\title{Endo-permutation modules as sources of simple modules}
\author{Nadia Mazza\thanks{This work is a part of a doctoral thesis 
in preparation at the University of Lausanne, under the supervision 
of Prof. Jacques Th\'evenaz}\\
Institut de 
Math\'ematiques\\
Universit\'e de Lausanne\\
CH-1015 
Lausanne\\
}


\begin{abstract} The source of a simple $kG$-module, for a finite 
$p$-solvable group $G$ and an algebraically closed field $k$ of prime 
characteristic $p$, is an endo-permutation module (see~\cite{Pu1} 
or~\cite{Th}). L. Puig has proved, more precisely, that this source 
must be isomorphic to the cap of an endo-permutation module of the 
form $\bigotimes_{Q/R\in\cal S}\Ten^P_Q\Inf^Q_{Q/R}(M_{Q/R})$, where 
$M_{Q/R}$ is an indecomposable torsion endo-trivial module with 
vertex $Q/R$, and $\cal S$ is a set of cyclic, quaternion and 
semi-dihedral sections of the vertex of the simple $kG$-module. At 
present, it is conjectured that, if the source of a simple module is 
an endo-permutation module, then it should have this shape. In this 
paper, we are going to give a method that allow us to realize 
explicitly the cap of any such indecomposable module as the source of 
a simple module for a finite $p$-nilpotent group.
\end{abstract}