On surfaces with pg=q=2 and non-birational bicanonical maps

The present paper is devoted to the classification of irregular surfaces of general type with p(g) = q = 2 and non-birational bicanonical map. The main result is that, if S is such a surface and if S is minimal with no pencil of curves of genus 2, then S is a double cover of a principally polarized abelian surface (A, Theta), with Theta irreducible. The double cover S --> A is branched along a divisor B is an element of \2Theta\, having at most double points and so K-S(2) = 4.