Minimality criteria for convergent power series over Zp and rational maps with good reduction on the projective line over Qp

초록

In this paper, we first characterize the minimality criterion for a convergent power series f on Z(p) in terms of its coefficients for the cases p = 2 or 3. For an arbitrary prime p >= 5, 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 fora rational map of at least degree 2 with good reduction on the projective line P-1 (Q(p)) over Q(p). This criterion enables us to obtain a complete description of minimal conditions for such a map on P-1(Q(p)) in terms of its coefficients for p = 2 or 3. For an arbitrary prime p >= 5, we present a method of characterizing minimal rational maps phi of degree >= 2 on P-1 (Q(p)), provided that the prescribed conditions for the reduction of phi on P-1(F-p) to be transitive are known.

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