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Asvin
Asvin
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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:

$$|A(\mathbb F_q)| = 1 + \sum_{\ell} (a_\ell - 1)$$$$|A(\mathbb F_q)| = \prod_{\ell} (a_\ell)$$ 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.

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:

$$|A(\mathbb F_q)| = 1 + \sum_{\ell} (a_\ell - 1)$$ 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.

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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Asvin
Asvin
  • 7.7k
  • 2
  • 21
  • 52

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:

$$|A(\mathbb F_q)| = \sum_{\ell} (a_\ell - 1)$$$$|A(\mathbb F_q)| = 1 + \sum_{\ell} (a_\ell - 1)$$ 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.

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:

$$|A(\mathbb F_q)| = \sum_{\ell} (a_\ell - 1)$$ 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.

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:

$$|A(\mathbb F_q)| = 1 + \sum_{\ell} (a_\ell - 1)$$ 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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Asvin
Asvin
  • 7.7k
  • 2
  • 21
  • 52

A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field

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:

$$|A(\mathbb F_q)| = \sum_{\ell} (a_\ell - 1)$$ 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.