算術動態系統中丟翻圖問題之研究;On Diophantine Problems in the Arithmetic of Dynamical Systems

Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/57046

Title:	算術動態系統中丟翻圖問題之研究;On Diophantine Problems in the Arithmetic of Dynamical Systems
Authors:	夏良忠
Contributors:	中央大學數學系
Keywords:	數學類;前週期點;Manin-Mumford 猜想;阿貝爾叢集;阿貝爾子叢集;有理映射;正則高度;preperiodic points;Manin-Mumford conjecture;abelian varieties;abelian subvarieties
Date:	2010-09-01
Issue Date:	2012-10-01 15:10:27 (UTC+8)
Publisher:	行政院國家科學委員會
Abstract:	Abstract. In this project, we propose to study some Diophantine problems arising from the iterations of a rational function in one variable. Let K be a number eld (or a global eld in general). and let ' : P1 ! P1 be a rational map de ned over K with d = deg '(z) 2: Let 2 P1(K) be a xed rational point which is not a totally rami ed point of '2. Let " > 0 be given and let 2 P1(K) such that its '-orbit is ini nite. For any nite subset S MK containing all the archimedean places of K; the set ô€€€';S( ; ; ") of quasi-(S; ")-integral points with respect to in O'( ) is a nite set [25]. We [14] obtained a bound for the cardinality of this nite set. It is expected that there exists a uniform bound for the cardinality of the set of quasi-(S; ")-integral points provided that ' satis es some suitable minimal conditions. In this project, we propose to test such uniform bound and other related questions for the two (one parameter) quadratic families : 'c(z) = z2 + c and c(z) = z + c=z where c 2 K is a parameter for the family. There are several related questions that need to be studied: (a) minimal conditions (minimal models) for ', (b) lower bounds for the canonical heights of points in P1(K) whose '-orbit is in nite and (c) the dependence on the parameter 2 P1(K): Related to these two families, we'll also study the dynatomic curves Y dyn 1 (n) arising from the n- periodic points of formal period n: These curves are de ned by n(c; z) = 0 which are equations for the periodic points of formal periodi equal to n: In paritcular, we'll invetigate the group of cuspidal divisors and correspondingly, the group of dynmaical units of the coordinate ring K[Y dyn 1 (n)] of Y dyn 1 (n): ; 研究期間 9908 ~ 10007
Relation:	財團法人國家實驗研究院科技政策研究與資訊中心
Appears in Collections:	[Department of Mathematics] Research Project

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