Modules for Elementary Abelian p-groups

Let E congruent to (Z/p)(r) (r >= 2) be an elementary abelian p-group and let k be an algebraically closed field of characteristic p. A finite dimensional kE-module M is said to have constant Jordan type if the restriction of M to every cyclic shifted subgroup of kE has the same Jordan canonical form. I shall begin by discussing theorems and conjectures which restrict the possible Jordan canonical form. Then I shall indicate methods of producing algebraic vector bundles on projective space from modules of constant Jordan type. I shall describe realisability and non-realisability theorems for such vector bundles, in terms of Chern classes and Frobenius twists. Finally, I shall discuss the closely related question: can a module of small dimension have interesting rank variety? The case p odd behaves throughout these discussions somewhat differently to the case p = 2.