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Let $A$ be an abelian variety over a finite field $k$ and $\ell$ a prime number. Fix a natural number $n\in \mathbb{N}$ and denote by $k_{n}$ the field $k(A[\ell^{n}](\overline k))$. Let $Q\in A[\ell^{\infty}](\overline k):=\bigcup_{m\in \mathbb{N}}A[\ell^{m}](\overline k)$ be an arbitrary point.

My question is the following:

Suppose $Q\notin A(k_{n})$. Is there an element $\sigma \in \operatorname{Gal}(\overline k/k_{n})$ such that $Q^{\sigma}-Q \in A(k_{n})\setminus \{0\}$?

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  • $\begingroup$ Any element of the Galois group which fixes $Q$. $\endgroup$ Commented Jun 26 at 6:24
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    $\begingroup$ I am not sure you are asking the right question. You might be rather interested in Prop. 3 in people.maths.ox.ac.uk/rossler/mypage/pdf-files/mma.pdf, which actually works over any perfect field if the prime number $p$ which appears there is prime to the characteristic. $\endgroup$ Commented Jun 26 at 10:09
  • $\begingroup$ @DamianRössler, Prof. Rössler, thank you very much! This result is a special case of Lemma 1 in Boxall's paper, "Autour d'un problème de Coleman," which you mentioned to me earlier. However, Boxall did not include a proof for this lemma. Instead, he established a more general result, in which the integer $\ell$ is not necessarily a prime number. On the other hand, it seems that Proposition 3 in your paper also works for any natural numbers prime to the characteristic (i.e., not only for prime numbers). $\endgroup$ Commented Jun 27 at 6:00
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    $\begingroup$ @aya Yes this is correct! $\endgroup$ Commented Jun 27 at 16:30

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