  Let G be a reductive group over an algebraically closed field of
characteristic p, and let u in G be a unipotent element of order p. Suppose
that p is a good prime for G. We show in this paper that there is a
homomorphism phi:SL_2/k --> G whose image contains u. This result was first
obtained by D. Testerman (J. Algebra, 1995) using case considerations for each
type of simple group (and using, in some cases, computer calculations with
explicit representatives for the unipotent orbits).
  The proof we give is free of case considerations (except in its dependence on
the Bala-Carter theorem). Our construction of phi generalizes the construction
of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in
particular, phi is obtained by reduction modulo P from a homomorphism of group
schemes over a valuation ring in a number field. This permits us to show
moreover that the weight spaces of a maximal torus of phi(SL_2/k) on Lie(G) are
``the same as in characteristic 0''; the existence of a phi with this property
was previously obtained, again using case considerations, by Lawther and
Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G.
Seitz (Invent. Math. 2000).
