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Let $A \to S$ be an abelian scheme of relative dimension $g$ over an algebraically closed field of characteristic $p$. We have the following short exact sequence on $S$ $$0 \to R^i\pi_*(\mathbb{Z}_l) \xrightarrow{\cdot l} R^i\pi_*(\mathbb{Z}_l) \to R^i \pi_*(\mu_l) \to 0. $$ Taking cohomology for $i=2g-1$ gives $$ 0 \to H^0(S, R^{2g-1}\pi_*(\mathbb{Z}_l)) \xrightarrow{\cdot l} H^0(S, R^{2g-1}\pi_*(\mathbb{Z}_l)) \to H^0(S, R^{2g-1}\pi_*(\mu_l))\\ \xrightarrow{c} H^1(S, R^{2g-1}\pi_*(\mathbb{Z}_l)) \xrightarrow{\cdot l} H^1(S, R^{2g-1}\pi_*(\mathbb{Z}_l)) \to \ldots $$

The group $H^0(S, R^{2g-1}\pi_*(\mu_l))$ can be identified with the set of $l$-torsion sections

I have two questions:

  1. Is it true that the connecting map $c$ maps a torsion section $s$ to the image of the difference of the cohomology class of this torsion section with that of the zero section (i.e. [s]-[0]) in $H^1(S, R^{2g-1}\pi_*(\mathbb{Z}_l))$, in the Leray filtration of $H^{2g}(A, \mathbb{Z}_l)$?

  2. Is there a similar description for $p$-torsions using crystalline cohomology?

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  • $\begingroup$ When $g$ equals $1$, the sheaf $R^{g-1}\pi_*(\mu_\ell)$ equals $\mu_\ell$. If $S$ is integral, $H^0(S,\mu_\ell)$ is just $\mu_\ell$. How is $\mu_\ell$ identified with the set of $\ell$-torsion sections of $A\to S$ in this case? $\endgroup$
    – Jason Starr
    Commented Jul 31, 2017 at 14:14
  • $\begingroup$ @JasonStarr Sorry it is a typo! It should be 2g-1. $\endgroup$
    – N. Zahti
    Commented Jul 31, 2017 at 14:32
  • $\begingroup$ I think that the connecting map $c$ is the cycle class map into Deligne cohomology. So you might find a reference by searching for "Deligne cohomology" and "Abelian scheme". $\endgroup$
    – Jason Starr
    Commented Jul 31, 2017 at 16:49

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