Bordism of elementary abelian groups via inessential Brown-Peterson homology

We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $\Omega_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $\Omega^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{Z}/p)^n$. We apply our results to the Gromov-Lawson-Rosenberg conjecture for atoral manifolds whose fundamental groups are elementary abelian of odd order.