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If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property?

For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an extension $L/K$ with $[L:K]\le B$ such that $A(L)$ is Zariski-dense in $A$.

Remarks:

  1. Of course, for $n=1$ we can take $B=2$. For arbitrary $n$, is it even possible to choose $B$ depending only on $n$?

  2. This is reminiscent of this previous MO question.

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    $\begingroup$ I think we could follow the same construction as for elliptic curves to yield in fact a $B$ depending only on $n$: 1. Reduce to the case that $A$ is simple; 2. Present $A$ as a branched cover $\pi: A \to \mathbb{P}_K^n$ with degree bounded only in terms of $n$; 3. Observe that $\pi^{-1}(\mathbb{P}^n(K))$ contains a non-torsion point $P$; 4. Then $[K(P):K] \leq \deg{\pi}$, and the group $\langle P \rangle \subset A(K(P))$ is Zariski-dense. $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 3, 2015 at 18:33
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    $\begingroup$ @VesselinDimitrov: How do you carry out step 2.? $\endgroup$
    – O-Ren Ishii
    Commented Sep 3, 2015 at 18:40
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    $\begingroup$ Yes, this argument only gives a $B$ that depends on $n$ and on the minimum degree of a polarization of $A$ over $K$. $\endgroup$
    – Vesselin Dimitrov
    Commented Sep 3, 2015 at 19:19
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    $\begingroup$ @VesselinDimitrov: For abelian varieties $A$ and $A'$, and subsets $S \subset A(L)$ and $S' \subset A'(L)$, the closure of $S \times S'$ in $(A \times A')_L$ is $\overline{S} \times \overline{S}'$. So to ensure $A(L)$ is Zariski-dense in $A_L$ (surely that is what is meant, not "Zariski-dense in $A$", right?) it suffices that $L$ works the same way for $(A \times A^{\vee})^4$. Hence, by Zarhin's trick (and allowing any dimension) it seems we can assume there is a principal polarization. Am I overlooking something? $\endgroup$
    – grghxy
    Commented Sep 3, 2015 at 22:02
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    $\begingroup$ By a theorem of Zarhin, if $A'$ is the dual of $A$, then $(A\times A')^4$ admits a principal polarization. So the principally polarized case (in dimension $8n$) implies the general case. $\endgroup$
    – Laurent Moret-Bailly
    Commented Sep 3, 2015 at 22:02

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