If I recall the sum appeared when I was looking at some questions from p-adic transcendence theory. We cannot prove the p-adic four exponentials conjecture, but I tried to show that for a given set of 4 numbers the statement of the conjecture holds for almost all p. One stumbling block was to estimate the sum given above. I don't remember the details.

I am sorry for being vague. I stopped doing mathematics two years ago and I have forgotten much. I guess I mean "average" in the sense that the sum $\sum_{p} p^{1-\epsilon}/\gamma_p$ converges for any $\epsilon > 0$.

@Qing Liu: I am sorry if the question was not stated well. I know that for a general semi-abelian scheme what I ask is false. I am only interested in the case when the scheme is an extension of an abelian scheme by a torus.

I have seen a generalization of this statement in several papers on 1-motives, however no proof or reference is given there. For example: Deligne [Hodge III, 10.1.10] or M. Raynaud, [1-Motifs et Monodromie Géométrique, 3.1]. This is why I think that it should be true.

The statement should be true for an arbitrary scheme but I would be happy with an answer for $S= Spec R$ when $R$ is the ring of integers of a finite $\mathbb{Q}_p$-extension.

That is exactly what I was looking for. Thanks a lot!