Unisingular subgroups of symplectic group $Sp_{2n}(2)$ for $2n<250$

Abstract

A linear group is called unisingular if every element of it has eigenvalue 1. A certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements. A more special, but an important question is on the existence of such subgroups in the symplectic groups of particular degree. We answer this question for almost all degrees $2n<250$, specifically, the question remains open only 7 values of $n$. Additionally, the paper contains results of general nature on the structure of unisingular irreducible linear groups.