  In this paper, we classify the irreducible representations of the
trigonometric Cherednik algebras of rank 1 in characteristic p > 0. There are
two cases. One is the "quantum" case, where "Planck's constant" is nonzero and
generic irreducible representations have dimension 2p. In this case, smaller
representations exist if and only if the "coupling constant" k is in F_p;
namely, if 0 <= k <= p-1, then there exist irreducible representations of
dimensions p-k and p+k. The other case is the "classical" case, where "Planck's
constant" is zero and generic irreducible representations have dimension 2p. In
that case, one-dimensional representations exist if and only if the "coupling
constant" k is zero.
