  This paper continues the study of certain two-dimensional Galois
representations attached to modular forms (mod p) via a construction due to
Deligne. In particular, we prove a criterion for determining when the
representation is split when restricted to a decomposition group at p.While
this problem was already well understood for "nonexceptional" modular forms,
here we specifically address the "exceptional" case. Although the problem is
not solved in all generality, it is reasonable to hope that this proof will in
fact provide the framework of a proof in the more general case.
