Timeline for Abel-Jacobi map isomorphism galois representations

3 events

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Dec 5, 2013 at 6:20	comment	added	user76758		Of course, to argue by passage to char. 0 you have to assume $\ell$ is not the characteristic (so restrict to abelian connected Galois covers with degree not divisible by the characteristic).
Dec 5, 2013 at 5:58	comment	added	user76758		This is true over any separably closed field with any $\ell$. The key is that it is true with coefficients in any finite abelian group: every abelian connected finite Galois cover of $X$ arises by $\phi$-pullback from a unique such cover of the Jacobian $J$. Milne discusses this with references in his article on Jacobians in the book "Arithmetic Geometry". Or use deformation of curves and smooth/proper base change to reduce to char. 0, and then to $\mathbf{C}$ where it reduces to determining the analytification of $(J, \phi)$ in terms of that of $X$.
Dec 5, 2013 at 5:34	history	asked	LMN	CC BY-SA 3.0