On some notions of good reduction for endomorphisms of the projective line. (English) Zbl 1285.14033
Based on the investigation by L. Szpiro and T. J. Tucker [Pure Appl. Math. Q. 4, No. 3, 715–728 (2008; Zbl 1168.14020)], the authors consider an endomorphism \(\Phi\) of the projective line \(\mathbb{P}^1_{\overline{\mathbb{Q}}}\) over the algebraic closure of \(\mathbb{Q}\), of degree \(\geq 2\) defined over a number field \(K\). Let \(v\) be a non-archimedean valuation of \(K\). L. Szpiro and T. J. Tucker introduced the notions of an endomorphism having critically good reduction, as opposed to simple good reduction, and investigated the relations between both notions.
The authors in this article improve the investigation, and show that for an endomorphism \(\Phi\) of the projective line of degree \(\geq 2\) such that the reduced map \(\Phi_v\) is separable at a finite place \(v\) of \(K\), the following are equivalent: (a) \(\Phi\) has critically good reduction at \(v\) and (b) \(\Phi\) has simple good reduction at \(v\) and \(\#\Phi(\mathcal{R}_\Phi)=\#(\Phi(\mathcal{R}_\Phi))_v\), where \(\mathcal{R}_\Phi\) is the set of ramification points of the map \(\Phi\).
The authors in this article improve the investigation, and show that for an endomorphism \(\Phi\) of the projective line of degree \(\geq 2\) such that the reduced map \(\Phi_v\) is separable at a finite place \(v\) of \(K\), the following are equivalent: (a) \(\Phi\) has critically good reduction at \(v\) and (b) \(\Phi\) has simple good reduction at \(v\) and \(\#\Phi(\mathcal{R}_\Phi)=\#(\Phi(\mathcal{R}_\Phi))_v\), where \(\mathcal{R}_\Phi\) is the set of ramification points of the map \(\Phi\).
Reviewer: Shigeki Matsutani (Kanagawa)
MSC:
| 14H25 | Arithmetic ground fields for curves |
Citations:
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