Proof of Theorem 4.1.1

Proof. [Proof of Theorem 4.1.1] We argue as in the proof of [AS02, Thm. 3.1]. The construction of the map (1) is similar to the one in the proof of [AS02, Lem. 3.6]. We have the commutative diagram

$$\xymatrix{ 0 \ar[r] & B[\ell] \ar[r]\ar[d] & B \ar[d]\ar[r]^{\ell... ... B \ar[d]^{\pi}\ar[r] & 0 \\ 0 \ar[r] & A \ar[r] & C \ar[r] & Q \ar[r] & 0, }$$

where $\psi : B \rightarrow Q$ is the composition of the inclusion $B \hookrightarrow C$ with the quotient map $C \rightarrow Q$ , and the existence of the morphism $\pi : B \rightarrow Q$ follows from the inclusion $B[\ell] \subset {\mathrm{Ker}}(\psi) = A \cap B$ . By naturality for the long exact sequence of Galois cohomology we obtain the following commutative diagram with exact rows and columns

$$\xymatrix{ & M_0\ar[d] & M_1\ar[d]& M_2\ar[d]\\ 0 \ar[r] & B(K)/... ...r]\ar[d] & Q(K) \ar[r] & {{\mathrm{Vis}}_C(\H ^1(K,A))} \ar[r] & 0\\ & M_3. }$$

Here, $M_0$ , $M_1$ and $M_2$ denote the kernels of the corresponding vertical maps and $M_3$ denotes the cokernel of the first map. Since $R$ preserves $A$ , $B$ , and $B[\ell]$ , all objects in the diagram are $R$ -module and the morphisms of abelian varieties are also $R$ -module homomorphisms.

The snake lemma yields an exact sequence

$$0 \to M_0\rightarrow M_1 \rightarrow M_2 \rightarrow M_3.$$

By hypothesis, $B(K)[\mathfrak{m}]=0$ , so $N_0={\mathrm{Ker}}(B(K) \to C(K)/A(K))$ has no  $\mathfrak{m}$ torsion. Noting that $B(K)[\ell] \subset N_0$ , it follows that $M_0=N_0/(B(K)[\ell])$ has no  $\mathfrak{m}$ torsion either, by Lemma 4.2.2. Also, $M_1[\mathfrak{m}]=0$ again since $B(K)[\mathfrak{m}]=0$ .

By the long exact sequence on Galois cohomology, the quotient $C(K)/B(K)$ is isomorphic to a subgroup of $Q(K)$ and by hypothesis $Q(K)[\mathfrak{m}]=0$ , so $(C(K)/B(K))[\mathfrak{m}]=0$ . Since $Q$ is isogenous to $A$ and $A(K)$ is finite and $C(K)/B(K) \hookrightarrow Q(K)$ , we see that $C(K)/B(K)$ is finite. Thus $M_3$ is a quotient of the finite $R$ -module $C(K)/B(K)$ , which has no $\mathfrak{m}$ -torsion, so Lemma 4.2.2 implies that $M_3[\mathfrak{m}]=0$ . The same lemma implies that $M_1/M_0$ has no $\mathfrak{m}$ -torsion, since it is a quotient of the finite module $M_1$ , which has no $\mathfrak{m}$ -torsion. Thus, we have an exact sequence

$$0 \to M_1/M_0 \to M_2 \to M_3 \to 0,$$

and both of $M_1/M_0$ and $M_3$ have trivial $\mathfrak{m}$ -torsion. It follows by Lemma 4.2.2, that $M_2[\mathfrak{m}]=0$ . Therefore, we have an injective morphism of $R/\mathfrak{m}$ -vector spaces

$$\varphi : (B(K)/\ell{}B(K))[\mathfrak{m}]\hookrightarrow {\mathrm{Vis}}_C(H^1(K,A))[\mathfrak{m}].$$

It remains to show that for any $x \in B(K)$ , we have $\varphi (x) \in {\mathrm{Vis}}_C({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, A))$ , i.e., that $\varphi (x)$ is locally trivial.

For real archimedian places tex2html_wrap_inline$v$ the cohomology group tex2html_wrap_inline$H^1(K_v/K_v,A)$ is trivial. For complex archimedian places, every cohomology class has order 2 since tex2html_wrap_inline$Gal(K_v/K) &cong#cong;Gal(C/R) &cong#cong;Z/2Z$ and the order of any cohomology class divides the order of the group []. Since tex2html_wrap_inline$res_v(&phiv#varphi;(&pi#pi;(x)))$ is also tex2html_wrap_inline$&ell#ell;$-torsion and tex2html_wrap_inline$&ell#ell;$ is odd (since tex2html_wrap_inline$1&le#leq;e < &ell#ell;- 1$), then tex2html_wrap_inline$res_v(&phiv#varphi;(&pi#pi;(x))) = 0$.

Let tex2html_wrap_inline$v$ be a non-archimedian place for which chartex2html_wrap_inline$(v) &ne#ne;&ell#ell;$. If tex2html_wrap_inline$m = c_B,v$ denotes the Tamagawa number at tex2html_wrap_inline$v$ for tex2html_wrap_inline$B$, then the reduction of tex2html_wrap_inline$mx$ lands in the identity component of the closed fiber of the Néron model of tex2html_wrap_inline$B$. The field tex2html_wrap_inline$K_v^ur$ is the fraction field of a strictly Henselian discrete valuation ring, so we can apply Proposition  to obtain a point tex2html_wrap_inline$z &isin#in;B(K^ur_v)$, such that tex2html_wrap_inline$mx = &ell#ell;z$. The cohomology class tex2html_wrap_inline$res_v(&pi#pi;(mx))$ is represented by the 1-cocycle tex2html_wrap_inline$&xi#xi;: Gal(K_v/K_v) &rarr#rightarrow;A(K_v^ur )$, given by tex2html_wrap_inline$&sigma#sigma;&map#mapsto;&sigma#sigma;(z)-z &isin#in;A(K_v^ur)$. It follows that tex2html_wrap_inline$[&xi#xi;]$ is an unramified cohomology class, i.e., tex2html_wrap_inline$[&xi#xi;] &isin#in; H^1(K_v^ur / K_v ,A(K_v^ur))$, i.e., tex2html_wrap_inline$res_v(&pi#pi;(mx))$ is unramified.

We proceed exactly as in Section 3.5 of [AS05]. In both cases char$(v)\neq \ell$ and char$(v)=\ell$ we arrive at the conclusion that the restriction of $\varphi (x)$ to $\H ^1(K_v,A)$ is an element $c\in \H ^1(K_v^{{\mathrm{ur}}}/K_v, A(K_v^{{\mathrm{ur}}}))$ . (Note that in the case char$(v)\neq \ell$ the proof uses our hypothesis that $\ell\nmid \char93 \Phi_{B,v}(k_v)$ .) By [Mil86, Prop I.3.8], there is an isomorphism

$$\H ^1(K_v^{{\mathrm{ur}}}/K_v, A(K_v^{{\mathrm{ur}}})) \cong \H ^1(\overline{k}_v/k_v, \Phi_{A,v}(\overline{k}_v)).$$ (2)

We will use our hypothesis that

$$\Phi_{A,v}(k_v)[\mathfrak{m}] = \Phi_{B,v}(k_v)[\ell] = 0$$

for all $v$ of bad reduction to deduce that the image of $\varphi$ lies in ${\mathrm{Vis}}_C({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, A))[\mathfrak{m}]$ . Let $d$ denote the image of $c$ in $\H ^1(\overline{k}_v/k_v, \Phi_{A,v}(\overline{k}_v))$ . The construction of $d$ is compatible with the action of $R$ on Galois cohomology, since (as is explained in the proof of [Mil86, Prop. I.3.8]) the isomorphism (2) is induced from the exact sequence of ${\mathrm{Gal}}(K_v^{{\mathrm{ur}}}/K_v)$ -modules

$$0 \to \mathcal{A}^0(K_v^{{\mathrm{ur}}}) \to \mathcal{A}(K_v^{{\mathrm{ur}}}) \to \Phi_{A,v}(\overline{k}_v)\to 0,$$

where $\mathcal{A}$ is the Néron model of $A$ and $\mathcal{A}^0$ is the subgroup scheme whose generic fiber is $A$ and whose closed fiber is the identity component of $\mathcal{A}_{k_v}$ . Since $\varphi (x)\in \H ^1(K,A)[\mathfrak{m}]$ , it follows that

$$d \in \H ^1(\overline{k}_v/k_v, \Phi_{A,v}(\overline{k}_v))[\mathfrak{m}].$$

Lemma 4.2.5, our hypothesis that $\Phi_{A,v}(k_v)[\mathfrak{m}]= 0$ , and that

$$\H ^1(\overline{k}_v/k_v, \Phi_{A,v}(\overline{k}_v)) = \varinjlim \H ^1({\mathrm{Gal}}(k_v'/k_v),\Phi_{A,v}(k_v'))),$$

together imply that $\H ^1(\overline{k}_v/k_v, \Phi_{A,v}(\overline{k}_v))[\mathfrak{m}]= 0$ , hence $d=0$ . Thus $c=0$ , so $\varphi (x)$ is locally trivial, which completes the proof. $\qedsymbol$