  We complete the first step in a two-part program proposed by Baker, Grigsby,
and the author to prove that Berge's construction of knots in the three-sphere
which admit lens space surgeries is complete. The first step, which we prove
here, is to show that a knot in a lens space with a three-sphere surgery has
simple (in the sense of rank) knot Floer homology. The second (conjectured)
step involves showing that, for a fixed lens space, the only knots with simple
Floer homology belong to a simple finite family. Using results of Baker, we
provide evidence for the conjectural part of the program by showing that it
holds for a certain family of knots. Coupled with work of Ni, these knots
provide the first infinite family of non-trivial knots which are characterized
by their knot Floer homology. As another application, we provide a Floer
homology proof of a theorem of Berge.
