Consideration of congruences for classical modular forms
naturally leads to investigating smooth irreducible representations of
GL(2,Q_p) on vector spaces over a field C of positive characteristic l.
The case where l=p has been considered only recently. We study more
generally such smooth irreducible representations for a reductive group
G over a local non-Archimedean field F of residual characteristic p. We
examine the basic issues: admissibility, existence of central character,
use of the Hecke algebra, analysis of parabolically induced representations.