# On the $\lambda$-invariant of Selmer groups arising from certain quadratic
twists of Gross curves

Jianing Li

###### Abstract.

Let $q$ be a prime with $q\equiv 7\ \mathrm{mod}\ 8$, and let
$K={\mathbb{Q}}(\sqrt{-q})$. Then $2$ splits in $K$, and we write
${\mathfrak{p}}$ for either of the primes $K$ above $2$. Let $K_{\infty}$ be
the unique ${\mathbb{Z}}_{2}$-extension of $K$ unramified outside
${\mathfrak{p}}$ with $n$-th layer $K_{n}$. For certain quadratic and
biquadratic extensions ${\mathfrak{F}}/K$, we prove a simple exact formula for
the $\lambda$-invariant of the Galois group of the maximal abelian 2-extension
unramified outside ${\mathfrak{p}}$ of the field
${\mathfrak{F}}_{\infty}={\mathfrak{F}}K_{\infty}$. Equivalently, our result
determines the exact ${\mathbb{Z}}_{2}$-corank of certain Selmer groups over
${\mathfrak{F}}_{\infty}$ of a large family of quadratic twists of the higher
dimensional abelian variety with complex multiplication, which is the
restriction of scalars to $K$ of the Gross curve with complex multiplication
defined over the Hilbert class field of $K$. We also give applications on the
associated Selmer groups over $K_{n}$ in the case $\lambda=1$.

###### Key words and phrases:

Iwasawa theory, Selmer groups, elliptic curves

###### 2010 Mathematics Subject Classification:

11R23, 11G05

##  1\. Introduction

The aim of the present paper is to prove a simple exact formula for the
$\lambda$-invariant of a certain classical Iwasawa module at the prime $p=2$.
Our result can be viewed as an analogue of a result of Kida [9] and Ferrero
[5]. However, there is a fundamental difference in that, unlike Kida’s and
Ferrero’s work, the Iwasawa module we consider has a simple interpretation in
terms of classical infinite descent theory on a family of abelian varieties
with complex multiplication arising from certain quadratic twists of the Gross
[8] family of elliptic curves with complex multiplication.

We first explain our result in terms of classical Iwasawa theory. Let $q$ be
any prime with $q\equiv 7\ \mathrm{mod}\ 8$, and let
$K={\mathbb{Q}}(\sqrt{-q})$. Write ${\mathcal{O}}_{K}$ for the ring of
integers of $K$. Then $2$ splits in $K$, say
$2{\mathcal{O}}_{K}={\mathfrak{p}}{\mathfrak{p}}^{*}$. By class field theory,
there is a unique ${\mathbb{Z}}_{2}$-extension $K_{\infty}/K$ which is
unramified outside ${\mathfrak{p}}$, and we write $K_{n}$ for the unique
intermediate field with $[K_{n}:K]=2^{n}$, and ${\mathcal{O}}_{K_{n}}$ for the
ring of integers of $K_{n}$. Throughout, we assume that ${\mathfrak{p}}$
corresponds to the embedding $\iota_{\mathfrak{p}}:K\to{\mathbb{Q}}_{2}$
determined by $\iota_{\mathfrak{p}}(\sqrt{-q})\equiv 1\bmod
4{\mathbb{Z}}_{2}$. For each prime ${\mathfrak{w}}$ of $K$, we write
${\mathrm{ord}}_{\mathfrak{w}}$ for the normalized additive valuation at
${\mathfrak{w}}$. Also, put ${\mathfrak{q}}=\sqrt{-q}{\mathcal{O}}_{K}$.

In all that follows, $R$ will denote any element of ${\mathcal{O}}_{K}$, which
will always be assumed to satisfy the following:-

Twisting Hypothesis We have $R\equiv 1\ \mathrm{mod}\ 4{\mathcal{O}}_{K}$,
$({\mathfrak{q}},R)=1$, and ${\mathrm{ord}}_{{\mathfrak{w}}}(R)$ is odd for
each prime ${\mathfrak{w}}$ of $K$ dividing $R$.

We then define

(1.1) $F=K(\sqrt{-\sqrt{-q}R}),\quad F^{\prime}=K(\sqrt{\sqrt{-q}R}),\quad
D=K(\sqrt{-1})\quad\text{ and }\quad J=FF^{\prime}=FD.$

For any finite extension $T/K$, let

(1.2) $T_{n}=TK_{n},\quad T_{\infty}=TK_{\infty}.$

Further, let $M(T_{\infty})$ be the maximal abelian $2$-extension of
$T_{\infty}$, which is unramified outside the primes of $T_{\infty}$ lying
above ${\mathfrak{p}}$. Then the main result of this paper is as follows.

######  Theorem 1.1.

Let $s_{\infty}(R)$ be the number of primes of $K_{\infty}$ dividing
$\sqrt{-q}R$, where $R$ satisfies the above Twisting Hypothesis. Then (i)
$X(F_{\infty})$ is a free finitely generated ${\mathbb{Z}}_{2}$-module of
exact rank $s_{\infty}(R)-1$; and (ii) if $q\equiv 7\bmod 16$, then
$X(F^{\prime}_{\infty})$ (resp. $X(J_{\infty})$) is a free finitely generated
${\mathbb{Z}}_{2}$-module of exact rank $s_{\infty}(R)$ (resp.
$2s_{\infty}(R)-1$).

The number $s_{\infty}(R)$ is finite and can be computed explicitly as
follows. Let $h$ be the class number of $K$, which is odd. Suppose
${\mathfrak{w}}$ is a prime of $K$ distinct from ${\mathfrak{p}}$. Then
${\mathfrak{w}}^{h}=(w)$ is principal. Adjust the sign of $w$ so that
$\iota_{\mathfrak{p}}(w)\equiv 1\bmod 4{\mathbb{Z}}_{2}$. Then, by class field
theory (see the proof of Lemma 2.3 of [4] or of Lemma 3.3 of [3]), the number
of primes of $K_{\infty}$ dividing ${\mathfrak{w}}$ is

$2^{{\mathrm{ord}}_{2}(\iota_{\mathfrak{p}}(w)-1)-2}.$

It follows that if $q\equiv 7\bmod 16$ and $R=1$, we have $s_{\infty}(R)=1$
whence $X(J_{\infty})$ is a free ${\mathbb{Z}}_{2}$-module of rank $1$. This
recovers [4, Theorem 1.1]. It was proven in [10] that $X(F_{\infty})\neq 0$
when $q\equiv 15\bmod 16$ and $R=1$, but the methods there are different from
those of the present paper. The fact that $X(F_{\infty})$ is a free finitely
generated ${\mathbb{Z}}_{2}$-module of ${\mathbb{Z}}_{2}$-rank at most
$s_{\infty}(R)-1$ has been proven when $q=7$ in [1], and the argument extends
easily to the more general case considered here. In particular, it can be
proved in this way that $X(D_{\infty})=0$ when $q\equiv 7\bmod 16$ (see [4,
Proposition 2.2] or Proposition 3.2).

As we shall explain in §4, Theorem 1.1 has an equivalent formulation in terms
of a certain Selmer group arising from infinite descent on the $h$-dimensional
abelian variety which is the twist by the quadratic extension $K(\sqrt{R})/K$
of the abelian variety $B/K$; here $B/K$ is the restriction of scalars from
the Hilbert class field $H$ of $K$ to $K$ of the Gross elliptic curve $A/H$
with complex multiplication by ${\mathcal{O}}_{K}$ [8]. It seems that no such
clear cut result giving the exact ${\mathbb{Z}}_{2}$-corank of Selmer groups
of a large family of quadratic twists of an abelian variety over a
${\mathbb{Z}}_{2}$-extension of the base field was known previously. In §5,
shall we discuss more on the case $s_{\infty}(R)=2$ where $R$ is a square free
rational integer satisfying the Twisting Hyptheosis.

The author is deeply grateful to John Coates for discussing this problem and
for his careful reading and polishing of the manuscript. Thanks also goes to
Yongxiong Li for helpful discussions and comments.

##  2\. Prelimaries

For this section alone, we consider a more general situation than in the rest
of the paper. Take $p$ to be an arbitrary prime number, and take $K$ to be any
imaginary quadratic field in which $p$ splits. We fix one of the primes
${\mathfrak{p}}$ of $K$ above lying above $p$. By class field theory, there is
a unique ${\mathbb{Z}}_{p}$-extension $K_{\infty}$ of $K$ which is unramified
outside ${\mathfrak{p}}$. If $W/K$ is any finite extension, we define
$W_{\infty}=WK_{\infty}$. Let $M(W_{\infty})$ be the maximal abelian
$p$-extension of $W_{\infty}$, which is unramified outside the primes lying
above ${\mathfrak{p}}$, and put
$X(W_{\infty})={\mathrm{Gal}}(M(W_{\infty})/W_{\infty})$. Again by class field
theory, we have:-

######  Lemma 2.1.

There are only finitely many primes of $W_{\infty}$ lying above each prime of
$W$.

For each finite extension ${\mathfrak{W}}/W$, we define
$S_{\infty}({\mathfrak{W}}/W)$ to be the set of primes of $W_{\infty}$, which
do not lie above ${\mathfrak{p}}$, and which are ramified in
${\mathfrak{W}}_{\infty}$. Of course, $S_{\infty}({\mathfrak{W}}/W)$ is a
finite set by Lemma 2.1. Put

(2.1) $s_{\infty}({\mathfrak{W}}/W)=\\#(S_{\infty}({\mathfrak{W}}/W)).$

Assume that ${\mathfrak{W}}/W$ is a cyclic extension of degree $p$ with
${\mathfrak{W}}\not\subset W_{\infty}$. Thus
${\mathfrak{W}}_{\infty}/W_{\infty}$ is cyclic of degree $p$, and we put
$\Delta={\mathrm{Gal}}({\mathfrak{W}}_{\infty}/W_{\infty})$, As usual,
$X({\mathfrak{W}}_{\infty})$ has its natural structure as a module over the
group ring ${\mathbb{Z}}_{p}[\Delta]$. The maximal ideal of the local ring
${\mathbb{Z}}_{p}[\Delta]$ is $(p,\delta-1)$, and its residue field is the
finite field ${\mathbb{F}}_{p}$ with $p$ elements; here $\delta$ is any
generator of $\Delta$. If $X$ is a ${\mathbb{Z}}_{p}[\Delta]$-module, put
$X_{\Delta}=X/(\delta-1)X$.

######  Lemma 2.2.

Let ${\mathfrak{W}}/W$ be a cyclic extension of degree $p$ with
${\mathfrak{W}}\not\subset F_{\infty}$. Then (i) $X({\mathfrak{W}}_{\infty})$
is finitely generated as a ${\mathbb{Z}}_{p}$-module if and only if
$X(W_{\infty})$ is finitely generated as a ${\mathbb{Z}}_{p}$-module, (ii) if
$X(W_{\infty})=0$, then $(X({\mathfrak{W}}_{\infty}))_{\Delta}$ is an
${\mathbb{F}}_{p}$-vector space of dimension at most
$s_{\infty}({\mathfrak{W}}/W)-1$, and (iii) the restriction map induces an
isomorphism
$(X({\mathfrak{W}}_{\infty})\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p})_{\Delta}\cong
X(W_{\infty})\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}$.

###### Proof.

If $X({\mathfrak{W}}_{\infty})$ is a finitely generated
${\mathbb{Z}}_{p}$-module, then so is its quotient
${\mathrm{Gal}}({\mathfrak{W}}_{\infty}M(W_{\infty})/{\mathfrak{W}}_{\infty})$.
Since $[{\mathfrak{W}}_{\infty}:W_{\infty}]$ is finite, it follows that the
abelian group
${\mathrm{Gal}}({\mathfrak{W}}_{\infty}M(W_{\infty})/W_{\infty})$ is finitely
generated over ${\mathbb{Z}}_{p}$, and hence so is its quotient
$X(W_{\infty})$. Conversely, assume that $X(W_{\infty})$ is finitely generated
over ${\mathbb{Z}}_{p}$, and let $T$ denote the maximal abelian extension of
$W_{\infty}$ contained in $M({\mathfrak{W}}_{\infty})$. Then we have an
isomorphism

(2.2)
$X({\mathfrak{W}}_{\infty})_{\Delta}\cong{\mathrm{Gal}}(T/{\mathfrak{W}}_{\infty}).$

Now $M(W_{\infty})$ is a subfield of $T$, and
${\mathrm{Gal}}(T/M(W_{\infty}))$ is generated by the inertial subgroups of
the primes in $S_{\infty}({\mathfrak{W}}/W)$ in the extension $T/W_{\infty}$.
Such an inertial subgroup is of order $p$ since we are assuming that
${\mathfrak{W}}_{\infty}/W_{\infty}$ is of degree $p$. Thus
${\mathrm{Gal}}(T/M(W_{\infty}))$ is an ${\mathbb{F}}_{p}$-vector space, and

(2.3) $\dim_{{\mathbb{F}}_{p}}{\mathrm{Gal}}(T/M(W_{\infty}))\leq
s_{\infty}({\mathfrak{W}}/W).$

Since we are assuming that $X(W_{\infty})$ is finitely generated over
${\mathbb{Z}}_{p}$, and the set $S_{\infty}({\mathfrak{W}}/W)$ is finite, it
follows easily that ${\mathrm{Gal}}(T/{\mathfrak{W}}_{\infty})$ is finitely
generated over ${\mathbb{Z}}_{p}$. Thus, in view of (2.2),
$X({\mathfrak{W}}_{\infty})/(p,\delta-1)X({\mathfrak{W}}_{\infty})$ is a
finite dimensional ${\mathbb{F}}_{p}$-vector space. Since
$X({\mathfrak{W}}_{\infty})$ is a compact ${\mathbb{Z}}_{p}[\Delta]$-module,
it follows from Nakayama’s lemma that $X({\mathfrak{W}}_{\infty})$ is a
finitely generated ${\mathbb{Z}}_{p}[\Delta]$-module, whence it is also
finitely generated as a ${\mathbb{Z}}_{p}$-module. This completes the proof of
(i). For (ii), when $X(W_{\infty})=0$, (2.3) becomes
$\dim_{{\mathbb{F}}_{p}}{\mathrm{Gal}}(T/W_{\infty})\leq
s_{\infty}({\mathfrak{W}}/W)$, whence
$\dim_{{\mathbb{F}}_{p}}{\mathrm{Gal}}(T/{\mathfrak{W}}_{\infty})\leq
s_{\infty}({\mathfrak{W}}/W)-1$. Recalling (2.2), the assertion (ii) follows.
To prove (iii), we first note that (2.2) implies that we have the exact
sequence

(2.4) $1\to
X({\mathfrak{W}}_{\infty})_{\Delta}\to{\mathrm{Gal}}(T/W_{\infty})\to{\mathrm{Gal}}({\mathfrak{W}}_{\infty}/W_{\infty})\to
1.$

We also have the exact sequence

(2.5) $1\to{\mathrm{Gal}}(T/M(W_{\infty}))\to{\mathrm{Gal}}(T/W_{\infty})\to
X(W_{\infty})\to 1.$

Since ${\mathrm{Gal}}({\mathfrak{W}}_{\infty}/W_{\infty})$ and
${\mathrm{Gal}}(T/M(W_{\infty}))$ are both finite, it follows that

$(X({\mathfrak{W}}_{\infty})_{\Delta})\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}\cong
X(W_{\infty})\otimes_{{\mathbb{Z}}_{p}}{\mathbb{Q}}_{p}.$

This completes the proof. ∎

######  Corollary 2.3.

If ${\mathfrak{W}}/W$ is a finite Galois extension of $p$-power degree, then
$X({\mathfrak{W}}_{\infty})$ is a finitely generated ${\mathbb{Z}}_{p}$-module
if and only if $X(W_{\infty})$ is a finitely generated
${\mathbb{Z}}_{p}$-module.

This is immediate because any finite $p$-group is solvable.

######  Lemma 2.4.

Let $\Delta$ be a cyclic group of order $p$. Let Y be a free
${\mathbb{Z}}_{p}$-module of finite rank, which is also a $\Delta$-module.
Assume that $Y_{\Delta}=({\mathbb{Z}}/p{\mathbb{Z}})^{r}$ for some integer
$r\geq 0$. Then (i) $N_{\Delta}(Y)=0$, where
$N_{\Delta}=\sum_{\tau\in\Delta}\tau$, whence $Y$ is a
${\mathbb{Z}}_{p}[\Delta]/(N_{\Delta})$-module, and (ii) $Y$ is a quotient of
${\mathbb{Z}}^{(p-1)r}_{p}$ for any prime $p$, and
$Y\cong{\mathbb{Z}}^{r}_{2}$ when $p=2$.

###### Proof.

The proof, which the authors attribute to Sharifi, is entirely similar to the
case $p=2$ given in [3, Lemma 2.8]. ∎

Combining Lemmas 2.2 and 2.4, we immediately obtain:-

######  Lemma 2.5.

Let ${\mathfrak{W}}/W$ be a cyclic extension of degree $p$. Assume that
$X(W_{\infty})=0$, and that $X({\mathfrak{W}}_{\infty})$ is a torsion free
${\mathbb{Z}}_{p}$-module. Then $X({\mathfrak{W}}_{\infty})$ is a finitely
generated ${\mathbb{Z}}_{p}$-module of rank at most
$(p-1)(s_{\infty}({\mathfrak{W}}/W)-1)$.

For a finitely generated ${\mathbb{Z}}_{p}$-module $Y$, we write $t_{p}(Y)$
for its ${\mathbb{Z}}_{p}$-rank. In our applications, we only need the
following lemma for the prime $p=2$.

######  Lemma 2.6.

Assume that $p=2$, and let $W$ be any finite extension of $K$ such that
$X(W_{\infty})$ is a finitely generated ${\mathbb{Z}}_{2}$-module. Take
${\mathfrak{M}}$ to be any Galois extension of $W$ with
${\mathrm{Gal}}({\mathfrak{M}}/W)\cong{\mathrm{Gal}}({\mathfrak{M}}_{\infty}/W_{\infty})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2}$.
Then

(2.6)
$t_{2}(X({\mathfrak{M}}_{\infty}))+2t_{2}(X(W_{\infty}))=\sum_{i=1}^{3}t_{2}(X({\mathfrak{W}}_{i,\infty})),$

where ${\mathfrak{W}}_{1},{\mathfrak{W}}_{2},{\mathfrak{W}}_{3}$ are the three
quadratic extensions of $W$ contained in ${\mathfrak{M}}$.

###### Proof.

Since we are assuming that $X(W_{\infty})$ is a finitely generated
${\mathbb{Z}}_{2}$-module, the same is true for $X({\mathfrak{M}}_{\infty})$
because ${\mathrm{Gal}}({\mathfrak{M}}/W)$ is a $2$-group. Let
$V=X({\mathfrak{M}}_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2}$, so
that the ${\mathbb{Q}}_{2}$-dimension of $V$ is equal to
$t_{2}(X({\mathfrak{M}}_{\infty}))$. Put
$\Omega={\mathrm{Gal}}({\mathfrak{M}}_{\infty}/W_{\infty})$. Then

$V=\oplus_{\chi}V^{e_{\chi}},$

where the sum runs over the characters $\chi$ of $\Omega$, and
$e_{\chi}=\sum_{\sigma\in\Omega}\chi(\sigma)\sigma$. Thus we must compute the
${\mathbb{Q}}_{2}$-dimension of each $V^{e_{\chi}}$. Note that if $Y$ is any
vector space over ${\mathbb{Q}}_{2}$ and $g$ is the non-identity element of a
group $\Delta$ of order $2$ acting on Y, then plainly

(2.7) $Y=Y^{1+g}\oplus Y^{1-g},\,\,(Y)_{\Delta}=Y^{1+g}.$

Suppose first that $\chi$ is the trivial character of $\Omega$, so that
$e_{\chi}=(1+\sigma)(1+\tau)$, where, of course, $\sigma$ and $\tau$ are
distinct elements $\Omega$ of exact order $2$. By the remark 2.7 and (iii) of
Lemma 2.2 applied to the group $<\sigma>$ of order 2, we conclude that
$V^{1+\sigma}=X(T_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2}$, where
$T_{\infty}$ is the fixed field of $\sigma$. Thus

(2.8)
$V^{\chi}=(X(T_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2})^{1+\tau}=X(W_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2};$

here the second equality follows from applying the remark 2.7 and (iii) of
Lemma 2.2 to the quadratic extension $T_{\infty}/W_{\infty}$. Hence in this
case the ${\mathbb{Q}}_{2}$-dimension of $V^{\chi}$ is equal to the
${\mathbb{Q}}_{2}$-dimension of
$X(W_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2}$. Suppose next that
$\chi$ is a nontrivial character of $\Omega$, say with $\chi(\sigma)=1$ and
$\chi(\tau)=-1$, so that $e_{\chi}=(1+\sigma)(1-\tau)$. If $T$ is now the
fixed field of ${\mathrm{Ker}}(\chi)=\\{1,\sigma\\}$, we again conclude from
the remark 2.7 and (iii) of Lemma 2.2 that
$V^{1+\sigma}=X(T_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2}$, whence

(2.9)
$V^{e_{\chi}}\cong(X(T_{\infty})\otimes_{{\mathbb{Z}}_{2}}{{\mathbb{Q}}}_{2})^{1-\tau}.$

But applying (iii) of Lemma 2.2 to the extension $T_{\infty}/W_{\infty}$, we
conclude that

$X(T_{\infty})\otimes_{{\mathbb{Z}}_{2}}{{\mathbb{Q}}}_{2}/(X(T_{\infty})\otimes_{{\mathbb{Z}}_{2}}{{\mathbb{Q}}}_{2})^{1-\tau}=X(W_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2},$

whence it follows that the ${\mathbb{Q}}_{2}$-dimension of $V^{e_{\chi}}$ is
equal to the ${\mathbb{Q}}_{2}$-dimension of
$X(T_{\infty})\otimes_{{\mathbb{Z}}_{2}}{{\mathbb{Q}}}_{2}$ minus the
${\mathbb{Q}}_{2}$-dimension of
$X(W_{\infty})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{Q}}_{2}$. Since this holds
for each of the three non-trivial characters of $\Omega$, the proof of Lemma
2.6 is now complete. ∎

##  3\. Proof of Theorem 1.1

We now return to the situation discuused in $\S 1$. Thus from now on
$K={\mathbb{Q}}(\sqrt{-q})$, where $q$ is any prime with $q\equiv 7\
\mathrm{mod}\ 8$, and again the fields $F,F^{\prime},D,J$ are defined by
(1.1). As always, ${\mathfrak{p}}$ will denote one of the primes of $K$ lying
above 2, and we again fix the sign of $\sqrt{-q}$ so that $\sqrt{-q}\equiv 1\
\mathrm{mod}\ {\mathfrak{p}}^{2}$. Recall that, for each finite exrension
$W/K$, the integer $s_{\infty}(W/K)$ is defined by (2.1) in $\S 2$. Again $R$
will denote an arbitrary element of ${\mathcal{O}}_{K}$ satisfying the
Twisting Hypothesis, and $s_{\infty}(R)$ will denote the number of primes of
$K_{\infty}$ dividing $\sqrt{-q}R$.

######  Lemma 3.1.

We have $s_{\infty}(F/K)=s_{\infty}(R)$ for all primes $q\equiv 7\
\mathrm{mod}\ 8$. Moreover, If $q\equiv 7\bmod 16$, we also have
$s_{\infty}(F^{\prime}/K)=s_{\infty}(R)+1$ and $s_{\infty}(D/K)=1$.

###### Proof.

The first assertion is clear because the primes of $K_{\infty}$, not lying
above ${\mathfrak{p}}$, which ramify in $F_{\infty}$ are precisely the primes
dividing $\sqrt{-q}R$. Note that ${\mathfrak{p}}^{*}$ is ramified in
$F^{\prime}/K$ and in $D/K$ by our choice of sign. Then the second assertion
follows from the fact that ${\mathfrak{p}}^{*}$ is inert in $K_{\infty}$ when
$q\equiv 7\bmod 16$ (see [4, Proposition 2.4]). ∎

######  Proposition 3.2 (Choi-Coates[1]).

(i) $X(F_{\infty})$ is a free finitely generated ${\mathbb{Z}}_{2}$-module of
rank at most $s_{\infty}(R)-1$, and (ii) if $q\equiv 7\bmod 16$, then
$X(D_{\infty})=0$ and $X(F^{\prime}_{\infty})$ is a free finitely generated
${\mathbb{Z}}_{2}$-module of rank at most $s_{\infty}(R)$.

###### Proof.

It is easy to see $X(K_{\infty})=0$. (for example see [3, Lemma 3.2]). Then by
Lemma 2.2 $X(F_{\infty})$, $X(D_{\infty})$ and $X(F^{\prime}_{\infty})$ are
all finitely generated over ${\mathbb{Z}}_{2}$. Moreover, by Greenberg’s
theorem [7], they are torsion-free whence are free of finite rank. The
assertions then follow from Lemma 3.1 and Lemma 2.5. ∎

######  Proposition 3.3.

Let ${\mathfrak{C}}_{n}$ be the ray class group of $K_{n}$ modulo
$({\mathfrak{p}}^{*})^{2}{\mathcal{O}}_{K_{n}}$. Then ${\mathfrak{C}}_{n}$ has
odd order for all $n\geq 0$.

Before giving the proof, we recall Chevalley’s formula for ray class groups
(see [11, §4], for example). Suppose that $L/k$ is any cyclic extension of
number fields with Galois group $G$, and write ${\mathcal{O}}_{L}$ (resp.
${\mathcal{O}}_{k}$) for the ring of integers of $L$ (resp. $k$). For our
application, we may assume that $k$ has no real places. If ${\mathfrak{m}}$ is
an integral ideal in $k$, let ${\mathrm{Cl}}^{\mathfrak{m}}_{L}$ (resp.
${\mathrm{Cl}}^{\mathfrak{m}}_{k}$) denote the ray class group of $L$ (resp.
$k$) modulo ${\mathfrak{m}}{\mathcal{O}}_{L}$ (resp. ${\mathfrak{m}}$).
Clearly ${\mathrm{Cl}}^{\mathfrak{m}}_{L}$ is a $G$-module, and
$({\mathrm{Cl}}^{{\mathfrak{m}}}_{L})^{G}$ will denote, as always, its
$G$-invariants. For a prime $v$ (resp. $w$) of $k$ (resp. $L$), write
${\mathcal{O}}_{v}$ (resp. ${\mathcal{O}}_{w}$) for the ring of integers of
the completion $k_{v}$ (resp. $L_{w}$). Let $L^{\mathfrak{m}}$ denote the set
of all $a\in L^{\times}$ such that
${\mathrm{ord}}_{w}(a-1)\geq{\mathrm{ord}}_{w}({\mathfrak{m}})$ for all places
$w$ of $L$ dividing ${\mathfrak{m}}$. Let
${\mathcal{O}}^{\times}_{k,{\mathfrak{m}}}$ be the subgroup of the unit group
${\mathcal{O}}_{k}^{\times}$ of $k$ consisting of all units $a$ with
${\mathrm{ord}}_{v}(a-1)\geq{\mathrm{ord}}_{v}({\mathfrak{m}})$ at all places
$v$ of $k$ dividing ${\mathfrak{m}}$. Then Chevalley’s formula with respect to
$L/k$ and the ideal ${\mathfrak{m}}$ is as follows:-

(3.1)
$\frac{\\#(({\mathrm{Cl}}^{{\mathfrak{m}}}_{L})^{G})}{\\#({\mathrm{Cl}}^{\mathfrak{m}}_{k})}=\frac{\prod\limits_{v\mid{\mathfrak{m}}}[1+{\mathfrak{m}}{\mathcal{O}}_{v}:N_{L_{w}/k_{v}}(1+{\mathfrak{m}}{\mathcal{O}}_{w})]\cdot\prod\limits_{v\nmid{\mathfrak{m}}}e_{v}}{[L:k][{\mathcal{O}}^{\times}_{k,{\mathfrak{m}}}:{\mathcal{O}}^{\times}_{k,{\mathfrak{m}}}\cap
N_{L/k}(L^{\mathfrak{m}})]};$

here $N_{L/k}$ (resp. $N_{L_{w}/k_{v}}$) is the norm map for $L/k$ (resp.
$L_{w}/k_{v}$), and $e_{v}$ is the ramification index of $v$ in $L$. In the
first product, $w$ is an arbitrary place of $L$ above $v$ and the group
$N_{L_{w}/k_{v}}(1+{\mathfrak{m}}{\mathcal{O}}_{w})$ is independent of the
choice of $w$.

######  Lemma 3.4.

If $k$ is a finite unramified extension of ${\mathbb{Q}}_{2}$, then for every
$m\geq 2$ we have
$N_{k/{\mathbb{Q}}_{2}}(1+2^{m}{\mathcal{O}}_{k})=1+2^{m}{\mathbb{Z}}_{2}$,
where ${\mathcal{O}}_{k}$ is the ring of integers in $k$.

###### Proof.

Let $\mathrm{Tr}_{k/{\mathbb{Q}}_{2}}$ be the trace map from $k$ to
${\mathbb{Q}}_{2}$. Let $\log$ be the usual $2$-adic logarithm map. It
commutes with the Galois action. For $m\geq 2$, we then have a commutative
diagram:

${1+2^{m}{\mathcal{O}}_{k}}$${2^{m}{\mathcal{O}}_{k}}$${1+2^{m}{\mathbb{Z}}_{2}}$${2^{m}{\mathbb{Z}}_{2}.}$$\scriptstyle{\log}$$\scriptstyle{N_{k/{\mathbb{Q}}_{2}}}$$\scriptstyle{\mathrm{Tr}_{k/{\mathbb{Q}}_{2}}}$$\scriptstyle{\log}$

The two horizontal maps are isomorphisms as $m\geq 2$. The right vertical map
is a surjection because $k/{\mathbb{Q}}_{2}$ is unramified. The lemma then
follows. ∎

###### Proof of Proposition 3.3.

We apply (3.1) with respect to the cyclic extension
$(K_{n}/K,{\mathfrak{m}})$, with ${\mathfrak{m}}=({\mathfrak{p}}^{*})^{2}$. As
${\mathcal{O}}^{\times}_{K,{\mathfrak{m}}}=\\{1\\}$, then, writing
$v={\mathfrak{p}}^{*}$, the formula gives

(3.2)
$\\#(({\mathfrak{C}}_{n})^{{\mathrm{Gal}}(K_{n}/K)})=\\#({\mathfrak{C}}_{0})\cdot[1+{\mathfrak{m}}{\mathcal{O}}_{v}:N(1+{\mathfrak{m}}{\mathcal{O}}_{w})]=\\#({\mathfrak{C}}_{0}).$

The second equality follows from Lemma 3.4. Using the fact that the class
number of $K$ is odd, and that the units of $K$ are $\\{\pm 1\\}$, it is
readily verified that $\\#({\mathfrak{C}}_{0})$ is odd. Therefore
$\\#({\mathfrak{C}}_{n})$ is odd by Nakayama’s lemma, since
${\mathrm{Gal}}(K_{n}/K)$ is cyclic of order $2^{n}$. ∎

###### Proof of Theorem 1.1.

Take $n$ to be an integer which is sufficiently large to ensure that there are
precisely $s_{\infty}(R)$ primes of $K_{n}$ dividing $\sqrt{-q}R$. These
primes are then all inert in $K_{\infty}$. To simplify notation, we shall
write $s=s_{\infty}(R)$ in the rest of this proof. We have

(3.3)
$\sqrt{-q}R{\mathcal{O}}_{K_{n}}={\mathfrak{a}}_{1}\cdots{\mathfrak{a}}_{s},$

where each ${\mathfrak{a}}_{i}$ is the power of some prime ideal of $K_{n}$,
and $({\mathfrak{a}}_{i},{\mathfrak{a}}_{j})=1$ when $i\neq j$. Note that the
exponents to which these prime ideals occur in (3.3) are all odd by the
definition of $R$ and the fact $({\mathfrak{q}},R)=1$. Write $h^{\prime}$ for
the order of ${\mathfrak{C}}_{n}$, which is odd by Proposition 3.3. Thus there
exist $\alpha_{i}\in{\mathcal{O}}_{K_{n}}$ such that

(3.4) ${\mathfrak{a}}^{h^{\prime}}_{i}=(\alpha_{i})\quad\text{ and
}\quad\alpha_{i}\equiv
1\bmod({\mathfrak{p}}^{*})^{2}{\mathcal{O}}_{K_{n}}\quad\text{ for }1\leq
i\leq s-1.$

We define

(3.5)
$\alpha_{s}=\frac{(-\sqrt{-q}R)^{h^{\prime}}}{\alpha_{1}\cdots\alpha_{r-1}}.$

Note that, thanks to the oddness of $h^{\prime}$ and our choice of the sign,
we also have $\alpha_{s}\equiv
1\bmod({\mathfrak{p}}^{*})^{2}{\mathcal{O}}_{K_{n}}$. Again because
$h^{\prime}$ is odd, it follows that, for $1\leq i\leq s$, the extension
$K_{n}(\sqrt{\alpha_{i}})/K_{n}$ is quadratic and is ramified at the unique
prime dividing $\alpha_{i}$. Moreover, this extension is unramified at the
primes above ${\mathfrak{p}}^{*}$, since it is obtained by adjoining a root of
$x^{2}-x+\frac{1-\alpha_{i}}{4}$ and this polynomial is separable modulo each
prime of $K_{n}$ above ${\mathfrak{p}}^{*}$. Define

(3.6) $T=K_{\infty}(\sqrt{\alpha_{1}},\cdots,\sqrt{\alpha_{s}}).$

Now $K_{\infty}(\sqrt{\alpha_{i}})/K_{\infty}$ is ramified at the unique prime
dividing $\alpha_{i}$, while it is unramified at the primes dividing
$\alpha_{j}$ for $j\neq i$, whence it follows easily that
$[T:K_{\infty}]=2^{s}$, and

${\mathrm{Gal}}(T/K_{\infty})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{s}.$

Thanks again to the oddness of $h^{\prime}$, it follows from (3.5) that

(3.7) $F_{\infty}\subset T\quad\text{
and}\quad{\mathrm{Gal}}(T/F_{\infty})\cong({\mathbb{Z}}/2{\mathbb{Z}})^{s-1}.$

The extension $T/F_{\infty}$ is unramified at primes not dividing
${\mathfrak{p}}{\mathfrak{a}}_{1}\cdots{\mathfrak{a}}_{s}$, since
$T/K_{\infty}$ is. Moreover, $T/F_{\infty}$ is also unramified at the prime
dividing ${\mathfrak{a}}_{i}$ for $1\leq i\leq s$, since from (3.6) the
ramification index of this prime in $T/K_{\infty}$ is clearly equal to $2$,
but $F_{\infty}/K_{\infty}$ is also ramified at this prime by Lemma 3.1. This
proves $T\subset M(F_{\infty})$. Hence $X(F_{\infty})$ has
$({\mathbb{Z}}/2{\mathbb{Z}})^{s-1}$ as a quotient, and so its
${\mathbb{Z}}_{2}$-rank must be at least $s-1$. Combining this with the upper
bound given by Proposition 3.2, the proof of assertion (i) of Theorem 1.1 is
complete.

Assume for the rest of the proof that $q\equiv 7\bmod 16$. By Proposition 3.2,
it suffices to construct $s+1$ independent quadratic extensions of
$F^{\prime}_{\infty}$ contained in $M(F^{\prime}_{\infty})$. Define

(3.8)
$T^{\prime}=T(\sqrt{-1})=K_{\infty}(\sqrt{-1},\sqrt{\alpha_{1}},\cdots,\sqrt{\alpha_{s}}).$

Note that $T^{\prime}\supset F^{\prime}_{\infty}$. Since
$F^{\prime}(\sqrt{-1})/F^{\prime}$ is unramified at ${\mathfrak{p}}$, we have
$F^{\prime}_{\infty}(\sqrt{-1})\subset M(F^{\prime}_{\infty})$. Moreover, by
the same argument as in the first paragraph, we have $T\subset
M(F^{\prime}_{\infty})$. Thus $T^{\prime}\subset M(F^{\prime}_{\infty})$, and
so it remains to show that $[T^{\prime}:F^{\prime}_{\infty}]=2^{s}$. Note that
$T/K_{\infty}$ is unramified at the unique prime above ${\mathfrak{p}}^{*}$,
but $K_{\infty}(\sqrt{-1})/K_{\infty}$ is ramified at this prime. It follows
that $T^{\prime}\neq T$ whence $[T^{\prime}:K_{\infty}]=2^{s+1}$ and therefore
$[T^{\prime}:F^{\prime}_{\infty}]=2^{s}$. This proves
$X(F^{\prime}_{\infty})\cong{\mathbb{Z}}^{s}_{2}$, as required.

Finally, since $J/K$ is a biquadratic extension, with $F,F^{\prime}$ and $D$
being the nontrivial intermediate fields, it follows from Lemma 2.6 and
Proposition 3.2 that $X(J_{\infty})\cong{\mathbb{Z}}^{2s-1}_{2}$. This
completes the proof of Theorem 1.1. ∎

##  4\. Equivalent formulation of Theorem 1.1 in terms of infinite descent on
certain abelian varieties with complex multiplication

The aim of this section is to briefly explain an equivalent formulation of
Theorem 1.1 in terms of infinite descent on a family of quadratic twists of a
certain higher dimensional abelian variety with complex multiplication. We
omit detailed proofs, and refer the reader to [8],[3] for further explanations
of the background material. Fix an embedding of $K$ in ${\mathbb{C}}$. Let
$H=K(j({\mathcal{O}}_{K}))$ be the Hilbert class field of $K$, where $j$ is
the classical modular function, and let $A/{\mathbb{Q}}(j({\mathcal{O}}_{K}))$
be the Gross elliptic curve with complex multiplication by ${\mathcal{O}}_{K}$
(see [8], Chap. 5). Thus $A$ has minimal discriminant $(-q^{3})$, and $A$ is
isogenous over $H$ to all of its conjugates. Let

$B=\mathrm{Res}^{H}_{K}A$

be the $h$-dimensional abelian variety over $K$ which is the restriction of
scalars of $A$ from $H$ to $K$. Let $\mathscr{B}={\mathrm{End}}_{K}(B)$, and
$\mathscr{T}=\mathscr{B}\otimes{\mathbb{Q}}$, so that $\mathscr{T}$ is a CM
field of degree $h$ over $K$. Then $\mathscr{B}$ is an order in $\mathscr{T}$,
which is ramified over ${\mathcal{O}}_{K}$ at precisely the primes dividing
$h$ (see [8], Theorem 15.2.5). In particular, since $h$ is odd, the primes
${\mathfrak{p}}$, and ${\mathfrak{p}}^{*}$ are both unramified in
$\mathscr{T}$. Now the torsion subgroup of $B(K)$ is
${\mathcal{O}}_{K}/2{\mathcal{O}}_{K}$, and the action of $\mathscr{B}$ on
this torsion subgroup gives an ${\mathcal{O}}_{K}$-algebra surjection from
$\mathscr{B}$ onto ${\mathcal{O}}_{K}/2{\mathcal{O}}_{K}$, whose kernel is the
product of two conjugate primes ${\mathfrak{P}},{\mathfrak{P}}^{*}$ of
$\mathscr{B}$ lying above ${\mathfrak{p}},{\mathfrak{p}}^{*}$, respectively.
These primes are both unramified in $\mathscr{B}$, and have residue fields
equal to the field ${\mathbb{F}}_{2}$ with $2$ elements. Our equivalent
formulation of Theorem 1.1 will be in terms of the arithmetic of the abelian
variety $B^{(R)}/K$, which is the twist of $B$ by the quadratic extension
$K(\sqrt{R})/K$. For each $n\geq 1$, let $B^{(R)}_{{\mathfrak{P}}^{n}}$ be the
Galois module of ${\mathfrak{P}}^{n}$-division points on $B^{(R)}$. Then we
have the following interpretation of the fields defined in $\S 1$.

######  Theorem 4.1.

We have

$\displaystyle F=K(B^{(R)}_{{\mathfrak{P}}^{2}}),\quad
F^{\prime}=K(B^{(R)}_{{{\mathfrak{P}}^{*}}^{2}}),\quad
J=FF^{\prime}=K(B^{(R)}_{{\mathfrak{P}}^{2}},\sqrt{-1})$ $\displaystyle
F_{\infty}=K(B^{(R)}_{{\mathfrak{P}}^{\infty}}),\quad
F^{\prime}_{\infty}=K(B^{(R)}_{{{\mathfrak{P}}^{*}}^{\infty}}),\quad
J_{\infty}=K(B^{(R)}_{{\mathfrak{P}}^{\infty}},\sqrt{-1}).$

Moreover, $B^{(R)}$ has good reduction everywhere over $F$ and $F^{\prime}$.

We omit the detailed proofs, which are similar to those given in [3] for the
abelian variety $B$ (see [3, §2 and Lemma 7.11] and also see [1, Lemma 2.1,
2.2]).

We next recall the definition of the ${\mathfrak{P}}^{\infty}$-Selmer group of
$B^{(R)}$ over an algebraic extension of $K$. Take any non-zero endomorphism
$\pi$ in $\mathscr{B}$ such that the ideal factorization of $\pi$ in the ring
of integers of $\mathscr{T}$ is equal to ${\mathfrak{P}}^{r}$ for some integer
$r\geq 1$. Let $L$ be any algebraic extension of $K$. Then for each integer
$n\geq 1$, the group ${\mathrm{Sel}}_{\pi^{n}}(B^{(R)}/L)$ is defined by

${\mathrm{Sel}}_{\pi^{n}}(B^{(R)}/L)={\mathrm{Ker}}(H^{1}(L,B^{(R)}_{\pi^{n}})\to\prod_{v}H^{1}(L_{v},B^{(R)})),$

where $v$ runs over all finite places of $L$, and $L_{v}$ is the compositum of
the completions at $v$ of all finite extensions of $K$ contained in $L$.
Passing to the inductive limit over all $n\geq 1$, and noting that
$B^{(R)}_{\pi^{\infty}}=B^{(R)}_{{\mathfrak{P}}^{\infty}}$ as Galois modules,
we then define the Selmer group
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/L)$ to be the inductive
limit of the ${\mathrm{Sel}}_{\pi^{n}}(B^{(R)}/L)$, so that:-

${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/L)={\mathrm{Ker}}(H^{1}(L,B^{(R)}_{{\mathfrak{P}}^{\infty}})\to\prod_{v}H^{1}(L_{v},B^{(R)})).$

Since $B^{(R)}$ has good reduction everywhere over $F$, and
$F_{\infty}=K(B^{(R)}_{{\mathfrak{P}}^{\infty}})$, it is then easily seen that
we have (see [3, Theorem 3.9] and [2]):-

######  Theorem 4.2.

As Galois modules, we have

${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})={\mathrm{Hom}}(X(F_{\infty}),B^{(R)}_{{\mathfrak{P}}^{\infty}}),\,\,\,{\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/J_{\infty})={\mathrm{Hom}}(X(J_{\infty}),B^{(R)}_{{\mathfrak{P}}^{\infty}}),$

Noting that
$B^{(R)}_{{\mathfrak{P}}^{\infty}}=\mathscr{T}_{\mathfrak{P}}/\mathscr{B}_{\mathfrak{P}}={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$
as abelian groups, we immediately obtain the following equivalent form of
Theorem 1.1.

######  Theorem 4.3.

As abelian groups, we have

${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})=({\mathbb{Q}}_{2}/{\mathbb{Z}}_{2})^{s_{\infty}(R)-1},\,\,\,{\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/J_{\infty})=({\mathbb{Q}}_{2}/{\mathbb{Z}}_{2})^{2s_{\infty}(R)-1},$

for all $R$ satisfying the Twisting Hypothesis, and where $s_{\infty}(R)$ is
the number of primes of $K_{\infty}$ dividing $\sqrt{-q}R$.

Let ${\mathfrak{F}}$ denote either of the fields $F$ or $J$. We recall that
the Tate-Shafarevich group $\hbox{\cyr X}(B^{(R)}/{\mathfrak{F}}_{\infty})$ of
$B^{(R)}/{\mathfrak{F}}_{\infty}$ is defined by

$\hbox{\cyr
X}(B^{(R)}/{\mathfrak{F}}_{\infty})={\mathrm{Ker}}(H^{1}({\mathfrak{F}}_{\infty},B^{(R)})\to\prod_{v}H^{1}({\mathfrak{F}}_{\infty,v},B^{(R)})).$

It is a $\mathscr{B}$-module and we write $\hbox{\cyr
X}(B^{(R)}/{\mathfrak{F}}_{\infty})({\mathfrak{P}}^{\infty})$ for its
${\mathfrak{P}}$-primary subgroup. Note also that the group
$B^{(R)}({\mathfrak{F}}_{\infty})$ of ${\mathfrak{F}}_{\infty}$-rational
points of $B^{(R)}$ has a natural structure as an $\mathscr{B}$-module, and we
have the exact sequence

(4.1) $0\to
B^{(R)}({\mathfrak{F}}_{\infty})\otimes_{\mathscr{B}}(\mathscr{T}_{\mathfrak{P}}/\mathscr{B}_{\mathfrak{P}})\to{\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/{\mathfrak{F}}_{\infty})\to\hbox{\cyr
X}(B^{(R)}/{\mathfrak{F}}_{\infty})({\mathfrak{P}}^{\infty})\to 0.$

Thus Theorem 4.3 can be reformulated as follows. Define
$g_{{\mathfrak{F}}_{\infty}}(R)$ to be the $\mathscr{T}$-dimension of
$B^{(R)}({\mathfrak{F}}_{\infty})\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$. Now
$\hbox{\cyr X}(B^{(R)}/{\mathfrak{F}}_{\infty})({\mathfrak{P}}^{\infty})$ is a
divisible group thanks to Theorem 4.2, the fact that
$X({\mathfrak{F}}_{\infty})$ is a free finitely generated
${\mathbb{Z}}_{2}$-module, and the exact sequence 4.1, whence we define
$e_{{\mathfrak{F}}_{\infty}}(R)$ to be its ${\mathbb{Z}}_{2}$-corank.

######  Theorem 4.4.

For all $R$ satisfying the Twisting Hypothesis, we have (i)
$g_{F_{\infty}}(R)+e_{F_{\infty}}(R)=s_{\infty}(R)-1$, and (ii)
$g_{J_{\infty}}(R)+e_{J_{\infty}}(R)=2s_{\infty}(R)-1$, where, as always,
$s_{\infty}(R)$ is the number of primes of $K_{\infty}$ dividing $\sqrt{-q}R$.

Since $s_{\infty}(R)$ is very easy to calculate explicitly, it seems to us
that no simple explicit general result like Theorems 4.3 and 4.4 was known
previously in the Iwasawa theory of elliptic curves. As an illustration, we
apply these theorems to one numerical example taken from [1]. Take $q=7$, so
that $B$ is the elliptic curve $X_{0}(49)$, with equation

(4.2) $y^{2}+xy=x^{3}-x^{2}-2x-1.$

Take $R=741=3.13.19$, with $E=X_{0}(49)^{(R)}$, whence $s_{\infty}(R)=4$, and
so we now know by Theorem 4.4 that $g_{F_{\infty}}(R)+e_{F_{\infty}}(R)=3$. It
is shown in [1] that $E(K)\otimes{\mathbb{Q}}$ has dimension $2$ over $K$.
Hence we must either have $g_{F_{\infty}}(R)=3$ and $\hbox{\cyr
X}(E/F_{\infty})({\mathfrak{p}}^{\infty})=0$ or $g_{F_{\infty}}(R)=2$ and
$\hbox{\cyr
X}(E/F_{\infty})({\mathfrak{p}}^{\infty})={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$.
Also, by Theorem 4.4, we have $g_{J_{\infty}}(R)+e_{J_{\infty}}(R)=7$. It
seems likely from root number calculations that $E(D)\otimes{\mathbb{Q}}$ has
dimension at least 3 over $K$, suggesting that $e_{J_{\infty}}(R)\leq 4$. It
would be very interesting to determine the exact value of the $K$-dimension of
both $E(F_{\infty})\otimes{\mathbb{Q}}$ and $E(J_{\infty})\otimes{\mathbb{Q}}$
in this numerical example.

##  5\. An application of Theorem 1.1

In this last section, we briefly discuss the situation when $R$ is a square
free rational integer satisfying the Twisting Hypothesis, and having the
property that $s_{\infty}(R)=2$ (the case when $s_{\infty}(R)=1$ is already
discussed fully in [4]). Then the following lemma is clear from the remarks
made immediately after the statement of Theorem 1.1.

######  Lemma 5.1.

Assume that $R$ is a square free rational integer satisfying the Twisting
Hypothesis. Then $s_{\infty}(R)=2$ if and only if one of the following three
cases hold:- (i) we have $q\equiv 15\ \mathrm{mod}\ 32$ and $R=1$, (ii) we
have $q\equiv 7\ \mathrm{mod}\ 16$ and $R=r$, where $r$ is a prime with
$r\equiv 5\ \mathrm{mod}\ 8$, which is inert in $K$, or (iii) we have $q\equiv
7\ \mathrm{mod}\ 16$ and $R=-r$, where $r$ is a prime with $r\equiv 3\
\mathrm{mod}\ 8$, which is inert in $K$.

By Theorem 4.3, we know that
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$
as an abelian group whenever $s_{\infty}(R)=2$. For each $n\geq 0$, recall
that $K_{n}$ is the $n$-th layer of the ${\mathbb{Z}}_{2}$-extension
$K_{\infty}/K$.

######  Theorem 5.2.

In case (i) of Lemma 5.1, we have that
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B/K)$ is finite,
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B/K_{n})$ has
${\mathbb{Z}}_{2}$-corank at most $1$ for all $n\geq 1$, and
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B/K_{n})$ has
${\mathbb{Z}}_{2}$-corank equal to $1$ for some $n\geq 1$ if and only if
$L(B/K_{n},1)=0$. In case (ii) of Lemma 5.1, we have that
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K)$ is finite,
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$ has
${\mathbb{Z}}_{2}$-corank at most 1 for all $n\geq 1$, and
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$ has
${\mathbb{Z}}_{2}$-corank equal to 1 for some $n\geq 1$ if and only if
$L(B^{(R)}/K_{n},1)=0$. In case (iii) of Lemma 5.1, we have that
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$ has
${\mathbb{Z}}_{2}$-corank equal to $1$ and $L(B^{(R)}/K_{n},1)=0$ for all
$n\geq 0$.

We now outline the proof of this theorem, again referring to [3] for similar
detailed arguments. We begin by recalling an elementary purely algebraic
lemma, which holds for any $R$ in ${\mathcal{O}}_{K}$ satisfying the Twisting
Hypothesis. Put $\Gamma={\mathrm{Gal}}(F_{\infty}/F)$,
$\Gamma_{n}={\mathrm{Gal}}(F_{\infty}/F_{n})$, and fix any topological
generator $\gamma$ of $\Gamma$. Let $\Lambda(\Gamma)$ be the Iwasawa algebra
of $\Gamma$ with coefficients in ${\mathbb{Z}}_{2}$. As usual, we identify
$\Lambda(\Gamma)$ with the formal power series ring ${\mathbb{Z}}_{2}[[T]]$ by
mapping $\gamma$ to $1+T$. Since $X(F_{\infty})$ is a finitely generated
${\mathbb{Z}}_{2}$-module, it is clearly a torsion $\Lambda(\Gamma)$-module,
and we write $c_{F_{\infty}}(T)$ for its characteristic power series. By
Theorem 4.1, we have a character $\kappa:\Gamma\to
1+{\mathfrak{P}}^{2}\mathscr{B}_{\mathfrak{P}}=1+4{\mathbb{Z}}_{2}$ giving the
action of $\Gamma$ on $B^{(R)}_{{\mathfrak{P}}^{\infty}}$ and we define
$u=\kappa(\gamma)$.

######  Lemma 5.3.

For each $R$ in ${\mathcal{O}}_{K}$ satisfying the Twisting Hypothesis,
$({\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty}))^{\Gamma_{n}}$
is infinite for some integer $n\geq 0$ if and only if
$c_{F_{\infty}}(u\zeta-1)=0$ for some $2^{n}$-th power root of unity $\zeta$.

###### Proof.

Put $\gamma_{n}=\gamma^{2^{n}}$, so that $\gamma_{n}$ is a topological
generator of $\Gamma_{n}$, $\gamma_{n}=(1+T)^{2^{n}}$ and
$\kappa(\gamma_{n})=u^{2^{n}}$. Write $X=X(F_{\infty})$ in this proof for
simplicity. By Theorem 4.2, we have

$({\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty}))^{\Gamma_{n}}={\mathrm{Hom}}(X/(\gamma_{n}-\kappa(\gamma_{n}))X,B^{(R)}_{{\mathfrak{P}}^{\infty}}).$

Thus, the left hand side is finite if and only if
$X/(\gamma_{n}-\kappa(\gamma_{n}))X$ is finite. Note that

$\gamma_{n}-\kappa(\gamma_{n})=(1+T)^{2^{n}}-u^{2^{n}}=\prod_{\zeta}(T-(u\zeta-1)),$

where $\zeta$ runs over all the $2^{n}$-th power of roots unity. Since
$c_{F_{\infty}}(T)$ is the characteristic power series of $X$, it follows that
$X/(\gamma_{n}-\kappa(\gamma_{n}))X$ is finite if and only if
$c_{F_{\infty}}(T)$ is coprime to $T-(u\zeta-1)$ for each $\zeta$. This proves
Lemma 5.3. ∎

In addition, we have the following arithmetic lemma. For each algebraic
extension $L/K$, we define

${\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/L)={\mathrm{Ker}}(H^{1}(L,B^{(R)}_{{\mathfrak{P}}^{\infty}})\to\prod_{v\nmid{\mathfrak{p}}}H^{1}(L_{v},B^{(R)})).$

Recall that $F_{n}=FK_{n}$ for $n\geq 0$.

######  Lemma 5.4.

For each $R$ in ${\mathcal{O}}_{K}$ satisfying the Twisting Hypothesis, and
for all $n\geq 0$, ${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$
has the same ${\mathbb{Z}}_{2}$-corank as
${\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{n})=({\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty}))^{\Gamma_{n}}$.

###### Proof.

Since $B^{(R)}$ has good reduction everywhere over $F$, entirely similar
arguments to those given in §3 of [3] show that

(5.1)
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})={\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty}),$

and that $\Delta$ acts trivially on
${\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})$, where
$\Delta={\mathrm{Gal}}(F_{\infty}/K_{\infty})$. Using again the fact that
$B^{(R)}$ has good reduction everywhere over $F$, the same argument as in the
proof of Proposition 3.11 of [3] shows that

(5.2)
$({\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty}))^{\Gamma_{n}}={\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{n})$

for all $n\geq 0$. Thus to complete the proof, it suffices to show that the
natural map

(5.3)
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})\to({\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{n}))^{\Delta}={\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{n})$

has finite kernel and cokernel for all $n\geq 0$. But this follows easily from
the fact that $\Delta$ is of order 2, combined with a similar argument using
Tate local duality for the abelian variety $B^{(R)}/K$, which has good
reduction at the prime ${\mathfrak{p}}$, as that given in the proof of Theorem
3.12 of [3]. ∎

For each $n\geq 0$, let $L(B^{(R)}/F_{n},s)$ (resp. $L(B^{(R)}/K_{n},s)$) be
the complex $L$-series of $B^{(R)}/F_{n}$ (resp. $B^{(R)}/K_{n}$). Let
$\chi_{n}$ be the non-trival character of ${\mathrm{Gal}}(F_{n}/K_{n})$, and
write $L(B^{(R)}/K_{n},\chi_{n},s)$ for the twist of $L(B^{(R)}/K_{n},s)$ by
$\chi_{n}$, so that we have

(5.4) $L(B^{(R)}/F_{n},s)=L(B^{(R)}/K_{n},s)L(B^{(R)}/K_{n},\chi_{n},s).$

######  Theorem 5.5.

For all $R$ in ${\mathcal{O}}_{K}$ satisfying the Twisting Hypothesis, and all
$n\geq 0$, we have $L(B^{(R)}/K_{n},\chi_{n},1)\neq 0$, and so
$L(B^{(R)}/F_{n},1)\neq 0$ if and only if $L(B^{(R)}/K_{n},1)\neq 0$.

###### Proof.

This theorem, which is valid for all primes $q\equiv 7\ \mathrm{mod}\ 8$ is an
equivalent form of Theorem 1.5 of [3]. Put $\beta=\sqrt{-q}$. Now
$L(B^{(R)}/K_{n},\chi_{n},s)$ is the complex $L$-series of the twist of
$B^{(R)}$ by the quadratic extension $F_{n}/K_{n}$. As
$F_{n}=K_{n}(\sqrt{-\beta R})$, it follows that the twist of $B^{(R)}$ by
$F_{n}/K_{n}$ is just $B^{(-\beta)}/K_{n}$. Put $H_{n}=HK_{n}$, where we
recall that $H$ is the Hilbert class field of $K$. Then the complex $L$-series
of $B^{(-\beta)}/K_{n}$ is the same as the complex $L$-series
$L(A^{(-\beta)}/H_{n},s)$ of $A^{(-\beta)}/H_{n}$, where $A/H$ is the basic
Gross ${\mathbb{Q}}$-curve defined over $H$. Since Theorem 1.5 of [3] asserts
that $L(A^{(-\beta)}/H_{n},1)\neq 0$ for all $n\geq 0$, the proof is complete.
∎

The following deep theorem is a consequence of the main conjecture of Iwasawa
theory for the ${\mathbb{Z}}_{2}$-extension $F_{\infty}/F$. As above,
$c_{F_{\infty}}(T)$ will denote a characteristic power series of the
$\Gamma$-module $X(F_{\infty})$.

######  Theorem 5.6.

Let $n\geq 0$ be any non-negative integer, and let $R$ in ${\mathcal{O}}_{K}$
satisfy the Twisting Hypothesis. Then $c_{F_{\infty}}(u\zeta-1)\neq 0$ for all
$2^{n}$-th roots of unity $\zeta$ if and only if $L(B^{(R)}/F_{n},1)\neq 0$.

Combining this result with Theorem 5.5 and Lemmas 5.3 and 5.4, we immediately
obtain

######  Corollary 5.7.

For all $n\geq 0$, we have
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$ is finite if and
only if $L(B^{(R)}/K_{n},1)\neq 0$.

Of course, this is not the place to give a detailed proof of this main
conjecture. We simply say that similar methods, using the Euler system of
elliptic units, to those used to prove Theorem 7.13 in §7 of [3], can be
generalized to prove the desired main conjecture for $B^{(R)}/F_{\infty}$,
from which the above theorem follows as an immediate corollary.

###### Proof.

We now prove Theorem 5.2. We first suppose we are in case (i), so that
$q\equiv 15\ \mathrm{mod}\ 32$ and $R=1$. By a theorem of Rohrlich [12], we
have $L(B/K,1)=L(A/H,1)\neq 0$. Thus, by Corollary 5.7,
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B/K)$ is finite, and, for all $n\geq
1$, ${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B/K_{n})$ is finite if and only
if $L(B/K_{n},1)\neq 0$. Moreover, as $s_{\infty}(R)=2$, it follows from
Theorem 4.3 that
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B/F_{\infty})={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$
as an abelian group, and so Lemma 5.4 implies that
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B/K_{n})$ has
${\mathbb{Z}}_{2}$-corank at most 1 for all $n\geq 1$. This completes the
proof of case (i). Next suppose we are in the case (ii), so that $q\equiv 7\
\mathrm{mod}\ 16$ and $R=r$, with $r\equiv 5\ \mathrm{mod}\ 8$ and $r$ inert
in $K$. Then Theorem 1.3 of [3] shows that $L(B^{(R)}/K,1)=L(A^{(R)}/H,1)\neq
0$. Again invoking Corollary 5.7, it follows that
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K)$ is finite, and, for all
$n\geq 1$, ${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$ is finite
if and only if $L(B^{(R)}/K_{n},1)\neq 0$. But
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$
as an abelian group, whence, again by Lemma 5.4,
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$ has
${\mathbb{Z}}_{2}$-crank at most $1$ for all $n\geq 1$, and the proof of case
(ii) is complete. Finally, suppose we are in case (iii), so that $q\equiv 7\
\mathrm{mod}\ 16$ and $R=-r$, with $r\equiv 3\ \mathrm{mod}\ 8$ and $r$ inert
in $K$. Now $B^{(R)}$ is in fact defined over ${\mathbb{Q}}$, and
$L(B^{(R)}/K,s)=L(B^{(R)}/{\mathbb{Q}},s)^{2}$, where
$L(B^{(R)}/{\mathbb{Q}},s)$ is the complex $L$-series of
$B^{(R)}/{\mathbb{Q}}$ (see §18 of [8]). Write

$\Phi(B^{(R)}/{\mathbb{Q}},s))=(-qr/(2\pi))^{hs}\Gamma(s)^{h}L(B^{(R)}/{\mathbb{Q}},s),$

where we recall that $h$ is the class number of $K$. Now, since $h$ is odd and
$2$ splits in $K$, Theorem 19.1.1 of [8] shows that
$\Phi(B^{(R)}/{\mathbb{Q}},s)$ satisfies the functional equation

$\Phi(B^{(R)}/{\mathbb{Q}},s)=-\Phi(B^{(R)}/{\mathbb{Q}},2-s).$

Hence $\Phi(B^{(R)}/{\mathbb{Q}},s))$ must have a zero of odd order at $s=1$,
and so $L(B^{(R)}/{\mathbb{Q}},1)=0$, whence also $L(B^{(R)}/K,1)=0$, which,
in turn, clearly implies that $L(B^{(R)}/F,1)=0$, and $L(B^{(R)}/K_{n},1)=0$
for all $n\geq 0$. Now
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$
as an abelian group, and
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})^{\Gamma}$ is
infinite because $L(B^{(R)}/F,1)=0$. Hence, since the only infinite subgroup
of ${\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$ is the whole group, it follows that

(5.5)
${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})^{\Gamma}={\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{\infty})={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}.$

Thus, by Lemma 5.4, for all $n\geq 0$, we have
${\mathrm{Sel}}^{\prime}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/F_{n})={\mathbb{Q}}_{2}/{\mathbb{Z}}_{2}$,
whence ${\mathrm{Sel}}_{{\mathfrak{P}}^{\infty}}(B^{(R)}/K_{n})$ has
${\mathbb{Z}}_{2}$-corank 1. This completes the proof of case (iii), and so of
the whole theorem. ∎

## References

  * [1] J. Choi, J. Coates, Iwasawa theory of quadratic twists of $X_{0}(49)$. Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 1, 19-28.
  * [2] J. Coates, Infinite descent on elliptic curves with complex multiplication, Arithmetic and geometry, vol. I, Progress in Mathematics 35 (eds M. Artin and J. Tate; Birkhauser Boston, Boston, MA, 1983) 107-137.
  * [3] J. Coates, Y. Li, Non-vanishing theorems for central L-values of some elliptic curves with complex multiplication, Proc. Lond. Math. Soc. (3) 121 (2020), no. 6, 1531-1578.
  * [4] J. Coates, J. Li, Y. Li, Classical Iwasawa theory and infinite descent on a family of abelian varieties, Selecta Math. (N.S.) 27 (2021), no. 2, Paper No. 28, 36 pp.
  * [5] B. Ferrero, The cyclotomic ${\mathbb{Z}}_{2}$-extension of imaginary quadratic fields, Amer. J. Math. 102 (1980), no. 3, 447-459.
  * [6] G. Gras, Classes généralisées invariantes, J. Math. Soc. Japan 46 (1994), no. 3, 467-476.
  * [7] R. Greenberg, On the structure of certain Selmer groups, Invent. Math. 47 (1978), no. 1, 85-99.
  * [8] B. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics, 776. Springer, Berlin, 1980.
  * [9] Y. Kida, On cyclotomic ${\mathbb{Z}}_{2}$-extensions of imaginary quadratic fields, Tohoku Math. J. (2) 31 (1979), no. 1, 91-96.
  * [10] J. Li, On the $2$-adic logarithm of units of certain totally imaginary quartic fields, to appear in Asian J. Math.
  * [11] J. Li, C. Yu, The Chevalley-Gras formula over global fields, J. Théor. Nombres Bordeaux. 32 (2020), no. 2, 525-544.
  * [12] D. Rohrlich, The non-vanishing of certain Hecke $L$-functions at the centre of the critical strip, Duke Math. J. 47 (1980), 223-232.

CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and
Technology of China,
Hefei, Anhui 230026, China.
<EMAIL_ADDRESS>


11institutetext: Nanjing University of Science and Technology 22institutetext:
University of Science and Technology of China 33institutetext: Sun Yat-sen
University 44institutetext: National University of Singapore
44email<EMAIL_ADDRESS>
# Polyp-DAM: Polyp segmentation via depth anything model

Zhuoran Zheng 11 Chen Wu 22 Wei Wang 33 Yeying Jin 44 Xiuyi Jia 11

###### Abstract

Recently, large models (Segment Anything model) came on the scene to provide a
new baseline for polyp segmentation tasks. This demonstrates that large models
with a sufficient image level prior can achieve promising performance on a
given task. In this paper, we unfold a new perspective on polyp segmentation
modeling by leveraging the Depth Anything Model (DAM) to provide depth prior
to polyp segmentation models. Specifically, the input polyp image is first
passed through a frozen DAM to generate a depth map. The depth map and the
input polyp images are then concatenated and fed into a convolutional neural
network with multiscale to generate segmented images. Extensive experimental
results demonstrate the effectiveness of our method, and in addition, we
observe that our method still performs well on images of polyps with noise.
The URL of our code is https://github.com/zzr-idam/Polyp-DAM.

###### Keywords:

Large models Polyp segmentation Depth anything model Depth map Convolutional
neural network.

##  1 Introduction

The appearance of polyps in the intestine is a precursor to rectal cancer
[17]. Therefore, the detection of polyps is of great importance for the early
diagnosis and treatment of rectal cancer. However, manual screening by
physicians usually leads to the problem of underdetection and misdetection. So
far, automated polyp segmentation methods provide physicians with accurate
decision-making.

Currently, several automated polyp segmentation methods based on deep learning
have been developed and made significant progress [32, 20, 5, 23, 34, 2, 11,
33]. In particular, the large model-based approach [2, 11, 33] (SAM) provides
a new baseline for polyp segmentation. The outstanding performance of these
methods on the polyp problem benefits from the role of large models that carry
a large number of prior image levels. However, SAM-based methods require large
computational resources to fine-tune polyp segmentation datasets, and the
fine-tuning results are fraught with uncertainty. To address this problem, we
use the image-level prior (see Figure 1) provided by the Depth Anything Model
(DAM) [28, 9] to explicitly construct a polyp segmentation model, which does
not require fine-tuning of the larger model.

---
Figure 1: This figure shows the result of processing a polyp image using DAM,
which distinguishes the polyp (foreground) from the rest of the image
(background) very well.
---
Figure 2: This figure shows the process of processing our global module. Here,
the transformation of the feature map dimension is followed by filtering using
a $1\times 1$ convolution, similar to performing an attention operation on
this dimension.

In this paper, we propose a simple but effective polyp segmentation method,
named Polyp-DAM. Specifically, first, the input polyp image is performed by
the DAM to obtain a depth image corresponding to the polyp. Depth images and
polyp images are sampled to four different sizes and fed into our designed
multi-scale MixNet (M2ixNet). M2ixNet is inspired by the vanilla MLP-Mixer
[21], but different from it, the global module encodes the feature maps of
images from the perspectives of the width $W$, height $H$, and channel $C$
(see Figure 2). Through dimensional transformation operations (permute), the
information encoded in each feature map is associated and fused. Note that
dimension transformation operations do not introduce additional parameters,
which means that by permuting the feature maps from different views, the
global module achieves efficient modeling of long-range dependency.
Furthermore, to efficiently model the local features of the feature map, we
design a local feature capture module. The local module filters the feature
maps produced by the global module through a 3D convolution for local modeling
(here the feature map is viewed as a grid). Finally, the four feature maps
output by M2ixNet with different scales are fused by a 2D convolution to
obtain a mask corresponding to a polyp image.

  * •

To the best of our knowledge, we are the first to introduce the DAM prior to
the polyp image segmentation task. The DAM can assist the segmentation model
in distinguishing between foreground and background.

  * •

We propose M2ixNet, which efficiently performs the segmentation of polyp
images through global versus local modeling. It is worth noting that the
number of parameters in our network is only 0.47M without considering DAM.

  * •

Extensive experiments demonstrate that our method outperforms state-of-the-art
approaches with better quantitative and qualitative evaluation.

##  2 Related Work

UNet [16] is a classical image segmentation model with a symmetric encoding-
decoding architecture, and it can efficiently fuse shallow and deep features
of an image. Based on its good segmentation performance, some polyp
segmentation methods [29, 14, 27, 15] adopt the U-shape network. In addition,
some methods use attention mechanisms and carefully designed strategies to
improve segmentation accuracy [8, 25, 31, 34, 24, 30, 3, 19]. Recently, since
the success of the Transformer in multiple fields, some approaches [4, 23, 5,
2, 11, 33] have replaced the backbone of networks from CNN with Transformer
and obtained good results. In this paper, we focus on obtaining accurate polyp
image segmentation results using an image-level prior.

---
Figure 3: The structure of our method. First, we obtain the depth map of the
input polyp image via DAM. Next, the input image and depth map are bilinearly
downsampled into $256\times 256$, $128\times 128$, $64\times 64$, and the
original resolution is fed into M2ixNet. Finally, our network outputs four
masks of different sizes and four sizes generated on GT for learning. It is
worth noting that our network used in the evaluation on the benchmark is the
image at the model output original resolution.

##  3 Method

An overview of Polyp-DAM is shown in Fig. 3. For a given polyp image
$\mathbf{X}\in\mathbb{R}^{H\times W\times 3}$, we start by projecting
$\mathbf{X}$ onto a 4-level resolution image
{$\mathbf{X}_{o},\mathbf{X}_{64},\mathbf{X}_{128},\\\ \mathbf{X}_{256}$} by
using the bilinear interpolation. Then, we input $\mathbf{X}$ into the DAM to
generate a depth map $\mathbf{D}\in\mathbb{R}^{H\times W\times 3}$, which is
also projected onto the image at 4 levels of resolution
{$\mathbf{D}_{o},\mathbf{D}_{64},\mathbf{D}_{128},\mathbf{D}_{256}$}. Finally,
these depth maps of different resolutions with corresponding polyp images are
input into the M2ixNet to get an accurate mask $\mathbf{M}$. Here, different
scales of polyp information are input into a convolutional layer ($3\times 3$
convolution kernel) in pairs respectively, followed by output feature maps up-
sampled to the resolution of the original image. It is worth noting that the
accurate mask $\mathbf{M}$ was also mapped to 4 resolutions to do loss in the
training stage for the GT $\mathbf{O}$.

###  3.1 Global Module of M2ixNet

The polyp images at different scales and the corresponding depth images are
convolved to obtain a feature map $\mathbf{F}_{0}\in\mathbb{R}^{H\times
W\times 12}$ by concatenating them on the channel domain, respectively. After
that, $\mathbf{F}_{0}$ is fed into the global module G initially. G is
designed from [21]. We try to capture the long-range dependence of polyp
images using a small number of operators. Specifically, we first adjust the
normalized input features’ resolution, then perform some dimension
transformation operations on them. Given the input feature $\mathbf{F}_{0}$,
this procedure can be written as:

$\displaystyle\mathbf{F}_{1}$ $\displaystyle=\text{IP}(\mathbf{F}_{0}),$ (1)
$\displaystyle\hat{\mathbf{F}}_{1}$
$\displaystyle=\underbrace{\text{Permute}(\text{ReLU}(\text{Conv}_{1\times
1}(\mathbf{F}_{1})))}_{\times 3},$ (2)

where $\hat{\mathbf{F}}_{t}$ is intermediate result. $\text{IP}(\cdot)$
corresponds to the interpolation operation, $\text{Conv}_{1\times 1}(\cdot)$
is a $1\times 1$ convolution, $\text{ReLU}(\cdot)$ represents ReLU function,
$\text{Permute}(\cdot)$ denotes the dimension transformation operation and
$\times 3$ denotes three operations in sequence. It is worth noting that
before executing the last dimension transformation operation, we replace the
ReLU function with the Sigmoid function.

Afterward, We use interpolation to adjust the feature $\hat{\mathbf{F}}_{t}$
to its original resolution to estimate the attention map and adaptively
modulate the input $\mathbf{F}_{0}$ according to the estimated attention via
element-wise product. This process can be written as:

$\mathbf{F}_{g}=\text{IP}(\hat{\mathbf{F}}_{1})\odot\mathbf{F}_{0},$ (3)

where $\mathbf{F}_{g}$ are the final output features and $\odot$ represents
element-wise product.

###  3.2 Local Module of M2ixNet

For local feature extraction, we mainly use a 3D convolution to perform on the
input feature map. Specifically, given the inputs $\mathbf{F}_{0}$ and
$\mathbf{F}_{g}$, they are expanded by one dimension thus obtaining
$\hat{\mathbf{F}}_{0}\in\mathbb{R}^{1\times H\times W\times C}$ and
$\hat{\mathbf{F}}_{g}\in\mathbb{R}^{1\times H\times W\times C}$. Next,
$\hat{\mathbf{F}}_{0}$ and $\hat{\mathbf{F}}_{g}$ are concatenated in the
newly expanded dimensions to obtain a tensor $\mathbf{B}\in\mathbb{R}^{2\times
H\times W\times C}$. Then, we use a 3D convolution to extract the features of
$\mathbf{B}$ to yield a new feature map $\mathbf{F}_{2}\in\mathbb{R}^{H\times
W\times C}$(here the input channel of the 3D convolution is 2 and the output
channel is 1). Finally, $\mathbf{F}_{2}$ passes through a convolutional block
(including two convolutional layers with $3\times 3$ convolutional kernels) to
obtain a feature map that is summed with $\mathbf{F}_{0}$ performing the
corresponding element. The above process can be written:

$\mathbf{F}_{l}=\text{ConvB}(\text{Conv3D}(\text{Cat}(\text{UnSqu}(\mathbf{F}_{0}),\text{UnSqu}(\mathbf{F}_{g}))))+\mathbf{F}_{0},$
(4)

where $\mathbf{F}_{l}$ are the final output features, UnSqu denotes the
expanded dimension, ConvB denotes the convolution block and $+$ represents the
element-wise sum.

In this paper, we use 32 pairs of global and local modules. Among other
things, we use a lot of shortcuts between modules. At the output of the
network, a 2D convolution is used to squeeze the channels of the feature map
to 1.

###  3.3 Loss Function

Our loss function can be formulated as :

$\mathcal{L}=\mathcal{L}_{IoU}^{w}+\mathcal{L}^{w}_{BCE},$ (5)

where $\mathcal{L}_{IoU}^{w}$ and $\mathcal{L}^{w}_{BCE}$ represent the
weighted intersection over union loss and weighted binary cross entropy loss
[26], which are adopted for the global and local restriction. Deep supervision
is performed on the four stage outputs ($i.e.,$ $\mathbf{M}_{o}$,
$\mathbf{M}_{64}$, $\mathbf{M}_{128}$ and $\mathbf{M}_{256}$). Each output is
up-sampled to the same size as the ground truth
$\mathbf{O}\in\\{\mathbf{O}_{o},\mathbf{O}_{64},\mathbf{O}_{128},\mathbf{O}_{256}\\}$.
The total loss can be formulated as:

$\mathcal{L}_{total}=\sum\limits_{i=1}^{i=4}\mathcal{L}(\mathbf{M}_{i},\mathbf{O}_{i})$
(6)

In this paper, we use the output of $\mathbf{M}_{o}$ as the segmented image
generated by the model. Note that, except for the original image resolution
$\mathbf{M}_{o},\mathbf{O}_{o}$, all other images are obtained by bilinear
interpolation.

##  4 Experiment

###  4.1 Implementation Details

We implement our model in the PyTorch 2.0 framework. The GPU shader is NVIDIA
TITAN RTX 3090 with 24G RAM. During training, all input images are uniformly
resized to $352\times 352$ and a multi-scale training strategy [8] is
employed. The network is trained for 200 epochs with a batch size of 16. For
optimizer, we use the AdamW [12] to update parameters. The learning rate is
1e-4 and the weight decay is 1e-4.

###  4.2 Datasets and Metrics

To evaluate the performance of the proposed M2ixNet, we follow the popular
experimental setups [8], which adopt 5 publicly available benchmarks: Kvasir-
SEG [10], CVC-ClinicDB [1], ColonDB [20], Endoscene [22] and ETIS [18]. The
training set comprises 900 images from Kvasir-SEG and 550 from CVC-ClinicDB.
The remaining 100 images from Kvasir-SEG and 62 images from CVC-ClinicDB are
used as test sets. Since they are already seen during training, we use them to
assess the learning ability of our model. We perform testing using ColonDB,
Endoscene, and ETIS, as these datasets are not used during the training stage,
they are utilized to measure the generalization ability of our model.

Following previous work [8, 4], We employ six widely-used metrics for
quantitative evaluation, including Mean Dice Similarity Coefficient (mDice),
mean Intersection over Union (mIoU), Mean Absolute Error (MAE), weighted
F-measure ($F_{\beta}^{w}$) [13], S-measure ($S_{\alpha}$) [6], and max
E-measure ($E_{\xi}^{max}$) [7]. mDice and mIoU are similarity measures at the
regional level, while MAE and $F_{\beta}^{w}$ are utilized to evaluate the
pixel-level accuracy. $S_{\alpha}$ and $E_{\xi}^{max}$ are used to analyze
global-level similarity. The lower value is better for the MAE and the higher
is better for others.

Table 1: Quantitative results on Kvasir-SEG [10] and CVC-ClinicDB [1] dataset. Bold indicates the best performance. |  | Kvasir-SEG
---|---|---
method | Year | mDice | mIoU | $F_{\beta}^{w}$ | $S_{\alpha}$ | $E_{\xi}^{max}$ | MAE
UNet [16] | 2015 | 0.818 | 0.746 | 0.794 | 0.858 | 0.893 | 0.055
UNet++ [35] | 2018 | 0.821 | 0.744 | 0.808 | 0.862 | 0.909 | 0.048
PraNet [8] | 2020 | 0.898 | 0.841 | 0.885 | 0.915 | 0.948 | 0.030
ACSNet [31] | 2020 | 0.898 | 0.838 | 0.882 | 0.920 | 0.952 | 0.032
SANet [25] | 2021 | 0.904 | 0.847 | 0.892 | 0.915 | 0.953 | 0.028
MSNet [32] | 2021 | 0.902 | 0.847 | 0.891 | 0.923 | 0.954 | 0.029
SSFormer [23] | 2022 | 0.925 | 0.878 | 0.921 | 0.931 | 0.969 | 0.017
DCRNet [29] | 2022 | 0.846 | 0.772 | 0.807 | 0.882 | 0.917 | 0.053
Polyp-sam [11] | 2023 | 0.900 | 0.860 | - | - | - | -
M2ixNet(Ours) | 2024 | 0.929 | 0.881 | 0.929 | 0.935 | 0.971 | 0.014
|  | CVC-ClinicDB
method | Year | mDice | mIoU | $F_{\beta}^{w}$ | $S_{\alpha}$ | $E_{\xi}^{max}$ | MAE
UNet [16] | 2015 | 0.823 | 0.755 | 0.811 | 0.890 | 0.953 | 0.019
UNet++ [35] | 2018 | 0.794 | 0.729 | 0.785 | 0.873 | 0.931 | 0.022
PraNet [8] | 2020 | 0.899 | 0.849 | 0.896 | 0.937 | 0.979 | 0.009
ACSNet [31] | 2020 | 0.882 | 0.826 | 0.873 | 0.928 | 0.959 | 0.011
SANet [25] | 2021 | 0.916 | 0.859 | 0.909 | 0.940 | 0.976 | 0.012
MSNet [32] | 2021 | 0.915 | 0.867 | 0.912 | 0.947 | 0.978 | 0.008
SSFormer [23] | 2022 | 0.916 | 0.873 | 0.924 | 0.937 | 0.971 | 0.007
DCRNet [29] | 2022 | 0.869 | 0.800 | 0.832 | 0.919 | 0.968 | 0.023
Polyp-sam [11] | 2023 | 0.920 | 0.870 | - | - | - | -
M2ixNet (Ours) | 2024 | 0.941 | 0.891 | 0.934 | 0.955 | 0.989 | 0.005
Table 2: Quantitative results on ColonDB [20], Endoscene [22] and ETIS [18]) dataset. Bold indicates the best performance. |  | CVC-ColonDB
---|---|---
method | Year | mDice | mIoU | $F_{\beta}^{w}$ | $S_{\alpha}$ | $E_{\xi}^{max}$ | MAE
UNet [16] | 2015 | 0.504 | 0.436 | 0.491 | 0.710 | 0.781 | 0.059
UNet++ [35] | 2018 | 0.481 | 0.408 | 0.467 | 0.693 | 0.763 | 0.061
PraNet [8] | 2020 | 0.712 | 0.640 | 0.699 | 0.820 | 0.872 | 0.043
ACSNet [31] | 2020 | 0.716 | 0.649 | 0.697 | 0.830 | 0.851 | 0.039
SANet [25] | 2021 | 0.752 | 0.669 | 0.725 | 0.837 | 0.875 | 0.043
MSNet [32] | 2021 | 0.747 | 0.668 | 0.733 | 0.837 | 0.883 | 0.042
SSFormer [23] | 2022 | 0.772 | 0.697 | 0.766 | 0.844 | 0.883 | 0.036
DCRNet [29] | 2022 | 0.661 | 0.576 | 0.613 | 0.767 | 0.828 | 0.110
Polyp-sam [11] | 2023 | 0.894 | 0.843 | - | - | - | -
M2ixNet (Ours) | 2024 | 0.820 | 0.855 | 0.799 | 0.869 | 0.935 | 0.021
|  | Endoscene
method | Year | mDice | mIoU | $F_{\beta}^{w}$ | $S_{\alpha}$ | $E_{\xi}^{max}$ | MAE
UNet [16] | 2015 | 0.710 | 0.627 | 0.684 | 0.843 | 0.875 | 0.022
UNet++ [35] | 2018 | 0.707 | 0.624 | 0.687 | 0.839 | 0.898 | 0.018
PraNet [8] | 2020 | 0.871 | 0.797 | 0.843 | 0.925 | 0.972 | 0.010
ACSNet [31] | 2020 | 0.863 | 0.787 | 0.825 | 0.923 | 0.968 | 0.013
SANet [25] | 2021 | 0.888 | 0.815 | 0.859 | 0.928 | 0.972 | 0.008
MSNet [32] | 2021 | 0.862 | 0.796 | 0.846 | 0.927 | 0.953 | 0.010
SSFormer [23] | 2022 | 0.887 | 0.821 | 0.869 | 0.929 | 0.962 | 0.007
DCRNet [29] | 2022 | 0.753 | 0.670 | 0.689 | 0.854 | 0.900 | 0.025
Polyp-sam [11] | 2023 | 0.905 | 0.860 | - | - | - | -
M2ixNet (Ours) | 2024 | 0.895 | 0.861 | 0.879 | 0.941 | 0.978 | 0.005
|  | ETIS
method | Year | mDice | mIoU | $F_{\beta}^{w}$ | $S_{\alpha}$ | $E_{\xi}^{max}$ | MAE
UNet [16] | 2015 | 0.398 | 0.335 | 0.366 | 0.684 | 0.740 | 0.036
UNet++ [35] | 2018 | 0.401 | 0.343 | 0.390 | 0.683 | 0.776 | 0.035
PraNet [8] | 2020 | 0.628 | 0.567 | 0.600 | 0.794 | 0.841 | 0.031
ACSNet [31] | 2020 | 0.578 | 0.509 | 0.530 | 0.754 | 0.764 | 0.059
SANet [25] | 2021 | 0.750 | 0.654 | 0.685 | 0.849 | 0.897 | 0.015
MSNet [32] | 2021 | 0.720 | 0.650 | 0.675 | 0.846 | 0.881 | 0.020
SSFormer [23] | 2022 | 0.767 | 0.698 | 0.736 | 0.863 | 0.891 | 0.016
DCRNet [29] | 2022 | 0.509 | 0.432 | 0.437 | 0.714 | 0.787 | 0.053
Polyp-sam [11] | 2023 | 0.905 | 0.861 | - | - | - | -
M2ixNet (Ours) | 2024 | 0.891 | 0.866 | 0.753 | 0.882 | 0.933 | 0.009
---
Figure 4: Our method exhibits the best visual results.

###  4.3 Experiments with State-of-the-art Methods

We compare the proposed D2LNet with previous state-of-the-art methods,
including Polyp-sam [11], DCRNet [29], SSFormer [23], MSNet [32], SANet [25],
ACSNet [31], PraNet [8], UNet++ [35], UNet [16]. For a fair comparison, we use
their open-source codes to evaluate the same datasets or use the predictions
provided by themselves.

Quantitative Comparison. As can be seen in Table 1 and Table 2, M2ixNet
achieves the best scores across five datasets on almost all metrics.

Visual Comparison. Figure 4 shows the visualization results of our model and
the compared models. Our proposed model significantly outperforms other
methods with better segmentation results, which are the closest to the ground
truth. We discover that our model possesses the following advantages: Our
model can adapt to different lighting conditions, such as the overexposure
observed in the third and sixth images in Figure 4. This is thanks to the role
of the image-level prior that DAM brings into play.

Table 3: Ablation study for M2ixNet on the Kvasir-SEG [10] dataset. | mDice | mIoU | $F_{\beta}^{w}$ | $S_{\alpha}$ | $E_{\xi}^{max}$ | MAE
---|---|---|---|---|---|---
w/o DAM | 0.874 | 0.840 | 0.882 | 0.899 | 0.930 | 0.049
w/o multi-scale | 0.911 | 0.876 | 0.920 | 0.910 | 0.969 | 0.020
M2ixNet | 0.929 | 0.881 | 0.929 | 0.935 | 0.971 | 0.014

###  4.4 Ablation Study

We conduct an ablation study to validate the effectiveness of the modules
designed in our M2ixNet. To validate the effectiveness of the DAM, we removed
the role of depth map (w/o DAM). To analyze the effectiveness of the multi-
scale input information, We use only the original polyp images as input (w/o
multi-scale). Table 3 demonstrates the performance of the model without the
DAM and multi-scale drops sharply on Kvasir datasets.

##  5 Discussion and Conclusion

We try to feed the depth maps generated by DAM as supplementary information to
the UNet and UNet++ models, and on the Kvasir-SEG [10] dataset, their
performance improves by 5% and 8%, respectively. Especially on extremely
exposed datasets, DAM is also capable of generating high-quality depth maps to
assist polyp segmentation models.

In this paper, we use the DAM-assisted polyp segmentation model to achieve
accurate polyp segmentation. Furthermore, our proposed M2ixNet shows efficient
segmentation. Experimental results demonstrate the superiority of our method.

## References

  * [1] Bernal, J., Sánchez, F.J., Fernández-Esparrach, G., Gil, D., Rodríguez, C., Vilariño, F.: WM-DOVA maps for accurate polyp highlighting in colonoscopy: Validation vs. saliency maps from physicians. CMIG 43, 99–111 (2015)
  * [2] Biswas, R.: Polyp-sam++: Can a text guided sam perform better for polyp segmentation? arXiv preprint arXiv:2308.06623 (2023)
  * [3] Chi, L., Jiang, B., Mu, Y.: Fast fourier convolution. Advances in Neural Information Processing Systems 33, 4479–4488 (2020)
  * [4] Dong, B., Wang, W., Fan, D.P., Li, J., Fu, H., Shao, L.: Polyp-pvt: Polyp segmentation with pyramid vision transformers. arXiv preprint arXiv:2108.06932 (2021)
  * [5] Duc, N.T., Oanh, N.T., Thuy, N.T., Triet, T.M., Dinh, V.S.: Colonformer: An efficient transformer based method for colon polyp segmentation. IEEE Access 10, 80575–80586 (2022)
  * [6] Fan, D.P., Cheng, M.M., Liu, Y., Li, T., Borji, A.: Structure-measure: A new way to evaluate foreground maps. In: Proceedings of the IEEE international conference on computer vision. pp. 4548–4557 (2017)
  * [7] Fan, D.P., Gong, C., Cao, Y., Ren, B., Cheng, M.M., Borji, A.: Enhanced-alignment measure for binary foreground map evaluation. arXiv preprint arXiv:1805.10421 (2018)
  * [8] Fan, D.P., Ji, G.P., Zhou, T., Chen, G., Fu, H., Shen, J., Shao, L.: Pranet: Parallel reverse attention network for polyp segmentation. In: International conference on medical image computing and computer-assisted intervention. pp. 263–273. Springer (2020)
  * [9] Huang, Y., Cui, B., Bai, L., Guo, Z., Xu, M., Ren, H.: Endo-4dgs: Distilling depth ranking for endoscopic monocular scene reconstruction with 4d gaussian splatting. arXiv preprint arXiv:2401.16416 (2024)
  * [10] Jha, D., Smedsrud, P.H., Riegler, M.A., Halvorsen, P., de Lange, T., Johansen, D., Johansen, H.D.: Kvasir-seg: A segmented polyp dataset. In: MMM. pp. 451–462 (2020)
  * [11] Li, Y., Hu, M., Yang, X.: Polyp-sam: Transfer sam for polyp segmentation. arXiv preprint arXiv:2305.00293 (2023)
  * [12] Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101 (2017)
  * [13] Margolin, R., Zelnik-Manor, L., Tal, A.: How to evaluate foreground maps? In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 248–255 (2014)
  * [14] Patel, K., Bur, A.M., Wang, G.: Enhanced u-net: A feature enhancement network for polyp segmentation. In: 2021 18th Conference on Robots and Vision (CRV). pp. 181–188. IEEE (2021)
  * [15] Poudel, S., Lee, S.W.: Deep multi-scale attentional features for medical image segmentation. Applied Soft Computing 109, 107445 (2021)
  * [16] Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation. In: Navab, N., Hornegger, J., III, W.M.W., Frangi, A.F. (eds.) MICCAI 2015. pp. 234–241. LNCS, Springer (2015), https://doi.org/10.1007/978-3-319-24574-4_28
  * [17] Shussman, N., Wexner, S.D.: Colorectal polyps and polyposis syndromes. Gastroenterology report 2(1), 1–15 (2014)
  * [18] Silva, J., Histace, A., Romain, O., Dray, X., Granado, B.: Toward embedded detection of polyps in WCE images for early diagnosis of colorectal cancer. CARS 9, 283–293 (2014)
  * [19] Suvorov, R., Logacheva, E., Mashikhin, A., Remizova, A., Ashukha, A., Silvestrov, A., Kong, N., Goka, H., Park, K., Lempitsky, V.: Resolution-robust large mask inpainting with fourier convolutions. In: Proceedings of the IEEE/CVF winter conference on applications of computer vision. pp. 2149–2159 (2022)
  * [20] Tajbakhsh, N., Gurudu, S.R., Liang, J.: Automated polyp detection in colonoscopy videos using shape and context information. IEEE TMI 35(2), 630–644 (2015)
  * [21] Tolstikhin, I.O., Houlsby, N., Kolesnikov, A., Beyer, L., Zhai, X., Unterthiner, T., Yung, J., Steiner, A., Keysers, D., Uszkoreit, J., et al.: Mlp-mixer: An all-mlp architecture for vision. Advances in neural information processing systems 34, 24261–24272 (2021)
  * [22] Vázquez, D., Bernal, J., Sánchez, F.J., Fernández-Esparrach, G., López, A.M., Romero, A., Drozdzal, M., Courville, A., et al.: A benchmark for endoluminal scene segmentation of colonoscopy images. JHE 2017 (2017)
  * [23] Wang, J., Huang, Q., Tang, F., Meng, J., Su, J., Song, S.: Stepwise feature fusion: Local guides global. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 110–120. Springer (2022)
  * [24] Wang, K.N., He, Y., Zhuang, S., Miao, J., He, X., Zhou, P., Yang, G., Zhou, G.Q., Li, S.: Ffcnet: Fourier transform-based frequency learning and complex convolutional network for colon disease classification. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 78–87. Springer (2022)
  * [25] Wei, J., Hu, Y., Zhang, R., Li, Z., Zhou, S.K., Cui, S.: Shallow attention network for polyp segmentation. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part I 24. pp. 699–708. Springer (2021)
  * [26] Wei, J., Wang, S., Huang, Q.: F3net: fusion, feedback and focus for salient object detection. In: Proceedings of the AAAI conference on artificial intelligence. vol. 34, pp. 12321–12328 (2020)
  * [27] Wu, H., Zhong, J., Wang, W., Wen, Z., Qin, J.: Precise yet efficient semantic calibration and refinement in convnets for real-time polyp segmentation from colonoscopy videos. In: Proceedings of the AAAI Conference on Artificial Intelligence. vol. 35, pp. 2916–2924 (2021)
  * [28] Yang, L., Kang, B., Huang, Z., Xu, X., Feng, J., Zhao, H.: Depth anything: Unleashing the power of large-scale unlabeled data. arXiv preprint arXiv:2401.10891 (2024)
  * [29] Yin, Z., Liang, K., Ma, Z., Guo, J.: Duplex contextual relation network for polyp segmentation. In: 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI). pp. 1–5. IEEE (2022)
  * [30] Yu, H., Zheng, N., Zhou, M., Huang, J., Xiao, Z., Zhao, F.: Frequency and spatial dual guidance for image dehazing. In: European Conference on Computer Vision. pp. 181–198. Springer (2022)
  * [31] Zhang, R., Li, G., Li, Z., Cui, S., Qian, D., Yu, Y.: Adaptive context selection for polyp segmentation. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd International Conference, Lima, Peru, October 4–8, 2020, Proceedings, Part VI 23. pp. 253–262. Springer (2020)
  * [32] Zhao, X., Zhang, L., Lu, H.: Automatic polyp segmentation via multi-scale subtraction network. In: MICCAI. pp. 120–130 (2021)
  * [33] Zhou, T., Zhang, Y., Zhou, Y., Wu, Y., Gong, C.: Can sam segment polyps? arXiv preprint arXiv:2304.07583 (2023)
  * [34] Zhou, T., Zhou, Y., He, K., Gong, C., Yang, J., Fu, H., Shen, D.: Cross-level feature aggregation network for polyp segmentation. Pattern Recognition 140, 109555 (2023)
  * [35] Zhou, Z., Rahman Siddiquee, M.M., Tajbakhsh, N., Liang, J.: Unet++: A nested u-net architecture for medical image segmentation. In: Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support: 4th International Workshop, DLMIA 2018, and 8th International Workshop, ML-CDS 2018, Held in Conjunction with MICCAI 2018, Granada, Spain, September 20, 2018, Proceedings 4. pp. 3–11. Springer (2018)