The classification of torsion endo-trivial modules

The main purpose of this paper is to show that, if G is a finite p-group which is not cyclic, quaternion or semi-dihedral, then the group of endo-trivial G-modules is detected on restriction to elementary abelian p-subgroups of rank 2. Consequently, the torsion subgroup of the group of endo-trivial modules is trivial for any finite p-group, except in the known cases of cyclic, quaternion and semi-dihedral groups. This requires a large amount of group cohomology, recent bounds for the number of Bocksteins in Serre's theorem, Carlson's recent theorem expressing Serre's theorem in terms of modules, and finally the theory of varieties attached to modules. A detection theorem for the torsion subgroup of the Dade group of all endo-permutation modules is deduced from the main theorem. The structure of this torsion subgroup can then be fully described when p is odd (using the results of Bouc-Thévenaz).