We consider cubic polynomials f(z)=z^3+az+b defined over the function field
C(L), with a marked point of period N and multiplier L. In the case N=1, there
are infinitely many such objects, and in the case N>2, only finitely many. The
case N=2 has particularly rich structure, and we are able to describe all such
cubic polynomials defined over the field obtained by adjoining to C the mth
roots of L, for all L.