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Let X be an irreducible symplectic variety defined over a number field K.Assume either that X has Picard number at least two or that X has even secondBetti number. We prove that there exist a finite algebraic field extension L/Kand a density 1 set S of non-archimedean places of L such that the reduction ofX at any place in S has nonzero Hasse-Witt invariant.

Invariants de Hasse-Witt des r\'eductions de certaines vari\'et\'es symplectiques irr\'eductibles