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(edited for clarity)

In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 curves over a base curve X, then (to compute the cohomology of X with coefficients in the relevant local system) you have to pass to a 125-fold covering of X and compute the Jacobian of that beast."

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.

(edited for clarity)

In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 curves over a base curve X, then (to compute the cohomology of X with coefficients in the relevant local system) you have to pass to a 125-fold covering of X and compute the Jacobian of that beast."

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.

In a comment on a response to this question, 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.

(edited for clarity)

In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 curves over a base curve X, then (to compute the cohomology of X with coefficients in the relevant local system) you have to pass to a 125-fold covering of X and compute the Jacobian of that beast."

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.