# Base change and Iwasawa Main Conjectures for ${\rm GL}_{2}$

Ashay Burungale  University of Texas at Austin, USA<EMAIL_ADDRESS>,
Francesc Castella  University of California Santa Barbara, South Hall, Santa
Barbara, CA 93106, USA<EMAIL_ADDRESS>and  Christopher Skinner  Princeton
University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, $p$ an odd prime of
good ordinary reduction such that $E[p]$ is an irreducible
$G_{\mathbb{Q}}$-module, and $K$ an imaginary quadratic field with all primes
dividing $Np$ split. We prove Iwasawa Main Conjectures for the
$\mathbb{Z}_{p}$-cyclotomic and $\mathbb{Z}_{p}$-anticyclotomic deformation of
$E$ over $\mathbb{Q}$ and $K$ respectively, dispensing with any of the
ramification hypotheses on $E[p]$ in previous works. Using base change, the
proofs are based on Wan’s divisibility towards a three-variable main
conjecture for $E$ over a quartic CM field containing $K$.

As an application, we prove cases of the two-variable main conjecture for $E$
over $K$. The one-variable main conjectures imply the $p$-part of the
conjectural Birch and Swinnerton-Dyer formula if
$\operatorname{ord}_{s=1}L(E,s)\leq 1$. They are also an ingredient in the
proof of Kolyvagin’s conjecture and its cyclotomic variant in our joint work
with Grossi [BCGS23].

###### Contents

  1. 1 Introduction
  2. 2 Main Conjecture over CM fields
  3. 3 Main Conjectures over quartic CM fields
  4. 4 Main Conjectures over imaginary quadratic fields
  5. 5 Base change

##  1\. Introduction

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ an odd prime of good ordinary
reduction for $E$, and $K$ an imaginary quadratic field. In this paper we
study Iwasawa theory of $E$ over the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$ and the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$, proving
corresponding Iwasawa Main Conjectures (cf. Theorems 1.1.2, 1.2.2 and 1.2.3).

###  1.1. Cyclotomic Main Conjecture

Let $\mathbb{Q}_{\infty}$ be the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$, put $\Gamma={\rm Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$, and let
$\Lambda=\mathbb{Z}_{p}[\\![\Gamma]\\!]$ be the cyclotomic Iwasawa algebra.

We consider the classical Selmer group ${\rm
Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})=\mathop{\mathchoice{\varlim@\displaystyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\rightarrowfill@\scriptscriptstyle}}}\displaylimits_{n}{\rm
Sel}_{p^{\infty}}(E/\mathbb{Q}_{n})$, where $\mathbb{Q}_{n}$ is the subfield
of $\mathbb{Q}_{\infty}$ with $[\mathbb{Q}_{n}:\mathbb{Q}]=p^{n}$. Its
Pontryagin dual

$\mathfrak{X}_{\rm ord}(E/\mathbb{Q}_{\infty}):={\rm
Hom}_{\mathbb{Z}_{p}}({\rm
Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p})$

is a finitely generated $\Lambda$-module. Let
${\mathcal{L}}_{p}(E/\mathbb{Q})\in\Lambda\otimes\mathbb{Q}_{p}$ be the
$p$-adic $L$-function attached to $E$ by Mazur–Swinnerton-Dyer [MSD74]. In
[Maz72], Mazur conjectured the following.

######  Conjecture 1.1.1 (Mazur’s Main Conjecture).

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

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

as ideals in $\Lambda$.

Note that implicit in Conjecture 1.1.1 is the integrality statement
$\mathcal{L}_{p}(E/\mathbb{Q})\in\Lambda$; this is most well-understood under
the assumption that $p$ is odd and

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

(see [GV00, Prop. 3.1]) where
$G_{\mathbb{Q}}:=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is the
absolute Galois group of $\mathbb{Q}$. (We similarly use $G_{L}$ to denote the
absolute Galois of a number field $L$.) Let $T$ be the $p$-adic Tate module of
$E$. In [Kat04], Kato proved the $\Lambda$-torsionness of $\mathfrak{X}_{\rm
ord}(E/\mathbb{Q}_{\infty})$ and the inclusion
$p^{c}\cdot\mathcal{L}_{p}(E/\mathbb{Q})\in{\rm
ch}_{\Lambda}(\mathfrak{X}_{\rm ord}(E/\mathbb{Q}_{\infty}))$ for some $c\geq
0$, with $c=0$ when $T$ has “large” image. Let $N$ be the conductor of $E$.
Assuming further that

(mult) there exists a prime $q\|N$ such that $E[p]$ is ramified at $q$,

the converse divisibility, and hence Conjecture 1.1.1 was proved by
Skinner–Urban [SU14].

Our main result towards Conjecture 1.1.1 removes the hypothesis (mult):

######  Theorem 1.1.2.

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ a prime of good ordinary
reduction for $E$. If $p>3$ satisfies (irrQ), then $\mathfrak{X}_{\rm
ord}(E/\mathbb{Q}_{\infty})$ is $\Lambda$-torsion, with

${\rm ch}_{\Lambda_{\mathbb{Q}}}\left(\mathfrak{X}_{\rm
ord}(E/\mathbb{Q}_{\infty})\right)=\bigl{(}\mathcal{L}_{p}(E/\mathbb{Q})\bigr{)}$

in $\Lambda\otimes\mathbb{Q}_{p}$. If in addition

(im) there exists an element $\sigma\in G_{\mathbb{Q}}$ fixing
$\mathbb{Q}(\mu_{p^{\infty}})$ such that $T/(\sigma-1)T\simeq\mathbb{Z}_{p}$,

then the equality holds in $\Lambda_{\mathbb{Q}}$, and hence Conjecture 1.1.1
holds.

######  Remark 1.1.3.

  1. (i)

For non-CM curves the condition (im) holds for all sufficiently large primes
$p$ by Serre’s open image theorem [Ser72]. In fact, it is expected that $p\geq
37$ suffices.

  2. (ii)

The only prior result towards Conjecture 1.1.1 without assuming the hypothesis
(mult) is due to Wan [Wan15], based on Eisenstein congruence on the unitary
group ${\rm GU}(2,2)$ over CM fields. However, it is conditional on a
$p$-integral comparison of certain automorphic periods, which still remains
open. Our proof of Theorem 1.1.2 relies on a main result of [Wan15], but
sidesteps the period comparison.

###  1.2. Anticyclotomic Main Conjectures

Assume that the discriminant $D_{K}<0$ satifies

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

Moreover, assume that $K$ satisfies the _Heegner hypothesis_ :

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

and that

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

for $v$ the prime of $K$ above $p$ induced by an embedding
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$, which we fix
throughout.

Let $K_{\infty}^{-}/K$ be the anticyclotomic $\mathbb{Z}_{p}$-extension,
$\Gamma_{K}^{-}={\rm Gal}(K_{\infty}^{-}/K)$, and
$\Lambda_{K}^{-}=\mathbb{Z}_{p}[\\![\Gamma_{K}^{-}]\\!]$ the anticyclotomic
Iwasawa algebra. In view of (Heeg) and the $p$-ordinarity hypothesis, the
Kummer images of Heegner points of $p$-power conductor give rise to a
$\Lambda_{K}^{-}$-adic class

$\kappa_{1}^{\rm
Heeg}\in\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T}),$

where
$\mathbf{T}=\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{n}{\rm
Ind}_{K_{n}^{-}/K}(T)$, with $K_{n}^{-}$ the subfield of $K_{\infty}^{-}$ with
$[K_{n}^{-}:K]=p^{n}$, and
$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})\subset\mathrm{H}^{1}(K,\mathbf{T})$
is the compact ordinary Selmer group111See e.g. [CGLS22, §4.1] for a review of
the construction, whose notations we largely follow. interpolating the
classical Selmer groups
$\mathop{\mathchoice{\varlim@\displaystyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\leftarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\leftarrowfill@\scriptscriptstyle}}}\displaylimits_{m}{\rm
Sel}_{p^{m}}(E/K_{n}^{-})$ as $n$ varies. Let $\mathfrak{X}_{\rm
ord}(E/K_{\infty}^{-})$ be the Pontryagin dual of ${\rm
Sel}_{p^{\infty}}(E/K_{\infty}^{-})=\mathop{\mathchoice{\varlim@\displaystyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\textstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptstyle{\rightarrowfill@\scriptscriptstyle}}{\varlim@\scriptscriptstyle{\rightarrowfill@\scriptscriptstyle}}}\displaylimits_{n}{\rm
Sel}_{p^{\infty}}(E/K_{\infty}^{-})$. The formulation of a Main Conjecture in
this setting is due to Perrin-Riou [PR87].

######  Conjecture 1.2.1 (Heegner point Main Conjecture).

$\mathfrak{X}_{\rm ord}(E/K_{\infty}^{-})$ and
$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$ have both
$\Lambda_{K}^{-}$-rank one, and

${\rm ch}_{\Lambda_{K}^{-}}\bigl{(}\mathfrak{X}_{\rm
ord}(E/K_{\infty}^{-})_{\rm tor}\bigr{)}={\rm
ch}_{\Lambda_{K}^{-}}\bigl{(}\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})/(\bm{\kappa}_{1}^{\rm
Heeg})\bigr{)}^{2}$

as ideals in $\Lambda_{K}^{-}$.

The first general results towards it are due to Bertolini [Ber95] and Howard
[How04]. These works established the rank statements in Conjecture 1.2.1, and
the latter proved the divisibility “$\supseteq$” if

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

The first cases of the opposite divisibility, and hence of Conjecture 1.2.1,
are a consequence of the main result [Wan20] of Wan, which employs Eisenstein
congruences on the unitary group ${\rm GU}(3,1)$. In addition to (sur), it
requires that $N$ be square-free and a ramification condition on $E[p]$.

Our different approach dispenses with any of the ramification hypotheses,
leading to the following result.

######  Theorem 1.2.2.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$, $p$ be a prime of
good ordinary reduction for $E$, and $K$ an imaginary quadratic field
satisfying (disc), (Heeg), and (spl). If $p>3$ satisfies (irrQ), then both
$\mathfrak{X}_{\rm ord}(E/K_{\infty}^{-})$ and
$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})$ have
$\Lambda_{K}^{-}$-rank one, and

${\rm ch}_{\Lambda_{K}^{-}}\bigl{(}\mathfrak{X}_{\rm
ord}(E/K_{\infty}^{-})_{\rm tor}\bigr{)}={\rm
ch}_{\Lambda_{K}^{-}}\bigl{(}\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,\mathbf{T})/(\bm{\kappa}_{1}^{\rm
Heeg})\bigr{)}^{2}$

in $\Lambda_{K}^{-}\otimes\mathbb{Q}_{p}$. If further $p>3$ satisfies (sur),
then the equality holds in $\Lambda_{K}^{-}$ and hence Conjecture 1.2.1 holds.

Such an equality has applications to the Birch and Swinnerton-Dyer conjecture,
especially to a $p$-converse to the Gross–Zagier and Kolyvagin theorem (cf.
[Ski20, Wan21, Cas17, BT20, BST21]).

In light of the $\Lambda_{K}^{-}$-adic analogue of the $p$-adic Waldspurger
formula of [BDP13] (see [CH18]) Conjecture 1.2.1 is equivalent to the
prediction that the $p$-adic $L$-function $\mathcal{L}_{p}^{\rm BDP}(E/K)$
constructed in _op. cit._ generates the characteristic ideal of the
anticyclotomic Selmer group $\mathfrak{X}_{\rm Gr}(E/K_{\infty}^{-})$ whose
classes are locally trivial (resp. unrestricted) at the primes above
$\overline{v}$ (resp. $v$). Hence, Theorem 1.2.2 also yields the following.

######  Theorem 1.2.3.

Let $(E,p,K)$ be as in Theorem 1.2.2. If $p>3$ satisfies (irrQ), then
$\mathfrak{X}_{\rm Gr}(E/K_{\infty}^{-})$ is $\Lambda_{K}^{-}$-torsion, and

${\rm ch}_{\Lambda_{K}^{-}}\bigl{(}\mathfrak{X}_{\rm
Gr}(E/K_{\infty}^{-})\bigr{)}=\bigl{(}\mathcal{L}_{p}^{\rm BDP}(E/K)\bigr{)}$

in $\Lambda_{K}^{-,{\rm ur}}\otimes\mathbb{Q}_{p}$. If further $p>3$ satisfies
(sur), then the equality of characteristic ideals holds in
$\Lambda_{K}^{-,{\rm ur}}$.

###  1.3. Application to the Birch and Swinnerton-Dyer formula

A consequence of Theorems 1.1.2 and 1.2.3 is the following.

######  Corollary 1.3.1.

Let $E/\mathbb{Q}$ be a non-CM elliptic curve. Let $p>3$ be a prime of good
ordinary reduction such that (irrQ) and (im) hold. If
$\operatorname{ord}_{s=1}L(E,s)=r\in\\{0,1\\}$, then

$\bigg{|}\frac{L^{(r)}(E,1)}{{\rm
Reg}(E)\cdot\Omega_{E}}\bigg{|}_{p}^{-1}=\bigg{|}\\#\text{\wncyr{W}}(E)\prod_{\ell\nmid\infty}c_{\ell}(E)\bigg{|}_{p}^{-1}$

and hence the $p$-part of the conjectural BSD formula for $E$ is true.

###### Proof.

In the case $r=0$, this follows from Theorem 1.1.2, the interpolation property
of $\mathcal{L}_{p}(E/\mathbb{Q})$ at the trivial character, and [Gre99, Thm.
4.1]. Similarly, for a suitably chosen imaginary quadratic field $K$, the
result for $r=1$ follows from Theorem 1.2.3, the formula of [BDP13] for the
value of $\mathcal{L}_{p}^{\rm BDP}(E/K)$ at the trivial character, [JSW17,
Thm. 3.3.1], and the $r=0$ result for the $K$-quadratic twist of $E$. (See
also [Cas24, §1] for a more detailed review of these arguments.) ∎

######  Remark 1.3.2.

The condition (im) in Corollary 1.3.1 excludes only finitely many primes $p$
(cf. Remark 1.1.3(i)).

###  1.4. On the two-variable Main Conjectures

Our approach to the above theorems also gives a proof of the two-variable
Iwasawa Main Conjectures for $E/K$ under some additional hypothesis on $E[p]$.
For the precise statement, consider the set of “vexing primes” $\ell$ for
$E[p]$:

$\mathcal{V}:=\bigl{\\{}\ell\equiv-1\,({\rm mod}\,p)\;\,|\;\,\textrm{
$\bar{\rho}_{E}|_{D_{\ell}}$ is irreducible and $\bar{\rho}_{E}|_{I_{\ell}}$
is reducible}\bigr{\\}},$

where $I_{\ell}\subset D_{\ell}$ are inertia and decomposition groups at
$\ell$, respectively.

Let $K_{\infty}/K$ denote the $\mathbb{Z}_{p}^{2}$-extension of $K$, and put
$\Gamma_{K}={\rm Gal}(K_{\infty}/K)$ and
$\Lambda_{K}=\mathbb{Z}_{p}[\\![\Gamma_{K}]\\!]$. Let $\mathfrak{X}_{\rm
ord}(E/K_{\infty})$ be the Pontryagin dual of the Selmer group ${\rm
Sel}_{p^{\infty}}(E/K_{\infty})$, and let $\mathcal{L}_{p}^{\rm
PR}(E/K)\in\Lambda_{K}$ be the two-variable $p$-adic Rankin $L$-series
constructed by Perrin-Riou [PR88] (normalized as in [CGS23, §1.2]).

######  Theorem 1.4.1.

Let $(E,p,K)$ be as in Theorem 1.2.2. Assume that $\mathcal{V}=\emptyset$. If
$p>3$ satisfies (irrQ), then $\mathfrak{X}_{\rm ord}(E/K_{\infty})$ is
$\Lambda_{K}$-torsion, with

${\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
ord}(E/K_{\infty})\bigr{)}=\bigl{(}\mathcal{L}_{p}^{\rm PR}(E/K)\bigr{)}$

in $\Lambda_{K}\otimes\mathbb{Q}_{p}$. If further $p>3$ satisfies (sur), then
the equality of characteristic ideals holds in $\Lambda_{K}$.

######  Remark 1.4.2.

Since the global root number of $E$ over $K$ equals $-1$ (cf. (Heeg)), Theorem
1.4.1 complements the results on the two-variable Iwasawa Main Conjecture in
[SU14].

###  1.5. About the proofs

The key new idea is to base change $E$ to a quartic CM field $M$ containing
$K$ for which the main result of [Wan15] towards a three-variable Main
Conjecture applies, and utilize the two-variable zeta element associated to
$E$ over $K$ as recently constructed in [BSTW23].

More precisely, the result of [Wan15] yields the divisibility

(1.1) $\bigl{(}{\mathcal{L}}_{p}^{\rm PR}(E/K)\cdot{\mathcal{L}}_{p}^{\rm
PR}(E^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
ord}(E/K_{\infty})\bigr{)}\cdot{\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
ord}(E^{F}/K_{\infty})\bigr{)}$

in $\Lambda_{K}\otimes\mathbb{Q}_{p}$, where $E^{F}$ is the quadratic twist of
$E$ for the real subfield $F$ contained in $M$. In view of the two-variable
zeta elements of [BSTW23] and their explicit reciprocity laws, this translates
into the divisibility

(1.2) $\bigl{(}{\mathcal{L}}_{p}^{\rm Gr}(E/K)\cdot{\mathcal{L}}_{p}^{\rm
Gr}(E^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
Gr}(E/K_{\infty})\bigr{)}\cdot{\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
Gr}(E^{F}/K_{\infty})\bigr{)}$

in $\Lambda_{K}^{\rm ur}\otimes\mathbb{Q}_{p}$, where $\mathcal{L}_{p}^{\rm
Gr}(E^{\cdot}/K)\in\Lambda_{K}^{\rm
ur}:=\Lambda_{K}\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{Z}_{p}^{\rm ur}$ is a
two-variable $p$-adic Rankin $L$-series specializing to $\mathcal{L}_{p}^{\rm
BDP}(E/K)$ under the natural projection $\Lambda_{K}^{\rm
ur}\rightarrow\Lambda_{K}^{-,{\rm
ur}}:=\Lambda_{K}^{-}\hat{\otimes}_{\mathbb{Z}_{p}}\mathbb{Z}_{p}^{\rm ur}$,
and $\mathfrak{X}_{\rm Gr}(E/K_{\infty})$ is the counterpart of
$\mathfrak{X}_{\rm Gr}(E/K_{\infty}^{-})$ over $K_{\infty}/K$. In view of the
vanishing of the Iwasawa $\mu$-invariant of $\mathcal{L}_{p}^{\rm BDP}(E/K)$
proved in [Hsi14, Bur17] following ideas in [Hid10], the divisibilities (1.1)
and (1.2) both hold integrally. The proof of Theorem 1.1.2 then follows from
(1.1) (for a suitably chosen $K$) by descending to the cyclotomic
$\mathbb{Z}_{p}$-extension $K_{\infty}^{+}/K$ and appealing to Kato’s work
[Kat04]. Similarly, the proof of Theorem 1.2.3 (and hence of Theorem 1.2.2)
follows from (1.2) by descending to $K_{\infty}^{-}/K$ and appealing to the
Kolyvagin system bound developed in [CGLS22, CGS23] applied to the Heegner
point Euler system. Without any restriction on $\mathcal{V}$, we thus arrive
at the equality

(1.3) $\bigl{(}{\mathcal{L}}_{p}^{\rm PR}(E/K)\cdot{\mathcal{L}}_{p}^{\rm
PR}(E^{F}/K)\bigr{)}={\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
ord}(E/K_{\infty})\bigr{)}\cdot{\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
ord}(E^{F}/K_{\infty})\bigr{)}$

(and likewise for (1.2)), and assuming $\mathcal{V}=\emptyset$ we separate the
two factors, concluding the proof of Theorem 1.4.1.

######  Remark 1.5.1.

An Euler system for $E/K$ extending222dispensing with the need in [LLZ15] to
twist by a non-Eisenstein and $p$-distinguished Hecke character the
construction in [BSTW23, LLZ15] would give rise to a divisibility

${\rm ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
ord}(E/K_{\infty})\bigr{)}\supset\bigl{(}\mathcal{L}_{p}^{\rm
PR}(E/K)\bigr{)},$

possibly after inverting $p$. With this divisibility in hand, the proof of
Theorem 1.4.1 follows from (1.3) without the additional hypothesis
$\mathcal{V}=\emptyset$.

######  Remark 1.5.2.

In the main text we prove Theorems 1.1.2, 1.2.2, 1.2.3, and 1.4.1 for any
weight two elliptic newform with good ordinary reduction at $p$.

We conclude this Introduction by noting that when $p>3$ satisfies (irrQ),
Theorems 1.1.2 and 1.2.3 are one of the key ingredients333A similar input when
(irrQ) is not satisfied is provided by the main results of [CGS23]. in the
proof of Kolyvagin’s conjecture and its analogue for Kato’s Euler systems in a
joint work of the authors with Grossi [BCGS23]. When $p>3$ satisfies (sur),
Theorems 1.1.2 and 1.2.3 are also used in [BCGS23] to prove the refinement of
Kolyvagin’s conjecture and its cyclotomic analog formulated by W. Zhang
[Zha14] and C.-H. Kim [Kim22], respectively.

###  1.6. Acknowledgements

The authors thank Giada Grossi, Haruzo Hida, Ye Tian, and Xin Wan for helpful
communications. During the preparation of this paper, A.B. was partially
supported by the NSF grants DMS-2303864 and DMS-2302064; F.C. was partially
supported by the NSF grants DMS-2101458 and DMS-2401321; C.S. was partially
supported by the Simons Investigator Grant #376203 from the Simons Foundation
and by the NSF grant DMS-1901985.

##  2\. Main Conjecture over CM fields

In this section we briefly recall the formulation of the Iwasawa Main
Conjecture over CM fields $M/F$ at the base of the proof of our main results.
Let $F$ be a totally real field of degree $d=[F:\mathbb{Q}]$. Let $g\in
S_{2}(\Gamma_{0}({\mathfrak{n}}))$ be a Hilbert modular newform over $F$ of
parallel weight $2$. Let $p$ be a prime with

(ur) $p\nmid D_{F}$,

where $D_{F}$ denotes the discriminant of $F/\mathbb{Q}$. For a prime
$\lambda$ of the Hecke field $F_{g}$ over $p$, let
$\rho_{g}:G_{F}\rightarrow{\rm GL}_{2}(F_{g,\lambda})$ be the associated
Galois representation and $V_{g}:=V_{g,\lambda}$ the underlying
$F_{g,\lambda}$-vector space. Let $T_{g}\subset V_{g}$ be a $G_{F}$-stable
${\mathcal{O}}:={\mathcal{O}}_{F_{g,\lambda}}$-lattice and,

$\bar{\rho}_{g}:G_{F}\rightarrow{\rm GL}_{2}(\bar{\mathbb{F}}_{p})$

the corresponding residual representation. Suppose that $g$ is ordinary at
each prime $w$ of $F$ over $p$, which we abbreviate as $p$ being a prime of
ordinary reduction for $g$. Let $0\subset{\rm Fil}_{w}^{+}(V_{g})\subset
V_{g}$ be the associated filtration of $F_{g,\lambda}[G_{F_{w}}]$-modules and
put ${\rm Fil}_{w}^{+}(T_{g})=T_{g}\cap{\rm Fil}_{w}^{+}(V_{g})$.

Let $M$ be a CM quadratic extension of $F$ such that

(splF) any prime of $F$ above $p$ splits in $M$.

Denote by $\Gamma_{M}^{-}$ (resp. $\Gamma_{M}^{+}$) the Galois group of the
anticyclotomic $\mathbb{Z}_{p}^{d}$-extension $M_{\infty}/M$ (resp. cyclotomic
$\mathbb{Z}_{p}$-extension $M_{\infty}^{+}/M$). Let
$M_{\infty}=M_{\infty}^{-}M_{\infty}^{+}$ be the compositum, and put
$\Gamma_{M}=\operatorname{Gal}(M_{\infty}/M)$ and
$\Lambda_{M}={\mathcal{O}}[\\![\Gamma_{M}]\\!]$.

#### Selmer groups

We consider the ${\mathcal{O}}[G_{M}]$-module

$\mathcal{M}_{g}=T_{g}\otimes_{\mathbb{Z}_{p}}\Lambda_{M}^{\vee},$

where $G_{M}$ acts on $\Lambda_{M}$ and $\Lambda_{M}^{\vee}$ via
$\Psi:G_{M}\twoheadrightarrow\Gamma_{M}\hookrightarrow\Lambda_{M}^{\times}$
and $\Psi^{-1}$, respectively. For every prime $w$ of $M$ above $p$, put
$\mathcal{M}_{g,w}^{+}={\rm{Fil}}_{w}^{+}(T_{g})\otimes_{\mathbb{Z}_{p}}\Lambda_{M}^{\vee}$.
Let $\Sigma$ be a finite set of places of $M$ containing the primes above
${\mathfrak{n}}p\infty$, let $M^{\Sigma}$ be the maximal extension of $M$
unramified outside $\Sigma$, and define the _ordinary Selmer group_ by

$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(M,\mathcal{M}_{g}):=\ker\bigg{\\{}\mathrm{H}^{1}(M^{\Sigma}/M,\mathcal{M}_{g})\rightarrow\prod_{w\in\Sigma,w\nmid
p}\mathrm{H}^{1}(M_{w},\mathcal{M}_{g})\times\prod_{w|p}\mathrm{H}^{1}(I_{w},\mathcal{M}_{g}/\mathcal{M}_{g,w}^{+})\bigg{\\}}.$

We put $\mathfrak{X}_{\rm
ord}(g/M_{\infty})=\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(M,\mathcal{M}_{g})^{\vee}$
to denote the Pontryagin dual, and for any subextension $N$ of $M_{\infty}/M$
let $\mathfrak{X}_{\rm ord}(g/N)$ be the analogously defined Selmer group with
${\rm Gal}(N/M)$ in place of $\Gamma_{M}$.

We shall also consider the _Greenberg Selmer group_

$\mathrm{H}^{1}_{\mathcal{F}_{\rm
Gr}}(M,\mathcal{M}_{g}):=\ker\bigg{\\{}\mathrm{H}^{1}(M^{\Sigma}/M,\mathcal{M}_{g})\rightarrow\prod_{w\in\Sigma,w\nmid
p}\mathrm{H}^{1}(M_{w},\mathcal{M}_{g})\times\mathrm{H}^{1}(I_{\overline{v}},\mathcal{M}_{g})\bigg{\\}}$

and its Pontryagin dual $\mathfrak{X}_{\rm Gr}(g/M_{\infty})$, as well as
their variants for any $N$ as above.

####  $p$-adic $L$-functions

Assume that the prime $p$ is odd and unramified in $F$. Let $M/F$ be a CM
quadratic field satisfying (splF) and such that

($\Delta$) $M$ is not contained in $H_{F}$, and any prime ramified in
$F/\mathbb{Q}$ splits in $M$.

Let

${\mathcal{L}}_{p}(g/M)\in\Lambda_{M}\otimes\mathbb{Q}_{p}$

be the associated $(d+1)$-variable $p$-adic $L$-function as in [Wan15, §7.3],
which interpolates the central $L$-values $L^{\rm alg}(g/M\otimes\chi,1)$ as
$\chi$ varies over finite order characters of $\Gamma_{M}$ (cf. [Wan15, Thm.
82(i)]). If an underlying Hecke algebra is Gorenstein, then [Wan15, Thm.
82(ii)] shows the inclusion ${\mathcal{L}}_{p}(g/M)\in\Lambda_{M}$.

#### Iwasawa Main Conjecture

######  Conjecture 2.0.1.

Let $g\in S_{2}(\Gamma_{0}({\mathfrak{n}}))$ be a Hilbert modular newform over
a totally real field $F$ and $p$ an odd prime unramified in $F$ and good
ordinary for $g$. Let $M/F$ be a CM quadratic extension satisfying (splF) and
such that

(irrM) $\bar{\rho}_{g}$ is irreducible as $G_{M}$-representation.

Then $\mathfrak{X}_{\rm ord}(g/M_{\infty})$ is $\Lambda_{M}$-torsion, with

${\rm ch}_{\Lambda_{M}}\bigl{(}\mathfrak{X}_{\rm
ord}(g/M_{\infty})\bigr{)}=\bigl{(}\mathcal{L}_{p}(g/M)\bigr{)}.$

######  Remark 2.0.2.

Without conditions (ur), and (irrM), the conjecture is still expected to hold,
with the equality of characteristic ideals being possibly up to tensoring with
$\mathbb{Q}_{p}$.

##  3\. Main Conjectures over quartic CM fields

We describe a consequence of a result [Wan15] towards Conjecture 2.0.1 which
will be central to the proofs of main results.

######  Theorem 3.0.1 (Wan).

Let $g\in S_{2}(\Gamma_{0}(N))$ be an elliptic newform and let $p>3$ be a
prime of good ordinary reduction for $g$. Let $F$ be a real quadratic field
with $(pN,D_{F})=1$, and let $g_{F}$ denote the base-change of $g$ to $F$. Let
$M/F$ be a CM quadratic extension satisfying (splF), ($\Delta$), and
$(N{\mathcal{O}}_{F},D_{M/F})=(1)$. Write

$N{\mathcal{O}}_{F}={\mathfrak{n}}^{+}{\mathfrak{n}}^{-},$

with ${\mathfrak{n}}^{+}$ (resp. ${\mathfrak{n}}^{-}$) divisible only by
primes which are split (resp. inert) in $M/F$. Suppose that:

  * (i)

$\bar{\rho}_{g_{F}}=\bar{\rho}_{g}|_{G_{F}}$ satisfies (irrM).

  * (ii)

Hypothesis 3.0.2 holds.

  * (iii)

${\mathfrak{n}}^{-}$ is the squarefree product of an even number of primes.

  * (iv)

$\bar{\rho}_{g_{F}}$ is ramified at every prime dividing ${\mathfrak{n}}^{-}$.

Then we have the divisibility

$\bigl{(}{\mathcal{L}}_{p}(g_{F}/M)\bigr{)}\supset{\rm
char}_{\Lambda_{M}}\bigl{(}\mathfrak{X}_{\rm ord}(g/M_{\infty})\bigr{)}$

in $\Lambda_{M}$.

###### Proof.

By [Wan15, Thm. 3], we have the divisibility

$\bigl{(}{\mathcal{L}}_{p}({\bf{g}}_{F}/M)\bigr{)}\supset{\rm
char}_{{\mathbb{I}}[\\![\Gamma_{M}]\\!]}\bigl{(}\mathfrak{X}_{\rm
ord}({\bf{g}}/M_{\infty})\bigr{)}$

in ${\mathbb{I}}[\\![\Gamma_{M}]\\!]$, where ${\bf{g}}_{F}$ denotes the
parallel weight Hida family passing through $g_{F}$ (cf. [Hid88], [Hid89]) and
${\mathbb{I}}$ is its coefficient ring. Since $\mathfrak{X}_{\rm
ord}({\bf{g}}/M_{\infty})$ specializes to $\mathfrak{X}_{\rm
ord}(g/M_{\infty})$ under the map induced by the specialization
${\mathbb{I}}\rightarrow{\mathcal{O}}$ corresponding to the $p$-ordinary
stabilization of $g_{F}$ (cf. [SU14, (3.5)]), and ${\mathcal{L}}_{p}(g_{F}/M)$
is defined as an analogous specialization of
${\mathcal{L}}_{p}({\bf{g}}_{F}/M)$, the assertion follows. ∎

######  Hypothesis 3.0.2.

  * (H1)

$\bar{\rho}_{g}|_{G_{F(\zeta_{p})}}$ is absolutely irreducible, and for $p=5$
the following case is excluded: the projective image $\bar{G}$ of
$\bar{\rho}_{g}|_{G_{F}}$ is isomorphic to ${\rm PGL}_{2}(\mathbb{F}_{p})$ and
the mod $p$ cyclotomic character factors through $G_{F}\rightarrow\bar{G}^{\rm
ab}\simeq\mathbb{Z}/2\mathbb{Z}$ (in particular $[F(\zeta_{5}):F]=2$).

  * (H2)

There is a minimal modular lifting of $\bar{\rho}_{g}|_{G_{F}}$ (cf. [Fuj06,
Def. 6.11]).

  * (H3)

For any finite place $v$ of $F$, if $\bar{\rho}_{g}|_{G_{F_{v}}}$ is
absolutely irreducible and $\bar{\rho}_{g}|_{I_{F_{v}}}$ is absolutely
reducible, then $q_{v}\not\equiv-1\pmod{p}$.

Here $I_{F_{v}}\subset G_{F_{v}}$ are inertia and decomposition groups at $v$,
respectively, and $q_{v}$ denotes the size of the residue field of $v$.

######  Remark 3.0.3.

  * (i)

Hypothesis (H1) implies that a certain Hecke algebra is Gorenstein (cf.
[Fuj06, Thm. 11.1]), hence one has ${\mathcal{L}}_{p}(g_{F}/M)\in\Lambda_{M}$.
(Note that here $\bar{\rho}_{g_{F}}$ is automatically $p$-distinguished in the
sense of _loc. cit._.)

  * (ii)

Under (H3), the exceptional case $0_{E}$ in [Fuj06, p. 16] does not occur, and
hence the results of [Fuj06] apply to the setting of [Wan15, §7–9] (cf.
[Fuj06, p. 57]).

  * (iii)

The case excluded by (H1) does not occur for $g$ corresponding to elliptic
curves (cf. [Fuj06, Prop. 9.8]).

##  4\. Main Conjectures over imaginary quadratic fields

We collect some known results in the direction of Conjecture 2.0.1 (and some
variants) in the case $F=\mathbb{Q}$. Although some of these results are known
under weaker hypotheses, here we shall assume that

(irrK) $\bar{\rho}_{g}$ is irreducible as $G_{K}$-representation,

where $K=M$ is imaginary quadratic inthis section.

We refer the reader to $\S\S{1.2}$ and ${1.4}$ of [CGS23] for a review of the
construction and interpolation property of the two-variable $p$-adic
$L$-functions $\mathcal{L}_{p}^{\rm PR}(g/K)\in\Lambda_{K}$ and
$\mathcal{L}_{p}^{\rm Gr}(g/K)\in\Lambda_{K}^{\rm ur}$ appearing below.

###  4.1. Two-variable Main Conjectures

######  Conjecture 4.1.1.

Let $g\in S_{2}(\Gamma_{0}(N))$ be a newform and $p>2$ a prime of good
ordinary reduction for $g$. Let $K$ be an imaginary quadratic field satisfying
(irrK). Then $\mathfrak{X}_{\rm ord}(g/K_{\infty})$ is $\Lambda_{K}$-torsion,
with

$\bigl{(}{{\mathcal{L}}}_{p}^{\rm PR}(g/K))={\rm
ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm ord}(g/K_{\infty})\bigr{)}.$

Note that it follows from the comparison of $p$-adic $L$-functions in [Wan15,
Prop. 84] that Conjecture 4.1.1 is nothing but Conjecture 3.0.1 in the case
$F=\mathbb{Q}$.

######  Conjecture 4.1.2.

Let $g\in S_{2}(\Gamma_{0}(N))$ be a newform and $p>2$ a prime of good
reduction for $g$. Let $K$ be an imaginary quadratic field satisfying (spl).
Then $\mathfrak{X}_{\rm Gr}(g/K_{\infty})$ is $\Lambda_{K}$-torsion, with

$\bigl{(}{{\mathcal{L}}}_{p}^{\rm Gr}(g/K)\bigr{)}={\rm
ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g/K_{\infty})\bigr{)}$

as ideals in $\Lambda_{K}^{\rm ur}$.

Underlying the following key result from [BSTW23] is a refinement of the
Beilinson–Flach classes of [KLZ17] and their explicit reciprocity laws,
allowing us to pass between the two preceding two-variable main conjectures.

######  Proposition 4.1.3.

Let $g\in S_{2}(\Gamma_{0}(N))$ be an elliptic newform, and $p\nmid 2N$ an
ordinary prime for $g$. Let $K$ be an imaginary quadratic field satisfying
(spl), $(D_{K},N)=1$, and (irrK). Then the following are equivalent:

  * (i)

$\mathfrak{X}_{\rm ord}(g/K_{\infty})$ is $\Lambda_{K}$-torsion, with

$\bigl{(}{{\mathcal{L}}}_{p}^{\rm PR}(g/K))\supset{\rm
ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm
ord}(g/K_{\infty})\bigr{)}\quad\textrm{in $\Lambda_{K}\otimes\mathbb{Q}_{p}$.
}$

  * (ii)

$\mathfrak{X}_{\rm Gr}(g/K_{\infty})$ is $\Lambda_{K}$-torsion, with

$\bigl{(}{{\mathcal{L}}}_{p}^{\rm Gr}(g/K)\bigr{)}\supset{\rm
ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g/K_{\infty})\bigr{)}\quad\textrm{in
$\Lambda_{K}^{\rm ur}\otimes\mathbb{Q}_{p}$.}$

The same conclusion holds for the opposite divisibilities, and before
inverting $p$. In particular, Conjecture 4.1.1 and Conjecture 4.1.2 are
equivalent.

###### Proof.

This is shown in [BSTW23] (cf. [CGS23, Prop. 3.2.1] or [Cas24, §3.3]) building
on a pair of four-term exact sequence coming from Poitou–Tate duality. ∎

Taking the direct sum of two pairs of four-term exact sequence as appeared in
the proof of Theorem 4.1.3 one deduces the following.

######  Corollary 4.1.4.

Let $g\in S_{2}(\Gamma_{0}(N))$ and $g^{\prime}\in
S_{2}(\Gamma_{0}(N^{\prime}))$ be newforms, $p\nmid 2NN^{\prime}$ an ordinary
prime for both $g$ and $g^{\prime}$, and $K$ an imaginary quadratic field
satisfying (spl), $(D_{K},NN^{\prime})=1$, and such that (irrK) holds for both
$g$ and $g^{\prime}$. Then the following are equivalent:

  * (i)

$\bigl{(}{\mathcal{L}}_{p}^{\rm PR}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
PR}(g^{\prime}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g^{\prime}/K_{\infty}))$.

  * (ii)

$\bigl{(}{\mathcal{L}}_{p}^{\rm Gr}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
Gr}(g^{\prime}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g^{\prime}/K_{\infty}))$.

Moreover, the same holds for the opposite divisibility.

###  4.2. Anticyclotomic Main Conjectures

A refinement of Kolyvagin’s methods (in the style of [How04] and [Nek07])
developed in [CGLS22, CGS23] yields the first part of the following result.

######  Theorem 4.2.1.

Let $g\in S_{2}(\Gamma_{0}(N))$ be an elliptic newform and $p$ an odd prime of
good ordinary reduction for $g$. Let $K$ be an imaginary quadratic field
satisfying (disc), (Heeg), and (irrK). Then the following hold:

  * (a)

Both $\mathfrak{X}_{\rm ord}(g/K_{\infty}^{-})$ and
$\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,T_{g}\otimes\Lambda_{K}^{-})$ have
$\Lambda_{K}^{-}$-rank one, and

${\rm ch}_{\Lambda_{K}^{-}}\bigl{(}\mathfrak{X}_{\rm
ord}(g/K_{\infty}^{-})_{\rm tor}\bigr{)}\supset{\rm
ch}_{\Lambda_{K}^{-}}\bigl{(}\mathrm{H}^{1}_{\mathcal{F}_{\Lambda}}(K,T_{g}\otimes\Lambda_{K}^{-})/(\bm{\kappa}_{1}^{\rm
Heeg})\bigr{)}^{2}\quad\textrm{in $\Lambda_{K}^{-}\otimes\mathbb{Q}_{p}$}.$

  * (b)

If $K$ also satisfies (spl), then $\mathfrak{X}_{\rm Gr}(g/K_{\infty}^{-})$ is
$\Lambda_{K}^{-}$-torsion, and

${\rm ch}_{\Lambda_{K}^{-}}(\mathfrak{X}_{\rm
Gr}(g/K_{\infty}^{-}))\Lambda_{K}^{-,{\rm ur}}\supset(\mathcal{L}_{p}^{\rm
BDP}(g/K))\quad\textrm{in $\Lambda_{K}^{-,{\rm ur}}\otimes\mathbb{Q}_{p}$.}$

Moreover, if (sur) holds, then both divisibilities hold integrally.

###### Proof.

Part (a) is contained in [CGS23, Thm. 5.5.2], and part (b) then follows from
[BCK21, Thm. 5.2]. Under (sur), part (a) follows from [How04, Thm. B], and
part (b) again from [BCK21, Thm. 5.2]. ∎

The following vanishing result following from Hida’s methods will also play an
important role our arguments.

######  Proposition 4.2.2.

Let $g\in S_{2}(\Gamma_{0}(N))$ be an elliptic newform with good reduction at
$p>2$, and suppose $K$ is an imaginary quadratic field satisfying (disc),
(Heeg), (spl), and (irrK). Then

$\mu({\mathcal{L}}_{p}^{\rm PR}(g/K))=\mu({\mathcal{L}}_{p}^{\rm
BDP}(g/K))=0.$

###### Proof.

By [Hsi14, Thm. B], ${\mathcal{L}}_{p}^{\rm BDP}(g/K)$ has vanishing
$\mu$-invariant. Since a direct comparison of the interpolation properties
shows that the projection of ${\mathcal{L}}_{p}^{\rm PR}(g/K)$ to
$\Lambda_{K}^{-,{\rm ur}}$ generates the same ideal as ${\mathcal{L}}_{p}^{\rm
BDP}(g/K)$ (see [CGS23, Prop. 1.4.5]), the result follows. ∎

##  5\. Base change

Let $g\in S_{2}(\Gamma_{0}(N))$ be an elliptic newform, and $p>3$ a good
ordinary prime for $g$ such that (irrQ) holds. Let $M/F$ be a CM quadratic
extension of a real quadratic field $F$ for the form $M=FK$ with $K$ an
imaginary quadratic field. For the $\mathbb{Z}_{p}^{2}$-extension
$\tilde{K}_{\infty}:=FK_{\infty}$ of $M$, put
$\tilde{\Lambda}_{K}:={\mathcal{O}}[\\![\operatorname{Gal}(\tilde{K}_{\infty}/M)]\\!]\simeq\Lambda_{K}$
and let

$\pi_{K}:\Lambda_{M}\rightarrow\Lambda_{K}$

denote the map arising from the projection
$\operatorname{Gal}(M_{\infty}/M)\twoheadrightarrow\operatorname{Gal}(\tilde{K}_{\infty}/M)$.

######  Lemma 5.0.1.

We have the divisibility

$\pi_{K}\bigl{(}{\rm ch}_{\Lambda_{M}}(\mathfrak{X}_{\rm
ord}(g/M_{\infty}))\bigr{)}\supset{\rm
ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm ord}(g/K_{\infty})\bigr{)}\cdot{\rm
ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm ord}(g^{F}/K_{\infty})\bigr{)}$

in $\Lambda_{K}$, where $g^{F}$ is the twist of $g$ by the quadratic character
corresponding to $F/\mathbb{Q}$.

###### Proof.

All the references are to [SU14]. It follows readily from Shapiro’s lemma, and
Propositions 3.7 and 3.6 that the restriction map
$\mathrm{H}^{1}(\tilde{K}_{\infty},T_{g})\rightarrow\mathrm{H}^{1}(M_{\infty},T_{g})[I_{K}]$,
where $I_{K}=\ker(\pi_{K})$, induces isomorphisms

$\displaystyle\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(M,\mathcal{M}_{g})[I_{K}]$
$\displaystyle\simeq\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(M,T_{g}\otimes\tilde{\Lambda}_{K}^{\vee})$
$\displaystyle\simeq\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(M,T_{g}\otimes\Lambda_{K}^{\vee})\oplus\mathrm{H}^{1}_{\mathcal{F}_{\rm
ord}}(M,T_{g}\otimes\chi_{F}\otimes\Lambda_{K}^{\vee})$

where $\eta_{F}$ is the non-trivial character of
$\operatorname{Gal}(F/\mathbb{Q})\simeq\operatorname{Gal}(M/K)$ (so
$T_{g}\otimes\eta_{F}\simeq T_{g^{F}}$). Since by Corollary 3.8 we have the
divisibility

$\pi_{K}\bigl{(}{\rm ch}_{\Lambda_{M}}(\mathfrak{X}_{\rm
ord}(g/M_{\infty}))\bigr{)}\supset{\rm
ch}_{\Lambda_{K}}\bigl{(}\mathfrak{X}_{\rm ord}(g/M_{\infty})/I_{K}\bigr{)}$

in $\Lambda_{M}/I_{K}\simeq\Lambda_{K}$, the result follows. ∎

######  Lemma 5.0.2.

In the setting of Lemma 5.0.1, we have

$\bigl{(}\pi_{K}({\mathcal{L}}_{p}(g_{F}/M))\bigr{)}=({\mathcal{L}}_{p}^{\rm
PR}(g/K)\cdot{\mathcal{L}}_{p}^{\rm PR}(g^{F}/K)\bigr{)}$

in $\Lambda_{K}\otimes\mathbb{Q}_{p}$.

###### Proof.

This just follows from a direct comparison of the interpolation properties
(cf. [Wan21, Prop. 84]). ∎

###  5.1. Proofs of main results

We can now complete the proof of the results stated in §1. We begin with the
following key.

######  Proposition 5.1.1.

Let $g\in S_{2}(\Gamma_{0}(N))$ be an elliptic newform, and $p\geq 5$ a prime
of good ordinary reduction for $g$. Let $K$ be an imaginary quadratic field
satisfying (disc), (Heeg), (spl), and (irrK). Suppose $F$ is a real quadratic
field of discriminant $D_{F}$ satisfying the following hypotheses:

  * (i)

$p$ is inert in $F$.

  * (ii)

$D_{F}$ is odd and every prime dividing $D_{F}$ splits in $K$.

  * (iii)

Every prime $\ell|N$ is inert in $F$ if $\ell\equiv-1\pmod{p}$, and is split
in $F$ otherwise.

  * (iv)

$\eqref{irr_M}$ holds for $M=FK$.

  * (v)

$\bar{\rho}_{g}|_{G_{F(\zeta_{p})}}$ is irreducible.

  * (vi)

If $p=5$, then $F\neq\mathbb{Q}(\zeta_{5})^{+}$.

Then we have the divisibilities

$\bigl{(}{\mathcal{L}}_{p}^{\rm PR}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
PR}(g^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g^{F}/K_{\infty}))\quad\textrm{in $\Lambda_{K}$,}$

and

$\bigl{(}{\mathcal{L}}_{p}^{\rm Gr}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
Gr}(g^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g^{F}/K_{\infty}))\quad\textrm{in $\Lambda_{K}^{\rm ur}$.}$

###### Proof.

Note that the CM quadratic extension $M/F$ and the residual representation
$\bar{\rho}_{g}|_{G_{F}}$ satisfy the hypotheses of Theorem 3.0.1. Indeed, the
conditions $(pN,D_{F})=1$ and $(N\mathcal{O}_{F},D_{M/F})=1$ are clear, and
hypotheses (i)–(iii) and (Heeg) imply (splF) and ($\Delta$). Further (iii)
implies ${\mathfrak{n}}^{-}={\mathcal{O}}_{F}$ and so the hypotheses (iii) and
(iv) of Theorem 3.0.1 are vacuous. Hypothesis 3.0.2 is readily verified: (H1)
follows by (v) and (vi), (H2) follows as in [Wan21, Thm. 103] (indeed, by our
choice of $F$, the base change to $F$ of a minimal modular lifting of
$\bar{\rho}_{g}$ gives a minimal modular lifting of $\bar{\rho}_{g_{F}}$), and
(H3) also follows by condition (iii) on $F$.

Now Theorem 3.0.1 leads to the divisibility

$\bigl{(}\pi_{K}({\mathcal{L}}_{p}(g_{F}/M))\bigr{)}\supset\pi_{K}\bigl{(}{\rm
ch}_{\Lambda_{M}}\bigl{(}\mathfrak{X}_{\rm ord}(g/M_{\infty}))\bigr{)}$

in $\Lambda_{K}$, and so by Lemmas 5.0.1 and 5.0.2, we have

$\bigl{(}{\mathcal{L}}_{p}^{\rm PR}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
PR}(g^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g/K))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm ord}(g^{F}/K))$

in $\Lambda_{K}\otimes\mathbb{Q}_{p}$. In turn Proposition 4.1.3 (and its
proof) implies

(5.1) $\bigl{(}{\mathcal{L}}_{p}^{\rm Gr}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
Gr}(g^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g/K))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g^{F}/K))$

in $\Lambda_{K}^{\rm ur}\otimes\mathbb{Q}_{p}$. Since
$\mu({\mathcal{L}}_{p}^{\rm Gr}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
Gr}(g^{F}/K))=0$ by Proposition 4.2.2, the divisibility (5.1) holds integrally
in $\Lambda_{K}^{\rm ur}$. By again appealing to Proposition 4.1.3, the proof
concludes. ∎

######  Remark 5.1.2.

In the case $p=5$, if $g$ corresponds to an elliptic curve $E/\mathbb{Q}$,
then the condition $F\neq\mathbb{Q}(\zeta_{5})^{+}$ is inessential (cf. Remark
3.0.3(iii)).

The existence of $F$ satisfying the conditions in Proposition 5.1.1 is easily
verified:

######  Lemma 5.1.3.

Let $(g,p,K)$ be as in Proposition 5.1.1. Then there exists a real quadratic
field $F$ satisfying ${\rm(i)}$–${\rm(vi)}$.

###### Proof.

Note that (i), (ii), and (iii) are independent splitting conditions which hold
for a positive proportion of real quadratic fields $F$. In view of (irrK), if
$FK$ is not a subfield of the splitting field of $\rho_{g}|_{G_{K}}$, then
$\eqref{irr_M}$ holds for $M=FK$. For such $F$, if
$\bar{\rho}_{g}|_{G_{F(\zeta_{p})}}$ is reducible, then
$\bar{\rho}_{g}|_{G_{F}}$ is induced by an index $2$ subgroup $G_{L}$ of
$G_{F}$ which contains $G_{F(\zeta_{p})}$; but this forces $p=5$ and
$F=\mathbb{Q}(\zeta_{5})^{+}$, so the result follows. ∎

###### Proof of Theorem 1.1.2.

Pick an imaginary quadratic field $K$ and a real quadratic field $F$ as in
Lemma 5.1.3. Then by Proposition 5.1.1,

(5.2) $\bigl{(}{\mathcal{L}}_{p}^{\rm PR}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
PR}(g^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g^{F}/K_{\infty})).$

By [CGS23, Prop. 1.2.4] and Propositions 3.6 and 3.9 in [SU14], taking the
image under the maps induced by the projection
$\pi_{+}:\Gamma_{K}\twoheadrightarrow\Gamma_{K}^{+}$, from (5.2) we get the
divisibilities

(5.3) $\displaystyle\bigl{(}{\mathcal{L}}_{p}(g/\mathbb{Q})$
$\displaystyle\cdot{\mathcal{L}}_{p}(g^{K}/\mathbb{Q})\cdot{\mathcal{L}}_{p}(g^{F}/\mathbb{Q})\cdot{\mathcal{L}}_{p}(g^{FK}/\mathbb{Q})\bigr{)}$
$\displaystyle\supset\pi_{+}\bigl{(}{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
ord}(g^{F}/K_{\infty}))\bigr{)}$ $\displaystyle\supset{\rm
ch}_{\Lambda}(\mathfrak{X}_{\rm ord}(g/\mathbb{Q}_{\infty}))\cdot{\rm
ch}_{\Lambda}(\mathfrak{X}_{\rm ord}(g^{K}/\mathbb{Q}_{\infty}))\cdot{\rm
ch}_{\Lambda}(\mathfrak{X}_{\rm ord}(g^{F}/\mathbb{Q}_{\infty}))\cdot{\rm
ch}_{\Lambda}(\mathfrak{X}_{\rm ord}(g^{FK}/\mathbb{Q}_{\infty}))$

in $\Lambda_{K}^{+}\simeq\Lambda$. On the other hand, by [Kat04] we have the
divisibility

(5.4) $\bigl{(}{\mathcal{L}}_{p}(g/\mathbb{Q})\bigr{)}\subset{\rm
ch}_{\Lambda}(\mathfrak{X}_{\rm ord}(g/\mathbb{Q}_{\infty})),$

and similarly for the twists $g^{K}$, $g^{F}$, and $g^{FK}$. Since a proper
divisibility in (5.4) would contradict (5.3), the result follows. ∎

###### Proof of Theorem 1.2.2 and Theorem 1.2.3.

Pick a real quadratic field $F$ as in Lemma 5.1.3. Then by Theorem 5.1.1,

(5.5) $\bigl{(}{\mathcal{L}}_{p}^{\rm Gr}(g/K)\cdot{\mathcal{L}}_{p}^{\rm
Gr}(g^{F}/K)\bigr{)}\supset{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g^{F}/K_{\infty})).$

By [CGS23, Prop. 1.4.5] and [JSW17, Cor. 3.4.2], taking the image under the
maps induced by the projection
$\pi_{-}:\Gamma_{K}\twoheadrightarrow\Gamma_{K}^{-}$, from (5.5) we get the
divisibilities

$\displaystyle\bigl{(}{\mathcal{L}}_{p}^{\rm
BDP}(g/K)\cdot{\mathcal{L}}_{p}^{\rm BDP}(g^{F}/K)\bigr{)}$
$\displaystyle\supset\pi_{-}\bigl{(}{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}^{-}}(\mathfrak{X}_{\rm
Gr}(g^{F}/K_{\infty}))\bigr{)}$ $\displaystyle\supset{\rm
ch}_{\Lambda_{K}^{-}}(\mathfrak{X}_{\rm Gr}(g/K_{\infty}^{-}))\cdot{\rm
ch}_{\Lambda_{K}^{-}}(\mathfrak{X}_{\rm Gr}(g^{F}/K_{\infty}^{-}))$

in $\Lambda_{K}^{-,{\rm ur}}$. Together with Theorem 4.2.1, this concludes the
proof. ∎

###### Proof of Theorem 1.4.1.

Pick an imaginary quadratic field $K$ and two different real quadratic fields
$F,F^{\prime}$ as in Lemma 5.1.3. For each of the two pairs $(K,F)$,
$(K,F^{\prime})$, the equalities of characteristic ideals of Theorem 1.2.3,
the divisibility (5.5), and the nonvanishing of the $p$-adic $L$-functions
$\mathcal{L}_{p}^{\rm BDP}(g^{\cdot}/K)$ yields the equalities

$\displaystyle\bigl{(}{\mathcal{L}}_{p}^{\rm
Gr}(g/K)\cdot{\mathcal{L}}_{p}^{\rm Gr}(g^{F}/K)\bigr{)}$ $\displaystyle={\rm
ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g/K_{\infty}))\cdot{\rm
ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g^{F}/K_{\infty})),$
$\displaystyle\bigl{(}{\mathcal{L}}_{p}^{\rm
Gr}(g/K)\cdot{\mathcal{L}}_{p}^{\rm Gr}(g^{F^{\prime}}/K)\bigr{)}$
$\displaystyle={\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g^{F^{\prime}}/K_{\infty})),$

and therefore

$\bigl{(}{\mathcal{L}}_{p}^{\rm Gr}(g/K)^{2}\cdot{\mathcal{L}}_{p}^{\rm
Gr}(g^{F}/K)\cdot{\mathcal{L}}_{p}^{\rm Gr}(g^{F^{\prime}}/K)\bigr{)}={\rm
ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g/K_{\infty}))^{2}\cdot{\rm
ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g^{F}/K_{\infty}))\cdot{\rm
ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm Gr}(g^{F^{\prime}}/K_{\infty})).$

When $\mathcal{V}=\emptyset$, the argument in the proof of Theorem 1.2.3
applies for any real quadratic $F_{0}$ satisfying conditions (i)-(ii) and
(iv)-(vi) in Lemma 5.1.3, but not necessarily (iii). Thus taking $F_{0}$ to be
the third real quadratic field in the compositum $FF^{\prime}$, as above we
obtain the equality

$\bigl{(}{\mathcal{L}}_{p}^{\rm Gr}(g^{F}/K)\cdot{\mathcal{L}}_{p}^{\rm
Gr}(g^{F^{\prime}}/K)\bigr{)}={\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g^{F}/K_{\infty}))\cdot{\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g^{F^{\prime}}/K_{\infty})).$

The combination of the last two equalities immediately gives
$({\mathcal{L}}_{p}^{\rm Gr}(g/K))={\rm ch}_{\Lambda_{K}}(\mathfrak{X}_{\rm
Gr}(g/K_{\infty}))$, which together with Proposition 4.1.3 yields the result.
∎

## References

  * [BCGS23] Ashay Burungale, Francesc Castella, Giada Grossi, and Christopher Skinner.  Non-vanishing of Kolyvagin systems and Iwasawa theory.  2023\.  preprint, arXiv:2312.09301.
  * [BCK21] Ashay Burungale, Francesc Castella, and Chan-Ho Kim.  A proof of Perrin-Riou’s Heegner point main conjecture.  Algebra Number Theory, 15(7):1627–1653, 2021.
  * [BDP13] Massimo Bertolini, Henri Darmon, and Kartik Prasanna.  Generalized Heegner cycles and $p$-adic Rankin $L$-series.  Duke Math. J., 162(6):1033–1148, 2013.  With an appendix by Brian Conrad.
  * [Ber95] Massimo Bertolini.  Selmer groups and Heegner points in anticyclotomic $\mathbb{Z}_{p}$-extensions.  Compositio Math., 99(2):153–182, 1995.
  * [BST21] Ashay A. Burungale, Christopher Skinner, and Ye Tian.  The Birch and Swinnerton-Dyer conjecture: a brief survey.  In Nine mathematical challenges—an elucidation, volume 104 of Proc. Sympos. Pure Math., pages 11–29. Amer. Math. Soc., Providence, RI, [2021] ©2021.
  * [BSTW23] Ashay Burungale, Christopher Skinner, Ye Tian, and Xin Wan.  Zeta elements for elliptic curves and applications.  preprint, 2023.
  * [BT20] Ashay A. Burungale and Ye Tian.  $p$-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin.  Invent. Math., 220(1):211–253, 2020.
  * [Bur17] Ashay A. Burungale.  On the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$, II: Shimura curves.  J. Inst. Math. Jussieu, 16(1):189–222, 2017.
  * [Cas17] Francesc Castella.  $p$-adic heights of Heegner points and Beilinson-Flach classes.  J. Lond. Math. Soc. (2), 96(1):156–180, 2017.
  * [Cas24] Francesc Castella.  On the Iwasawa theory of elliptic curves at Eisenstein primes.  2024\.  preprint, arXiv:2404.12644.
  * [CGLS22] Francesc Castella, Giada Grossi, Jaehoon Lee, and Christopher Skinner.  On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes.  Invent. Math., 227:517–580, 2022.
  * [CGS23] Francesc Castella, Giada Grossi, and Christopher Skinner.  Mazur’s main conjecture at Eisenstein primes.  2023\.  preprint, arXiv:2303.04373.
  * [CH18] Francesc Castella and Ming-Lun Hsieh.  Heegner cycles and $p$-adic $L$-functions.  Math. Ann., 370(1-2):567–628, 2018.
  * [Fuj06] Kazuhiro Fujiwara.  Deformation rings and Hecke algebras in the totally real case, 2006\.
  * [Gre99] Ralph Greenberg.  Iwasawa theory for elliptic curves.  In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in Math., pages 51–144. Springer, Berlin, 1999\.
  * [GV00] Ralph Greenberg and Vinayak Vatsal.  On the Iwasawa invariants of elliptic curves.  Invent. Math., 142(1):17–63, 2000.
  * [Hid88] Haruzo Hida.  On $p$-adic Hecke algebras for ${\rm GL}_{2}$ over totally real fields.  Ann. of Math. (2), 128(2):295–384, 1988.
  * [Hid89] Haruzo Hida.  On nearly ordinary Hecke algebras for ${\rm GL}(2)$ over totally real fields.  In Algebraic number theory, volume 17 of Adv. Stud. Pure Math., pages 139–169. Academic Press, Boston, MA, 1989.
  * [Hid10] H. Hida.  The Iwasawa $\mu$-invariant of $p$-adic Hecke $L$-functions.  Ann. of Math. (2), 172(1):41–137, 2010.
  * [How04] Benjamin Howard.  The Heegner point Kolyvagin system.  Compos. Math., 140(6):1439–1472, 2004.
  * [Hsi14] Ming-Lun Hsieh.  Special values of anticyclotomic Rankin-Selberg $L$-functions.  Doc. Math., 19:709–767, 2014.
  * [JSW17] Dimitar Jetchev, Christopher Skinner, and Xin Wan.  The Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one.  Camb. J. Math., 5(3):369–434, 2017.
  * [Kat04] Kazuya Kato.  $p$-adic Hodge theory and values of zeta functions of modular forms.  Astérisque, 295:117–290, 2004.
  * [Kim22] Chan-Ho Kim.  The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves.  2022\.  preprint, arXiv:2203.12159.
  * [KLZ17] Guido Kings, David Loeffler, and Sarah Livia Zerbes.  Rankin-Eisenstein classes and explicit reciprocity laws.  Camb. J. Math., 5(1):1–122, 2017.
  * [LLZ15] Antonio Lei, David Loeffler, and Sarah Livia Zerbes.  Euler systems for modular forms over imaginary quadratic fields.  Compos. Math., 151(9):1585–1625, 2015.
  * [Maz72] Barry Mazur.  Rational points of abelian varieties with values in towers of number fields.  Invent. Math., 18:183–266, 1972.
  * [MSD74] Barry Mazur and Peter Swinnerton-Dyer.  Arithmetic of Weil curves.  Invent. Math., 25:1–61, 1974.
  * [Nek07] Jan Nekovář.  The Euler system method for CM points on Shimura curves.  In $L$-functions and Galois representations, volume 320 of London Math. Soc. Lecture Note Ser., pages 471–547. Cambridge Univ. Press, Cambridge, 2007.
  * [PR87] Bernadette Perrin-Riou.  Fonctions $L$ $p$-adiques, théorie d’Iwasawa et points de Heegner.  Bull. Soc. Math. France, 115(4):399–456, 1987.
  * [PR88] Bernadette Perrin-Riou.  Fonctions $L$ $p$-adiques associées à une forme modulaire et à un corps quadratique imaginaire.  J. London Math. Soc. (2), 38(1):1–32, 1988.
  * [Ser72] Jean-Pierre Serre.  Propriétés galoisiennes des points d’ordre fini des courbes elliptiques.  Invent. Math., 15:259–331, 1972.
  * [Ski20] Christopher Skinner.  A converse to a theorem of Gross, Zagier, and Kolyvagin.  Ann. of Math. (2), 191(2):329–354, 2020.
  * [SU14] Christopher Skinner and Eric Urban.  The Iwasawa main conjectures for $\rm GL_{2}$.  Invent. Math., 195(1):1–277, 2014.
  * [Wan15] Xin Wan.  The Iwasawa main conjecture for Hilbert modular forms.  Forum Math. Sigma, 3:Paper No. e18, 95, 2015.
  * [Wan20] Xin Wan.  Iwasawa main conjecture for Rankin-Selberg $p$-adic $L$-functions.  Algebra Number Theory, 14(2):383–483, 2020.
  * [Wan21] Xin Wan.  Heegner Point Kolyvagin System and Iwasawa Main Conjecture.  Acta Math. Sin. (Engl. Ser.), 37(1):104–120, 2021.
  * [Zha14] Wei Zhang.  The Birch–Swinnerton-Dyer conjecture and Heegner points: a survey.  In Current developments in mathematics 2013, pages 169–203. Int. Press, Somerville, MA, 2014.