.
1 Introduction
Throughout this article,
$p\ge 5$
is a prime. Let k be a number field and
$k_\infty /k$
be a
$\mathbb {Z}_p$
-extension. Let
$k_n$
be the unique subextension of degree
$p^n$
and
$h_n$
be the p-class number of
$k_n$
. Iwasawa proved in his seminal paper [Reference Iwasawa6] the asymptotic formula
$$\begin{align*}v_p(h_n)=\mu p^n+\lambda n+\nu \quad n\gg 0,\end{align*}$$
for invariants
$\mu ,\lambda \ge 0$
and
$\nu \in \mathbb {Z}$
.
In light of this key result, the analysis of arithmetic objects along
$\mathbb {Z}_p$
-extensions has become a central topic in modern Iwasawa theory. Mazur [Reference Mazur10] generalized Iwasawa’s ideas and applied them to elliptic curves with good ordinary reduction at all primes above p. He showed that – if the p-primary Selmer group over
$k_\infty $
is cotorsion over the Iwasawa algebra of
$\operatorname {\mathrm {Gal}}(k_\infty /k)$
– there is an asymptotic formula
where 
is the Tate–Shafarevich group of E over
$k_n$
(assuming that it is finite). A crucial step in his argument is a so-called control theorem. For supersingular primes, this control theorem is no longer valid. Let
$E/\mathbb {Q}$
be an elliptic curve supersingular at p and let
$\mathbb {Q}_\infty $
be the cyclotomic
$\mathbb {Z}_p$
-extension of
$\mathbb {Q}$
. As
$p\ge 5$
, this implies that
$a_p=0$
by the Hasse bound [Reference Silverman11, Chapter V, Theorem 1.1]. Kobayashi [Reference Kobayashi7] constructed plus/minus Selmer groups that satisfy a control theorem. From his control theorem, he was able to derive the following asymptotic formula:
where
$r=\text {rank}(E(\mathbb {Q}_\infty ))<\infty $
and
$\phi $
denotes the Euler
$\phi $
-function. The invariants
$\mu ^\pm $
and
$\lambda ^\pm $
are the Iwasawa invariants of the Pontryagin duals of the plus/minus Selmer groups.
Instead of the cyclotomic
$\mathbb {Z}_p$
-extension, we consider the anticyclotomic
$\mathbb {Z}_p$
-extension for the rest of the article. We keep the assumption that
$E/\mathbb {Q}$
is an elliptic curve and that p is a supersingular prime. Let K be an imaginary quadratic field. Let
$K_\infty /K$
be the anticyclotomic
$\mathbb {Z}_p$
-extension, i.e., the
$\mathbb {Z}_p$
-extension on which
$\operatorname {\mathrm {Gal}}(K/\mathbb {Q})$
acts as
$-1$
, and let
$K_n$
be the unique subextension of degree
$p^n$
. Assume that K satisfies the generalized Heegner hypothesis: Let
$N=N_1N_2$
be the conductor of E, where
$N_1$
and
$N_2$
are coprime and
$N_2$
is square-free. Assume that all primes dividing
$pN_1$
are split in K and that all primes dividing
$N_2$
are inert in K. In particular, we assume that p splits in K. In this setting, the rank of
$E(K_n)$
is unbounded and the plus/minus Selmer groups are no longer cotorsion. Nevertheless, there is – under the assumption that 
is finite and that the representation
$$\begin{align*}\rho \colon G_{\mathbb{Q}}\to \text{Aut}(T_p(E))\end{align*}$$
is surjective – an asymptotic formula [Reference Lei, Lim and Müller8]:
These Iwasawa invariants are no longer the ones of the plus/minus Selmer groups.
The Heegner hypothesis excludes the case of CM elliptic curves with complex multiplication by
$\mathcal {O}_K$
as primes of good supersingular reduction are inert in K. The aim of the present article is to consider this case. Let
$\varepsilon \in \{\pm 1\}$
be the root number of
$E/\mathbb {Q}$
. Then, the
$-\varepsilon $
-Selmer group is cotorsion while the
$\varepsilon $
-Selmer group is not. Burungale, Kobayashi, and Ota [Reference Burungale, Kobayashi and Ota3, Theorem 1.1] prove that for all n large enough such that
$(-1)^n=-\varepsilon ,$
one has
for some invariants
$\mu ,\lambda \ge 0$
.
The invariants occurring in the asymptotic formula of Burungale–Kobayashi–Ota come from the fine Selmer groups, the
$-\varepsilon $
Selmer group, and a finitely generated
$\mathbb {Z}_p$
-module A independent of n.
Let
$\Lambda $
be the Iwasawa algebra of
$\Gamma =\operatorname {\mathrm {Gal}}(K_\infty /K)$
over the ring
$\mathcal {O}$
, the ring of integers of
$K_p$
, where
$K_p$
denotes the completion of K at p. Define
$$\begin{align*}\omega^+_n(T)=\prod_{1\le k\le n, k\text{ even}}\Phi_k(T+1)\quad \omega_n^-=T\prod_{1\le k\le n,k \text{ odd}}\Phi_k(T+1),\end{align*}$$
where
$\Phi _k$
denotes the
$p^k$
-th cyclotomic polynomial. Our main result covers the remaining steps.
Theorem 1 Assume that 
and the fine Selmer group
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
are finite for all n. Then, for all n large enough and such that
$(-1)^n=\varepsilon ,$
one has
The integers
$\mu $
and
$\lambda $
are the Iwasawa invariants of the fine Tate–Shafarevich groups.
As an immediate corollary, we obtain the following theorem.
Theorem 2 Assume that 
and
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
are finite for all n. Then, for all n large enough, one has
The invariants are the ones from (1.1) and Theorem 1, respectively. Note that one expects
$\mu ^\pm =0$
in this setting.
Remark 1.1 The condition that
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite is equivalent to the statement that the characteristic ideal of
$\operatorname {\mathrm {Sel}}^0(E/K_\infty )^\vee $
is corpime to
$\omega _n^{-\varepsilon }$
.
If one assumes that 
is finite, this is equivalent to
$$\begin{align*}f_n=\frac{\operatorname{\mathrm{rank}}(E(K_n))-\operatorname{\mathrm{rank}}(E(K_{n-1}))}{2\phi(p^n)}\le 1,\end{align*}$$
for all n such that
$(-1)^n=-\varepsilon $
. It is known that
$f_n=0$
for all such n large enough [Reference Greenberg5].
The central idea of the proof is to decompose 
into plus and minus Tate–Shafarevich groups whose intersection is the fine Tate–Shafarevich group. Using control theorems for the respective Selmer groups, we will then derive the above asymptotic formula. This approach differs from the one presented in [Reference Burungale, Kobayashi and Ota2]. In loc. cit, the authors relate the growth of 
to the cokernel of
$$\begin{align*}\operatorname{\mathrm{Sel}}(E/K_{n-1})\to \operatorname{\mathrm{Sel}}(E/K_n).\end{align*}$$
If
$(-1)^n=-\varepsilon ,$
this cokernel is finite and compuatable in terms of Iwasawa invariants. In the case
$(-1)^n=\varepsilon $
, this cokernel is of corank
$\phi (p^n)$
for all n. We have thus to apply different methods and need the additional assumption that
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite for all n.
The fine Tate–Shafarevich groups do not only play a central role in our proofs, but are also of independent interest and we are able to derive an asymptotic formula for them.
Theorem 3 Let
$\kappa ^0(E/K_n)$
be the fine Tate–Shafarevich group of E over
$K_n$
. For all
$n\gg 0,$
we have
$$\begin{align*}v_p(\vert \kappa^0(E/K_n)\vert)=\lambda n+p^n\mu +\nu,\end{align*}$$
for
$\mu ,\lambda \ge 0$
and
$\nu \in \mathbb {Z}$
.
Note that Theorem 3 is a generalization of the results in [Reference Lim9]. Loc. cit. only considers the cases of good ordinary reduction (Theorem 1.7) and of
$E(K_\infty )$
being of finite rank (Theorem 1.6). Both conditions are not satisfied for supersingular elliptic curves and the anticyclotomic
$\mathbb {Z}_p$
-extension.
2 Plus/Minus Selmer groups
Let K be an imaginary quadratic field and let p be prime that is inert in K. Let
$E/\mathbb {Q}$
be an elliptic curve that has complex multiplication by
$\mathcal {O}_K$
. Let
$K_\infty /K$
be the anticyclotomic
$\mathbb {Z}_p$
-extension and let
$K_n$
be the intermediate fields. Let
$\varepsilon $
be the root number of E. Let
$\tau $
be a topological generator of
$\operatorname {\mathrm {Gal}}(K_\infty /K)$
, let
$T=\tau -1$
and 
. Throughout the article, we assume that 
is finite for all n.
Let
$\Xi $
denote the set of Dirichlet characters of
$\operatorname {\mathrm {Gal}}(K_\infty /K)$
. Let
$\Xi ^+$
be the subset of non-trivial characters whose order is an even power of p and let
$\Xi ^-$
be the set of characters whose order is an odd power of p and the trivial character. Let
$K_{n,p}$
be the localization of
$K_n$
at the unique prime above p in
$K_n$
. We denote by
$\widehat {E}$
the formal group of E at p and by
$\log $
its formal logarithm. For any character
$\chi \in \Xi $
and any
$x\in \widehat {E}(K_{n,p}),$
we define
$$\begin{align*}\lambda_\chi(x)=p^{-m}\sum_{\sigma\in \operatorname{\mathrm{Gal}}(K_{m,p}/K_p)}\log(x^\sigma)\chi^{-1}(\sigma),\end{align*}$$
where
$\chi $
is a character factoring through
$\operatorname {\mathrm {Gal}}(K_{m,p}/K)$
and
$m\ge n$
. Let
$$\begin{align*}\widehat{E}(K_{n,p})^\pm=\{x\in \widehat{E}(K_{n,p})\mid \lambda_\chi(x)=0\quad \forall \chi\in \Xi^\mp\}.\end{align*}$$
Let
$\Sigma $
be the set of primes dividing the conductor of E and p. Let
$K_\Sigma $
be the maximal Galois extension of K unramified outside
$\Sigma $
.
Definition 2.1 We define
$$ \begin{align*}&\operatorname{\mathrm{Sel}}(E/K_n)\\&=\ker\left(H^1(K_\Sigma/K_n,E[p^\infty])\to \prod_{v\in \Sigma,(v,p)=1}H^1(K_{n,v},E[p^\infty])\times \frac{H^1(K_{n,p},E[p^\infty])}{\widehat{E}(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p}\right).\end{align*} $$
We define the plus/minus Selmer groups
$$ \begin{align*}&\operatorname{\mathrm{Sel}}^\pm(E/K_n)\\&=\ker\left(H^1(K_\Sigma/K_n,E[p^\infty])\to \prod_{v\in \Sigma,(v,p)=1}H^1(K_{n,v},E[p^\infty])\times \frac{H^1(K_{n,p},E[p^\infty])}{\widehat{E}^\pm(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p}\right)\end{align*} $$
as well as the fine Selmer group
$$\begin{align*}\operatorname{\mathrm{Sel}}^0(E/K_n)=\ker\left(H^1(K_\Sigma/K_n,E[p^\infty])\to \prod_{v\in \Sigma}H^1(K_{n,v},E[p^\infty])\right).\end{align*}$$
For
$*\in \{0,+,-\},$
we define
$$\begin{align*}\mathcal{M}^*(E/K_n)=\operatorname{\mathrm{Sel}}^*(E/K_n)\cap (E(K_n)\otimes \mathbb{Q}_p/\mathbb{Z}_p).\end{align*}$$
The intersection is taken in
$H^1(K_\Sigma /K_n,E[p^\infty ])$
and
$E(K_n)\otimes \mathbb {Q}_p/\mathbb {Z}_p$
is a subgroup after applying the Kummer map. We furthermore define
$$\begin{align*}\kappa^*(E/K_n)=\frac{\operatorname{\mathrm{Sel}}^*(E/K_n)}{\mathcal{M}^*(E/K_n)}.\end{align*}$$
Let
$\operatorname {\mathrm {Sel}}^*(E/K_\infty )=\varinjlim _n\operatorname {\mathrm {Sel}}^*(E/K_n)$
.
Remark 2.1 By [Reference Burungale, Kobayashi and Ota3, Lemma 2.2],
$H^1(K_{n,v},E[p^\infty ])=0$
for all v coprime to p. Thus, one can omit the conditions at primes away from p in the definition of Selmer groups.
In the following, we will analyze the
$\varepsilon $
-Selmer groups.
Lemma 2.2
$\left (\frac {\operatorname {\mathrm {Sel}}^\varepsilon (E/K_\infty )}{\operatorname {\mathrm {Sel}}^0(E/K_\infty )}\right )$
has
$\Lambda $
-corank one.
Proof By [Reference Burungale, Kobayashi and Ota3, Proposition 3.4],
$\operatorname {\mathrm {Sel}}^0(E/K_\infty )$
is
$\Lambda $
-cotorsion. By [Reference Agboola and Howard1, Theorem 3.6],
$\operatorname {\mathrm {Sel}}^\varepsilon (E/K_\infty )$
has
$\Lambda $
-corank one. Both results together imply the desired result.
Lemma 2.3
$\left (\frac {\operatorname {\mathrm {Sel}}^\varepsilon (E/K_\infty )}{\operatorname {\mathrm {Sel}}^0(E/K_\infty )}\right )^\vee \cong \Lambda $
.
Proof By [Reference Burungale, Kobayashi and Ota3, Theorem 3.2 and Lemma 3.3], we have
$$\begin{align*}(\widehat{E}^\pm(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee\cong \omega^{\mp}_n\Lambda_n,\end{align*}$$
where
$\Lambda _n=\mathcal {O}[\operatorname {\mathrm {Gal}}(K_n/K)]$
. Taking the projective limit, we obtain
$$\begin{align*}(\widehat{E}^\pm(K_{\infty,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p)^\vee\cong \Lambda.\end{align*}$$
By definition
$$\begin{align*}\left (\frac{\operatorname{\mathrm{Sel}}^\varepsilon(E/K_\infty)}{\operatorname{\mathrm{Sel}}^0(E/K_\infty)}\right)\hookrightarrow \widehat{E}^\varepsilon(K_\infty)\otimes \mathbb{Q}_p/\mathbb{Z}_p. \end{align*}$$
This implies that we have a natural surjection
$$\begin{align*}\Lambda \to \left (\frac{\operatorname{\mathrm{Sel}}^\varepsilon(E/K_\infty)}{\operatorname{\mathrm{Sel}}^0(E/K_\infty)}\right)^\vee.\end{align*}$$
As the latter module has
$\Lambda $
-rank one by Lemma 2.2, this map is actually an isomorphism.
As an immediate corollary, we obtain the following.
Corollary 2.4 The natural map
$$\begin{align*}\operatorname{\mathrm{Sel}}^\varepsilon(E/K_\infty)\to \widehat{E}^\varepsilon(K_{\infty,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p\end{align*}$$
is a surjection.
For all
$n\ge 0,$
we define
$$\begin{align*}e_n=\frac{\text{rank}_{\mathcal{O}_K}(E(K_n))-\text{rank}_{\mathcal{O}_K}(E(K_{n-1}))}{ \phi(p^n)} .\end{align*}$$
Lemma 2.5 Assume that
$(-1)^n=\varepsilon $
. Then,
$e_n\ge 1$
.
Proof By [Reference Agboola and Howard1, Theorem 5.2], there is an injective homomorphism with finite cokernel
$$\begin{align*}\operatorname{\mathrm{Sel}}^\varepsilon(E/K_n)[\omega_n^\varepsilon]\to \operatorname{\mathrm{Sel}}^\varepsilon(E/K_\infty)[\omega_n^\varepsilon].\end{align*}$$
By Lemma 2.2,
$\operatorname {\mathrm {Sel}}^\varepsilon (E/K_\infty )$
has a quotient that is isomorphic to
$\Lambda ^\vee $
. It follows that
$\operatorname {\mathrm {Sel}}^\varepsilon (E/K_n)^\vee \otimes \mathbb {Q}_p$
contains a submodule isomorphic to
$\Lambda /(\omega _n^\varepsilon /\omega _{n-1}^\varepsilon )\otimes \mathbb {Q}_p$
. As the Tate–Shafarevich group is assumed to be finite for all n, this implies that
$E(K_n)\otimes \mathbb {Q}_p$
contains a submodule isomorphic to
$\Lambda /(\omega _n^\varepsilon /\omega _{n-1}^\varepsilon )\otimes \mathbb {Q}_p$
. As
$E(K_{n-1})$
is annihilated by
$\omega _{n-1}$
, we obtain the desired result.
Corollary 2.6 The natural homomorphisms
$$\begin{align*}\operatorname{\mathrm{Sel}}^\varepsilon (E/K_n)\to \widehat{E}^\varepsilon(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p\end{align*}$$
and
$$\begin{align*}\mathcal{M}^\varepsilon (E/K_n)\to \widehat{E}^\varepsilon(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p\end{align*}$$
are surjective. In particular,
$$\begin{align*}\frac{\operatorname{\mathrm{Sel}}^\varepsilon(E/K_n)}{\operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1})+\operatorname{\mathrm{Sel}}^0(E/K_n)}\cong \frac{\widehat{E}^\varepsilon(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p}{\widehat{E}^\varepsilon(K_{n-1,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p}\cong \frac{\mathcal{M}^\varepsilon(E/K_n)}{\mathcal{M}^\varepsilon(E/K_{n-1})+\mathcal{M}^0(E/K_n).}\end{align*}$$
Proof By [Reference Burungale, Kobayashi and Ota3, Theorem 3.2(1)],
$\widehat {E}^\varepsilon (K_{n,p})\otimes \mathbb {Q}_p\cong \mathbb {Q}_p[X]/\omega _n^\varepsilon $
. As
$e_n\ge 1$
for all n with
$(-1)^n=\varepsilon $
,
$E(K_n)\otimes \mathbb {Q}_p$
contains a subrepresentation isomorphic to
$\mathbb {Q}_p[X]/\omega _n^\varepsilon $
. Thus,
$\widehat {E}^\varepsilon (K_{n,p})\otimes \mathbb {Q}_p$
lies in the image of the natural homomorphism
$$\begin{align*}E(K_n)\otimes \mathbb{Q}_p\to \widehat{E}(K_{n,p})\otimes \mathbb{Q}_p.\end{align*}$$
It follows that
$$\begin{align*}\operatorname{\mathrm{Sel}}^\varepsilon(E/K_n)\to \widehat{E}^\varepsilon(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p\end{align*}$$
and
$$\begin{align*}\mathcal{M}^\varepsilon(E/K_n)\to \widehat{E}^\varepsilon(K_{n,p})\otimes \mathbb{Q}_p/\mathbb{Z}_p\end{align*}$$
have finite cokernels. As the right-hand side is divisible, the homomorphism has to be surjective.
The remainder of the section is dedicated to prove the existence of a short exact sequence
Proposition 2.7 We have two short exact sequences
$$\begin{align*}0\to \operatorname{\mathrm{Sel}}^0(E/K_n)\to \operatorname{\mathrm{Sel}}^+(E/K_n)\oplus \operatorname{\mathrm{Sel}}^-(E/K_n)\to \operatorname{\mathrm{Sel}}(E/K_n)\to 0\end{align*}$$
and
$$\begin{align*}0\to \mathcal{M}^0(E/K_n)\to \mathcal{M}^+(E/K_n)\oplus \mathcal{M}^-(E/K_n)\to E(K_n)\otimes \mathbb{Q}_p/\mathbb{Z}_p\to 0.\end{align*}$$
Proof For the first claim, it suffices to show that
$$\begin{align*}\operatorname{\mathrm{Sel}}^+(E/K_n)+\operatorname{\mathrm{Sel}}^-(E/K_n)=\operatorname{\mathrm{Sel}}(E/K_n).\end{align*}$$
As
$\widehat {E}(K_{n,p})\otimes \mathbb {Q}_p/\mathbb {Z}_p=(\widehat {E}^+(K_{n,p})\otimes \mathbb {Q}_p/\mathbb {Z}_p)\oplus (\widehat {E}^-(K_{n,p})\otimes \mathbb {Q}_p/\mathbb {Z}_p)$
(c.f. [Reference Burungale, Kobayashi and Ota3, Theorem 3.2]), Corollary 2.6 implies that indeed
$$\begin{align*}\text{im}(\operatorname{\mathrm{Sel}}^+(E/K_n)+\operatorname{\mathrm{Sel}}^-(E/K_n))=\text{im}(\operatorname{\mathrm{Sel}}(E/K_n)),\end{align*}$$
where
$\text {im}(\cdot )$
denotes the image inside
$E(K_{n,p})\otimes \mathbb {Q}_p/\mathbb {Z}_p$
. As
$\operatorname {\mathrm {Sel}}^0(E/K_n)\subset \operatorname {\mathrm {Sel}}^\pm (E/K_n)$
, the first claim follows.
The second claim can be proved similarly.
As an immediate corollary, we obtain the following.
Corollary 2.8 We have a short exact sequence
Proof Consider the following commutative diagram:
The vertical maps are all injective. The claim now follows from the snake lemma.
3 Plus/Minus Tate–Shafarevich groups
In view of Corollary 2.8, it suffices to find asymptotic formulas for
$\kappa ^*(E/K_n)$
for
$*\in \{\pm ,0\}$
. In the present section, we will concentrate on the comparison of
$\kappa ^\varepsilon (E/K_n)$
and
$\kappa ^0(E/K_n)$
.
Definition 3.1 Let
$*\in \{0,+,-\}$
we denote by
$$\begin{align*}\alpha^*_{n,n-1}\colon \kappa^*(E/K_{n-1})\to \kappa^*(E/K_n)\end{align*}$$
the natural map. We define
$$\begin{align*}\kappa^*_{n,n-1}=\frac{\kappa^*(E/K_n)}{\text{im}(\alpha^*_{n,n-1})}.\end{align*}$$
Analogously, we define 
.
Lemma 3.1 The natural homomorphism
$$\begin{align*}\Phi_n\colon\kappa^0_{n,n-1}\to \kappa^\varepsilon_{n,n-1}\end{align*}$$
is an isomorphism.
Proof By definition,
$$\begin{align*}\operatorname{\mathrm{coker}}(\Phi_n)\cong \frac{\operatorname{\mathrm{Sel}}^\varepsilon(E/K_n)}{\operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1})+\mathcal{M}^\varepsilon(E/K_n)+\operatorname{\mathrm{Sel}}^0(E/K_n)}=0\end{align*}$$
by Corollary 2.6. It remains to show that
$\Phi _n$
is injective.
The kernel of
$\Phi _n$
is given by the image of
$(\mathcal {M}^\varepsilon (E/K_n)+\operatorname {\mathrm {Sel}}^\varepsilon (E/K_{n-1}))\cap \operatorname {\mathrm {Sel}}^0(E/K_n)$
in
$\kappa ^0_{n,n-1}$
. Note that
$$ \begin{align*} &(\mathcal{M}^\varepsilon(E/K_n)+\operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1}))\cap \operatorname{\mathrm{Sel}}^0(E/K_n)\\&\subset (\mathcal{M}^\varepsilon(E/K_{n-1})+\mathcal{M}^0(E/K_n)+\operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1}))\cap \operatorname{\mathrm{Sel}}^0(E/K_n)\\ &=(\mathcal{M}^0(E/K_n)+\operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1}))\cap \operatorname{\mathrm{Sel}}^0(E/K_n)\\ &=(\operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1})\cap \operatorname{\mathrm{Sel}}^0(E/K_n))+\mathcal{M}^0(E/K_n)=\operatorname{\mathrm{Sel}}^0(E/K_{n-1})+\mathcal{M}^0(E/K_n), \end{align*} $$
where the first inclusion follows from the following fact. Let
$a\in \mathcal {M}^\varepsilon (E/K_n)$
,
$b\in \operatorname {\mathrm {Sel}}^\varepsilon (E/K_{n-1})$
and
$a+b\in \operatorname {\mathrm {Sel}}^0(E/K_n)$
. Then,
$\text {im} (a)\in \widehat {E}^\varepsilon (K_{n-1,p})\otimes \mathbb {Q}_p/\mathbb {Z}_p$
, which implies by Corollary 2.6 that
$a\in \mathcal {M}^0(E/K_n)+\mathcal {M}^\varepsilon (E/K_{n-1})$
.
By definition, the last term in the above equation has trivial image in
$\kappa ^0_{n,n-1}$
. Thus,
$\Phi _n$
is indeed injective.
The next lemma is a preparation to prove the following exact sequence:
Lemma 3.2 Assume that
$(-1)^n=\varepsilon $
. The natural homomorphism
is injective.
Proof Consider the natural homomorphism
The kernel is given by
$$ \begin{align*} &\operatorname{\mathrm{Sel}}^{\varepsilon}(E/K_n)\cap (E(K_n)\otimes \mathbb{Q}_p/\mathbb{Z}_p+\operatorname{\mathrm{Sel}}(E/K_{n-1}))\\&=(E(K_{n-1})\otimes\mathbb{Q}_p/\mathbb{Z}_p+\mathcal{M}^\varepsilon(E/K_n)+\operatorname{\mathrm{Sel}}(E/K_{n-1}))\cap \operatorname{\mathrm{Sel}}^\varepsilon(E/K_n)\\&=(\operatorname{\mathrm{Sel}}(E/K_{n-1})+\mathcal{M}^\varepsilon(E/K_n))\cap\operatorname{\mathrm{Sel}}^\varepsilon(E/K_n)\\&=\operatorname{\mathrm{Sel}}^\varepsilon(E/K_n)\cap \operatorname{\mathrm{Sel}}(E/K_{n-1})+\mathcal{M}^\varepsilon(E/K_n)\\&=\operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1})+\mathcal{M}^\varepsilon(E/K_n), \end{align*} $$
which has a trivial image in
$\kappa ^\varepsilon _{n,n-1}$
.
Proposition 3.3 There is an exact sequence
Proof Consider the following commutative diagram:
where the rows are exact by Corollary 2.8. Applying the snake lemma, we obtain
The left most homomorphism is injective by Lemma 3.1, which implies the desired short exact sequence.
Corollary 3.4 We have
3.1 Estimating
$\kappa ^0_{n,n-1}$
Before we continue to estimate 
and
$\kappa ^{-\varepsilon }_{n,n-1}$
, we first determine
$\kappa ^0_{n,n-1}$
.
Lemma 3.5 The natural homomorphism
$$\begin{align*}\kappa^\varepsilon(E/K_{n-1})\to \kappa^\varepsilon(E/K_n)\end{align*}$$
is injective.
Proof Consider the natural map
$$\begin{align*}\operatorname{\mathrm{Sel}}^\varepsilon(K_{n-1}/E)\to \kappa^\varepsilon(K_n/E).\end{align*}$$
Its kernel is given by
$$ \begin{align*} &\mathcal{M}^\varepsilon(E/K_n)\cap \operatorname{\mathrm{Sel}}^\varepsilon(E/K_{n-1})=(\mathcal{M}^\varepsilon(E/K_{n-1})+\operatorname{\mathrm{Sel}}^0(E/K_{n-1}))\cap \mathcal{M}^\varepsilon(E/K_n)\\&=\mathcal{M}^\varepsilon(E/K_{n-1})+\mathcal{M}^0(E/K_{n-1})=\mathcal{M}^\varepsilon(E/K_{n-1}), \end{align*} $$
where the first equality follows from Corollary 2.6. As the image of
$\mathcal {M}^\varepsilon (E/K_{n-1})$
in
$\kappa ^\varepsilon (E/K_{n-1})$
is trivial, the claim follows.
As an immediate consequence of Lemmas 3.1 and 3.5, we obtain the following.
Corollary 3.6 The homomorphism
$$\begin{align*}\kappa^0(E/K_{n-1})\to \kappa^0(E/K_n)\end{align*}$$
is injective.
To prove an asymptotic formula for
$\kappa ^0(E/K_n)$
, we need a control theorem for
$\operatorname {\mathrm {Sel}}^0(E/K_n)$
and
$\mathcal {M}(E/K_n)$
. We write
$\Gamma _n$
for
$\Gamma ^{p^n}=\operatorname {\mathrm {Gal}}(K_n/K)$
.
Theorem 3.7 The natural homomorphisms
$$\begin{align*}\operatorname{\mathrm{Sel}}^0(E/K_n)\to \operatorname{\mathrm{Sel}}^0(E/K_\infty)^{\Gamma_n}\end{align*}$$
and
$$\begin{align*}\mathcal{M}^0(E/K_n)\to \mathcal{M}^0(E/K_\infty)^{\Gamma_n}\end{align*}$$
are injective with uniformly bounded cokernels.
Proof The injectivity follows from the inflation restriction exact sequence and the fact that
$E(K_n)[p]=0$
. The fact that the first map has uniformly bounded cokernel follows from [Reference Lim9, Theorem 1.1]. To show the boundedness of the cokernels for the second homomorphism, consider the following commutative diagram:
By Corollary 3.6,
$c_n$
is injective. We therefore obtain an injection
$\operatorname {\mathrm {coker}}(a_n)\to \operatorname {\mathrm {coker}}(b_n)$
. As the latter group is uniformly bounded independent of n, the same is true for
$\operatorname {\mathrm {coker}}(a_n)$
.
Recall that
$\operatorname {\mathrm {Sel}}^0(E/K_\infty )$
is
$\Lambda $
-cotorsion. As
$\kappa ^0(E/K_\infty )$
is a quotient of
$\operatorname {\mathrm {Sel}}^0(E/K_\infty )$
,
$\kappa ^0(E/K_\infty )$
is
$\Lambda $
-cotorsion as well. In particular, its Pontryagin dual
$\kappa ^0(E/K_\infty )^\vee $
is a finitely generated torsion
$\Lambda $
-module.
Theorem 3.8 Let
$\mu $
and
$\lambda $
be the Iwasawa invariants of
$\kappa ^0(E/K_\infty )^\vee $
. Then, for all n large enough, we have
$$\begin{align*}v_p(\vert \kappa^0_{n,n-1}\vert)=\lambda +\mu \phi(p^n). \end{align*}$$
Proof Using standard arguments in Iwasawa theory, Theorem 3.7 implies that there are invariants
$\lambda ',\lambda ",\mu '$
, and
$\mu "$
such that
$$\begin{align*}v_p(\vert\operatorname{\mathrm{coker}}(\operatorname{\mathrm{Sel}}^0(E/K_{n-1})\to \operatorname{\mathrm{Sel}}^0(E/K_{n}))\vert)=\lambda'+\mu'\phi(p^n)\end{align*}$$
and
$$\begin{align*}v_p(\vert \operatorname{\mathrm{coker}}(\mathcal{M}^0(E/K_{n-1})\to \mathcal{M}^0(E/K_n))\vert )=\lambda"+\mu"\phi(p^n)\end{align*}$$
for
$n\gg 0$
. By Corollary 3.6, we furthermore have an exact sequence
$$ \begin{align*}0\to \operatorname{\mathrm{coker}}(\mathcal{M}^0(E/K_{n-1})\to \mathcal{M}^0(E/K_n))\to \\\to\operatorname{\mathrm{coker}}(\operatorname{\mathrm{Sel}}^0(E/K_{n-1})\to \operatorname{\mathrm{Sel}}^0(E/K_{n}))\to \kappa^0_{n,n-1}\to 0,\end{align*} $$
which implies
$$\begin{align*}v_p(\vert \kappa^0_{n,n-1}\vert)=\lambda'-\lambda"+(\mu'-\mu")\phi(p^n).\end{align*}$$
Let
$F'$
and
$F"$
be the characteristic ideals of
$\operatorname {\mathrm {Sel}}^0(E/K_\infty )^\vee $
and
$\mathcal {M}^0(E/K_\infty )$
. Choose
$n_0$
such that
$\gcd (F',\omega _n)=\gcd (F', \omega _{n_0})$
and
$\gcd (F",\omega _n)=\gcd (F",\omega _{n_0})$
for all
$n\ge n_0$
. Let
$G'=\gcd (F',\omega _{n_0})$
and
$G"=\gcd (F",\omega _{n_0})$
. Then, we have
$\lambda '=\lambda (F')-\lambda (G')$
as well as
$\lambda "=\lambda (F")-\lambda (G")$
. As we are assuming that 
is finite foll all n,
$\operatorname {\mathrm {Sel}}^0(E/K_n)$
and
$\mathcal {M}^0(E/K_n)$
have the same corank for all n. Thus,
$G'=G"$
. This implies
$\lambda '-\lambda "=\lambda $
and
$\mu '-\mu "=\mu $
.
3.2 Estimating the kernels
To obtain an asymptotic formula for 
we do not only need to understand the cokernels 
but also the kernels of the natural maps 
. It turns out that these maps are injective as we will prove in Proposition 3.10.
Lemma 3.9 Assume that
$(-1)^n=\varepsilon $
. For all n large enough, we have that
$$\begin{align*}\mathcal{M}^{-\varepsilon}(E/K_n)=\mathcal{M}^{-\varepsilon}(E/K_{n-1})+\mathcal{M}^0(E/K_n). \end{align*}$$
Proof We have a natural isomorphism
$$\begin{align*}\frac{E(K_n)\otimes \mathbb{Q}_p/\mathbb{Z}_p}{E(K_{n-1})\otimes \mathbb{Q}_p/\mathbb{Z}_p}\to \frac{\mathcal{M}^\varepsilon(E/K_n)}{\mathcal{M}^{\varepsilon}(E/K_{n-1})}.\end{align*}$$
Note that
$E(K_{n-1})\otimes \mathbb {Q}_p/\mathbb {Z}_p=\mathcal {M}^\varepsilon (E/K_{n-1})+\mathcal {M}^{-\varepsilon }(E/K_{n-1})$
.
By definition,
$$ \begin{align*} \mathcal{M}^{-\varepsilon}(E/K_n)&= \mathcal{M}^{-\varepsilon}(E/K_n)\cap (E(K_{n-1})\otimes \mathbb{Q}_p/\mathbb{Z}_p+\mathcal{M}^\varepsilon(E/K_n))\\&=\mathcal{M}^{-\varepsilon}(E/K_{n})\cap (\mathcal{M}^\varepsilon(E/K_n)+\mathcal{M}^{-\varepsilon}(E/K_{n-1}))\\&=\mathcal{M}^{-\varepsilon}(E/K_{n-1})+(\mathcal{M}^{-\varepsilon}(E/K_n)\cap \mathcal{M}^\varepsilon(E/K_n))\\&=\mathcal{M}^{-\varepsilon}(E/K_{n-1})+\mathcal{M}^0(E/K_n).\\[-34pt] \end{align*} $$
Proposition 3.10 The natural homomorphism
is injective for all n large enough.
Proof If
$(-1)^n=-\varepsilon ,$
this follows from [Reference Burungale, Kobayashi and Ota3, Lemma 4.5 and Remark 4.6]. Assume now that
$(-1)^n=\varepsilon $
. By Corollary 2.8, it suffices to show that
$\kappa ^\pm (E/K_{n-1})\to \kappa ^\pm (E/K_n)$
is injective. For the
$\varepsilon $
part, this is Lemma 3.5. It remains to consider the kernel of
$\kappa ^{-\varepsilon }(E/K_{n-1})\to \kappa ^{-\varepsilon }(E/K_n)$
. Consider the natural map
$$\begin{align*}\psi_n^{-\varepsilon}\colon \operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_{n-1})\to \kappa^{-\varepsilon}(E/K_n).\end{align*}$$
The kernel is given by
$\operatorname {\mathrm {Sel}}^{-\varepsilon }(E/K_{n-1})\cap \mathcal {M}^{-\varepsilon }(E/K_n)$
. By Lemma 3.9, we know
$$\begin{align*}\mathcal{M}^{-\varepsilon}(E/K_n)=\mathcal{M}^{-\varepsilon}(E/K_{n-1})+\mathcal{M}^0(E/K_n).\end{align*}$$
Thus,
$$ \begin{align*} \operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_{n-1})\cap \mathcal{M}^{-\varepsilon}(E/K_n)&=\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_{n-1})\cap (\mathcal{M}^{-\varepsilon}(E/K_{n-1})+\mathcal{M}^0(E/K_n))\\&=\mathcal{M}^{-\varepsilon}(E/K_{n-1})+(\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_{n-1})\cap\mathcal{M}^0(E/K_n))\\&=\mathcal{M}^{-\varepsilon}(E/K_{n-1})+(\operatorname{\mathrm{Sel}}^{0}(E/K_{n-1})\cap \mathcal{M}^0(E/K_n)).\end{align*} $$
Corollary 3.6 implies that
$\operatorname {\mathrm {Sel}}^0(E/K_{n-1})\cap \mathcal {M}^0(E/K_n)=\mathcal {M}^0(E/K_{n-1})$
. Thus,
$$\begin{align*}\mathcal{M}^{-\varepsilon}(E/K_{n-1})+(\operatorname{\mathrm{Sel}}^{0}(E/K_{n-1})\cap\mathcal{M}^0(E/K_n))=\mathcal{M}^{-\varepsilon}(E/K_{n-1}).\end{align*}$$
Therefore,
$\ker (\psi _n^{-\varepsilon })=\mathcal {M}^{-\varepsilon }(E/K_{n-1})$
, which implies that
$$\begin{align*}\kappa^{-\varepsilon}(E/K_{n-1})\to \kappa^{-\varepsilon}(E/K_n)\end{align*}$$
is indeed injective.
Corollary 3.11 For
$n\gg 0,$
we have
$$\begin{align*}\operatorname{\mathrm{Sel}}(E/K_{n-1})\cap (E(K_n)\otimes \mathbb{Q}_p/\mathbb{Z}_p)=E(K_{n-1})\otimes \mathbb{Q}_p/\mathbb{Z}_p. \end{align*}$$
3.3 Estimating
$\kappa ^{-\varepsilon }_{n,n-1}$
In this section, we always assume that
$(-1)^n=\varepsilon $
. Before we can analyze
$\kappa ^{-\varepsilon }_{n,n-1}$
, we first need the following result on signed Selmer groups.
Lemma 3.12 Assume that the
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite for all n. The natural maps
$$\begin{align*}\frac{\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_n)}{\operatorname{\mathrm{Sel}}^0(E/K_n)}[\omega_n^{-\varepsilon}]\to \frac{\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_\infty)}{\operatorname{\mathrm{Sel}}^0(E/K_\infty)}[\omega_n^{-\varepsilon}]\end{align*}$$
are injective with uniformly bounded cokernel.
Proof Consider the exact sequence
$$ \begin{align*}H^1(K_\Sigma/K_\infty,E[p^\infty])[\omega_n^{-\varepsilon}]\to \left(\frac{H^1(K_\Sigma/K_\infty,E[p^\infty])}{\operatorname{\mathrm{Sel}}^0(E/K_\infty)}\right)[\omega_n^{-\varepsilon}]\\\to \operatorname{\mathrm{Sel}}^0(E/K_\infty)/\omega_n^{-\varepsilon}\operatorname{\mathrm{Sel}}^0(E/K_\infty). \end{align*} $$
If
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite for all n, the characteristic ideal of
$\operatorname {\mathrm {Sel}}^0(E/K_\infty )$
is coprime to
$\omega ^{-\varepsilon }_n$
for all n. In particular,
$\operatorname {\mathrm {Sel}}^0(E/K_\infty )/\omega _n^{-\varepsilon }\operatorname {\mathrm {Sel}}^0(E/K_n)$
is uniformly bounded. It follows that the natural homomorphism
$$\begin{align*}H^1(K_\Sigma/K_\infty,E[p^\infty])[\omega_n^{-\varepsilon}]\to \frac{H^1(K_\Sigma/K_\infty,E[p^\infty])}{\operatorname{\mathrm{Sel}}^0(E/K_\infty)}[\omega_n^{-\varepsilon}]\end{align*}$$
is injective with uniformly bounded cokernel. In particular,
$$\begin{align*}\frac{H^1(K_\Sigma/K_n,E[p^\infty])}{\operatorname{\mathrm{Sel}}^0(E/K_n)}[\omega_n^{-\varepsilon}]\to \frac{H^1(K_\Sigma/K_\infty, E[p^\infty])}{\operatorname{\mathrm{Sel}}^0(E/K_\infty)}[\omega_n^{-\varepsilon}]\end{align*}$$
is injective with uniformly bounded cokernel. Consider the following commutative diagram:
The right vertical map is injective (this can be proved as in [Reference Kobayashi7, Theorem 9.2] using the Coleman maps defined by [Reference Burungale, Kobayashi and Ota4]). The middle vertical map is injective with uniformly bounded cokernel. Thus, the left vertical map is injective with uniformly bounded cokernel.
Lemma 3.13 Assume that
$(-1)^n=\varepsilon $
and that
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite for all n. Then, we have
$$\begin{align*}\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_n)=\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_{n-1})+\operatorname{\mathrm{Sel}}^0(E/K_n).\end{align*}$$
Proof As
$\omega _n^{-\varepsilon }=\omega _{n-1}^{-\varepsilon }$
, Lemma 3.12 implies that
$$\begin{align*}\frac{\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_n)}{\operatorname{\mathrm{Sel}}^0(E/K_n)}[\omega_n^{-\varepsilon}]=\frac{\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_{n-1})}{\operatorname{\mathrm{Sel}}^0(E/K_{n-1})}[\omega_{n-1}^{-\varepsilon}].\end{align*}$$
As
$\frac {\operatorname {\mathrm {Sel}}^{-\varepsilon }(E/K_m)}{\operatorname {\mathrm {Sel}}^0(E/K_m)}$
is annihilated by
$\omega _m^{-\varepsilon }$
for all m, we obtain that
$$\begin{align*}\frac{\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_n)}{\operatorname{\mathrm{Sel}}^0(E/K_n)}=\frac{\operatorname{\mathrm{Sel}}^{-\varepsilon}(E/K_{n-1})}{\operatorname{\mathrm{Sel}}^0(E/K_{n-1})}.\end{align*}$$
In particular,
$\operatorname {\mathrm {Sel}}^{-\varepsilon }(E/K_n)=\operatorname {\mathrm {Sel}}^{-\varepsilon }(E/K_{n-1})+\operatorname {\mathrm {Sel}}^0(E/K_n)$
.
As an immediate corollary, we obtain the following.
Corollary 3.14 Assume that
$(-1)^n=\varepsilon $
and that
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite for all n. The natural homomorphism
$$\begin{align*}\kappa^{0}_{n,n-1}\to \kappa^{-\varepsilon}_{n,n-1}\end{align*}$$
is surjective.
3.4 Estimating
In this section, we put the results from previous sections together to obtain an estimate for 
and to derive an asymptotic formula for 
.
Theorem 3.15 Assume that
$(-1)^n=\varepsilon $
and that
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite for all n. Then, we have
Proof By Corollaries 3.4 and 3.14, we obtain
On the other hand, Lemmas 3.1 and 3.2 imply that there is a chain of injective homomorphisms
which implies
As a direct consequence of the above analysis, we obtain the following.
Theorem 3.16 Assume that
$\operatorname {\mathrm {Sel}}^0(E/K_n)[\omega _n^{-\varepsilon }]$
is finite for all n. For all n large enough, we have
Proof This is a direct consequence of [Reference Burungale, Kobayashi and Ota3, Theorem 1.1], Theorem 3.15, Proposition 3.10, and Theorem 3.8.
Acknowledgments
The author would like to thank Ben Forrás, Antonio Lei, Meng Fai Lim, and Andreas Nickel for helpful comments on an earlier draft of this article. The author would furthermore like to thank the anonymous referee for their helpful suggestions.
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