Proof.

1. (1) By definition, $V_{\infty }^{*}/(\gamma ^{p^n}-1)$ is isomorphic to $\varprojlim _m H^1(\Psi _m, T^{\otimes -1}(1))/(\gamma ^{p^n}-1)$ . In light of the inflation-restriction sequence and [Reference Rubin47, Prop. 4.5 (ii)], it follows that $\varprojlim _m H^1(\Psi _m, T^{\otimes -1}(1))/(\gamma ^{p^n}-1)$ is isomorphic to the Pontryagin dual of

   $$ \begin{align*}H^0(\Psi_n, H^1(\Psi_{\infty}, {\mathscr {F}}[\pi^{\infty}])) \cong H^1(\Psi_n, {\mathscr {F}}[\pi^{\infty}]).\end{align*} $$
   The local duality thus implies (2.10).

2. (2) Note that $H^1_{\mathrm {f}}(\Psi _{n}, T)$ coincides with the Kummer image of ${\mathscr {F}}({\mathfrak {m}}_n)$ in $H^1(\Psi _{n}, T)$ , and that $H^1_{\mathrm {f}}(\Psi _{n}, T^{\otimes -1}(1))$ coincides with the kernel of $\exp ^*_{\Psi _n}: H^1_{}(\Psi _{n}, T^{\otimes -1}(1)) \to \Psi _n$ . Hence, by the commutative diagram (2.8), the proof concludes.

Proof. The first assertion is a simple consequence of Theorem 2.1.

Note that $T^{\otimes -1}(1) \cong T^{\tau }$ as an ${\mathcal {O}}[G_{\Phi }]$ -module, where $T^{\tau }$ denotes conjugation of T by the complex conjugate. Then, by [Reference Rubin47, Prop. 4.5], $H^1(\Psi _n, T)$ and $H^1(\Psi _n, T^{\otimes -1}(1))\cong H^1(\Psi _n, T)$ are ${\mathcal {O}}$ -free, and we have natural identifications

$$ \begin{align*} H^1(\Psi_n, T)\kern1.2pt{\otimes}\kern1.2pt \Phi/{\mathcal {O}} \kern1.2pt{=}\kern1.2pt H^1(\Psi_n, T \kern1.2pt{\otimes}\kern1.2pt \Phi/{\mathcal {O}}), \ \, H^1(\Psi_n, T^{\otimes -1}(1))\kern1.2pt{\otimes}\kern1.2pt \Phi/{\mathcal {O}} \kern1.2pt{=}\kern1.2pt H^1(\Psi_n, T^{\otimes -1}(1) \kern1.2pt{\otimes}\kern1.2pt \Phi/{\mathcal {O}}). \end{align*} $$

Hence, the local duality induces

$$ \begin{align*} H^1(\Psi_n, T) \cong {\mathrm{Hom}}_{{\mathcal {O}}}(H^1(\Psi_n, T^{\otimes -1}(1) \otimes \Phi/{\mathcal {O}}), \Phi/{\mathcal {O}}) ={\mathrm{Hom}}_{{\mathcal {O}}}(H^1(\Psi_n, T^{\otimes -1}(1)),{\mathcal {O}}), \end{align*} $$

where the isomorphism arises from the perfect pairing

$$ \begin{align*}H^1(\Psi_n, T)\times H^1(\Psi_n, T^{\otimes -1}(1) \otimes \Phi/{\mathcal {O}}) \to \Phi/{\mathcal {O}}.\end{align*} $$

It follows that the map $H^1(\Psi _n, T) \to {\mathrm {Hom}}_{{\mathcal {O}}}(H^1(\Psi _n, T^{\otimes -1}(1)),{\mathcal {O}})$ induced by $(\ \ , \ \ )_n$ is an isomorphism. By replacing $T $ with $T^{\otimes -1}(1)$ , the map $H^1(\Psi _n, T^{\otimes -1}(1)) \to {\mathrm {Hom}}_{{\mathcal {O}}}(H^1(\Psi _n, T),{\mathcal {O}})$ induced by $(\ \ , \ \ )_n$ is also an isomorphism, and hence, $(\ \ , \ \ )_n$ is perfect.

The assertion (3) then follows from the fact that $H^1_{\mathrm {f}}(\Psi _{n}, T)\otimes \mathbb {Q}_p$ is the orthogonal complement of $H^1_{\mathrm {f}}(\Psi _{n}, T^{\otimes -1}(1))\otimes \mathbb {Q}_p$ under the base change of $(\ \ , \ \ )_n$ , and vice versa.

Proof. It suffices to show that for a finite character $\chi $ of ${\mathrm {Gal}}(\Psi _n/\Phi )$ , the image $\chi (c_n^{\pm })$ of $c_n^{\pm }$ under the natural map $H^1(\Psi _n, T) \to H^1(\Phi , V (\chi ))$ lies in the finite part $H^1_{\mathrm {f}}(\Phi , V(\chi )),$ where $V(\chi ):=T\otimes \mathbb {Q}_p(\chi )$ denotes the twist of $T\otimes \mathbb {Q}_p$ by $\chi $ .

If $\chi \in \Xi ^{\pm }$ , then $\chi (\omega _{n}^{\mp })=0,$ and so $\chi (c_n^{\pm }) \in H^1_{\mathrm {f}}(\Phi , V(\chi )).$ If $\chi \in \Xi ^{\mp },$ then $\delta _{\chi }(v_{\pm })=0$ . Now, by Lemma 2.2 and (2.8), the image of $v_{\pm ,n}$ under

$$ \begin{align*}H^1(\Psi_n, V^{\otimes -1}(1))\to H^1(\Phi, V(\chi)^{\otimes -1}(1))\end{align*} $$

lies in the finite part, and it gives rise to a generator of $H^1_{\mathrm {f}}(\Phi , V(\chi )^{\otimes -1}(1))$ over $\Phi (\mathrm {Im}(\chi ))$ . Since $H^1_{\mathrm {f}}(\Phi , V(\chi ))$ is the orthogonal complement of $H^1_{\mathrm {f}}(\Phi , V(\chi )^{\otimes -1}(1))$ with respect to the local duality, it thus follows that $\chi (c_n^{\pm })$ lies in the finite part.

Proof.

1. (1) First, note that an element x of $H^1(\Psi _n, T)$ is determined by the two elements $(x, v_{+,n}^{\sigma })_{\Lambda _n}$ and $(x, v_{-,n}^{\sigma })_{\Lambda _n} \in \Lambda _n$ .

   By definition,

   $$ \begin{align*} (\mathrm{Tr}_{n+1/{n}}c_{n+1}^{\pm}, v_{\pm,n})_{\Lambda_n}=(c_{n}^{\pm}, v_{\pm,n})_{\Lambda_n}=(\mathrm{Res}_{n, n-1}c_{n-1}^{\pm}, v_{\pm,n})_{\Lambda_n}=0. \end{align*} $$

   As $(c_{n}^{\pm }, v_{\mp ,n})_{\Lambda _n}=\omega _n^{\mp },$ it suffices to show that

   $$ \begin{align*} (\mathrm{Tr}_{n+1/{n}}c_{n+1}^{\pm}, v_{\mp,n})_{\Lambda_n}=(c_{n-1}^{\pm}, v_{\mp,n})_{\Lambda_n}=\omega_n^{\mp}. \end{align*} $$

   Since $\omega _{n+1}^{\mp }=\omega _{n}^{\mp },$ we have

   $$ \begin{align*} (\mathrm{Tr}_{n+1/{n}}c_{n+1}^{\pm}, v_{\mp,n})_{\Lambda_n}\equiv (c_{n+1}^{\pm}, v_{\mp,n})_{\Lambda_{n+1}} \equiv\omega_n^{\mp} \bmod (\gamma^{p^n}-1). \end{align*} $$

   Since $\omega _{n}^{\mp }=\omega _{n-1}^{\mp }\Phi _{n}(\gamma )$ and $\{v_{\mp ,n}\}_n$ is norm compatible,

   $$ \begin{align*} \kern-20pt (c_{n-1}^{\pm}, v_{\mp,n})_{\Lambda_n}&\equiv \sum_{\sigma \in {\mathrm{Gal}}(\Psi_n/\Phi)}(c_{n-1}^{\pm}, v^{\sigma}_{\mp, n-1})_{n-1}\sigma \equiv \omega_{n-1}^{\mp}\sum_{\sigma \in {\mathrm{Gal}}(\Psi_n/\Phi)}(v_{\pm, n-1}^{\perp}, v^{\sigma}_{\mp, n-1})_{n-1}\sigma \\ & \equiv \omega_{n-1}^{\mp}\sum_{\sigma \in {\mathrm{Gal}}(\Psi_n/\Psi_{n-1})}\sigma \equiv \omega_{n-1}^{\mp}\Phi_{n-1}(\gamma)=\omega_n^{\mp}. \end{align*} $$

   Therefore, the assertion follows.

2. (2) This is a simple consequence of (1).

Proof. By definition,

(2.14) $$ \begin{align} \sum_{\sigma \in {\mathrm{Gal}}(\Psi_n/\Phi)}(c_n^{\pm}, v_{\mp,n}^{\sigma})_n\sigma=\omega_n^{\mp} \in \Lambda_n. \end{align} $$

1. (a) Let $x \in \mathscr {F}(\mathfrak {m}_n)$ and consider

   $$ \begin{align*} \sum_{\sigma \in \mathrm{Gal}(\Psi_n/\Phi)} (x, v_{\mp,n}^\sigma)_n \sigma, \end{align*} $$
   which livesFootnote ³ in $\omega ^{\mp }_{n}\Lambda _n$ . Thus, by (2.14), there exists $h_{\pm }(\gamma ) \in \Lambda _n$ such that
   $$ \begin{align*} \sum_{\sigma \in \mathrm{Gal}(\Psi_n/\Phi)} (x-h_{\pm}(\gamma)c_n^{\pm}, v_{\mp,n}^\sigma)_n \sigma=0. \end{align*} $$
   Now, for $y:=h_+(\gamma )c_n^++h_-(\gamma )c_n^-$ , we have
   $$ \begin{align*} \sum_{\sigma \in \mathrm{Gal}(\Psi_n/\Phi)} (x-y, u^\sigma)_n \sigma=0 \end{align*} $$
   for arbitrary $u \in H^1(\Psi _n, T^{\otimes -1}(1))$ . Hence, $x=y$ .

2. (b) If $x \in {\mathscr {F}}({\mathfrak {m}}_n)^{\pm }$ , then

   $$ \begin{align*}\sum_{\sigma \in \mathrm{Gal}(\Psi_n/\Phi)} (x, v_{\pm,n}^\sigma)_n \sigma=0.\end{align*} $$
   Thus, we may choose $h_{\mp }$ as above to be $0$ .

Proof. By definition and [Reference Kato30, Prop. 15.9], $z_\chi $ lies in the image of

$$ \begin{align*} \varprojlim_m H^1(K({\mathfrak{f}} p^m), T^{\otimes -1}(1)) \to H^1(K_n, T^{\otimes -1}(1)), \end{align*} $$

where $K({\mathfrak {f}} p^m)$ denotes the ray class field of K of conductor ${\mathfrak {f}} p^m$ .

For a prime $v\nmid p$ of $K_n$ and a prime $w \mid v$ of $\cup _{m\ge 1}K({\mathfrak {f}} p^m)$ , note that the completion of $\cup _mK({\mathfrak {f}} p^m)$ at w contains the maximal pro-p unramified extension of $K_{n,v}$ , and so ${\mathrm {loc}}_{v}(z_{\chi })\in H^1_{\mathrm {f}}(K_{n,v}^{} ,T^{\otimes -1}(1)).$ Since $L(\varphi \chi ^{-1},1)=0$ , by the explicit reciprocity law (3.3) and (3.5),

$$ \begin{align*}\exp^*_{K_n}({\mathrm{loc}}_{p}(z_{\chi}))=0.\end{align*} $$

As the Bloch–Kato subgroup $\oplus _{v\mid p} H^1_{\mathrm {f}}(K_{n,v}, T^{\otimes -1}(1))$ coincides with the kernel of $\exp ^*_{K_n}$ , the proof concludes.

Proof. Note that $T^{\otimes -1}(1)(\chi ) \cong T(\chi ^{-1})^{\tau }$ as an ${\mathcal {O}}[G_{K}]$ -module, and so the first assertion follows.

As for the irreducibility, in light of the proof of [Reference Kato30, Lem. 15.20], it suffices to show that for a finite character $\chi $ of ${\mathrm {Gal}}(K_{\infty }/K)$ , there exists an integral ideal ${\mathfrak {b}}$ of K relatively prime to $p{\mathfrak {f}}$ such that

(3.6) $$ \begin{align} (\varphi\chi^{-1}({\mathfrak{b}})-\varphi\chi^{-1}(\bar{{\mathfrak{b}}}))N_{K/{\mathbb {Q}}}({\mathfrak{b}})\in {\mathcal {O}}^{\times}. \end{align} $$

We put ${\mathfrak {b}}=(1+f \sqrt {-D_K}){\mathcal {O}}_K$ for $f=N_{K/{\mathbb {Q}}}({\mathfrak {f}}) \in {\mathbb {Z}}.$ As p is inert in K, note that p is relatively prime to $N_{K/{\mathbb {Q}}}({\mathfrak {b}})=1+f^2D_K$ , i.e. $N_{K/{\mathbb {Q}}}({\mathfrak {b}})\in {\mathcal {O}}^{\times }$ . Since $p\nmid fD_K$ and $1+f \sqrt {-D_K} \equiv 1\ \mathrm {mod}\ {\mathfrak {f}}$ , we have $\varphi ({\mathfrak {b}})-\varphi (\bar {{\mathfrak {b}}}) =2f\sqrt {-D_K}\in {\mathcal {O}}^{\times }$ , and so $\varphi ({\mathfrak {b}})\gamma _{{\mathfrak {b}}}-\varphi (\bar {{\mathfrak {b}}})\gamma _{\bar {{\mathfrak {b}}}} \in \Lambda ^{\times }$ , from which (3.6) follows. (Here, $\gamma _{{\mathfrak {a}}}\in G^{-}$ denotes the element which corresponds via the Artin map to an integral ideal ${\mathfrak {a}}$ of K relatively prime to $p{\mathfrak {f}}$ .)

Proof. Since $\mathrm {ord}_{s=1}L(f_{\chi },s)=r$ , by the main result of [Reference Bump, Friedberg and Hoffstein8], there exists an imaginary quadratic field L such that

1. (i) ${\mathrm {ord}}_{s=1}L(f_{\chi /L},s)=1$ and

2. (ii) the pair $(f_{\chi }, L)$ satisfies the Heegner hypothesis.

Then, the Gross–Zagier formula [Reference Gross and Zagier28], [Reference Yuan, Zhang and Zhang55] implies that the Heegner point $y_{L} \in A_{\chi }(L)$ is non-torsion, and so the assertion is due to Kolyvagin [Reference Kolyvagin35] (see also [Reference Nekovář40]). As for the “in particular” part, note that $A_{\chi }({\mathbb {Q}})$ is p-torsion-free by Lemma 3.4.

Proof. Let $B_n$ denote the Weil restriction ${\mathrm {Res}}_{K_n/K}(A_{/K_n})$ of A over $K_{n}$ . By considering the Galois action on valued points of $B_n$ , note that the Galois group ${\mathrm {Gal}}(K_{n}/K)$ embeds into $\mathrm {End}\,B_n$ , which in turn implies

$$ \begin{align*}\mathrm{End}\,B_n=(\mathrm{End}\,A) [{\mathrm{Gal}}(K_{n}/K)]\end{align*} $$

as algebras (cf. [Reference Diem and Naumann23, Thm. 3]).

In light of the decomposition ${\mathbb {Q}}[{\mathrm {Gal}}(K_{n}/K)] \cong {\mathbb {Q}}[\gamma ]/(\Phi _{p^n}(\gamma )) \times {\mathbb {Q}}[{\mathrm {Gal}}(K_{n-1}/K)]$ and factorisation of the underlying L-functions, we have an isogeny

(3.8) $$ \begin{align} B_n \sim A_n \times B_{n-1} \end{align} $$

of abelian varieties over K. Here, $A_n$ is the abelian variety defined as a product of copies of $A_{\chi }$ with $\dim (A_n)=[F:{\mathbb {Q}}](p^n-p^{n-1})$ . Note that the set of K-rational points is given by

$$ \begin{align*} A(K_n)\otimes {\mathbb {Q}}\cong \left(A_n(K)\otimes {\mathbb {Q}} \right) \oplus \left(A(K_{n-1})\otimes {\mathbb {Q}} \right). \end{align*} $$

Now, we consider ${\mathrm {Gal}}(K_{n}/K)$ -action which leads to

(3.9) $$ \begin{align} (\gamma^{p^{n-1}}-1) A(K_n)\otimes {\mathbb {Q}} \cong A_n(K)\otimes {\mathbb {Q}}, \end{align} $$

and

(3.10)

In light of the Gross–Zagier formula [Reference Gross and Zagier28], [Reference Yuan, Zhang and Zhang55] and Proposition 3.5, we have

(3.11)

and so

(3.12)

Hence, in conjunction with (3.10), it follows that

(3.13)

where $\chi _1$ ranges over the conjugates of $\chi $ .

1. (a) Recall the short exact sequence

   
   Now, as is p-torsion-free, (3.13) readily implies the second asserted equality of part (a).

   Since p is unramified in K, observe $K[\gamma ]/\Phi _{p^n}(\gamma )$ is a field with $\mathrm {dim}_{K}(K[\gamma ]/ \Phi _{p^n}(\gamma ))=p^{n}-p^{n-1}$ . Naturally, $(\gamma ^{p^{n-1}}-1)A(K_n)\otimes {\mathbb {Q}}$ is a $K[\gamma ]/\Phi _{p^n}(\gamma )$ -vector space. So, in view of (3.9) and (3.12), it follows that

   $$ \begin{align*} (\gamma^{p^{n-1}}-1)A(K_n)\otimes {\mathbb {Q}} \cong (K [\gamma]/(\Phi_{p^n}(\gamma)))^{\oplus [F:{\mathbb {Q}}]}. \end{align*} $$
   Hence, the evaluation at $\chi $ yields the first asserted equality of part (a).

2. (b) In view of (3.13), we have

   
   i.e. part (b) holds.

Proof. This is a simple consequence of Theorem 3.9 (a) and Remark 3.8.

Proof. The non-vanishing of $\delta _{\chi ^{-1}}(v_{\varepsilon })$ is a consequence of Theorem 2.1 and [Reference Rubin47, Lem. 10.1]. Hence, the assertion follows by (2.8) and (3.3).

Proof. As for the first assertion, by Nakayama’s lemma, it suffices to show that $(v_{+,0},v_{-,0})_0 \in {\mathcal {O}}^{\times }.$

In view of Theorem 2.1,

(3.16) $$ \begin{align} V_{\infty}^{*}/(\gamma-1)=V_{\infty}^{*,+}/(\gamma-1)\oplus V_{\infty}^{*,-}/(\gamma-1)= {\mathcal {O}} v_{+,0}\oplus {\mathcal {O}} v_{-,0}, \end{align} $$

where $v_{\pm ,0}$ is the image of $v^{\pm }$ in $V_{\infty }^{*}/(\gamma -1)$ . Note that $H^1_{\mathrm {f}}(\Phi , T) \cong {\mathscr {F}}({\mathfrak {m}}_0) \cong {\mathcal {O}}$ , and so by (3.16),

$$ \begin{align*}H^1_{\mathrm{f}}(\Phi, T)=V^{*,-}_{\infty}/(\gamma-1).\end{align*} $$

As $\delta _{0}(v_{-})=0$ , observe $v_{-,0} \in H^1_{\mathrm {f}}(\Phi , T)$ by Lemma 2.2. Recall $(\ \ , \ \ )_0$ is perfect and $ H^1_{\mathrm {f}}(\Phi , T)$ is a maximal isotropic subgroup. Hence, (3.16) implies that $(v_{+,0}, v_{-,0})_0 \in {\mathcal {O}}^{\times }$ .

We now consider the second assertion. For any $\chi \in \Xi ^{\mp }$ , note that $\delta _{\chi }(v_{\pm })=0,$ i.e. the image of $v_{\pm }$ under $\varprojlim _{m}H^1(\Phi _m, T) \to H^1(\Phi , V(\chi ^{-1}))$ lies in the finite part $H^1_{\mathrm {f}}(\Phi , V(\chi ^{-1}))$ . Since $H^1_{\mathrm {f}}(\Phi , V(\chi ^{-1}))$ is the orthogonal complement of itself under $(\ \ , \ \ )_n$ , we have $\chi \left ((v_{\pm },v_{\pm })_{\Lambda }\right )=0$ . This implies $(v_{\pm },v_{\pm })_{\Lambda }=0$ as $\Xi ^{\mp }$ is an infinite set.

Proof.

1. (a) By Definition 3.13, note that

   $$ \begin{align*} \sum_{\sigma \in {\mathrm{Gal}}(K_n^{}/K)}(v^{\perp}_{-\varepsilon, n}, z_n^{\sigma})_n\sigma=\mathscr{L}(\gamma^{-1}) \ \mathrm{mod}\ (\gamma^{p^n}-1), \end{align*} $$
   where $z_{n}$ also denotes ${\mathrm {loc}}_{{\mathfrak {p}}}(z_{n})$ .

   In view of the explicit reciprocity law of Wiles (cf. [Reference Wiles53]) and [Reference Rubin47, Lem. 5.5], we have

   $$ \begin{align*} \kern-25pt \sum_{\sigma \in {\mathrm{Gal}}(K_n^{}/K)}(v^{\perp}_{-\varepsilon, n}, z_n^{\sigma})_n\chi^{-1}(\sigma) &= ( v_{-\varepsilon,n}^{\perp}, \sum_{\sigma} z_{n}^{\sigma}\chi^{-1}(\sigma) )_{n}\\ &= \mathrm{Tr}_{\Psi_n(\mathrm{Im}(\chi))/\Phi(\mathrm{Im}(\chi))}\left[\exp^*(v_{-\varepsilon,n}^{\perp}), \log\left( \sum_{\sigma} z_{n}^{\sigma}\chi^{-1}(\sigma)\right)\right]\\ &= \left[\sum_{\sigma \in {\mathrm{Gal}}(\Psi_n/\Phi)}\exp^*(\sigma v_{-\varepsilon,n}^{\perp})\chi(\sigma), \log\left( \sum_{\sigma} z_{n}^{\sigma}\chi^{-1}(\sigma)\right)\right]\\ &=\delta_{\chi^{-1}}(v_{-\varepsilon})\cdot \lambda_{E}(z_{\chi^{-1}}) \end{align*} $$
   (see also [Reference Rubin47, p. 413]). Here,
   $$ \begin{align*}[\ \ , \ \ ] :D_{{\mathrm{dR}}}(V)\otimes_{\Phi} \Psi_n \times D_{{\mathrm{dR}}}(V^{\otimes -1}(1))\otimes_{\Phi}\Psi_n \to {\mathcal {O}}[1/p]\otimes_{\Phi}\Psi_n\end{align*} $$
   denotes the natural pairing, $\exp ^*$ is the dual exponential map, and the last equality follows from (2.9) and the fact that $\sigma v_{-\varepsilon , n}^{\perp }$ corresponds to $\sigma ^{-1} v_{-\varepsilon , n}$ under (3.15). Hence,
   (3.17) $$ \begin{align} \mathscr{L}(\chi(\gamma))= \delta_{\chi^{-1}}(v_{-\varepsilon}^{}) \cdot \lambda_{E}(z_{\chi^{-1}}) \end{align} $$

   However,

   $$ \begin{align*} \sum_{\sigma\in {\mathrm{Gal}}(K_n^{}/K)}(\sigma c_n^{\varepsilon}, v_{-\varepsilon,n})_n\sigma =\omega^{-\varepsilon}_n(\gamma^{}) \bmod (\gamma^{p^n}-1), \end{align*} $$
   and so
   $$ \begin{align*}\delta_{\chi^{-1}}(v_{-\varepsilon})\cdot p^n\lambda_{E,\chi^{-1}}(c_n^{\varepsilon})= \omega^{-\varepsilon}_n(\chi(\gamma)). \end{align*} $$
   Hence, (3.17) concludes the proof.

2. (b) This just follows by letting $n=0$ and in the above argument.

Proof. The following is based on Iwasawa main conjectures [Reference Agboola and Howard2], [Reference Burungale, Kobayashi and Ota12], to which we refer for notation.

Let $\varepsilon $ denote the sign of the root number $W(\varphi )$ . By [Reference Agboola and Howard2, Thm. 3.6], the Selmer group $\mathscr {X}_{\infty }^{\varepsilon }$ has $\Lambda $ -rank one. Let $\mathscr {X}_{\mathrm {str}}$ be the Pontryagin dual of the strict Selmer group $S_{\mathrm {{str}}}:=\mathrm {Sel}_{\mathrm {str}}(K_{\infty }, V/T)$ (cf. [Reference Agboola and Howard2, §2] or (4.1) below). Let $\mathscr {X}_{\mathrm {rel}}$ be defined analogously.

In light of Proposition 3.5 and [Reference Agboola and Howard2, Thm. 5.2], the latter being the main conjectureFootnote ⁶ , the $\chi ^{-1}$ -specialization of $\mathrm {char}_{\Lambda }(\mathscr {X}_{\infty ,{\mathrm {tor}}}^{\varepsilon })$ is non-zero, where $\mathrm {char}_{\Lambda }(\cdot )$ denotes the characteristic ideal. Observe that Proposition 3.5 also implies that the $\chi ^{-1}$ -specialization of $\mathrm {char}_{\Lambda }(\mathscr {X}_{\mathrm {str}})$ is non-zero and then so is the $\chi ^{-1}$ -specialization of $\mathrm {char}_{\Lambda }(\mathscr {X}_{\mathrm {rel}, \Lambda -\mathrm {tor}})$ (cf. [Reference Agboola and Howard2, Thm. 4.1]). Hence, in view of [Reference Agboola and Howard2, (4.1), (4.4)], it follows that $z_{\chi ^{-1}}$ is non-torsion, and the proof concludes by Theorem 3.16 (b).

Proof. This is [Reference Kato30, (15.16.1)].

Proof. Fix an isomorphism $\mathrm {Ind}^K_{{\mathbb {Q}}}(V) \cong V_{L_{{\mathfrak {p}}}}(f)$ of $G_{\mathbb {Q}}$ -representations and let

(4.2) $$ \begin{align} H^{1}(K,V) \cong H^{1}({\mathbb {Q}},V_{L_{{\mathfrak{p}}}}(f)) \end{align} $$

be the induced identification. Let

$$ \begin{align*}z_{f} \in H^{1}({\mathbb {Q}},V(f))\end{align*} $$

be a Beilinson–Kato element as in [Reference Kato30, Thm. 12.5], which depends on a choice of an element in $H^1(A({\mathbb {C}}), {\mathbb {Q}})\cong F$ (cf. §B.2).

Since the root number of f is $-1$ , $L(f,1)=0$ , and so ${\mathrm {loc}}_p(z_f) \in H^1_{\mathrm {f}}({\mathbb {Q}}_p, V(f))$ by Kato’s reciprocity law [Reference Kato30, Thm. 12.5 (1)]. Now in view of Theorems 3.16 and 4.2, it follows that

(4.3)

up to an element in $L^\times $ . Hence, Theorem 4.3 concludes the proof.

Proof. By [Reference Gross27, Thm. 15.2.5], A is a CM abelian varietyFootnote ⁷ defined over ${\mathbb {Q}}$ which is simple over K. Since

$$ \begin{align*}L(A_{/{\mathbb {Q}}},s)=\prod_{\sigma: F\hookrightarrow {\mathbb {C}}} L(f^{\sigma},s)\end{align*} $$

(cf. [Reference Gross27, Thm. 18.1.7]), the assertion follows.

Proof. We first consider the case (i).

Since $L(\bar {\varphi },1)/\Omega $ is a p-adic unit, note that $\varepsilon (\varphi )=+1$ and

$$ \begin{align*}\mathscr{L}\in\Lambda^{\times}.\end{align*} $$

For $\chi \in \Xi ^{+}$ , we then have $ L(\varphi \chi ,1)\neq 0 $ and so ${\mathrm {rank}}_{\mathcal O_{\chi }}E(K_{n})^{\chi ^{-1}}=0$ by Theorem 3.9 (a) where n denotes $\max \{0, \mathrm {ord}_p({\mathfrak {c}}_{\chi })-1\}$ . Now, let $\chi \in \Xi ^{-}$ . In view of Theorem 3.16,

$$ \begin{align*}{\mathrm{loc}}_{{\mathfrak{p}}}(z_{\chi^{-1}})\neq 0. \end{align*} $$

Hence, the image of z in $S_{\mathrm {{rel}}}$ is a $\Lambda _{{\mathfrak {q}}}$ -basis for $S_{\mathrm {{rel}},{\mathfrak {q}}}$ up to tensoring with ${\mathbb {Q}}_p$ , where $S_{\mathrm {{rel}}}$ denotes the relaxed compact Selmer group and ${\mathfrak {q}}$ the prime ideal of $\Lambda $ corresponding to the $\chi ^{-1}$ -specialization. In turn, $\mathscr {X}_{\mathrm {{str},{\mathfrak {q}}}}$ is finite by [Reference Agboola and Howard2, Prop. 3.3] and then so is $H^{2}(\mathcal O_{K}[\frac {1}{p}],T(\chi ))$ . Hence, Tate’s Euler characteristic formula implies

$$ \begin{align*}{\mathrm{rank}}_{\mathcal O_{\chi}}H^{1}(\mathcal O_{K}[\frac{1}{p}],T(\chi))=1.\end{align*} $$

Since $z_{\chi ^{-1}}\in H^1_{\mathrm {{f}}}(K,T(\chi ))$ by Lemma 3.3, we conclude that ${\mathrm {rank}}_{\mathcal O_{\chi }}H^{1}_{\mathrm {{f}}}(\mathcal O_{K}[\frac {1}{p}],T(\chi ))=1$ . The assertion follows from this.

The case (ii) is similarly proven by using Theorem 4.8.

Proof. By [Reference Kato30, Lem. 15.11 (2)], there is a unique isomorphism

(4.5) $$ \begin{align} \mathrm{Ind}^{{\mathbb {C}}}_{{\mathbb {R}}}H^1(E({\mathbb {C}}), {\mathbb {Q}})\otimes_K L \cong H^1(A({\mathbb {C}}), {\mathbb {Q}}) \otimes_{F} L \end{align} $$

of $L[\mathrm {Gal}({\mathbb {C}}/{\mathbb {R}})]$ -modules such that the following diagram

(4.6)

commutes. Here, $H^1(A({\mathbb {C}}), {\mathbb {Q}})$ is regarded as a $\mathrm {Gal}({\mathbb {C}}/{\mathbb {R}})$ -module via the complex conjugation on $A({\mathbb {C}})$ , $\mathrm {per}_{\varphi }$ is the period map induced by that of E (cf. [Reference Kato30, §15.8]) and the right vertical map is the base change of (4.5) via $L \subseteq {\mathbb {C}}$ .

Recall that $z\in \varprojlim _n H^1(K_n, T)$ is the image of an element $\mathbf {z}^{\mathrm {ell}} \in \varprojlim _n H^1(K({\mathfrak {f}} p^n), T)$ associated to $b_E$ as in [Reference Kato30, Prop. 15.9] under the corestriction map

$$ \begin{align*} \varprojlim_n H^1(K({\mathfrak{f}} p^n), T) \to \varprojlim_n H^1(K_n, T), \end{align*} $$

where ${\mathfrak {f}}$ denotes the conductor of $\varphi $ and $K({\mathfrak {f}} p^n)$ the ray class field of K of conductor ${\mathfrak {f}} p^n$ (cf. [Reference Kato30, p. 254]). Let ${\mathbb {Q}}_n$ be the n-th layer of the cyclotomic ${\mathbb {Z}}_p$ -extension ${\mathbb {Q}}_{\infty }$ of ${\mathbb {Q}}$ . Let

$$ \begin{align*}z^{\mathrm{ell}}=(z_{n}^{\mathrm{ell}}) \in \varprojlim_n H^1({\mathbb {Q}}_n, T(f))\otimes L_{{\mathfrak{p}}}\end{align*} $$

denote the image of $\mathbf {z}^{\mathrm {ell}}$ under

$$ \begin{align*} \varprojlim_n H^1(K({\mathfrak{f}} p^n) ,T)\otimes L_{{\mathfrak{p}}} \to \varprojlim_n H^1(K\otimes {\mathbb {Q}}_{n}, T)\otimes L_{{\mathfrak{p}}} = \varprojlim_n H^1({\mathbb {Q}}_{n}, T(f))\otimes L_{{\mathfrak{p}}}. \end{align*} $$

Here, the first map is induced by the corestriction maps with respect to $K\otimes {\mathbb {Q}}_{n} \subseteq K({\mathfrak {f}} p^n)$ , and the equality is a consequence of (4.4) and Shapiro’s lemma.

Note that $b_E=\Omega ^{-1}\mathrm {per}_{\varphi }(\omega )$ maps to $\Omega ^{-1}\mathrm {per}_f(\omega _A)$ under (4.5) and $\mathbf {z}^{\mathrm {ell}}$ is associated to $b_E$ . Thus, in light of (4.6) and [Reference Kato30, (15.16.1)], it follows that $z^{\mathrm {ell}}$ coincides with the system of Beilinson–Kato elements associated to

$$ \begin{align*}\Omega^{-1}\mathrm{per}_f(\omega_A) \in H^1(A({\mathbb {C}}), {\mathbb {Q}})\otimes_F L.\end{align*} $$

Since ${\mathbb {Q}}_{\infty }$ is a totally real field, the last assertion in [Reference Kato30, Thm. 12.5 (1)] implies that $z^{\mathrm {ell}}$ also coincides with the system of Beilinson–Kato elements associated to

$$ \begin{align*}\Omega^{-1}2^{-1}(1+\iota)\mathrm{per}_f(\omega_A),\end{align*} $$

and that $\mathfrak {z}_f$ coincides with the one associated to

$$ \begin{align*}b_A^{+}:=2^{-1}(1+\iota)b_A=\Omega_f^{-1}2^{-1}(1+\iota)\mathrm{per}_f(\omega_A)\end{align*} $$

(cf. (B.3)), where $\iota $ denotes the involution induced by the complex conjugation on $A({\mathbb {C}})$ .

Therefore, we have

$$ \begin{align*}z^{\mathrm{ell}}_{0}=u z_{f,0}.\end{align*} $$

Since $z_0= z^{\mathrm {ell}}_{0}$ in $H^1(K, V)=H^1({\mathbb {Q}}, V_{L_{{\mathfrak {p}}}}(f))$ , the proposition follows.

Proof. By (4.4) and Shapiro’s lemma, we have an identification

(4.7) $$ \begin{align} H^1(K, V)=H^1({\mathbb {Q}}, V_{L_{{\mathfrak{p}}}}(f)). \end{align} $$

Let $T(f)$ be a Galois stable ${\mathcal {O}}_{F_{{\mathfrak {p}}}}$ -lattice of $V(f)$ .

Let

(4.8) $$ \begin{align} {\mathfrak{z}}_{f}=(z_{f,n}) \in \varprojlim_n H^{1}({\mathbb {Q}}_n,T(f)) \otimes {\mathbb {Q}}_p \end{align} $$

be the Beilinson–Kato element associated to $b_A$ as in [Reference Kato30, Thm. 12.5 (1)] (since our $V_{L_{{\mathfrak {p}}}}(f)$ is a Tate twist of that in [Reference Kato30], ${\mathfrak {z}}_f$ is the corresponding twist of $\mathbf {z}_{b_A}^{(p)}$ as in [Reference Kato30]). In particular, ${\mathfrak {z}}_{f}$ satisfies the explicit reciprocity law (B.4).

Note that $L(f,1)=0$ , and so by Proposition 4.12,

(4.9) $$ \begin{align} z_{0}=uz_{f,0} \in H^{1}_{\mathrm{{f}}}(K, V\otimes L_{{\mathfrak{p}}})=H^{1}_{\mathrm{{f}}}({\mathbb {Q}}, V_{L_{{\mathfrak{p}}}}(f)). \end{align} $$

Hence, Theorem 4.8 is a consequence of Theorems 3.16 (b) and 4.3.

Proof. This is a consequence of Theorems 4.2 and 4.3.

Proof. We have $A({\mathbb {Q}}_p)\otimes _{{\mathcal {O}}_F} F_{{\mathfrak {p}}}/{\mathcal {O}}_{F_{\mathfrak {p}}} \cong F_{{\mathfrak {p}}}/{\mathcal {O}}_{F_{\mathfrak {p}}}$ . So the exact sequence

(4.10) $$ \begin{align} 0\to {\mathrm{Sel}}_{\mathrm{str}}({\mathbb {Q}}, A[{\mathfrak{p}}^{\infty}]) \to {\mathrm{Sel}}_{{\mathfrak{p}}^{\infty}}(A) \to A({\mathbb {Q}}_p)\otimes_{{\mathcal {O}}_F} F_{{\mathfrak{p}}}/{\mathcal {O}}_{F_{\mathfrak{p}}} \end{align} $$

and the finiteness of ${\mathrm {Sel}}_{\mathrm {str}}({\mathbb {Q}}, A[{\mathfrak {p}}^{\infty }])$ imply that

(4.11) $$ \begin{align} \mathrm{corank}_{{\mathcal {O}}_{F,{\mathfrak{p}}}}{\mathrm{Sel}}_{{\mathfrak{p}}^{\infty}}(A)\le 1. \end{align} $$

ObserveFootnote ⁹

(4.12) $$ \begin{align} \begin{aligned} {\mathrm{Sel}}_{\mathrm{str}}(K_{\infty}, V/T)^{\vee}\otimes {\mathbb {Q}}_p/(\gamma-1)&\cong {\mathrm{Sel}}_{\mathrm{str}}(K, V/T)^{\vee}\otimes {\mathbb {Q}}_p\\&\cong {\mathrm{Sel}}_{\mathrm{str}}({\mathbb {Q}}, A[{\mathfrak{p}}^{\infty}])^{\vee}\otimes_{{\mathcal {O}}_{F_{{\mathfrak{p}}}}} L_{{\mathfrak{p}}}\\ &= \{0\}. \end{aligned} \end{align} $$

Thus, $z_0 \in H^1(K,T) $ is non-torsion by [Reference Agboola and Howard2, Prop. 3.3 (ii)].

We now show

(4.13) $$ \begin{align} 0\neq{\mathrm{loc}}_{{\mathfrak{p}}}(z_0) \in H^1(K_{{\mathfrak{p}}}, V). \end{align} $$

To begin, ${\mathrm {Sel}}_{\mathrm {str}}(K, V/T)$ is finite by (4.12). Put

$$ \begin{align*}p^a=|{\mathrm{Sel}}_{\mathrm{str}}(K, V/T)|.\end{align*} $$

Since $z_0 \in H^1(K,T) $ is non-torsion, pick an integer m such that $ p^{-m+a} z_0 \in H^1(K,V)$ does not lie in the image of $H^1(K, T)$ . Suppose that ${\mathrm {loc}}_{{\mathfrak {p}}}(z_0) =0 \in H^1(K_{{\mathfrak {p}}}, V)$ . Then, as $z_0$ is unramified outside ${\mathfrak {p}}$ , the image $w_m$ of $p^{-m}z_0$ in $H^1(K,V/T)$ lies in ${\mathrm {Sel}}_{\mathrm {str}}(K, V/T)$ and so does $p^aw_m$ . However,

$$ \begin{align*}p^aw_m=0\in H^1(K, V/T).\end{align*} $$

This contradiction yields (4.13).

• • The case $W(\varphi )=+1$ . In view of (3.3) and (4.13), we have $L(\varphi , 1)\neq 0$ . Hence, by the theorem of Coates-Wiles [Reference Coates and Wiles20] and Rubin [Reference Rubin46],

  $$ \begin{align*} \mathrm{ord}_{s=1}L(A,s)=[F: {\mathbb {Q}}]\cdot \mathrm{corank}_{{\mathcal {O}}_{F,{\mathfrak{p}}}}{\mathrm{Sel}}_{{\mathfrak{p}}^{\infty}}(A)=0. \end{align*} $$

• • The case $W(\varphi )=-1$ . In view of (4.13) and Lemma 3.3, the image of $p^{-m}z_0$ in $H^1(K,V/T)$ gives a non-zero element of ${\mathrm {Sel}}(K, V/T)$ for a sufficiently large m. By (4.11), it then follows that

  $$ \begin{align*}\mathrm{corank}_{{\mathcal {O}}_{F,{\mathfrak{p}}}}{\mathrm{Sel}}_{{\mathfrak{p}}^{\infty}}(A)=1.\end{align*} $$
  Hence, the assertion is a consequence of Proposition 4.16.

Proof.

1. (a) In view of the assumption and (4.10), ${\mathrm {Sel}}_{\mathrm {str}}({\mathbb {Q}}, A[{\mathfrak {p}}^{\infty }])$ is finite. So, the assertion directly follows from Theorem 4.17.

2. (b) Recall the exact sequence

   (4.14)

   
   So, by the assumption, we have $A({\mathbb {Q}})\otimes _{{\mathcal {O}}_{F}} F_{{\mathfrak {p}}}/{\mathcal {O}}_{F_{{\mathfrak {p}}}} \cong F_{{\mathfrak {p}}}/{\mathcal {O}}_{F_{{\mathfrak {p}}}}$ . Hence,
   $$ \begin{align*}{\mathrm{loc}}_{{\mathfrak{p}}}: A({\mathbb {Q}})\otimes_{{\mathcal {O}}_{F}} F_{{\mathfrak{p}}}/{\mathcal {O}}_{F_{{\mathfrak{p}}}} \to A({\mathbb {Q}}_p)\otimes_{{\mathcal {O}}_{F}} F_{{\mathfrak{p}}}/{\mathcal {O}}_{F_{{\mathfrak{p}}}}\end{align*} $$
   is a non-zero map.