Let $G$ be an abelian variety defined over a ring $R$. Is there a natural number $n$ such that, for any field $k$ over $R$ and any $G\_k$-torsor $T$, there exists an extension $L/k$ of degree $n$ for which $T(L)$ is not empty?



Triviality of torsors after a field extension of bounded degree
This is not even true for elliptic curves over $\mathbb Q$ or $\mathbb Q\_p$. For example, by Tate duality the group $H^1(\mathbb Q\_p,E)$ is dual to $E(\mathbb Q\_p)$, which is an infinite group. This shows that the period can be arbitrarily large, and then one can use the fact that the period divides the index, to show that the index can be arbitrarily large.


