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Let's assume we have a $G$-torsor $X_{1} \to X_{2}$, where $G$ is a finite abelian group, and both $X_{1}$ and $X_{2}$ are defined over $\text{Spec}(\mathbb{Z})$. Is there an easy way to compute $\#X_{2}(\mathbb{F}_{p^{n}})$ in term of $\#X_{1}(\mathbb{F}_{p^{n}})$? Any references to maybe help me work something like this out?

I'd like to know the above question in some amount of generality. But for example, if $A_{1}$ and $A_{2}$ are two "$k$-isogenous" abelian varieties over a finite field $k$, then

$$\#A_{1}(K) = \#A_{2}(K)$$

for all finite extensions $K$ over $k$. To see this result, see Corollary 2.3 here (http://virtualmath1.stanford.edu/~conrad/mordellsem/Notes/L03.pdf). I'm not sure I fully understand what "$k$-isogenous" means. If $A_{1}$ and $A_{2}$ are isogenous over $\mathbb{Q}$, are they isogenous over $\mathbb{F}_{p^{n}}$ for all prime powers?

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    $\begingroup$ $\#G \#X_2(\mathbb{F}_q) = \sum \#X_1^{\sigma}(\mathbb{F}_q)$ where the $X_1^{\sigma}$ are the twists of $X_1$ over $\mathbb{F}_q$. I wouldn't expect much more than that in general. $\endgroup$
    – Felipe Voloch
    Commented Nov 29, 2018 at 7:28
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    $\begingroup$ "$k$-isogenous" = there is an isogeny between them defined over $k$. $\endgroup$
    – Chris Wuthrich
    Commented Nov 29, 2018 at 9:22
  • $\begingroup$ @FelipeVoloch Thanks for your comment. I'm a little out of my comfort zone here, so maybe I can ask if my idea of twists is correct. A twist $X_{1}^{\sigma}$ should be $X \times \sigma /G$ for a $G$-torsor $\sigma \to \text{Spec}(\mathbb{F}_{p^{n}})$. Is that product and quotient the usual ones, or is there something subtle here? Finally, the sum you write is over $\sigma \in H^{1}(\mathbb{F}_{p^{n}}, G)$, correct? $\endgroup$
    – Benighted
    Commented Nov 29, 2018 at 22:13
  • $\begingroup$ Your notation in the first question of your comment is a bit ambiguous but I think it's correct. For the second question, yes, the twists are described by the $H^1$. $\endgroup$
    – Felipe Voloch
    Commented Nov 30, 2018 at 5:38

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