On some notions of good reduction for endomorphisms of the projective line
| hal.structure.identifier | Department of Mathematics [Basel] | |
| dc.contributor.author | CANCI, Jung Kyu | |
| hal.structure.identifier | Institut fur Mathematik | |
| dc.contributor.author | PERUGINELLI, Giulio | |
| hal.structure.identifier | Scuola Normale Superiore di Pisa [SNS] | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | TOSSICI, Dajano | |
| dc.date.accessioned | 2024-04-04T02:22:32Z | |
| dc.date.available | 2024-04-04T02:22:32Z | |
| dc.date.created | 2010 | |
| dc.date.issued | 2012 | |
| dc.identifier.issn | 0025-2611 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/189659 | |
| dc.description.abstractEn | Let $\varphi$ be an endomorphism of $\mathbb{P}^1_{\overline{\Q}}$ defined over a number field $K$. Given a discrete valuation $v$ of $K$, we consider here two notions of good reduction of $\varphi$ at $v$, called Standard Good Reduction (S.G.R., for short) and Critically Good Reduction (C.G.R.). If we consider the reduced map $\varphi_v$, in general its degree is smaller or equal to the degree of $\varphi$. We say that the map $\varphi$ has S.G.R. at $v$ if the degree of the reduced map $\varphi_v$ is equal to the degree of $\varphi$. This notion is frequently used in the study of arithmetical dynamical systems, allowing to reduce a global problem to a local problem. Another notion of good reduction has been recently introduced by Szpiro and Tucker to prove a finitess result about equivalence classes of endomorphisms of the projective line. We say that $\varphi$ has C.G.R. at $v$ if every pair of ramification points of $\varphi$ do not coincide modulo $v$ and the same holds for every pair of branch points. As an application of their result, Szpiro and Tucker showed that their theorem implies the well-known Shafarevich-Faltings theorem about the finiteness of the isomorphism classes of elliptic curves defined over a number field $K$ having good reduction outside a prescribed finite set of discrete valuations of $K$. Szpiro and Tucker already in their paper showed with same examples that these two notions are not equivalent. We prove here that if $\varphi$ has C.G.R. at $v$ and the reduced map $\varphi_v$ is separable, then $\varphi$ has S.G.R. at $v$. | |
| dc.language.iso | en | |
| dc.publisher | Springer Verlag | |
| dc.subject.en | good reduction | |
| dc.subject.en | endomorphism projective line | |
| dc.subject.en | ramification point | |
| dc.subject.en | branch locus | |
| dc.subject.en | separability | |
| dc.title.en | On some notions of good reduction for endomorphisms of the projective line | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1007/s00229-012-0573-y | |
| dc.subject.hal | Mathématiques [math]/Théorie des nombres [math.NT] | |
| bordeaux.journal | Manuscripta mathematica | |
| bordeaux.page | 18 pages | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-00795633 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-00795633v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Manuscripta%20mathematica&rft.date=2012&rft.spage=18%20pages&rft.epage=18%20pages&rft.eissn=0025-2611&rft.issn=0025-2611&rft.au=CANCI,%20Jung%20Kyu&PERUGINELLI,%20Giulio&TOSSICI,%20Dajano&rft.genre=article |
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