mathrm{adm}}$.

######  Lemma 5.2.

For each $\ell\in\mathcal{P}^{\mathrm{adm}}$, the decomposition $T\simeq
R\oplus R(1)$ induces a decomposition

$\mathrm{H}^{1}(K_{\ell},T)\simeq\mathrm{H}^{1}_{\mathrm{ur}}(K_{\ell},T)\oplus\mathrm{H}^{1}_{\mathrm{ord}}(K_{\ell},T)$

in which each summand is free of rank one over $R$ and is maximal isotropic
under the local Tate pairing.

###### Proof.

See [How06, Lemma 2.2.1]. ∎

A Selmer structure $\mathcal{F}$ is cartesian if for every quotient
$T/\mathfrak{m}^{i}T$ of $T$, every place $v\in\Sigma$, and any generator
$\pi\in\mathfrak{m}$, the isomorphism

$\times\pi^{\mathrm{length}(R)-i}:T/\mathfrak{m}^{i}T\to T[\mathfrak{m}^{i}]$

induces an isomorphism

$\mathrm{H}^{1}_{\mathcal{F}}(K_{v},T/\mathfrak{m}^{i}T)\simeq\mathrm{H}^{1}_{\mathcal{F}}(K_{v},T[\mathfrak{m}^{i}]).$

######  Assumption 5.3.

Throughout this section, we make the following assumptions:

  1. (1)

The residual representation $T/\mathfrak{m}T$ is absolutely irreducible.

  2. (2)

$\mathcal{F}$ is cartesian.

For any square-free integer $a\cdot b$ and any
$c\in\mathcal{N}^{\mathrm{adm}}$, the Selmer structure
$(\mathcal{F}^{a}_{b}(c),\Sigma_{\mathcal{F}^{a}_{b}(c)})$ is defined by

  * •

$\Sigma_{\mathcal{F}^{a}_{b}(c)}=\Sigma\cup\left\\{q,\textrm{ a
prime}:q\textrm{ divides }a\cdot b\cdot c\right\\}$,

  * •

$\mathrm{H}^{1}_{\mathcal{F}^{a}_{b}(c)}(K_{v},T)=\mathrm{H}^{1}_{\mathcal{F}}(K_{v},T)$
for $v$ prime to $a\cdot b\cdot c$, and

  * •

$\mathrm{H}^{1}_{\mathcal{F}^{a}_{b}(c)}(K_{v},T)=\left\\{\begin{array}[]{ll}\mathrm{H}^{1}(K_{v},T)&\textrm{
if }v|a\\\ 0&\textrm{ if }v|b\\\
\mathrm{H}^{1}_{\mathrm{ord}}(K_{v},T)&\textrm{ if }v|c\end{array}\right.$.

If any one of $a$, $b$, $c$ is the empty product, we omit it from the
notation.

######  Lemma 5.4.

The Selmer structure $\mathcal{F}(n)$ is cartesian for any
$n\in\mathcal{N}^{\mathrm{adm}}$. For any choice of generator
$\pi\in\mathfrak{m}$ and any $0\leq i\leq\mathrm{length}(R)$, the composition

$\textstyle{T/\mathfrak{m}^{i}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\times\mathrm{length}(R)-i}}$$\textstyle{T[\mathfrak{m}^{i}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$

induces isomorphisms

$\mathrm{Sel}_{\mathcal{F}(n)}(K,T/\mathfrak{m}^{i}T)\simeq\mathrm{Sel}_{\mathcal{F}(n)}(K,T[\mathfrak{m}^{i}])\simeq\mathrm{Sel}_{\mathcal{F}(n)}(K,T)[\mathfrak{m}^{i}].$

###### Proof.

See [How06, Lemma 2.2.6]. ∎

######  Proposition 5.5.

For any $n\in\mathcal{N}^{\mathrm{adm}}$, there is a non-canonical isomorphism

$\mathrm{Sel}_{\mathcal{F}(n)}(K,T)\simeq R^{e(n)}\oplus M_{n}\oplus M_{n}$

with $e(n)\in\\{0,1\\}$.

###### Proof.

See [How06, Proposition 2.2.7]. ∎

######  Definition 5.6.

Let $\mathcal{N}^{\mathrm{def}}\subseteq\mathcal{N}^{\mathrm{adm}}$ be the
subset for which $e(n)=0$ and
$\mathcal{N}^{\mathrm{ind}}\subseteq\mathcal{N}^{\mathrm{adm}}$ be the subset
for which $e(n)=1$. For $n\in\mathcal{N}^{\mathrm{adm}}$, the stub submodule
is defined by

$\mathrm{Stub}_{n}=\left\\{\begin{array}[]{ll}\mathfrak{m}^{\mathrm{length}(M_{n})}\cdot
R&\textrm{ when }n\in\mathcal{N}^{\mathrm{def}},\\\
\mathfrak{m}^{\mathrm{length}(M_{n})}\cdot\mathrm{Sel}_{\mathcal{F}(n)}(K,T)&\textrm{
when }n\in\mathcal{N}^{\mathrm{ind}},\end{array}\right.$

with $M_{n}$ as in Proposition 5.5. Note that $\mathrm{Stub}_{n}$ is a cyclic
$R$-module for every $n\in\mathcal{N}^{\mathrm{adm}}$.

######  Proposition 5.7.

Let $n\ell\in\mathcal{N}^{\mathrm{adm}}$. There exist non-negative integers
$a$, $b$ with $a+b=\mathrm{length}(R)$ such that the diagram of inclusions

---
$\textstyle{\mathrm{Sel}_{\mathcal{F}^{\ell}(n)}(K,T)}$$\textstyle{\mathrm{Sel}_{\mathcal{F}(n)}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{\mathrm{Sel}_{\mathcal{F}(n\ell)}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathrm{Sel}_{\mathcal{F}_{\ell}(n)}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{a}$

where the labels on the arrows are the length of the respective quotients,
i.e.

  * •

$a=\mathrm{length}\left(\dfrac{\mathrm{Sel}_{\mathcal{F}^{\ell}(n)}(K,T)}{\mathrm{Sel}_{\mathcal{F}(n\ell)}(K,T)}\right)=\mathrm{length}\left(\dfrac{\mathrm{Sel}_{\mathcal{F}(n)}(K,T)}{\mathrm{Sel}_{\mathcal{F}_{\ell}(n)}(K,T)}\right)$.

  * •

$b=\mathrm{length}\left(\dfrac{\mathrm{Sel}_{\mathcal{F}^{\ell}(n)}(K,T)}{\mathrm{Sel}_{\mathcal{F}(n)}(K,T)}\right)=\mathrm{length}\left(\dfrac{\mathrm{Sel}_{\mathcal{F}(n\ell)}(K,T)}{\mathrm{Sel}_{\mathcal{F}_{\ell}(n)}(K,T)}\right)$.

All four quotients are cyclic $R$-modules and

$\textstyle{a=\mathrm{length}\left(\mathrm{loc}_{\ell}\left(\mathrm{Sel}_{\mathcal{F}(n)}(K,T)\right)\right),}$$\textstyle{b=\mathrm{length}\left(\mathrm{loc}_{\ell}\left(\mathrm{Sel}_{\mathcal{F}(n\ell)}(K,T)\right)\right).}$

###### Proof.

See [How06, Proposition 2.2.9]. ∎

######  Corollary 5.8.

Fix $n\in\mathcal{N}^{\mathrm{adm}}$ and $e(n)$ be as in Proposition 5.5. The
integer

$\rho(n)=\mathrm{dim}_{R/\mathfrak{m}}\left(\mathrm{Sel}_{\mathcal{F}(n)}(K,T/\mathfrak{m}T)\right)$

satisfies $e(n)\equiv\rho(n)\pmod{2}$, and for any
$\ell\in\mathcal{P}^{\mathrm{adm}}$ primes to $n$,

$\displaystyle\rho(n\ell)$
$\displaystyle=\rho(n)+1\Leftrightarrow\mathrm{loc}_{\ell}\left(\mathrm{Sel}_{\mathcal{F}(n)}(K,T/\mathfrak{m}T)\right)=0,$
$\displaystyle\rho(n\ell)$
$\displaystyle=\rho(n)-1\Leftrightarrow\mathrm{loc}_{\ell}\left(\mathrm{Sel}_{\mathcal{F}(n)}(K,T/\mathfrak{m}T)\right)\neq
0.$

In particular, for $n\ell\in\mathcal{N}^{\mathrm{adm}}$,
$n\in\mathcal{N}^{\mathrm{def}}$ if and only if
$n\ell\in\mathcal{N}^{\mathrm{ind}}$.

###### Proof.

See [How06, Corollary 2.2.10 and Remark 2.2.11]. ∎

######  Corollary 5.9.

Suppose $n\ell\in\mathcal{N}^{\mathrm{adm}}$, and let $a$ and $b$ be as in
Proposition 5.7. Then

$\mathrm{length}(M_{n})=\left\\{\begin{array}[]{ll}\mathrm{length}(M_{n\ell})+a&\textrm{
when }n\in\mathcal{N}^{\mathrm{def}},\\\ \mathrm{length}(M_{n\ell})-b&\textrm{
when }n\in\mathcal{N}^{\mathrm{ind}}.\end{array}\right.$

###### Proof.

See [How06, Corollary 2.2.12]. ∎

######  Corollary 5.10.

Suppose $n\ell\in\mathcal{N}^{\mathrm{adm}}$. There exists an isomorphism of
$R$-modules

$\begin{array}[]{ll}\mathrm{loc}_{\ell}(\mathrm{Stub}_{n})\simeq\mathrm{Stub}_{n\ell}&\textrm{
if }n\in\mathcal{N}^{\mathrm{ind}},\\\
\mathrm{loc}_{\ell}(\mathrm{Stub}_{n\ell})\simeq\mathrm{Stub}_{n}&\textrm{ if
}n\in\mathcal{N}^{\mathrm{def}}.\end{array}$

###### Proof.

See [How06, Corollary 2.2.13]. ∎

####  5.1.3. Bipartite Euler systems

We keep Assumption 5.3 as well as the following one.

######  Assumption 5.11.

For any non-zero $c\in\mathrm{H}^{1}(K,T/\mathfrak{m}T)$, there are infinitely
many primes $\ell\in\mathcal{P}^{\mathrm{adm}}$ such that
$\mathrm{loc}_{\ell}(c)\neq 0$.

######  Definition 5.12.

A bipartite Euler system for $(T,\mathcal{F},\mathcal{P}^{\mathrm{adm}})$ is a
pair of families

$\textstyle{\boldsymbol{\kappa}^{\mathrm{bip}}=\left\\{\kappa^{\mathrm{bip}}_{n}\in\mathrm{Sel}_{\mathcal{F}(n)}(K,T):n\in\mathcal{N}^{\mathrm{ind}}\right\\},}$$\textstyle{\boldsymbol{\lambda}^{\mathrm{bip}}=\left\\{\lambda^{\mathrm{bip}}_{n}\in
R:n\in\mathcal{N}^{\mathrm{def}}\right\\}}$

related by the first and second explicit reciprocity laws:

  1. (1)

For $n\ell\in\mathcal{N}^{\mathrm{ind}}$, there exists an isomorphism of
$R$-modules

$R/\lambda^{\mathrm{bip}}_{n}R\simeq\mathrm{H}^{1}_{\mathrm{ord}}(K_{\ell},T)/R\cdot\mathrm{loc}_{\ell}(\kappa^{\mathrm{bip}}_{n\ell}).$

  2. (2)

For $n\ell\in\mathcal{N}^{\mathrm{def}}$, there exists an isomorphism of
$R$-modules

$R/\lambda^{\mathrm{bip}}_{n\ell}R\simeq\mathrm{H}^{1}_{\mathrm{ur}}(K_{\ell},T)/R\cdot\mathrm{loc}_{\ell}(\kappa^{\mathrm{bip}}_{n}).$

######  Remark 5.13.

A bipartite Euler system is non-trivial if $\lambda^{\mathrm{bip}}_{n}\neq 0$
for some $n\in\mathcal{N}^{\mathrm{def}}$. Indeed, this definition of
bipartite Euler systems is “of odd type” following the original definition.
Since any even type bipartite Euler system is always trivial [How06,
Proposition 2.3.4], we ignore the odd-even type classification.

######  Lemma 5.14.

For any $n\in\mathcal{N}^{\mathrm{adm}}$ and any cyclic $R$-submodule
$C\subseteq\mathrm{Sel}_{\mathcal{F}(n)}(K,T)$, there exist infinitely many
$\ell\in\mathcal{P}^{\mathrm{adm}}$ such that the map
$\mathrm{loc}_{\ell}:C\to\mathrm{H}^{1}_{\mathrm{ur}}(K,T)$ is injective. If
$C$ is free of rank one over $R$, then for any such $\ell$, we have
isomorphism $\mathrm{loc}_{\ell}:C\simeq\mathrm{H}^{1}_{\mathrm{ur}}(K,T)$.

###### Proof.

See [How06, Lemma 2.3.3]. This is based on Assumption 5.11. ∎

######  Proposition 5.15.

Fix a bipartite Euler system
$(\boldsymbol{\lambda}^{\mathrm{bip}},\boldsymbol{\kappa}^{\mathrm{bip}})$ for
$(T,\mathcal{F},\mathcal{P}^{\mathrm{adm}})$. Let $k=\mathrm{length}(R)$ and
$M_{n}$ be as in Proposition 5.5.

  1. (1)

If $\lambda^{\mathrm{bip}}_{n}\neq 0$ for some
$n\in\mathcal{N}^{\mathrm{def}}$, then $\mathfrak{m}^{k-1}\cdot M_{n}=0$.

  2. (2)

If $\kappa^{\mathrm{bip}}_{n}\neq 0$ for some
$n\in\mathcal{N}^{\mathrm{ind}}$, then $\mathfrak{m}^{k-1}\cdot M_{n}=0$.

###### Proof.

See [How06, Proposition 2.3.5]. ∎

We say that a bipartite Euler system
$(\boldsymbol{\lambda}^{\mathrm{bip}},\boldsymbol{\kappa}^{\mathrm{bip}})$ for
$(T,\mathcal{F},\mathcal{P}^{\mathrm{adm}})$ is free if for every
$n\in\mathcal{N}^{\mathrm{ind}}$, there exists a free $R$-submodule
$C_{n}\in\mathrm{Sel}_{\mathcal{F}(n)}(K,T)$ of rank one containing
$\kappa^{\mathrm{bip}}_{n}$.

######  Theorem 5.16.

For any free bipartite Euler system
$(\boldsymbol{\lambda}^{\mathrm{bip}},\boldsymbol{\kappa}^{\mathrm{bip}})$ for
$(T,\mathcal{F},\mathcal{P}^{\mathrm{adm}})$, we have

  * •

$\lambda^{\mathrm{bip}}_{n}\in\mathrm{Stub}_{n}$ for every
$n\in\mathcal{N}^{\mathrm{def}}$, and

  * •

$\kappa^{\mathrm{bip}}_{n}\in\mathrm{Stub}_{n}$ for every
$n\in\mathcal{N}^{\mathrm{ind}}$.

Equivalently, the $R$-module $M_{n}$ of Proposition 5.5 satisfies

$\mathrm{length}(M_{n})\leq\left\\{\begin{array}[]{ll}\mathrm{length}(M_{n\ell})+a&\textrm{
when }n\in\mathcal{N}^{\mathrm{def}},\\\ \mathrm{length}(M_{n\ell})-b&\textrm{
when }n\in\mathcal{N}^{\mathrm{ind}}.\end{array}\right.$

###### Proof.

See [How06, Theorem 2.3.7]. ∎

######  Theorem 5.17.

We make the following assumptions:

  1. (1)

The residual representation $T/\mathfrak{m}T$ is absolutely irreducible.

  2. (2)

$\mathcal{F}$ is cartesian.

  3. (3)

For any non-zero $c\in\mathrm{H}^{1}(K,T/\mathfrak{m}T)$, there exist
infinitely many $\ell\in\mathcal{P}^{\mathrm{adm}}$ such that
$\mathrm{loc}_{\ell}(c)\neq 0$.

Suppose that we are given a non-trivial free bipartite Euler system for
$(T,\mathcal{F},\mathcal{P}^{\mathrm{adm}})$. Then there exists a unique
integer $\delta$, independently of $n\in\mathcal{N}^{\mathrm{adm}}$, with the
property that

  1. (1)

$\lambda^{\mathrm{bip}}_{n}$ generates
$\mathfrak{m}^{\delta}\mathrm{Stub}_{n}$ for every
$n\in\mathcal{N}^{\mathrm{def}}$, and

  2. (2)

$\kappa^{\mathrm{bip}}_{n}$ also generates
$\mathfrak{m}^{\delta}\mathrm{Stub}_{n}$ for every
$n\in\mathcal{N}^{\mathrm{ind}}$.

Furthermore, $\delta$ is given by

$\displaystyle\delta$
$\displaystyle=\mathrm{min}\left\\{\mathrm{ind}(\lambda^{\mathrm{bip}}_{n},R):n\in\mathcal{N}^{\mathrm{def}}\right\\}$
$\displaystyle=\mathrm{min}\left\\{\mathrm{ind}(\kappa^{\mathrm{bip}}_{n},\mathrm{Sel}_{\mathcal{F}(n)}(K,T)):n\in\mathcal{N}^{\mathrm{ind}}\right\\}.$

###### Proof.

See [How06, Theorem 2.5.1]. ∎

####  5.1.4. The structure of Selmer groups

Write $k=\mathrm{length}(R)<\infty$ and

$\mathrm{Sel}_{\mathcal{F}}(K,T)\simeq\mathrm{Sel}_{\mathcal{F}^{*}}(K,T^{*})\simeq
M_{1}\oplus M_{1}$

and

$M_{1}\simeq\bigoplus_{i\geq 1}R/\mathfrak{m}^{d_{i}}R$

with $d_{1}\geq d_{2}\geq\cdots$ by using Proposition 5.5. Here,
$\mathcal{F}^{*}$ is the dual Selmer structure to $\mathcal{F}$ and
$T^{*}=\mathrm{Hom}(T,\mu_{p^{\infty}})$. As in Proposition 5.7, all stub
submodules are cyclic $R$-modules. Thus, Theorem 5.17 shows that

$\displaystyle\partial^{(r)}(\boldsymbol{\lambda}^{\mathrm{bip}})$
$\displaystyle=\mathrm{min}\left\\{\mathrm{ind}(\lambda^{\mathrm{bip}}_{n},R):n\in\mathcal{N}^{\mathrm{def}}\textrm{
with (even) }\nu(n)=r\right\\},$
$\displaystyle=\mathrm{min}\left\\{k,\delta+\mathrm{length}(M_{n}):n\in\mathcal{N}^{\mathrm{def}}\textrm{
with (even) }\nu(n)=r\right\\}.$

When $r=0$, we have equality

$\partial^{(0)}(\boldsymbol{\lambda}^{\mathrm{bip}})=\mathrm{min}\left\\{k,\delta+\mathrm{length}(M_{1})\right\\}=\mathrm{min}\left\\{k,\delta+\sum_{i\geq
1}d_{i}\right\\}.$

By using Lemma 5.14, there exists a prime
$\ell_{1}\in\mathcal{P}^{\mathrm{adm}}$ such that the restriction of the
natural map

$\mathrm{loc}_{\ell_{1}}:\mathrm{Sel}_{\mathcal{F}}(K,T)\to\mathrm{H}^{1}_{\mathrm{ur}}(K_{\ell_{1}},T)$

to one of the $R/\mathfrak{m}^{d_{1}}R$-components is injective. Then we have
exact sequence

$\textstyle{\mathrm{Sel}_{\mathcal{F}(\ell_{1})}(K,T)}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{\mathcal{F}_{\ell_{1}}}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{\mathcal{F}}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}_{\ell_{1}}}$$\textstyle{\mathrm{H}^{1}_{\mathrm{ur}}(K_{\ell_{1}},T).}$

By the choice of $\ell_{1}$, we have

$\mathrm{Sel}_{\mathcal{F}_{\ell_{1}}}(K,T)\simeq\left(\bigoplus_{i\geq
2}R/\mathfrak{m}^{d_{i}}R\right)\oplus M_{1}$

Since
$\mathrm{length}\left(\mathrm{loc}_{\ell_{1}}(\mathrm{Sel}_{\mathcal{F}}(K,T))\right)=d_{1}$,
Proposition 5.7 implies that

$\mathrm{length}\left(\dfrac{\mathrm{Sel}_{\mathcal{F}(\ell_{1})}(K,T)}{\mathrm{Sel}_{\mathcal{F}_{\ell_{1}}}(K,T)}\right)=k-d_{1}.$

Then by Proposition 5.5, we have

$\mathrm{Sel}_{\mathcal{F}(\ell_{1})}(K,T)\simeq R\oplus M_{\ell_{1}}\oplus
M_{\ell_{1}}$

with $M_{\ell_{1}}\simeq\bigoplus_{i\geq 2}R/\mathfrak{m}^{d_{i}}R$.

By using Lemma 5.14 again, there exists a prime
$\ell_{2}\in\mathcal{P}^{\mathrm{adm}}$ such that the restriction of the
natural map

$\mathrm{loc}_{\ell_{2}}:\mathrm{Sel}_{\mathcal{F}(\ell_{1})}(K,T)\to\mathrm{H}^{1}_{\mathrm{ur}}(K_{\ell_{2}},T)$

to the $R$-component is an isomorphism. Applying the same argument, we obtain
exact sequence

$\textstyle{\mathrm{Sel}_{\mathcal{F}(\ell_{1}\ell_{2})}(K,T)}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}_{\mathcal{F}(\ell_{1})_{\ell_{2}}}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\mathrm{Sel}_{\mathcal{F}(\ell_{1})}(K,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{loc}_{\ell_{2}}}$$\textstyle{\mathrm{H}^{1}_{\mathrm{ur}}(K_{\ell_{2}},T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Thus we have

$\mathrm{Sel}_{\mathcal{F}(\ell_{1}\ell_{2})}(K,T)\simeq
M_{\ell_{1}\ell_{2}}\oplus M_{\ell_{1}\ell_{2}}$

with $M_{\ell_{1}\ell_{2}}\simeq M_{\ell_{1}}\simeq\bigoplus_{i\geq
2}R/\mathfrak{m}^{d_{i}}R$. It shows that

$\partial^{(2)}(\boldsymbol{\lambda}^{\mathrm{bip}})=\mathrm{min}\left\\{k,\delta+\mathrm{length}(M_{\ell_{1}\ell_{2}})\right\\}=\mathrm{min}\left\\{k,\delta+\sum_{i\geq
2}d_{i}\right\\}.$

By repeating this argument, we obtain that

$\partial^{(r)}(\boldsymbol{\lambda}^{\mathrm{bip}})=\mathrm{min}\left\\{k,\delta+\sum_{i\geq
r/2+1}d_{i}\right\\}$

for even $r$. In particular, we recover $d_{i}$’s from
$\boldsymbol{\lambda}^{\mathrm{bip}}$ as follows.

$\displaystyle d_{1}$
$\displaystyle=\partial^{(0)}(\boldsymbol{\lambda}^{\mathrm{bip}})-\partial^{(2)}(\boldsymbol{\lambda}^{\mathrm{bip}})$
$\displaystyle d_{2}$
$\displaystyle=\partial^{(2)}(\boldsymbol{\lambda}^{\mathrm{bip}})-\partial^{(4)}(\boldsymbol{\lambda}^{\mathrm{bip}})$
$\displaystyle d_{3}$
$\displaystyle=\partial^{(4)}(\boldsymbol{\lambda}^{\mathrm{bip}})-\partial^{(6)}(\boldsymbol{\lambda}^{\mathrm{bip}})$
$\displaystyle\vdots$ $\displaystyle d_{i}$
$\displaystyle=\partial^{(2(i-1))}(\boldsymbol{\lambda}^{\mathrm{bip}})-\partial^{(2i)}(\boldsymbol{\lambda}^{\mathrm{bip}})$

for $i\geq 1$. To sum up, we have the following statement.

######  Theorem 5.18.

Suppose that we are given a non-trivial free bipartite Euler system for
$(T,\mathcal{F},\mathcal{P}^{\mathrm{adm}})$; in particular,
$\boldsymbol{\lambda}^{\mathrm{bip}}\neq 0$. Write
$\mathrm{Sel}_{\mathcal{F}}(K,T)\simeq\bigoplus_{i\geq
1}\left(R/\mathfrak{m}^{d_{i}}R\right)^{\oplus 2}$ with $d_{1}\geq
d_{2}\geq\cdots$. Then

$\partial^{(r)}(\boldsymbol{\lambda}^{\mathrm{bip}})=\mathrm{min}\left\\{k,\delta+\sum_{i\geq
r/2+1}d_{i}\right\\}$

for even $r\geq 0$.

######  Remark 5.19.

This is an analogue of [MR04, Proposition 4.5.8] for the bipartite Euler
system setting.

###  5.2. Elliptic curves

We keep all the working hypotheses in $\S$2.1 with odd $\nu(N^{-})$.

####  5.2.1. Applying $\S$5.1 to elliptic curves

Let $T$ be the $p$-adic Tate module of $E$ (cf. Remark 5.1). We assume that
$E[p]$ is absolutely irreducible as a $G_{K}$-module and $N^{-}$ is a square-
free product of an odd number of primes. We fix the self-dual Selmer structure
$\mathcal{F}=\mathcal{F}_{\mathrm{cl}}$ as the classical Selmer structure.
Then for every $k\geq 1$, the triple
$(T/p^{k}T,\mathcal{F},\mathcal{P}^{\mathrm{adm}}_{k})$ satisfies Assumptions
5.3 and 5.11 given in $\S$5.1 thanks to [How06, Lemma 3.3.4].

For the actual construction of the bipartite Euler system for elliptic curves,
the following assumptions are further needed.

######  Assumption 5.20.

  1. (1)

$E[p]$ is absolutely irreducible as a $G_{\mathbb{Q}}$-module,

  2. (2)

$E$ has a good reduction at $p$, and

  3. (3)

If a prime $q$ divides $N^{-}$ and $q\equiv\pm 1\pmod{p}$, then $E[p]$ is
ramified at $q$.

####  5.2.2. Weak level raising

Let $f=\sum_{n\geq 1}a_{n}(f)q^{n}$ be the newform corresponding to $E$. Let
$B_{N^{-}}$ be the definite quaternion algebra over $\mathbb{Q}$ of
discriminant $N^{-}$ and $R_{N^{+}}$ be an Eichler order of $B_{N^{-}}$ of
level $N^{+}$. By using the Jacquet–Langlands correspondence, we have the
corresponding quaternionic automorphic form

$f_{N^{-}}:B^{\times}_{N^{-}}\backslash\widehat{B}^{\times}_{N^{-}}/\widehat{R}^{\times}_{N^{+}}\to\mathbb{Z}_{p}$

with normalization $f_{N^{-}}\not\equiv 0\pmod{p}$ where
$\widehat{(-)}=(-)\otimes\widehat{\mathbb{Z}}$.

Let $n\in\mathcal{N}^{\mathrm{def}}_{1}$. Let $\mathbb{T}^{N^{-}}(N^{+})$ and
$\mathbb{T}^{nN^{-}}(N^{+})$ be the full Hecke algebras over $\mathbb{Z}_{p}$
faithfully acting on the $N^{-}$-new subspace of
$S_{2}(\Gamma_{0}(N),\mathbb{Z}_{p})$ and the $nN^{-}$-new subspace of
$S_{2}(\Gamma_{0}(nN),\mathbb{Z}_{p})$, respectively. We define a
$\mathbb{Z}_{p}$-algebra homomorphism

$\pi_{f_{N^{-}}}:\mathbb{T}^{N^{-}}(N^{+})\to\mathbb{Z}_{p}$

by $\pi_{f_{N^{-}}}(T_{q})=a_{q}(f)$ if $q\nmid N$ and
$\pi_{f_{N^{-}}}(U_{q})=a_{q}(f)$ if $q\mid N$.

Let $\mathfrak{m}_{f_{N^{-}}}\subseteq\mathbb{T}^{N^{-}}(N^{+})$ be the
maximal ideal corresponding to $f_{N^{-}}$.

######  Theorem 5.21.

We keep Assumption 5.20 so that $\mathfrak{m}_{f_{N^{-}}}$ is non-Eisenstein.

  1. (1)

For each $n\in\mathcal{N}^{\mathrm{def}}$, there exists a surjective
homomorphism

$\pi_{f_{nN^{-}}}:\mathbb{T}^{nN^{-}}(N^{+})\to\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}$

such that $\pi_{f_{nN^{-}}}(T_{q})=\pi_{f_{N^{-}}}(T_{q})=a_{q}(f)$ if $q\nmid
nN$, $\pi_{f_{nN^{-}}}(U_{q})=\pi_{f_{N^{-}}}(U_{q})=a_{q}(f)$ if $q\mid N$ ,
and $\pi_{f_{nN^{-}}}(U_{q})=\epsilon_{q}$ if $q\mid n$ where
$\epsilon_{q}=\pm 1$ satisfying $a_{q}(f)=\epsilon_{\ell}\cdot(\ell+1)$ in
$\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}$.

  2. (2)

For each $n\in\mathcal{N}^{\mathrm{def}}$, there exists a mod
$I^{\mathrm{adm}}_{n}$ quaternionic $\mathbb{T}^{nN^{-}}(N^{+})$-eigenform

$f_{nN^{-}}:B^{\times}_{nN^{-}}\backslash\widehat{B}^{\times}_{nN^{-}}/\widehat{R}^{\times}_{N^{+}}\to\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}$

such that the induced map on the Hecke algebra is $\pi_{f_{nN^{-}}}$ and
$f_{nN^{-}}\not\equiv 0\pmod{p}$.

###### Proof.

See [BD05, Theorems 5.15 and 9.3]. See [PW11, Theorem 6.2] how Assumption 5.20
is used. See also [CH15, Definition 4.1]. ∎

####  5.2.3. Construction of $\boldsymbol{\lambda}^{\mathrm{bip}}$

Let $n\in\mathcal{N}^{\mathrm{def}}_{1}$. Let $H$ be the Hilbert class field
of $K$ and identify $\mathrm{Gal}(H/K)\simeq
K^{\times}\backslash\widehat{K}^{\times}/\widehat{\mathcal{O}}^{\times}_{K}$
via the global class field theory. An embedding $\Phi:K\hookrightarrow B$
induces map
$\widehat{\Phi}:K^{\times}\backslash\widehat{K}^{\times}/\widehat{\mathcal{O}}^{\times}_{K}\to
B^{\times}_{nN^{-}}\backslash\widehat{B}^{\times}_{nN^{-}}/\widehat{R}^{\times}_{N^{+}}$
and then $\mathrm{Gal}(H/K)$ acts on
$B^{\times}_{nN^{-}}\backslash\widehat{B}^{\times}_{nN^{-}}/\widehat{R}^{\times}_{N^{+}}$
by left translation with $\widehat{\Phi}$. We define
$\lambda^{\mathrm{bip}}_{n}$ by the toric period integral

$\lambda^{\mathrm{bip}}_{n}=\sum_{\sigma\in\mathrm{Gal}(H/K)}f_{nN^{-}}(\sigma\cdot\varsigma^{(0)})\in\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}$

where $\varsigma^{(0)}$ is the Gross point of conductor 1 reviewed in Appendix
A.

####  5.2.4. Construction of $\boldsymbol{\kappa}^{\mathrm{bip}}$

Let $n\in\mathcal{N}^{\mathrm{ind}}_{1}$. Since $\nu(N^{-})$ is odd, we always
have $\nu(n)\geq 1$. Choose one admissible prime $\ell$ dividing $n$ with
$I^{\mathrm{adm}}_{n/\ell}=I^{\mathrm{adm}}_{\ell}$. For an
$I^{\mathrm{adm}}_{n/\ell}$-admissible form $f_{nN^{-}/\ell}$ and
$I^{\mathrm{adm}}_{\ell}$-admissible prime $\ell$, there exists a cohomology
class
$\kappa_{f_{nN^{-}/\ell}}(\ell)\in\mathrm{H}^{1}(K,T/I^{\mathrm{adm}}_{n}T)$
coming from Heegner points on Shimura curve $X^{nN^{-}}(N^{+})$. See [CH15,
$\S$4.3] for the precise construction of $\kappa_{f_{nN^{-}/\ell}}(\ell)$.
More precisely, we have

$\kappa_{f_{nN^{-}/\ell}}(\ell)\in\mathrm{Sel}_{\mathcal{F}_{\mathrm{cl}}(n)}(K,T/I^{\mathrm{adm}}_{n}T)$

thanks to [CH15, Proposition 4.7], and we let

$\kappa^{\mathrm{bip}}_{n}=\kappa_{f_{nN^{-}/\ell}}(\ell).$

For our purpose, we need to check the following statement.

######  Lemma 5.22.

The divisibility of $\kappa_{f_{nN^{-}/\ell}}(\ell)$ is independent of the
choice of $\ell$.

###### Proof.

Let $q\in\mathcal{P}^{\mathrm{adm}}_{1}$ with
$I^{\mathrm{adm}}_{n}=I^{\mathrm{adm}}_{nq}$. By using the second explicit
reciprocity law [BD05, Theorem 4.2], we have equality

$\mathrm{loc}_{q}(\kappa_{f_{nN^{-}/\ell}}(\ell))=\lambda^{\mathrm{bip}}_{nq}$

under the isomorphism
$\mathrm{H}^{1}_{\mathrm{ur}}(K_{q},T/I^{\mathrm{adm}}_{n}T)\simeq\mathbb{Z}_{p}/I^{\mathrm{adm}}_{nq}\mathbb{Z}_{p}$.
Since $\lambda^{\mathrm{bip}}_{nq}$ itself is defined without using
$\kappa_{f_{nN^{-}/\ell}}(\ell)$, the divisibility of
$\lambda^{\mathrm{bip}}_{nq}$ is independent of the choice of $\ell$. Thus,
the conclusion follows. ∎

###  5.3. Structure theorem

######  Theorem 5.23.

Suppose that $\boldsymbol{\lambda}^{\mathrm{bip}}\neq 0$. Then we have
isomorphism

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

###### Proof.

We closely follow [MR04, Theorem 5.2.12]. Write

$\mathrm{Sel}(K,E[p^{\infty}])\simeq(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\oplus
r}\oplus\bigoplus_{i\geq
r+1}\left(\mathbb{Z}/p^{e_{i}}\mathbb{Z}\right)^{\oplus 2}$

with $e_{r+1}\geq e_{r+2}\geq\cdots$. Let $e_{1}=\cdots e_{r}=\infty$. For
$k>0$, we have

$\mathrm{Sel}(K,E[p^{k}])\simeq(\mathbb{Z}/p^{k}\mathbb{Z})^{\oplus
r}\oplus\bigoplus_{i\geq
r+1}\left(\mathbb{Z}/p^{\mathrm{min}\\{k,e_{i}\\}}\mathbb{Z}\right)^{\oplus
2}.$

Let $j\geq 2k$ be an integer. Then the pair

$\displaystyle\boldsymbol{\lambda}^{\mathrm{bip},(k)}$
$\displaystyle=\left\\{\lambda^{\mathrm{bip},(k)}_{n}=\lambda^{\mathrm{bip}}_{n}\pmod{p^{k}}\in\mathbb{Z}/p^{k}\mathbb{Z}:n\in\mathcal{N}^{\mathrm{def}}_{j}\right\\},$
$\displaystyle\boldsymbol{\kappa}^{\mathrm{bip},(k)}$
$\displaystyle=\left\\{\kappa^{\mathrm{bip},(k)}_{n}=\kappa^{\mathrm{bip}}_{n}\pmod{p^{k}}\in\mathrm{H}^{1}_{\mathcal{F}_{\mathrm{cl}}(n)}(K,T/p^{k}T):n\in\mathcal{N}^{\mathrm{ind}}_{j}\right\\}$

is a free bipartite Euler system for
$(T/p^{k}T,\mathcal{F}_{\mathrm{cl}},\mathcal{P}^{\mathrm{adm}}_{j})$ thanks
to [How06, Lemma 3.3.6].

By Theorem 5.18, we have

(5.1)
$\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip},(k)})=\mathrm{min}\left\\{k,\delta+\sum_{i\geq
s+1}\mathrm{min}\\{k,e_{i}\\}\right\\}$

for every $s\geq 0$.

Fix $s\geq 0$, and let
$h=\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip}})$. Then there exists
$n\in\mathcal{N}^{\mathrm{def}}_{1}$ (necessarily in
$\mathcal{N}^{\mathrm{def}}_{h+1}\cap\mathcal{N}^{\mathrm{def}}_{j}$) with
$\nu(n)=2s$ such that $\lambda^{\mathrm{bip}}_{n}\not\in
p^{h+1}\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}$. By Theorem 5.16 and
Definition 5.6, we have
$\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip},(h+1)})\leq h$. By (5.1),
it follows that $\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip},(k)})\leq
h=\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip}})$.

On the other hand, since $\lambda^{\mathrm{bip}}_{n}\in
p^{h}\mathbb{Z}_{p}/I^{\mathrm{adm}}_{n}\mathbb{Z}_{p}$ for every
$n\in\mathcal{N}^{\mathrm{def}}_{1}$ with $\nu(n)=2s$, we have
$\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip},(k)})\geq h$ for every
$k\geq h$. Thus, we have
$\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip}})=\mathrm{sup}\left\\{\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip},(k)}):k\geq
1\right\\}$. Also, $\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip},(k)})$
is a non-decreasing function on $k$ by (5.1). To sum up, we have

$\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip}})=\lim_{k\to\infty}\partial^{(2s)}(\boldsymbol{\lambda}^{\mathrm{bip},(k)}).$

In the limit process, we may choose $j=2k$ for every $k$. The conclusion
follows from this limit and (5.1). ∎

######  Remark 5.24.

Since $\boldsymbol{\lambda}^{\mathrm{bip}}$ does not admit the action of
complex conjugation (cf. [Zha14b, $\S$6.1]), we do not expect to observe the
structure of each $\mathrm{Sel}(K,E[p^{\infty}])^{\pm}$ from
$\boldsymbol{\lambda}^{\mathrm{bip}}$. However, the corank of each
$\mathrm{Sel}(K,E[p^{\infty}])^{\pm}$ can be still detected by considering the
different variation of $\lambda^{\mathrm{bip}}_{1}$ along anticyclotomic
Kolyvagin primes [Swe].

##  6\. Structural comparisons

###  6.1. Decomposition of Selmer groups

Let $K$ be an imaginary quadratic field and $c\in\mathrm{Gal}(K/\mathbb{Q})$
the complex conjugation. Since $\mathrm{Gal}(K/\mathbb{Q})$ has order 2 and
$p>3$ is odd, we have decomposition

(6.1)
$\mathrm{Sel}(K,E[p^{\infty}])=\mathrm{Sel}(K,E[p^{\infty}])^{+}\oplus\mathrm{Sel}(K,E[p^{\infty}])^{-}$

where $\mathrm{Sel}(K,E[p^{\infty}])^{\pm}$ is the $c$-eigenspace with
eigenvalue $\pm 1$, respectively. If $E(K)[p]=0$, then we have isomorphisms

(6.2)
$\textstyle{\mathrm{Sel}(K,E[p^{\infty}])^{+}\simeq\mathrm{Sel}(\mathbb{Q},E[p^{\infty}]),}$$\textstyle{\mathrm{Sel}(K,E[p^{\infty}])^{-}\simeq\mathrm{Sel}(\mathbb{Q},E^{K}[p^{\infty}]).}$

###  6.2. The structural Gross–Zagier formula

Suppose that $\nu(N^{-})$ is even. Recall (4.1)

$\mathrm{Sel}(K,E[p^{\infty}])^{\pm}_{/\mathrm{div}}\simeq\bigoplus_{i\geq
1}\left(\mathbb{Z}/p^{a^{\pm}_{i}}\mathbb{Z}\right)^{\oplus 2}$

where $a^{\pm}_{1}\geq a^{\pm}_{2}\geq\cdots$.

####  6.2.1. When $r^{+}(E/K)>r^{-}(E/K)$

We first assume that
$W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1}=+1$.
Under the working hypotheses in $\S$2.1, by using Theorem 4.1, we have

  * •

$r^{+}(E/K)=\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1$,

  * •

$\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-r^{-}(E/K)\geq 0$ and is
even, and

  * •

$r^{+}(E/K)>r^{-}(E/K)$.

Applying Theorem 4.3, we also have

$\displaystyle a^{+}_{i}$
$\displaystyle=\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i)}(\boldsymbol{\kappa}^{\mathrm{Heeg}}),$
$\displaystyle
a^{-}_{i+(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-r^{-}(E/K))}$
$\displaystyle=\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-2)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})$

for $i\geq 1$. In particular, (6.2) implies

$\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E))=\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1,$

and the comparison between Theorem 3.1 and Theorem 4.3 via (6.1) implies

$\displaystyle\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E))+2(i-1))}(\widetilde{\boldsymbol{\delta}}(E))-\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E))+2i)}(\widetilde{\boldsymbol{\delta}}(E))$
$\displaystyle=2\cdot\left(\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})\right)$

for $i\geq 1$. If we further assume that
$\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})=r^{-}(E/K))=\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))$,
we have equality

$\displaystyle\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))+2(i-1))}(\widetilde{\boldsymbol{\delta}}(E^{K}))-\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))+2i)}(\widetilde{\boldsymbol{\delta}}(E^{K}))$
$\displaystyle=2\cdot\left(\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-2)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg}})\right)$

for $i\geq 1$. The collection of all the above formulas can be regarded as a
structural refinement of Gross–Zagier formula. Theorem 2.3 follows from these
formulas and Corollary 4.2.

######  Remark 6.1.

Since $\widetilde{\boldsymbol{\delta}}(E)$ and
$\widetilde{\boldsymbol{\delta}}(E^{K})$ are the images of
$\boldsymbol{\kappa}^{\mathrm{Kato}}(E)$ and
$\boldsymbol{\kappa}^{\mathrm{Kato}}(E^{K})$ under the dual exponential maps
(as explained in Remark 2.4), respectively, these formulas can be understood
as a structural refinement of Perrin-Riou’s conjecture.

####  6.2.2. When $r^{-}(E/K)>r^{+}(E/K)$

We now assume that
$W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg}})+1}=-1$.
Applying the argument in $\S$6.2.1, we obtain the exactly same result except
the changes of sign and the role of $E$ and $E^{K}$.

###  6.3. The structural Waldspurger formula

Suppose that $\nu(N^{-})$ is odd. Under the working hypotheses in $\S$2.1, the
comparison between Theorem 3.1 and Theorem 5.23 via (6.1) implies isomorphism

(6.3)
$\displaystyle\begin{split}&(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{\oplus\mathrm{ord}(\boldsymbol{\lambda}^{\mathrm{bip}})}\oplus\bigoplus_{i\geq
1}\left(\mathbb{Z}/p^{\partial^{(\mathrm{ord}(\boldsymbol{\lambda}^{\mathrm{bip}})+2(i-1))}(\boldsymbol{\lambda}^{\mathrm{bip}})-\partial^{(\mathrm{ord}(\boldsymbol{\lambda}^{\mathrm{bip}})+2i)}(\boldsymbol{\lambda}^{\mathrm{bip}})}\mathbb{Z}\right)^{\oplus
2}\\\ \simeq\
&\left(\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\oplus\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E))}\oplus\bigoplus_{j\geq
1}\left(\mathbb{Z}/p^{(\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E))+2(j-1))}(\widetilde{\boldsymbol{\delta}}(E))-\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E))+2j)}(\widetilde{\boldsymbol{\delta}}(E)))/2}\mathbb{Z}\right)^{\oplus
2}\\\
&\oplus\left(\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\oplus\mathrm{ord}(\widetilde{\boldsymbol{\delta}})(E^{K})}\oplus\bigoplus_{k\geq
1}\left(\mathbb{Z}/p^{(\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))+2(k-1))}(\widetilde{\boldsymbol{\delta}}(E^{K}))-\partial^{(\mathrm{ord}(\widetilde{\boldsymbol{\delta}}(E^{K}))+2k)}(\widetilde{\boldsymbol{\delta}}(E^{K})))/2}\mathbb{Z}\right)^{\oplus
2},\end{split}$

and the isomorphism itself can be regarded as a structural Waldspurger
formula. Theorem 2.5 immediately follows from (6.3).

##  7\. The non-triviality of Kolyvagin systems and the main conjectures

In this section, we investigate all the non-triviality questions mentioned in
§2.5 following the strategy given in [Kim]. More precisely, we relate each
non-triviality question to the localization of the corresponding main
conjecture at the height one prime ideal induced from the trivial character.

###  7.1. The non-triviality of $\widetilde{\boldsymbol{\delta}}$

We keep all the settings in $\S$2.1. Let $\mathbb{Q}_{\infty}$ be the
cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ and
$\Lambda^{\mathrm{cyc}}=\mathbb{Z}_{p}\llbracket\mathrm{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})\rrbracket\simeq\mathbb{Z}_{p}\llbracket
X\rrbracket$ the cyclotomic iwasawa algebra.

Let
$\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)=\varprojlim_{m}\mathrm{H}^{1}(\mathbb{Q}_{m},T)$
be the first Iwasawa cohomology group of $T$ over $\mathbb{Q}_{\infty}$ where
$\mathbb{Q}_{m}$ is the cyclic subextension of $\mathbb{Q}$ of degree $p^{m}$
in $\mathbb{Q}_{\infty}$. Let $\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$ be
the $\Lambda^{\mathrm{cyc}}$-adic Kato’s Kolyvagin system following the
convention given in [Kim], and we have
$\kappa^{\mathrm{Kato},\infty}_{1}\in\mathrm{H}^{1}_{\mathrm{Iw}}(\mathbb{Q},T)$.
Denote by $\mathrm{Sel}_{0}(\mathbb{Q}_{\infty},E[p^{\infty}])$ the $p$-strict
Selmer group of $E[p^{\infty}]$ over $\mathbb{Q}_{\infty}$. Write
$(-)^{\vee}=\mathrm{Hom}(-,\mathbb{Q}_{p}/\mathbb{Z}_{p})$.

In [Kim], the following statement is proved.

######  Theorem 7.1.

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

  1. (1)

$\boldsymbol{\kappa}^{\mathrm{Kato},\infty}$ does not vanish modulo
$\mathfrak{P}$.

  2. (2)

The localization of the Iwasawa main conjecture at $\mathfrak{P}$ holds; in
other words,

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

By applying Theorem 7.1 to $\mathfrak{P}=X\Lambda^{\mathrm{cyc}}$, we easily
obtain the following statement.

######  Corollary 7.2.

We keep all the settings in $\S$2.1. If the localization of the Iwasawa main
conjecture at $X\Lambda^{\mathrm{cyc}}$ holds, then
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ is non-trivial and
$\widetilde{\boldsymbol{\delta}}$ is non-zero.

###### Proof.

We give a sketch only here, and we refer to [Kim] for details. Let

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

and
$p^{t}\cdot\widetilde{\boldsymbol{\delta}}=\left\\{p^{t}\cdot\widetilde{\delta}_{n}\in\mathbb{Z}_{p}/I^{\mathrm{cyc}}_{n}\mathbb{Z}_{p}\right\\}_{n\in\mathcal{N}^{\mathrm{cyc}}_{1}}$.
Then it is easy to see that $p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ is
non-zero if and only if $\widetilde{\boldsymbol{\delta}}$ is non-zero. It
suffices to show that $\boldsymbol{\kappa}^{\mathrm{Kato}}$ is non-trivial if
and only if $p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ is non-zero under our
working hypotheses. If $p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ is
identically zero, then $\boldsymbol{\kappa}^{\mathrm{Kato}}$ becomes a
Kolyvagin system of core rank zero, so it is trivial. If
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ is trivial, then
$p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ is identically zero since
$p^{t}\cdot\widetilde{\boldsymbol{\delta}}$ occurs as the image of
$\boldsymbol{\kappa}^{\mathrm{Kato}}$ under the dual exponential map. ∎

######  Remark 7.3.

In particular, the Iwasawa main conjecture inverting $p$ is strong enough to
obtain the non-triviality of $\widetilde{\boldsymbol{\delta}}$.

###  7.2. The non-triviality of $\boldsymbol{\kappa}^{\mathrm{Heeg}}$

We keep all the settings in $\S$2.1 with even $\nu(N^{-})$.

######  Assumption 7.4.

We also assume that

  1. (1)

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

  2. (2)

$a_{p}(E)\not\equiv 1\pmod{p}$ if $p$ splits in $K$,

  3. (3)

$a_{p}(E)\not\equiv\pm 1\pmod{p}$ if $p$ is inert in $K$, and

  4. (4)

the discriminant of $K$ is odd and $\neq-3$.

Condition (2) and (3) are called the non-anomalous condition.

Let $K_{\infty}$ be the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$ and
$\Lambda^{\mathrm{ac}}=\mathbb{Z}_{p}\llbracket\mathrm{Gal}(K_{\infty}/K)\rrbracket\simeq\mathbb{Z}_{p}\llbracket
X\rrbracket$ the anticyclotomic iwasawa algebra. Denote by $K_{m}$ the cyclic
subextension of $K$ of degree $p^{m}$ in $K_{\infty}$. Let $\alpha$ be the
unit root of $X^{2}-a_{p}(E)X+p$ and
$\kappa^{\mathrm{Heeg},\alpha,\infty}_{1}$ the $p$-stabilized Heegner point
over $K_{\infty}$ with $U_{p}$-eigenvalue $\alpha$ [How04]. It is known that
$\kappa^{\mathrm{Heeg},\alpha,\infty}_{1}\neq 0$ thanks to the work of Cornut
and Vatsal [Cor02, Vat03].

Although we do not go through the precise definitions of various Selmer
structures here, all the details can be found in [How04].

######  Theorem 7.5 (Howard).

We keep all the assumptions in $\S$2.1 with even $\nu(N^{-})$ and Assumption
7.4. Then
$\widehat{\mathrm{Sel}}(K_{\infty},T)=\varprojlim_{m}\mathrm{Sel}(K_{m},T)$
and $\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}$ are
$\Lambda^{\mathrm{ac}}$-modules of rank one, and there exists a finitely
generated torsion $\Lambda^{\mathrm{ac}}$-module $M_{\infty}$ such that

  1. (1)

$\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\sim\Lambda\oplus
M_{\infty}\oplus M_{\infty}$,

  2. (2)

$\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(M_{\infty}\right)=\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(M_{\infty}\right)^{\iota}$,
and

  3. (3)

$\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(\dfrac{\widehat{\mathrm{Sel}}(K_{\infty},T)}{\kappa^{\mathrm{Heeg},\alpha,\infty}_{1}}\right)\subseteq\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(M_{\infty}\right)$

where $\sim$ means a pseudo-isomorphism of $\Lambda^{\mathrm{ac}}$-modules,
$(-)^{\iota}:\Lambda^{\mathrm{ac}}\to\Lambda^{\mathrm{ac}}$ is the involution
defined by $\gamma\mapsto\gamma^{-1}$ for
$\gamma\in\mathrm{Gal}(K_{\infty}/K)$.

###### Proof.

See [How04, Theorem 2.2.10]. ∎

######  Remark 7.6.

The Heegner point main conjecture à la Perrin-Riou [PR87] is the equality of
(3) of Theorem 7.5. See [Wan21, BCK21, CW22] for the recent developments on
this conjecture.

######  Theorem 7.7.

We keep all the assumptions in $\S$2.1 with even $\nu(N^{-})$ and Assumption
7.4. Let $\mathfrak{P}$ be a height one prime of $\Lambda^{\mathrm{ac}}$. Then
the following statements are equivalent.

  1. (1)

$\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,\infty}$ is non-trivial modulo
$\mathfrak{P}$.

  2. (2)

$\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(\dfrac{\widehat{\mathrm{Sel}}(K_{\infty},T)}{\kappa^{\mathrm{Heeg},\alpha,\infty}_{1}}\right)\right)=\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(M_{\infty}\right)\right)$.

###### Proof.

Our argument mainly follows [How04, Theorem 2.2.10] but with some
modification.

Let $f_{\Lambda^{\mathrm{ac}}}$ be a generator of
$\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(\dfrac{\widehat{\mathrm{Sel}}(K_{\infty},T)}{\kappa^{\mathrm{Heeg},\alpha,\infty}_{1}}\right)$.
We first suppose that $\mathfrak{P}\neq p\Lambda^{\mathrm{ac}}$ and write
$\mathfrak{P}=(g)$ where $g\in\Lambda^{\mathrm{ac}}$. Define
$\mathfrak{P}_{M}=(g+p^{M})\Lambda^{\mathrm{ac}}$. By taking $M$ sufficiently
large, we may assume the following statements.

  * •

$\mathfrak{P}_{M}$ is a height one prime ideal of $\Lambda^{\mathrm{ac}}$.

  * •

$\Lambda^{\mathrm{ac}}/\mathfrak{P}_{M}\simeq\Lambda^{\mathrm{ac}}/\mathfrak{P}$
as rings.

  * •

$\mathfrak{P}_{M}$ and $f_{\Lambda^{\mathrm{ac}}}$ are relatively prime, so we
have $\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}_{1}\neq 0$.

  * •

Both natural maps

$\textstyle{\widehat{\mathrm{Sel}}(K_{\infty},T)/\mathfrak{P}_{M}\to\mathrm{Sel}(K,T_{\mathfrak{P}_{M}}),}$$\textstyle{\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})\to\mathrm{Sel}(K_{\infty},E[p^{\infty}])[\mathfrak{P}_{M}]}$

have finite kernel and cokernel which are bounded by a constant depending only
on $[S_{\mathfrak{P}_{M}}:\Lambda^{\mathrm{ac}}/\mathfrak{P}_{M}]$ but not on
$\mathfrak{P}_{M}$ itself where $S_{\mathfrak{P}_{M}}$ is the integral closure
of $\Lambda^{\mathrm{ac}}/\mathfrak{P}_{M}$ [How04, Proposition 2.2.8].

Following the argument of [MR04, Theorem 5.3.10], the equality
$(\mathfrak{P}_{M},\mathfrak{P}^{n})=(\mathfrak{P}_{M},p^{Mn})$ implies that

$\displaystyle\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,T_{\mathfrak{P}_{M}})/S_{\mathfrak{P}_{M}}\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}_{1}\right)$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}}\left(\Lambda^{\mathrm{ac}}/(f_{\Lambda^{\mathrm{ac}}},\mathfrak{P}_{M}\right)$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}}\left(\Lambda^{\mathrm{ac}}/(\mathfrak{P}^{\mathrm{ord}_{\mathfrak{P}}(f_{\Lambda^{\mathrm{ac}}})},\mathfrak{P}_{M}\right)$
$\displaystyle=M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}(\Lambda^{\mathrm{ac}}/\mathfrak{P}_{M})\cdot\mathrm{ord}_{\mathfrak{P}}(f_{\Lambda^{\mathrm{ac}}})$

up to $O(1)$ as $M$ varies. Here, $O(1)$ means that the differences are
constant independent of $M$. Similarly, we have

$\displaystyle 2\cdot\mathrm{length}_{\mathbb{Z}_{p}}(M_{\mathfrak{P}_{M}})$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})_{/\mathrm{div}}\right)$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}}\left((\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}/\mathfrak{P}_{M})_{\mathbb{Z}_{p}\textrm{-tors}}\right)$
$\displaystyle=M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}(\Lambda^{\mathrm{ac}}/\mathfrak{P}_{M})\cdot\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left((\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee})_{\Lambda^{\mathrm{ac}}\textrm{-tors}}\right)\right)$

up to $O(1)$ as $M$ varies where
$M_{\mathfrak{P}_{M}}=M_{\infty}\otimes_{\Lambda^{\mathrm{ac}}}S_{\mathfrak{P}_{M}}$.
We also have equalities

$\displaystyle\mathrm{length}_{S_{\mathfrak{P}_{M}}}\left(\mathrm{Sel}(K,T_{\mathfrak{P}_{M}})/S_{\mathfrak{P}_{M}}\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}_{1}\right)=\partial^{(0)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})})=\partial^{(0)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})$

where the first equality follows from the definitions and the second equality
follows from the non-anomalous condition. Moreover,
$\partial^{(i)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})})=\partial^{(i)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})$
for all $i\geq 0$ under the non-anomalous condition.

Write

$\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})^{\pm}_{/\mathrm{div}}\simeq\bigoplus_{i\geq
1}\left(S_{\mathfrak{P}_{M}}/\mathfrak{m}^{a^{\pm}_{i}}_{S_{\mathfrak{P}_{M}}}S_{\mathfrak{P}_{M}}\right)^{\oplus
2}$

where $\mathfrak{m}_{S_{\mathfrak{P}_{M}}}$ is the maximal ideal of
$S_{\mathfrak{P}_{M}}$ and $a^{\pm}_{1}\geq a^{\pm}_{2}\geq\cdots$. Although
we consider Kolyvagin systems at height one prime $\mathfrak{P}_{M}$, the
corresponding Selmer structure is self-dual thanks to [How04, Lemma 2.1.1].
Thus, we are able to apply Kolyvagin’s structure theorem (Theorem 4.3, Remark
4.4, and [Kol91]) to this setting. Therefore, we obtain

$\displaystyle
a^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})+1}}_{i}$
$\displaystyle=\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})+2i)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})}),$
$\displaystyle
a^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})}}_{i}$
$\displaystyle=\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})+2i-2)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})-\partial^{(\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})+2i-1)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})$

for $i\geq 1$, and

$\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})-r^{W(E/\mathbb{Q})\cdot(-1)^{\mathrm{ord}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})}}(E/K)=0$

since $\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}_{1}\neq 0$. Thus, we
obtain

$\mathrm{length}_{S_{\mathfrak{P}_{M}}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})_{/\mathrm{div}}\right)=2\cdot\left(\partial^{(0)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})-\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})\right)$

and also write
$j_{\mathfrak{P}_{M}}=\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})$.

Combining all the above computations, we obtain equalities

(7.1) $\displaystyle\begin{split}&2\cdot
M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})\cdot\mathrm{ord}_{\mathfrak{P}}(f_{\Lambda^{\mathrm{ac}}})\\\
&=2\cdot\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,T_{\mathfrak{P}_{M}})/S_{\mathfrak{P}_{M}}\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}_{1}\right)\\\
&=2\cdot\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot\mathrm{length}_{S_{\mathfrak{P}_{M}}}\left(\mathrm{Sel}(K,T_{\mathfrak{P}_{M}})/S_{\mathfrak{P}_{M}}\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}_{1}\right)\\\
&=2\cdot\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot\partial^{(0)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})\\\
&=\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot\left(\mathrm{length}_{S_{\mathfrak{P}_{M}}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})_{/\mathrm{div}}\right)-2\cdot\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},(\mathfrak{P}_{M})})\right)\\\
&=\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot\left(\mathrm{length}_{S_{\mathfrak{P}_{M}}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})_{/\mathrm{div}}\right)-2\cdot
j_{\mathfrak{P}_{M}}\right)\\\
&=\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})_{/\mathrm{div}}\right)-2\cdot\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot
j_{\mathfrak{P}_{M}}\end{split}$

up to $O(1)$ as $M$ varies, and

(7.2)
$\displaystyle\begin{split}&\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})_{/\mathrm{div}}\right)\\\
&=M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})\cdot\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left((\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee})_{\Lambda^{\mathrm{ac}}\textrm{-tors}}\right)\right)\end{split}$

up to $O(1)$ as $M$ varies again.

We prove (2) $\Rightarrow$ (1) first. Suppose that

$\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left((\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee})_{\Lambda^{\mathrm{ac}}\textrm{-tors}}\right)\right)=2\cdot\mathrm{ord}_{\mathfrak{P}}(f_{\Lambda^{\mathrm{ac}}}).$

Combining (7.1) with (7.2), we obtain that
$2\cdot\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot
j_{\mathfrak{P}_{M}}$ is a constant as $M$ varies, so $j_{\mathfrak{P}_{M}}$
is also a constant as $M$ varies. Since

$\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P})}\equiv\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}\pmod{p^{M}}$

for every $M\geq 1$, we obtain
$\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P})})=j_{\mathfrak{P}}<\infty$,
so $\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P})}$ is non-trivial.

Now we prove (1) $\Rightarrow$ (2). Suppose that
$\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P})}$ is non-trivial, so
$\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P})})=j_{\mathfrak{P}}<\infty$.
By using the congruence

$\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P})}\equiv\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{M})}\pmod{p^{M}}$

again, $j_{\mathfrak{P}_{M}}$ is bounded as $M$ varies. Combining (7.1) with
(7.2) again, we obtain

$\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left((\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee})_{\Lambda^{\mathrm{ac}}\textrm{-tors}}\right)\right)=2\cdot\mathrm{ord}_{\mathfrak{P}}(f_{\Lambda^{\mathrm{ac}}}).$

When $\mathfrak{P}=p\Lambda^{\mathrm{ac}}$, the same argument works by taking
$\mathfrak{P}_{M}=X^{M}+p$. ∎

######  Corollary 7.8.

We keep all the assumptions in $\S$2.1 with even $\nu(N^{-})$ and Assumption
7.4.

  1. (1)

If the localization of the Heegner point main conjecture at
$X\Lambda^{\mathrm{ac}}$ holds, then $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is
non-trivial, i.e. the Kolyvagin conjecture holds.

  2. (2)

If $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial modulo $p$, then the
Heegner point main conjecture holds.

  3. (3)

We further assume that $p$ splits in $K$. Then the following statements are
equivalent.

    1. (a)

$\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial modulo $p$.

    2. (b)

The Heegner point main conjecture holds and $p$ does not divide any Tamagawa
factor at a prime dividing $N^{+}$.

###### Proof.

  1. (1)

Since the divisibilities of $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ and
$\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha}$ are equivalent under the non-
anomalous assumption, the conclusion follows easily from Theorem 7.7.

  2. (2)

In the same manner, Theorem 7.7 implies that the mod $p$ non-triviality of
$\boldsymbol{\kappa}^{\mathrm{Heeg}}$ $\Rightarrow$ the Heegner point main
conjecture.

  3. (3)

We first show that (b) $\Rightarrow$ (a). We follow the strategy of [Zan] and
give a sketch only. Let $\chi$ be a finite order character on
$\mathrm{Gal}(K_{\infty}/K)$ and
$\mathfrak{P}_{\chi}\subseteq\Lambda^{\mathrm{ac}}$ the height one prime ideal
defined by the kernel of $\chi:\Lambda^{\mathrm{ac}}\to\mathcal{O}_{\chi}$
where $\mathcal{O}_{\chi}$ is the discrete valuation ring generated by the
values of $\chi$ over $\mathbb{Z}_{p}$. Let
$\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})}$ be the
$\chi$-specialization of $\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,\infty}$
and we may assume that
$\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})}_{1}\neq 0$ by varying
$\chi$. Applying Kolyvagin’s structure theorem (Theorem 4.3, [Kol91]) to this
setting, we obtain

$\mathrm{length}_{\mathcal{O}_{\chi}}M_{\chi}=\partial^{(0)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})})-\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})})$

where
$M_{\chi}=M_{\infty}\otimes_{\Lambda^{\mathrm{ac}},\chi}\mathcal{O}_{\chi}$.
Note that we do not see any error term (“the $\chi$-twisted version of
$\mathrm{Err}$” in Remark 4.4) since
$\kappa^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})}_{1}\neq 0$. We now assume
that the Heegner point main conjecture. Since $p$ splits in $K$, it is
equivalent to the BDP main conjecture [BCK21, Theorem 5.2]. Evaluating the BDP
$p$-adic $L$-function at $\chi^{-1}$, we obtain the following formula from
[Zan, Theorem 4.4.3], which is the $\chi$-twisted version of [BCK21,
Proposition A.3]

$2\cdot\mathrm{length}_{\mathcal{O}_{\chi}}M_{\chi}=2\cdot\partial^{(0)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})})-\sum_{v|N^{+}}\mathrm{ord}_{p}\left(c_{v}((T_{\mathfrak{P}_{\chi}})^{*}/K)\right)$

where $c_{v}((T_{\mathfrak{P}_{\chi}})^{*}/K)$ is the Tamagawa factor of
$(T_{\mathfrak{P}_{\chi}})^{*}=\mathrm{Hom}(T_{\mathfrak{P}_{\chi}},\mu_{p^{\infty}})$
over $K$ at $v$ and $v$ runs over places of $K$. Thus, we obtain

$2\cdot\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})})=\sum_{v|N^{+}}\mathrm{ord}_{p}\left(c_{v}((T_{\mathfrak{P}_{\chi}})^{*}/K)\right).$

Under our assumptions (especially on the non-anomalous one), [Zan, Proposition
3.1.3] shows that
$\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha})=0$ if and
only if
$\partial^{(\infty)}(\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha,(\mathfrak{P}_{\chi})})=0$.
Also, [Zan, Proposition 4.3.5] implies that
$\sum_{v|N^{+}}\mathrm{ord}_{p}\left(c_{v}(E/K)\right)=0$ if and only if
$\sum_{v|N^{+}}\mathrm{ord}_{p}\left(c_{v}((T_{\mathfrak{P}_{\chi}})^{*}/K)\right)=0$.
Since the mod $p$ non-vanishing properties of
$\boldsymbol{\kappa}^{\mathrm{Heeg}}$ and
$\boldsymbol{\kappa}^{\mathrm{Heeg},\alpha}$ are equivalent under the non-
anomalous assumption, $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is non-trivial
modulo $p$.

For the direction (a) $\Rightarrow$ (b), we know that the mod $p$ non-
triviality of $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ $\Rightarrow$ the Heegner
point main conjecture by using (2). By using the same argument of the (b)
$\Rightarrow$ (a) part, any Tamagawa factor at a prime dividing $N^{+}$ is not
divisible by $p$ if and only if $\boldsymbol{\kappa}^{\mathrm{Heeg}}$ is not
divisible by $p$. The conclusion follows.

∎

######  Remark 7.9.

  1. (1)

Corollary 7.8.(1) says that a small piece of the Heegner point main conjecture
is strong enough to obtain the Kolyvagin conjecture, i.e. the non-triviality
of $\boldsymbol{\kappa}^{\mathrm{Heeg}}$.

  2. (2)

Corollary 7.8.(2) significantly simplifies the proof of the main result of
[BCK21]. See also [Zan]. If we further assume that
$\kappa^{\mathrm{Heeg}}_{1}\neq 0$, then $p$ does not divide any Tamagawa
factor at a prime dividing $N^{+}$ thanks to the work of Jetchev [Jet08].

  3. (3)

Corollary 7.8.(3) is also proved in [Zan].

######  Remark 7.10.

It seems that most argument generalizes to the $\pm$-Iwasawa-theoretic setting
when $E$ has supersingular reduction at $p$. However, $p$ should split in $K$
in this case because the formulation of the anticyclotomic $\pm$-Iwasawa
theory for non-CM elliptic curves at an inert prime is still mysterious.

###  7.3. The non-triviality of $\boldsymbol{\lambda}^{\mathrm{bip}}$

We keep all the settings in $\S$2.1 with odd $\nu(N^{-})$.

######  Assumption 7.11.

We also assume that

  1. (1)

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

  2. (2)

$a_{p}(E)\not\equiv 1\pmod{p}$ if $p$ splits in $K$, and

  3. (3)

$a_{p}(E)\not\equiv\pm 1\pmod{p}$ if $p$ is inert in $K$.

We adapt most notation from the previous subsection.

Let $\lambda^{\mathrm{bip},\alpha,\infty}_{1}\in\Lambda^{\mathrm{ac}}$ be the
$p$-stabilized Bertolini–Darmon’s theta element for $E$ over $K_{\infty}$
[BD05]. It is known that $\lambda^{\mathrm{bip},\alpha,\infty}_{1}\neq 0$
thanks to the work of Vatsal [Vat02].

In [How06, $\S$3.2], the mod $p^{k}$ version of $\Lambda^{\mathrm{ac}}$-adic
bipartite Euler systems is illustrated. As in the proof of Theorem 5.23,
$\Lambda^{\mathrm{ac}}$-adic bipartite Euler systems are defined by taking the
inverse limit with respect to $k$. Denote by
$\boldsymbol{\lambda}^{\mathrm{bip},\alpha,\infty}$ the definite part of the
$\Lambda^{\mathrm{ac}}$-adic bipartite Euler system.

######  Theorem 7.12 (Bertolini–Darmon).

We keep all the assumptions in $\S$2.1 with odd $\nu(N^{-})$ and Assumption
7.11. Then $\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}$ is a finitely
generated torsion $\Lambda^{\mathrm{ac}}$-module, and

$\left(\lambda^{\mathrm{bip},\alpha,\infty}_{1}\cdot\lambda^{\mathrm{bip},\alpha,\infty,\iota}_{1}\right)\subseteq\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right).$

###### Proof.

See [BD05, PW11, CH15, KPW17]. See also [How06, Theorem 3.2.3.(b)]. ∎

######  Remark 7.13.

The anticyclotomic main conjecture for elliptic curves à la Bertolini–Darmon
is the equality of the inequality in Theorem 7.12.

######  Lemma 7.14.

We keep all the assumptions in $\S$2.1 with odd $\nu(N^{-})$ and Assumption
7.11. Then
$\left(\lambda^{\mathrm{bip},\alpha,\infty}_{1}\right)=\left(\lambda^{\mathrm{bip},\alpha,\infty,\iota}_{1}\right)$
in $\Lambda^{\mathrm{ac}}$.

###### Proof.

See [BD05, Lemma 1.5]. ∎

######  Theorem 7.15.

We keep all the assumptions in $\S$2.1 with odd $\nu(N^{-})$ and Assumption
7.11. Let $\mathfrak{P}$ be a height one prime of $\Lambda^{\mathrm{ac}}$.
Then the following statements are equivalent.

  1. (1)

$\boldsymbol{\lambda}^{\mathrm{bip},\alpha,\infty}$ is non-trivial modulo
$\mathfrak{P}$.

  2. (2)

$2\cdot\mathrm{ord}_{\mathfrak{P}}\left(\lambda^{\mathrm{bip},\alpha,\infty}_{1}\right)=\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right)\right)$.

By Lemma 7.14, the second statement is exactly the localization of
anticyclotomic main conjecture for elliptic curves at $\mathfrak{P}$.

###### Proof.

The direction (1) $\Rightarrow$ (2) is [How06, Theorem 3.2.3.(c)], so we focus
on the converse. Like the proof of Theorem 7.7, our argument mainly follows
[How06, $\S$3.4] again but with some modification.

We first suppose that $\mathfrak{P}\neq p\Lambda^{\mathrm{ac}}$ and write
$\mathfrak{P}=(g)$ where $g\in\Lambda^{\mathrm{ac}}$. Define
$\mathfrak{P}_{M}=(g+p^{M})\Lambda^{\mathrm{ac}}$. By taking $M$ sufficiently
large, we may assume the following statement.

  * •

$\mathfrak{P}_{M}$ is a height one prime ideal of $\Lambda^{\mathrm{ac}}$.

  * •

$\Lambda^{\mathrm{ac}}/\mathfrak{P}_{M}\simeq\Lambda^{\mathrm{ac}}/\mathfrak{P}$
as rings.

  * •

$\mathfrak{P}_{M}$ and $\lambda^{\mathrm{bip},\alpha,\infty}_{1}$ are
relatively prime, so $\lambda^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})}_{1}\neq
0$.

  * •

The natural map

$\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})\to\mathrm{Sel}(K_{\infty},E[p^{\infty}])[\mathfrak{P}_{M}]$

has finite kernel and cokernel which are bounded by a constant depending only
on $[S_{\mathfrak{P}_{M}}:\Lambda^{\mathrm{ac}}/\mathfrak{P}_{M}]$ but not on
$\mathfrak{P}_{M}$ itself. [How06, Proposition 3.3.1].

Following the argument of [MR04, Theorem 5.3.10], we obtain

$\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})\right)=M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}S_{\mathfrak{P}_{M}}\cdot\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right)$

up to $O(1)$ as $M$ varies. We also have

$\mathrm{length}_{\mathbb{Z}_{p}}\left(S_{\mathfrak{P}_{M}}/S_{\mathfrak{P}_{M}}\cdot\lambda^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})}_{1}\right)=M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}S_{\mathfrak{P}_{M}}\cdot\mathrm{ord}_{\mathfrak{P}}\left(\lambda^{\mathrm{bip},\alpha,\infty}_{1}\right)$

up to $O(1)$ as $M$ varies. By considering a variant of self-dual Selmer
structures as in [How06, $\S$2.6], we are able to apply Theorem 5.23 to this
setting. Thus, we have

$\displaystyle
M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}S_{\mathfrak{P}_{M}}\cdot\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right)$
$\displaystyle=\mathrm{length}_{\mathbb{Z}_{p}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})\right)$
$\displaystyle=\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot\mathrm{length}_{S_{\mathfrak{P}_{M}}}\left(\mathrm{Sel}(K,(T_{\mathfrak{P}_{M}})^{*})\right)$
$\displaystyle=\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot
2\cdot\left(\partial^{(0)}(\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})})-\partial^{(\infty)}(\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})})\right)$
$\displaystyle\leq
2\cdot\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot\partial^{(0)}(\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})})$
$\displaystyle=2\cdot\dfrac{\mathrm{rk}_{\mathbb{Z}_{p}}(S_{\mathfrak{P}_{M}})}{e(S_{\mathfrak{P}_{M}}/\mathbb{Z}_{p})}\cdot\mathrm{length}_{S_{\mathfrak{P}_{M}}}\left(S_{\mathfrak{P}_{M}}/S_{\mathfrak{P}_{M}}\cdot\lambda^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})}_{1}\right)$
$\displaystyle=2\cdot\mathrm{length}_{\mathbb{Z}_{p}}\left(S_{\mathfrak{P}_{M}}/S_{\mathfrak{P}_{M}}\cdot\lambda^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})}_{1}\right)$
$\displaystyle=2\cdot
M\cdot\mathrm{rk}_{\mathbb{Z}_{p}}S_{\mathfrak{P}_{M}}\cdot\mathrm{ord}_{\mathfrak{P}}\left(\lambda^{\mathrm{bip},\alpha,\infty}_{1}\right)$

up to $O(1)$ as $M$ varies, and the inequality in the middle is an equality if
and only if
$\partial^{(\infty)}(\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})})$
is a constant as $M$ varies.

If we assume
$2\cdot\mathrm{ord}_{\mathfrak{P}}\left(\lambda^{\mathrm{bip},\alpha,\infty}_{1}\right)=\mathrm{ord}_{\mathfrak{P}}\left(\mathrm{char}_{\Lambda^{\mathrm{ac}}}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right)\right)$,
then the inequality in the middle becomes an equality, so
$\partial^{(\infty)}(\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})})$
is a constant as $M$ varies. Since

$\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P})}\equiv\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P}_{M})}\pmod{p^{M}},$

we have
$\partial^{(\infty)}(\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P})})<\infty$,
so $\boldsymbol{\lambda}^{\mathrm{bip},\alpha,(\mathfrak{P})}$ is non-trivial,
By the non-anomalous assumption,
$\boldsymbol{\lambda}^{\mathrm{bip},(\mathfrak{P})}$ is also non-trivial.

When $\mathfrak{P}=p\Lambda^{\mathrm{ac}}$, the same argument works by taking
$\mathfrak{P}_{M}=X^{M}+p$. ∎

######  Corollary 7.16.

We keep all the assumptions in $\S$2.1 with odd $\nu(N^{-})$ and Assumption
7.11. If the localization of the anticyclotomic main conjecture for elliptic
curves at $X\Lambda^{\mathrm{ac}}$ holds, then
$\boldsymbol{\lambda}^{\mathrm{bip}}$ is non-trivial.

######  Remark 7.17.

  1. (1)

Theorem 7.15 refines [How06, Theorem 3.2.3.(c)]. In particular, under our
working hypotheses, if $\boldsymbol{\lambda}^{\mathrm{bip}}$ is non-trivial
modulo $p$, then the anticyclotomic main conjecture for elliptic curves holds.

  2. (2)

Remark 7.10 applies to the bipartite Euler system setting equally.

##  Appendix A Construction of the Gross point of conductor 1

We quickly review the construction of the Gross point of conductor 1, which is
used for the construction of $\lambda^{\mathrm{bip}}_{n}$ in $\S$5. Some
notation might be slightly different from those in the main text. See [CH15,
CH18, Kim19] for details.

Let $K$ be the imaginary quadratic field of discriminant $-D_{K}<0$. Define

$\vartheta:=\left\\{\begin{array}[]{ll}\dfrac{D_{K}-\sqrt{-D_{K}}}{2}&\textrm{
if }2\nmid D_{K}\\\ \dfrac{D_{K}-2\sqrt{-D_{K}}}{4}&\textrm{ if }2\mid
D_{K}\end{array}\right.$

so that $\mathcal{O}_{K}=\mathbb{Z}+\mathbb{Z}\vartheta$. Let $B_{nN^{-}}$ be
the definite quaternion algebra over $\mathbb{Q}$ of discriminant $nN^{-}$.
Then there exists an embedding of $K$ into $B_{nN^{-}}$ [Vig80]. More
explicitly, we choose a $K$-basis $(1,J)$ of $B_{nN^{-}}$ so that
$B_{nN^{-}}=K\oplus K\cdot J$ such that $\beta:=J^{2}\in\mathbb{Q}^{\times}$
with $\beta<0$, $J\cdot t=\overline{t}\cdot J$ for all $t\in K$,
$\beta\in\left(\mathbb{Z}^{\times}_{q}\right)^{2}$ for all $q\mid pN^{+}$, and
$\beta\in\mathbb{Z}^{\times}_{q}$ for all $q\mid D_{K}$. Fix a square root
$\sqrt{\beta}\in\overline{\mathbb{Q}}$ of $\beta$. Fix an isomorphism

$i:=\prod
i_{q}:\widehat{B}^{(nN^{-})}_{nN^{-}}\simeq\mathrm{M}_{2}(\mathbb{A}^{(nN^{-}\infty)})$

as follows.

  * •

For each finite place $q\mid N^{+}p$, the isomorphism
$i_{q}:B_{nN^{-},q}\simeq\mathrm{M}_{2}(\mathbb{Q}_{q})$ is defined by

$\textstyle{{i_{q}(\vartheta)=\left(\begin{matrix}\mathrm{trd}(\vartheta)&-\mathrm{nrd}(\vartheta)\\\
1&0\end{matrix}\right)},}$$\textstyle{{i_{q}(J)=\sqrt{\beta}\cdot\left(\begin{matrix}-1&\mathrm{trd}(\vartheta)\\\
0&1\end{matrix}\right)}}$

where $\mathrm{trd}$ and $\mathrm{nrd}$ are the reduced trace and the reduced
norm on $B$, respectively.

  * •

For each finite place $q\nmid pN^{+}$, the isomorphism
$i_{q}:B_{nN^{-},q}\simeq\mathrm{M}_{2}(\mathbb{Q}_{q})$ is chosen so that
$i_{q}\left(\mathcal{O}_{K}\otimes\mathbb{Z}_{q}\right)\subseteq\mathrm{M}_{2}(\mathbb{Z}_{q})$.

Under the fixed isomorphism $i$, for any rational prime $q$, the local Gross
point $\varsigma_{q}\in B^{\times}_{nN^{-},q}$ is defined as follows:

  * •

$\varsigma_{q}:=1$ in $B^{\times}_{nN^{-},q}$ for $q\nmid pN^{+}$.

  * •

$\varsigma_{q}:=\frac{1}{\sqrt{D_{K}}}\cdot\left(\begin{matrix}\vartheta&\overline{\vartheta}\\\
1&1\end{matrix}\right)\in\mathrm{GL}_{2}(K_{\mathfrak{q}})=\mathrm{GL}_{2}(\mathbb{Q}_{q})$
for $q\mid N^{+}$ with $q=\mathfrak{q}\overline{\mathfrak{q}}$ in
$\mathcal{O}_{K}$.

  * •

$\varsigma^{(0)}_{p}=\left(\begin{matrix}\vartheta&-1\\\
1&0\end{matrix}\right)\in\mathrm{GL}_{2}(K_{\mathfrak{p}})=\mathrm{GL}_{2}(\mathbb{Q}_{p})$
when $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits in $K$.

  * •

$\varsigma^{(0)}_{p}=\left(\begin{matrix}0&1\\\
-1&0\end{matrix}\right)\in\mathrm{GL}_{2}(K_{p})=\mathrm{GL}_{2}(\mathbb{Q}_{p^{2}})$
when $p$ is inert in $K$.

The Gross point $\varsigma^{(0)}$ of conductor $1$ on
$\widehat{B}^{\times}_{nN^{-}}$ is defined by

$\varsigma^{(0)}:=\varsigma^{(0)}_{p}\times\prod_{q\neq
p}\varsigma_{q}\in\widehat{B}^{\times}_{nN^{-}}.$

The action of $\widehat{K}^{\times}$ on $\varsigma^{(0)}$ is defined by left
translation with embedding $K\hookrightarrow B$.

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