Minimality criteria for 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 $\text{Zp}$ in terms of its coefficients for the cases $p=2$ or $3$. For an arbitrary prime $p\ge 5$, 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 for a rational map of at least degree 2 with good reduction on the projective line ${\mathbb{P}}^1(\text{Qp})$ over $\text{Qp}$. This criterion enables us to obtain a complete description of minimal conditions for such a map on ${\mathbb{P}}^1(\text{Qp})$ in terms of its coefficients for $p=2$ or $3$. For an arbitrary prime $p\ge 5$, we present a method of characterizing minimal rational maps $\varphi$ of degree $\ge 2$ on ${\mathbb{P}}^1(\text{Qp})$, provided that the prescribed conditions for the reduction of $\varphi$ on ${\mathbb{P}}^1({\text{F}}_p)$ to be transitive are known.