2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60520Let 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 λand 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 λand the conductor of R are odd and coprime.14 pages; LaTeX2e source; to be published in Bulletin of the LMSNumber Theory11G10 (Primary) 11G15, 14K22 (Secondary)text