Rings of invariants for the three-dimensional modular representations of elementary abelian $p$-groups of rank four

Abstract

We show that the rings of invariants for the three-dimensional modular representations of an elementary abelian $p$-group of rank four are complete intersections with embedding dimension at most five. Our results confirm the conjectures of Campbell, Shank and Wehlau (Transform. Groups 18 (2013), 1–22) for these representations.