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Suppose $A$ is an abelian variety over $\overline{\mathbb{Q}}$ of dimension $g$, such that $A$ is isomorphic to all of its Galois conjugates. Note that I'm not including any polarization data.

Can I conclude that $A$ descends to $\mathbb{Q}$, or at least to a field $K$ of bounded degree over $\mathbb{Q}$?

Thanks!

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    $\begingroup$ Your assumption is that the field of moduli is $\mathbb Q$, but it is insufficient to ensure that the field of definition is $\mathbb Q$. There is an obstruction in a $H^2$. $\endgroup$
    – Niels
    Commented Jul 27, 2017 at 12:04
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    $\begingroup$ Yeah, right! My question is how serious the obstruction is basically. $\endgroup$
    – jacob
    Commented Jul 27, 2017 at 12:37

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1.Theorem (Shimura 1971, "On the Field of Rationality for an Abelian Variety"): If $g$ = dim $A$ is odd and $Aut(A)=\{\pm1\}$ then $A$ has a model over its field of moduli. Proof: $\{1\} = H^1(\overline{\mathbb{Q}}^*) \rightarrow H^2(\pm1) \rightarrow H^2(\overline{\mathbb{Q}}^*)$ and $\{\pm1\}$ acts faithfully on $H^0(A_{\overline{\mathbb{Q}}}, \Omega^g) = \overline{\mathbb{Q}}$ since $g$ is odd, so the image of the obstruction in $H^2(Aut(H^0(A_{\overline{\mathbb{Q}}}, \Omega^g))) = H^2(\overline{\mathbb{Q}}^*)$ can be identified as the obstruction to the descent of a one-dimenstional $\overline{\mathbb{Q}}$ vector space to $\mathbb{Q}$, which is therefore trivial.

2.Of interest:

Maus 1973, "On the Field of Moduli of an Abelian Variety with Complex Multiplication"

Shimura 1982, "Models of an Abelian Variety with Complex Multiplication Multiplication over Small Fields"

Shioda 1977, "Some Remarks on Abelian Varieties"

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  • $\begingroup$ I checked the paper, and it looks like Shimura's theorem deals with polarized abelian varieties. $\endgroup$
    – jacob
    Commented Jul 27, 2017 at 18:50
  • $\begingroup$ 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). $\endgroup$
    – David Lampert
    Commented Jul 27, 2017 at 20:36
  • $\begingroup$ 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. $\endgroup$
    – jacob
    Commented Jul 28, 2017 at 0:53
  • $\begingroup$ 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 $H^2(Aut(A))$. $\endgroup$
    – David Lampert
    Commented Jul 28, 2017 at 1:40

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