Abstract
Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $\pi_1(X,x)$ acting on the étale cohomology groups $H^*(Y_x,\mathbb{F}_\ell)$ are the reduction modulo-$\ell$ of those of $\pi_1(X,x)$ acting on $H^*(Y_x,\mathbb{Z}_\ell)$ for $\ell $ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $\mathbb{F}_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture — namely, that $\pi_1(X,x)$ acts semisimply on $H^*(Y_x,\mathbb{F}_\ell)$ for $\ell\gg 0$ — is equivalent to the condition that the image of $\pi_1(X,x)$ acting on $H^*(Y_x,\mathbb{Q}_\ell)$ is `almost maximal’ (in a precise sense; what we call `almost hyperspecial’) with respect to the group of $\mathbb{Q}_\ell$-points of its Zariski closure. Ultimately, we prove the geometric variant with $\mathbb{F}_\ell$-coefficients of the Grothendieck-Serre semisimplicity conjecture.
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@BOOK{St67, zblnumber = {1196.22001},
mrkey = {0466335},
mrnumber = {0466335},
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title = {Lectures on {C}hevalley Groups},
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pages = {iii+277},
year = {1968},
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issn = {0003-486X},
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title = {Reductive groups over local fields},
pages = {29--69},
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} -
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@article{Zar14,
author = {Zarhin, {\relax Yu. G}.},
title={Abelian varieties over fields of finite characteristic},
JOURNAL={Cent. Eur. J. Math.},
VOLUME={12},
YEAR={2014},
NUMBER={5},
PAGES={659--674},
mrnumber = {3165576},
DOI = {10.2478/s11533-013-0370-1},
ZBLNUMBER = {1296.11068},
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[Zarhin] . Y. G. Zarhin, "Endomorphisms of abelian varieties and points of finite order in characteristic $P$," Mat. Zametki, vol. 21, iss. 6, pp. 737-744, 1977.
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