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Let $k$ a field with $char(k)=p>0$, separable closure $k^{sep}$ and $f:X\rightarrow Y$ a smooth projective morphism of smooth variety over $k$.

1)Is it true that there exists a (EDIT) dense open susbset $U$ of $Y$ such that $R^{i}f_*\mathbb Q_p$ is a lisse sheaf over $U$?

2)Assume $Y=spec(k)$ and $k$ finitely generated. Then $R^{i}f_{*}\mathbb Q_p$ correspond to the representation $Gal(k^{sep}|k)\rightarrow GL(H^i(X_{k^{sep}}, \mathbb Q_p))$. Is it possible that this representation is trivial or almost trivial (EDIT i.e it has finite image)?

When specialized to abelian varieties the second question becomes:

2')Is it possible that the action of the absolute Galois group of $k$ on the $p$ Tate module is trivial or almost trivial?

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  • $\begingroup$ For $2'$, no for $k$ a finitely generated field unless the $p$-adic Tate module itself vanishes. All eigenvalues of $\operatorname{Frob}_q$ on the $p$-adic Tate module are roots of the characteristic polynomial of Frobenius and hence are complex numbers of norm $\sqrt{q}\neq 1$ and so the Galois action is nontrivial. $\endgroup$
    – Will Sawin
    Commented Apr 5, 2017 at 15:44
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    $\begingroup$ This is true for the $\ell$-adic Tate modules when $\ell\neq p$ and they are essentially the Weil conjectures. Is it true also for the p-adic Tate module? $\endgroup$
    – Emiliano Ambrosi
    Commented Apr 5, 2017 at 15:52
  • $\begingroup$ For 1), it might suffice to find a $U$ on which all $R^i f_* \mathbb{F}_p$ are locally constant constructible. For this, we can use the Artin-Schreier sequence $0\to \mathbb{F}_p\to \mathcal{O}_X\to \mathcal{O}_X\to 0$. Shrinking $Y$, we can assume that all $R^i f_* \mathcal{O}_X$ are locally constant and (hence) commute with base change. Now we have to prove that for a perfect complex of $\mathcal{O}_Y$-modules $K$ endowed with a Frobenius-semilinear $\phi: K\to K$, the cone of ${\rm id}-\phi : K\to K$ has constructible cohomology. EDIT: nevermind, nfdc23 just gave an excellent answer below! $\endgroup$
    – Piotr Achinger
    Commented Apr 5, 2017 at 15:54
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    $\begingroup$ Weil proved that Frobenius satisfies a polynomial equation in the ring of endomorphisms of the abelian variety whose roots are complex numbers of norm $\sqrt{q}$. This implies the same statement for both the $\ell$-adic and $p$-adic Tate modules, as well as for crystalline cohomology and maybe even cohomology theories not yet discovered. $\endgroup$
    – Will Sawin
    Commented Apr 5, 2017 at 21:12

1 Answer 1

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First let's consider constructible abelian sheaves for the etale topology. Higher direct images under proper morphisms carry constructible abelian sheaves to constructible abelian sheaves. The point is that for proper curves over separably closed fields the finiteness and cohomological vanishing beyond dimension 2 hold even for $p$-torsion in characteristic $p>0$ since one can use the Artin-Schreier sequence in place of Kummer theory to make contact with coherent cohomology (this is explained in Milne's book on etale cohomology, for example). Consequently, the usual fibration method to prove preservation of constructibility for torsion orders invertible on the base also works in general (i.e., without hypotheses on torsion orders) in the proper case.

The proof that a (constructible) $\mathbf{Z}_{\ell}$-sheaf becomes lisse upon restriction to members of a stratification is rather soft and so works without any assumption on $\ell$, and by the preceding the usual proof (such as 12.15 in Chapter I of the book by Freitag and Kiehl on etale cohomology) that ${\rm{R}}^if_{!}$ carries constructible $\mathbf{Z}_{\ell}$-sheaves to constructible $\mathbf{Z}_{\ell}$-sheaves works for proper $f$ without any hypothesis on $\ell$. Question (1) (for which I assume you meant for $U$ to be dense in $Y$, so not empty for example) therefore has an affirmative answer for ${\rm{R}}^i f_{\ast}(\mathbf{Z}_p)$ for any proper map $f:X \rightarrow Y$ between noetherian schemes and any prime $p$, so tautologically also with $\mathbf{Q}_p$ in place of $\mathbf{Z}_p$.

Question (2) has a rather negative answer for any meaningful notion of "almost trivial"; even for a non-isotrivial pencil of ordinary elliptic curves the resulting representation into $\mathbf{Z}_p^{\times}$ typically has open image. So without a more precise meaning assigned to "almost trivial", one can't really say anything more (and it isn't explained why question (2) would be expected to possibly have an affirmative answer).

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  • $\begingroup$ Thank you very much! for 2) for me almost trivial means that the image of the representation is finite (I will edit). Have you a reference for the fact that "a non-isotrivial pencil of ordinary elliptic curves the resulting representation into Z_p^* typically has open image."? My worries were about the possible fact that an abelian variety over kk could have all the pp power torsion defined over purely inseparable extensions, and hence that the action of the galois could be trivial on it. Thank you! $\endgroup$
    – Emiliano Ambrosi
    Commented Apr 5, 2017 at 16:17
  • $\begingroup$ If $h:Y' \rightarrow Y$ is a universal homeomorphism of schemes (equivalently: surjective, radiciel, integral) then $U \rightsquigarrow U \times_Y Y'$ is an equivalence of etale sites, and in particular if $F$ is an abelian etale sheaf on $Y$ then ${\rm{H}}^i(Y,F) \rightarrow {\rm{H}}^i(Y',h^*F)$ is an isomorphism. By sheafifying, higher direct images along any map $X \rightarrow Y$ are "unaffected" by base change along $h$. In particular, your question (2) is the same over the perfect closure of $k$ as over $k$, and over the perfect closure the connected-etale sequence splits, etc. $\endgroup$
    – nfdc23
    Commented Apr 5, 2017 at 21:02
  • $\begingroup$ Sorry for my late answer. I was wondering what happens for the elliptic curves without p-torsion defined over the separable closure of K. I have the impression that in this situation the action of the Galois group is trivial on the p-adic Tate module. What it is wrong? They are all isotrivial? $\endgroup$
    – Emiliano Ambrosi
    Commented Oct 14, 2017 at 10:38

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