Minimality criteria for convergent power series over Z p and rational maps with good reduction on the projective line over Q p

Abstract

In this paper, we first characterize the minimality criterion for a convergent power series f on in terms of its coefficients for the cases p = 2 or 3. For an arbitrary prime , the minimality criterion of such a series can be obtained explicitly provided that the prescribed minimal conditions for the reduction of f modulo p are found. Second, we provide the minimality criterion for a rational map of at least degree 2 with good reduction on the projective line over . This criterion enables us to obtain a complete description of minimal conditions for such a map on in terms of its coefficients for p = 2 or 3. For an arbitrary prime , we present a method of characterizing minimal rational maps ϕ of degree on , provided that the prescribed conditions for the reduction of ϕ on to be transitive are known.