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Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \neq 0$ in $\mathbb F_q$ if necessary.

Is there a Lefschetz style trace formula relating $A_\ell$ to the trace of the Frobenius on some cohomology groups? If we let $a_\ell = |A_\ell|$, then note that:

$$|A(\mathbb F_q)| = \prod_{\ell} (a_\ell)$$ where the right hand side sums over all primes and only finitely many terms are non zero. The left hand side has an interpretation in terms of the action of the Frobenius on the Etale cohomology groups and it would be great if any such relation also lifted to the level of cohomology groups.

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  • $\begingroup$ Shouldn't the formula involve a product of the $a_\ell$'s? Finite abelian groups are products of the $\ell$-primary parts, and the cardinality of a finite set multiplies under products. Once you make that switch, you can recover $a_\ell$ as from the $\ell$-adic valuation of the trace of Frobenius. $\endgroup$
    – Anonymous
    Commented Mar 31, 2020 at 16:44
  • $\begingroup$ Thanks for the correction! Indeed, I did consider the valuations and it is useful in my context but I would still be interested in a lift of this equality to the level of cohomology. $\endgroup$
    – Asvin
    Commented Mar 31, 2020 at 21:14

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