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 For supersingular primes, this control theorem is no longer valid.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

(1.1)

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.