let $k$ be a global field, that is, a number field of finite degree over $\bbb q$ or the function field of a smooth projective curve $c$ over a finite field $f$. let $x$ be a smooth projective variety over $k$, and let $k$ be the maximal cyclotomic extension of $k$, obtained by adjoining all roots of unity. if $x$ is an abelian variety, a famous theorem, due to ribet in the number field case and lang-neron in the function field case when $x$ has trace zero over the constant subfield of $k$, asserts that the torsion subgroup of the mordell-weil group of $x$ over $k$ is finite. denoting by $k^{sep}$ a separable closure of $k$, this result is equivalent to finiteness of the fixed part by $g=gal(k^{sep}/k)$ of the etale cohomology group $h^1(x_{k^{sep}},\bbb q/\bbb z)$, where we ignore the $p$-part in positive characteristic $p$. in a recent paper, roessler-szamuely generalize this result to all odd cohomology groups. the trace zero assumption in the function field case is replaced by a ''large variation'' assumption on the characteristic polynomials of frobenius acting on the cohomology of the fibres of a morphism $f: \mathcal{x}\to c$ from a smooth projective variety $\mathcal{x}$ over a finite field to $c$ with generic fibre $x$. in this talk, i will discuss the case of even degree, proving some positive results in the number field case and negative results in the function field case.