2 Lubin–Tate extensions and formal groups attached to abelian varieties

Let $\{ k_n\}_{n\geq 0}$ with ${\mathbb Q}_p=k_0\subset k_1 \subset k_2 \subset \cdots \subset k_\infty =\bigcup _n k_n$ be a tower of fields with $G_n=\mathrm {Gal}(k_n/{\mathbb Q}_n)\cong {\mathbb Z}/p^n{\mathbb Z}$ such that $k_\infty $ is a totally ramified ${\mathbb Z}_p$ -extension of ${\mathbb Q}_p$ . Let $L_{n+1}:=k_n(\mu _p)$ for $n\geq 0$ and $L_\infty =\bigcup _n L_n=k_\infty (\mu _p)$ . Here, if M is a field, we denote by $M(\mu _p)$ the extension of M obtained by adjoining the p th roots of unity in some fixed algebraic closure of M. It then follows that $L_\infty $ is a ${\mathbb Z}_p^\times $ -extension of ${\mathbb Q}_p$ , and the group of its universal norms is generated by a uniformiser $\pi $ of ${\mathbb Z}_p$ such that $\mathrm {ord}_p(({\pi }/{p})-1)>0$ . In particular, we may choose $\pi $ so that $k_n$ coincides with the completion of $K_n$ at a prime above p.

Following [Reference Iovita and PollackIP06, Section 4.1], we fix a one-dimensional Lubin–Tate formal group ${\mathscr F}$ of height one over ${\mathbb Q}_p$ , with parameter $\pi $ and the lift of Frobenius

$$ \begin{align*} {\mathfrak f}(X):= \pi X+ \sum_{i=2}^p \binom{p}{i}\ X^i \in {\mathbb Z}_p[[X]]. \end{align*} $$

For every n, we have $L_n={\mathbb Q}_p({\mathscr F}[\pi ^n])$ , where ${\mathscr F}[\pi ^n]$ denotes the $\pi ^n$ -torsion of the formal group ${\mathscr F}$ .

We fix a $\mathbb {Z}$ -basis of $\mathcal {O}_F$ , which induces an injective ring homomorphism $\Phi :\mathcal {O}_F\longrightarrow M_{g\times g}({\mathbb Z})$ . Let $C_r=\Phi (a_r)$ , where $r\in \{q,q^2\}$ and q is a prime. Define the formal Dirichlet series

$$ \begin{align*} \sum_{n=1}^\infty C_n n^{-s} =\prod_q(I_g-C_qq^{-s}+C_{q^2}q^{1-2s})^{-1},\end{align*} $$

where $C_n\in M_{g\times g}(\mathbb {Z})$ .

Theorem 2.1. The formal group $\widehat {A}$ of the abelian variety A at p is isomorphic over ${\mathbb Z}$ to the formal group whose logarithm is given by

$$ \begin{align*} \log_{\widehat{A}}({\mathbf{x}})=\sum_{n\ge 1}\frac{C_n}{n}{\mathbf{x}}^n, \end{align*} $$

where ${\mathbf {x}}=(X_1,\ldots ,X_g)^t$ is a column vector of g variables and ${\mathbf {x}}^{n}$ denotes $(X^{n}_1,\ldots , X_g^{n} )^t$ .

Consider the sequence $\{x_k\}$ of $g\times g$ matrices defined by $x_{-1}=0$ , $x_0=I_g$ and

(2-1) $$ \begin{align} px_k-C_p x_{k-1}+x_{k-2}=0\quad\text{for } k\geq 1, \end{align} $$

where $C_p=\Phi (a_p)$ . In particular, $p^k x_k$ has integral coefficients.

Definition 2.2. We define ${\mathfrak f}^{(0)}(X)=X$ and, if $k\ge 1$ is an integer, we define ${\mathfrak f}^{(k)}$ recursively by

$$ \begin{align*} {\mathfrak f}^{(k)}(X)={\mathfrak f}({\mathfrak f}^{(k-1)}(X)). \end{align*} $$

Furthermore, we define

$$ \begin{align*} \ell({\mathbf{x}})=\sum_{k\ge0}x_k{\mathfrak f}^{(k)}({\mathbf{x}}), \end{align*} $$

where ${\mathfrak f}^{(k)}({\mathbf {x}}):=({\mathfrak f}^{(k)}(X_1),\ldots ,{\mathfrak f}^{(k)}(X_g) )^t$ .

Proof. Since A is supersingular at p, the entries of $C_p$ are divisible by p. Consequently, $x_1$ is defined over $\mathbb {Z}$ . By induction, we can prove that the entries of $x_{k}$ are in $p^{-\lfloor k/2\rfloor }\mathbb {Z}$ using (2-1).

The lemma would follow from the inclusion

$$ \begin{align*} {\mathfrak f}^{(t)}(X+pY)-{\mathfrak f}^{(t)}(X)\in p^{t+1}{\mathbb Z}_p[X,Y], \end{align*} $$

which can be proved using induction. Indeed, if $t=0$ , the inclusion is immediate. Suppose that it holds for some $t\ge 0$ . Let us write

$$ \begin{align*}{\mathfrak f}^{(t)}(X+pY)={\mathfrak f}^{(t)}(X)+p^{t+1}S(X,Y).\end{align*} $$

Then

$$ \begin{align*} {\mathfrak f}^{(t+1)}(X+pY)&={\mathfrak f}({\mathfrak f}^{(t)}(X)+p^{t+1}S(X,Y))\\ &=\pi({\mathfrak f}^{(t)}(X)+p^{t+1}S(X,Y))+\sum_{i=2}^p\binom{p}{i}\ ({\mathfrak f}^{(t)}(X)+p^{t+1}S(X,Y))^i\\ &\equiv \pi{\mathfrak f}^{(t)}(X)+\sum_{i=2}^p\binom{p}{i}\ ({\mathfrak f}^{(t)}(X))^i={\mathfrak f}^{(t+1)}(X)\,\pmod {\,p^{t+2}\ {\mathbb Z}_p[X,Y]} \end{align*} $$

as $\pi \in p{\mathbb Z}_p$ . This gives the desired inclusion for all $t\ge 0$ by induction.

Proof. In the light of Theorem A.6, it is enough to show that the two formal groups have the same Honda type $u(T):=T^2-C_pT+p$ . In particular, we need to show that

$$ \begin{align*} u*\ell\equiv u*\log_{\hat{A}}\equiv0\,\pmod {\,p\ {\mathbb Z}_p[[X]]}, \end{align*} $$

where $u*\ell $ and $ u*\log _{\hat {A}}$ are given as in Definition A.4. Note that ${\mathfrak f}^{(k)}(X)\equiv X^{p^k} \,\pmod {\,p} $ . We then deduce that

$$ \begin{align*} u*\ell&=\ell({\mathbf{x}}^{p^2})-C_p\ell({\mathbf{x}}^{p})+p\ell({\mathbf{x}})\\ &=\sum_{k\ge0}x_k {\mathfrak f}^{(k)}({\mathbf{x}}^{p^2})-C_p\sum_{k\ge0}x_k {\mathfrak f}^{(k)}({\mathbf{x}}^p)+p\sum_{k\ge0}x_k {\mathfrak f}^{(k)}({\mathbf{x}}) \\ &\equiv \sum_{k\ge2}(px_k-C_px_{k-1}+x_{k-2}) {\mathbf{x}}^{p^{k+2}} +p{\mathbf{x}} \,\pmod {\,p}\\ &\equiv 0 \,\pmod {\,p}. \end{align*} $$

It then follows from Theorem A.6(i) that $\hat \ell $ is a formal group over ${\mathbb Z}_p$ . The proof for $u*\log _{\hat {A}}$ is similar and relies on the fact that $C_{mp^k}=C_mC_{p^k}$ whenever $p\nmid m$ .

For the rest of this section, we identify these two formal groups.

3 Construction of local points over Lubin–Tate extensions

Choose a $\pi $ -sequence $\{e_n\}_{n\geq 0}$ in $L_\infty $ , that is, $e_n\in {\mathscr F}[\pi ^n]\setminus {\mathscr F}[\pi ^{n-1}]$ such that ${\mathfrak f}(e_n)=e_{n-1}$ for all $n\geq 1$ and $e_0=0$ .

Let $u_1\in \mathbb {Z}^{\oplus g}$ be the column vector that represents $1\in \mathcal {O}_F$ with respect to the basis chosen for the ring homomorphism $\Phi $ . In particular, the orbit of $u_1$ under the action of $\mathcal {O}_F$ (induced by $\Phi $ ) is $\mathbb {Z}^{\oplus g}$ . After a change of basis if necessary, we can take $u_1=(1,\ldots ,1)^t$ . Indeed, we can choose an integral basis of F that is of the form $\{1,\alpha _2,\ldots ,\alpha _g\}$ . Then $\{1-\sum _{i=2}^g\alpha _i, \alpha _2,\ldots ,\alpha _g\}$ is another integral basis of F. The vector representing $1$ with respect to this basis is then equal to $u_1$ .

Let $\epsilon \in (p{\mathbb Z}_p)^g$ be such that

$$ \begin{align*} \ell(\epsilon)= (1+p-C_p)^{-1}(pu_1). \end{align*} $$

Note that we have written $1+p$ for the $g\times g$ scalar matrix whose diagonal entries are all equal to $1+p$ . As $a_p$ is a nonunit of $\mathcal {O}_F$ , the matrix $C_p$ is p-adically nilpotent. Thus, $1+p-C_p$ is p-adically unipotent and belongs to $\mathrm {GL}_g({\mathbb Z}_p)$ . We regard $\epsilon $ as an element of $\widehat {A}({\mathbb Q}_p)$ .

Definition 3.1. We define $c_0=\epsilon \in \widehat {A}({\mathbb Q}_p)$ , and

$$ \begin{align*} c_n:=\epsilon [+]_{\widehat{A}} (e_n u_1)\in\widehat{A}(L_n)\quad\text{for all } n\geq 1, \end{align*} $$

where $[+]_{\widehat {A}}$ is the addition of $\widehat {A}$ .

We furthermore define $d_n$ to be the image of $c_{n+1}$ under the trace map $\widehat {A}(L_{n+1})\rightarrow \widehat {A}(k_n)$ for $n\ge 0$ .

Fix a prime v of F lying above p and let $F_v$ denote the completion of F at v with the integer ring $\mathcal {O}_v$ . Note that $\Phi $ induces an $\mathcal {O}_v$ -action on $\widehat {A}(M)$ if M is a finite extension of ${\mathbb Q}_p$ . Under this action, $c_0=\epsilon $ is a generator of the $\mathcal {O}_v$ -module $\widehat {A}({\mathbb Q}_p)$ . Given finite extensions $M_1\supset M_2\supset {\mathbb Q}_p$ , let $\mathrm {Tr}_{M_1/M_2}$ be the trace map $M_1\longrightarrow M_2$ . By an abuse of notation, we use the same notation to denote the trace map $\widehat {A}(M_1)\longrightarrow \widehat {A}(M_2)$ with respect to the group law on $\widehat {A}$ .

Proof. Our proof follows closely the calculations in [Reference Iovita and PollackIP06, Section 4]. Let $n\ge 2$ . Note that the minimal polynomial of $e_n$ over $L_{n-1}$ is ${\mathfrak f}(X)-e_{n-1}$ ; we have $\mathrm {Tr}_{L_n/L_{n-1}}(e_n)=-p$ . Thus,

$$ \begin{align*} &\ell(\mathrm{Tr}_{L_n/L_{n-1}}( c_{n}))\\ &\quad =\mathrm{Tr}_{L_n/L_{n-1}}( \ell (c_{n}))\\ &\quad = \mathrm{Tr}_{L_n/L_{n-1}}\bigg((1+p-C_p)^{-1}(pu_1)+\sum_{k\geq 0} x_k ( {\mathfrak f}^{(k)}(e_{n})u_1)\bigg) \\ &\quad = (1+p-C_p)^{-1}(p^2u_1)-pu_1+p\sum_{k\geq1} x_k ( e_{n-k}u_1) \\ &\quad = C_p(1+p-C_p)^{-1}(pu_1)-(1+p-C_p)^{-1}(pu_1)+\sum_{k\geq1} (C_px_{k-1}-x_{k-2}) ( e_{n-k}u_1) \\ &\quad = C_p(1+p-C_p)^{-1}(pu_1)-(1+p-C_p)^{-1}(pu_1) \\ & \qquad + C_p\sum_{k\geq1} x_{k-1} ( e_{n-k}u_1) - \sum_{k\geq1} x_{k-2} ( e_{n-k}u_1)\\ &\quad = C_p(1+p-C_p)^{-1}(pu_1)-(1+p-C_p)^{-1}pu_1 \\ & \qquad+ C_p\sum_{k\geq0} x_{k} ( e_{n-1-k}u_1) - \sum_{k\geq0} x_{k} ( e_{n-2-k}u_1 )\\ &\quad =C_p\ell(c_{n-1})-\ell(c_{n-2}). \end{align*} $$

Similarly, as the minimal polynomial of $e_1$ over ${\mathbb Q}_p$ is ${\mathfrak f}(X)/X$ , we have ${\mathrm {Tr}_{L_1/{\mathbb Q}_p}(e_1)=-p}$ . This allows us to deduce

$$ \begin{align*} \ell(\mathrm{Tr}_{L_1/{\mathbb Q}_p}(c_1))=(C_p-2)\ell(c_{0}). \end{align*} $$

Recall that $d_n=\mathrm {Tr}_{L_{n+1}/k_n}(c_{n+1})$ for $n\geq 1$ . Taking norms from $L_{n+1}$ to $k_n$ , we deduce that

$$ \begin{align*} \mathrm{Tr}_{k_n/k_{n-1}}(d_{n})=C_p d_{n-1}-d_{n-2}, \quad\text{for } n\geq 2, \end{align*} $$

and $d_0=(C_p-2)c_0$ . This gives

$$ \begin{align*} \mathrm{Tr}_{k_1/{\mathbb Q}_p}(d_1)=C_p d_0-(C_p-2)^{-1}(p-1)d_0, \end{align*} $$

as required.

Proof. We have chosen $c_0$ to be an $\mathcal {O}_v$ -generator of $\widehat {A}(L_0)$ . Suppose that $c_{n}$ and $c_{n-1}$ generate $\widehat {A}(L_n)$ as an $\mathcal {O}_v[\mathrm {Gal}(L_n/{\mathbb Q}_p)]$ -module. We show that this implies $c_{n}$ and $c_{n+1}$ generate $\widehat {A}(L_{n+1})$ as an $\mathcal {O}_v[\mathrm {Gal}(L_{n+1}/{\mathbb Q}_p)]$ -module. Since $c_{n-1}= C_pc_n-\mathrm {Tr}_{L_{n+1}/L_n}( c_{n+1} ) $ , we have

$$ \begin{align*} \widehat{A}(L_n)\subseteq \mathcal{O}_v[\mathrm{Gal}(L_n/{\mathbb Q}_p)] c_n+\mathcal{O}_v[\mathrm{Gal}(L_{n+1}/{\mathbb Q}_p)]c_{n+1}. \end{align*} $$

Therefore, it suffices to show the following equality:

$$ \begin{align*} \frac{\mathcal{O}_v[\mathrm{Gal}(L_{n+1}/{\mathbb Q}_p)]c_{n+1}+\widehat{A}(L_n)}{\widehat{A}(L_n)}=\widehat{A}(L_{n+1})/\widehat{A}(L_n). \end{align*} $$

As $\ell $ is injective on $\widehat {A}(L_{n+1})$ by Lemma 2.5, it suffices to show that $\ell (c_{n+1})$ generates the quotient $\ell ({\mathfrak m}_{n+1}^{\oplus g})/\ell ({\mathfrak m}_{n}^{\oplus g})$ , where ${\mathfrak m}_n$ denotes the maximal ideal of the integer ring of $L_n$ .

Let $\mathbf {y}=(y_1,\ldots ,y_g)\in \widehat {A}(L_{n+1})$ . Then [Reference Iovita and PollackIP06, Proposition 4.4] tells us that each $y_i$ can be expressed as $\sum _{\sigma \in \mathrm {Gal}(L_{n+1}/L_n)} b_\sigma e_{n+1}^\sigma +c$ , where $b_\sigma \in {\mathbb Z}_p$ and $c\in {\mathfrak m}_n $ . Since ${\mathfrak f}^{(s)}(X)\equiv X^{p^s} \,\pmod {\,p} $ for all $s\ge 1$ , we have

$$ \begin{align*} {\mathfrak f}^{(s)}(y_i)\equiv \sum_{\sigma\in\mathrm{Gal}(L_{n+1}/L_n)} b_\sigma (e_{n+1}^\sigma)^{p^s}+c^{p^s}\equiv\sum_{\sigma\in\mathrm{Gal}(L_{n+1}/L_n)} b_\sigma {\mathfrak f}^{(s)}(e_{n+1}^\sigma)+{\mathfrak f}^{(s)}(c)\,\pmod {\,p\ \mathcal{O}_{L_{n+1}}}. \end{align*} $$

Furthermore, ${\mathfrak f}^{(s)}(e_{n+1}^\sigma )\in {\mathfrak m}_n$ . Therefore, for all $k\ge 1$ , setting $s=k-\lfloor k/2\rfloor $ and applying Lemma 2.3 with $t=\lfloor k/2\rfloor $ , we deduce that

$$ \begin{align*} x_k\cdot{\mathfrak f}^{(k)}(\mathbf{y})=x_k\cdot{\mathfrak f}^{(t)}({\mathfrak f}^{(s)}(\mathbf{y}))\in ({\mathfrak m}_{n+1}+L_n)^{\oplus g}. \end{align*} $$

Consequently, we deduce that

$$ \begin{align*} \ell( \widehat{A}(L_{n+1}))\subseteq ({\mathfrak m}_{n+1}+L_n)^{\oplus g}. \end{align*} $$

This implies that

$$ \begin{align*} \ell({\mathfrak m}_{n+1}^{\oplus g})/\ell({\mathfrak m}_{n}^{\oplus g}) \cong {\mathfrak m}_{n+1}^{\oplus g}/{\mathfrak m}_{n}^{\oplus g}. \end{align*} $$

Recall that $c_{n+1}$ generates ${\mathfrak m}_{n+1}/{\mathfrak m}_{n}$ as a ${\mathbb Z}_p[\mathrm {Gal}(L_{n+1}/{\mathbb Q}_p)]$ -module and $\ell (c_{n+1})\equiv c_{n+1}u_1\,\pmod {\,{\mathfrak m}_n^{\oplus g}}$ . Furthermore, as an $\mathcal {O}_v$ -module, $u_1$ generates ${\mathbb Z}_p^{\oplus g}$ . Hence, we deduce that $\ell (c_{n+1})$ generates $\ell ({\mathfrak m}_{n+1}^{\oplus g})/\ell ({\mathfrak m}_{n}^{\oplus g})$ as an $\mathcal {O}_v[\mathrm {Gal}(L_{n+1}/{\mathbb Q}_p)]$ -module.

Since $[L_{n+1} : k_n]$ is prime to p, the map $\mathrm {Tr}_{L_{n+1}/k_n} : \widehat {A} (L_{n+1}) \longrightarrow \widehat {A}(k_n)$ is surjective. Indeed, we have that the space $\widehat {A}(L_{n+1})$ decomposes as a direct sum of eigenspaces:

$$ \begin{align*} \widehat{A}(L_{n+1})=\bigoplus_{\chi}\widehat{A}(L_{n+1})^{\chi}, \end{align*} $$

where $\chi $ runs over the distinct characters of $\mathrm {Gal}(L_{n+1}/k_n)$ , and $\widehat {A}(L_{n+1})^{\chi }$ is the eigenspace corresponding to the character $\chi $ . For $\chi \neq \mathbf {1}$ and $a\in \widehat {A}(L_{n+1})^\chi $ , we have

$$ \begin{align*} \mathrm{Tr}_{L_{n+1}/k_n}(a)=\sum_{\sigma\in \mathrm{Gal}(L_{n+1}/k_n)} \chi(\sigma)a = \bigg(\sum_{\sigma\in \mathrm{Gal}(L_{n+1}/k_n)} \chi(\sigma) \bigg) a=0. \end{align*} $$

On the other hand, for $\chi =\mathbf {1}$ and $a\in \widehat {A}(k_{n})$ , we have

$$ \begin{align*} \mathrm{Tr}_{L_{n+1}/k_n}(a)=\sum_{\sigma\in \mathrm{Gal}(L_{n+1}/k_n)} \chi(\sigma)a=(p-1)a. \end{align*} $$

Since $p-1$ is prime to p, the multiplication by the $p-1$ map is surjective on $\widehat {A}(k_n)$ . Thus, the trace map is surjective on $\widehat {A}(L_{n+1})$ .

Since $c_n$ and $c_{n-1}$ generate $\widehat {A}(L_{n+1})$ as a $\mathcal {O}_v[\mathrm {Gal}(L_{n+1}/{\mathbb Q}_p)]$ -module, we have that $d_n$ and $d_{n-1}$ generate $\widehat {A}(k_n)$ as an $\mathcal {O}_v[\mathrm {Gal}(k_{n}/{\mathbb Q}_p)]$ -module for $n \geq 1$ .

4 Signed Coleman maps

Write $T_v$ for the v-adic Tate module of A, which is a free $\mathcal {O}_v$ -module of rank two equipped with a continuous $G_{\mathbb Q}$ -action. Write $H_{{\mathbf f}}^1(k_n,T_v)$ for the image of the Kummer map $\widehat {A}(k_{n}) \longrightarrow H^1(k_{n},T_v)$ , and similarly $H_{{\mathbf f}}^1(L_n,T_v)$ for $\widehat {A}(L_{n})$ .

For $n\geq 0$ , let $\omega _n=(1+X)^{p^n}-1=\prod _{0\le i\le n}\Phi _i$ , where $\Phi _{i}$ is the $p^i$ th cyclotomic polynomial in $1+X$ . Set $\Lambda _{v}:=\lim _{\longleftarrow{n}} \mathcal {O}_v[G_n]$ so that $\Lambda _{v,n}:= \Lambda _{v} /(\omega _n)=\mathcal {O}_v[G_n]$ , where we have $G_n=\mathrm {Gal}(k_n/{\mathbb Q}_n)$ . Define the matrices

$$ \begin{align*} \mathcal{C}_i(X):=\begin{bmatrix} a_p& 1\\ -\Phi_i&0 \end{bmatrix} \end{align*} $$

and

$$ \begin{align*} \mathcal{H}_n:=\mathcal{C}_n(X)\cdots \mathcal{C}_1(X).\end{align*} $$

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

$$ \begin{align*} \langle-,-\rangle_{n}: \widehat{A}(k_{n}) \times H^1(k_{n},T_v) \longrightarrow \mathcal{O}_v \end{align*} $$

denote the local pairing (A is self-dual). We define $H_{\mathrm {Iw}}^i(k_{\infty },T_v)=\lim _{\longleftarrow{n}} H^i(k_{n},T_v)$ , where the transition maps are given by the corestriction maps. Suppose that ${\mathbb d}=(d_n)_{n\geq 0}$ is a primitive Q-system for $T_v$ given as in Proposition 4.1. Define

(4-1) $$ \begin{align} \mathrm{Col}_{v,n}: H^1_{\mathrm{Iw}}(k_\infty,T_v)&\longrightarrow \Lambda_{v,n}\notag\\ z&\mapsto \sum_{\sigma\in G_n} \langle z^{\sigma^{-1}},d_n\rangle_{n}\cdot\sigma. \end{align} $$

Theorem 4.2. There exist unique $\Lambda _v$ -morphisms (called the -Coleman maps)

such that

where $\xi _{n-1}:\Lambda _{v,n-1}\longrightarrow \Lambda _{v,n}$ is the norm map. Further, the Coleman maps are surjective onto $\Lambda _{v}$ .

Proposition 4.3. Let $J_{v}:=\{ (g_1,g_2)\in \Lambda _{v}^{\oplus 2} \mid (p-1)g_1(0)=(2-a_p)g_2(0) \}$ . Then we have

Proof. On taking $n=1$ in Theorem 4.2, we have

Combined with Theorem 3.2(ii), we deduce that

Thus,

Combining these two congruences gives

In particular, , as required.

Proof. Let ${\mathbb F}_v$ denote the residue field of $\mathcal {O}_v$ and fix a uniformiser $\varpi $ . We write $\Omega =\Lambda _v/(\varpi )$ . Let $\underline {\mathrm {Col}}:H^1_{\mathrm {Iw}}(k_\infty ,T_v)\longrightarrow \Omega ^{\oplus 2}$ denote the map given by modulo $\varpi $ and write $\underline {J_v}$ for the image of $J_v\subset \Omega ^{\oplus 2}$ under the reduction map. It follows immediately from Proposition 4.3 that

$$ \begin{align*} \mathrm{Image}(\underline{\mathrm{Col}})\subseteq\underline{J_v}. \end{align*} $$

We claim that the theorem would follow from the equality $\mathrm {Image}(\underline {\mathrm {Col}})=\underline {J_v}$ . Indeed, if this equality holds, given any $x\in J_v$ , we can write $x=y+pz$ for some and $z\in \Lambda _v^{\oplus 2}$ . This implies that $x-y=pz\in J_v\cap p\Lambda _v^{\oplus 2}$ . A direct calculation shows that this intersection is $pJ_v$ . Consequently, we deduce that . Applying Nakayama’s lemma, we can then deduce that .

We verify the equality $\mathrm {Image}(\underline {\mathrm {Col}})=\underline {J_v}$ in the remainder of the proof. Since $\Omega ^{\oplus 2}/\underline {J_v}\cong {\mathbb F}_v$ , it is enough to show that the cokernel of $\underline {\mathrm {Col}}$ is isomorphic to ${\mathbb F}_v$ . We prove this following a similar strategy to [Reference Kurihara and PollackKP07, the proof of Proposition 1.2].

Let $\underline {\widehat {A}(k_n)}^\star $ denote the ${\mathbb F}_v[G_n]$ -submodule of $\underline {\widehat {A}(k_n)}:=\widehat {A}(k_n)/(\varpi )$ generated by $d_n$ . We can check that $\underline {\widehat {A}(k_n)}^\star \cap \underline {\widehat {A}(k_{n-1})}^\star =\widehat {A}({\mathbb Q}_p)$ for all $n\ge 1$ . Combining with Proposition 3.3 gives the short exact sequence

(4-2) $$ \begin{align} 0\rightarrow \underline{\widehat{A}({\mathbb Q}_p)}\rightarrow \underline{\widehat{A}(k_n)}^\star\oplus \underline{\widehat{A}(k_{n-1})}^\star\rightarrow \underline{\widehat{A}(k_n)}\rightarrow 0 \end{align} $$

as in [Reference KobayashiKob03, Proposition 8.12].

For , let $\mathcal {P}^\natural $ be the image of $\ker (\mathrm {Col}_v^\natural )$ in $H^1_{\mathrm {Iw}}(k_\infty ,T_v)/(\varpi )$ . Since the matrix $\mathcal {C}_i(X)$ is congruent to $[\begin {smallmatrix} 0&1\\ -\Phi _i&0 \end {smallmatrix}]$ (modulo $\varpi $ ), it follows from Theorem 4.2 that

where $(-)^\perp $ denotes the orthogonal complement under the pairing

$$ \begin{align*} H^1_{\mathrm{Iw}}(k_\infty,T_v)/(\varpi)\times \widehat{A}(k_\infty)/(\varpi)\rightarrow{\mathbb F}_v \end{align*} $$

induced from the local Tate duality

$$ \begin{align*} H^1_{\mathrm{Iw}}(k_\infty,T_v)\times (\widehat{A}(k_\infty)\otimes_{\mathcal{O}_F} F_v/\mathcal{O}_v)\rightarrow F_v/\mathcal{O}_v \end{align*} $$

after identifying $\widehat {A}(k_\infty )\otimes _{\mathcal {O}_F} F_v/\mathcal {O}_v$ with $H^1_{\mathrm {Iw}}(k_\infty ,A[v^\infty ])$ (which we can do since $H^1_{\mathrm {Iw}}(k_\infty ,A[v^\infty ])/\widehat {A}(k_\infty )\otimes _{\mathcal {O}_F} F_v/\mathcal {O}_v$ is the Pontryagin dual of the universal norm $\lim _{\longleftarrow{n}} \widehat {A}(k_n)$ , which is zero when A is supersingular at p; see [Reference SchneiderSch87, Theorem 1]).

Letting $n\rightarrow \infty $ in (4-2) gives the short exact sequence

By duality, we deduce that the cokernel of $\underline {\mathrm {Col}}$ is isomorphic to $\underline {\widehat {A}({\mathbb Q}_p)}^\vee \cong {\mathbb F}_v$ , as desired.

5 Signed Selmer groups

Recall that $K_\infty $ is a ${\mathbb Z}_p$ -extension of K such that each prime ${\mathfrak p}_1,\ldots ,{\mathfrak p}_d$ of K above p is totally ramified in $K_\infty $ , and that $K_n \subset K_\infty $ denotes the unique subextension such that $[K_n : K] = p^n$ , for an integer $n \geq 0$ .

For $i\in \{1,\ldots , d\}$ , write $K_{{\mathfrak p}_i}$ for the completion of K at ${\mathfrak p}_i$ , which is isomorphic to ${\mathbb Q}_p$ , and $K_{\infty ,{\mathfrak p}_i}$ for the ${\mathbb Z}_p$ -extension of $K_{{\mathfrak p}_i}$ corresponding to $k_\infty $ . Let $K_{n,{\mathfrak p}_i}$ denote the unique subextension of $K_{\infty ,{\mathfrak p}_i}/K_{{\mathfrak p}_i}$ that is of degree $p^n$ . For every place v of F lying above p, Theorem 4.2 gives the signed Coleman maps

$$ \begin{align*} \mathrm{Col}^{\bullet}_{v,i}: H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i},T_v)\longrightarrow \Lambda_{v}, \end{align*} $$

where and ${\mathfrak p}_i\in \{{\mathfrak p}_1,\ldots ,{\mathfrak p}_d\}$ .

Definition 5.1. For $n\geq 0$ , , and $i\in \{1,\ldots ,d\}$ , define $H^1_\bullet (K_{n,{\mathfrak p}_i},T_v)\subset H^1(K_{n,{\mathfrak p}_i},T_v)$ to be the image of $\ker \mathrm {Col}_{v,i}^{\bullet }$ under the natural projection $H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i},T_v)\longrightarrow H^1(K_{n,{\mathfrak p}_i},T_v)$ .

Fix a uniformiser $\varpi $ of $F_v$ . Define $H^1_\bullet (K_{n,{\mathfrak p}_i},A[\varpi ^\infty ])\subset H^1(K_{n,{\mathfrak p}_i},A[\varpi ^\infty ])$ to be the orthogonal complement of $H^1_\bullet (K_{n,{\mathfrak p}_i},T_v)$ under the local Tate pairing

$$ \begin{align*} H^1(K_{n,{\mathfrak p}_i},T_v)\times H^1(K_{n,{\mathfrak p}_i},A[\varpi^\infty]) \longrightarrow F_v/\mathcal{O}_v. \end{align*} $$

Let $\lambda $ be a prime of $K_n$ lying above a rational prime $\ell $ . Write $H^1_{{\mathbf f}}(K_{n,\lambda }, A[\varpi ^\infty ])\subset H^1(K_{n,\lambda }, A[\varpi ^\infty ])$ for the image of the Kummer map $A(K_{n,\lambda })\otimes _{\mathcal {O}_F} F_v/\mathcal {O}_v \longrightarrow H^1(K_{n,\lambda }, A[\varpi ^\infty ])$ .

The singular quotient at $\ell $ is given by

$$ \begin{align*} H^1_{/{\mathbf f}}(K_{n,\ell}, A[\varpi^\infty]):=\bigoplus_{\lambda\mid \ell}\dfrac{H^1(K_{n,\lambda}, A[\varpi^\infty])}{H^1_{{\mathbf f}}(K_{n,\lambda}, A[\varpi^\infty])}, \end{align*} $$

where the direct sum runs over all primes of $K_n$ above $\ell $ .

Definition 5.2. For , we define the signed Selmer group of A over $K_n$ by

$$\begin{align*}\mathrm{Sel}_v^{\mathbf{s}}(A/K_n):=\ker\bigg(H^1(K_n,A[v^\infty])\longrightarrow \prod_{\ell\nmid p} H^1_{/{\mathbf f}}\ (K_{m,\ell}, A[\varpi^\infty]) \prod_{i=1}^d \dfrac{H^1(K_{n,{\mathfrak p}_i}, A[\varpi^\infty])}{H^1_{s_i}(K_{n,{\mathfrak p}_i}, A[\varpi^\infty])} \bigg). \end{align*}$$

Further, define $\mathrm {Sel}^{\mathbf {s}}_v(A/K_\infty )=\lim\nolimits_{\longrightarrow{n}} \mathrm {Sel}^{\mathbf {s}}_v(A/K_n)$ .

If $\mathrm {Sel}_{v}(A/K_\infty )$ denotes the classical $v^\infty $ -Selmer group, then we have that $\mathrm {Sel}^{\mathbf {s}}_v(A/K_\infty )\subset \mathrm {Sel}_{v}(A/K_\infty )$ . It follows from [Reference ManinMan71, Theorem 4.5] that $\mathrm {Sel}_{v}(A/K_\infty )$ is cofinitely generated over $\Lambda _{v}$ . Hence, so is $\mathrm {Sel}^{\mathbf {s}}_v(A/K_\infty )$ . We consider the following hypothesis.

For the remainder of this paper, we assume that Hypothesis 5.3 holds.

Let $\Sigma $ denote a fixed finite set of primes of K containing p, the ramified primes of $K/{\mathbb Q}$ , the archimedean place, and all the bad reduction primes of A. Write $K_\Sigma $ for the maximal algebraic extension of K that is unramified outside $\Sigma $ . For any (possibly infinite) extension $K \subseteq K' \subseteq K_\Sigma $ , write $G_\Sigma (K') = \mathrm {Gal}(K_\Sigma /K')$ .

For $i\in \{1,2\}$ , we define $H^i_{\mathrm {Iw},\Sigma }(K_\infty ,T_v)=\lim _{\longleftarrow} H^i(G_\Sigma (K_n),T_v)$ , where the transition maps are given by the corestriction maps. Note that $H^i_{\mathrm {Iw},\Sigma }(K_\infty ,T_v)$ is independent of the choice of $\Sigma $ (see [Reference RubinRub00, Corollary B.3.5], [Reference Mazur and RubinMR04, Lemma 5.3.1] and [Reference KobayashiKob03, Proposition 7.1]). We will drop the subscript $\Sigma $ from the notation for simplicity and simply write $H^i_{\mathrm {Iw}}(K_\infty ,T_v)$ .

We recall the definition of fine Selmer groups:

Definition 5.6. For $0\leq n \leq \infty $ , we define the fine Selmer group

$$ \begin{align*} \mathrm{Sel}_v^0(A/K_n):=\ker( \mathrm{Sel}_{v}(A/K_n)\longrightarrow H^1(k_n, A[\varpi^\infty]) ). \end{align*} $$

Equivalently, we have

$$ \begin{align*} \mathrm{Sel}_v^0(A/K_n):=\ker\bigg( H^1(G_\Sigma(K_n),A[\varpi^\infty])\longrightarrow \prod_{{\mathfrak q}_n\in \Sigma(K_n)} H^1(K_{n,{\mathfrak q}_n}, A[\varpi^\infty]) \bigg), \end{align*} $$

where $\Sigma (K_n)$ denotes the set of primes of $K_n$ lying above $\Sigma $ .

The Pontryagin duals of $\mathrm {Sel}_{v}(A/K_\infty )$ , $\mathrm {Sel}^{\mathbf {s}}_v(A/K_\infty )$ and $ \mathrm {Sel}^0_v(A/K_\infty )$ are denoted by $\mathcal {X}_v(A/K_\infty )$ , $\mathcal {X}^{\mathbf {s}}_v(A/K_\infty )$ and $\mathcal {X}^0_v(A/K_\infty )$ , respectively. By [Reference Perrin-RiouPR00, Proposition A.3.2], for , we have the Poitou–Tate exact sequence

(5-1) $$ \begin{align} H^1_{\mathrm{Iw}}(K_\infty,T_v) \longrightarrow \prod_{i=1}^d \dfrac{H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i}, T_v)}{H^1_{s_i}(K_{\infty,{\mathfrak p}_i}, T_v)} \longrightarrow\mathcal{X}^{\mathbf{s}}_v(A/K_\infty) \longrightarrow \mathcal{X}^0_v(A/K_\infty) \longrightarrow 0, \end{align} $$

and, for any $n\geq 0$ .

(5-2) $$ \begin{align} H^1(G_\Sigma(K_n),T_v) \longrightarrow \prod_{i=1}^d \dfrac{H^1(K_{n,{\mathfrak p}_i}, T_v)}{A(K_{n,{\mathfrak p}_i})\otimes_{\mathcal{O}_F}\mathcal{O}_v} \longrightarrow \mathcal{X}_v(A/K_\infty) \longrightarrow \mathcal{X}^0_v(A/K_\infty) \longrightarrow 0. \end{align} $$

We conclude this section with the following statement on the $\Lambda _v$ -module structure of $H^i_{\mathrm {Iw}}(K_\infty ,T_v)$ .

Proof. It follows from Hypothesis 5.3 and the Poitou–Tate exact sequence (5-1) that $\mathcal {X}^0_v(A/K_\infty )$ is $\Lambda _v$ -torsion. By [Reference Perrin-RiouPR00, Proposition A.3.2], we have the exact sequence

$$ \begin{align*} 0 \longrightarrow \mathcal{X}^0_v(A/K_\infty) \longrightarrow H^2_{\mathrm{Iw}}(K_\infty,T_v) \longrightarrow \prod_{w\in \Sigma} H^2_{\mathrm{Iw}}(K_{\infty,w},T_v). \end{align*} $$

By [Reference Loeffler and ZerbesLZ14, Theorem A.2], we see that each $H^2_{\mathrm {Iw}}(K_{\infty ,w},T_v)$ is $\Lambda _{v}$ -torsion, whence $H^2_{\mathrm {Iw}}(K_\infty ,T_v)$ is a torsion $\Lambda _{v}$ -module. It can be proven similarly to [Reference Perrin-RiouPR00, Proposition 1.3.2(i) $\Rightarrow$ (ii)] that assertion (i) implies (ii).

6 Definition and basic properties of the Kobayashi rank

Let $E/{\mathbb Q}_p$ be a finite extension with ring of integers $\mathcal {O}$ , and let e denote the ramification index of $E/{\mathbb Q}_p$ . Following [Reference KobayashiKob03, Section 10] and [Reference Lei, Loeffler and ZerbesLLZ17, Section 4], we define the Kobayashi ranks as follows.

Definition 6.1. Let $(M_n)_{n\geq 1}$ be a projective system of finitely generated ${\mathbb Z}_p$ -modules. Given an integer $n\ge 1$ , if $\pi _n:M_n\longrightarrow M_{n-1}$ has finite kernel and cokernel, we define

$$ \begin{align*} \nabla M_n:=\mathrm{length}_{\mathcal{O}}(\ker\pi_n)-\mathrm{length}_{\mathcal{O}}(\mathrm{coker}\; \pi_n)+\mathrm{rank}_{\mathcal{O}}M_{n-1}. \end{align*} $$

For $n\ge 1$ , let $\zeta _{p^n}$ be a primitive $p^n$ th root of unity and set $\epsilon _n=\zeta _{p^n}-1$ . Recall that $\omega _n=(1+X)^{p^n}-1$ .

Write $p^f$ for the size of the residue field of E. If N is a finite $\mathcal {O}$ -module, then ${|N| = p^{f\cdot \mathrm {length}_{\mathcal {O}}(N)}}$ . Using the following lemma we can relate the growth in the size of a tower of finite $\mathcal {O}$ -modules and Kobayashi ranks.

7 Proof of Theorem A

The proof of Theorem A is divided into a number of steps. Following [Reference KobayashiKob03, Section 10], we define, for each integer $n\ge 0$ ,

$$ \begin{align*} \mathcal{Y}_v(A/K_n)&:=\mathrm{coker}\ \bigg( H^1(G_\Sigma(K_n),T_v) \longrightarrow \bigoplus_{i=1}^d \dfrac{H^1(K_{n,{\mathfrak p}_i}, T_v)}{A(K_{n,{\mathfrak p}_i})\otimes_{\mathcal{O}_F}\mathcal{O}_v}\bigg). \end{align*} $$

7.1 Calculating Kobayashi ranks of certain auxiliary modules

We introduce auxiliary modules in preparation for the calculation of $\nabla \mathcal {Y}_v(A/K_n)$ . For , the composition

$$ \begin{align*} \mathrm{Col}_{\mathbf{s}} : H^1_{\mathrm{Iw}}(K_\infty,T_v) \xrightarrow{\; (\mathrm{loc}_{{\mathfrak p}_i})_{1\leq i\leq d}\;} \bigoplus_{i=1}^d H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i},T_v) \xrightarrow{\; (\mathrm{Col}_{v,i}^{s_i})_{1\leq i\leq d}\;} \Lambda_{v} ^{\oplus d} \end{align*} $$

is a $\Lambda _v$ -homomorphism between two $\Lambda _v$ -modules of rank d (see Proposition 5.7).

Lemma 7.1. The localisation map

$$ \begin{align*} \mathrm{loc}_p:=(\mathrm{loc}_{{\mathfrak p}_1},\ldots,\mathrm{loc}_{{\mathfrak p}_d} ) : H^1_{\mathrm{Iw}}(K_\infty,T_v) \longrightarrow \bigoplus_{i=1}^d H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i},T_v) \end{align*} $$

is injective.

Proof. For , (5-1) gives the exact sequence

(7-1) $$ \begin{align} H^1_{\mathrm{Iw}}(K_\infty,T_v) \longrightarrow \bigoplus_{i=1}^d \dfrac{H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i}, T_v)}{H^1_{s_i}(K_{\infty,{\mathfrak p}_i}, T_v)} \longrightarrow\mathcal{X}^{\mathbf{s}}_v(A/K_\infty). \end{align} $$

It follows from Theorem [Reference Perrin-RiouPR94, Proposition 3.2.1] and Theorem 4.2 that the middle term of the exact Sequence (7-1) is of $\Lambda _v$ -rank d. Since the $\Lambda _v$ -module $H^1_{\mathrm {Iw}}(K_\infty ,T_v)$ is of rank d (Proposition 5.7), the kernel of the first map in (7-1) is $\Lambda _v$ -torsion as $\mathcal {X}^{\mathbf {s}}_v(A/K_\infty )$ is $\Lambda _v$ -torsion by Hypothesis 5.3. On the other hand, an argument similar to [Reference Lei and LimLL22, Lemma 2.6] shows that the $\Lambda _v$ -module $H^1_{\mathrm {Iw}}(K_\infty ,T_v)$ is torsion-free, which implies that the kernel of the first map in (7-1) is trivial. Consequently, the kernel of $\mathrm {loc}_p$ is trivial.

Proposition 7.2. There exist a family of elements $c_1,\ldots ,c_d \in H^1_{\mathrm {Iw}}(K_\infty ,T_v) $ such that

$$ \begin{align*} \frac{H^1_{\mathrm{Iw}}(K_\infty,T_v)}{\langle c_1,\ldots,c_d \rangle} \end{align*} $$

is $\Lambda _v$ -torsion, and $\mathrm {loc}_{{\mathfrak p}_j}(c_i)=0$ whenever $i\neq j$ .

Proof. Let $R_i$ denote the pre-image of $ H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i}, T_v)\subset \bigoplus _{i=1}^d H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i}, T_v)$ under $\mathrm {loc}_p$ . It follows from Lemma 7.1 that $\bigoplus _{i=1}^d R_i$ is isomorphic to a submodule of $H^1_{\mathrm {Iw}}(K_\infty ,T_v)$ . Further, by an abuse of notation, we can identify $\bigoplus _{i=1}^d R_i$ with its image under $\mathrm {loc}_p$ as a submodule of $\bigoplus _{i=1}^d H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i}, T_v)$ , where $R_i$ is regarded as a $\Lambda _v$ -submodule of $H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i}, T_v)$ of rank at most two.

In the proof of Lemma 7.1 we see that, for each $i\in \{1,\ldots ,d\}$ , the cokernel of the composition

$$ \begin{align*} H^1_{\mathrm{Iw}}(K_\infty,T_v) \xrightarrow{\;\;\mathrm{loc}_{{\mathfrak p}_i}\;\;} H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i}, T_v) \xrightarrow{\;\;\mathrm{Col}_{v,i}^{s_i}\;\;}\Lambda_v \end{align*} $$

is $\Lambda _v$ -torsion. It then follows that the $\Lambda _v$ -rank of $R_i$ cannot be zero. Therefore, the $\Lambda _v$ -module $\bigoplus _{i=1}^d R_i$ has rank at least d. Since $H^1_{\mathrm {Iw}}(K_\infty ,T_v)$ is of $\Lambda _v$ -rank d, we deduce that the $\Lambda _v$ -rank of $\bigoplus _{i=1}^d R_i$ is exactly d. Thus, the $\Lambda _v$ -rank of each $R_i$ is one, and we may pick $c_i\in R_i$ so that $ {H^1_{\mathrm {Iw}}(K_\infty ,T_v)}/{\langle c_1,\ldots ,c_d \rangle }$ is $\Lambda _v$ -torsion. Further, since ${R_i\cap R_j=0}$ whenever $i\neq j$ , we see that $\mathrm {loc}_{{\mathfrak p}_j}(c_i)=0$ whenever $i\neq j$ .

Let and $H_n^\flat $ denote the entries of the first row of the matrix $\mathcal {H}_n$ . We then see that $\mathcal {H}_n$ is of the form

Since $\det \mathcal {H}_n=\omega _n/X$ , we have

(7-2)

Recall that $\zeta _{p^n}$ denotes a primitive $p^n$ th root of unity and $\epsilon _n=\zeta _{p^n}-1$ . By [Reference Lei, Loeffler and ZerbesLLZ17, Proposition 4.6], we have, for $n\ge 1$ ,

and

$$ \begin{align*}\mathrm{ord}_{\epsilon_n}(H_n^\flat(\epsilon_n))=\begin{cases} \displaystyle\phi(p^n)\ \bigg( {\sum_{k=1}^{{n-1}/{2}}}\dfrac{1}{p^{2k-1}} \bigg) & \text{ if } n \text{ is odd,}\\[7pt] \displaystyle \phi(p^n)\ \bigg( r_p + {\sum_{k=1}^{({n}/{2})-1}}\dfrac{1}{p^{2k}} \bigg) & \text{ if } n \text{ is even,} \end{cases}\end{align*} $$

where $r_p=\mathrm {ord}_p(a_p)$ , $\phi (p^n)=p^n-p^{n-1}$ .

Definition 7.3. For $n\geq 1$ , $i\in \{1,\ldots , d\}$ , and $u\in \mathcal {O}_v^\times $ , we define

$$ \begin{align*} \underline{\textrm{Col}}_{v,n,i}^{u}:H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i}, T_v) \longrightarrow \Lambda_{v,n} \end{align*} $$

to be

In what follows, we write ${\textrm {Col}}_{v,n,i}$ for the map given by (4-1), after taking $k_\infty $ to be  $K_{\infty ,{\mathfrak p}_i}$ .

Lemma 7.4. For each $n\ge 1$ and $i\in \{1,\ldots ,d\}$ , the map

$$ \begin{align*} & {\underline{\mathrm{Col}}}_{v,n,i}: H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i},T_v)\to \prod_{m=0}^n\mathcal{O}_v[X]/(\Phi_m)\\ & \quad z\mapsto(\mathrm{Col}_{v,n,i}(z))_{0\le m\le n} \end{align*} $$

induces an injection on

$$ \begin{align*}H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i},T_v):=\dfrac{H^1(K_{n,{\mathfrak p}_i}, T_v)}{A(K_{n,{\mathfrak p}_i})\otimes_{\mathcal{O}_F}\mathcal{O}_v}.\end{align*} $$

Proof. As $A(k_\infty )[p^\infty ]=0$ (see the proof of Lemma 2.5), the projection $H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i}, T_v)\to H^1(K_{n,{\mathfrak p}_i},T_v)$ is surjective, resulting in the isomorphism

$$ \begin{align*} H^1_{\mathrm{Iw}}(K_{\infty,{\mathfrak p}_i},T_v)/{(\omega_n)}\cong H^1(K_{n,{\mathfrak p}_i},T_v). \end{align*} $$

In particular, $\underline {\mathrm {Col}}_{v,n,i}$ induces a map on $H^1(K_{n,{\mathfrak p}_i},T_v)$ .

Let $\theta $ be a character of $G_n=\mathrm {Gal}(K_{n,{\mathfrak p}_i}/K_{{\mathfrak p}_i})$ . We write $e_\theta \in F_v[G_n]$ for the idempotent element of $\theta $ . Let $0\le m\le n$ be the integer such that $\theta $ factors through $G_m$ but not $G_{m-1}$ . Then $e_\theta \cdot z $ belongs to the image of $e_\theta \cdot A(K_{n,{\mathfrak p}_i})\otimes _{\mathcal {O}_F}\mathcal {O}_v$ if and only if $\mathrm {Col}_{v,m,i}(z)\equiv 0\,\pmod {\,\Phi _m}$ since $\mathrm {Col}_{v,m,i}$ is defined by pairings with $d_n$ , and $d_n$ generates $\widehat {A}(K_{m,{\mathfrak p}_i})/\widehat {A}(K_{m-1,{\mathfrak p}_i})$ as an $\mathcal {O}_v[G_m]$ -module (see the proof of Proposition 3.3). Therefore, $z\in \ker \underline {\mathrm {Col}}_{v,n,i}$ if and only if $e_\theta \cdot z\in e_\theta \cdot A(K_{n,{\mathfrak p}_i})\otimes _{\mathcal {O}_F}\mathcal {O}_v$ for all $\theta $ . This tells us that the kernel of $\underline {\mathrm {Col}}_{v,n,i}$ is the image of $ A(K_{n,{\mathfrak p}_i})\otimes _{\mathcal {O}_F}\mathcal {O}_v$ in $H^1(K_{n,{\mathfrak p}_i},T_v)$ , as desired.

Lemma 7.5. For $n\geq 1$ and $i\in \{1,\ldots ,d\}$ , we have the following assertions.

1. (i) $\omega _{n-1}\Lambda _v\subset \mathrm {Image}(\mathrm {Col}_{v,n,i}^{u})$ for any choice of u.

2. (ii) There exist infinitely many $u \in \mathcal {O}_v^\times $ such that $ \mathrm {Col}_{v,n,i}^{u}$ induces an injection

   
   with finite cokernel.

Proof. By Theorem 4.4, there exists $z\in H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i},T_v)$ such that and . For such z, we have

by (7-2). Therefore, assertion (i) holds.

We now prove assertion (ii). Recall from Theorem 4.2 that

which implies

$$ \begin{align*} \mathrm{Col}_{v,n,i}^u=\mathrm{Col}_{v,n,i}-u\xi_{n-1}\mathrm{Col}_{v,n-1,i}. \end{align*} $$

By the Chinese remainder theorem, there is an injection $\Lambda _{v,n}\hookrightarrow \prod _{m=0}^n\mathcal {O}[X]/(\Phi _m)$ . For all $1\le m\le n$ , we have

$$ \begin{align*} \mathcal{C}_r(X)\equiv\begin{bmatrix} a_p&1\\-p&0 \end{bmatrix}\,\pmod {\,\omega_m},\quad m<r\le n. \end{align*} $$

Thus,

which in turn implies that

$$ \begin{align*} \mathrm{Col}_{v,n,i}^u\equiv\begin{bmatrix} 1&u \end{bmatrix} \begin{bmatrix} a_p&1\\-p&0 \end{bmatrix}^{n-m} \begin{bmatrix} \mathrm{Col}_{v,m,i}\\ 0 \end{bmatrix} \,\pmod {\,\Phi_m}. \end{align*} $$

Therefore, there are infinitely many choices of u such that $\mathrm {Col}_{v,n,i}^u$ is congruent to a nonzero multiple of $\mathrm {Col}_{v,m,i}$ (modulo $\Phi _m$ ). A similar argument shows that the same is true for $m=0$ . In particular, there exist infinitely many u such that this holds for all $0\le m\le n$ . Once such u is picked, Lemma 7.4 tells us that $\mathrm {Col}_{v,n,i}^u$ induces an injection on $H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i},T_v)$ .

Finally, as $ H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i}, T_v)$ and $\Lambda _{v,n}$ are both of rank $p^n$ over $\mathcal {O}_v$ , the cokernel is finite.

Corollary 7.6. For $i\in \{1,\ldots ,d\}$ , let $z_i\in H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i},T_v)$ and $u\in \mathcal {O}_v^\times $ , satisfying Lemma 7.5(ii). For $n\geq 1$ , write $M_{n,i}$ for the $\Lambda _{v,n}$ -module generated by the natural image of z in $H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i}, T_v)$ . Suppose that the natural projection

$$ \begin{align*} \pi_i: \frac{H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i}, T_v)}{M_{i,n}} \longrightarrow \frac{H^1_{/{\mathbf f}}(K_{n-1,{\mathfrak p}_i}, T_v)}{M_{i,n-1}} \end{align*} $$

has finite kernel for some n. Then we have

$$ \begin{align*} \nabla\bigg(\frac{H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i}, T_v)}{M_{i,n}} \bigg)=\mathrm{ord}_{\epsilon_n}(\mathrm{Col}_{v,n,i}^{u}(z)(\epsilon_n)). \end{align*} $$

Proof. The proof is identical to the proof of [Reference Lei and LimLL22, Corollary 4.5].

Proposition 7.7. For $n\hspace{-0.75pt}\geq\hspace{-0.75pt} 1$ , let $z\hspace{-0.75pt}\in\hspace{-0.75pt} H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i},T_v)$ and $u\hspace{-0.75pt}\in\hspace{-0.75pt} \mathcal {O}_v^\times $ , satisfying Lemma 7.5(ii), be such that $\mathrm {Col}_{v,n,i}^{u}(z)(\epsilon _n)\neq 0$ . Then we have

$$ \begin{align*} \mathrm{ord}_{\epsilon_n}(\mathrm{Col}_{v,n,i}^{u}(z)(\epsilon_n)) = \mathrm{ord}_{\epsilon_n}( H_n^{s_i}(\epsilon_n))+ \mathrm{ord}_{\epsilon_n}( \mathrm{Col}_{v,i}^{s_i}(z)(\epsilon_n)), \end{align*} $$

where depends on the parity of n.

Proof. See [Reference Lei and LimLL22, Proposition 4.6], which is a special case of [Reference Lei, Loeffler and ZerbesLLZ17, Proposition 4.6 and Corollary 4.8].

From now on, we fix ${\mathbf c}=(c_1,\ldots ,c_{d})$ satisfying the properties given by Proposition 7.2.

Definition 7.8. Write $M_{\mathbf c}$ for the $\Lambda _v$ -module generated by $ c_1,\ldots ,c_{d}$ . We define, for each integer $n\ge 0$ , the $\mathcal {O}_v$ -modules

$$ \begin{align*} \mathcal{Y}_v'(A/K_n)&:=\mathrm{coker}\ \bigg( H^1_{\mathrm{Iw}}(K_\infty,T_v)_{\Gamma_n} \longrightarrow \bigoplus_{i=1}^d H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i},T_v)\bigg),\\ \mathcal{Y}_v^{\prime\prime}(A/K_n)&:=\mathrm{coker}\ \bigg((M_{\mathbf c})_{\Gamma_n} \longrightarrow \bigoplus_{i=1}^dH^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i},T_v)\bigg). \end{align*} $$

Proposition 7.9. For $n\gg 0$ , $\nabla \mathcal {Y}^{\prime \prime }_v(A/K_n)$ is defined. Moreover, for each n that is sufficiently large, there exists that depends only on the parity of n such that

$$ \begin{align*} \nabla\mathcal{Y}^{\prime\prime}_v(A/K_n)= F_v({\mathbf{s}},n)+ \nabla\bigg(\frac{\Lambda_v^{\oplus d}}{\mathrm{Col}_{\mathbf{s}}(M_{\mathbf c})}\bigg)_{\Gamma_n}, \end{align*} $$

where

where ( $a_\flat $ ) is the number of ( $s_i=\flat $ ).

Proof. The proof is similar to the proof of [Reference Lei and LimLL22, Proposition 5.3]. By definition, we have

(7-3) $$ \begin{align} \mathcal{Y}^{\prime\prime}_v(A/K_n)= \bigoplus_{i=1}^d\frac{H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i},T_v)}{(M_i)_{\Gamma_n}}, \end{align} $$

where $M_i$ denotes the image of $M_{\mathbf c}$ in $H^1_{\mathrm {Iw}}(K_{\infty ,{\mathfrak p}_i},T_v)$ . Let $\pi _i$ denote the natural projection

$$ \begin{align*} \pi_i: \frac{H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i}, T_v)}{(M_{i})_{\Gamma_n}} \longrightarrow \frac{H^1_{/{\mathbf f}}(K_{n-1,{\mathfrak p}_i}, T_v)}{(M_{i})_{\Gamma_{n-1}}}. \end{align*} $$

We deduce from Corollary 7.6 that $\ker \pi _i$ is finite if and only if $\mathrm {ord}_{\epsilon _n}(\mathrm {Col}_{v,n,i}^{u}(c_i)(\epsilon _n))$ does not vanish at $\epsilon _n$ .

Let be given as in Proposition 7.7, which depends only on the parity of n. It follows from the exact sequence (7-1) and Hypothesis 5.3 that $\mathrm {Col}_{v,n,i}^{u}(c_i)$ is a nonzero element of $\Lambda _{v,n}$ . In particular, for $n\gg 0$ , we have $\mathrm {Col}_{v,n,i}^{u}(c_i)(\epsilon _n)\neq 0$ . It then follows that $\ker \pi _i$ is finite and $\nabla ({H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i},T_v)}/{(M_i)_{\Gamma _n}} )$ is defined. It follows from Proposition 7.7 and Corollary 7.6 that

$$ \begin{align*} \nabla\bigg( \frac{H^1_{/{\mathbf f}}(K_{n,{\mathfrak p}_i},T_v)}{(M_i)_{\Gamma_n}} \bigg) = \mathrm{ord}_{\epsilon_n}( H_n^{s_i}(\epsilon_n)) + \mathrm{ord}_{\epsilon_n}(\mathrm{Col}_{v,i}^{s_i}(c_i)(\epsilon_n)). \end{align*} $$

The assertion of the proposition now follows from Lemma 6.2 applied to the direct sum (7-3).

Effectively, there are two possible choices for the sign ${\mathbf {s}}$ : one that applies to all sufficiently large even values of n, and another that applies to all sufficiently large odd values of n.

Remark 7.10. If p is an odd prime and $E/{\mathbb Q}$ is a p-supersingular elliptic curve with $a_p(E) = 0$ , then we have

$$ \begin{align*} F_v({\mathbf{s}},n)= \begin{cases} \displaystyle\phi(p^n)\cdot\bigg( {\sum_{k=1}^{({n-1})/{2}}}\dfrac{1}{p^{2k-1}}\bigg) & \text{ if } n \text{ is odd,}\\ \displaystyle\phi(p^n) \cdot \bigg( {\sum_{k=1}^{{n}/{2}}}\dfrac{1}{p^{2k-1}} \bigg) & \text{ if } n \text{ is even.} \end{cases} \end{align*} $$

Corollary 7.11. For $n\gg 0$ , $\nabla \mathcal {Y}_v(A/K_n)$ is defined and is equal to

$$ \begin{align*} \nabla\mathcal{Y}_v(A/K_n)= F_v({\mathbf{s}},n)+\nabla (\mathrm{coker} (\mathrm{Col}_{\mathbf{s}}))_{\Gamma_n}, \end{align*} $$

where and $F_v({\mathbf {s}},n)$ are given in Proposition 7.9.

Proof. Consider the short exact sequences

$$ \begin{align*} 0&\longrightarrow \frac{H^1_{\mathrm{Iw}}(K_\infty,T_v)_{\Gamma_n}}{(M_{\mathbf c})_{\Gamma_n}}\longrightarrow \bigg(\frac{\Lambda_v^{\oplus d}}{\mathrm{Col}_{\mathbf{s}}(M_{\mathbf c})} \bigg)_{\Gamma_n} \longrightarrow (\mathrm{coker} (\mathrm{Col}_{\mathbf{s}}))_{\Gamma_n} \longrightarrow 0,\\ 0&\longrightarrow \frac{H^1_{\mathrm{Iw}}(K_\infty,T_v)_{\Gamma_n}}{(M_{\mathbf c})_{\Gamma_n}}\longrightarrow \mathcal{Y}^{\prime\prime}_v(A/K_n)\longrightarrow\mathcal{Y}^{\prime}_v(A/K_n)\longrightarrow 0. \end{align*} $$

Furthermore, $({H^1_{\mathrm {Iw}}(K_\infty ,T_v)_{\Gamma _n}})/({(M_{\mathbf c})_{\Gamma _n}})$ is isomorphic to $({H^1_{\mathrm {Iw}}(K_\infty ,T_v)}/{M_{\mathbf c}})_{\Gamma _n}$ . Since ${H^1_{\mathrm {Iw}}(K_\infty ,T_v)}/{M_{\mathbf c}}$ is $\Lambda _v$ -torsion, we see that $\nabla ({H^1_{\mathrm {Iw}}(K_\infty ,T_v)_{\Gamma _n}}/{(M_{\mathbf c})_{\Gamma _n}})$ is defined for $n\gg 0$ . It then follows from Proposition 7.9 and Lemma 6.2 that $\nabla \mathcal {Y}_v'(A/K_n)$ is defined and is equal to

$$ \begin{align*} \nabla \frac{H^1_{\mathrm{Iw}}(K_\infty,T_v)_{\Gamma_n}}{(M_{\mathbf c})_{\Gamma_n}}= \nabla \mathcal{Y}^{\prime\prime}_v(A/K_n)-\nabla\mathcal{Y}^{\prime}_v(A/K_n). \end{align*} $$

It follows from Proposition 7.9 that $\nabla ({\Lambda _v^{\oplus d}}/{\mathrm {Col}_{\mathbf {s}}(M_{\mathbf c})} )_{\Gamma _n} $ is defined for $n\gg 0$ . Thus, Proposition 6.2 tells us that

$$ \begin{align*} \nabla(\mathrm{coker} (\mathrm{Col}_{\mathbf{s}}))_{\Gamma_n}= \nabla \bigg(\frac{\Lambda_v^{\oplus d}}{\mathrm{Col}_{\mathbf{s}}(M_{\mathbf c})} \bigg)_{\Gamma_n} -\nabla \frac{H^1_{\mathrm{Iw}}(K_\infty,T_v)_{\Gamma_n}}{(M_{\mathbf c})_{\Gamma_n}}. \end{align*} $$

By an adaptation of [Reference KobayashiKob03, Proposition 10.6(i)] and [Reference Lei and LimLL22, Proposition 3.6] to our setting, we have, for $n\gg 0$ ,

$$ \begin{align*} \nabla \mathcal{Y}_v(A/K_n)= \nabla \mathcal{Y}^{\prime}_v(A/K_n). \end{align*} $$

The assertion of the corollary now follows from combining the last three equalities.

7.2 Final steps of the proof of Theorem A

We are now ready to conclude the proof of Theorem A. Recall that $\mathcal {X}_v(A/K_\infty )$ and $\mathcal {X}^0_v(A/K_\infty )$ denote the Pontryagin duals of $\mathrm {Sel}_{v}(A/K_\infty )$ and $ \mathrm {Sel}^0_v(A/K_\infty )$ , respectively.

Lemma 7.12. The natural restriction map

$$ \begin{align*} \mathrm{Sel}^0_v(A/K_n) \longrightarrow \mathrm{Sel}^0_v(A/K_\infty)^{\Gamma_n} \end{align*} $$

is injective and has finite cokernel. Furthermore, the cardinality of the cokernel stabilises as $n\rightarrow \infty $ .

Proof. The assertion follows from [Reference LimLim20, Theorem 3.3].

Lemma 7.13. Assume that Hypothesis 5.3 holds. Let $n\geq 0$ be an integer.

1. (i) We have the short exact sequence

   $$ \begin{align*} 0 \longrightarrow \mathcal{Y}_v(A/K_n) \longrightarrow \mathcal{X}_v(A/K_n) \longrightarrow \mathcal{X}^0_v(A/K_n) \longrightarrow 0. \end{align*} $$

2. (ii) For $n\gg 0$ , $\nabla \mathcal {X}^0_v(A/K_n)$ is defined and satisfies the equality

   $$ \begin{align*} \nabla \mathcal{X}^0_v(A/K_n)= \nabla \mathcal{X}^0_v(A/K_\infty)_{\Gamma_n}. \end{align*} $$

Proof. Assertion (i) follows from the exact sequence (5-2) and the definition of $\mathcal {Y}_v(A/K_n)$ . We saw in the proof of Proposition 5.7 that $\mathcal {X}^0_v(A/K_\infty )$ is $\Lambda _{v}$ -torsion. By Lemma 7.12, the kernel and cokernel of the natural map

$$ \begin{align*} \mathcal{X}^0_v(A/K_\infty)_{\Gamma_n} \longrightarrow \mathcal{X}^0_v(A/K_n) \end{align*} $$

are finite with cardinality independent of n when n is sufficiently large. Combining this with Lemma 6.3(ii), assertion (ii) follows.

Let $e_v$ denote the ramification index of $F_v/{\mathbb Q}_p$ .

Proposition 7.14. Suppose that Hypothesis 5.3 holds. For $n\gg 0$ , we have

$$ \begin{align*} \nabla \mathcal{X}_v(A/K_n)= F_v({\mathbf{s}},n)+e_v\lambda_{v,{\mathbf{s}}} + (p^n-p^{n-1})\mu_{v,{\mathbf{s}}}, \end{align*} $$

where and $F_v({\mathbf {s}},n)$ are given in Proposition 7.9, which depend on the parity of n.

Proof. Let be given as in Proposition 7.9. We have the following Poitou–Tate exact sequence of $\Lambda _v$ -torsion modules:

$$ \begin{align*} 0\longrightarrow \mathrm{coker}(\mathrm{Col}_{\mathbf{s}}) \longrightarrow \mathcal{X}_v^{\mathbf{s}}(A/K_\infty) \longrightarrow \mathcal{X}_v^0(A/K_\infty) \longrightarrow0. \end{align*} $$

By [Reference KobayashiKob03, Lemma 10.3], we obtain the six-term exact sequence

$$ \begin{align*} & 0\longrightarrow \mathrm{coker}(\mathrm{Col}_{\mathbf{s}})^{\Gamma_n} \longrightarrow \mathcal{X}_v^{\mathbf{s}}(A/K_\infty)^{\Gamma_n} \longrightarrow \mathcal{X}_v^0(A/K_\infty)^{\Gamma_n} \\ & \quad \longrightarrow (\mathrm{coker}(\mathrm{Col}_{\mathbf{s}}))_{\Gamma_n} \longrightarrow \mathcal{X}_v^{\mathbf{s}}(A/K_\infty)_{\Gamma_n} \longrightarrow \mathcal{X}_v^0(A/K_\infty)_{\Gamma_n} \longrightarrow0. \end{align*} $$

By [Reference Lei and LimLL22, Lemma 3.5], the Kobayashi ranks of the first three terms vanish for $n\gg 0$ . Hence, it follows from Lemma 6.2(i) that, for $n\gg 0$ , we have

$$ \begin{align*} \nabla\mathcal{X}_v^{\mathbf{s}}(A/K_\infty)_{\Gamma_n} = \nabla (\mathrm{coker}(\mathrm{Col}_{\mathbf{s}}))_{\Gamma_n}+\nabla \mathcal{X}_v^0(A/K_\infty)_{\Gamma_n}. \end{align*} $$

Combining this equality with Lemma 7.13 and Corollary 7.11, we deduce that

$$ \begin{align*} \nabla\mathcal{X}_v(A/K_\infty)_{\Gamma_n} =F_v({\mathbf{s}},n)+\nabla \mathcal{X}_v^{\mathbf{s}}(A/K_\infty)_{\Gamma_n}. \end{align*} $$

The assertion of the proposition now follows from Lemma 6.3(ii) under Hypothesis 5.3.

Theorem 7.15. Let A be an abelian variety of $\mathrm {GL}_2$ -type over ${\mathbb Q}$ associated to a weight two newform with trivial nebentype. Let K be a number field and p be an odd prime satisfying Hypothesis 1.1. Furthermore, assume that Hypothesis 5.3 holds. Then the following statements hold.

1. (i) $\mathrm {rank}_{\mathcal {O}_F} A(K_n)$ is bounded independently of n.

2. (ii) Assume further that the v-part of the $\mathcal {O}_F$ -module is finite for all $n\geq 0$ and let . Define

   $$ \begin{align*} r_\infty=\sup_{n\ge0}\{ \mathrm{rank}_{\mathcal{O}_F} A(K_n)\}. \end{align*} $$
   Then, for $n\gg 0$ , we have
   $$ \begin{align*} e_n-e_{n-1}= F_v({\mathbf{s}},n) +(p^n-p^{n-1})\mu_{v,{\mathbf{s}}} + e_v\lambda_{v,{\mathbf{s}}} -r_\infty. \end{align*} $$

Proof. Since $\nabla \mathcal {X}_v(A/K_n)$ is defined when n is sufficiently large by Proposition 7.14, the kernel and cokernel of the natural map $\mathcal {X}_v(A/K_{n+1})\rightarrow \mathcal {X}_v(A/K_n)$ are finite for such n. In particular, $\mathrm {rank}_{\mathcal {O}_v}\mathcal {X}_v(A/K_n)$ is bounded independently of n. Hence, the first assertion follows from the well-known exact sequence

Taking duals, we have

(7-4)

It follows from the exact sequence (7-4) and Lemma 6.2 that

Assertion (i) tells us that $\nabla A(K_n)\otimes _{\mathcal {O}_F} \mathcal {O}_v=r_\infty $ for $n\gg 0$ . Thus, we deduce from Proposition 7.14 that

As and are assumed to be finite, we have

which is equal to $e_n -e_{n-1}$ . Thus, assertion (ii) follows.