Invariant dimensions and maximality of geometric monodromy action

Let X be a smooth separated geometrically connected variety over F_q and f:Y-> X a smooth projective morphism. We compare the invariant dimensions of the l-adic representation V_l and the F_l-representation \bar V_l of the geometric étale fundamental group of X arising from the sheaves R^wf_*Q_l and R^wf_*Z/lZ respectively. These invariant dimension data is used to deduce a maximality result of the geometric monodromy action on V_l whenever \bar V_l is semisimple and l is sufficiently large. We also provide examples for \bar V_l to be semisimple for l>>0.

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