\title{Endotrivial modules for nilpotent restricted Lie algebras}
\author{David J. Benson}
\address{Institute of Mathematics, Fraser Noble Building, Univeristy
  of Aberdeen, Aberdeen AB24 3UE, United Kingdom}
\author{Jon F. Carlson}
\address{Department of Mathematics, University of Georgia, Athens GA
  30602, USA}
\thanks{The second author was partially supported by
Simons Foundation Grant 054813-01.}
\subjclass[2010]{17B50}

\begin{document}

\begin{abstract}
Let $\g$ be a finite dimensional nilpotent $p$-restricted Lie algebra
over a field $k$ of characteristic $p$. For $p\ge 5$, we show that
every endotrivial $\g$-module is a direct sum of a syzygy of the trivial
module and a projective module. The proof includes a theorem that 
the intersection of the maximal linear subspaces of
the null cone of a nilpotent restricted $p$-Lie algebra for $p \ge 5$ 
has dimension at least two.
We give an example to show that the statement 
about endotrivial modules is false in characteristic two. In characteristic
three, another example shows that our proof fails, 
and we do not know a characterization of the endotrivial modules
in this case.
\end{abstract}