Let $A$ be a principally polarised abelian variety over $\mathbb{C}$ of dimension $g$. Let $K$ be the field of moduli of $A$.

Proposition. $A$ has a model defined over an extension $L$ of $K$ such that $[L:K]$ is bounded above by a function of $g$.

This follows from the fact that "principally polarised abelian varieties of dimension $g$ with full level-$3$ structure" have a fine moduli space $\mathcal{A}_g(3)$. Hence the proposition holds with the bound being the degree of the forgetful map $\mathcal{A}_g(3) \to \mathcal{A}_g$.

Is the proposition written down anywhere that I could cite? I assume it is well-known but I have not been able to find it. A referee has asked me to provide a reference.

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    $\begingroup$ Why not just include the proof as you have it written in your question? That will be much more useful to the reader than a reference (which they probably will not lookup, and then just end up confused unless they know already that it's as easy as your proof makes it sound). $\endgroup$ – John Pardon May 28 '13 at 17:54
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    $\begingroup$ Actually, you don't even need the existence of a fine moduli space (which, after all, is quite hard to prove). The fact that a polarized abelian variety with full level-3 structure has no automorphisms implies that it is defined over its field of moduli, which is of bounded degree over the field of moduli of the polarized abelian variety itself. $\endgroup$ – anon May 29 '13 at 1:47
  • $\begingroup$ Yes, I shall just include the proof (with anon's simplification). I was concerned it would distract the reader from the main ideas of the proof in which I use the Proposition but I suppose that it is short enough not to worry about this, especially if I avoid the heavy machinery of a fine moduli space. $\endgroup$ – Martin Orr May 29 '13 at 11:36

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