Timeline for Do Abelian varieties isomorphic to all their conjugates descend?

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Jul 28, 2017 at 1:40	comment	added	David Lampert		It's from the long exact Galois cohomology sequence of $1 \rightarrow \{\pm1\} \rightarrow \overline{\mathbb{Q}}^* \rightarrow \overline{\mathbb{Q}}^* \rightarrow 1$. The descent obstruction is in $H^2(Aut(A))$.
Jul 28, 2017 at 0:53	comment	added	jacob		Ah, of course, you're right. Can you explain your exact sequence at the beginning and where the obstruction lives exactly? I ultimately care about A=E^r, so it has a gigantic automorphism group unfortunately.
Jul 27, 2017 at 20:36	comment	added	David Lampert		Doesn't the argument apply to unpolarized varieties? Shimura refers to generic polarized varieties, and a polarization is required to define a moduli variety. Note Shioda's "Question 2" with the example of elliptic curves $E_1,E_2$ defined over $\mathbb{Q}(i)$ such that $E_1 \times E_1 \ncong E_1 \times E_2$ over $\mathbb{C}$ but the reductions are isomorphic over $\mathbb{F}_{p^2}$ for all $p \equiv 3$ mod(4).
Jul 27, 2017 at 18:50	comment	added	jacob		I checked the paper, and it looks like Shimura's theorem deals with polarized abelian varieties.
Jul 27, 2017 at 18:37	history	edited	David Lampert	CC BY-SA 3.0	deleted 109 characters in body
Jul 27, 2017 at 18:32	history	answered	David Lampert	CC BY-SA 3.0