Abstract: The cohomology of arithmetic groups has connections to many areas of mathematics such as number theory and diffeomorphism groups. Classifying spaces of congruence subgroups of symplectic groups have an algebro-geometric interpretation as the moduli space of principally polarized abelian varieties with level structures. These congruence subgroups Sp_2n(Z,L) are the kernel of the mod-L reduction map Sp_2n(Z) to Sp_2n(Z / L). By work of Borel-Serre, H^i(Sp_2n(Z / L)) vanishes for i > n^2. I will report on lower bounds in the top degree i = n^2. The key tools in the proof are the theory of Steinberg modules and highly connected simplicial complexes.