Let $E\to B$ be a locally trivial fibration of curves of genus >1 (everyhting is compact and over $\mathbb{C}$). I read that this fibration becomes trivial "after a finite unramified base change". Could anyone explain what does it mean, i.e. what is the exact construction?

  • $\begingroup$ It becomes trivial after you perform a base change by the finite cover of symplectic bases for the $3$-torsion of your Abelian scheme. $\endgroup$ – Jason Starr Apr 14 '17 at 11:22
  • $\begingroup$ You glue such a bundle together by transition maps: automorphisms of the fiber. But the automorphism group is finite. So after a suitable cover, you are gluing together by the identity automorphism. $\endgroup$ – Ben McKay Apr 16 '17 at 11:53

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