Degrees of polarizations on an abelian surface with real multiplication

Let F be a real quadratic field, and let R be an order in F . Suppose given a polarized abelian surface (A,λ) defined over a number field k with a symmetric action of R defined over k. This paper considers varying A within the k-isogeny class of A to reduce the degree of λ and the conductor of R. It is proved, in particular, that there is a k-isogenous principally polarized abelian surface with an action of the full ring of integers of F when F has class number 1 and the degree of λ and the conductor of R are odd and coprime.