Scarcity of finite orbits for rational functions over a number field

Abstract

Let φ be a an endomorphism of degree d ≥ 2 of the projective line, defined over a number field K. Let S be a finite set of places of K, including the archimedean places, such that φ has good reduction outside of S. The article presents two main results: the first result is a bound on the number of K-rational preperiodic points of φ in terms of the cardinality of the set S and the degree d of the endomorphism φ. This bound is quadratic in terms of d which is a significant improvement to all previous bounds on the number of preperiodic points in terms of the degree d. For the second result, if we assume that there is a K-rational periodic point of period at least two, then there exists a bound on the number of K-rational preperiodic points of φ that is linear in terms of the degree d.

Cycles for rational maps over global function fields with one prime of bad reduction

“…Some authors characterize bad reduction as absence of potentially good reduction as in [1], while others (e.g. [5]) say that a map has bad 1.1]. Similar techniques have been applied to find bounds for the cardinality of finite orbits in terms of the number of bad primes, even for global function fields, as in [4].…”

“…Let F q denote the full constant field of K and S the set of primes of bad reduction: in this case O * S = F * q . Let L be the finite normal extension given by Lemma 4.4: L can be explicitly constructed, provided that one knows the ideal class group of O S (the reader is referred to the discussion before [5,Lemma 4.4] for details), but in general we do not even know its degree over K. The group O * S is by definition the radical of F * q in L * , i.e. O * S = F * , where F is the full constant field of L. The aim of this section is to prove Theorem 5.1, which can be regarded as a generalization of equation (2.1).…”

“…The next result is of a different nature: in this case we consider points with coordinates in a global field K. Let S be a finite set of primes of K; by putting together local information about p-adic distances for all primes outside S we establish a connection between points with certain properties and solutions of unit equations. The n-points lemmas are central tools in the work of Canci, Troncoso and Vishkautsan over number fields (see [5] and [6]) where they are applied in combination with bounds for the number of solutions of the resulting unit equations. The three points lemma we present here is a qualitative version of [5,Corollary 4.2]; its proof can be adapted from that of [5,Lemma 4.1] but we include it here for the reader's convenience.…”

“…The n-points lemmas are central tools in the work of Canci, Troncoso and Vishkautsan over number fields (see [5] and [6]) where they are applied in combination with bounds for the number of solutions of the resulting unit equations. The three points lemma we present here is a qualitative version of [5,Corollary 4.2]; its proof can be adapted from that of [5,Lemma 4.1] but we include it here for the reader's convenience. Lemma 6.3 (Three points lemma).…”

“…Canci, Troncoso e Vishkautsan in [5] give a bound for preperiodic points of rational functions over number fields that is quadratic in d and exponential in s. 1 This result depends on a reduction to unit equations with a number of solutions uniformly bounded by […”