Proof of Theorem 3.1

Proof. [Proof of Theorem 3.1] The proof proceeds in two steps. The first step is to use the hypothesis that $B[n] \subset A$ to produce a map $B(K)/n B(K)\rightarrow \Vis_J(H^1(K,A))[n]$. This was done in Section 3.3. The second step is to perform a local analysis at each place $v$ of $K$ in order to prove that the image of this map consists of locally-trivial cohomology classes. We divide this local analysis into three cases:

1. When $v$ is real archimedian, we use that $\gcd(2,n)=1$. (We know that for any $p\mid n$ we have $p>2$ because $1\leq e_p<p-1$, by assumption.)
2. When $\gcd(\ch(v),n)=1$, we use the result of Section 3.2 and a relationship between unramified cohomology and the cohomology of a component group.
3. When $\gcd(\ch(v),n)\neq 1$, for each prime $p\mid n$, the reduction of $J$ is abelian and by hypothesis $e_p<p-1$, so we can apply an exactness theorem from [BLR90].

We now deduce that the image of $B(K)/nB(K)$ in $H^1(K,A)$ lies in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$. Fix an element $x\in B(K)$. To show that $\pi(x)\in {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$, it suffices to show that $\res_v(\pi(x))=0$ for all places $v$ of $K$.

Case 1. $\mathbf{v}$ real archimedian: At a real archimedian place $v$, the restriction $\res_v(\pi(x))$ is killed by $2$ and the odd $n$, hence $\res_v(\pi(x))=0$.

Case 2. $\gcd(\ch(v),n)=1$: Suppose that $\gcd(\ch(v),n)=1$. Let $m=c_{B,v}= \Phi_{B,v}(\mathbf{F}_v)$ be the Tamagawa number of $B$ at $v$. The reduction of $m x$ lies in the identity component of the closed fiber $\mathcal{B}_{\mathbf{F}_v}$ of the Néron model of $B$ at $v$, so by Lemma 3.4, there exists $z \in B(K_v^{\ur})$ such that $n z= m x$. Thus the cohomology class $\res_v(\pi(m x))$ is defined by a cocycle that sends $\sigma \in \Gal(\overline{K_v}/K_v)$ to $\sigma(z) - z \in A(K_v^{\ur})$ (see diagram (3.2) for the definition of $\pi$). In particular, $\res_v(\pi(m x))$ is unramified at $v$. By [Mil86, Prop. 3.8],

$$H^1(K_v^{\ur}/K_v,A(K_v^{\ur})) =H^1(K_v^{\ur}/K_v,\Phi_{A,v}(\overline{\mathbf{F}}_v)),$$

where $\Phi_{A,v}$ is the component group of $A$ at $v$. The Herbrand quotient of a finite module is $1$ (see, e.g., [Ser79, VIII.4.8]), so

$$\char93 \Phi_{A,v}(\mathbf{F}_v) = \char93 H^1(K_v^{\ur}/K_v,\Phi_{A,v}(\overline{\mathbf{F}}_v)).$$

Thus the order of $\res_v(\pi(m x))$ divides both $\char93 \Phi_{A,v}(\mathbf{F}_v)$ and $n$. Since by assumption $\gcd(\char93 \Phi_{A,v}(\mathbf{F}_v), n)=1$, it follows that $\res_v(\pi(m x))=0$, hence $m\res_v(\pi(x))=0$. Again, since the order of $\pi(x)$ divides $n$, and $\gcd(n,m)=1$, we have $\res_v(\pi(x))=0$.

Case 3. $\gcd(\ch(v),n)=p\neq 1$: Suppose that $\ch(v)=p\mid n$. Let $R$ be the ring of integers of $K_v^{\ur}$, and let $\mathcal{A}$, $\mathcal{J}$, and $\mathcal{C}$ be the Néron models of $A$, $J$, and $C$, respectively. Since $e_p<p-1$ and $J$ has abelian reduction at $v$ (since $p\nmid N$), by [BLR90, Thm. 7.5.4(iii)], the induced sequence $0\rightarrow \mathcal{A}\rightarrow \mathcal{J}\stackrel{\phi}{\rightarrow } \mathcal{C}\rightarrow 0$ is exact, which means that $\phi$ is faithfully flat and surjective with scheme-theoretic kernel $\mathcal{A}$. Since $\phi$ is faithfully flat with smooth kernel, $\phi$ is smooth (see, e.g., [BLR90, 2.4.8]). By Lemma 3.6, $\mathcal{J}(R) \rightarrow \mathcal{C}(R)$ is a surjection; i.e., $J(K_v^{\ur}) \rightarrow C(K_v^{\ur})$ is a surjection.

So $\res_v(\pi(x))$ is unramified, and again by [Mil86, Prop. 3.8],

$$H^1(K_v^{\ur}/K_v,A) \cong H^1(K_v^{\ur}/K_v,\Phi_{A,v}(\overline{\mathbf{F}}_v)).$$

But $H^1(K_v^{\ur}/K_v,\Phi_{A,v}(\overline{\mathbf{F}}_v))=\{0\}$, since $\Phi_{A,v}(\overline{\mathbf{F}}_v)$ is trivial, as $A$ has good reduction at $v$ (because $p\nmid N$). Thus $\res_v(\pi(x))=0$. $\qedsymbol$