\author{David J. Benson}
\title{Modules for elementary abelian $p$-groups}

\begin{abstract}
Let $E\cong(\bZ/p)^r$ $(r\ge 2$)
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 $p=2$.
\end{abstract}