2 Notation and preliminaries

2.1 Iwasawa algebras and projections

Recall from the introduction that F is a totally real field of degree d and K is a CM extension of F. The compositum of all $\mathbb {Z}_p$ -extensions of K is denoted by $K_\infty $ and $\Gamma _\infty =\operatorname {Gal}(K_\infty /K)\cong \mathbb {Z}_p^{\oplus (d+1)}$ . Let $K_{\mathrm {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_{\mathrm {ac}}$ the anticyclotomic extension of K inside $K_\infty $ . Write

$$ \begin{align*} \Gamma_\infty=\overline{\langle \sigma_0,\sigma_1,\ldots,\sigma_d\rangle}, \end{align*} $$

where $\operatorname {Gal}(K_{\mathrm {cyc}}/K)$ is generated by $\sigma _0$ and $\operatorname {Gal}(K_{\mathrm {ac}}/K)$ is generated by $\sigma _1,\ldots ,\sigma _d$ . This gives rise to the isomorphism

sending $\sigma _i$ to $X_i-1$ .

Given a $\mathbb {Z}_p$ -extension $\mathcal {K}/K$ with $\Gamma _{\mathcal {K}}=\operatorname {Gal}(\mathcal {K}/K)$ , we have a natural projection

$$ \begin{align*} \pi_{\mathcal{K}}:\Gamma_\infty \longrightarrow \Gamma_{\mathcal{K}}, \end{align*} $$

whose kernel is isomorphic to $\mathbb {Z}_p^{\oplus d}$ . We can write

$$ \begin{align*} \ker\pi_{\mathcal{K}}=\bigg\{\kern-2pt\prod_{i=0}^d\sigma_i^{c_i}:\sum_{i=0}^da_ic_i=0\bigg\} \end{align*} $$

for some $a_i\in \mathbb {Z}_p$ not all zero. This allows us to identify the set of $\mathbb {Z}_p$ -extensions of K with $\mathbb {P}^d(\mathbb {Z}_p)$ . After scaling if necessary, we may assume that $c_j\in \mathbb {Z}_p^\times $ for some $j=j(\mathcal {K})$ . Then, for all $i\in \{0,\ldots ,d\}$ ,

$$ \begin{align*} \pi_{\mathcal{K}}(\sigma_i)=\pi_{\mathcal{K}}(\sigma_j^{{c_i}/{c_j}}). \end{align*} $$

In particular, we see that $\Gamma _{\mathcal {K}}$ is topologically generated by $\pi _{\mathcal {K}}(\sigma _{j(\mathcal {K})})$ . We shall denote this element by $\sigma _{\mathcal {K}}$ and write $X_{\mathcal {K}}=\sigma _{\mathcal {K}}-1$ .

Set . The natural extension of $\pi _{\mathcal {K}}$ to $\Lambda _\infty \rightarrow \Lambda _{\mathcal {K}}$ (which we still denote by the same symbol) can be realised as

Therefore,

$$ \begin{align*} \frac{d\pi_K(f)}{d X_{\mathcal{K}}}=\sum_{i=0}^d \frac{c_i}{c_{j(\mathcal{K})}}(1+X_{\mathcal{K}})^{{c_i}/{c_{j(\mathcal{K})}}-1}\frac{\partial f}{\partial X_i}((1+X_{\mathcal{K}})^{{c_0}/{c_{j(\mathcal{K})}}}-1,\ldots,(1+X_{\mathcal{K}})^{{c_d}/{c_{j(\mathcal{K})}}}-1), \end{align*} $$

which tells us that

(2.1) $$ \begin{align} \frac{d\pi_K(f)}{d X_{\mathcal{K}}}\bigg|_{X_{\mathcal{K}}=0}=\sum_{i=0}^d \frac{c_i}{c_{j(\mathcal{K})}}\cdot\frac{\partial f}{\partial X_i}(0,\ldots,0). \end{align} $$

Note that if $\mathcal {K}=K_{\mathrm {cyc}}$ , then $j(K_{\mathrm {cyc}})=0$ , corresponding to $(1:0:\cdots :0)\in \mathbb {P}^d(\mathbb {Z}_p)$ . We write $\pi _{\mathrm {cyc}}$ for $\pi _{K_{\mathrm {cyc}}}$ , which is given by $f(X_0,X_1,\ldots ,X_d)\mapsto f(X_0,0,\ldots ,0)$ . Furthermore, we write for the corresponding Iwasawa algebra.

Throughout, we often consider $\mathcal {K}$ to be a nonanticyclotomic $\mathbb {Z}_p$ -extension of K, that is, $\mathcal {K}\not \subseteq K_{{\mathrm {ac}}}$ . Such $\mathcal {K}$ correspond to $(c_0:\cdots :c_d)\in \mathbb {P}^d(\mathbb {Z}_p)$ , where $c_0\ne 0$ .

2.2 Control theorems and rank growth in $\mathbb {Z}_{p}$ -extensions of number fields

Let $\mathsf {A}/F$ be a simple abelian variety of $\mathrm {GL}_2$ -type and level $\mathfrak {N}$ over a totally real field F with potentially good ordinary reduction at all primes above p. Let $\Sigma (F)$ be a finite set of primes in F containing $\mathfrak {p}$ and all primes of bad reduction for $\mathsf {A}$ ; in other words, $\Sigma (F) \supseteq \{\mathfrak {p} \} \cup \{v \colon v \mid \mathfrak {N}\}$ . For any field $L/F$ , define $\Sigma (L)$ to be the set of places of L lying above those in $\Sigma (F)$ and write $G_\Sigma (L)$ for the Galois group of the maximal extension of L that is unramified outside of $\Sigma (L)$ . For any $v \in \Sigma (F)$ and any finite extension $L/F$ , write

$$ \begin{align*} J_v(\mathsf{A}/L)= \bigoplus_{w\mid v} H^1(L_w,\mathsf{A})[\mathfrak{p}^\infty]. \end{align*} $$

When $\mathcal {L}/L$ is an infinite extension of L, set

$$ \begin{align*} J_v(\mathsf{A}/\mathcal{L}) = \varinjlim_{L\subseteq L' \subseteq \mathcal{L}} J_v(\mathsf{A}/L'). \end{align*} $$

Definition 2.1. Let $\mathsf {A}/F$ be a simple abelian variety of $\mathrm {GL}_2$ -type over a totally real field F with potentially good ordinary reduction at all primes above p. Let $\Sigma (F)$ be any finite set of primes containing those dividing $\mathfrak {p}\mathfrak {N}$ . For any extension $L/F$ , define the Selmer group

$$ \begin{align*} \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/L):=\ker \bigg( H^1(G_{\Sigma}(L),\mathsf{A}[\mathfrak{p}^\infty]) \longrightarrow \prod_{v \in \Sigma} J_v(\mathsf{A}/L) \bigg). \end{align*} $$

We now recall the statement of Mazur’s control theorem, which allows us to study the growth behaviour of Selmer groups in $\mathbb {Z}_p$ -extensions.

Theorem 2.2 (Mazur’s control theorem).

Fix an odd prime p. Let F be a totally real number field and let $\mathsf {A}/F$ be an abelian variety of $\mathrm {GL}_2$ -type which has potentially ordinary reduction at all primes of F lying over p. Let $K/F$ be a quadratic extension of F which is a CM field. Let $\mathcal {K}$ be any $\mathbb {Z}_p$ -extension of K and $\mathcal {K}_n$ denote the nth layer of this extension with $\operatorname {Gal}(\mathcal {K}_n/K) \simeq \mathbb {Z}/p^n\mathbb {Z}$ . Then, the kernel and cokernel of the natural map

$$ \begin{align*} \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}_n) \longrightarrow \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K})^{\operatorname{Gal}(\mathcal{K}/\mathcal{K}_n)} \end{align*} $$

are finite and of bounded order independent of n.

Proof. This is proved in the same way as [Reference Greenberg10, Proposition 5.1]; see also [Reference Murty and Ouyang27, Theorem 2].

Corollary 2.3. With the assumptions of Theorem 2.2, set $r {\kern-1pt}={\kern-1pt} \mathrm {corank}_{\Lambda _{\mathcal {K}}}(\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K}))$ . Then, as $n\to \infty $ ,

$$ \begin{align*} \mathrm{corank}_{\mathcal{O}_{\mathfrak{p}}}(\mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}_n)) = rp^n + O(1). \end{align*} $$

In particular, if $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K)$ is finite, then as $n\to \infty $ ,

$$ \begin{align*} \mathrm{corank}_{\mathcal{O}_{\mathfrak{p}}}(\mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}_n)) = O(1). \end{align*} $$

Proof. The arguments for both assertions are standard. They are recorded in [Reference Greenberg, Conrad and Rubin9, Corollaries 4.9 and 4.12] for $\mathbb {Z}_p$ -coranks of $p^\infty $ -Selmer groups of elliptic curves, but the proofs go through for the current setting.

3 $\mathfrak {p}$ -adic L-functions

Throughout this section, we assume hypothesis (GHH⁺ ) holds. We review results of Disegni [Reference Disegni6] that will be used in our proof of Theorem 1.2. For each place of F lying above p, we fix a level 0 additive character (see [Reference Disegni6, end of page 1993]).

Theorem 3.1 [Reference Disegni6, Theorem A].

Let $\mathfrak {p}$ be a place of F lying above p where $\mathsf {A}$ has potentially good ordinary reduction. There exists a unique $\mathfrak {p}$ -adic L-function ${L_{\mathfrak {p}}(\mathsf {A})\in F_{\mathfrak {p}}\otimes \Lambda _\infty }$ such that for all finite characters $\chi $ of $\Gamma _\infty $ ,

$$ \begin{align*} L_{\mathfrak{p}}(\mathsf{A})(\chi)=C_\chi\cdot \frac{L(\mathsf{A}/K,\chi,1)}{\Omega} \end{align*} $$

for some constant $C_\chi $ and a period $\Omega $ .

Here, ${L(\mathsf {A}/K,\chi ,1)}/{\Omega }$ is an algebraic number, regarded as an element of $\overline {\mathbb {Q}_p}$ through $\iota _{\mathfrak {p}}$ . Many authors, including [Reference Disegni6], refer to $L_{\mathfrak {p}}$ as a p-adic L-function, but to highlight the dependence on $\mathfrak {p}\mid p$ , we refer to it as a $\mathfrak {p}$ -adic L-function.

The sign of the functional equation of $L(\mathsf {A},K,\chi ,s)$ at $s=1$ is constant for all finite characters $\chi $ of $\operatorname {Gal}(K_{\mathrm {ac}}/K)$ (see [Reference Disegni6, top of page 1999]). In particular, since $\mathsf {A}$ is assumed to be self-dual, $L_{\mathfrak {p}}(\mathsf {A})(\chi )=0$ for all such $\chi $ under our running hypotheses. If we consider $L_{\mathfrak {p}}(\mathsf {A})$ as a power series in $X_0,X_1,\ldots , X_d$ , we have $L_{\mathfrak {p}}(\mathsf {A})(0,X_1,\ldots ,X_d)=0$ . Consequently, we can expand $L_{\mathfrak {p}}(\mathsf {A})$ as a power series in $X_0$ with coefficients in $\Lambda _{\mathrm {ac}}\bigotimes _{\mathbb {Z}_p}\mathbb {Q}_p$ , where . More specifically,

(3.1) $$ \begin{align} L_{\mathfrak{p}}(\mathsf{A})=G_{\mathfrak{p}}(\mathsf{A})X_0+O(X_0^2), \end{align} $$

where $G_{\mathfrak {p}}(A)\in \Lambda _{\mathrm {ac}}\otimes \mathbb {Q}_p$ .

Proposition 3.2. Let $\mathcal {K}$ be a nonanticyclotomic $\mathbb {Z}_p$ -extension of K. If $G_{\mathfrak {p}}(\mathsf {A}) (0,\ldots ,0)\ne 0$ , then

$$ \begin{align*} \frac{d\pi_K(L_{\mathfrak{p}}(\mathsf{A}))}{d X_{\mathcal{K}}}\bigg|_{X_{\mathcal{K}}=0}\ne0. \end{align*} $$

Proof. Recall that $\mathcal {K}$ corresponds to a $\mathbb {Z}_p$ -extension such that $(c_0:\cdots :c_d)\in \mathbb {P}^d(\mathbb {Z}_p)$ with $c_0\ne 0$ . It follows from (2.1) and (3.1) that

$$ \begin{align*} \frac{d\pi_K(L_{\mathfrak{p}}(\mathsf{A}))}{d X_{\mathcal{K}}}\bigg|_{X_{\mathcal{K}}=0}=\frac{c_0}{c_{j(\mathcal{K})}}\cdot G_{\mathfrak{p}}(\mathsf{A})(0,\ldots,0)+\sum_{i=1}^d\frac{c_i}{c_{j(\mathcal{K})}}\cdot\left(\frac{\partial G_{\mathfrak{p}}(A)}{\partial X_i}X_0\right)(0,\ldots,0), \end{align*} $$

which is a nonzero multiple of $G_{\mathfrak {p}}(\mathsf {A})(0,\ldots ,0)$ . Hence, the lemma follows.

By [Reference Disegni6, Theorem C(4)], $G_{\mathfrak {p}}(\mathsf {A})$ is described via the $\Lambda _{\mathrm {ac}}$ -adic heights of the Heegner points on $\mathsf {A}$ defined over finite sub-extensions of $K_{\mathrm {ac}}/K$ . In particular, $G_{\mathfrak {p}}(\mathsf {A})(0,\ldots ,0)$ is a nonzero multiple of the $\mathfrak {p}$ -adic height of the Heegner point $z_{\mathrm {Heeg}}$ attached to $\mathsf {A}/K$ . The reader is referred to [Reference Disegni6, Section 1.1] for the definition of $z_{\mathrm {Heeg}}$ (where the character $\chi $ in [Reference Disegni6] is taken to be the trivial character).

We use the notation $\langle -,-\rangle _K$ to denote the p-adic height associated with $\mathfrak {p}$ over K, as given in [Reference Disegni7, Section 1.3]. We consider the hypothesis:

1. (HGT) $\langle z_{\mathrm {Heeg}},z_{\mathrm {Heeg}}\rangle _K\ne 0$ .

From Proposition 3.2, we can deduce the following result.

Corollary 3.3. Let $\mathcal {K}$ be a nonanticyclotomic $\mathbb {Z}_p$ -extension of K. If hypothesis (HGT) holds, then

$$ \begin{align*} \frac{d\pi_K(L_{\mathfrak{p}}(\mathsf{A}))}{d X_{\mathcal{K}}}\bigg|_{X_{\mathcal{K}}=0}\ne0. \end{align*} $$

In particular, $\pi _{\mathcal {K}}(L_{\mathfrak {p}}(\mathsf {A}))\ne 0$ .

4 Nonexistence of nontrivial pseudonull submodules

In this section, we review a special case of a result of Greenberg [Reference Greenberg, Loeffler and Zerbes13, Proposition 4.1.1] regarding sufficient conditions for $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee $ to admit no nontrivial pseudonull submodule.

Throughout this section, we keep the notation introduced previously. We further assume:

1. (CYC) $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_{\mathrm {cyc}})^\vee $ is $\Lambda _{\mathrm {cyc}}$ -torsion.

In view of [Reference Hachimori and Ochiai14, Lemma 2.6] (which is based on the ideas in [Reference Balister and Howson1]), we can conclude that hypothesis (CYC) implies that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee $ is $\Lambda _\infty $ -torsion.

Let $\mathcal {T}=T_{\mathfrak {p}}(\mathsf {A})\otimes \Lambda _\infty ^\iota $ , where $\iota $ is the involution on $\Lambda _\infty $ sending a group-like element to its inverse, and set $\mathcal {D}=\mathcal {T}\bigotimes _{\Lambda _\infty } \Lambda _\infty ^\vee $ . Throughout this section, we will be consistent with the notation of [Reference Greenberg, Loeffler and Zerbes13, Section 2.1] as much as possible.

The condition RFX( $\mathcal {D}$ ) asserts that $\mathcal {T}$ is a reflexive $\Lambda _\infty $ -module; in our setting, this condition holds since $\mathcal {T}$ is free over $\Lambda _\infty $ . In our context, the condition LEO( $\mathcal {D}$ ) asserts that

$$ \begin{align*} \ker\bigg(H^2(K_\Sigma/K,\mathcal{D})\longrightarrow \prod_{v\in \Sigma}H^2(K_v,\mathcal{D})\bigg) \end{align*} $$

is a cotorsion $\Lambda (G_K)$ -module. From [Reference Greenberg11, Theorem 3], there is an isomorphism of $\Lambda (G_K)$ -modules $H^2(K_\Sigma /K,\mathcal {D})\cong H^2(K_\Sigma /K_\infty , \mathsf {A}[\mathfrak {p}^\infty ])$ .

Since our hypotheses imply that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee $ is $\Lambda _\infty $ -torsion, by arguments analogous to [Reference Ochi and Venjakob31, Theorem 3.2],

(4.1) $$ \begin{align} \mathrm{rank}_{\Lambda_\infty}H^1(K_{\Sigma}/K_\infty, \mathsf{A}[\mathfrak{p}^\infty])^\vee -\mathrm{rank}_{\Lambda_\infty}H^2(K_\Sigma/K_\infty,\mathsf{A}[\mathfrak{p}^\infty])^\vee = r_2(K) \times r, \end{align} $$

where r is the $\mathcal {O}_{\mathfrak {p}}$ -corank of $\mathsf {A}[\mathfrak {p}^\infty ]$ . By combining [Reference Coates and Greenberg5, Proposition 4.8] and [Reference Ochi and Venjakob31, Theorem 4.1],

(4.2) $$ \begin{align} \mathrm{rank}_{\Lambda_\infty}\bigoplus_{\ell\in \Sigma}J_\ell(\mathsf{A} /K_\infty)^\vee = \mathrm{rank}_{\Lambda_\infty}\bigoplus_{\substack{w\mid \ell \\ \ell\in \Sigma}} H^1(K_{\infty,w}, \mathsf{A})[\mathfrak{p}^{\infty}]^\vee = r_2(K) \times r. \end{align} $$

A standard argument with the Poitou–Tate exact sequence implies that the module $H^2(K_S/K_\infty , \mathsf {A}[\mathfrak {p}^\infty ])$ is co-torsion over $\Lambda _\infty $ and the module $H^1(K_S/K_\infty , \mathsf {A}[\mathfrak {p}^\infty ])$ is of corank $r_2(K) \times r$ . In particular, condition LEO( $\mathcal {D}$ ) holds.

The condition CRK $(\mathcal {D},\mathcal {L})$ asserts that

$$ \begin{align*} \mathrm{corank}_{\Lambda_\infty} H^1(K_\Sigma/K_\infty,\mathsf{A}[\mathfrak{p}^\infty]) & =\mathrm{corank}_{\Lambda_\infty} \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K_\infty)+\mathrm{corank}_{\Lambda_\infty}\bigoplus_{\ell\in \Sigma}J_\ell(\mathsf{A}/K_\infty) \\ & = \mathrm{corank}_{\Lambda_\infty}\bigoplus_{\ell\in \Sigma}J_\ell(\mathsf{A}/K_\infty) \end{align*} $$

since we use hypothesis (CYC). The desired equality follows from (4.1) and (4.2).

We now consider the conditions LOC $_v^{(i)}(\mathcal {D})$ , $i=1,2$ . Write $\mathcal {T}^*=\mathrm {Hom}(\mathcal {D},\mu _{p^\infty })$ . The conditions assert that for $v\in \Sigma (K)$ , we have $(\mathcal {T}^*)^{G_{K_v}}=0$ and $\mathcal {T}^*/(\mathcal {T}^*)^{G_{K_v}}$ is a reflexive $\Lambda _\infty $ -module, respectively. Since $p\ne 2$ , we have $(\mathcal {T}^*)^{G_{K_v}}=0$ when v is an Archimedean prime. However, if v is a non-Archimedean prime, it does not split completely in $K_\infty $ . It follows from [Reference Greenberg12, Lemma 5.2.2] that $(\mathcal {T}^*)^{G_{K_v}}=0$ . This guarantees that condition LOC $_v^{(1)}(\mathcal {D})$ holds for all $v\in \Sigma (K)$ . Finally, as $\mathcal {T}^*$ is a free $\Lambda _\infty $ -module, condition LOC $_v^{(2)}(\mathcal {D})$ also holds for all $v\in \Sigma (K)$ .

We can now state the result due to Greenberg under the condition:

1. (TOR) $\mathsf {A}(K)$ has no $\mathfrak {p}$ -torsion.

5 Proof of Theorem 1.2

5.1 Preliminary results on Selmer groups

Let $\mathsf {A}$ be a simple self-dual modular abelian variety of $\mathrm {GL}_2$ -type over a totally real field F and let $K/F$ be a CM field.

Under hypotheses (ORD), (CYC) and (TOR), the structure theorem of finitely generated $\Lambda _\infty $ -modules (as given in [Reference Bourbaki2, Ch. VII, Section 4, Théorème 5]) combined with Proposition 4.1 asserts the existence of the short exact sequence

(5.1) $$ \begin{align} 0 \longrightarrow \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K_\infty)^\vee \longrightarrow \bigoplus_{i=1}^m\frac{\Lambda_\infty}{I_i} \longrightarrow N \longrightarrow 0, \end{align} $$

where $I_1,\ldots ,I_m $ are principal ideals of $\Lambda _{\infty }$ and $\prod _{i=1}^mI_i=I = \mathrm {char}_{\Lambda _\infty }(\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee )$ is the characteristic ideal of $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee $ as a $\Lambda _\infty $ -module and N is a pseudonull $\Lambda _\infty $ -module.

From the discussion in Section 3, there is a unique $\mathfrak {p}$ -adic L-function $L_{\mathfrak {p}}(\mathsf {A})\in F_{\mathfrak {p}} \otimes \Lambda _{\infty }$ associated with the abelian variety $\mathsf {A}$ , which is an analytic object. However, Theorem 1.2 is an assertion involving Selmer groups which are algebraic objects. To bridge this gap, we assume one inclusion of the Iwasawa main conjecture:

1. (h-IMC) $L_{\mathfrak {p}}(\mathsf {A}) \in \mathrm {char}_{\Lambda _\infty }(\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee )=\prod _{i=1}^mI_i$ .

Implicitly, hypothesis (h-IMC) asserts that $L_{\mathfrak {p}}(\mathsf {A})$ lies inside $\Lambda _\infty $ . In what follows, set $H_{\mathcal {K}} \simeq \operatorname {Gal}(K_\infty /\mathcal {K})\simeq \mathbb {Z}_p^{\oplus d}$ . When $\mathcal {K} = K_{\mathrm {cyc}}$ , we abbreviate $H_{K_{\mathrm {cyc}}} = H_{\mathrm {cyc}}$ .

Lemma 5.1. Let $\mathcal {K}$ be a nonanticyclotomic $\mathbb {Z}_p$ -extension of K. If hypotheses (HGT), (ORD), (GHH⁺ ), (TOR) and (h-IMC) hold, then $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )_{H_{\mathcal {K}}}^\vee $ is a finitely generated torsion $\Lambda _{\mathcal {K}}$ -module.

Proof. The short exact sequence (5.1) induces the exact sequence

$$ \begin{align*} H_1(H_{\mathcal{K}}, N) \longrightarrow \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K_\infty)^{\vee}_{H_{\mathcal{K}}} \longrightarrow\bigoplus_{i=1}^m \frac{\Lambda_{\mathcal{K}}}{\pi_{\mathcal{K}}(I_i)} \longrightarrow H_0(H_{\mathcal{K}}, N) \longrightarrow 0. \end{align*} $$

The inclusion of assumption (h-IMC) combined with Corollary 3.3 implies that ${\pi _{\mathcal {K}}(I)\ne 0}$ . In particular, ${\Lambda _{\mathcal {K}}}/{\pi _{\mathcal {K}}(I_i)}$ is $\Lambda _{\mathcal {K}}$ -torsion for $i=1,\ldots ,m$ . This surjectivity of the last arrow in the exact sequence above implies that $H_0(H_{\mathcal {K}}, N)$ is $\Lambda _{\mathcal {K}}$ -torsion. By [Reference Lim24, Proposition 2.3], we conclude that $H_1(H_{\mathcal {K}}, N)$ is $\Lambda _{\mathcal {K}}$ -torsion. Hence, we conclude that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )_{H_{\mathcal {K}}}^\vee $ is also a $\Lambda _{\mathcal {K}}$ -torsion module.

Proposition 5.2. Write $\mathcal {K}/K$ to denote a $\mathbb {Z}_p$ -extension and let $H_{\mathcal {K}}$ denote the Galois group $\operatorname {Gal}(K_\infty /\mathcal {K})\simeq \mathbb {Z}_p^{\oplus d}$ . Suppose that hypotheses (TOR) and (ORD) are satisfied. Then, the restriction map

$$ \begin{align*} \alpha: \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}) \longrightarrow \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K_\infty)^{H_{\mathcal{K}}} \end{align*} $$

is injective. When $\mathcal {K} = K_{\mathrm {cyc}}$ , the cokernel of $\alpha $ is a torsion $\Lambda _{\mathrm {cyc}}$ -module. In particular, $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )_{H_{\mathrm {cyc}}}^\vee $ is a torsion $\Lambda _{\mathrm {cyc}}$ -module if and only if $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_{\mathrm {cyc}})^\vee $ is a torsion $\Lambda _{\mathrm {cyc}}$ -module.

We note that this result does not require any hypothesis on the reduction type at primes $v\mid p$ .

Proof. We begin by recalling the fundamental diagram

where the vertical maps are given by restriction. To prove the first assertion, we study the leftmost downward arrow and obtain the exact sequence

$$ \begin{align*} 0 \longrightarrow \ker({\alpha}) \longrightarrow \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}) \longrightarrow \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K_\infty)^{H_{\mathcal{K}}}. \end{align*} $$

Note that

$$ \begin{align*} \ker(\alpha) \hookrightarrow \ker(\beta) = H^1(H_{\mathcal{K}}, \mathsf{A}(K_\infty)[\mathfrak{p}^\infty]), \end{align*} $$

where the description of $\ker (\beta )$ comes from inflation-restriction. Since our assumptions imply that $\mathsf {A}(K)[\mathfrak {p}]=0$ , it follows that $\mathsf {A}(K_\infty )[\mathfrak {p}]=0$ (see, for example, [Reference Neukirch, Schmidt and Wingberg30, I.6.13]). Thus, $\ker (\alpha )=\ker (\beta )=0$ . The result is now immediate from our assumption that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )_{H_{\mathcal {K}}}^\vee $ is a $\Lambda _{\mathcal {K}}$ -torsion module.

Next, we observe that the same argument yields

$$ \begin{align*} \mathrm{coker}(\beta) = H^2(H_{\mathcal{K}}, \mathsf{A}(K_\infty)[\mathfrak{p}^\infty]) =0. \end{align*} $$

To complete the proof, it suffices to show that

$$ \begin{align*} \mathrm{coker}(\alpha) = \ker(\gamma) = \prod_{v} \ker(\gamma_v) = \text{torsion } \Lambda_{\mathrm{cyc}}\text{-module}. \end{align*} $$

Note that $v\mid \mathfrak {p}$ is deeply ramified in the sense of Coates–Greenberg. Imitating the proof of [Reference Coates and Greenberg5, Proposition 4.8], when $v\mid \mathfrak {p}$ has good ordinary reduction,

$$ \begin{align*} \ker(\gamma) = \prod \ker(\gamma_v) = \prod_{\substack{v \in \Sigma(\mathcal{K})\\ v \nmid \mathfrak{p}}} H^1(H_{w},\mathsf{A}(K_{\infty,w})[\mathfrak{p}^\infty]) \times \prod_{\substack{v \in \Sigma(\mathcal{K})\\ v\mid \mathfrak{p}}} H^1(H_{w},\widetilde{\mathsf{A}}(k_{w})[\mathfrak{p}^\infty]), \end{align*} $$

where $w=w_v$ denotes any place of $K_\infty $ lying above v, $H_{w}$ denotes the decomposition group of w inside $H_{\mathcal {K}}$ , $\widetilde {\mathsf {A}}$ denotes the reduction of $\mathsf {A}$ modulo w and $k_{w}$ denotes the residue field of $K_\infty $ at w.

For the remainder of the argument, we assume that $\mathcal {K} = K_{\mathrm {cyc}}$ . First, we consider the case $v\nmid \mathfrak {p}$ . In this case, the local map $\gamma _v$ is simply the identity map: for any $v \in \Sigma (K_{\mathrm {cyc}})$ such that $v\nmid \mathfrak {p}$ and any place w of $K_\infty $ lying above v, we see that $K_{\infty ,w}=K_{\mathrm {cyc},v}$ is the unique $\mathbb {Z}_p$ -extension of $K_{\mathfrak {p}'}$ , where $\mathfrak {p}'$ is the place of K lying below v. In other words, $\ker (\gamma _v)=0$ .

When $v\mid \mathfrak {p}$ and $\mathfrak {p}$ is a prime of good ordinary reduction, it suffices to know that $\widetilde {\mathsf {A}}(k_w)[\mathfrak {p}^\infty ]$ itself is cotorsion (since it is of finite corank over $\mathcal {O}_{\mathfrak {p}}$ ). However, when $\mathfrak {p}$ is a prime of potentially ordinary reduction, $\ker (\gamma _v) = H^1(H_w, D)$ , where $D = \mathsf {A}[\mathfrak {p}^\infty ]/C$ and C is a formal group (over a base extension). In any case, the kernel lies inside $H^1(K_{\infty ,v},\mathsf {A}[\mathfrak {p}^\infty ])$ . As a $\Lambda _{\mathrm {cyc}}$ -module, we note that $\mathsf {A}[\mathfrak {p}^\infty ]$ is still cotorsion. So, $H^1(K_{\infty ,v},\mathsf {A}[\mathfrak {p}^\infty ])$ , and hence $\ker (\gamma _v)$ , is $\Lambda _{\mathrm {cyc}}$ -cotorsion. Here, we crucially use the fact that $v\mid \mathfrak {p}$ is finitely decomposed in the cyclotomic $\mathbb {Z}_p$ -extension.

5.2 The (non)-growth of Selmer coranks

We are now ready to conclude the proof of Theorem 1.2, which is divided into two steps. The first step is to show that the Selmer corank is bounded in a nonanticyclotomic $\mathbb {Z}_p$ -extension of K using the preliminary results from the previous sections. The second step is to show that the Selmer corank grows, as predicted, in an anticyclotomic $\mathbb {Z}_p$ -extension of K. The main input will be results of Nekovář [Reference Nekovář28, Reference Nekovář, Burns, Buzzard and Nekovář29] on the Heegner point Euler system and the parity conjecture.

Theorem 5.3. Let $\mathsf {A}$ be a simple modular self-dual abelian variety of $\mathrm {GL}_2$ -type over a totally real field F with trivial central character and let K be a totally imaginary extension of F, satisfying hypotheses (ORD), (GHH⁺ ), (TOR), (h-IMC) and (HGT). If $\mathcal {K}$ is a nonanticyclotomic $\mathbb {Z}_p$ -extension of K and $\mathcal {K}_n$ is the nth layer of the $\mathbb {Z}_p$ -extension, then $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K}_n)^\vee $ is bounded as $n\rightarrow \infty $ .

Proof. In light of Theorem 2.2 and Corollary 2.3, it suffices to prove that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K})^\vee $ is $\Lambda _{\mathcal {K}}$ -torsion. By Proposition 5.2, there is a surjection

$$ \begin{align*} \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K_\infty)^\vee_{H_{\mathcal{K}}}\twoheadrightarrow\mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}) ^\vee. \end{align*} $$

Furthermore, it follows from Lemma 5.1 that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee _{H_{\mathcal {K}}}$ is $\Lambda _{\mathcal {K}}$ -torsion. Hence, we deduce that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K})^\vee $ is $\Lambda _{\mathcal {K}}$ -torsion, as desired.

Theorem 5.4. Let $\mathsf {A}$ be a simple modular self-dual abelian variety of $\mathrm {GL}_2$ -type over a totally real field F with trivial central character and let K be a totally imaginary extension of F, satisfying hypotheses (ORD), (GHH⁺ ) and (HGT). Let $\mathcal {K}$ be a $\mathbb {Z}_p$ -extension of K that lies inside $K_{\mathrm {ac}}$ and set $\mathcal {K}_n$ to denote the nth layer of the $\mathbb {Z}_p$ -extension $\mathcal {K}/K$ . Then,

$$ \begin{align*} \mathrm{corank}_{\mathcal{O}_{\mathfrak{p}}}\mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}_n)=p^n+O(1). \end{align*} $$

Proof. Under hypothesis (HGT), $z_{\mathrm {Heeg}}$ is not a torsion element of $\mathsf {A}(K)$ . In particular, the main theorem of [Reference Nekovář, Burns, Buzzard and Nekovář29] implies that is finite and the $\mathcal {O}_{\mathfrak {p}}$ -module generated by $z_{\mathrm {Heeg}}$ inside $\mathsf {A}(K)$ is an $\mathcal {O}_{\mathfrak {p}}$ -module of rank one. Therefore, $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K)$ is of corank one over $\mathcal {O}_{\mathfrak {p}}$ . As the restriction map

$$ \begin{align*} \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K)\to\mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K})^{\Gamma_{\mathcal{K}}} \end{align*} $$

has finite kernel and cokernel by Theorem 2.2, it follows that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K})^\vee $ is of rank one or zero over $\Lambda _{\mathcal {K}}$ . However, since the root number of $\mathsf {A}$ over $\mathcal {K}_n$ is $-1$ by hypothesis (GHH⁺ ), it follows from [Reference Nekovář28, Theorem 0.4] that the $\mathcal {O}_{\mathfrak {p}}$ -corank of $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K}_n)$ is unbounded as $n\to \infty $ . By Corollary 2.3, the $\Lambda _{\mathcal {K}}$ -corank of $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K})$ is one and $\mathrm {corank}_{\mathcal {O}_{\mathfrak {p}}}\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K}_n)=p^n+O(1)$ , as desired.

6 Proof of Theorem 1.3

The main goal of this section is to prove Theorem 1.3. We adopt the method of [Reference Gajek-Leonard, Hatley, Kundu and Lei8], removing the nonanomalous hypothesis and streamlining the use of the control theorem. Instead of hypothesis (HGT) in the previous section, we consider the following hypotheses.

1. (S-C) The $\Lambda _\infty $ -module $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_\infty )^\vee $ is a direct sum of cyclic $\Lambda _\infty $ modules.

2. (RK) The order of vanishing of $\mathrm {char}_{\Lambda _{\mathrm {cyc}}}\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K_{\mathrm {cyc}})^\vee $ at $X_0$ is at most one.

Proof. In case (i), it follows from Theorem 2.2 that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/K)$ is finite; hence, $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K})^\vee $ is $\Lambda _{\mathcal {K}}$ -torsion. By Corollary 2.3, we conclude that the $\mathcal {O}_{\mathfrak {p}}$ -rank of $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K}_n)^\vee $ is bounded as $n\rightarrow \infty $ .

For the remainder of the proof, we consider case (ii). In particular, hypothesis (S-C) holds, which means that the pseudonull module N in (5.1) is trivial. Let $f_i\in \Lambda _\infty $ be a generator of $I_i$ . Let us write

$$ \begin{align*} f_i=\sum_{k=0}^\infty G_{i,k}(X_1,\ldots, X_d)X_0^k, \end{align*} $$

where . By Proposition 5.2, there is a surjection

(6.1) $$ \begin{align} ( \mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/K_\infty)^\vee)_{H_{\mathcal{K}}}\simeq\bigoplus_{i=1}^m\frac{\Lambda_{\mathcal{K}}}{\left(\pi_{\mathcal{K}}(f_i)\right)}\twoheadrightarrow\mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K})^\vee. \end{align} $$

Thus, $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K})^\vee $ is $\Lambda _{\mathcal {K}}$ -torsion if $\pi _{\mathcal {K}}(f_i)\ne 0$ for all $i=1,\ldots , m$ .

When $\mathcal {K}=K_{\mathrm {cyc}}$ , Proposition 5.2 says that the kernel in (6.1) is $\Lambda _{\mathrm {cyc}}$ -torsion. Thus, by invoking assumption (CYC),

$$ \begin{align*} \pi_{\mathrm{cyc}}(f_i)=\sum_{k=0}^\infty G_{i,k}(0,\ldots, 0)X_0^k\ne0 \end{align*} $$

for $i\in \{1,\ldots ,m\}$ . In particular, there is an integer $m(i)$ such that $G_{i,m(i)}(0,\ldots , 0)\ne 0$ , with $G_{i,k}(0,\ldots , 0)= 0$ for all $k< m(i)$ .

Let $\mathcal {K}$ be a $\mathbb {Z}_p$ -extension of K which is not an anticyclotomic extension. Recall that $\mathcal {K}$ corresponds to $(c_0:\cdots :c_d)\in \mathbb {P}^d(\mathbb {Z}_p)$ with $c_0\ne 0$ . In particular,

$$ \begin{align*} \pi_{\mathcal{K}}(X_0)=(1+X_{\mathcal{K}})^{{c_0}/{c_j(\mathcal{K})}}-1=\frac{c_0}{c_j(\mathcal{K})}X_{\mathcal{K}}+O(X_{\mathcal{K}}^2). \end{align*} $$

Therefore,

$$ \begin{align*} & \pi_{\mathcal{K}}(f_i)=\sum_{k\ge m(i)}G_{i,k}((1+X_{\mathcal{K}})^{{c_1}/{c_j(\mathcal{K})}}-1,\ldots, (1+X_{\mathcal{K}})^{{c_d}/{c_j(\mathcal{K})}}-1)\pi_{\mathcal{K}}(X_0)^k\\ &\quad =G_{i,m(i)}((1+X_{\mathcal{K}})^{{c_1}/{c_j(\mathcal{K})}}-1,\ldots, (1+X_{\mathcal{K}})^{{c_d}/{c_j(\mathcal{K})}}-1)\bigg(\frac{c_0}{c_j(\mathcal{K})}X_{\mathcal{K}}\bigg)^{m(i)}+O(X_{\mathcal{K}}^{m(i)+1})\\ &\quad =G_{i,m(i)}(0,\ldots, 0)\bigg(\frac{c_0}{c_j(\mathcal{K})}X_{\mathcal{K}}\bigg)^{m(i)}+O(X_{\mathcal{K}}^{m(i)+1}) \ne0. \end{align*} $$

It follows that $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K})^\vee $ is $\Lambda _{\mathcal {K}}$ -torsion and, from Corollary 2.3, the $\mathcal {O}_{\mathfrak {p}}$ -rank of $\mathrm {Sel}_{\mathfrak {p}^\infty }(\mathsf {A}/\mathcal {K}_n)^\vee $ is bounded as $n\rightarrow \infty $ .

Theorem 6.2. Let $\mathsf {A}$ be as in Theorem 6.1. Suppose that hypotheses (ORD), (GHH⁺ ), (CYC) and (RK) hold. If $\mathcal {K}$ is a $\mathbb {Z}_p$ -extension of K that lies inside $K_{\mathrm {ac}}$ , then

$$ \begin{align*} \mathrm{corank}_{\mathcal{O}_{\mathfrak{p}}}\mathrm{Sel}_{\mathfrak{p}^\infty}(\mathsf{A}/\mathcal{K}_n)=p^n+O(1). \end{align*} $$