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We determine the irreducible 2-modular representations of the symplecticgroup $G=Sp_{2n}(2)$ whose restriction to every abelian subgroup has a trivialconstituent. A similar result is obtained for maximal tori of $G$. There issignificant information on the existence of eigenvalue 1 of elements of $G$ ina given irreducible representation of $G$.

Abelian subgroups and semisimple elements in $2$-modular representations of the symplectic group $Sp_{2n}(2)$