Etale cohomology of (semi)abelian varieties

This is joint work with Alexander Petrov. We give an elementary calculation of the first potentially non-zero differential of the Hochschild-Serre spectral sequence computing the étale cohomology of a (semi)abelian variety. We give an example of a genus two curve over the field of rational numbers for which this differential is non-zero, using the non-divisibility by 2 of the Bockstein image of its torsor of theta-characteristics. We also point out that this spectral sequence degenerates at the second page for the Jacobians of curves with trivial Albanese torsor.