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

We determine the irreducible 2-modular representations of the symplectic group $G=S{p}_{2n}(2)$ whose restriction to every abelian subgroup has a trivial constituent. A similar result is obtained for maximal tori of $G$. There is significant information on the existence of eigenvalue 1 of elements of $G$ in a given irreducible representation of $G$.