# Refined conjectures on Fitting ideals of Selmer groups over
$\mathbf{Z}_{p}^{2}$-extensions

Cédric Dion  https://orcid.org/0000-0002-5593-7824 Département de
Mathématiques et de Statistique, Université Laval, Pavillion Alexandre-Vachon,
1045 Avenue de la Médecine, Québec, QC, Canada G1V 0A6
<EMAIL_ADDRESS>
###### Abstract.

Let $p>3$ be a prime number and $K$ be an imaginary quadratic field where $p$
splits. Let $K_{\infty}$ be the $\mathbf{Z}_{p}^{2}$-extension of $K$ and let
$K_{n}$ be a finite subextension of $K_{\infty}/K$. Let $E$ be an elliptic
curve with good ordinary reduction at $p$. Under some hypotheses, we show that
the Mazur–Tate element attached to $E$ over $K_{n}$ by S. Haran generates the
Fitting ideal of the dual Selmer group of $E$ over $K_{n}$.

###### Key words and phrases:

Elliptic curves, Iwasawa theory, Mazur–Tate elements

###### 2020 Mathematics Subject Classification:

Primary: 11R23; Secondary: 11G05, 11R20

##  1\. Introduction

Let $E$ be elliptic curve with good ordinary reduction at an odd prime $p$ and
consider $\mathbf{Q}_{\infty}$, the cyclotomic $\mathbf{Z}_{p}$-extension of
$\mathbf{Q}$. Let $\Sigma$ be a finite set of places of $\mathbf{Q}$ including
$p$, $\infty$ and the bad reduction primes of $E$. If
$\mathbf{Q}^{(n)}/\mathbf{Q}$ is a finite subextension inside
$\mathbf{Q}_{\infty}$, we attach to $E$ its Selmer group over
$\mathbf{Q}^{(n)}$ defined by the kernel

${\mathrm{Sel}}(\mathbf{Q}^{(n)},E[p^{\infty}])\colonequals\ker\left(H^{1}(\mathbf{Q}_{\Sigma}/\mathbf{Q}^{(n)},E[p^{\infty}])\to\prod_{v}\frac{H^{1}(\mathbf{Q}_{v}^{(n)},E[p^{\infty}])}{E(\mathbf{Q}_{v}^{(n)})\otimes\mathbf{Q}_{p}/\mathbf{Z}_{p}}\right)$

where $\mathbf{Q}_{\Sigma}$ is the maximal extension of $\mathbf{Q}$
unramified outside $\Sigma$, $v$ runs over all the finite places of
$\mathbf{Q}^{(n)}$ above the places in $\Sigma$ and $\mathbf{Q}_{v}^{(n)}$ is
the completion of $\mathbf{Q}^{(n)}$ at $v$. On the analytic side, let $f$ be
the modular form attached to $E$ by modularity. Denote by $\mu_{p^{n}}$ the
set of $p^{n}$-th roots of unity and consider the Galois group
$G^{\prime}_{n+1}\colonequals{\mathrm{Gal}}(\mathbf{Q}(\mu_{p^{n+1}})/\mathbf{Q})/\\{\pm
1\\}\cong(\mathbf{Z}/p^{n+1}\mathbf{Z})^{\times}/\\{\pm 1\\}$. Let
$\sigma_{a}$ denote the element corresponding to
$a\in(\mathbf{Z}/p^{n+1}\mathbf{Z})^{\times}/\\{\pm 1\\}$. Then, one can
define an element in the group ring $\mathbf{Z}_{p}[G^{\prime}_{n+1}]$ by the
formula

$\theta^{\prime}_{n+1}(f)\colonequals\sum_{a\in(\mathbf{Z}/p^{n+1}\mathbf{Z})^{\times}/\\{\pm
1\\}}\left[\frac{a}{p^{n+1}}\right]^{+}\cdot\sigma_{a}$

where $\left[\frac{a}{b}\right]^{+}$ is a modular symbol scaled by an
appropriate choice of period. The Mazur–Tate element $\theta_{n}(f)$ of $f$ at
$\mathbf{Q}^{(n)}$ is defined to be the image of $\theta^{\prime}_{n+1}(f)$
inside
$\Lambda_{n}\colonequals\mathbf{Z}_{p}[{\mathrm{Gal}}(\mathbf{Q}^{(n)}/\mathbf{Q})]$.
While the Iwasawa main conjecture relates the Pontryagin dual of the Selmer
group of $E$ over $\mathbf{Q}_{\infty}$ with the $p$-adic $L$-function of $E$,
refined Iwasawa theory aims to relate algebraic and analytic objects over
$\mathbf{Q}^{(n)}$. If $M$ is a finitely presented $\Lambda_{n}$-module, one
studies its Fitting ideal $\mathrm{Fitt}_{\Lambda_{n}}(M)$ (see section 6.1
for the definition). Mazur and Tate [17] gave the following conjecture:

######  Conjecture 1.1 (Weak Mazur–Tate conjecture).

Assume that $a_{p}\not\equiv 1\bmod p$, then

$\theta_{n}(f)\in\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Hom}_{\mathbf{Z}_{p}}({\mathrm{Sel}}(\mathbf{Q}^{(n)},E[p^{\infty}]),\mathbf{Q}_{p}/\mathbf{Z}_{p})\right).$

A stronger form of this conjecture formulated by Kurihara [15] asserts that
$\theta_{n}(f)$ in fact generates the whole Fitting ideal.

######  Conjecture 1.2 (Strong Mazur–Tate conjecture).

Assume that $a_{p}\not\equiv 1\bmod p$ and that $p$ does not divide the
Tamagawa number of $E$, then

$(\theta_{n}(f))=\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Hom}_{\mathbf{Z}_{p}}({\mathrm{Sel}}(\mathbf{Q}^{(n)},E[p^{\infty}]),\mathbf{Q}_{p}/\mathbf{Z}_{p})\right).$

The strong form of the Mazur–Tate conjecture was proven by Kim and Kurihara in
[14].

######  Theorem 1.3 (Kim-Kurihara, Strong Mazur–Tate conjecture).

Let $E$ be an elliptic curve over $\mathbf{Q}$ with good ordinary reduction at
an odd prime $p$. Let $\overline{\rho}$ be the residual Galois representation
attached to $E$. Assume that $\overline{\rho}$ is surjective if $E$ is non-CM.
Assume that $a_{p}(E)\not\equiv 1\bmod p$ and that $p$ does not divide the
Tamagawa number of $E$. Further assume that the Iwasawa main conjecture over
$\mathbf{Q}_{\infty}$ holds. Then,

$(\theta_{n}(f))=\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Hom}_{\mathbf{Z}_{p}}({\mathrm{Sel}}(\mathbf{Q}^{(n)},E[p^{\infty}]),\mathbf{Q}_{p}/\mathbf{Z}_{p})\right).$

In this note, we investigate Mazur–Tate type conjectures when $\mathbf{Q}$ is
replaced by an imaginary quadratic field $K$ and the cyclotomic
$\mathbf{Z}_{p}$-extension $\mathbf{Q}_{\infty}$ is replaced by $K_{\infty}$,
the unique $\mathbf{Z}_{p}^{2}$-extension of $K$. The setup is the following:
We fix a prime number $p>3$ and still consider an elliptic curve $E$ defined
over $\mathbf{Q}$ with good ordinary reduction at $p$ and conductor $N_{E}$.
We let $K$ be an imaginary quadratic field where
$(p)={\mathfrak{p}}{\mathfrak{q}}$ splits and we require that both the class
number of $K$ and the discriminant of $K$ are corpime to $p$. If
$\mathfrak{f}$ is a modulus of $K$, let $K(\mathfrak{f})$ denote the ray class
field of $K$ of conductor $\mathfrak{f}$. By using an adelic description of
modular symbols, Haran [10] constructed analogues of Mazur–Tate elements at
$K({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})$ for integers $n,m\geq-1$. These
elements will be denoted by $\Theta_{n,m}$ and live in the group ring
$\mathbf{Z}_{p}[{\mathrm{Gal}}(K({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})/K)/{\mathrm{Gal}}(K(p)/K)]$
after scaling by a period. Let
$K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})$ be the subfield of
$K({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})$ fixed by
${\mathrm{Gal}}(K(p)/K)$. Put
$\Lambda\colonequals\mathbf{Z}_{p}\llbracket{\mathrm{Gal}}(K_{\infty}/K)\rrbracket$.
To link the Mazur–Tate element $\Theta_{n,m}$ with the Selmer group of $E$
over $K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})$, we will need a
two-variable Iwasawa main conjecture.

######  Conjecture 1.4 (Iwasawa main conjecture).

Let $L_{p}^{\mathrm{Hi}}(E/K)$ be Hida’s two-variable $p$-adic $L$-function
attached to $E$ over $K_{\infty}$. Then,
${\mathrm{Sel}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}),E[p^{\infty}])$
is a cotorsion $\Lambda$-module and

$(L_{p}^{\mathrm{Hi}}(E/K))=\mathrm{Char}_{\Lambda}\left(\mathrm{Hom}_{\mathbf{Z}_{p}}({\mathrm{Sel}}(K_{\infty},E[p^{\infty}]),\mathbf{Q}_{p}/\mathbf{Z}_{p})\right)$

as ideals in $\Lambda$.

Let
$\Lambda_{n,m}\colonequals\mathbf{Z}_{p}[{\mathrm{Gal}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})/K)]$.
Our theorem is the following:

######  Theorem 1.5.

Assume that the Iwasawa main conjecture 1.4 holds. Also suppose that
$a_{p}\not\equiv 1\bmod p$, $E(K)[p]$ is trivial and $E(K_{v})[p]$ is trivial
for every prime $v$ where $E$ has bad reduction. Then, for $n+m\geq 1$, we
have

$(\Theta_{n,m})=\mathrm{Fitt}_{\Lambda_{n,m}\otimes\mathbf{Q}_{p}}\left(\mathrm{Hom}_{\mathbf{Z}_{p}}({\mathrm{Sel}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}),E[p^{\infty}]),\mathbf{Q}_{p}/\mathbf{Z}_{p})\right)$

as ideals in $\Lambda_{n,m}\otimes\mathbf{Q}_{p}$.

We end this section by reviewing previous works on Mazur–Tate type
conjectures. In their work [14], Kim and Kurihara also showed that a modified
version of the weak Mazur–Tate conjecture, namely that

$(\theta_{n}(f),\nu_{n,n-1}(\theta_{n-1}(f))\subseteq\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Hom}_{\mathbf{Z}_{p}}({\mathrm{Sel}}(\mathbf{Q}^{(n)},E[p^{\infty}]),\mathbf{Q}_{p}/\mathbf{Z}_{p})\right),$

also holds for elliptic curves with supersingular reduction at $p$ under mild
hypotheses. Here, $\nu_{n,n-1}$ is a trace map $\Lambda_{n-1}\to\Lambda_{n}$.
Moreover, they give sufficient conditions so that the inclusion becomes an
equality. Bley and Marcias Castillo [1, Theorem 2.12] also showed a form of
conjecture 1.2 for abelian varieties over a cyclic extension of a base number
field. Emerton, Pollack and Weston [7] proved a form of conjecture 1.1 for
Mazur–Tate elements coming from newforms using the $p$-adic Langland
correspondance and Kato’s zeta element. Kataoka also proved conjecture 1.1
over more general number field by using an equivariant point of view in his
PhD thesis. C.H. Kim proved an anticyclotomic analogue of conjecture 1.1 for
$p$-ordinary modular forms over a large class of cyclic ring class extensions
of an imaginary quadratic field [12] and for supersingular elliptic curves
along finite layers of the anticyclotomic $\mathbf{Z}_{p}$-extension of an
imaginary quadratic field [13]. See also [2] and the works of Ota [20], [21].

Acknowledgements: The author is deeply grateful to his PhD advisor Antonio Lei
for suggesting the problem and for numerous discussions on the topic. The
author also thanks Kâzım Büyükboduk, Bastien Desrochers, Sören Kleine, David
Loeffler and Katharina Müller for helpful discussions. This research is
supported by the Fond de Recherche du Québec Nature et Technologie’s 3rd cycle
scholarship.

##  2\. Notations

Fix an odd prime number $p>3$. Let $K$ be an imaginary quadratic number field
where $(p)={\mathfrak{p}}{\mathfrak{q}}$ splits. Let $K_{\infty}$ be the
compositum of all $\mathbf{Z}_{p}$-extensions of $K$ and let
$\Gamma\colonequals{\mathrm{Gal}}(K_{\infty}/K)\cong\mathbf{Z}_{p}^{2}$. We
will assume that the class number of $K$ is coprime to $p$. Given a modulus
$\mathfrak{f}$ of $K$, we denote by $K(\mathfrak{f})$ its ray class field of
conductor $\mathfrak{f}$. Let $S=\\{{\mathfrak{p}},{\mathfrak{q}}\\}$ denote
the set of primes above $p$ in $K$. If $\Sigma$ is any finite set of places of
$K$, let $G_{K,\Sigma}$ be the Galois group of the maximal algebraic extension
of $K$ unramified outside $\Sigma$. For $v\in S$, let $G_{v}$ denote the
Galois group of the extension $K(v^{\infty})\cap K_{\infty}/K$ and note that
$\Gamma\cong G_{\mathfrak{p}}\times G_{{\mathfrak{q}}}$. We will use the
symbol $\Lambda$ for the Iwasawa algebra
$\mathbf{Z}_{p}\llbracket\Gamma\rrbracket\colonequals\varprojlim_{H}\mathbf{Z}_{p}[\Gamma/H]$
where $H$ runs through the normal open subgroups of $\Gamma$. We fix
$\gamma_{\mathfrak{p}}$ (resp. $\gamma_{{\mathfrak{q}}}$) a topological
generator of $G_{\mathfrak{p}}$ (resp. $G_{{\mathfrak{q}}}$). Then, the
Iwasawa algebra $\Lambda$ may be identified with the power series ring
$\mathbf{Z}_{p}\llbracket\gamma_{\mathfrak{p}}-1,\gamma_{{\mathfrak{q}}}-1\rrbracket$.
For an integer $n\geq 0$, define $\omega_{n}(1+X)\colonequals(1+X)^{p^{n}}-1$.
For an integer $n\geq 1$, define
$\Phi_{n}(1+X)\colonequals\sum_{k=0}^{p-1}(1+X)^{kp^{n-1}}$ and
$\Phi_{0}(1+X)\colonequals 1+X$. Let $\iota:\Lambda\to\Lambda$ be the
involution induced by $\sigma\mapsto\sigma^{-1}$ for $\sigma\in\Gamma$. If $M$
is any $\Lambda$-module, we write $M^{\iota}$ for the same module $M$ but with
$\Lambda$-action given by $\lambda\cdot m\colonequals\iota(\lambda)m$ for all
$m\in M$ and $\lambda\in\Lambda$. Another equivalent definition can be given
by setting $M^{\iota}\colonequals M\otimes_{\Lambda,\iota}\Lambda$. This is a
$\Lambda$-module with operations given by
$\delta(m\otimes\delta^{\prime})=m\otimes\delta\delta^{\prime}$ and $(\delta
m)\otimes\delta^{\prime}=m\otimes\iota(\delta)\delta^{\prime}$ for all
$\delta,\delta^{\prime}\in\Lambda,m\in M$. Let $\Delta$ be a finite group and
$\eta$ a character of $\Delta$. Suppose that $p\nmid|\Delta|$ and let
$\mathcal{O}_{\Delta}$ be the $p$-adic completion of the extension of
${\mathbf{Z}}$ obtained be adding the $|\Delta|$-th roots of unity. We let
$e_{\eta}\colonequals\frac{1}{|\Delta|}\sum_{\sigma\in\Delta}\eta(\sigma)^{-1}\cdot\sigma\in\mathcal{O}_{\Delta}[\Delta]$
be the idempotent element corresponding to $\eta$.

Let $E$ be an elliptic curve defined over ${\mathbf{Q}}$ with good ordinary
reduction at $p$ and with conductor $N_{E}$ not divisible by $p$. As a
representation of $G_{\mathbf{Q}_{p}}$, the $p$-adic Tate module
$T\colonequals T_{p}E=\varprojlim_{n}E[p^{n}]$ is reducible and we have an
exact sequence of $\mathbf{Z}_{p}[G_{\mathbf{Q}_{p}}]$-modules

$0\to T^{+}\to T\to T^{-}\to 0$

where $T^{+}$ and $T^{-}$ are free of rank one over $\mathbf{Z}_{p}$ and
$T^{-}$ is unramified. Let
$A\colonequals\mathrm{Hom}_{\mathrm{cts}}(T,\mu_{p^{\infty}})\cong
E[p^{\infty}]$ where $\mu_{p^{\infty}}$ is the set of $p$-power roots of unity
and let
$A^{-}\colonequals\mathrm{Hom}_{\mathrm{cts}}(T^{-},\mu_{p^{\infty}})$. We let
$a_{p}$ be the quantity defined by $p+1-|E(\mathbb{F}_{p})|$. If $v$ is a
place of $K$, $c_{v}$ will denote the local Tamagawa number of $E$ at $v$
which is defined by the index $[E(K_{v}):E^{0}(K_{v})]$ where $E^{0}(K_{v})$
is the subgroup of points which have good reduction. We let $\mathrm{Tam}(E)$
denote the product of the local Tamagawa numbers $\prod_{v}c_{v}$.

If $L$ is a number field or a local field, we write $G_{L}$ for the Galois
group ${\mathrm{Gal}}(\overline{L}/L)$ where $\overline{L}$ is an algebraic
closure of $L$.

##  3\. Iwasawa main conjecture for $\mathrm{GL}_{2}$

In this section, we review the formulation of the Iwasawa main conjecture that
was proven by Castella and Wan in [6].

###  3.1. Selmer groups

We first review the construction of the Selmer groups appearing in the main
conjecture.

Let $F$ be a finite extension of $K$ contained in $K_{\infty}$ and let
$\Sigma$ be a finite set of places of $F$ containing the infinite places and
the primes dividing $N_{E}p$. Let $G_{F,\Sigma}$ be the Galois group of the
maximal extension of $F$ unramified outside $\Sigma$.

######  Definition 3.1.

Define the (strict) Greenberg Selmer group ${\mathrm{Sel}}_{\mathrm{Gr}}(F,A)$
over $F$ by

${\mathrm{Sel}}_{\mathrm{Gr}}(F,A)\colonequals\ker\left(H^{1}(G_{F,\Sigma},A)\to\prod_{w\nmid
p}H^{1}(I_{w},A)\times\prod_{w|p}H^{1}(F_{w},A^{-})\right)$

where $w$ runs over the places of $F$ and $I_{w}$ is the inertia subgroup at
$w$ inside ${\mathrm{Gal}}(\overline{F_{w}}/F_{w})$. Define the (strict)
Greenberg Selmer group ${\mathrm{Sel}}_{\mathrm{Gr}}(K_{\infty},A)$ over
$K_{\infty}$ by

${\mathrm{Sel}}_{\mathrm{Gr}}(K_{\infty},A)\colonequals\ker\left(H^{1}(K_{\infty},A)\to\prod_{w\nmid
p}H^{1}(I_{w},A)\times\prod_{w|p}H^{1}(K_{\infty,w},A^{-})\right)$

where $w$ runs over the places of $K_{\infty}$ and $I_{w}$ is the inertia
subgroup at $w$ inside ${\mathrm{Gal}}(\overline{K_{\infty,w}}/K_{\infty,w})$.

We shall consider the Pontryagin dual of the previously defined Selmer group

$\mathcal{X}_{\mathrm{Gr}}(K_{\infty},A)\colonequals\mathrm{Hom}_{\mathbf{Z}_{p}}({\mathrm{Sel}}_{\mathrm{Gr}}(K_{\infty},A),\mathbf{Q}_{p}/\mathbf{Z}_{p}).$

One may also consider the Kummer sequence

$0\to
E(F_{w})\otimes\mathbf{Q}_{p}/\mathbf{Z}_{p}\xrightarrow{k_{w}}H^{1}(F_{w},A)\to
H^{1}(F_{w},E(\overline{F_{w}}))_{\text{$p$-primary}}\to 0$

and define the classical Selmer group ${\mathrm{Sel}}_{\mathrm{class}}(F,A)$
over $F$ by

${\mathrm{Sel}}_{\mathrm{class}}(F,A)\colonequals\ker\left(H^{1}(F,A)\to\prod_{w|p}\frac{H^{1}(F_{w},A)}{\mathrm{Im}(k_{w})}\right).$

The classical Selmer over $K_{\infty}$ is then given by the direct limit

${\mathrm{Sel}}_{\mathrm{class}}(K_{\infty},A)\colonequals\varinjlim_{K\subseteq
F\subseteq K_{\infty}}{\mathrm{Sel}}_{\mathrm{class}}(F,A)$

where $F$ runs over the finite subextensions of $K_{\infty}/K$. The discussion
of [8, §2] shows that

(1) ${\mathrm{Sel}}_{\mathrm{Gr}}(F,A)={\mathrm{Sel}}_{\mathrm{class}}(F,A).$

The Greenberg Selmer group satisfies the following control theorem:

######  Proposition 3.2.

Let $F$ be a finite extension contained in $K_{\infty}/K$. Assume that
$E(K)[p]$ is trivial and that $a_{p}\not\equiv 1\bmod p$. For all prime $v$
where $E$ has bad reduction, assume that $E(K_{v})[p]$ is trivial. Then,

${\mathrm{Sel}}_{\mathrm{Gr}}(K_{\infty},A)^{{\mathrm{Gal}}(K_{\infty}/F)}\cong{\mathrm{Sel}}_{\mathrm{Gr}}(F,A).$

###### Proof.

Let $F$ be a finite extension contained in $K_{\infty}/K$. Note that the
condition $a_{p}\not\equiv 1\bmod p$ is equivalent to
$E(\mathbf{F}_{p})[p]=\\{0\\}$. Under the assumptions of the proposition, [9,
Proposition 5.6] gives

${\mathrm{Sel}}_{\mathrm{class}}(K_{\infty},A)^{{\mathrm{Gal}}(K_{\infty}/F)}\cong{\mathrm{Sel}}_{\mathrm{class}}(F,A).$

However, it follows from (1) that both definition of Selmer groups agree and
so the result follows. ∎

###  3.2. $p$-adic $L$-functions

Let $f$ be the weight $2$ modular form of level $\Gamma_{0}(N_{E})$ associated
to $E$ by modularity. Remark that the Fourier coefficients of $f$ are defined
over $\mathbf{Q}$. By our ordinarity hypothesis, there is a unique unit root
of the Hecke polynomial $X-a_{p}(E)+p$ which we denote by $\alpha$. Let
$f_{\alpha}$ be the $p$-stabilization of $f$ associated to the root $\alpha$.
Then, Hida defined a three-variable $p$-adic $L$-function
$L_{p}^{Hi}(\mathbf{f}/K)$ where $\mathbf{f}$ is a Hida family over
$\mathbb{I}$, some flat extension of a one-variable Iwasawa algebra (see [6,
Section 2.4] for a detailed construction). By taking $\mathbf{f}$ to be a Hida
family passing through $f_{\alpha}$, we may specialize
$L_{p}^{Hi}(\mathbf{f}/K)$ to a two-variable $p$-adic $L$-function
$L_{p}^{Hi}(f_{\alpha}/K)$. Since $f_{\alpha}$ is defined over
$\mathbf{Q}_{p}$, $L_{p}^{Hi}(f_{\alpha}/K)$ is an element of $\Lambda$.

###  3.3. The main conjecture

We now state the main result of [6]. Let
$\rho_{f_{\alpha}}:G_{\mathbf{Q}}\to\mathrm{GL}_{2}(\mathbf{Q}_{p})$ be the
Galois representation associated to $f_{\alpha}$ and
$\overline{\rho}_{f_{\alpha}}:G_{\mathbf{Q}}\to\mathrm{GL}_{2}(\mathbf{Z}/p\mathbf{Z})$
be the residual representation. Suppose that the quadratic field $K$ has
discriminant coprime to $N_{E}p$. Write $N_{E}=N^{+}N^{-}$ where $N^{+}$
(resp. $N^{-}$) is divisible only by primes which are split (resp. inert) in
$K$. We will assume that $K$ satisfies the generalized Heegner hypothesis
relative to $N_{E}$:

$N^{-}$ is the squarefree product of an even number of primes.

######  Theorem 3.3.

Suppose that $N_{E}$ is squarefree, $N^{-}\neq 1$ and that
$\overline{\rho}_{f}$ is ramified at every prime dividing $N^{-}$. Also assume
that $\mathbb{I}$ is regular. Then, $X_{\mathrm{Gr}}(K_{\infty},A)$ is
$\Lambda$-torsion and

$\mathrm{Char}_{\Lambda}(X_{\mathrm{Gr}}(K_{\infty},A))=(L_{p}^{\mathrm{Hi}}(f_{\alpha}/K))$

as ideals in $\Lambda\otimes\mathbf{Q}_{p}$.

##  4\. Two-variable Mazur–Tate elements

We review the main result of Haran [10] on the construction of Mazur–Tate
elements for $E$ over $K_{\infty}$ by following the exposition in [16].

We write $G_{\mathfrak{f}}\colonequals{\mathrm{Gal}}(K(\mathfrak{f})/K)$ for
the ray class group of $K$ modulo $\mathfrak{f}$ and
$G_{1}\colonequals{\mathrm{Gal}}(K(1)/K)$ for the Galois group of the Hilbert
class field of $K$. If $\mathfrak{m}$ and $\mathfrak{f}$ are two moduli such
that $\mathfrak{m}|\mathfrak{f}$, we have a natural surjection
$G_{\mathfrak{f}}\twoheadrightarrow G_{\mathfrak{m}}$. We let
$\mathrm{Norm}_{\mathfrak{m}}^{\mathfrak{f}}:\mathbf{Z}_{p}[G_{\mathfrak{f}}]\to\mathbf{Z}_{p}[G_{\mathfrak{m}}]$
be the induced map on the group rings. We define the trace map by

$\displaystyle\mathrm{Tr}_{\mathfrak{m}}^{\mathfrak{f}}:\mathbf{Z}_{p}[G_{\mathfrak{m}}]$
$\displaystyle\to\mathbf{Z}_{p}[G_{\mathfrak{f}}]$ $\displaystyle x$
$\displaystyle\mapsto\sum_{y\in\mathbf{Z}_{p}[G_{\mathfrak{f}}]\atop{\mathrm{Norm}^{\mathfrak{f}}_{\mathfrak{m}}(y)=x}}y.$

Let $v\in\\{{\mathfrak{p}},{\mathfrak{q}}\\}$. In the special case where
$\mathfrak{m}=v^{n-1}$ and $\mathfrak{f}=v^{n}$ with $n\geq 2$, remark that
$\mathrm{Tr}_{\mathfrak{m}}^{\mathfrak{f}}$ is given by multiplication by
$\Phi_{n-1}(\gamma_{v})$. Indeed, preimages of $x$ under the norm map take the
form $y\gamma_{v}^{kp^{n-1}}$ where $k=0,\ldots p-1$ and $y$ is a fixed
preimage of $x$.

Let $\Pi/K$ be the base-change to $K$ of the automorphic representation coming
from $f$. Since $p$ is split in $K$, the $v$-th Hecke eigenvalue
$a_{v}(\Pi/K)$ is equal to $a_{p}$ for both choices
$v={\mathfrak{p}},{\mathfrak{q}}$.

######  Theorem 4.1.

There exists a rank one $\mathbf{Z}$-submodule $L_{f}$ of $\mathbf{C}$, and
for each ideal $\mathfrak{f}$ of $K$ coprime to the conductor of $\Pi$ an
element

$\widetilde{\Theta}_{\mathfrak{f}}(\Pi)\in{\mathbf{Z}}[G_{\mathfrak{f}}]\otimes_{\mathbf{Z}}L_{f},$

where $G_{\mathfrak{f}}$ is the ray class group modulo $\mathfrak{f}$, such
that the following relations hold:

  1. (1)

If $\omega$ is a primitive ray class character of conductor $\mathfrak{f}$,
then

$\omega(\widetilde{\Theta}_{\mathfrak{f}}(\Pi))=\frac{L_{\mathfrak{f}}(\Pi,\omega,1)}{\tau(\omega)\cdot|\mathfrak{f}|^{1/2}\cdot(4\pi)^{2}}$

where $L_{\mathfrak{f}}(\Pi,\omega,s)$ denotes the $L$-function of $\Pi$
twisted by $\omega$, without the Euler factors at $\infty$ or at primes
dividing $\mathfrak{f}$, and $\tau(\omega)$ is the Gauss sum (normalized so
that $|\tau(\omega)|=1$);

  2. (2)

For a prime $v$ not dividing the conductor of $\Pi$, we have

$\mathrm{Norm}_{\mathfrak{f}}^{\mathfrak{f}v}\widetilde{\Theta}_{\mathfrak{f}v}(\Pi)=\begin{cases}\left(a_{v}(\Pi)-\sigma_{v}-\sigma_{v}^{-1}\right)\widetilde{\Theta}_{\mathfrak{f}}(\Pi)&\text{if
$v\nmid\mathfrak{f}$},\\\
a_{v}(\Pi)\widetilde{\Theta}_{\mathfrak{f}}-\mathrm{Tr}_{\mathfrak{f}/v}^{\mathfrak{f}}\widetilde{\Theta}_{\mathfrak{f}/v}(\Pi)&\text{if
$v|\mathfrak{f}$},\end{cases}$

where $\sigma_{v}$ denotes the class of $v$ in $G_{\mathfrak{f}}$.

###### Proof.

This is the main result of [10] combined with [16, Proposition 4.3] for the
fact that $L_{f}$ has rank one. ∎

Since we only consider $\Pi$ in this note, we remove it from the notation and
write $\widetilde{\Theta}_{\mathfrak{f}}$ for the Mazur–Tate element of
theorem 4.1.

######  Definition 4.2.

A $S$-refinement of $\Pi$ is a choice of roots
$\alpha_{v}\in\overline{\mathbf{Q}}$ for $v\in S$ of the Hecke polynomial
$X^{2}-a_{v}(\Pi)X+p\in\mathbf{Q}[X]$.

######  Definition 4.3.

When $p$ divides the ideal $\mathfrak{f}$, we define the $S$-refined
Mazur–Tate element by

$\widetilde{\Theta}_{\mathfrak{f}}^{S}\colonequals\alpha_{\mathfrak{p}}^{-\mathrm{ord}_{\mathfrak{p}}(\mathfrak{f})}\alpha_{\mathfrak{q}}^{-\mathrm{ord}_{{\mathfrak{q}}}(\mathfrak{f})}\left(\widetilde{\Theta}_{\mathfrak{f}}-\alpha_{\mathfrak{p}}^{-1}\mathrm{Tr}_{\mathfrak{f}/{\mathfrak{p}}}^{\mathfrak{f}}\widetilde{\Theta}_{\mathfrak{f}/{\mathfrak{p}}}-\alpha_{{\mathfrak{q}}}^{-1}\mathrm{Tr}_{\mathfrak{f}/{\mathfrak{q}}}^{\mathfrak{f}}\widetilde{\Theta}_{\mathfrak{f}/{\mathfrak{q}}}+\alpha_{\mathfrak{p}}^{-1}\alpha_{{\mathfrak{q}}}^{-1}\mathrm{Tr}_{\mathfrak{f}/{\mathfrak{p}}{\mathfrak{q}}}^{\mathfrak{f}}\widetilde{\Theta}_{\mathfrak{f}/{\mathfrak{p}}{\mathfrak{q}}}\right).$

If ${\mathfrak{q}}$ divides $\mathfrak{f}$, but ${\mathfrak{p}}$ does not, we
put

$\widetilde{\Theta}_{\mathfrak{f}}^{S}\colonequals\alpha_{\mathfrak{p}}^{-1}\alpha_{\mathfrak{q}}^{-\mathrm{ord}_{\mathfrak{q}}(\mathfrak{f})}(\alpha_{\mathfrak{p}}-\sigma_{\mathfrak{p}}-\sigma_{\mathfrak{p}}^{-1})\left(\widetilde{\Theta}_{\mathfrak{f}}-\alpha_{\mathfrak{q}}^{-1}\mathrm{Tr}_{\mathfrak{f}/{\mathfrak{q}}}^{\mathfrak{f}}\widetilde{\Theta}_{\mathfrak{f}/{\mathfrak{q}}}\right)$

and if ${\mathfrak{p}}$ divides $\mathfrak{f}$, but ${\mathfrak{q}}$ does not,
we put

$\widetilde{\Theta}_{\mathfrak{f}}^{S}\colonequals\alpha_{\mathfrak{p}}^{-\mathrm{ord}_{\mathfrak{p}}(\mathfrak{f})}\alpha_{\mathfrak{q}}^{-1}(\alpha_{\mathfrak{q}}-\sigma_{\mathfrak{q}}-\sigma_{\mathfrak{q}}^{-1})\left(\widetilde{\Theta}_{\mathfrak{f}}-\alpha_{\mathfrak{p}}^{-1}\mathrm{Tr}_{\mathfrak{f}/{\mathfrak{p}}}^{\mathfrak{f}}\widetilde{\Theta}_{\mathfrak{f}/{\mathfrak{p}}}\right).$

Finally, in the case where nether ${\mathfrak{p}}$ nor ${\mathfrak{q}}$
divides $\mathfrak{f}$, we put

$\widetilde{\Theta}_{\mathfrak{f}}^{S}\colonequals\left(\prod_{v\in
S}(1-\alpha_{v}^{-1}\sigma_{v})(1-\alpha_{v}^{-1}\sigma_{v}^{-1})\right)\widetilde{\Theta}_{\mathfrak{f}}.$

One can check that

$\mathrm{Norm}_{\mathfrak{f}}^{\mathfrak{f}v}\left(\widetilde{\Theta}_{\mathfrak{f}v}^{S}\right)=\widetilde{\Theta}_{\mathfrak{f}}^{S}$

for any $v\in S$. Then the elements
$\\{\widetilde{\Theta}_{\mathfrak{f}}^{S}:\mathfrak{f}|p^{\infty}\\}$ form a
ray class distribution modulo $p^{\infty}$ with values in
$\mathbf{Q}(\alpha_{\mathfrak{p}},\alpha_{{\mathfrak{q}}})\otimes_{\mathcal{O}_{\mathbf{Q}(\alpha_{\mathfrak{p}},\alpha_{{\mathfrak{q}}})}}L_{f}$
and growth parameter $(\alpha_{\mathfrak{p}},\alpha_{{\mathfrak{q}}})$ in the
sense of Loeffler [16]. Fix an embedding
$\mathbf{Q}(\alpha_{\mathfrak{p}},\alpha_{{\mathfrak{q}}})\hookrightarrow\overline{\mathbf{Q}_{p}}$
which is the same as fixing a prime $\mathfrak{e}$ above $p$. Then, the
completion
$\mathbf{Q}(\alpha_{\mathfrak{p}},\alpha_{{\mathfrak{q}}})_{\mathfrak{e}}$ is
equal to $\mathbf{Q}_{p}(\alpha_{\mathfrak{q}},\alpha_{\mathfrak{q}})$. But it
follows by Hensel’s lemma that
$\alpha_{\mathfrak{p}},\alpha_{\mathfrak{q}}\in\mathbf{Q}_{p}$, hence
$\mathbf{Q}(\alpha_{\mathfrak{p}},\alpha_{{\mathfrak{q}}})_{\mathfrak{e}}=\mathbf{Q}_{p}$.

######  Theorem 4.4.

If we have $r\colonequals\mathrm{ord}_{p}(\alpha_{\mathfrak{p}})<1$ and
$s\colonequals\mathrm{ord}_{p}(\alpha_{{\mathfrak{q}}})<1$, then there exists
a unique distribution $L_{p,\underline{\alpha}}(f)\in
D^{(r,s)}(G_{p^{\infty}},\mathbf{Q}_{p})\otimes_{\mathbf{Z}}L_{f}$ satisfying
the interpolation formula 111The exponent $1/2$ of $|\mathfrak{f}|$ is missing
in [16, Theorem 8].

$\displaystyle L_{p,\underline{\alpha}}(f)(\omega)=\left(\prod_{v\in
S}\alpha_{v}^{-\mathrm{ord}_{v}(\mathfrak{f})}\right)\cdot\left(\prod_{v\in
S,v\nmid\mathfrak{f}}(1-\alpha_{v}^{-1}\omega(v))(1-\alpha_{v}^{-1}\omega(v)^{-1})\right)\cdot\frac{L_{\mathfrak{f}}(\Pi,\omega,1)}{\tau(\omega)\cdot|\mathfrak{f}|^{1/2}\cdot(4\pi)^{2}}$

where

  * •

$\omega$ is a finite order ray class character on $G_{\mathfrak{f}}$ with
$\mathfrak{f}|p^{\infty}$,

  * •

$L_{\mathfrak{f}}(\Pi,\omega,s)$ denotes the $L$-function of $\Pi$ twisted by
$\omega$, without the Euler factors at $\infty$ or at primes dividing
$\mathfrak{f}$,

  * •

$\tau(\omega)$ is the Gauss sum.

###### Proof.

This is [16, Theorem 9] applied to the case where $\Pi$ is the base-change to
$K$ of the automorphic representation attached to $f$. A priori,
$L_{p,\underline{\alpha}}(f)$ takes value in
$\mathbf{Q}(\alpha_{\mathfrak{p}},\alpha_{{\mathfrak{q}}})_{\mathfrak{e}}\otimes_{\mathbf{Z}}L_{f}$,
but the discussion above shows that $L_{p,\underline{\alpha}}(f)\in
D^{(r,s)}(G_{p^{\infty}},\mathbf{Q}_{p})\otimes_{\mathbf{Z}}L_{f}$. ∎

In other words, $L_{p,\underline{\alpha}}(f)$ is the unique distribution
interpolating the $\widetilde{\Theta}_{\mathfrak{f}}^{S}$ when
$\mathfrak{f}|p^{\infty}$. Thus, if $\omega$ is a primitive ray class
character of conductor $\mathfrak{f}$ as above,
$L_{p,\underline{\alpha}}(f)(\omega)=\widetilde{\Theta}_{\mathfrak{f}}^{S}(\omega)$.
Under our $p$-ordinary assumption, there is a unique choice of roots
$\alpha_{{\mathfrak{p}}}$ and $\alpha_{{\mathfrak{q}}}$ (with
$\mathrm{ord}_{p}(\alpha_{{\mathfrak{p}}})=\mathrm{ord}_{p}(\alpha_{{\mathfrak{q}}})=0$)
satisfying the hypothesis of theorem 4.4. We shall make this choice and
suppress it from the notation. We obtain a measure
$L_{p}(f)\in\mathcal{M}(G_{p^{\infty}},\mathbf{Q}_{p})\otimes_{\mathbf{Z}_{p}}L_{f}$.
This corresponds, under the Amice transform, to an element of
$\mathbf{Z}_{p}\llbracket
G_{p^{\infty}}\rrbracket\otimes_{\mathbf{Z}_{p}}L_{f}$ which we also denote by
$L_{p}(f)$. Suppose that
(C.N) $p$ does not divide the class number of $K$.
Since $G_{p^{\infty}}\cong\Gamma\times\Delta$ where
$\Delta={\mathrm{Gal}}(K(p)/K)$ is a finite group with order not divisible by
$p$ by (C.N), we may define $L_{p,\Delta}(f)\colonequals e_{\chi_{0}}\cdot
L_{p}(f)\in\mathbf{Z}_{p}\llbracket\Gamma\rrbracket\otimes_{\mathbf{Z}_{p}}L_{f}$
where $\chi_{0}$ is the trivial character $\Delta\to\mathbf{Q}_{p}^{\times}$.
Finally, by choosing a generator $\Omega_{\mathrm{Loe}}$ of $L_{f}$, we view
$L_{p,\Delta}(f)$ as an element of $\mathbf{Z}_{p}\llbracket\Gamma\rrbracket$
and $\widetilde{\Theta}_{\mathfrak{f}}$ as an element of
$\mathbf{Z}_{p}[G_{\mathfrak{f}}]$. For any modulus $\mathfrak{f}|p^{\infty}$,
let $K^{\prime}(\mathfrak{f})$ be the largest Galois subextension of
$K(\mathfrak{f})$ with degree a power of $p$ and let
$G_{\mathfrak{f}}^{\prime}$ be ${\mathrm{Gal}}(K^{\prime}(\mathfrak{f})/K)$,
i.e. ${\mathrm{Gal}}(K(\mathfrak{f})/K)\cong
G_{\mathfrak{f}}^{\prime}\times\Delta$.

######  Definition 4.5.

We define the Mazur–Tate element $\Theta_{n,m}^{S}$ (resp. $\Theta_{n,m}$) at
$K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})$ to be the projection to
$\mathbf{Z}_{p}[G^{\prime}_{{\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}}]$ of
$\widetilde{\Theta}_{{\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}}^{S}$ (resp.
$\widetilde{\Theta}_{{\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}}$).

######  Lemma 4.6.

The element $1-\alpha_{v}^{-1}\sigma_{v}$ is invertible in
$\mathbf{Z}_{p}[G_{1}]$. Furthermore, the element
$1-\alpha_{v}^{-1}\sigma_{v}-\alpha_{v}^{-1}\sigma_{v}^{-1}$ is also
invertible in $\mathbf{Z}_{p}[G_{v^{n}}]$ ($n\geq 1$).

###### Proof.

Say that $G_{1}$ has $k$ elements $\\{g_{1},\ldots,g_{k}\\}$. Write
$\\{\chi_{1},\ldots,\chi_{k}\\}$ for the $k$ characters on $G_{1}$. A general
element of $\mathbf{Z}_{p}[G_{1}]$ takes the form $\sum_{i=1}^{k}a_{i}g_{i}$
with $a_{i}\in\mathbf{Z}_{p}$. Thus, we want to show that the equation with
unknowns $a_{i}$ ($i=1,\ldots,k$)

$(1-\alpha_{v}^{-1}\sigma_{v})\cdot\sum_{i=1}^{k}a_{i}g_{i}=1$

has a solution. Put in matrix form, this system of equations forms a
$G_{1}$-circulant matrix (see [11, §1.3] for a reference on the topic). In
[11, discussion after Corollary 2.4], it is shown that the determinant of such
matrices is given by

$\mathrm{det}=\prod_{i=1}^{k}\left(1-\alpha_{v}^{-1}\chi_{i}(\sigma_{v})\right).$

Since the $\chi_{i}(\sigma_{v})$ are $k$-th roots of unity, the product is
non-zero. Hence, an inverse certainly exists in $\mathbf{Q}_{p}[G_{1}]$. To
show integrality, notice that the solutions $a_{i}$ can be written as
$a_{i}=b_{i}/\mathrm{det}$ for some $b_{i}$ in $\mathbf{Z}_{p}$. Thus, if $p$
divided $\mathrm{det}$, it would also divide $1-\alpha_{v}^{-1}\sigma_{v}$
which is not the case. This shows that
$(1-\alpha_{v}^{-1}\sigma_{v})^{-1}\in\mathbf{Z}_{p}[G_{1}]$.

For $1-\alpha_{v}^{-1}\sigma_{v}-\alpha_{v}^{-1}\sigma_{v}^{-1}$, remark that
$G_{v^{n}}$ has $k(p-1)p^{n-1}$ elements. Say that these elements are
$\\{g_{1},\ldots,g_{k(p-1)p^{n-1}}\\}$ with associated characters
$\\{\chi_{1},\ldots,\chi_{k(p-1)p^{n-1}}\\}$. By loc. cit., the determinant of
the system of equations we now want to solve is given by

$\prod_{i=1}^{k(p-1)p^{n-1}}(1-\alpha_{v}^{-1}(\chi_{i}(\sigma_{v})-\chi_{i}(\sigma_{v})^{-1})).$

This last expression is zero if and only if one of the
$1-\alpha_{v}^{-1}(\chi_{i}(\sigma_{v})-\chi_{i}(\sigma_{v})^{-1})$ is zero.
This equivalent to

$\alpha_{v}=\chi_{i}(\sigma_{v})-\chi_{i}(\sigma_{v})^{-1}.$

Fix an embedding
$\iota_{p}:\overline{\mathbf{Q}}\hookrightarrow\mathbf{C}_{p}$. Since
$a_{v}\neq 0$, $\iota_{p}^{-1}(\alpha_{v})$ can not be a purely imaginary
complex number. But,
$\iota_{p}^{-1}(\chi_{i}(\sigma_{v})-\chi_{i}(\sigma_{v})^{-1})$ is purely
imaginary, hence we can not have equality. We conclude that
$1-\alpha_{v}^{-1}\sigma_{v}-\alpha_{v}^{-1}\sigma_{v}^{-1}$ is invertible in
$\mathbf{Q}_{p}[G_{v^{n}}]$. By the same argument as before, it is also
invertible in $\mathbf{Z}_{p}[G_{v^{n}}]$. ∎

######  Lemma 4.7.

For $n,m\geq 0$, we have
$\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}=C(n,m)\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}^{S}$
where $C(n,m)\in\mathbf{Z}_{p}[G_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}]$.

###### Proof.

We show this by strong two-variable induction on $n$ and $m$.

Base case: By definition 4.3, we have

$\widetilde{\Theta}_{(1)}\cdot\left(\prod_{v\in
S}(1-\alpha_{v}^{-1}\sigma_{v})(1-\alpha_{v}^{-1}\sigma_{v}^{-1})\right)=\widetilde{\Theta}_{(1)}^{S}$

as elements of $\mathbf{Z}_{p}[G_{1}]$. By lemma 4.6,
$C(0,0)^{-1}\colonequals\prod_{v\in
S}(1-\alpha_{v}^{-1}\sigma_{v})(1-\alpha_{v}^{-1}\sigma_{v}^{-1})$ is
invertible, hence
$\widetilde{\Theta}_{(1)}=C(0,0)\cdot\widetilde{\Theta}^{S}_{(1)}$.

Induction step: Suppose the result is true for
$\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}$ for all $n,m$ such
that $n+m<l$. We want to show it also holds for all $n,m$ such that $n+m=l+1$.
We distinguish two cases: if both $n,m$ are non zero or one of $m$ or $n$ is
zero. If $m\neq 0$ and $n\neq 0$, we have by definition that

$\displaystyle\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}^{S}=\alpha_{\mathfrak{p}}^{-n}\alpha_{\mathfrak{q}}^{-m}\Big{(}\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}-\alpha_{\mathfrak{p}}^{-1}\mathrm{Tr}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m}}^{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m}}$
$\displaystyle-\alpha_{\mathfrak{p}}^{-1}\mathrm{Tr}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m-1}}^{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m-1}}$
$\displaystyle+\alpha_{\mathfrak{p}}^{-1}\alpha_{\mathfrak{q}}^{-1}\mathrm{Tr}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m-1}}^{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m-1}}\Big{)}.$

By the induction hypothesis, we may replace the various
$\widetilde{\Theta}_{{\mathfrak{p}}^{a}{\mathfrak{q}}^{b}}$ by
$C(a,b)\widetilde{\Theta}_{{\mathfrak{p}}^{a}{\mathfrak{q}}^{b}}^{S}$ in the
right hand side of the last expression. Since
$\mathrm{Tr}_{\mathfrak{e}}^{\mathfrak{f}}\widetilde{\Theta}_{\mathfrak{e}}^{S}$
is a multiple of $\widetilde{\Theta}_{\mathfrak{f}}^{S}$ by some element of
$\mathbf{Z}_{p}[G_{\mathfrak{f}}]$, we find
$\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}=C(n,m)\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}^{S}$
with

(2)
$C(n,m)\colonequals\alpha_{\mathfrak{p}}^{n}\alpha_{\mathfrak{q}}^{m}+\alpha_{\mathfrak{p}}^{-1}h_{1}C(n-1,m)+\alpha_{\mathfrak{q}}^{-1}h_{2}C(n,m-1)-\alpha_{\mathfrak{p}}^{-1}\alpha_{\mathfrak{q}}^{-1}h_{3}C(n-1,m-1)$

where $h_{1},h_{2}$ and $h_{3}$ are some elements of
$\mathbf{Z}_{p}[G_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}]$. For the second
case, assume without loss of generality that $m=0$. By definition,

$\displaystyle\widetilde{\Theta}_{{\mathfrak{p}}^{n}}^{S}=\alpha_{\mathfrak{p}}^{-n}(1-\alpha_{\mathfrak{q}}^{-1}\sigma_{\mathfrak{q}}-\alpha_{\mathfrak{q}}^{-1}\sigma_{\mathfrak{q}}^{-1})\left(\widetilde{\Theta}_{{\mathfrak{p}}^{n}}-\alpha_{\mathfrak{p}}^{-1}\mathrm{Tr}_{{\mathfrak{p}}^{n-1}}^{{\mathfrak{p}}^{n}}\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}}\right).$

By the induction hypothesis,
$\mathrm{Tr}_{{\mathfrak{p}}^{n-1}}^{{\mathfrak{p}}^{n}}\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}}=C(n-1,0)\mathrm{Tr}_{{\mathfrak{p}}^{n-1}}^{{\mathfrak{p}}^{n}}\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}}^{S}$
and thus
$\mathrm{Tr}_{{\mathfrak{p}}^{n-1}}^{{\mathfrak{p}}^{n}}\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}}=C(n-1,0)h_{4}\widetilde{\Theta}_{{\mathfrak{p}}^{n}}$
for some element $h_{4}$ of $\mathbf{Z}_{p}[G_{{\mathfrak{p}}^{n}}]$. It
follows that
$\widetilde{\Theta}_{{\mathfrak{p}}^{n}}=C(n,0)\widetilde{\Theta}_{{\mathfrak{p}}^{n}}^{S}$
where

$C(n,0)\colonequals\alpha_{\mathfrak{p}}^{n}(1-\alpha_{\mathfrak{q}}^{-1}\sigma_{\mathfrak{q}}-\alpha_{\mathfrak{q}}^{-1}\sigma_{\mathfrak{q}}^{-1})^{-1}+\alpha_{\mathfrak{p}}^{-1}C(n-1,0)h_{4}.$

Note that we were able to invert
$1-\alpha_{\mathfrak{q}}^{-1}\sigma_{\mathfrak{q}}-\alpha_{\mathfrak{q}}^{-1}\sigma_{\mathfrak{q}}^{-1}$
by lemma 4.6. This complete the induction step. ∎

######  Proposition 4.8.

For $n+m\geq 1$, we have $\Theta_{n,m}=B(n,m)\Theta_{n,m}^{S}$ where
$B(n,m)\in\mathbf{Z}_{p}[G_{{\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}}^{\prime}]^{\times}$.

###### Proof.

To show this, we need to study more carefully the $h_{i}$’s appearing in the
proof of lemma 4.7. Consider $x\in
G_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m}}$. Let $y$ be a preimage of $x$ in
$G_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}$ and let $\gamma_{n}$ be a generator
of $G_{{\mathfrak{p}}^{n}}/G_{\mathfrak{p}}\cong\mathbf{Z}/p^{n-1}\mathbf{Z}$.
If we suppose that $n\geq 2$, we have

$\mathrm{Tr}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m}}^{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}x=\sum_{i=0}^{p-1}y\gamma_{n}^{p^{n-2}i}=y\cdot\Phi_{n-1}(\gamma_{n}).$

Since $\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}^{S}$ is a
preimage of $\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m}}^{S}$,
we find
$\mathrm{Tr}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m}}^{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}\widetilde{\Theta}_{{\mathfrak{p}}^{n-1}{\mathfrak{q}}^{m}}^{S}=\Phi_{n-1}(\gamma_{n})\cdot\widetilde{\Theta}_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}^{S}$.
Remark that, under the isomorphism $\mathbf{Z}_{p}\llbracket
G_{{\mathfrak{p}}^{\infty}}/G_{{\mathfrak{p}}{\mathfrak{q}}}\rrbracket\xrightarrow{\sim}\mathbf{Z}_{p}\llbracket
X\rrbracket$, $\Phi_{n-1}(\gamma_{n})$ is send to $\Phi_{n-1}(1+X)$ which has
constant term divisible by $p$. Let $\pi_{n,m}$ be the projection map
$G_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}\to
G_{{\mathfrak{p}}^{n}{\mathfrak{q}}^{m}}/G_{{\mathfrak{p}}{\mathfrak{q}}}$.
The upshot of this is that $p$ divides the constant term of all the
$\pi_{n,m}(h_{i})=h_{i}$’s provided that $n+m\geq 2$. Define
$B(n-1,m-1)\colonequals\pi_{n,m}C(n,m)$ (so that we have
$\Theta_{n,m}=B(n,m)\Theta_{n,m}^{S}$ by lemma 2). Write $b(n,m)$ for the
constant term of $B(n,m)$ viewed as an element of $\mathbf{Z}_{p}\llbracket
X,Y\rrbracket$ via the isomorphism $\mathbf{Z}_{p}\llbracket
G_{p^{\infty}}/G_{{\mathfrak{p}}{\mathfrak{q}}}\rrbracket\xrightarrow{\sim}\mathbf{Z}_{p}\llbracket
X,Y\rrbracket$. Then, the constant term of $B(n,m)$ is

$\alpha_{\mathfrak{p}}^{n-1}\alpha_{\mathfrak{q}}^{m-1}+p\left(\alpha_{\mathfrak{p}}^{-1}b(n-1,m)+\alpha_{\mathfrak{q}}^{-1}b(n,m-1)-p\alpha_{\mathfrak{p}}^{-1}\alpha_{\mathfrak{q}}^{-1}b(n-1,m-1)\right)$

or

$\alpha_{\mathfrak{p}}^{n-1}(1-\alpha_{\mathfrak{q}}^{-1}\pi_{n-1,0}\sigma_{\mathfrak{q}}-\alpha_{\mathfrak{q}}^{-1}\pi_{n-1,0}\sigma_{\mathfrak{q}}^{-1})^{-1}+p\alpha_{\mathfrak{p}}^{-1}b(n-1,0)$

depending on whether or not one of the $n,m$ is zero. In both cases, we find
that $B(n,m)$ is a unit. ∎

######  Remark 4.9.

In the statement of proposition 4.8, we need to exclude the pairs
$(n,m)\in\\{(0,0),(1,0),(0,1)\\}$. In these cases, the elements $h_{i}$ are
given by traces of the form
$\mathrm{Tr}_{{\mathfrak{p}}}^{{\mathfrak{p}}{\mathfrak{q}}}$. Since
$G_{{\mathfrak{p}}{\mathfrak{q}}}/G_{\mathfrak{p}}\cong\mathbf{Z}/(p-1)\mathbf{Z}$,
the trace map is not given by multiplication by a cyclotomic polynomial. It
may be possible that the $B(n,m)$ in those cases are still units, but one
would need to find a different argument.

######  Theorem 4.10.

For $n+m\geq 1$, we have $(\Theta_{n,m})=(\Theta_{n,m}^{S})$ as ideals in
$\mathbf{Z}_{p}[G_{{\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}}^{\prime}]$.

###### Proof.

Follows directly by proposition 4.8. ∎

##  5\. Comparing $p$-adic $L$-function

Recall that the period appearing in the interpolation formula for
$L_{p}^{\mathrm{Hi}}(f_{\alpha}/K)$ [3, Theorem 2.1] is the Petersson inner
product $\langle f,f\rangle_{N_{E}}$.

######  Proposition 5.1.

We have $(L_{p,\Delta}(f))=(L_{p}^{\mathrm{Hi}}(f_{\alpha}/K))$ as ideals in
$\Lambda\otimes\mathbf{Q}_{p}$. Furthermore, if the ratio of the periods
$\Omega_{\mathrm{Loe}}/\langle f,f\rangle_{N_{E}}$ is a $p$-adic unit, then
the equality holds in $\Lambda$.

###### Proof.

Comparing the interpolation formula 4.4 for $L_{p,\Delta}(f)$ and the one for
$L_{p}^{\mathrm{Hi}}(f_{\alpha}/K)$ [3, Theorem 2.1], one finds that the
quotient $L_{p,\Delta}(f)/L_{p}^{\mathrm{Hi}}(f_{\alpha}/K)$ is an element of
$\mathbf{Q}_{p}$ (remark that the absolute value of root number
$\tau(\psi_{p})$ of loc. cit. is $p^{(n+m)/2}$ whereas the absolute value of
the Gauss sum of $\psi$ is $1$). If $\Omega_{\mathrm{Loe}}/\langle
f,f\rangle_{N_{E}}\in\mathbf{Z}_{p}^{\times}$, then we get that
$L_{p,\Delta}(f)/L_{p}^{\mathrm{Hi}}(f_{\alpha}/K)\in\mathbf{Z}_{p}^{\times}$.
∎

##  6\. Selmer complexes

###  6.1. Fitting ideals

Let $R$ be a ring and $M$ be a finitely presented $R$-module. Let

$R^{s}\xrightarrow{h}R^{r}\to M\to 0$

be a presentation of $M$. The zeroth Fitting ideal of $M$ over $R$, denoted
$\mathrm{Fitt}_{R}(M)$, is the ideal of $R$ generated by the $r\times r$
minors of the matrix $h$. The Fitting ideal does not depend of the choice of
presentation for $M$.

######  Lemma 6.1.

Let $M$ be a finitely presented $R$-module. If $I\subset R$ is an ideal, then

$\mathrm{Fitt}_{R/I}(M/IM)=\pi\left(\mathrm{Fitt}_{R}(M)\right)$

where $\pi:R\to R/I$ is the natural quotient map.

###### Proof.

See [18, 4, Appendix]. ∎

######  Lemma 6.2.

If $R\to R^{\prime}$ is a ring map, then
$\mathrm{Fitt}_{R^{\prime}}(M\otimes_{R}R^{\prime})$ is the ideal of
$R^{\prime}$ generated by the image of $\mathrm{Fitt}_{R}(M)$.

###### Proof.

This follows from the fact that $\otimes_{R}R^{\prime}$ is right exact. ∎

###  6.2. Definition

######  Definition 6.3.

Let $R$ be a Noetherian ring. A complex $M^{\bullet}$ of $R$-modules is
perfect if there exists a quasi-isomorphism $P^{\bullet}\to M^{\bullet}$,
where $P^{\bullet}$ is a bounded complex of projective $R$-modules of finite
type. Denote by $D_{\mathrm{parf}}(R)$ the category of perfect $R$-modules.
One says that $M^{\bullet}\in D_{\mathrm{parf}}(R)$ has perfect amplitude
contained in $[a,b]$ (denoted $M^{\bullet}\in D_{\mathrm{parf}}^{[a,b]}(R)$)
if the complex $P^{\bullet}$ above can be chosen in such a way that $P^{i}=0$
for every $i<a$ and $i>b$.

######  Definition 6.4.

Suppose that $M^{\bullet}\in D_{\mathrm{parf}}(R)$ is represented by a complex

$M^{\bullet}=\cdots\to M^{0}\to M^{1}\to M^{2}\to\cdots$

of finitely generated free $R$-modules such that $M^{N}=M^{-N}=0$ for all
sufficiently large integers $N$. We define the Euler-Poincaré characteristic
$\chi(M^{\bullet})$ as

$\chi(M^{\bullet})\colonequals\sum_{n\in\mathbf{Z}}(-1)^{n}\mathrm{rank}_{R}(H^{n}(M)).$

###  6.3. Selmer complexes for elliptic curves

Recall that $\Sigma$ is a finite set of places of $K$ containing the places at
infinity and the ones dividing $N_{E}p$. Let $\Sigma^{\prime}$ be the set of
primes dividing $N_{E}$. For $v$ a prime in $K$, let
$I_{v}\subseteq{\mathrm{Gal}}(\overline{K_{v}}/K_{v})$ be the inertia group at
$v$. We denote by
$\widetilde{{\mathbf{R}}\Gamma}_{f}(G_{K,\Sigma},A,\Delta_{\text{Gr}})$ the
Selmer complex defined by using the Greenberg local conditions for $E$ [19]
and we denote its cohomology by
$\widetilde{H}_{f}^{i}(G_{K,\Sigma},A,\Delta_{\text{Gr}})$. We define
$\widetilde{H}^{i}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})$ to be
$\widetilde{H}_{f}^{i}(G_{K,\Sigma},T\otimes_{\mathbf{Z}_{p}}\Lambda^{\iota},\Delta_{\text{Gr}})$
where the local datum $\Delta_{\text{Gr}}$ is induced by the filtration
$T^{+}\otimes\Lambda^{\iota}\subset T\otimes\Lambda^{\iota}$. Then, Nekovář
duality theorem [19, 8.9.6.2] provides us with a canonical isomorphism

(3)
$\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})^{\iota}\cong\widetilde{H}_{f}^{1}(G_{K,\Sigma},A,\Delta_{\text{Gr}})^{\vee}.$

######  Proposition 6.5.

For every prime $v\in\Sigma^{\prime}$, suppose that $A^{G_{v}}=0$. Then, we
have

$\mathrm{Fitt}_{\Lambda}(\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}}))=\mathrm{char}_{\Lambda}(\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})).$

###### Proof.

By the definition of the Selmer complex
$\widetilde{{\mathbf{R}}\Gamma}_{f}(G_{K,\Sigma},A,\Delta_{\text{Gr}})$, we
have an isomorphism

$\widetilde{H}^{0}_{f}(G_{K,\Sigma},A,\Delta_{\mathrm{Gr}})\xrightarrow{\sim}\ker\left(H^{0}(G_{K,\Sigma},A)\to\prod_{v\in\Sigma}H^{0}(G_{K_{v}},A)\right).$

This shows that
$\widetilde{H}^{0}_{f}(G_{K,\Sigma},A,\Delta_{\mathrm{Gre}})=0$. Let $v$ be a
prime of $K$ and let $\mathrm{Tam}_{v}(T,(\varpi))$ be the local Tamagawa
factor as defined in [19, Definition 7.6.10.1]. If $A^{G_{v}}=0$, then we have
by [19, 7.6.10.12] that $\mathrm{Tam}_{v}(T,(\varpi))=0$. Thus, it follows by
[19, Proposition 9.7.7 (iii)] that

(4)
$\widetilde{{\mathbf{R}}\Gamma}_{f,\mathrm{Iw}}(G_{K,\Sigma},T,\Delta_{\text{Gr}})\in
D_{\mathrm{parf}}^{[1,2]}(\Lambda).$

Furthermore, [19, Theorem 8.9.15] gives us that the Euler-Poincaré
characteristic of
$\widetilde{{\mathbf{R}}\Gamma}_{f,\mathrm{Iw}}(G_{K,\Sigma},T,\Delta_{\text{Gr}})$
is

$\mathrm{rank}_{\Lambda}(T\otimes\Lambda^{\iota})-\sum_{v\in\\{{\mathfrak{p}},{\mathfrak{q}}\\}}[K_{v}:\mathbf{Q}_{p}]\mathrm{rank}_{\Lambda}(T_{v}^{+}\otimes\Lambda^{\iota}).$

By the definition of $T_{v}^{+}$ and the fact that $p$ splits in $K$, we get

(5)
$\chi(\widetilde{{\mathbf{R}}\Gamma}_{f,\mathrm{Iw}}(G_{K,\Sigma},T,\Delta_{\text{Gr}}))=0.$

By combining (4) and (5), there exists an exact sequence of $\Lambda$-modules

$0\to\widetilde{H}^{1}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})\to
P\to
P\to\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})\to
0,$

where $P$ is a free $\Lambda$-module of finite rank. The proof of [5,
Corollary 2.10] shows that
$\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})$ is
$\Lambda$-torsion if and only if
$\widetilde{H}^{1}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})=0$, in
which case

$\mathrm{Fitt}_{\Lambda}(\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}}))=\mathrm{char}_{\Lambda}(\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})).$

If $\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})$ is
not torsion,

$\mathrm{Fitt}_{\Lambda}(\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}}))=0=\mathrm{char}_{\Lambda}(\widetilde{H}^{2}_{f,\mathrm{Iw}}(K_{\infty}/K,T,\Delta_{\text{Gr}})).$

∎

###  6.4. Relation to Greenberg’s Selmer groups

######  Proposition 6.6.

Suppose that we have $A^{G_{v}}=0$ for every prime $v\in\Sigma^{\prime}$.
Then, we have

$\mathrm{Fitt}_{\Lambda}(\mathcal{X}_{\text{Gr}}(K_{\infty},A))=\mathrm{char}_{\Lambda}(\mathcal{X}_{\text{Gr}}(K_{\infty},A)).$

###### Proof.

As remarked in [4, Section 3.5], a slight generalization of [19, Proposition
9.6.6(iii)] gives a canonical isomorphism

$\widetilde{H}_{f}^{1}(G_{K,S},A,\Delta_{\text{Gr}})^{\vee}\cong\mathcal{X}_{\text{Gr}}(E/K_{\infty}).$

If $M$ is any finitely generated torsion $\Lambda$-module, we have
$\mathrm{Fitt}_{\Lambda}(M^{\iota})=\iota\mathrm{Fitt}_{\Lambda}(M)$. Indeed,
apply lemma 6.2 to the ring map $\iota:\Lambda\to\Lambda$. It also follows
from the definition of the characteristic ideal that
$\mathrm{char}_{\Lambda}(M^{\iota})=\iota\mathrm{char}_{\Lambda}(M)$ (see for
example the discussion of [23, §15.5]. Thus, the result follows by proposition
6.5 and the isomorphism (3). ∎

##  7\. Refined Mazur–Tate conjecture

We now prove the main result of this note.

Let $\mathfrak{f}={\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}$ be a modulus.
Since taking ${\mathrm{Gal}}(K_{\infty}/K^{\prime}(\mathfrak{f}))$-invariants
is the same as considering the part of
${\mathrm{Sel}}_{\mathrm{Gr}}(K_{\infty},A)$ killed by
$(\omega_{n}(\gamma_{{\mathfrak{p}}}-1),\omega_{m}(\gamma_{{\mathfrak{q}}}-1))$,
the control theorem (3.2) can be reformulated as

(6)
${\mathrm{Sel}}_{\mathrm{Gr}}(K_{\infty},A)[\omega_{n}(\gamma_{{\mathfrak{p}}}-1),\omega_{m}(\gamma_{{\mathfrak{q}}}-1)]\cong{\mathrm{Sel}}_{\mathrm{Gr}}(K^{\prime}(\mathfrak{f}),A)$

provided that $E(K)[p]$ is trivial, that $a_{p}\not\equiv 1\bmod p$ and that
$E(K_{v})[p]$ is trivial for all $v\in\Sigma^{\prime}$. Put
$\Lambda_{n,m}\colonequals\mathbf{Z}_{p}[{\mathrm{Gal}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1})/K)]$.

######  Theorem 7.1.

Assume that the Iwasawa main conjecture 1.4 holds. Assume that the ratio of
periods $\Omega_{\mathrm{Loe}}/\langle f,f\rangle_{N_{E}}$ is a $p$-adic unit.
Suppose that $E(K)[p]$ is trivial, that $E(K_{v})[p]$ is trivial for all
$v\in\Sigma^{\prime}$ and that $a_{p}\not\equiv 1\bmod p$. then,

$\mathrm{Fitt}_{\Lambda_{n,m}}(\mathcal{X}_{\mathrm{Gr}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}),A))=(\Theta_{n,m})$

as ideals in $\Lambda_{n,m}$ for $n+m\geq 1$.

###### Proof.

By the Iwasawa main conjecture 1.4, proposition 5.1 and proposition 6.6, we
have an equality of ideals

(7)
$\mathrm{Fitt}_{\Lambda}(\mathcal{X}_{\mathrm{Gr}}(K_{\infty},A))=\mathrm{Char}_{\Lambda}(\mathcal{X}_{\mathrm{Gr}}(K_{\infty},A))=(L_{p}^{\mathrm{Hi}}(f_{\alpha}/K))=(L_{p,\Delta}(f))$

inside $\Lambda$. Take the projection to $\Lambda_{n,m}$ on both side of (7)
and use lemma 6.1 to get

$(\Theta_{n,m}^{S})=\mathrm{Fitt}_{\Lambda_{n,m}}(\mathcal{X}_{\mathrm{Gr}}(K_{\infty},A)/(\omega_{n}(\gamma_{\mathfrak{p}}-1),\omega_{m}(\gamma_{\mathfrak{q}}-1))\mathcal{X}_{\mathrm{Gr}}(K_{\infty},A)).$

Finally, since Mazur–Tate elements and $p$-stabilized Mazur–Tate elements
generate the same ideal by theorem 4.10, we get

$\displaystyle(\Theta_{n,m})$
$\displaystyle=\mathrm{Fitt}_{\Lambda_{n,m}}({\mathrm{Sel}}_{\mathrm{Gr}}(K_{\infty},A)[\omega_{n}(\gamma_{{\mathfrak{p}}}-1),\omega_{m}(\gamma_{{\mathfrak{q}}}-1)]^{\vee})$
$\displaystyle=\mathrm{Fitt}_{\Lambda_{n,m}}({\mathrm{Sel}}_{\mathrm{Gr}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}),A)^{\vee})$
$\displaystyle=\mathrm{Fitt}_{\Lambda_{n,m}}(\mathcal{X}_{\mathrm{Gr}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}),A))$

where the second equality follows by Mazur’s control theorem (6). ∎

By using the proof of the Iwasawa main conjecture 3.3 given by Castella and
Wan in the indefinite setting, we obtain a formulation of the previous theorem
that is less conjectural.

######  Corollary 7.2.

Assume that

  * •

$N^{-}$ is the product of an even number of prime in $K$;

  * •

$N_{E}$ is squarefree;

  * •

$N^{-}\neq 1$;

  * •

$\overline{\rho}_{f}$ is ramified at every prime dividing $N^{-}$;

  * •

$\mathbb{I}$ is regular;

  * •

$E(K)[p]$ is trivial;

  * •

$E(K_{v})[p]$ is trivial for all $v\in\Sigma^{\prime}$;

  * •

$a_{p}\not\equiv 1\bmod p$.

Then,

$\mathrm{Fitt}_{\Lambda_{n,m}\otimes\mathbf{Q}_{p}}(\mathcal{X}_{\mathrm{Gr}}(K^{\prime}({\mathfrak{p}}^{n+1}{\mathfrak{q}}^{m+1}),A))=(\Theta_{n,m})$

as ideals in $\Lambda_{n,m}\otimes\mathbf{Q}_{p}$ for $n+m\geq 1$.

###### Proof.

The proof is the same as theorem 7.1, but now the equality of ideals

$\mathrm{Fitt}_{\Lambda}(\mathcal{X}_{\mathrm{Gr}}(K_{\infty},A))=\mathrm{Char}_{\Lambda}(\mathcal{X}_{\mathrm{Gr}}(K_{\infty},A))=(L_{p}^{\mathrm{Hi}}(f_{\alpha}/K))=(L_{p,\Delta}(f))$

takes place inside $\Lambda\otimes\mathbf{Q}_{p}$. ∎

######  Remark 7.3.

In the definite case, Skinner and Urban [22] proved a two-variable Iwasawa
main conjecture. Their result shows that a multiple of the Hida $p$-adic
$L$-function appearing in this note with some Euler factors removed is a
generator of the characteristic ideal of the dual $\Sigma$-primitive Selmer
group (see [22, Section 3.1.3] for the definition). To prove the strong
Mazur–Tate conjecture in this setting, one would need to compare the Fitting
ideal of the dual Selmer group with the one of the dual $\Sigma$-primitive
Selmer group.

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