Currently, I am reading D. Bertrand's paper Galois representation and transcendental numbers (here is the link to the paper), and I have a question about Theorem 2 (on Kummer theory of semi-abelian varieties).
Theorem 2. Let $G = A \times L$, and let $P$ be a point in $G(k)$. For all prime numbers $\ell$, the group $\chi_{(\ell)}(P)$ is commensurable with $T_\ell(G^o_P)$, and coincides with $T_\ell(G^o_P)$ when $\ell$ is sufficiently large.
The proof of Theorem 2 follows Ribet's method [60] (see also [37],[38]): since the representations $\rho_{(\ell)}$, $\rho_\ell$ are semi-simple and satisfy Tate's conjecture [30], and in view of the finiteness or vanishing of the cohomology groups $H^1(\mathcal{G}_{(\ell)}, T_\ell(G))$ ([62], [21], [66]), one is reduced to showing that an endomorphism $\alpha$ of $G_P$ sends $P$ to a highly divisible point in $G_P(k)$ if and only if $\alpha$ itself is highly divisble in $\text{End}(G_P$; in practice, this will mean that $\alpha = 0$. An effective version of this statement requires a precise description of the groups $G_P$, and we return to this point in Theorem 8 below.
He said: "one is reduced to showing that ...". I do not understand how to reduce the theorem to this claim. Do you have any hints or references? He also said that this Theorem can be proved using Ribet's method (in the paper Kummer theories on extensions of abelian varieties by tori, you can see the paper here), but as I understand, in Ribet's paper, he proves claims involving almost all primes, while Theorem 2 involves every prime.
Thank you.
Edit: I have found a paper M .Hindry here where he sketches a proof of a more general result (appendix 2 in the paper).