In a comment on a response to [this question][1], the following result is vaguely stated:  Suppose $X/\mathbb{Q}$ is a surface which fibers into genus 2 curves over a base curve $C/\mathbb{Q}$; then computing $H^2_{et}(X)$ is equivalent to computing $H^1_{et}$ of a 125-fold cover of $C$.  Would someone be willing to explain this further?  Given the importance of etale cohomology and the difficulties in working with it, I would really love to see the details of this spelled out.


  [1]: http://mathoverflow.net/questions/7318/etale-cohomology-and-l-adic-tate-modules