# On the Mordell-Weil Ranks of supersingular abelian varieties over
${\mathbb{Z}}_{p}^{2}$-extensions

Cédric Dion  Département de Mathématiques et de Statistique
Université Laval, Pavillion Alexandre-Vachon
1045 Avenue de la Médecine
Québec, QC
Canada G1V 0A6<EMAIL_ADDRESS>and  Jishnu Ray  School of
Mathematics, Tata Institute of Fundamental Research, Dr Homi Bhabha Road, Navy
Nagar, Colaba, Mumbai, Maharashtra 400005<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in
which $p$ splits. Let $A$ be an abelian variety defined over $K$ with
supersingular reduction at both primes above $p$ in $K$. Under certain
assumptions, we give a growth estimate for the Mordell–Weil rank of $A$ over
finite extensions inside the ${\mathbb{Z}}_{p}^{2}$-extension of $K$. In the
last section, written by Chris Williams, he includes some speculative remarks
on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the
multi-signed Selmer groups constructed in this paper.

###### Key words and phrases:

supersingular abelian varieties, Mordell–Weil ranks, Coleman maps, Selmer
groups.

###### 2010 Mathematics Subject Classification:

Primary: 11R23; Secondary: 11G10, 11R20

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Multi-Signed Selmer groups over ${\mathbb{Z}}_{p}^{2}$-extension
  4. 4 Coleman maps control the Mordell–Weil ranks
  5. 5 The bound on the Mordell–Weil rank
  6. 6 Speculative remarks on multi-signed $p$-adic $L$-functions for $\mathrm{GSp}(4)$ by Chris Williams

##  1\. Introduction

###  1.1. Work of Lei–Ponsinet

Let $p$ be an odd prime, $L$ be a number field and $L_{\mathrm{cyc}}$ be the
cyclotomic ${\mathbb{Z}}_{p}$-extension of $L$. Let
$L_{\mathrm{cyc}}=\cup_{n}L_{n}$ where
${\mathrm{Gal}}(L_{n}/L)\cong{\mathbb{Z}}/p^{n}{\mathbb{Z}}$. Let $A$ be an
abelian variety over $L$. Suppose $A$ has good ordinary reduction at the
primes above $p$. Then it is conjectured that the Selmer group
${\mathrm{Sel}}_{p}(A/L_{\mathrm{cyc}})$ is cotorsion as a
${\mathbb{Z}}_{p}[[{\mathrm{Gal}}(L_{\mathrm{cyc}}/L)]]$-module (follows by
the work of Kato [Kat04] and Rohrlich [Roh84] for elliptic curves defined over
${\mathbb{Q}}$ and $L/{\mathbb{Q}}$ abelian). Under this assumption Mazur's
control theorem on the Selmer groups implies that the Mordell–Weil rank
$A(L_{n})$ is bounded independently of $n$. However, this technique only works
under the ordinary assumption. If the abelian variety has supersingular
reduction at one of the primes above $p$, then the
${\mathrm{Sel}}_{p}(A/L_{\mathrm{cyc}})$ is not cotorsion and Mazur's control
theorem will not hold. Now suppose that $p$ is unramified in $L$ and $A$ has
supersingular reduction at all primes of $L$ above $p$. Then Büyükboduk and
Lei [BL17] constructed signed Selmer groups for $A$; these generalize plus and
minus Selmer groups of Kobayashi [Kob03] for $a_{p}=0$ and Sprung's
$\sharp/\flat$-Selmer groups for $p\mid a_{p}\neq 0$ [Spr12]. The former are
proved to be cotorsion for elliptic curves over ${\mathbb{Q}}$ and the latter
are conjectured to be cotorsion as well. In the case when $p\mid a_{p}\neq 0$,
at least one of the $\sharp/\flat$-Selmer groups is known to be cotorsion
[Spr12]. In the case of $a_{p}=0$, Kobayashi also proved an analogue of the
control theorem for $\pm$-Selmer groups. Combined, those result lead to a
bound independent of $n$ for the growth of the rank for supersingular elliptic
curves [Kob03, Corollary 10.2]. Under the asumption that the signed Selmer
groups of Büyükboduk and Lei are cotorsion, in [LP20], Lei and Ponsinet give
an explicit sufficient condition in terms of Coleman maps attached to $A$
which ensures that the rank of the Mordell–Weil group of the dual abelian
variety $A^{\vee}(L_{n})$ stays bounded as $n$ varies. Further they show that
when the Frobenius operator on the Dieudonné module at $p$ can be expressed as
certain block anti-diagonal matrix (cf. [LP20, Section 3.2.1]) then their
explicit condition is satisfied; hence their result applies to abelian
varieties of $GL_{2}$-type.

###  1.2. Work of Lei–Sprung

In [LS20] (see also [Spr] for a less technical exposition), Lei and Sprung
consider the special case of elliptic curves defined over ${\mathbb{Q}}$ and
the base field $F=K$, an imaginary quadratic field in which $p$ splits. The
Mordell–Weil rank of $E$ at the $n$-th layer of a ${\mathbb{Z}}_{p}$-extension
of $K$ is know to be of the form $ap^{n}+O(1)$ where $a$ is an integer called
the growth number. When $E$ has ordinary reduction at $p$, this growth number
$a$ is equal to $0$ for the cyclotomic ${\mathbb{Z}}_{p}$-extension of $K$ by
the works of Kato and Rohrlich. But works of [Ber95], [Nek12], [How04],
[Cor02] and [Vat03] exhibit cases where $a=1$ along the anticyclotomic
extension of $K$ when the root number of $E/K$ is $-1$. In the supersingular
$p\geq 5$ case, [LV15] also finds $a=1$ under some assumptions. So along the
anticyclotomic tower, the Mordell–Weil rank may be unbounded. Let $K_{\infty}$
be the ${\mathbb{Z}}_{p}^{2}$-extension of $K$ obtained as a compositum of the
cyclotomic and the anticyclotomic extension of $K$. Let $K_{n}$ be a subfield
of $K_{\infty}$ such that
${\mathrm{Gal}}(K_{n}/K)\cong({\mathbb{Z}}/p^{n}{\mathbb{Z}})^{2}$. Suppose
the prime $p$ is supersingular, Lei and Sprung constructed four
$\sharp/\flat$-Selmer groups for elliptic curves over the
$\mathbb{Z}_{p}^{2}$-extension $K_{\infty}$, generalizing works of Kim for the
case $a_{p}=0$ [Kim14]. Assume that $E$ is an elliptic curve over
${\mathbb{Q}}$ with conductor prime to $p$ and the class number of $K$ is
coprime to $p$. Suppose that one of the four $\sharp/\flat$-Selmer groups is
cotorsion, and when $a_{p}=0$, assumed that all the four signed Selmer groups
are cotorsion. Then the main result shown by Lei and Sprung is that
$\mathrm{rank}E(K_{n})=O(p^{n})$.

###  1.3. Our work in this article

Kazim Büyükboduk and Antonio Lei jointly started a program to generalize the
plus/minus theory of Kobayashi and Pollack in the case of motives crystalline
at $p$. This led to their series of joint papers [BL17], [BL15] and [BL21].
Note that [BL17] and [BL15] (including the work of Lei–Ponsinet mentioned
above) are over the cyclotomic ${\mathbb{Z}}_{p}$-extension and [BL21] is for
representations coming from modular forms. In this paper our results add to
that program giving us results for abelian varieties over a maximal abelian
pro-$p$ extension of a number field that is unramified outside $p$.

We restrict ourselves to the case of imaginary quadratic field $K$ where $p$
splits. But instead of working with elliptic curves we work with abelian
varieties with supersingular reduction at both primes above $p$. In this case,
we have at our disposal the signed Selmer groups of Büyükboduk and Lei [BL17]
which are defined only over the cyclotomic extension of $K$. The objectives in
our paper are threefold. We

  * •

define multi-signed Selmer groups for $A$ over the
${\mathbb{Z}}_{p}^{2}$-extension $K_{\infty}$. In order to achieve this we
construct multi-signed Coleman maps attached to the abelian variety $A$. This
gives us the local condition at primes above $p$.

  * •

give an explicit sufficient condition in terms of Coleman maps (hypothesis
(H-large)) mentioned in the text) attached to $A$ which ensures that the
Mordell–Weil rank is bounded by a function which is $O(p^{n})$ along the tower
$K_{n}$ (see theorem 5.5).

  * •

show that this explicit condition (H-large) is satisfied when the Frobenius on
the Dieudonné module at primes above $p$ can be expressed in a certain block
diagonal form. This special case can be thought of as an analogue for the case
$a_{p}=0$ for supersingular elliptic curves and happens for abelian varieties
of the $GL_{2}$-type (see section 4.3). When our signed Selmer groups for $A$
are cotorsion, we hence deduce $\mathrm{rank}A^{\vee}(K_{n})=O(p^{n})$ (see
theorem 5.6).

In order the state the main results more precisely, let us introduce some more
notation first. Let $A^{\vee}$ be the dual abelian variety and let
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ be the multi-signed
Selmer group attached to $A$ constructed in section 3.3. Let
$\mathcal{X}_{\underline{J}}(A^{\vee}/K_{\infty})$ denote the Pontryagin dual
$\mathrm{Hom}_{\mathrm{cts}}({\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_{p})$.
Let $\Lambda$ be the completed group ring
$\mathbb{Z}_{p}\llbracket{\mathrm{Gal}}(K_{\infty}/K)\rrbracket$. Then, our
main theorems are the following:

######  Theorem A (Theorem 5.5).

Suppose that $\mathcal{X}_{\underline{J}}(A^{\vee}/K_{\infty})$ is a torsion
$\Lambda$-module for some $\underline{J}$ and that hypothesis (H-large) is
satisfied. We have $\mathrm{rank}A^{\vee}(K_{n})=O(p^{n}).$

Let $T_{p}(A)$ be the Tate module of $A$. Suppose that
$p={\mathfrak{p}}{\mathfrak{p}}^{c}$ where ${\mathfrak{p}}$ and
${\mathfrak{p}}^{c}$ are prime ideals of $K$. Let
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T_{p}(A))$ (reps.
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}^{c}}(T_{p}(A))$) be the Dieudonné
module of $T_{p}(A)$ seen as a representation of
${\mathrm{Gal}}(\overline{K_{\mathfrak{p}}}/K_{\mathfrak{p}})$ (resp.
${\mathrm{Gal}}(\overline{K_{{\mathfrak{p}}^{c}}}/K_{{\mathfrak{p}}^{c}})$).
These modules are naturally equipped with an action of a Frobenius operator.
Let $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ be the matrices defined in
section 2.2 arising from the Frobenius action on
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T_{p}(A))$ and
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}^{c}}(T_{p}(A))$ respectively.

######  Theorem B (Theorem 5.6).

Suppose $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ are block anti-
diagonal matrices and the Pontryagin dual of the Selmer groups
${\mathrm{Sel}}_{\underline{I}_{0}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{1}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{0,1}}}(A^{{\vee}}/K_{\infty})$
and
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{1,0}}}(A^{{\vee}}/K_{\infty})$
are all $\Lambda$-torsion. Then $\mathrm{rank}A^{\vee}(K_{n})=O(p^{n}).$

The Selmer groups appearing in theorem B are all multi-signed Selmer groups
for particular choices of the indexing set $\underline{J}$ (see the discussion
before proposition 4.13).

Let the abelian variety $A$ be of dimension $g$ and such that
$C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ are block anti-diagonal. The
indexing set $\underline{J}:=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}})$ where
$J_{\mathfrak{p}}$ and $J_{{\mathfrak{p}}^{c}}$ are subsets of
$\\{1,...,2g\\}$ such that $|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$.
Note that the Selmer groups
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ depend not only on the
indexing set $\underline{J}$ but also on the choice of Hodge-compatible bases
of the Dieudonné modules $\mathbb{D}_{\mathrm{cris},{\mathfrak{p}}}(T_{p}(A))$
and $\mathbb{D}_{\mathrm{cris},{\mathfrak{p}}^{c}}(T_{p}(A))$ (see section 3.3
and remark 3.4). This is also the case for the multi-signed Selmer groups
defined in [BL17] and [LP20] for supersingular abelian varieties in the
cyclotomic case. When $2g=4$, upon fixing a Hodge-compatible bases, the
indexing set $\underline{J}$ has cardinality $70$. But all of these $70$
multi-signed Selmer groups are not canonically defined. Defining multi-signed
Selmer groups canonically is crucial in framing corresponding Iwasawa main
conjectures relating our multi-signed Selmer groups to canonically defined
$p$-adic $L$-functions in the analytic side. A new key observation made in
this paper is that the specific four multi-signed Selmer groups mentioned in
theorem B are independent of the choice of the Hodge-compatible bases and
hence canonically defined (see proposition 4.17). 111We thank David Loeffler
for suggesting this to us. Therefore, this gives a strong incentive for
regarding these particular multi-signed choices as ``more fundamental and
arithmetically interesting" than the other multi-signed Selmer groups. (In the
case of elliptic curves over the ${\mathbb{Z}}_{p}^{2}$-extension
$K_{\infty}$, they correspond to the usual four $\sharp/\flat$ signed Selmer
groups of Lei and Sprung, generalizing four $+/-$ signed Selmer groups of Kim
when $a_{p}=0$.) Apart from these four multi-signed Selmer groups, we also
show that the two other multi-signed Selmer groups corresponding to the
choices $\underline{J}=(\\{1,...,2g\\},\emptyset)$ and
$\underline{J}=(\emptyset,\\{1,...,2g\\})$ are also canonically defined and
independent of the choice of Hodge-compatible bases (see proposition 4.17).
These two Selmer groups should correspond to BDP type $p$-adic $L$-functions
(see section 6).

In section 2, we introduce our notations and recall the basic preliminaries on
$p$-adic Hodge theory that we will need throughout our article. The definition
of multi-signed Selmer groups and Coleman maps are given in section 3. In
section 4, under the hypothesis (H-large), we use multi-signed Coleman maps
and logarithmic matrices to study the growth of

$\mathcal{Y}_{n}^{\prime}\colonequals{\mathrm{Coker}}\left(H_{\mathrm{Iw}}^{1}(K_{\infty},T)_{\Gamma_{n}}\to\prod_{v|p}\frac{H^{1}(K_{n,v},T)}{H^{1}_{f}(K_{n,v},T)}\right)$

where $H^{1}_{f}(K_{n,v},T)$ is the Bloch-Kato local condition defined using
the kernel of the dual exponential map. To achieve this, we generalized the
work of [LP20] to the setting of $\mathbb{Z}_{p}^{2}$-extensions by using
inputs from [LS20]. We find that
$\mathrm{rank}_{\mathbb{Z}_{p}}(\mathcal{Y}_{n}^{\prime})=O(p^{n})$. In
section 5, we consider the short exact sequence

$0\to\mathcal{Y}_{n}\to\mathcal{X}_{n}\to\mathcal{X}_{n}^{0}\to 0$

where $\mathcal{X}_{n}$ is the Pontryagin dual of the classical $p$-Selmer
group of $A^{\vee}$ over $K_{n}$, $\mathcal{X}_{n}^{0}$ is the Pontryagin dual
of the fine Selmer group of $A^{\vee}$ and

$\mathcal{Y}_{n}\colonequals{\mathrm{Coker}}\left(H^{1}_{\Sigma}(K_{n},T)\to\prod_{v|p}\frac{H^{1}(K_{n,v},T)}{H^{1}_{f}(K_{n,v},T)}\right).$

Under the assumption that at least one of the Selmer groups
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ is $\Lambda$-cotorsion,
we show that $\mathrm{rank}_{\mathbb{Z}_{p}}(\mathcal{X}_{n}^{0})$ is also
$O(p^{n})$. Using the natural surjection
$\mathcal{Y}_{n}^{\prime}\to\mathcal{Y}_{n}$, we deduce that the growth of
$\mathcal{Y}_{n}$ is at most $O(p^{n})$. We are then able to conclude that the
Mordell-Weil rank of $A^{\vee}$ is $O(p^{n})$ along $K_{\infty}$.

One expects that there should be analytic analogues of all of the above,
corresponding to our (algebraic) results under appropriate Iwasawa main
conjectures. In section 6, written by Chris Williams, we give some speculative
remarks on the shape of the analytic theory.

In this paper we stick to the case of ${\mathbb{Z}}_{p}^{2}$-extension of an
imaginary quadratic field instead of working more generally over
${\mathbb{Z}}_{p}^{d}$-extension of a number field. Our guess is that the
growth condition will be $O(p^{(d-1)n})$, but we will need to make the
hypotheses that the multi-signed Selmer groups are cotorsion over the
corresponding Iwasawa algebra. Such hypotheses are currently only known for
elliptic curves under non-vanishing of signed $p$-adic $L$-functions over
${\mathbb{Z}}_{p}^{2}$-extension of an imaginary quadratic field (see remark
4.3). The analytic side for general abelian varieties are even more mysterious
for $d>2$ (see section 6), and in this case (the Euler system machinery is
also unavailable) as per our knowledge, there are currently no results in the
literature towards cotorsion-ness of signed Selmer groups. Even if we assume
that the signed Selmer groups are cotorsion, the present techniques in this
paper will not generalize verbatim for $d>2$. We also faced technical
difficulty generalizing (4.5) for general ${\mathbb{Z}}_{p}^{d}$-extensions.

## Acknowledgements

We are grateful to Antonio Lei for answering many of our questions. We thank
David Loeffler and Chris Williams for several useful conversations. We also
thank Kâzım Büyükboduk, Eknath Ghate, Mahesh Kakde, Chan-Ho Kim, Debanjana
Kundu, Meng Fai Lim, Filippo Nuccio, Gautier Ponsinet, Dipendra Prasad, Anwesh
Ray and Romyar Sharifi for comments and corrections that helped improving the
quality of the paper. The first named author's research is supported by the
Canada Graduate Scholarships – Doctoral program from the Natural Sciences and
Engineering Research Council of Canada. The second named author's research is
supported by the postdoctoral research grant from the Tata Institute of
Fundamental Research, Mumbai.

##  2\. Preliminaries

###  2.1. Local and Global setup

Let $p\geq 3$ be a fixed prime number for the rest of the paper. Let $K$ be an
imaginary quadratic field in which $p$ splits into the primes ${\mathfrak{p}}$
and ${\mathfrak{p}}^{c}$ of $K$. Here, $c$ denotes the complex conjugation. We
will always use ${\mathfrak{q}}$ to mean an element of the set
$\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$. Let $K_{\infty}$ be the unique
$\mathbb{Z}_{p}^{2}$-extension of $K$ with Galois group
$\Gamma\colonequals{\mathrm{Gal}}(K_{\infty}/K)$. If $\mathfrak{a}$ is an
ideal of $\mathcal{O}_{K}$, $K(\mathfrak{a})$ will denote the ray class field
of $K$ of conductor $\mathfrak{a}$. If $n\geq 0$ is an integer, write
$\Gamma_{n}\colonequals\Gamma^{p^{n}}$ and $K_{n}\colonequals
K_{\infty}^{\Gamma_{n}}=K(p^{n+1})^{{\mathrm{Gal}}(K(1)/K)}$. We make the
hypothesis that $K(1)\bigcap K_{\infty}=K$. It follows that the Galois group
$\Gamma$ is isomorphic to $G_{{\mathfrak{p}}}\times G_{{\mathfrak{p}}^{c}}$
where $G_{\mathfrak{q}}$ is the Galois group of the extension
$K({\mathfrak{q}}^{\infty})\bigcap K_{\infty}/K$. We fix topological
generators $\gamma_{\mathfrak{p}}$ and $\gamma_{{\mathfrak{p}}^{c}}$
respectively for these groups. Let
$\Lambda\colonequals\mathbb{Z}_{p}\llbracket\Gamma\rrbracket\cong\mathbb{Z}_{p}\llbracket\gamma_{\mathfrak{p}}-1,\gamma_{{\mathfrak{p}}^{c}}-1\rrbracket$.
Sometimes, we shall also use the notation $\Lambda(\Gamma)$ for $\Lambda$ when
we want to emphasize that $\Lambda$ is the Iwasawa algebra of the group
$\Gamma$. More generally, if $G$ is any profinite group, we denote by
$\Lambda(G)$ the completed group ring $\mathbb{Z}_{p}\llbracket G\rrbracket$.
Write $\Gamma_{\mathfrak{q}}$ for the decomposition group of ${\mathfrak{q}}$
in $\Gamma$. Let $\mu_{p^{n}}$ denote the set of $p^{n}$th roots of unity and
let $\mu_{p^{\infty}}\colonequals\bigcup_{n\geq 1}\mu_{p^{n}}$.

We let $F_{\infty}$ and $k_{\text{cyc}}$ be the unramified
$\mathbb{Z}_{p}$-extension and the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}_{p}$ respectively. Let $k_{\infty}$ be the compositum of
$F_{\infty}$ and $k_{\text{cyc}}$. For $n\geq 0$, $k_{n}$ and $F_{n}$ denote
the subextension of $k_{\text{cyc}}$ and $F_{\infty}$ such that
$[k_{n}:\mathbb{Q}_{p}]=p^{n}$ and $[F_{n}:\mathbb{Q}_{p}]=p^{n}$. Write
$\Gamma_{p}\colonequals{\mathrm{Gal}}(k_{\infty}/\mathbb{Q}_{p})\cong\mathbb{Z}_{p}^{2}$,
$\Gamma_{\text{ur}}\colonequals{\mathrm{Gal}}(F_{\infty}/\mathbb{Q}_{p})=\overline{\langle\sigma\rangle}\cong\mathbb{Z}_{p}$
and
$\Gamma_{\text{cyc}}\colonequals{\mathrm{Gal}}(k_{\text{cyc}}/\mathbb{Q}_{p})=\overline{\langle\gamma\rangle}\cong\mathbb{Z}_{p}$.
We identify $K_{\mathfrak{q}}$ with $\mathbb{Q}_{p}$, $\Gamma_{\mathfrak{q}}$
with $\Gamma_{p}$ and $G_{\mathfrak{q}}$ with $\Gamma_{\text{cyc}}$ via
$\gamma_{\mathfrak{q}}\mapsto\gamma$. Let $\mathcal{H}(\Gamma_{\text{cyc}})$
be the set of power series

$\sum_{n\geq 0}c_{n}\cdot(\gamma-1)^{n}$

with coefficients in $\mathbf{Q}_{p}$ such that $\sum_{n\geq 0}c_{n}X^{n}$
converges on the open unit disk.

###  2.2. $p$-adic Hodge Theory

For this subsection only, let $K$ be any number field where all primes above
$p$ are unramified. Note that for our purpose, $p$ will be totally split in
$K$ and so we can take $K_{v}=\mathbb{Q}_{p}$ in the following discussion
where $v$ is a prime of $K$ above $p$. Let $\mathcal{M}_{/K}$ be a motive
defined over $K$ in the sense of [FPR94] and $\mathcal{M}_{p}$ its $p$-adic
realization. Let $T$ be a $G_{K}$-stable $\mathbb{Z}_{p}$-lattice inside
$\mathcal{M}_{p}$. We shall denote by
$T^{\dagger}\colonequals\mathrm{Hom}(T,\mu_{p^{\infty}})$ the cartier dual of
$T$ and by $T^{\ast}(1)\colonequals\mathrm{Hom}(T,\mathbb{Z}_{p}(1))$ the Tate
dual of $T$. Suppose that

(H.crys) $\mathcal{M}_{p}$ is crystalline at all the primes $v$ above $p$ in
$K$.

For simplicity, suppose that the dimension of
$\mathrm{Ind}_{K/\mathbb{Q}}\mathcal{M}_{p}$ over $K_{v}$ is even. This will
be the case for example when the motive $\mathcal{M}$ is the motive associated
to an abelian variety. Let
$2g\colonequals\mathrm{dim}_{K_{v}}(\mathrm{Ind}_{K/\mathbb{Q}}\mathcal{M}_{p})$
and let
$g_{\pm}:=\mathrm{dim}_{K_{v}}(\mathrm{Ind}_{K/\mathbb{Q}}\mathcal{M}_{p})^{c=\pm
1}$. Let $v$ be a prime above $p$ in $K$ and let
$\mathbb{D}_{{\mathrm{cris}},v}(\mathcal{M}_{p})$ be
$(\mathbb{B}_{\mathrm{cris}}\otimes_{\mathbb{Q}_{p}}\mathcal{M}_{p})^{G_{K_{v}}}$
where $\mathbb{B}_{\mathrm{cris}}$ is the crystalline period ring defined by
Fontaine. It admits the structure of a filtered $\varphi$-module. Let
$\mathbb{A}_{K_{v}}^{+}\colonequals\mathcal{O}_{K_{v}}\llbracket\pi\rrbracket$.
Let $\mathbb{N}_{v}(T)$ be the Wach module of $T$ whose existence and
properties are shown in [Ber04, Proposition 2.1.1]. It is a free
$\mathbb{A}_{K_{v}}^{+}$-module of rank $2g$. Furthermore, the quotient
$\mathbb{N}_{v}(T)/\pi\mathbb{N}_{v}(T)$ is identified with a
$\mathcal{O}_{K_{v}}$-lattice of
$\mathbb{D}_{{\mathrm{cris}},v}(\mathcal{M}_{p})$. We denote by
$\mathbb{D}_{{\mathrm{cris}},v}(T)$ this $\mathcal{O}_{K_{v}}$-lattice. It is
equipped with a filtration of $\mathcal{O}_{K_{v}}$-modules
$\\{\mathrm{Fil}^{i}\mathbb{D}_{{\mathrm{cris}},v}(T)\\}_{i\in\mathbb{Z}}$ and
a Frobenius operator $\varphi$. If we suppose that

(H.HT) the Hodge–Tate weights of $\mathcal{M}_{p}$ are either $0$ or $1$,

the filtration takes the form

$\mathrm{Fil}^{i}\mathbb{D}_{{\mathrm{cris}},v}(T)=\begin{cases}0&\text{if
$i\geq 1$,}\\\ \mathbb{D}_{{\mathrm{cris}},v}(T)&\text{if
$i\leq-1$.}\end{cases}$

Note that
$\mathbb{D}_{{\mathrm{cris}},v}(V)=\mathbb{D}_{{\mathrm{cris}},v}(T)\otimes_{\mathcal{O}_{K_{v}}}K_{v}$.
We also make the following assumptions:

(H.Frob) The slopes of the Frobenius on the Dieudonné module
$\mathbb{D}_{{\mathrm{cris}},v}(\mathcal{M}_{p})$ lie inside $(0,-1]$ and that
$1$ is not an eigenvalue;

(H.P) $g_{+}=g_{-}$ ($=g$) and
$\mathrm{dim}_{K_{v}}\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},v}(\mathcal{M}_{p})=g$.

######  Definition 2.1.

Choose an $\mathcal{O}_{K_{v}}$-basis $\\{v_{1},\ldots,v_{2g}\\}$ of
$\mathbb{D}_{{\mathrm{cris}},v}(T)$ such that $\\{v_{1},\ldots,v_{g}\\}$ is a
$\mathcal{O}_{K_{v}}$-basis of
$\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},v}(T)$. Such a basis is called
Hodge-compatible.

The matrix of $\varphi$ with respect to this basis is of the form

$C_{\varphi,v}=C_{v}\left[\begin{array}[]{c|c}I_{g}&0\\\ \hline\cr
0&\frac{1}{p}I_{g}\end{array}\right]$

for some $C_{v}\in\mathrm{GL}_{2g}(\mathcal{O}_{K_{v}})$ and where $I_{g}$ is
the identity $g\times g$ matrix.

##  3\. Multi-Signed Selmer groups over ${\mathbb{Z}}_{p}^{2}$-extension

In this section, we define multi-signed Selmer groups for abelian varieties
over ${\mathbb{Z}}_{p}^{2}$-extension of an imaginary quadratic field
generalizing an earlier work of Büyükboduk and Lei [BL21].

Let $K$ an imaginary quadratic field where
$(p)={\mathfrak{p}}{\mathfrak{p}}^{c}$ splits. We write $K^{\text{cyc}}/K$ and
$K^{\text{ac}}/K$ for the cyclotomic and anticyclotomic
$\mathbb{Z}_{p}$-extensions contained in $K_{\infty}$ respectively.

###  3.1. Yager module

Let $L/\mathbb{Q}_{p}$ be a finite unramified extension. For
$x\in\mathcal{O}_{L}$, define

$y_{L/\mathbb{Q}_{p}}(x)\colonequals\sum_{\tau\in{\mathrm{Gal}}(L/\mathbb{Q}_{p})}\tau(x)\cdot\tau^{-1}\in\mathcal{O}_{L}[{\mathrm{Gal}}(L/\mathbb{Q}_{p})].$

Let $S_{L/\mathbb{Q}_{p}}$ be the image of $y_{L/\mathbb{Q}_{p}}$ in
$\mathcal{O}_{L}[{\mathrm{Gal}}(L/\mathbb{Q}_{p})]$. In [LZ14, Section 3.2],
it is shown that there is an isomorphism of
$\Lambda(\Gamma_{\text{ur}})$-modules

$\varprojlim_{\mathbb{Q}_{p}\subseteq L\subseteq
F_{\infty}}\mathcal{O}_{L}\cong
S_{F_{\infty}/\mathbb{Q}_{p}}\colonequals\varprojlim_{\mathbb{Q}_{p}\subseteq
L\subseteq F_{\infty}}S_{L/\mathbb{Q}_{p}}$

where the inverse limit is taken with respect to the trace maps on the left
and the projection maps
${\mathrm{Gal}}(L^{\prime}/\mathbb{Q}_{p})\to{\mathrm{Gal}}(L/\mathbb{Q}_{p})$
for $L\subseteq L^{\prime}$ on the right. By [LZ14, Proposition 3.2],
$\varprojlim_{\mathbb{Q}_{p}\subseteq L\subseteq F_{\infty}}\mathcal{O}_{L}$
is a free $\Lambda(\Gamma_{\text{ur}})$-module of rank one, thus the Yager
module $S_{F_{\infty}/\mathbb{Q}_{p}}$ is also free of rank one over
$\Lambda(\Gamma_{\text{ur}})$.

###  3.2. Interpolation property of Loeffler–Zerbes' big logarithm map

Let $\\{\Omega_{\mathbb{Q}_{p}}\\}$ be a basis of the Yager module
$S_{F_{\infty}/\mathbb{Q}_{p}}$. Recall that $T$ is a $G_{K}$-stable
$\mathbb{Z}_{p}$-lattice inside $\mathcal{M}_{p}$. Let
$\mathcal{L}_{T}^{\infty}$ be the two-variable big logarithm map of
Loeffler–Zerbes [LZ14]

$\mathcal{L}_{T}^{\infty}:H^{1}_{\mathrm{Iw}}(k_{\infty},T)\to\Omega_{\mathbb{Q}_{p}}\cdot\left(\mathcal{H}(\Gamma_{\text{cyc}})\widehat{\otimes}\Lambda(\Gamma_{\text{ur}})\right)\otimes_{\mathbb{Z}_{p}}\mathbb{D}_{\mathrm{cris}}(T).$

It is a morphism of $\Lambda(\Gamma_{p})$-modules. For $k_{n}$ a finite
subextension of $k_{\infty}$, let $\exp_{T}^{\ast}:H^{1}(k_{n},T)\to
k_{n}\otimes\mathbb{D}_{\text{cris}}(T)$ be the Bloch–Kato dual exponential
map. We say that the cyclotomic part of a finite order character
$\eta:\Gamma_{p}\to\overline{{\mathbb{Q}}}_{p}^{\times}$ is of conductor
$p^{n+1}$ if $\eta$ sends the topological generator $\gamma$ to a primitive
$p^{n}$-th root of unity. Then, the big logarithm map enjoys the following
interpolation property (see [LZ14, Theorem 4.15]):

######  Proposition 3.1.

Let $x\in H^{1}_{\mathrm{Iw}}(k_{\infty},T)$. Let $\eta$ be a character on
$\Gamma_{p}$ whose cyclotomic part is of conductor $p^{n+1}>1$. Then we have

$\Phi^{-n-1}\mathcal{L}_{T}^{\infty}(x)(\eta)=\varepsilon(\eta^{-1})\exp_{T}^{\ast}(e_{\eta}x)$

where $\varepsilon(\eta^{-1})$ is the $\varepsilon$-factor of $\eta^{-1}$,
$e_{\eta}$ is the idempotent corresponding to $\eta$ and $\Phi$ is the
operator which act as the arithmetic Frobenius $\sigma$ on $F_{m}$ and
$\varphi$ on $\mathbb{D}_{\mathrm{cris}}(T)$.

Let $2g$ denote the dimension of $\mathbb{D}_{\mathrm{cris}}(T)$ as a
$\mathbb{Z}_{p}$-module. Choose $\\{v_{i}\\}_{i=1}^{2g}$ a Hodge-compatible
$\mathbb{Z}_{p}$-basis of $\mathbb{D}_{\mathrm{cris}}(T)$ as in section 2.2.
Then, in [BL21, Section 2.3], the authors show the existence of Coleman maps

$\mathrm{Col}_{T,i}^{\infty}:H^{1}_{\mathrm{Iw}}(k_{\infty},T)\to\Omega_{\mathbb{Q}_{p}}\cdot\Lambda(\Gamma_{p})$

for $1\leq i\leq 2g$. By identifying
$\Omega_{\mathbb{Q}_{p}}\cdot\Lambda(\Gamma_{p})$ with $\Lambda(\Gamma_{p})$,
we omit $\Omega_{\mathbb{Q}_{p}}$ from the notation and see
$\mathrm{Col}_{T,i}$ as taking value in $\Lambda(\Gamma_{p})$. These Coleman
maps decompose the big logarithm map

$\mathcal{L}_{T}^{\infty}=(v_{1},\ldots,v_{2g})\cdot
M_{T}\cdot\begin{bmatrix}\mathrm{Col}_{T,1}^{\infty}\\\ \vdots\\\
\mathrm{Col}_{T,2g}^{\infty}\end{bmatrix}.$

The matrix $M_{T}$ is called a logarithmic matrix and is defined in the
following way: Let $\Phi_{p^{n}}(1+X)$ denote the $p^{n}$th cyclotomic
polynomial. For $n\geq 1$, first define the matrices

$C_{n}\colonequals\left[\begin{array}[]{c|c}I_{g}&0\\\ \hline\cr
0&\Phi_{p^{n}}(1+X)I_{g}\end{array}\right]C^{-1}$

and $M_{n}\colonequals(C_{\varphi})^{n+1}C_{n}\cdots C_{1}$ where $C$ and
$C_{\varphi}$ are defined as in section 2.2 with $v=p$. In [BL17], it is shown
that the sequence $\\{M_{n}\\}$ converges to a $2g\times 2g$ matrix with
entries in $\mathcal{H}(\Gamma_{\text{cyc}})$ which we call $M_{T}$. Let
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$. If we include the
primes ${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$ in our notation, then we have
the Coleman maps

(3.1)
$\mathrm{Col}_{T,{\mathfrak{q}},i}^{\infty}:H^{1}(K_{\mathfrak{q}},T\otimes\Lambda(\Gamma_{\mathfrak{q}})^{\iota})\rightarrow{\mathbb{Z}}_{p}[[\Gamma_{p}]]$

for $i\in\\{1,...,2g\\}$. They satisfy

$\mathcal{L}_{T,{\mathfrak{q}}}^{\infty}=(v_{1},\ldots,v_{2g})\cdot
M_{T,{\mathfrak{q}}}\cdot\begin{bmatrix}\mathrm{Col}_{T,{\mathfrak{q}},1}^{\infty}\\\
\vdots\\\ \mathrm{Col}_{T,{\mathfrak{q}},2g}^{\infty}\end{bmatrix}.$

For ${\mathfrak{q}}={\mathfrak{p}}$,
$M_{T,{\mathfrak{p}}}=\lim_{r\rightarrow\infty}M_{{\mathfrak{p}},r}$. Here
$M_{{\mathfrak{p}},r}\colonequals(C_{\varphi,{\mathfrak{p}}})^{r+1}C_{{\mathfrak{p}},r}\cdots
C_{{\mathfrak{p}},1}$,

$C_{{\mathfrak{p}},r}\colonequals\left[\begin{array}[]{c|c}I_{g}&0\\\
\hline\cr
0&\Phi_{p^{r}}(\gamma_{\mathfrak{p}})I_{g}\end{array}\right]C_{\mathfrak{p}}^{-1}.$

For ${\mathfrak{q}}={\mathfrak{p}}^{c}$ we have expressions
$M_{T,{\mathfrak{p}}^{c}}=\lim_{s\rightarrow\infty}M_{{\mathfrak{p}}^{c},s}$,
where
$M_{{{\mathfrak{p}}^{c}},s}\colonequals(C_{\varphi,{\mathfrak{p}}^{c}})^{s+1}C_{{\mathfrak{p}}^{c},s}\cdots
C_{{{\mathfrak{p}}^{c}},1}$,

$C_{{\mathfrak{p}}^{c},s}\colonequals\left[\begin{array}[]{c|c}I_{g}&0\\\
\hline\cr
0&\Phi_{p^{s}}(\gamma_{{\mathfrak{p}}^{c}})I_{g}\end{array}\right]C_{{\mathfrak{p}}^{c}}^{-1}.$

For any $r\geq 1$, $r$ an integer, define
$H_{{\mathfrak{p}},r}=C_{{\mathfrak{p}},r}\cdots C_{{\mathfrak{p}},1}$.
Similarly, for any integer $s\geq 1$, we define
$H_{{\mathfrak{p}}^{c},s}=C_{{\mathfrak{p}}^{c},s}\cdots
C_{{\mathfrak{p}}^{c},1}$.
Let $H_{r,s}$ be the block diagonal matrix where the blocks on the diagonal
are given by $H_{{\mathfrak{p}},r}$ and $H_{{\mathfrak{p}}^{c},s}$. Consider
the tuples $\underline{I}=(I_{\mathfrak{p}},I_{{\mathfrak{p}}^{c}})$ and
$\underline{J}=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}})$ where
$I_{\mathfrak{q}}\subseteq\\{1,...,2g\\}$ and
$J_{\mathfrak{q}}\subseteq\\{1,...,2g\\}$ for
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$. Let us suppose
that $|I_{\mathfrak{p}}|+|I_{{\mathfrak{p}}^{c}}|=2g$ and
$|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$.
We define $H_{\underline{I},\underline{J},r,s}$ to be the
$(\underline{I},\underline{J})^{th}$-minor of $H_{r,s}$. We also define
$\underline{I}_{0}\colonequals(I_{{\mathfrak{p}},0},I_{{\mathfrak{p}}^{c},0})$
where $I_{{\mathfrak{q}},0}=\\{1,\ldots,g\\}$.

###  3.3. Definitions and properties

The multi-signed Selmer group can be defined without the assumption that
$K(1)\cap K_{\infty}=K$. For this subsection only, we work in more generality
and drop this assumption. Suppose that there are $p^{t}$ primes above
${\mathfrak{p}}$ and ${\mathfrak{p}}^{c}$ in $K_{\infty}$. Fix a choice of
coset representatives $\gamma_{1},\ldots,\gamma_{p^{t}}$ and
$\delta_{1},\ldots,\delta_{p^{t}}$ for $\Gamma/\Gamma_{\mathfrak{p}}$ and
$\Gamma/\Gamma_{{\mathfrak{p}}^{c}}$ respectively. Since $p$ splits in $K$, we
can identify $K_{\mathfrak{q}}$ with $\mathbb{Q}_{p}$ and
$\Gamma_{\mathfrak{q}}$ with $\Gamma_{p}$. Consider the ``semi-local"
decomposition coming from Shapiro's lemma

$H^{1}(K_{\mathfrak{p}},T\otimes\Lambda^{\iota})=\bigoplus_{j=1}^{p^{t}}H^{1}(K_{\mathfrak{p}},T\otimes\Lambda(\Gamma_{\mathfrak{p}})^{\iota})\cdot\gamma_{j}\cong\bigoplus_{v|{\mathfrak{p}}}H^{1}_{\text{Iw}}(K_{\infty,v},T)$

where $v$ runs through the primes above ${\mathfrak{p}}$ in $K_{\infty}$. By
choosing a Hodge-compatible basis of
$\mathbb{D}_{\mathrm{cris},{\mathfrak{p}}}(T)$, define the Coleman map for $T$
at ${\mathfrak{p}}$ by

$\displaystyle\mathrm{Col}_{T,{\mathfrak{p}},i}^{k_{\infty}}:H^{1}(K_{\mathfrak{p}},T\otimes\Lambda^{\iota})$
$\displaystyle\to\Lambda(\Gamma)$ $\displaystyle
x=\sum_{j=1}^{p^{t}}x_{j}\cdot\gamma_{j}$
$\displaystyle\mapsto\sum_{j=1}^{p^{t}}\mathrm{Col}_{T,{\mathfrak{p}},i}^{\infty}(x_{j})\cdot\gamma_{j}$

for all $1\leq i\leq 2g$.

Let
$\mathcal{L}_{T,{\mathfrak{p}}}^{k_{\infty}}=\oplus_{j=1}^{p^{t}}\mathcal{L}_{T,{\mathfrak{p}}}^{\infty}\cdot\gamma_{j}$.
Define $\mathrm{Col}_{T,{\mathfrak{p}}^{c},i}^{k_{\infty}}$ and
$\mathcal{L}_{T,{\mathfrak{p}}^{c}}^{k_{\infty}}$ in an analogous way. Let
$J_{\mathfrak{q}}$ denote a subset of $\\{1,\ldots,2g\\}$ and let

$\displaystyle\mathrm{Col}_{T,J_{\mathfrak{q}}}:H^{1}(K_{\mathfrak{q}},T\otimes\Lambda^{\iota})$
$\displaystyle\to\bigoplus_{i=1}^{|J_{\mathfrak{q}}|}\Lambda(\Gamma)$
$\displaystyle z$
$\displaystyle\mapsto(\mathrm{Col}^{k_{\infty}}_{T,{\mathfrak{q}},i}(z))_{i\in
J_{\mathfrak{q}}}.$

Tate's local pairing induces a pairing

$\bigoplus_{v|{\mathfrak{q}}}H^{1}_{\mathrm{Iw}}(K_{\infty,v},T)\times\bigoplus_{v|{\mathfrak{q}}}H^{1}(K_{\infty,v},T^{\dagger})\to\mathbb{Q}_{p}/\mathbb{Z}_{p}$

for all places $v$ of $K_{\infty}$ above ${\mathfrak{q}}$. We define
$H^{1}_{J_{\mathfrak{q}}}(K_{\infty,{\mathfrak{q}}},T^{\dagger})\subseteq\bigoplus_{v|{\mathfrak{q}}}H^{1}(K_{\infty,v},T^{\dagger})$
as the orthogonal complement of $\ker\mathrm{Col}_{T,J_{\mathfrak{q}}}$ under
the previous pairing. If $L$ is a finite extension of $K_{\mathfrak{q}}$,
define

$H^{1}_{\mathrm{ur}}(L,\mathcal{M}_{p}^{\ast}(1))\colonequals\ker\left(H^{1}(L,\mathcal{M}_{p}^{\ast}(1))\to
H^{1}(L_{\mathrm{ur}},\mathcal{M}_{p}^{\ast}(1))\right)$

where $L_{\mathrm{ur}}$ is the maximal unramified extension of $L$. Let
$H^{1}_{\mathrm{ur}}(L,T^{\dagger})$ be the image of
$H^{1}_{\mathrm{ur}}(L,\mathcal{M}_{p}^{\ast}(1))$ under the natural map
$H^{1}(L,\mathcal{M}_{p}^{\ast}(1))\to H^{1}(L,T^{\dagger})$. Let

$H^{1}_{\mathrm{ur}}(K_{\infty,w},T^{\dagger})\colonequals\varinjlim_{\mathbb{Q}_{p}\subseteq
L\subseteq K_{\infty,w}}H^{1}_{\mathrm{ur}}(L,T^{\dagger})$

where $w$ is a place in $K_{\infty}$ and $L$ runs through finite extensions of
$\mathbb{Q}_{p}$ contained in $K_{\infty,w}$. Let
$\underline{J}\colonequals(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}})$ where as
before $J_{\mathfrak{p}}$ and $J_{{\mathfrak{p}}^{c}}$ are subsets of
$\\{1,\ldots,2g\\}$ such that
$|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$. Fix $\Sigma$ a finite set of
prime of $K$ containing the primes above $p$, the archimedean primes and the
primes of ramification of $T^{\dagger}$. Let $K_{\Sigma}$ be the maximal
extension of $K$ unramified outside $\Sigma$. Let $\Sigma^{\prime}$ be the set
of primes of $K_{\infty}$ lying above the primes in $\Sigma$. If $M$ is any
${\mathrm{Gal}}(K_{\Sigma}/K_{\infty})$-module, we denote
$H^{1}(K_{\Sigma}/K_{\infty},M)$ by $H^{1}_{\Sigma}(K_{\infty},M)$.

######  Definition 3.2.

Let

$\mathcal{P}_{\Sigma,\underline{J}}(T^{\dagger}/K_{\infty})\colonequals\prod_{w\in\Sigma^{\prime},w\nmid
p}\frac{H^{1}(K_{\infty,w},T^{\dagger})}{H^{1}_{\mathrm{ur}}(K_{\infty,w},T^{\dagger})}\times\prod_{{\mathfrak{q}}|p}\frac{\bigoplus_{v|{\mathfrak{q}}}H^{1}(K_{\infty,v},T^{\dagger})}{H^{1}_{J_{\mathfrak{q}}}(K_{\infty,{\mathfrak{q}}},T^{\dagger})}.$

The $\underline{J}$-Selmer group of $T^{\dagger}$ over $K_{\infty}$ is

${\mathrm{Sel}}_{\underline{J}}(T^{\dagger}/K_{\infty})\colonequals\ker\left(H^{1}_{\Sigma}(K_{\infty},T^{\dagger})\to\mathcal{P}_{\Sigma,\underline{J}}(T^{\dagger}/K_{\infty})\right).$

We define the fine Selmer group over $K_{\infty}$ as

${\mathrm{Sel}}_{p}^{0}(T^{\dagger}/K_{\infty}):={\mathrm{Ker}}\Big{(}H^{1}_{\Sigma}(K_{\infty},T^{\dagger})\rightarrow\prod_{v}H^{1}(K_{\infty,v},T^{\dagger})\Big{)}$

where $v$ runs through all the places of $K_{\infty}$.

######  Remark 3.3.

In the literature, ${\mathrm{Sel}}_{p}^{0}(T^{\dagger}/K_{\infty})$ is
sometime referred as the strict Selmer group instead.

Let $\mathcal{X}_{\underline{J}}(T^{\dagger}/K_{\infty})$ and
$\mathcal{X}_{0}(T^{\dagger}/K_{\infty})$ denote the Pontryagin dual of the
$\underline{J}$-Selmer group and fine Selmer group respectively. Recall that
${\mathrm{Sel}}_{\underline{J}}(T^{\dagger}/K_{\infty})$ (resp.
${\mathrm{Sel}}_{p}^{0}(T^{\dagger}/K_{\infty})$) is said to be
$\Lambda$-cotorsion if $\mathcal{X}_{\underline{J}}(T^{\dagger}/K_{\infty})$
(resp. $\mathcal{X}_{0}(T^{\dagger}/K_{\infty})$) is a torsion module over
$\Lambda$.

######  Remark 3.4.

Change of basis. The Selmer groups
${\mathrm{Sel}}_{\underline{J}}(T^{\dagger}/K_{\infty})$ depend both on the
indexing set $\underline{J}$ and the choice of the Hodge-compatible basis of
the Dieudonné module $\mathbb{D}_{\mathrm{cris},{\mathfrak{q}}}(T)$ (defined
after (H.P)). A change of Hodge-compatible basis will effect the Coleman maps
$\mathrm{Col}_{T,{\mathfrak{q}},i}^{\infty}$ in the same way as described in
[BL17, Section 2.4].

######  Remark 3.5.

Under the hypothesis $K(1)\cap K_{\infty}=K$, there is a unique prime above
${\mathfrak{q}}$ in $K_{\infty}$. In this case,

$H^{1}(K_{\mathfrak{q}},T\otimes\Lambda^{\iota})\cong
H^{1}_{\text{Iw}}(K_{\infty,v},T)$

where $v$ is the unique prime above ${\mathfrak{q}}$. Furthermore,
$\Gamma=\Gamma_{\mathfrak{q}}$ and we get Coleman maps

$\mathrm{Col}_{T,{\mathfrak{q}},i}^{k_{\infty}}:H^{1}_{\text{Iw}}(K_{\infty,v},T)\to\Lambda.$

###  3.4.  Poitou–Tate exact sequence

By [PR95, Proposition A.3.2], we have the following Poitou–Tate exact sequence

(3.2)
$H^{1}_{\text{Iw}}(K_{\infty},T)\to\frac{H^{1}(K_{\mathfrak{p}},T\otimes\Lambda^{\iota})}{\ker\mathrm{Col}_{T,J_{\mathfrak{p}}}}\oplus\frac{H^{1}(K_{{\mathfrak{p}}^{c}},T\otimes\Lambda^{\iota})}{\ker\mathrm{Col}_{T,J_{{\mathfrak{p}}^{c}}}}\to\mathcal{X}_{\underline{J}}(T^{\dagger}/K_{\infty})\to\mathcal{X}_{0}(T^{\dagger}/K_{\infty})\to
0,$

where $H^{i}_{\text{Iw}}(K_{\infty},T)$ is the inverse limit of
$H^{i}(K_{\Sigma}/K_{n},T)$ where $K\subseteq K_{n}\subseteq K_{\infty}$ and
the transition maps are the corestriction maps.

##  4\. Coleman maps control the Mordell–Weil ranks

Let $A$ be an abelian variety over $K$ of dimension $g$ having supersingular
reduction at all the primes in $K$ above $p$. Let $T=T_{p}(A)$ be the Tate
module of $A$ and $V=T_{p}(A)\otimes\mathbb{Q}_{p}$. Then $V=\mathcal{M}_{p}$
satisfies the hypotheses (H.crys), (H.HT), (H.Frob) and (H.P). Furthermore,
$T^{\dagger}=A^{\vee}[p^{\infty}]$ where $A^{\vee}$ denotes the dual abelian
variety.

###  4.1. Ranks of Iwasawa modules

Let $V$ be the ${\mathbb{Q}}_{p}$-representation associated to the
${\mathbb{Z}}_{p}$-lattice $T$. We say that $V$ satisfies weak Leopoldt's
conjecture $\mathrm{Leop}(K_{\infty},V)$ if the following equivalent
statements for some $T$ (and hence every $T$) are true.

######  Proposition 4.1.

The following statements are equivalent:

  1. (1)

$H^{2}_{\mathrm{Iw}}(K_{\infty},T)$ is $\Lambda$-torsion,

  2. (2)

$H^{2}(K_{\Sigma}/K_{\infty},T\otimes{\mathbb{Q}}_{p}/{\mathbb{Z}}_{p})=0.$

###### Proof.

See [LZ14, Proposition A.5]. ∎

We recall the fine Selmer group over $K_{\infty}$ is given by

${\mathrm{Sel}}_{p}^{0}(A^{\vee}/K_{\infty})={\mathrm{Ker}}\Big{(}H^{1}_{\Sigma}(K_{\infty},A^{\vee}[p^{\infty}])\rightarrow\prod_{v}H^{1}(K_{\infty,v},A^{\vee}[p^{\infty}])\Big{)}.$

Let $\mathcal{X}^{0}$ be the Pontryagin dual of this fine Selmer group over
$K_{\infty}$. Define

$\mathcal{I}\colonequals\\{\underline{J}=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}}):|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g\\},$

where $J_{\mathfrak{q}}\subseteq\\{1,...,2g\\}$ for
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$.

######  Lemma 4.2.

Suppose that there exists any $\underline{J}\in\mathcal{I}$ such that
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ is $\Lambda$-cotorsion.
Then $H^{1}_{\mathrm{Iw}}(K_{\infty},T)$ is of rank $2g$.

###### Proof.

By the Poitou–Tate exact sequence (3.2),
${\mathrm{Sel}}_{p}^{0}(A^{\vee}/K_{\infty})^{\vee}$ is a quotient of
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})^{\vee}$ and hence is
$\Lambda$-torsion. By [PR95, Proposition A.3.2], we have the exact sequence

$0\rightarrow{\mathrm{Sel}}_{p}^{0}(A^{\vee}/K_{\infty})^{\vee}\rightarrow
H^{2}_{\mathrm{Iw}}(K_{\infty},T)\rightarrow\prod_{v\in\Sigma}H^{2}_{\mathrm{Iw}}(K_{\infty,v},T).$

Now $H^{2}_{\mathrm{Iw}}(K_{\infty,v},T)$ is $\Lambda$-torsion for all $v$
(see [LZ14, Theorem A.2]). Therefore $H^{2}_{\mathrm{Iw}}(K_{\infty},T)$ is
$\Lambda$-torsion and hence $\mathrm{Leop}(K_{\infty},V)$ is true. Now since
$K$ is totally imaginary, [LZ14, Corollary A.8] shows that the rank of
$H^{1}_{\mathrm{Iw}}(K_{\infty},T)$ is of rank $2g$. ∎

We now discuss a partial result toward the converse of lemma 4.2. Suppose that
$\mathrm{Leop}(K_{\infty},V)$ is true. Then
$H^{1}_{\mathrm{Iw}}(K_{\infty},T)$ is of rank $2g$ over $\Lambda$ by [LZ14,
Theorem A.4]. Assume that the images of
$\mathrm{Col}_{T,{\mathfrak{q}},i}^{k_{\infty}}$ ($1\leq i\leq
2g,{\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$) are nonzero.
Thus, if $v$ and $w$ are the unique places above ${\mathfrak{p}}$ and
${\mathfrak{p}}^{c}$ in $K_{\infty}$ respectively,

(4.1) $\frac{H^{1}_{\mathrm{Iw}}(K_{\infty,v},T)\bigoplus
H^{1}_{\mathrm{Iw}}(K_{\infty,w},T)}{\ker\mathrm{Col}_{T,J_{{\mathfrak{p}}}}\bigoplus\ker\mathrm{Col}_{T,J_{{\mathfrak{p}}^{c}}}}$

is of rank $2g$ since $|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$ and
$H^{1}_{\mathrm{Iw}}(K_{\infty,v},T)\bigoplus
H^{1}_{\mathrm{Iw}}(K_{\infty,w},T)$ has rank $4g$ by [LZ14, Theorem A.2]. By
[Gre89, proof of proposition 6],
$\frac{H^{1}(K_{\infty,w},A^{\vee}[p^{\infty}])}{H^{1}_{\mathrm{ur}}(K_{\infty,w},A^{\vee}[p^{\infty}])}$
is $\Lambda$-cotorsion for $w\in\Sigma^{\prime}$ such that $w\nmid p$. We
deduce, by local Tate duality, that the $\Lambda$-module (4.1) has the same
rank as the Pontryagin dual of
$\mathcal{P}_{\Sigma,\underline{J}}(A^{\vee}[p^{\infty}]/K_{\infty})$. Thus
$\mathcal{P}_{\Sigma,\underline{J}}(A^{\vee}[p^{\infty}]/K_{\infty})$ has
$\Lambda$-corank $2g$ and it follows that
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ is the kernel of a
morphism between two modules of the same corank. This is a necessary (but not
sufficient) condition for
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ to be
$\Lambda$-cotorsion.

######  Remark 4.3.

Suppose that $A=E$ is an elliptic curve. In that case, the Coleman maps
defined in section 3.3 are the same as the $\sharp/\flat$-Coleman maps in
[LS20]

$\mathrm{Col}_{{\mathfrak{q}}}^{\sharp},\mathrm{Col}_{{\mathfrak{q}}}^{\flat}:H^{1}(K_{\mathfrak{q}},T_{p}(E)\otimes\Lambda^{\iota})\to\Lambda(\Gamma).$

For $\bullet,\circ\in\\{\sharp,\flat\\}$, it is then possible to construct
signed $p$-adic $L$-functions $\mathfrak{L}_{p}^{\bullet,\circ}(E/K)$. If we
suppose that $\mathfrak{L}_{p}^{\bullet,\circ}(E/K)$ is nonzero, then [CÇSS18,
Theorem 3.7] shows that the signed Selmer group
${\mathrm{Sel}}^{\bullet,\circ}(E/K_{\infty})$ is $\Lambda$-cotorsion.
Moreover, when $a_{p}=0$, the $\sharp/\flat$-Selmer groups generalise Kim's
$\pm/\pm$-Selmer groups [Kim14]. The cotorsion-ness of those Selmer groups is
known unconditionally by [LP19, Remark 8.5] in the case $\underline{J}=(1,1)$
and $\underline{J}=(2,2)$ corresponding to $+/+$ and $-/-$ Selmer groups.

In the next lemma, we give a condition over the cyclotomic extension which
will ensure that $H^{1}_{\mathrm{Iw}}(K_{\infty},T)$ is of rank $2g$.

######  Lemma 4.4.

Suppose that there exists any $\underline{J}\in\mathcal{I}$ such that
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K^{\mathrm{cyc}})$ is
${\mathbb{Z}}_{p}[[{\mathrm{Gal}}(K^{\mathrm{cyc}}/K)]]$-cotorsion. Then
$H^{1}_{\mathrm{Iw}}(K_{\infty},T)$ is of rank $2g$.

###### Proof.

Since $K$ is totally imaginary, in view of [LZ14, Corollary A.8], it is
sufficient to show that $\mathrm{Leop}(K_{\infty},V)$ is true. By the
discussion before Example A.7 of (loc.cit),
$\mathrm{Leop}(K^{\mathrm{cyc}},V)$ implies $\mathrm{Leop}(K_{\infty},V)$. But
[LP20, Lemma 2.4(2)] implies that
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K^{\mathrm{cyc}})$ being cotorsion as
a ${\mathbb{Z}}_{p}[[{\mathrm{Gal}}(K^{\mathrm{cyc}}/K)]]$-module is a
sufficient condition for $\mathrm{Leop}(K^{\mathrm{cyc}},V)$ to hold (see
Remark 4.5). ∎

######  Remark 4.5.

Note that [LP20, Lemma 2.4(2)] says that it needs to assume that
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K^{\mathrm{cyc}})$ is cotorsion for
all $\underline{J}\in\mathcal{I}$ in order to ensure
$\mathrm{Leop}(K^{\mathrm{cyc}},V)$ holds. But a careful analysis at their
proof reveals that only one $\underline{J}$ is sufficient.

######  Remark 4.6.

[BL17, Proposition 3.28] gives a sufficient condition for
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K(\mu_{p^{\infty}}))_{+}$ to be
cotorsion (the plus notation is defined before proposition 3.14 of [BL17]).

###  4.2. Bound using logarithmic matrices

Let $F$ be a $p$-adic local field. Consider the Bloch–Kato's dual exponential
map

$\mathrm{exp}^{*}:H^{1}(F,T)\rightarrow F\otimes\mathbb{D}_{\mathrm{cris}}(T)$

with kernel $H^{1}_{f}(F,T)$. Note that the image of the dual Bloch–Kato
exponential map $\mathrm{exp}^{*}$ lands inside
$F\otimes\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{cris}}(T)$ and
$\mathrm{Fil}^{0}\mathbb{D}_{\mathrm{cris}}(T)$ has ${\mathbb{Z}}_{p}$-rank
$g=g_{+}=g_{-}$ by (H.P). Denote

$H^{1}_{/f}(F,T)=H^{1}(F,T)/H^{1}_{f}(F,T)$

and

$H^{1}_{/f}(K_{n,p},T)=\prod_{v\mid p}H^{1}_{/f}(K_{n,v},T).$

For $n\geq 0$, we define

(4.2) $\displaystyle\mathcal{Y}^{\prime}_{n}$
$\displaystyle:={\mathrm{Coker}}\Big{(}H^{1}_{\mathrm{Iw}}(K_{\infty},T)_{\Gamma_{n}}\rightarrow\prod_{v\mid
p}H^{1}_{/f}(K_{n,v},T)\Big{)}.$

For $i\in[1,2g]$, let us suppose that we fix a family of classes
$c_{1},...,c_{2g}\in H^{1}_{\mathrm{Iw}}(K_{\infty},T)$ such that
$H^{1}_{\mathrm{Iw}}(K_{\infty},T)/\langle c_{1},\cdots,c_{2g}\rangle$ is
$\Lambda$-torsion. Their existence is guaranteed since
$H^{1}_{\mathrm{Iw}}(K_{\infty},T)$ is of rank $2g$ (under the hypothesis of
either lemma 4.2 or lemma 4.4). Let $\mathrm{loc}_{p,n}(c_{i})$ be the image
of $c_{i}$ in $H^{1}_{/f}(K_{n,p},T)$.

We write $W=\mu_{p^{\infty}}\times\mu_{p^{\infty}}$ and
${\mathbb{Z}}_{p}[w]={\mathbb{Z}}_{p}[w_{1},w_{2}]$ for $(w_{1},w_{2})\in W.$
If $F\in\Lambda$, we can evaluate $F$ at $w$ in the following way. We write
$F$ as a power series in $\gamma_{\mathfrak{p}}-1$ and
$\gamma_{{\mathfrak{p}}^{c}}-1$, say
$F_{0}(\gamma_{{\mathfrak{p}}}-1,\gamma_{{\mathfrak{p}}^{c}}-1)$. We then
define

$F(w)=F_{0}(w_{1}-1,w_{2}-1).$

For a $\Lambda$-module $M$, we write $M_{w}=M\otimes{\mathbb{Z}}_{p}[w]$ for
the ${\mathbb{Z}}_{p}[w]$-module induced by this evaluation map.

Let $o(w_{1})$ (resp. $o(w_{2})$) be the least $r_{1}$ (resp. $r_{2}$) such
that $w_{1}^{p^{r_{1}}}=1$ (resp. $w_{2}^{p^{r_{2}}}=1$). Then
${\mathbb{Z}}_{p}[w]$ is a free ${\mathbb{Z}}_{p}$-module of rank
$\varphi(p^{r})$ where $r=\max\\{o(w_{1}),o(w_{2})\\}$. Let $n\geq 1$ be an
integer. There are exactly $p^{2n}$ elements $w\in W$ such that $w^{p^{n}}=1$;
two such elements $w$ are called conjugate if there is an automorphism of the
algebraic closure of ${\mathbb{Q}}_{p}$ taking one to the other [CM81, Section
2, p. 238]. The number of conjugates of a given $w$ is the rank of
${\mathbb{Z}}_{p}[w]$.

######  Lemma 4.7.

Let $n\geq 1$ be an integer. We have

$\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n}-\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n-1}\leq\sum_{w}\mathrm{rank}_{{\mathbb{Z}}_{p}}\Big{(}\frac{H^{1}_{/f}(K_{n,p},T)}{\langle\mathrm{loc}_{p,n}(c_{1}),\cdots,\mathrm{loc}_{p,n}(c_{2g})\rangle}\Big{)}_{w}$

where the sum runs over conjugacy classes of $w\in W$ such that
$w^{p^{n}}=(1,1)$, but $w^{p^{n-1}}\neq(1,1)$.

###### Proof.

We will essentially follow the line of sketch given in [LS20, Corollary 5.7]
with some modifications. The Bloch–Kato dual exponential map implies

$\prod_{v\mid
p}H^{1}_{/f}(K_{n,v},T)\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}\cong\bigoplus_{v\mid
p}K_{n,v}^{\oplus g}.$

This gives

$\Big{(}H^{1}_{/f}(K_{n,p},T)\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}\Big{)}_{\Gamma_{n-1}}=H^{1}_{/f}(K_{n-1,p},T)\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}$

which implies

$\Big{(}H^{1}_{/f}(K_{n,p},T)\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}\Big{)}_{w}=\Big{(}H^{1}_{/f}(K_{n-1,p},T)\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}\Big{)}_{w}$

for $w^{p^{n-1}}=(1,1)$. Now suppose $M$ is a $\Lambda$-module. Then [CM81,
Lemma 2.7] implies that

$\mathrm{rank}_{{\mathbb{Z}}_{p}}M_{\Gamma_{n}}=\sum_{w}\mathrm{rank}_{{\mathbb{Z}}_{p}}M_{w}$

where the direct sum runs over the conjugacy classes of $w\in W$ such that
$w^{p^{n}}=(1,1)$. Applying this for $M=H^{1}_{/f}(K_{n,p},T)$, we deduce

$\mathrm{rank}_{{\mathbb{Z}}_{p}}H^{1}_{/f}(K_{n,p},T)=\mathrm{rank}_{{\mathbb{Z}}_{p}}H^{1}_{/f}(K_{n-1,p},T)+\sum_{w}\mathrm{rank}_{{\mathbb{Z}}_{p}}\Big{(}H^{1}_{/f}(K_{n,p},T)\Big{)}_{w},$

where the sum runs over conjugacy classes of $w\in W$ such that
$w^{p^{n}}=(1,1)$, but $w^{p^{n-1}}\neq(1,1)$. We obtain a similar
decomposition for $H^{1}_{\mathrm{Iw}}(K_{\infty},T)_{\Gamma_{n}}$. Let us
abbreviate $H^{1}_{/f}(K_{n,p},T)$ by $H^{1}_{n}$ and
$H^{1}_{\mathrm{Iw}}(K_{\infty},T)_{\Gamma_{n}}$ by $H^{1}_{\mathrm{Iw},n}$.
Functoriality of coinvariance and tensor products gives the following
commutative diagram

${H^{1}_{n}\otimes{\mathbb{Q}}_{p}}$${\cong}$${H^{1}_{n-1}\otimes{\mathbb{Q}}_{p}}$${\oplus}$${\Big{(}\oplus_{w\in
W}\big{(}H^{1}_{n}\big{)}_{w}\otimes{\mathbb{Q}}_{p}\Big{)}}$${H^{1}_{\mathrm{Iw},n}\otimes{\mathbb{Q}}_{p}}$${\cong}$${H^{1}_{\mathrm{Iw},n-1}\otimes{\mathbb{Q}}_{p}}$${\oplus}$${\Big{(}\oplus_{w\in
W}\big{(}H^{1}_{\mathrm{Iw},n}\big{)}_{w}\otimes{\mathbb{Q}}_{p}\Big{)}}$

where the sum on the right runs over conjugacy classes of $w\in W$ such that
$w^{p^{n}}=(1,1)$, but $w^{p^{n-1}}\neq(1,1)$. This implies

$\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n}-\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n-1}=\sum_{w}\mathrm{rank}_{{\mathbb{Z}}_{p}}(\mathcal{Y}^{\prime}_{n})_{w}.$

The lemma follows from noting that $\mathcal{Y}^{\prime}_{n}$ is a quotient of
$\frac{H^{1}_{/f}(K_{n,p},T)}{\langle\mathrm{loc}_{p,n}(c_{1}),\cdots,\mathrm{loc}_{p,n}(c_{2g})\rangle}.$
∎

Suppose $\theta$ be a character of $\Gamma$ sending
$(\gamma_{\mathfrak{p}},\gamma_{{\mathfrak{p}}^{c}})$ to $(w_{1},w_{2})\in W.$
Then evaluating an element of ${\mathbb{Z}}_{p}[[X,Y]]$ at $X=w_{1}-1$ and
$Y=w_{2}-1$ is equivalent to evaluating an element of $\Lambda$ at $\theta$.
Let $w=(w_{1},w_{2})\in W$ with $w_{1}$ a primitive $p^{r}$-th root of unity
and $w_{2}$ a primitive $p^{s}$-th root of unity, then the corresponding
character of $\Gamma$ has conductor
${\mathfrak{p}}^{r+1}({\mathfrak{p}}^{c})^{s+1}.$ Suppose $\theta$ be a
character of $\Gamma$ of conductor
$\mathfrak{p}^{r+1}(\mathfrak{p}^{c})^{s+1}$. When restricted to
$\Gamma_{\mathfrak{p}}$ (respectively $\Gamma_{\mathfrak{p}^{c}}$), the
character $\theta$ gives a character of $\Gamma_{p}$ whose cyclotomic part is
of conductor $p^{r+1}$ (respectively $p^{s+1}$)).

Recall that $\underline{J}=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}})$ where
$|J_{\mathfrak{p}}|+|J_{{\mathfrak{p}}^{c}}|=2g$.
Let $\mathbf{z}=z_{1}\wedge z_{2}\wedge\cdots\wedge
z_{2g}\in\bigwedge^{2g}\Big{(}\prod_{v\mid
p}H^{1}(K_{v},T\otimes{\mathbb{Z}}_{p}[[\Gamma_{p}]]^{\iota})\Big{)}.$ Then we
define

$\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{z})=\det\Big{(}\mathrm{Col}_{T,v,i}^{k_{\infty}}(z_{j})\Big{)}_{v\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\},1\leq
j\leq 2g,i\in J_{v}.}$

######  Lemma 4.8.

Let $\mathbf{z}=z_{1}\wedge z_{2}\wedge\cdots\wedge
z_{2g}\in\bigwedge^{2g}\Big{(}\prod_{v\mid
p}H^{1}(K_{v},T\otimes{\mathbb{Z}}_{p}[[\Gamma_{p}]]^{\iota})\Big{)}$ and let
$\theta$ be a character of $\Gamma$ of conductor
$\mathfrak{p}^{r+1}(\mathfrak{p}^{c})^{s+1}$. Write $e_{\theta}$ for the
idempotent corresponding to $\theta$. Then $e_{\theta}\cdot\mathbf{z}$ lies in
$\bigwedge^{2g}\Big{(}\prod_{v\mid p}H^{1}_{f}(K_{n,v},T)\Big{)}$ if and only
if

$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{z})(\theta)=0.$

###### Proof.

For $v\in\\{\mathfrak{p},\mathfrak{p}^{c}\\}$, we have the dual exponential
maps

$\mathrm{exp}_{v}^{*}:H^{1}(K_{n,v},T)\rightarrow
K_{n,v}\otimes\mathds{D}_{{\mathrm{cris}},v}(T).$

Define
${\mathrm{exp}}_{{\mathfrak{p}}}^{*}=\prod_{v\mid{\mathfrak{p}}}{\mathrm{exp}}_{v}^{*}$
and
${\mathrm{exp}}_{{\mathfrak{p}}^{c}}^{*}=\prod_{v\mid{\mathfrak{p}}^{c}}{\mathrm{exp}}_{v}^{*}$.
Then $e_{\theta}\cdot\mathbf{z}$ lies in $\bigwedge^{2g}\Big{(}\prod_{v\mid
p}H^{1}_{f}(K_{n,v},T)\Big{)}$ if and only if

$e_{\theta}\cdot\bigwedge_{1\leq j\leq
2g}\Big{(}{\mathrm{exp}}^{*}_{\mathfrak{p}}(z_{j})\times{\mathrm{exp}}^{*}_{{\mathfrak{p}}^{c}}(z_{j})\Big{)}=0.$

(Here ${\mathrm{exp}}^{*}_{\mathfrak{p}}(z_{j})$ is the dual Bloch–Kato
exponential map applied to the appropriate projection of $z_{j}$). Via the
interpolation property of the Loeffler–Zerbes' big logarithm map , this is
equivalent to

(4.3) $\bigwedge_{1\leq j\leq
2g}\Big{(}\Phi^{-r-1}\mathcal{L}_{T,{\mathfrak{p}}}^{k_{\infty}}(z_{j})\times\Phi^{-s-1}\mathcal{L}_{T,{\mathfrak{p}}^{c}}^{k_{\infty}}(z_{j})\Big{)}(\theta)=0.$

Applying $\Phi^{-r-1}$ to $\mathcal{L}_{T,{\mathfrak{p}}}^{k_{\infty}}(z_{j})$
we obtain

$\Phi^{-r-1}\mathcal{L}_{T,{\mathfrak{p}}}^{k_{\infty}}(z_{j})(\theta)=(v_{1},\cdots,v_{2g})C_{{\mathfrak{p}},r}\cdots
C_{{\mathfrak{p}},1}(\theta)\begin{bmatrix}\mathrm{Col}_{T,{\mathfrak{p}},1}^{k_{\infty}}(z_{j})^{\sigma^{-r-1}}(\theta)\\\
\vdots\\\
\mathrm{Col}_{T,{\mathfrak{p}},2g}^{k_{\infty}}(z_{j})^{\sigma^{-r-1}}(\theta)\end{bmatrix}.$

Similarly, applying $\Phi^{-s-1}$ to
$\mathcal{L}_{T,{\mathfrak{p}}^{c}}^{k_{\infty}}(z_{j})$ we obtain

$\Phi^{-s-1}\mathcal{L}_{T,{\mathfrak{p}}^{c}}^{k_{\infty}}(z_{j})(\theta)=(v_{1},\cdots,v_{2g})C_{{\mathfrak{p}}^{c},s}\cdots
C_{{\mathfrak{p}}^{c},1}(\theta)\begin{bmatrix}\mathrm{Col}_{T,{\mathfrak{p}}^{c},1}^{k_{\infty}}(z_{j})^{\sigma^{-s-1}}(\theta)\\\
\vdots\\\
\mathrm{Col}_{T,{\mathfrak{p}}^{c},2g}^{k_{\infty}}(z_{j})^{\sigma^{-s-1}}(\theta)\end{bmatrix}.$

Note that $H_{{\mathfrak{p}},r}(\theta)=C_{{\mathfrak{p}},r}\cdots
C_{{\mathfrak{p}},1}(\theta)$ and
$H_{{\mathfrak{p}}^{c},s}(\theta)=C_{{\mathfrak{p}}^{c},s}\cdots
C_{{\mathfrak{p}}^{c},1}(\theta)$. Therefore by taking wedge product we obtain
that (4.3) is equivalent to the vanishing of

$\sum_{\underline{I},\underline{J}\in\mathcal{I}}H_{\underline{I},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{z})(\theta).$

Now we claim that $H_{\underline{I},\underline{J},r,s}(\theta)$ is zero unless
$I=\underline{I}_{0}$ and $\underline{J}\in\mathcal{I}$.

If $i\in\\{g+1,...,2g\\}$, then the entire $i$-th row of
$C_{{\mathfrak{p}},r}$ (respectively $C_{{\mathfrak{p}}^{c},s}$) is divisible
by $\Phi_{p^{r}}(\gamma_{\mathfrak{p}})$ (respectively
$\Phi_{p^{s}}(\gamma_{{\mathfrak{p}}^{c}})$). That is, the lower half of
$H_{{\mathfrak{p}},r}$ and $H_{{\mathfrak{p}}^{c},s}$ are always zero. Hence,
in order for a $2g\times 2g$ minor to be nonzero we should take the upper
halves of $H_{{\mathfrak{p}},r}$ and $H_{{\mathfrak{p}}^{c},s}$. Hence the
claim follows. ∎

######  Lemma 4.9.

Let $\theta$ be a character of $\Gamma$ of conductor
${\mathfrak{p}}^{r+1}({\mathfrak{p}}^{c})^{s+1}$ and let
$w=(\theta(\gamma_{p}),\theta(\gamma_{{\mathfrak{p}}^{c}}))\in W$. If

$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)\neq
0$

then

$\mathrm{rank}_{{\mathbb{Z}}_{p}}\Big{(}\frac{H^{1}_{/f}(K_{n,p},T)}{\langle\mathrm{loc}_{p,n}(c_{1}),\cdots,\mathrm{loc}_{p,n}(c_{2g})\rangle}\Big{)}_{w}=0.$

Let ${\mathbb{Q}}_{p}(\theta)$ be the field obtained by adjoining the image of
$\theta$. When
$\mathrm{rank}_{{\mathbb{Z}}_{p}}\Big{(}\frac{H^{1}_{/f}(K_{n,p},T)}{\langle\mathrm{loc}_{p,n}(c_{1}),\cdots,\mathrm{loc}_{p,n}(c_{2g})\rangle}\Big{)}_{w}$
is nonzero, it is always bounded by
$2g\dim_{{\mathbb{Q}}_{p}}{\mathbb{Q}}_{p}(\theta).$

###### Proof.

For ${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$,

$\prod_{v\mid{\mathfrak{q}}}H^{1}_{/f}(K_{n,v},T)_{w}\otimes{\mathbb{Q}}_{p}\cong{\mathbb{Q}}_{p}(\theta)^{\oplus
g}.$

So

$\bigwedge^{2g}\Big{(}H^{1}_{/f}(K_{n,p},T)\Big{)}_{w}\otimes{\mathbb{Q}}_{p}=\bigwedge^{2g}\Big{(}\prod_{v\mid{\mathfrak{p}}}H^{1}_{/f}(K_{n,v},T)_{w}\times\prod_{v\mid{\mathfrak{p}}^{c}}H^{1}_{/f}(K_{n,v},T)_{w}\Big{)}\otimes{\mathbb{Q}}_{p}$

is a one dimensional ${\mathbb{Q}}_{p}(\theta)$-vector space. By lemma 4.8,

$\mathrm{rank}_{{\mathbb{Z}}_{p}}\Big{(}\frac{\bigwedge^{2g}H^{1}_{/f}(K_{n,p},T)}{\mathrm{loc}_{p,n}(c_{1})\wedge\cdots\wedge\mathrm{loc}_{p,n}(c_{2g})}\Big{)}_{w}>0$

if and only if

$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)=0.$

Therefore under the hypothesis of lemma 4.9,

$\mathrm{rank}_{{\mathbb{Z}}_{p}}\Big{(}\frac{\bigwedge^{2g}H^{1}_{/f}(K_{n,p},T)}{\mathrm{loc}_{p,n}(c_{1})\wedge\cdots\wedge\mathrm{loc}_{p,n}(c_{2g})}\Big{)}_{w}=0.$

Note that $\bigwedge^{2g}H^{1}_{/f}(K_{n,p},T)_{w}\otimes{\mathbb{Q}}_{p}$ is
a one dimensional ${\mathbb{Q}}_{p}(\theta)$-vector space and so the image of
$\mathrm{loc}_{p,n}(c_{1})\wedge\cdots\wedge\mathrm{loc}_{p,n}(c_{2g})$ in it
is nonzero, since the rank of the quotient is zero. This implies that
$\mathrm{rank}_{{\mathbb{Z}}_{p}}\Big{(}\frac{H^{1}_{/f}(K_{n,p},T)}{\langle\mathrm{loc}_{p,n}(c_{1}),\cdots,\mathrm{loc}_{p,n}(c_{2g})\rangle}\Big{)}_{w}=0.$
∎

We impose the following hypothesis whenever needed.

(H-large): For $r,s$ such that $|r-s|\gg 0$,
$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)\neq
0.$

######  Lemma 4.10.

Suppose the hypothesis (H-large) holds. Let $n\geq 1$ and $\Upxi_{n}$ be the
set of characters $\theta$ of $\Gamma$ which factor through $\Gamma_{n}$ but
not $\Gamma_{n-1}$ such that

$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)=0$

(with the conductor of $\theta$ being
${\mathfrak{p}}^{r+1}({\mathfrak{p}}^{c})^{s+1}$). Then the cardinality of
$\Upxi_{n}$ is bounded independent of $n$.

###### Proof.

If $\theta$ factors through $\Gamma_{n}$ but not $\Gamma_{n-1}$, then either
$r$ or $s$ equals $n$. By the hypothesis (H-large), there exists a fixed
integer $n_{0}$ (independent of $n$) such that if

$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)=0,$

then $|r-s|\leq n_{0}$. If either $r$ or $s$ equals $n$, then the number of
such $(r,s)$ is bounded and hence the result. ∎

######  Proposition 4.11.

Suppose the hypothesis (H-large) holds. Then
$\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n}=O(p^{n})$.

###### Proof.

Let $C_{n}$ be the cardinality of $\Upxi_{n}$. By lemmas 4.9 and 4.7,

$\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n}-\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n-1}\leq
2gC_{n}\varphi(p^{n}).$

Under the hypothesis (H-large), $C_{n}$ is less than a certain constant
independent of $n$. Summing the inequality over $n$, we deduce that

$\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n}-\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{1}\leq\sum_{i=2}^{n}C\varphi(p^{i})=O(p^{n}),$

where $C$ is some constant depending on $g$ and independent of $n$. ∎

######  Lemma 4.12.

Let $\underline{J}\in\mathcal{I}$ and suppose that
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})^{\vee}$ is
$\Lambda$-torsion. Then
$\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})\neq 0$.

###### Proof.

The fact that ${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})^{\vee}$ is
$\Lambda$-torsion combined with (3.2) and lemma 4.2 imply the desired result.
∎

###  4.3. Block anti-diagonal matrices

Suppose we assume that
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})^{\vee}$ is
$\Lambda$-torsion. Then it will imply that
$\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)\neq 0$ for
$|r-s|\gg 0$. But this is not enough to ensure that
$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)\neq
0$ since $H_{\underline{I}_{0},\underline{J},r,s}(\theta)$ could be zero. In
general, this is complicated to verify unless we are in some special cases
where we can explicitly write down the matrices $C_{\varphi,v}$. This special
case that we will be looking at can be thought of as an analogue for the case
$a_{p}=0$ for supersingular elliptic curves as explained below.

In this section, for each $v$, let us suppose that there exist a basis of
$\mathbb{D}_{{\mathrm{cris}},v}(T)$ such that the matrix $C_{v}$ is of the
form $\left[\begin{array}[]{c|c}0&*\\\ \hline\cr*&0\end{array}\right],$ where
* is a $g\times g$ matrix defined over ${\mathbb{Z}}_{p}$. This means
$\varphi(v_{i})\notin\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},v}(T)$ and
$\varphi^{2}(v_{i})\in\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},v}(T)$ for
all $i\in\\{1,...,g\\}$. Therefore, $C_{v,n}$ is of the form

$C_{v,n}=\left[\begin{array}[]{c|c}0&B_{v,1}\\\
\hline\cr\Phi_{p^{n}}(1+X)B_{v,2}&0\end{array}\right],$

for some invertible $g\times g$ matrices $B_{v,1}$ and $B_{v,2}$ defined over
${\mathbb{Z}}_{p}$. In the context of elliptic curve with $a_{p}=0$, we may
choose a basis $\\{\omega\\}$ for
$\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},v}(T)$ and extend it to a basis
$\\{\omega,\varphi(\omega)\\}$ of $\mathbb{D}_{{\mathrm{cris}},v}(T)$. With
this choice, the matrix $C_{v}$ is equal to the anti-diagonal matrix
$\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ [BL17, Appendix 4]. In particular,
the matrix $C_{v,n}$ will have the same structure as the matrix $C_{n}$
defined in [LS20, p. 192]. In this case, it can be shown [LS20, Proposition
5.12] that the hypothesis (H-large) is satisfied for such elliptic curves.

For all $n\geq 1$, we fix a primitive $p^{n}$-th root of unity $\zeta_{p^{n}}$
and we write $\varepsilon_{n}=\zeta_{p^{n}}-1$.

Recall that
$\underline{I}_{0}=(I_{{\mathfrak{p}},0},I_{{{\mathfrak{p}}^{c}},0})$ and
$I_{{\mathfrak{q}},0}=\\{1,...,g\\}$ for
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}.$ Let
$\underline{I}_{1}$ be the complement of $\underline{I}_{0}$. That is
$\underline{I}_{1}=(I_{{\mathfrak{p}},1},I_{{{\mathfrak{p}}^{c}},1})$ and
$I_{{\mathfrak{q}},1}=\\{g+1,...,2g\\}$ for
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}.$ Let
$\underline{I}_{\mathrm{mix}_{0,1}}=(I_{{\mathfrak{p}},0},I_{{\mathfrak{p}}^{c},1})$
and
$\underline{I}_{\mathrm{mix}_{1,0}}=(I_{{\mathfrak{p}},1},I_{{\mathfrak{p}}^{c},0})$.

######  Proposition 4.13.

Suppose $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ are block anti-
diagonal matrices and the Selmer groups
${\mathrm{Sel}}_{\underline{I}_{0}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{1}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{0,1}}}(A^{{\vee}}/K_{\infty})$
and
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{1,0}}}(A^{{\vee}}/K_{\infty})$
are all $\Lambda$-cotorsion. Then the hypothesis (H-large) holds.

###### Proof.

For any $k\geq 1$, let $\delta_{k}$ be the constant

$\delta_{k}=\begin{cases}\frac{\varepsilon_{1}}{\varepsilon_{2}}\cdot\frac{\varepsilon_{3}}{\varepsilon_{4}}\cdots\frac{\varepsilon_{k-2}}{\varepsilon_{k-1}}&\text{if
$k$ is odd}\\\
\frac{\varepsilon_{1}}{\varepsilon_{2}}\cdot\frac{\varepsilon_{3}}{\varepsilon_{4}}\cdots\frac{\varepsilon_{k-1}}{\varepsilon_{k}}&\text{if
$k$ is even.}\end{cases}$

Let $\theta$ be a character of $\Gamma$ of conductor
${\mathfrak{p}}^{r+1}({\mathfrak{p}}^{c})^{s+1}$. By [LP20, Lemma 3.5],

$H_{{\mathfrak{q}},k}(\zeta_{p^{k}}-1)=\begin{cases}\left[\begin{array}[]{c|c}0&\delta_{k}(B_{{\mathfrak{q}},1}B_{{\mathfrak{q}},2})^{\frac{k-1}{2}}B_{{\mathfrak{q}},1}\\\
\hline\cr 0&0\end{array}\right]&\text{if $k$ is odd,}\\\ &\\\
\left[\begin{array}[]{c|c}\delta_{k}(B_{{\mathfrak{q}},1}B_{{\mathfrak{q}},2})^{\frac{k}{2}}&0\\\
\hline\cr 0&0\end{array}\right]&\text{if $k$ is even,}\end{cases}$

where $k=r$ if ${\mathfrak{q}}={\mathfrak{p}}$ and $k=s$ if
${\mathfrak{q}}={\mathfrak{p}}^{c}$. Hence,

(4.4) $H_{\underline{I}_{0},\underline{J},r,s}(\theta)=0\text{ unless
}\begin{cases}\underline{J}=\underline{I}_{0}&\text{if $r$ is even and $s$ is
even,}\\\ \underline{J}=\underline{I}_{1}&\text{if $r$ is odd and $s$ is
odd,}\\\
\underline{J}=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}}),J_{\mathfrak{p}}=I_{{\mathfrak{p}},1},J_{{\mathfrak{p}}^{c}}=I_{{\mathfrak{p}}^{c},0}&\text{if
$r$ is odd and $s$ is even,}\\\
\underline{J}=(J_{\mathfrak{p}},J_{{\mathfrak{p}}^{c}}),J_{\mathfrak{p}}=I_{{\mathfrak{p}},0},J_{{\mathfrak{p}}^{c}}=I_{{\mathfrak{p}}^{c},1}&\text{if
$r$ is even and $s$ is odd.}\end{cases}$

Under the hypotheses on the Selmer groups being cotorsion $\Lambda$-modules,
we obtain $\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})\neq 0$ for
$\underline{J}=\underline{I}_{0},\underline{J}=\underline{I}_{1},\underline{J}=\underline{I}_{\mathrm{mix}_{0,1}}$
and $\underline{J}=\underline{I}_{\mathrm{mix}_{1,0}}$ depending on the parity
of $r$ and $s$. Therefore, when $|r-s|\gg 0$, by [LS20, Corollary 5.10], we
obtain

(4.5)
$\operatorname{val}_{p}\Big{(}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)\Big{)}=a+\frac{b}{\varphi(p^{r})}+\frac{c}{\varphi(p^{s})}.$

In particular, for $|r-s|\gg 0$,
$\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)\neq 0$ and
hence

$\sum_{\underline{J}\in\mathcal{I}}H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)=H_{\underline{I}_{0},\underline{J},r,s}\mathrm{Col}_{T,\underline{J}}^{k_{\infty}}(\mathbf{c})(\theta)\neq
0,$

where
$\underline{J}\in\\{\underline{I}_{0},\underline{I}_{1},\underline{I}_{\mathrm{mix}_{0,1}},\underline{I}_{\mathrm{mix}_{1,0}}\\}$
depending on the parity of $r$ and $s$ given in (4.4). This shows that
hypothesis (H-large) is satisfied.

∎

######  Remark 4.14.

Proposition 4.13 is a generalization of [LP20, Lemma 3.6] and it was already
assumed (and conjectured) there that the Selmer groups are cotorsion [LP20,
Conjecture 2.2].

As mentioned previously, the Selmer groups
${\mathrm{Sel}}_{\underline{J}}(A^{\vee}[p^{\infty}]/K_{\infty})$ not only
depends on the choice of subset $\underline{J}$ but also on the choice of
$\mathbb{Z}_{p}$-bases for $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T)$
and $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}^{c}}(T)$. However, as we will
see, the Selmer groups
${\mathrm{Sel}}_{\underline{I}_{0}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{1}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{0,1}}}(A^{{\vee}}/K_{\infty})$
and
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{1,0}}}(A^{{\vee}}/K_{\infty})$
are canonically defined when $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$
are block anti-diagonal matrices.

######  Definition 4.15.

Fix a Hodge-compatible $\mathbb{Z}_{p}$-basis
$\mathcal{B}_{\mathfrak{q}}=\\{v_{1},\ldots,v_{2g}\\}$ of
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$. Hypothesis (H.HT) implies
that $\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$ is a
direct summand of $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$ [BL17,
Remark 2.3]. Let
$N_{\mathfrak{q}}\subseteq\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$
denote the free $\mathbb{Z}_{p}$-module complementary to
$\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$ and generated
by $\\{v_{g+1},\ldots,v_{2g}\\}$. We say that a $\mathbb{Z}_{p}$-basis
$\mathcal{B}_{\mathfrak{q}}^{\prime}=\\{w_{1},\ldots,w_{2g}\\}$ is Hodge-
compatible with the basis $\mathcal{B}_{\mathfrak{q}}$ if
$\\{w_{1},\ldots,w_{g}\\}$ (resp. $\\{w_{g+1},\ldots,w_{2g}\\}$) generated the
submodule $\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$
(resp. $N_{\mathfrak{q}}$).

######  Lemma 4.16.

Let $\mathcal{B}_{\mathfrak{q}}$ and $\mathcal{B}_{\mathfrak{q}}^{\prime}$ be
two Hodge-compatible bases in the sense of definition 2.1. Suppose that
$C_{\mathfrak{q}}$ is a block anti-diagonal matrix with respect to the basis
$\mathcal{B}_{\mathfrak{q}}$. Then $\mathcal{B}_{\mathfrak{q}}^{\prime}$ is
Hodge-compatible with $\mathcal{B}_{\mathfrak{q}}$ in the sense of definition
4.15.

###### Proof.

Since $\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$ is a
direct summand of $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)$, we get
decompositions
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)=\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)\bigoplus
N_{\mathfrak{q}}$ and
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)=\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T)\bigoplus
N_{\mathfrak{q}}^{\prime}$ where $N_{\mathfrak{q}}$ is generated by
$\\{v_{g+1},\ldots,v_{2g}\\}$ and $N_{\mathfrak{q}}^{\prime}$ is generated by
$\\{w_{g+1},\ldots,w_{2g}\\}$. Since $C_{{\mathfrak{q}}}$ is block anti-
diagonal with respect to $\mathcal{B}_{\mathfrak{q}}$, we have that
$\mathrm{Span}\\{\varphi(v_{1}),\ldots,\varphi(v_{g})\\}\subseteq
N_{\mathfrak{q}}$ and since $\varphi$ is injective we in fact have equality.
By a similar argument,
$\mathrm{Span}\\{\varphi(v_{1}),\ldots,\varphi(v_{g})\\}=N_{\mathfrak{q}}^{\prime}$.
Thus $\\{w_{g+1},\ldots,w_{2g}\\}$ generates $N_{\mathfrak{q}}$. ∎

Let $\underline{I}_{\text{BDP}_{2g,\emptyset}}=(\\{1,...,2g\\},\emptyset)$ and
$\underline{I}_{\text{BDP}_{\emptyset,2g}}=(\emptyset,\\{1,...,2g\\})$. These
indices should correspond to Selmer groups related with BDP type $p$-adic
$L$-functions.

######  Proposition 4.17.

Let $\mathcal{B}_{\mathfrak{p}}$ and $\mathcal{B}_{{\mathfrak{p}}^{c}}$ be
fixed Hodge-compatible bases of
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T)$ and
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}^{c}}(T)$. Suppose that
$C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ are block anti-diagonal with
respect to these bases. Let
$\underline{J}\in\\{\underline{I}_{0},\underline{I}_{1},\underline{I}_{\mathrm{mix}_{0,1}},\underline{I}_{\mathrm{mix}_{1,0}},\underline{I}_{\text{BDP}_{2g,\emptyset}},\underline{I}_{\text{BDP}_{\emptyset,2g}}\\}$.
Then ${\mathrm{Sel}}_{\underline{J}}(A^{\vee}[p^{\infty}]/K_{\infty})$ does
not depend on the choice of Hodge-compatible bases for
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T)$ and
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}^{c}}(T)$.

###### Proof.

Write $\mathcal{B}_{\mathfrak{p}}=\\{v_{1},\ldots,v_{2g}\\}$ and let
$\mathcal{B}_{\mathfrak{p}}^{\prime}=\\{w_{1},\ldots,w_{2g}\\}$ be another
Hodge-compatible basis of $\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T)$.
Recall that Hodge-compatible means that $\\{v_{1},\ldots,v_{g}\\}$ and
$\\{w_{1},\ldots,w_{g}\\}$ are bases for the $\mathbb{Z}_{p}$-submodule
$\mathrm{Fil}^{0}\mathbb{D}_{{\mathrm{cris}},{\mathfrak{p}}}(T)$. Let
$B_{\mathfrak{p}}$ be the change of basis matrix from
$\mathcal{B}_{\mathfrak{p}}^{\prime}$ to $\mathcal{B}_{\mathfrak{p}}$. Then
the hypothesis on both basis forces $B_{\mathfrak{p}}$ to be block diagonal.
Write

$B_{\mathfrak{p}}=\left[\begin{array}[]{c|c}B_{1,1}&0\\\ \hline\cr
0&B_{2,2}\end{array}\right]$

where $B_{1,1},B_{2,2}\in{\mathrm{GL}}_{g}(\mathbb{Z}_{p})$. Let
$\mathrm{Col}_{T,{\mathfrak{p}},v_{i}}^{k_{\infty}}$ be the $i$-th Coleman map
as defined in section 3.3 with respect to the basis
$\mathcal{B}_{\mathfrak{p}}$. Let
$\mathrm{Col}_{T,{\mathfrak{p}},\mathcal{B}_{\mathfrak{p}}}^{k_{\infty}}$
denotes the vector of Coleman maps
$(\mathrm{Col}_{T,{\mathfrak{p}},v_{i}}^{k_{\infty}})_{i=1}^{2g}$ which we see
as a column vector. Define similarly
$\mathrm{Col}_{T,{\mathfrak{p}},\mathcal{B}_{\mathfrak{p}}^{\prime}}^{k_{\infty}}$
the column vector of Coleman maps defined with respect to
$\mathcal{B}_{\mathfrak{p}}^{\prime}$. Since $C_{\mathfrak{p}}$ is block anti-
diagonal, lemma 4.16 gives us that $\mathcal{B}_{\mathfrak{p}}^{\prime}$ is
Hodge-compatible with $\mathcal{B}_{\mathfrak{p}}$. Hence we can use [BL17,
Lemma 2.16] to deduce that both vectors of Coleman maps are related by the
linear transformation
$\mathrm{Col}_{T,{\mathfrak{p}},\mathcal{B}_{\mathfrak{p}}^{\prime}}^{k_{\infty}}=B\cdot\mathrm{Col}_{T,{\mathfrak{p}},\mathcal{B}_{\mathfrak{p}}}^{k_{\infty}}.$
Thus,

$\begin{bmatrix}\mathrm{Col}_{T,{\mathfrak{p}},w_{1}}^{k_{\infty}}\\\
\vdots\\\
\mathrm{Col}_{T,{\mathfrak{p}},w_{g}}^{k_{\infty}}\end{bmatrix}=B_{1,1}\begin{bmatrix}\mathrm{Col}_{T,{\mathfrak{p}},v_{1}}^{k_{\infty}}\\\
\vdots\\\
\mathrm{Col}_{T,{\mathfrak{p}},v_{g}}^{k_{\infty}}\end{bmatrix},\quad\begin{bmatrix}\mathrm{Col}_{T,{\mathfrak{p}},w_{g+1}}^{k_{\infty}}\\\
\vdots\\\
\mathrm{Col}_{T,{\mathfrak{p}},w_{2g}}^{k_{\infty}}\end{bmatrix}=B_{2,2}\begin{bmatrix}\mathrm{Col}_{T,{\mathfrak{p}},v_{g+1}}^{k_{\infty}}\\\
\vdots\\\ \mathrm{Col}_{T,{\mathfrak{p}},v_{2g}}^{k_{\infty}}\end{bmatrix}.$

This implies that
$(\mathrm{Col}_{T,{\mathfrak{p}},w_{i}}^{k_{\infty}}(z))_{i=1}^{g}=0$ if and
only if $(\mathrm{Col}_{T,{\mathfrak{p}},v_{i}}^{k_{\infty}}(z))_{i=1}^{g}=0$
since $B_{1,1}$ is invertible. The same goes for
$(\mathrm{Col}_{T,{\mathfrak{p}},w_{i}}^{k_{\infty}}(z))_{i=g+1}^{2g}$ and
$(\mathrm{Col}_{T,{\mathfrak{p}},v_{i}}^{k_{\infty}}(z))_{i=g+1}^{2g}$. We
conclude that $\ker\mathrm{Col}_{T,J_{\mathfrak{p}}}$ is independent of the
choice of basis if $J_{{\mathfrak{p}}}$ is $I_{{\mathfrak{p}},0}$,
$I_{{\mathfrak{p}},1}$ or $\\{1,...,2g\\}$. The same argument holds for
${\mathfrak{p}}$ replaced by ${\mathfrak{p}}^{c}$ and so the result follows. ∎

######  Remark 4.18.

It is easy to see that $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ remain
block anti-diagonal matrices upon a change of basis. This follows because
$B_{\mathfrak{q}}C_{{\mathfrak{q}}}B_{{\mathfrak{q}}}^{-1}$ is block anti-
diagonal if $B_{\mathfrak{q}}$ is block diagonal and $C_{\mathfrak{q}}$ is
block anti-diagonal for
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$.

##  5\. The bound on the Mordell–Weil rank

Let $\Sigma$ be a finite set of primes of $K$ dividing $p$, the archimedean
primes and the primes of bad reduction of $A^{\vee}$. Let $K_{\Sigma}$ be the
maximal extension of $K$ unramified outside $\Sigma$. Let
$H^{1}_{\Sigma}(K_{n},T)$ be the Galois cohomology group
$H^{1}({\mathrm{Gal}}(K_{\Sigma}/K_{n}),T)$.

For $n\geq 0$, we define

$\displaystyle\mathcal{Y}_{n}$
$\displaystyle:={\mathrm{Coker}}\Big{(}H^{1}_{\Sigma}(K_{n},T)\rightarrow\prod_{v\mid
p}H^{1}_{/f}(K_{n,v},T)\Big{)},$
$\displaystyle{\mathrm{Sel}}_{p}^{0}(A^{\vee}/K_{n})$
$\displaystyle:={\mathrm{Ker}}\Big{(}H^{1}_{\Sigma}(K_{n},A^{\vee}[p^{\infty}])\rightarrow\prod_{v}H^{1}(K_{n,v},A^{\vee}[p^{\infty}])\Big{)}.$

Let $\mathcal{X}_{n}^{0}$ and $\mathcal{X}_{n}$ be the dual of the fine Selmer
group ${\mathrm{Sel}}_{p}^{0}(A^{\vee}/K_{n})$ and the classical $p$-Selmer
group ${\mathrm{Sel}}_{p}(A^{\vee}/K_{n})$ respectively. The Poitou–Tate exact
sequence [PR95, Proposition A.3.2] gives

$H^{1}_{\Sigma}(K_{n},T)\rightarrow\prod_{v\mid
p}\frac{H^{1}(K_{n,v},T)}{H^{1}_{f}(K_{n,v},T)}\rightarrow\mathcal{X}_{n}\rightarrow\mathcal{X}_{n}^{0}\rightarrow
0.$

For $v\mid p$, as $H^{1}_{f}(K_{n,v},T)\cong
A(K_{n,v})\otimes{\mathbb{Z}}_{p}$ (ref. [BL17, Remark 3.27]), we obtain the
exact sequence,

(5.1)
$0\rightarrow\mathcal{Y}_{n}\rightarrow\mathcal{X}_{n}\rightarrow\mathcal{X}_{n}^{0}\rightarrow
0.$

######  Theorem 5.1.

Let $M$ be a finitely generated $\Lambda$-module of rank $r$. Then,

$\mathrm{rank}_{\mathbb{Z}_{p}}M_{\Gamma_{n}}=rp^{2n}+O(p^{n}).$

###### Proof.

This is [Har00, Theorem 1.10]. ∎

######  Lemma 5.2.

Let $w$ be a place above ${\mathfrak{q}}$ in $K_{\infty}$. We have that
$A^{\vee}(K_{\infty,w})[p^{\infty}]=0$.

###### Proof.

We follow the arguments in the proof of [LP20, Lemma 1.1]. Let
$\kappa_{\mathfrak{q}}$ denote the residue field of $K_{\mathfrak{q}}$. By
[Maz72, Lemma 5.11], the reduction map $A^{\vee}(K_{\mathfrak{q}})\to
A^{\vee}(\kappa_{\mathfrak{q}})$ induces an isomorphism on $p$-torsion points.
Thus, the supersingularity of $A^{\vee}$ at ${\mathfrak{q}}$ implies that
$A^{\vee}(K_{\mathfrak{q}})[p^{\infty}]=0$. Furthermore, for a place $w$ above
${\mathfrak{q}}$ in $K_{\infty}$, the decomposition group
${\mathrm{Gal}}(K_{\infty,w}/K_{\mathfrak{q}})$ is a pro-$p$ group. We deduce
that $A^{\vee}(K_{\infty,w})$ also has no $p$-torsion by an application of the
orbit-stabilizer theorem as shown by the following argument.

Let $v$ be a prime above ${\mathfrak{q}}$ in $K_{n}$. Suppose for the sake of
contradiction that $A^{\vee}(K_{n,v})[p]$ is not trivial. Write

$A^{\vee}(K_{n,v})[p]=\bigcup_{i}\mathrm{Orb}(x_{i})$

as a disjoint union of orbits under the action of
${\mathrm{Gal}}(K_{n,v}/K_{\mathfrak{q}})$. If $x_{i}\neq 0$,
${\mathrm{Gal}}(K_{n,v}/K_{\mathfrak{q}})$ cannot fix $x_{i}$ or else $x_{i}$
would be a $p$-torsion point in $A^{\vee}(K_{\mathfrak{q}})$. So, if
$x_{i}\neq 0$, then the orbit-stabilizer theorem

$|\mathrm{Orb}(x_{i})|=\frac{|{\mathrm{Gal}}(K_{n,v}/K_{\mathfrak{q}})|}{|\mathrm{Stab}(x_{i})|}$

implies that $p$ divides $|\mathrm{Orb}(x_{i})|$ since
${\mathrm{Gal}}(K_{n,v}/K_{\mathfrak{q}})$ is a $p$-group. Thus, all the
$|\mathrm{Orb}(x_{i})|$ are divisible by $p$ except $\mathrm{Orb}(0)$ which is
of cardinality $1$. Hence, $|A^{\vee}(K_{n,v})[p]|\equiv 1\bmod p$. This is a
contradiction, since $A^{\vee}(K_{n,v})[p]$ must contain a subgroup of order
$p$. We conclude that $A^{\vee}(K_{n,v})[p]=0$ and thus
$A^{\vee}(K_{\infty,w})[p]=\bigcup_{n}A^{\vee}(K_{n,v})[p]=0$. ∎

######  Remark 5.3.

If $w$ is a place above ${\mathfrak{q}}$ in $K(\mu_{p^{\infty}})$, then the
fact that the torsion part of $A^{\vee}(K(\mu_{p^{\infty}})_{w})$ is finite
was first showed by Imai in [Ima75].

######  Proposition 5.4.

Suppose ${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})^{\vee}$ is
$\Lambda$-torsion for some $\underline{J}\in\mathcal{I}$. Then, the
${\mathbb{Z}}_{p}$-rank of $\mathcal{X}_{n}^{0}$ is $O(p^{n}).$

###### Proof.

We follow the outline of the proof of [LS20, Proposition 5.5]. We begin by
considering the commutative diagram with exact rows

${0}$${{\mathrm{Sel}}_{p}^{0}(A^{\vee}/K_{n})}$${H^{1}(K_{n},A^{\vee}[p^{\infty}])}$${\prod_{v_{n}}H^{1}(K_{n,v_{n}},A^{\vee}[p^{\infty}])}$${0}$${{\mathrm{Sel}}_{p}^{0}(A^{\vee}/K_{\infty})^{\Gamma_{n}}}$${H^{1}(K_{\infty},A^{\vee}[p^{\infty}])^{\Gamma_{n}}}$${\prod_{w}H^{1}(K_{\infty,w},A^{\vee}[p^{\infty}])^{\Gamma_{n}},}$$\scriptstyle{\alpha_{n}}$$\scriptstyle{\beta_{n}}$$\scriptstyle{\prod_{\gamma_{n}}}$

where the vertical maps are restriction maps, the product in the first row
runs over all the places $v_{n}$ of $K_{n}$ and the product in the second row
over all the places $w$ of $K_{\infty}$. By lemma 5.2,
$A^{\vee}(K_{\infty,w})[p^{\infty}]=0$ and hence $\prod\gamma_{n}$ is an
isomorphism for all $v_{n}|p$ by the inflation-restriction exact sequence. As
the $p$-torsion global points inject into the $p$-torsion local points, by the
same argument, $\beta_{n}$ is also an isomorphism. If $v_{n}\nmid p$, [Gre03,
page 270] shows that $\ker\gamma_{n}$ is finite and of bounded order as
$K_{n}$ varies. The snake lemma then shows that $\ker\alpha_{n}$ and
$\operatorname{coker}\alpha_{n}$ are bounded independently of $n$. By the
proof of lemma 4.2, $\mathcal{X}^{0}$ is $\Lambda$-torsion. It follows from
theorem 5.1 and the control on $\alpha_{n}$ that

$\mathrm{rank}_{\mathbb{Z}_{p}}\mathcal{X}_{n}^{0}=\mathrm{rank}_{\mathbb{Z}_{p}}(\mathcal{X}^{0})_{\Gamma_{n}}=O(p^{n}).$

∎

######  Theorem 5.5.

Suppose that ${\mathrm{Sel}}_{\underline{J}}(A^{\vee}/K_{\infty})$ is
$\Lambda$-cotorsion for some $\underline{J}\in\mathcal{I}$ and hypothesis
(H-large) hold. Then the bound for the Mordell–Weil rank is given by
$\mathrm{rank}A^{\vee}(K_{n})=O(p^{n}).$

###### Proof.

Under hypothesis (H-large), proposition 4.11 implies
$\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n}=O(p^{n})$. It is
easy to see that there is a natural surjection
$\mathcal{Y}^{\prime}_{n}\rightarrow\mathcal{Y}_{n}$. In particular
$\mathrm{rank}_{Z_{p}}\mathcal{Y}_{n}\leq\mathrm{rank}_{{\mathbb{Z}}_{p}}\mathcal{Y}^{\prime}_{n}$.
Then the theorem follows from the exact sequence (5.1) and proposition 5.4. ∎

######  Theorem 5.6.

Suppose $C_{\mathfrak{p}}$ and $C_{{\mathfrak{p}}^{c}}$ are block anti-
diagonal matrices and the Selmer groups
${\mathrm{Sel}}_{\underline{I}_{0}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{1}}(A^{{\vee}}/K_{\infty})$,
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{0,1}}}(A^{{\vee}}/K_{\infty})$
and
${\mathrm{Sel}}_{\underline{I}_{\mathrm{mix}_{1,0}}}(A^{{\vee}}/K_{\infty})$
are all $\Lambda$-cotorsion. Then
$\mathrm{rank}_{{\mathbb{Z}}_{p}}(\mathcal{X}_{n})=O(p^{n})$. Hence the bound
for the Mordell–Weil rank is given by $\mathrm{rank}A^{\vee}(K_{n})=O(p^{n}).$

###### Proof.

The hypotheses imply that the hypothesis (H-large) holds by proposition 4.13.
Rest of the proof follows from that of theorem 5.5. ∎

Examples. Suppose that $A$ is an abelian variety defined over ${\mathbb{Q}}$
with good supersingular reduction at both prime over $p$ in $K$ such that the
algebra of ${\mathbb{Q}}$-endomorphisms of $A$ contains a number field $E$
with $[E:{\mathbb{Q}}]=\dim A$. Such abelian varieties are said to be of
${\mathrm{GL}}_{2}$-type. Suppose further that the ring of integers
$\mathcal{O}_{E}$ of $E$ is the ring of ${\mathbb{Q}}$-endomorphisms of $A$
and that $p$ is unramified in $E$. It follows that the $p$-adic Tate module of
$A$ splits into

$T\cong\bigoplus_{\ell|p}T_{\ell}(A)$

where the direct sum runs over all the prime $\ell$ of $E$ above $p$ and
$T_{\ell}(A)$ is a free $\mathcal{O}_{\ell}$-module of rank $2$ where
$\mathcal{O}_{\ell}$ is the completion of $\mathcal{O}_{E}$ at $\ell$. Then,
it was proved in [LP20, Section 3.3] that there exists a basis of
$\mathbb{D}_{{\mathrm{cris}},{\mathfrak{q}}}(T_{\ell}(A))$ where the action of
$\varphi$ is given a matrix of the form
$\begin{bmatrix}0&\frac{b_{\ell,{\mathfrak{q}}}}{p}\\\
1&\frac{a_{\ell,{\mathfrak{q}}}}{p}\end{bmatrix}$ for some
$a_{\ell,{\mathfrak{q}}}\in p\mathcal{O}_{\ell}$ and
$b_{\ell,{\mathfrak{q}}}\in\mathcal{O}_{\ell}^{\times}$. If we assume that
$a_{\ell,{\mathfrak{q}}}=0$ for all $\ell$ and
${\mathfrak{q}}\in\\{{\mathfrak{p}},{\mathfrak{p}}^{c}\\}$, both matrices
$C_{\mathfrak{q}}$ will be block anti-diagonal and theorem 5.6 holds.

######  Remark 5.7.

Let $A$ be an abelian variety over a number field $F$ with good ordinary
reduction at all the primes above $p$ in $F$. Let $F_{\infty}$ be a uniform
admissible $p$-adic extension of $F$ of dimension $d\geq 2$. In this context,
assume that the Selmer group over $F_{\infty}$ is cotorison over the
corresponding Iwasawa algebra. Under the strong hypothesis of the
$\mathfrak{M}_{H}(G)$-conjecture, Hung and Lim gives an explicit bound on the
growth of the Mordell–Weil rank of $A$ along $F_{\infty}$ (see [HL20, Theorem
3.1]). In the case of a supersingular elliptic curve, they also give an
explicit bound for the Mordell–Weil rank of $E$ along the
$\mathbb{Z}_{p}^{2}$-extension of a quadratic imaginary field where $p$ splits
(see [HL20, Theorem 6.3, Conjecture 2]). Under the same assumptions, in the
recent work [Ray21], A. Ray proved stronger bounds for the Mordell–Weil rank
of an elliptic curve with good ordinary reduction at all the primes above $p$
along the noncommutative extension $F(E[p^{\infty}])$ (see [Ray21, Theorem 2.5
and Remark 2.6]. Therefore, it is a natural question to generalize those
result in the supersingular case when $F_{\infty}$ is a
${\mathbb{Z}}_{p}^{d}$-extension over $F$ containing the cyclotomic extension
$F_{\mathrm{cyc}}$. One might also ask if it is possible to remove this
hypothesis on the $\mathfrak{M}_{H}(G)$-conjecture, or at least replace it
with something weaker. Another avenue to explore is the case of mixed
reduction type. Let $E$ be an elliptic curve defined over a number field $F$
where $p$ splits completely. Suppose that $E$ has good reduction at primes
above $p$. In [LL21], Lei and Lim constructed multi-signed Selmer groups where
they allow both ordinary or supersingular reduction at primes above $p$. Under
the assumption that at least one prime is supersingular for $E$ and that the
dual of the aforementioned Selmer groups are torsion over the appropriate
Iwasawa algebra, they show [LL21, Theorem 5.9] that the Mordell-Weil rank of
$E$ stays bounded along the cyclotomic $\mathbb{Z}_{p}$-extension of $F$. One
may ask if this can be generalized to abelian varieties with mixed reduction
type at primes above $p$ over more general extensions. These are our future
projects and needs further research.

##  6\. Speculative remarks on multi-signed $p$-adic $L$-functions for
$\mathrm{GSp}(4)$ by Chris Williams

We thank Chris Williams222CW would like to thank Antonio Lei and David
Loeffler for very informative conversations on this topic, though any and all
misconceptions are his own. for allowing us to include this in our paper.

There are a number of works in the literature proving signed Iwasawa main
conjectures relating signed Selmer groups to signed $p$-adic $L$-functions. As
such, it is natural to ask if the Selmer groups of the present paper have
corresponding $p$-adic $L$-functions. We finally make some (very) speculative
remarks in this direction.

###  6.1. Elliptic curves

In the case of elliptic curves over ${\mathbb{Q}}$ with good supersingular
reduction, there are two signed Selmer groups, and two `standard' $p$-adic
$L$-functions, i.e. those interpolating $L$-values in the standard (and only)
critical region. Via [Pol03] these give rise to two signed $p$-adic
$L$-functions, and the signed Iwasawa main conjectures [CÇSS18] relate all
possible Selmer groups and all possible $p$-adic $L$-functions.

Let $K$ be an imaginary quadratic field in which $p$ splits. For elliptic
curves over $K$ with good supersingular reduction at both primes above $p$,
$\mathcal{I}$ has size 6, containing the pairs

$(J_{\mathfrak{p}},J_{\mathfrak{p}^{c}})=(\varnothing,\\{1,2\\}),\
(\\{1\\},\\{1\\}),\ (\\{1\\},\\{2\\}),\ (\\{2\\},\\{1\\}),\
(\\{2\\},\\{2\\}),\ (\\{1,2\\},\varnothing).$

These are the examples of Proposition 4.17, and thus all 6 correspond to
canonical multi-signed Selmer groups in this case. On the analytic side, there
are only _4_ multi-signed $p$-adic $L$-functions, constructed in [Loe14, §5]
from the 4 standard $p$-adic $L$-functions of §4 _op. cit_. These correspond
to the 4 Selmer groups where $|J_{\mathfrak{p}}|=|J_{\mathfrak{p}^{c}}|=1$.
The discrepancy here arises as the standard $p$-adic $L$-functions do not
account for all possible regions where the $L$-function has critical values.
This phenomenon is discussed in [BDP13, Fig. 4.1] or [LLZ15, Fig. 6.1]. There
should be two additional two-variable $p$-adic $L$-functions of `BDP type',
interpolating the critical $L$-values in the `non-standard' region. When $E$
can be defined over ${\mathbb{Q}}$, i.e. when $\mathcal{F}$ is base-change,
one- and two-variable $p$-adic $L$-functions interpolating values in this
region were studied in [BDP13] and [Wan] respectively. The Selmer groups where
$J_{\mathfrak{p}}$ or $J_{\mathfrak{p}^{c}}$ is empty should be related to
these BDP type $p$-adic $L$-functions. Their construction for non-base-change
Bianchi modular forms, however, remains mysterious.

We briefly elaborate why there are four standard $p$-adic $L$-functions. Via
modularity, we expect that if it is not a $\mathbb{Q}$-curve, $E/K$
corresponds to a weight 2 cuspidal Bianchi newform $\mathcal{F}$ of level
$\Gamma_{0}(\mathfrak{n})$, for some ideal
$\mathfrak{n}\subset\mathcal{O}_{K}$ prime to $p\mathcal{O}_{K}$. There are
four $p$-refinements to level $\Gamma_{0}(p\mathfrak{n})$: there are
independently two refinements at $\mathfrak{p}$ and two at $\mathfrak{p}^{c}$,
corresponding to choices of roots of the Hecke polynomials at $\mathfrak{p}$
and $\mathfrak{p}^{c}$ (cf. [BSW21, §2.1]). This amounts to choosing one of
the (four) Hecke eigenspaces in $S_{2}(\Gamma_{0}(p\mathfrak{n}))$ where the
prime-to-$p$ Hecke operators act as they do on $\mathcal{F}$. Each
$p$-refinement yields a standard $p$-adic $L$-function.

Rephrasing, $\mathcal{F}$ generates an automorphic representation $\pi$ of
$\mathrm{GL}_{2}(\mathbb{A}_{K})$ whose local components $\pi_{\mathfrak{p}}$
and $\pi_{\mathfrak{p}^{c}}$ are both unramified principal series. A
$p$-refinement is a pair of (independent) choices of $U_{\mathfrak{p}}$ and
$U_{\mathfrak{p}^{c}}$ eigenvalues in the 2-dimensional spaces
$\pi_{\mathfrak{p}}^{\mathrm{Iw}_{\mathfrak{p}}}$ and
$\pi_{\mathfrak{p}^{c}}^{\mathrm{Iw}_{\mathfrak{p}^{c}}}$ of Iwahori-invariant
vectors.

Summarising, in the case of elliptic curves, all of the 6 elements of the set
$\mathcal{I}$ have attached multi-signed Selmer groups and $p$-adic
$L$-functions, which we expect to be related by suitable Iwasawa main
conjectures.

###  6.2. Abelian surfaces

Suppose now $2g=2$, i.e. $A$ is an abelian surface. Under the paramodularity
conjecture a suitably `generic' abelian surface $A/K$ should correspond to a
cuspidal automorphic representation $\Pi$ of
$\mathrm{GSp}_{4}({\mathbb{A}}_{K})$ (and this is known after base-change to
some finite extension by [BCGP]). If $A$ has good reduction at the primes
above $p$, then $\Pi_{\mathfrak{p}}$ and $\Pi_{\mathfrak{p}^{c}}$ will be
unramified.

In this case, the set $\mathcal{I}$ has size 70. By analogy to the elliptic
curves case, one might naively expect that there are 70 multi-signed Selmer
groups with 70 corresponding multi-signed $p$-adic $L$-functions. A key
difference in this case, however, is that unlike weight 2 classical and
Bianchi modular forms, where it exists $\Pi$ is _not cohomological_. This
causes some degeneration, which on the algebraic side is reflected in
Proposition 4.17 of the main text: only 6 elements of $\mathcal{I}$ lead to
canonically defined multi-signed Coleman maps that are independent of the
choice of Hodge-compatible basis. We now indicate how this degeneration occurs
on the analytic side.

####  6.2.1. Counting $p$-adic $L$-functions: cohomological case

First consider the cohomological case, which is the closest direct analogue to
the elliptic curves setting. Let $\Pi$ be a cohomological cuspidal automorphic
representation of $\mathrm{GSp}_{4}({\mathbb{A}}_{K})$ unramified at the
primes above $p$. As above, there should be standard $p$-adic $L$-functions
attached to all $p$-refinements of $\Pi$ to sufficiently deep level at $p$. In
this case, the _Panchishkin condition_ [Pan94] predicts that we should look
for Hecke eigensystems that arise in the Klingen-parahoric invariants of
$\Pi_{\mathfrak{p}}$ and $\Pi_{\mathfrak{p}^{c}}$. One may compute that the
Klingen-parahoric invariants of $\Pi_{\mathfrak{p}}$ are 4-dimensional (e.g.
[OST, Prop 4.3]). Given $\Pi$ is cohomological, we expect that this
4-dimensional space should be the direct sum of four 1-dimensional Hecke
eigenspaces, each with different Hecke eigenvalues. There should then be 16
standard $p$-adic $L$-functions for $\Pi$, corresponding to the 16 distinct
pairs of choices of Klingen-invariant Hecke eigensystems in
$\Pi_{\mathfrak{p}}$ and $\Pi_{\mathfrak{p}^{c}}$.

It is natural to expect higher-weight analogues of the constructions of the
present article, upon which we may ask which elements of $\mathcal{I}$ these
$p$-adic $L$-functions should correspond to. An alternative version of this
theory can be described after transferring $\Pi$ to an automorphic
representation $\pi$ of $\mathrm{GL}_{4}({\mathbb{A}}_{K})$ via [AS06], as
considered in [DJR20, §3.3] or [BSDG+, §5–7]. One sees that, for the
Asgari–Shahidi normalisations considered in [BSDG+], the 16 standard $p$-adic
$L$-functions should correspond to Selmer groups where $J_{\mathfrak{p}}$ and
$J_{\mathfrak{p}^{c}}$ are independently allowed to be one of
$\\{1,2\\},\\{1,3\\},\\{4,2\\},\\{4,3\\}$ (i.e. each can contain precisely one
of $\\{1,4\\}$ and precisely one of $\\{2,3\\}$). It is reasonable to expect
that from these, one can construct 16 multi-signed $p$-adic $L$-functions (see
§6.2.3).

This accounts for only 16 of the 70 elements in $\mathcal{I}$:

  * (a)

There are a further 20 elements of $\mathcal{I}$ where each of
$J_{\mathfrak{p}}$ and $J_{\mathfrak{p}^{c}}$ both have size 2, but are not of
the special shape above: these should correspond to $p$-refinements of the
$\mathrm{GL}_{4}$-representation $\pi$ that are not of `Shalika type', as in
[BSDG+, §5–7].

  * (b)

There are 16 elements of $\mathcal{I}$ where $|J_{\mathfrak{p}}|=1$ (so
$|J_{\mathfrak{p}^{c}}|=3$), and 16 when $|J_{\mathfrak{p}^{c}}|=1$.

  * (c)

Finally, there are 2 cases where either $J_{\mathfrak{p}}=\varnothing$ or
$J_{\mathfrak{p}^{c}}=\varnothing$.

All of this is reflected in the fact that $L(\Pi,s)$ now has many more
critical regions, analogous for example to [LZb, §2.3]. The standard $p$-adic
$L$-functions above and in (a) will only see the standard critical region. The
2 cases of type (c) should correspond to BDP-style $p$-adic $L$-functions, and
will interpolate values in another of the critical regions. One might expect
more types of $p$-adic $L$-functions, corresponding to type (b) elements of
$\mathcal{I}$ and interpolating $L$-values in the other regions. This hints at
a tantalising, but at present very mysterious, picture of Selmer groups and
$p$-adic $L$-functions in this setting.

####  6.2.2. Counting $p$-adic $L$-functions: non-cohomological case

Now suppose $\Pi$ is not cohomological, attached to an abelian variety $A$
with supersingular reduction at each prime above $p$. This causes degeneracy
in the above picture.

In the picture of [LZb, §2.3], a number of the critical regions degenerate
away to nothing: that is, some of the critical regions are empty in the non-
cohomological case. (This is an analogue, for example, of the fact that the
standard critical region is empty for weight 1 modular forms). In particular,
in the type (b) cases above there is no hope of constructing $p$-adic
$L$-functions with interpolative properties for non-cohomological $\Pi$.

There is also degeneracy in the 16 standard $p$-adic $L$-functions. In the
supersingular case, the Frobenius eigenvalues at $p$ can only be
$\pm\sqrt{p}$, each appearing twice. The 4-dimensional Klingen-invariants in
$\Pi_{\mathfrak{p}}$ hence cannot split into four disjoint 1-dimensional Hecke
eigenspaces, but only as a direct sum of two 2-dimensional Hecke eigenspaces,
putting us in the so-called `irregular' setting. For $\mathrm{GL}_{2}$ over
${\mathbb{Q}}$ and $K$, the irregular setting was for example studied in
[BW21]. As a result, there should only be 4 standard $p$-adic $L$-functions in
this case, corresponding to independent choices of eigenvalue $\pm\sqrt{p}$ at
$\mathfrak{p}$ and $\mathfrak{p}^{c}$.

These 4 standard $p$-adic $L$-functions should then have multi-signed
analogues, corresponding to the 4 canonical Selmer groups in Proposition 4.17
with $|J_{\mathfrak{p}}|=|J_{\mathfrak{p}^{c}}|=2$.

This picture holds in higher dimensions: for cohomological $\Pi$, the number
of standard $p$-adic $L$-functions for $\mathrm{GSp}_{2g}$ will grow with $g$,
but in the case of supersingular abelian varieties, there will only ever be 2
choices (again either $\pm\sqrt{p}$) of $p$-refinement for each of
$\mathfrak{p}$ and $\mathfrak{p}^{c}$, reflected on the algebraic side in the
four `standard' cases in Proposition 4.17.

The other cases of Proposition 4.17, where $J_{\mathfrak{p}}$ or
$J_{\mathfrak{p}^{c}}$ is empty, provide good evidence that there exist two-
variable BDP-style $p$-adic $L$-functions attached to abelian varieties.

####  6.2.3. What is known?

For automorphic representations of $\mathrm{GSp}_{4}$ over $K$, even the
existence of standard $p$-adic $L$-functions remains conjectural. We briefly
mention some related work when the base field is ${\mathbb{Q}}$. For
cohomological representations $\Pi$ on $\mathrm{GSp}(4)/{\mathbb{Q}}$,
standard $p$-adic $L$-functions for $\Pi$ have been constructed (after
transfer to $\pi$ on $\mathrm{GL}(4)$) for Klingen-invariant eigensystems that
correspond to non-ordinary, _non-critical_ $p$-refinements for
$\mathrm{GL}(4)$ (see [BSDW]). When $\pi$ is `good supersingular' at $p$, and
there are two non-critical slope $p$-refinements $\tilde{\pi}_{1}$ and
$\tilde{\pi}_{2}$ of $\pi$, recent work of [Roc] and [LR] constructs two
signed $p$-adic $L$-functions attached to $\pi$. They use the unbounded
$p$-adic $L$-functions $L_{p}(\tilde{\pi}_{1})$ and $L_{p}(\tilde{\pi}_{2})$
of [BSDW], and prove that

$L_{p}(\tilde{\pi}_{1})\pm
L_{p}(\tilde{\pi}_{2})=L_{p}^{\pm}\times\mathrm{log}_{\pi}^{\pm},$

where $L_{p}^{\pm}$ is a measure and $\mathrm{log}_{\pi}^{\pm}$ is Pollack's
(unbounded) $\pm$-logarithmic distribution. The signed $p$-adic $L$-functions
are then the measures $L_{p}^{\pm}$.

One might expect the existence of two more (critical slope) signed $p$-adic
$L$-functions in this case. This is presently not known, but one can guess at
the shape of this theory. There are two further critical slope $p$-refinements
of $\Pi$, transferring to critical slope $p$-refinements $\tilde{\pi}_{3}$ and
$\tilde{\pi}_{4}$ of $\pi$ via the process in [BSDG+, §5]. In the
`supersingular' setting above, these refinements can be shown to have the same
slope for the $U_{p}$ operator defined by $\mathrm{diag}(p1_{n},1_{n})$;
hence, if $\tilde{\pi}_{3}$ and $\tilde{\pi}_{4}$ are both non-critical (cf.
[BSDW, Rem. 3.15]), then the corresponding $p$-adic $L$-functions constructed
in Theorem 6.23 _op. cit_. have the same growth property. One might reasonably
expect (in line with [Roc, LR]) that $L_{p}(\tilde{\pi}_{3})\pm
L_{p}(\tilde{\pi}_{4})$ is of the form [measure]± $\times$ [explicit unbounded
logarithmic distribution]±. However, unlike in [Roc, LR], it is not possible
to deduce this solely using the interpolative properties of
$L_{p}(\tilde{\pi}_{3})$ and $L_{p}(\tilde{\pi}_{4})$ alone: the growth is too
big, so there are insufficient critical $L$-values to uniquely determine
$L_{p}(\tilde{\pi}_{3})\pm L_{p}(\tilde{\pi}_{4}).$

The above constructions fundamentally use that $\Pi$ (hence $\pi$) appears in
the Betti cohomology of locally symmetric spaces, and break down for non-
cohomological $\Pi$. To construct $p$-adic $L$-functions for such $\Pi$, one
strategy is to deform from cohomological to non-cohomological weight along an
eigenvariety. In the present setting, this appears extremely difficult, as the
points are non-regular, non-cohomological, and non-ordinary. In particular:

  * –

One must obtain good control over the $\mathrm{GSp}_{4}$-eigenvariety at non-
regular, non-cohomological points, analogous to the study done in [BDP19] for
$\mathrm{GL}_{2}/{\mathbb{Q}}$.

  * –

In the $\mathrm{GSp}_{4}$ setting, the relevant points will not be ordinary,
meaning one cannot study them using Hida families. Instead one must use
Coleman families, which do not exist over the whole weight space but only over
smaller neighbourhoods.

  * –

For $\mathrm{GSp}_{4}/K$, since $\mathrm{GSp}_{4}(\mathbb{C})$ does not admit
discrete series, the picture is even worse. In general, it is not expected
that the $p$-refinements of $A/K$ will vary in classical families in the
eigenvariety; this is a folklore conjecture [Urb11, Intro.], a general
analogue of [CM09]. In this scenario deformation from cohomological weight is
impossible.

For $A/{\mathbb{Q}}$ ordinary, a $p$-adic $L$-function attached to its
(unique) ordinary $p$-refinement is given in [LZa, Prop. 3.3]. However, the
method of proof – using higher Hida theory, via the coherent cohomology of
Shimura varieties – will not apply to the analogous setting over $K$, where
the corresponding locally symmetric spaces are not even algebraic varieties.

Summarising, for general supersingular $A$ defined over an imaginary quadratic
field $K$ with $p$ split, constructing the 4 conjectured standard $p$-adic
$L$-functions attached to $A/K$ seems completely out of reach with present
methods.

Finally, consider the degenerate case where $A/K$ is a product of elliptic
curves $E_{1}\times\cdots\times E_{g}$, each $E_{i}$ defined over $K$. The
Galois representation attached to $A$ will just be a direct sum of the Tate
modules of the $E_{i}$, and its $L$-function will be a product of the
$L(E_{i},s)$. Good candidates for the multi-signed $p$-adic $L$-functions in
this case would be products of the multi-signed $p$-adic $L$-functions
attached to $E_{i}$ constructed in [Loe14, Prop. 5.2], where we take the same
choice in $\\{++,+-,-+,--\\}$ at each of the $E_{i}$.

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