CP symmetry and symplectic modular invariance

We analyze CP symmetry in symplectic modular-invariant supersymmetric theories. We show that for genus g >= 3 the definition of CP is unique, while two independent possibilities are allowed when g <= 2. We discuss the transformation properties of moduli, matter multiplets and modular forms in the Siegel upper half plane, as well as in invariant subspaces. We identify CP-conserving surfaces in the fundamental domain of moduli space. We make use of all these elements to build a CP and symplectic invariant model of lepton masses and mixing angles, where known data are well reproduced and observable phases are predicted in terms of a minimum number of parameters.