On Automorphic Forms of Small Weight for Fake Projective Planes

On the projective plane there is a unique cubic root of canonical bundle and this acyclic. fake planes such exists if are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any must be In present note give two short proofs statement show acyclicity some other line bundles on with at least $9$ automorphisms. Similarly to our earlier work employ simple representation theory for non-abelian finite groups. The first proof based observation non-linearizable respect abelian group, then it should linearized by finite, \emph{non-abelian}, Heisenberg group. For second proof, also demonstrate vanishing odd Betti numbers class covers, use linearization an auxiliary as well.