What is a standard reference for Kummer theory of semi-Abelian varieties ? I need a complete exposition with detailed proofs. Also in prime characteristic, although I am not sure what the statement there exactly is.

Below I give an example of what I mean by Kummer theory in zero characteristic.

Let $G = A\times L$ be a product of an Abelian variety by a torus L so that after a finite extension of k it satisfies Poincare's complete reducibility theorem (as a variety over k).

Let $l$ be a prime. Let $G_l=\{ x \in > G(\bar k) : \exists n l^nx=0\}$ be the $l$-torsion of G. For a point $P\in > G(\bar k)$, let $G_P$ be the smallest algebraic subgroup of G containing P, i.e. Zariski closure of subgroup ${\Bbb Z}P$ of $G$, and let $G_P$ be its connected component through the origin,

$$\zeta_l(P) : Gal(\bar > k/k(G_l,P))\longrightarrow T_l(A\times > L)$$ $$\sigma \mapsto > \sigma(P_l)-P_l$$ where $P_l$ is a compatible sequence of division points, $P_1 = P$. By Kummer theory I mean the statement that ([Bertrand, Theorem 2]) the image of of the map $\prod \zeta_l$ is of finite index in $NT(G_P^o)$ for a large $N$.

Above is taken from Bertrand, D.: Galois representations and transcendental numbers. In: New advances in transcendence theory (Durham, 1986), Cambridge University Press, Cambridge (1988); but proofs there are a little too sketchy for me. 13. Ribet, K.: Kummer theory on extensions of varieties by tori. Duke Math. J. 46(4), (1979) has a slightly weaker statement.