K-RATIONAL PREPERIODIC POINTS AND HYPERSURFACES ON PROJECTIVE SPACE By Sebastian Ignacio Troncoso Naranjo A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics — Doctor of Philosophy 2017 ABSTRACT K-RATIONAL PREPERIODIC POINTS AND HYPERSURFACES ON PROJECTIVE SPACE By Sebastian Ignacio Troncoso Naranjo The present thesis has two main parts. In the first one, we study bounds for the number of rational preperiodic points of an endomorphism of P1 . Let K be a number field and φ be an endomorphism of P1 over K of degree d ≥ 2. Let S be the set of places of bad reduction for φ (including the archimedean places). Let Per(φ, K), PrePer(φ, K), and Tail(φ, K) be the set of K-rational periodic, preperiodic, and purely preperiodic points of φ, respectively. If we assume that | Per(φ, K)| ≥ 4 (resp. | Tail(φ, K)| ≥ 3), we prove bounds for | Tail(φ, K)| (resp. | Per(φ, K)|) that depend only on the number of places of bad reduction |S| (and not on the degree d). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when | Per(φ, K)| < 4 (resp. | Tail(φ, K)| < 3). The key tool involved in these results is a bound for the number of solutions of S-unit equations. Using bounds for the number of solutions of the celebrated Thue-Mahler equation, we obtain bounds for | Per(φ, K)| and | Tail(φ, K)| in terms of the number of places of bad reduction |S| and the degree d of the rational function φ. Bounds obtained in this way are a significant improvement to previous result given by J. Canci and L. Paladino. In the second part of the thesis, we study the set of K-rational purely preperiodic hypersurfaces of Pn of a given degree for an endomorphism of Pn . Let φ be an endomorphism of Pn over K, S be the set of places of bad reduction for φ and HTail(φ, K, e) be the set of K-rational purely preperiodic hypersurfaces of Pn of degree e. We give a strong arithmetic relation between K-rational purely preperiodic hypersurfaces and K-rational periodic points. If we consider N = e+n − 1 and assume that φ has at e least 2N + 1 K-rational periodic points such that no N + 1 of them lie in a hypersurface of degree e then we give an effective bound on a large subset of HTail(φ, K, e) depending on e and the number of places of bad reduction |S|. Finally, we prove that the set HTail(φ, K, e) is finite if we assume that φ is an endomorphism of P2 . To my daughter Emma, my son Sebastian and my wife Yira. I love you all. I would not be here without you. iv ACKNOWLEDGMENTS I have received a tremendous amount of support during my time at Michigan State University. I am truly grateful for my advisor Dr. Aaron Levin, there are not words to describe how thankful I am to be your student. During my studies, I was able to dedicate more time to research because I was partially funded by my advisor grants. Additionally, I received a great deal of support from Michigan State University, which has helped me in completing this project: College of Natural Science, Department of Mathematics, Becas Chile and Dissertation Completion Fellowship. Many professors help me to obtain my doctorate degree. In particular, I want to thank Herminia Ochsenius, Martin Berz, Rajesh Kulkarni, and Aaron Levin. My friends, both in and out of the math department, played a key role throughout this process. I especially want to thank Casey Machen, Charlotte Ure, Hector Moreno, and Robert Auffarth. My family is the core of my life. There were always present to support me, love me and keep me sane. I want to thanks my parents Emma Naranjo, Rodrigo Sarria, Marco Troncoso, Denise Cancino, my sisters Natalia Troncoso and Paula Troncoso, my wife Yira Feliciano, and my children Emma Troncoso and Sebastian Troncoso. Those not mentioned here are not forgotten. I am thankful for everyone else who has helped me along the way. v TABLE OF CONTENTS KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 Background Material 2.1 Basics of arithmetic dynamics 2.2 Definitions and results for P1 2.3 General results . . . . . . . . . . . . . . . . . . . . vii 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 18 22 Chapter 3 Arithmetic Dynamics on P1 3.1 Main propositions on P1 . . . . . . . 3.2 S-unit equation approach . . . . . . 3.2.1 Proof of Theorem 3.2.1 . . . . 3.2.2 Proof of Theorem 3.2.2 . . . . 3.2.3 Examples . . . . . . . . . . . 3.3 Thue-Mahler approach . . . . . . . . 3.3.1 Proof of Theorem 3.3.1 . . . 3.3.2 Proof of Theorem 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 29 30 35 37 38 39 46 Chapter 4 Arithmetic Dynamics on Pn 4.1 Main definitions and propositions . . 4.2 Effective results . . . . . . . . . . . . 4.3 Finiteness . . . . . . . . . . . . . . . 4.3.1 Finiteness on Pn . . . . . . . 4.3.2 Finiteness on P2 . . . . . . . 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 48 53 56 57 58 61 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 vi KEY TO SYMBOLS 1. N the set of natural numbers. 2. N0 the set of non-negative integers. 3. Z, Q, R, C the set of integers, rational, real and complex numbers respectively. 4. ⊂ means subset. 5. means a proper subset. 6. |A| the cardinality of a set A. 7. i = (i0 , . . . , in ) ∈ Nn+1 an n + 1-dimensional multi-index. 0 8. |i| = i0 + · · · + in 9. X = (X0 , . . . , Xn ) where X0 , . . . , Xn are n + 1 variables. i 10. Xi = X00 · · · Xnin 11. R∗ the group of units of a ring R. 12. K a number field. ¯ an algebraic closure of K. 13. K 14. O the ring of integers of K. 15. p a non-zero prime ideal of O. 16. vp the p-adic valuation on K corresponding to the prime ideal p (we always assume vp to be normalized so that vp (K ∗ ) = Z). 17. If the context is clear, we will also use vp (I) for the p-adic valuation of a fractional ideal I of K. 18. S a fixed finite set of places of K including all archimedean places. 19. |S| = s the cardinality of S. 20. OS = {x ∈ K : vp (x) ≥ 0 for every prime ideal p ∈ / S} the ring of S-integers. 21. OS∗ = {x ∈ K : vp (x) = 0 for every prime ideal p ∈ / S} the group of S-units. • Let φ be an endomorphism of Pn defined over K. vii 22. Per(φ, K) the set of K-rational periodic points. 23. Tail(φ, K) the set of K-rational tail points. 24. PrePer(φ, K) the set of K-rational preperiodic points. 25. HPer(φ, K, e) the set of K-rational periodic hypersurfaces of degree e. 26. HTail(φ, K, e) the set of K-rational tail hypersurfaces of degree e. 27. HPrePer(φ, K, e) the set of K-rational preperiodic hypersurfaces of degree e. 28. HPer(φ, K) the set of K-rational periodic hypersurfaces. 29. HTail(φ, K) the set of K-rational tail hypersurfaces. 30. HPrePer(φ, K) the set of K-rational preperiodic hypersurfaces. viii Chapter 1 Introduction Let S be a set and φ : S → S a function mapping the set S to itself. A (discrete) dynamical system is a pair consisting of the set S and the function φ. We denote by φn the nth iterate of φ under composition and by φ0 the identity map. The orbit of P ∈ S under φ is the set Oφ (P ) = {φn (P ) : n ≥ 0}. The set S could be simply a set with no additional structure but most frequently we study dynamics when the set S has some additional structure. In arithmetic dynamics we are interested when the set S is an arithmetic set such as Z, Q, number fields K, quasiprojective variety, K-rational points, etc. and the function φ is a polynomial, a rational map, an endomorphism, etc. In this arithmetic context the Principal Goal of Dynamics is to classify the points P in S according to the behavior of their orbits Oφ (P ) when S is an arithmetic set. Let K be a number field. Our study in arithmetic dynamics will be when S is PN (K) and φ an endomorphism of PN of degree d ≥ 2. A point P ∈ PN (K) is called periodic under φ if there is an integer n > 0 such that φn (P ) = P . It is called preperiodic under φ if there is an integer m ≥ 0 such that φm (P ) is periodic. A point that is preperiodic but not periodic is called a tail point. Let Tail(φ, K), Per(φ, K) and PrePer(φ, K) be the sets of K-rational tail, periodic and preperiodic points of φ, respectively. A first objective of this thesis is to study the cardinality of the sets Tail(φ, K), Per(φ, K) 1 and PrePer(φ, K). We start by asking • Is the set of K-rational preperiodic points finite or infinite?. • If finite, can we give an effective bound?. Northcott [Nor50] proved in 1950 that the total number of K-rational preperiodic points of φ is finite. In fact, from Northcott’s proof, an explicit bound can be found in terms of the coefficients of φ, the number field K and the dimension N . Even when Northcott answered both questions, a bound for PrePer(φ, K) in terms of only a few basic parameters is desired. In 1994, Morton and Silverman [MS94] conjectured the celebrated Uniform Boundedness Conjecture (UBC) which predicts the existence of such a bound depending only on d, the dimension of the projective space and the degree of K. Conjecture 1.0.1 (Uniform Boundedness Conjecture). Let K be a number field with [K : Q] = D, and let φ be an endomorphism of PN , defined over K. Let d ≥ 2 be the degree of φ. Then there is C = C(D, N, d) such that φ has at most C preperiodic points in PN (K). This conjecture is an extremely strong uniformity conjecture. For example, the UBC on maps of degree 4 on P1 defined over Q implies Mazur’s theorem that the torsion subgroup of an elliptic curve E/Q is bounded independently of E. More generally, the UBC for maps of degree 4 on P1 defined over K implies Merel’s theorem that the size of the torsion subgroup of an elliptic curve over a number field K is bounded only in terms of the degree of [K : Q]. The conjecture can also be applied to Latt`es maps and abelian varieties, for more detail see [Fak01], [Maz77] and [Mer96]. Poonen [Poo98] later stated a sharper version of the conjecture for the special case of quadratic polynomials over Q. Since every such quadratic polynomial map is conjugate to 2 a polynomial of the form ψc (z) = z 2 + c with c ∈ Q we can state Poonen’s conjecture as follows: Conjecture 1.0.2 (Poonen’s conjecture). Let ψc ∈ Q[z] be a polynomial of degree 2 of the form ψc (z) = z 2 + c with c ∈ Q. Then | PrePer(ψc , Q)| ≤ 9. Even though Poonen’s Conjecture is arguably the simplest case of the UBC, a proof of Poonen’s Conjecture seems to be very far off at this time. If we consider polynomials of the form ψc (z) = z 2 + c with c ∈ Q, B. Hutz and P. Ingram [HI13] have shown that Poonen’s conjecture holds when the numerator and denominator of c don’t exceed 108 . For more information on quadratic rational functions see [BCH+ 14], [Can10], [FHI+ 09], [Man07], [MN06], [Poo98]. Even though the UBC or Poonen’s conjecture are impossible to prove at the moment, if we allow the bound from the UBC to depend on one more parameter, then effective results can be given in the case of P1 . In the first half of the thesis we work in the case N = 1, so from now we assume that φ is an endomorphism of P1 . Let S be the set of places of K at which φ has bad reduction, including all archimedean places of K. The nonarchimedean places of bad reduction are those in which the degree of the reduction of φ in the residue field decreases. In other words, a place is said to be a place of good reduction if φ has a good behavior in the residue field associated with the place. Then the extra parameter needed to give effective results is the cardinality of S. The first main result of this thesis [[Tro], Corollary 1.3.] gives a bound for | PrePer(φ, K)| 3 in terms of the number of places of bad reduction |S| and the degree of the rational function φ. This bound significantly improves a previous bound given by J. Canci and L. Paladino [CP16]. In the second result, assuming that | Tail(φ, K)| ≥ 3 (resp. | Per(φ, K)| ≥ 4 ), we prove bounds for | Per(φ, K)| (resp. | Tail(φ, K)| ) that depend only on the number of places of bad reduction |S| and [K : Q] (and not on the degree of φ). We show that the hypotheses of this result are sharp. Example 3.2.4 and Example 3.2.5 give counterexamples to any possible result of this form when | Tail(φ, K)| < 3 (resp. | Per(φ, K)| < 4). Theorem 1.0.3. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. (a) If there are at least three K-rational tail points of φ then | Per(φ, K)| ≤ 216|S| + 3. (b) If there are at least four K-rational periodic points of φ then | Tail(φ, K)| ≤ 4(216|S| ). Using the previous theorem, we can deduce a bound for | PrePer(φ, K)| in terms of |S| and the degree of φ for any endomorphism of P1 . Corollary 1.0.4. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. Then 4 3 (a) | Per(φ, K)| ≤ 216|S|d + 3. 3 (b) | Tail(φ, K)| ≤ 4(216|S|d ). 3 (c) | PrePer(φ, K)| ≤ 5(216|S|d ) + 3. These bounds depend, ultimately, on a reduction to S-unit equations. Using a reduction to Thue-Mahler equations instead, we obtain a better bound for | Tail(φ, K)| in terms of |S| and d. Theorem 1.0.5. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. Then | Tail(φ, K)| ≤ d max (5 · 106 (d3 + 1))|S|+4 , 4(264(|S|+3) ) . To get a similar bound for | Per(φ, K)| we need to assume that φ has at least one Krational tail point. Under this assumption, using Theorem 1.0.3 and results about ThueMahler equations, we can get: Theorem 1.0.6. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. If φ has at least one K-rational tail point then | Per(φ, K)| ≤ max (5 · 106 (d − 1))|S|+3 , 4(2128(|S|+2) ) + 1. While the work described so far was being carried out, Canci and Vishkautsan [CV] proved a bound for | Per(φ, K)|, just assuming that φ has good reduction outside S. Their 5 bound on | Per(φ, K)| is roughly of the order of d216|S| + 22187|S| where d ≥ 2 is the degree of φ. Now let’s go through previous bounds for | PrePer(φ, K)| which are relevant for our work. In 2007, Canci [Can07] proved for rational functions with good reduction outside S that the length of finite orbits is bounded by: 12 e10 (|S| + 1)8 (log(5(|S| + 1)))8 |S| . (1.1) Note that this bound depends only on the cardinality of S. In Canci’s recent work (2014) with Paladino [CP16] a sharper bound for the length of finite orbits was found: max (216|S|−8 + 3) [12|S| log(5|S|)][K:Q] , [12(|S| + 2) log(5|S| + 5)]4[K:Q] . (1.2) In our work we are interested in the number of K-rational tail points and K-rational periodic points, | Tail(φ, K)| and | Per(φ, K)| respectively. The bounds mentioned in (1.1) and (1.2) can be used to deduce bounds on | PrePer(φ, K)|. For instance, if we assume that every finite orbit has cardinality given by (1.1) and using that every point could have at most d preimages under φ we obtain a bound for | PrePer(φ, K)| 8|S| that is roughly of the order of d(|S| log |S|) where d ≥ 2 is the degree of φ. Similarly, the 16|S| (|S| log(|S|)[K:Q] bound deduced from (1.2) is roughly of the order of d2 , where d ≥ 2 is the degree of φ. These bounds are polynomial in the degree of φ, however they will be rather large in terms of |S|. In 2007, Benedetto [Ben07] proved for the case of polynomial maps of degree d ≥ 2 that 6 | PrePer(φ, K)| is bounded by O(|S| log |S|), where S is the set of places of K at which φ has 2 bad reduction, including all archimedean places of K. The big-O is essentially d −2d+2 for log d large |S|. Results in positive characteristic have also been found. For instance, in 2007 Ghioca [Ghi07] proved a bound for the number of torsion points of a Drinfeld module. In this case, torsion points are preperiodic points under the action of an additive polynomial of degree larger than one. Another result in characteristic different from 0 is the work of Canci and Paladino [CP16] which gives a bound for the length of finite orbits under an endomorphism of P1 . The second part of this thesis provides quantitative and finiteness results for the set of K-rational tail curves of degree e for a given endomorphism of P2 . Compared to the 1-dimensional case, a primary difficulty in proving higher-dimensional results comes from the limited availability of arithmetic tools in higher dimensions. Indeed, arithmetic tools used frequently in the one-dimensional setting include Siegel’s theorem, Faltings’ theorem, and Roth’s theorem. Higher-dimensional conjectural generalizations of these results remain largely open, even for surfaces (e.g., Bombieri-Lang conjecture, Vojta’s conjecture). A secondary difficulty comes from the more complicated geometry possible in higher dimensions. For instance, general position conditions (which appear, for example, in Vojta’s conjecture) are rather trivial and uninteresting on curves. For these reasons, any progress towards the UBC in higher dimensions is highly valuable. Even though the UBC in PN is very hard there are some results on the literature. For instance, Hutz [Hut15] provides an algorithm to find Q-rational preperiodic points for endomorphisms of Pn . His techniques may be used to find a bound for the cardinality of the set of Q-rational periodic points, depending on the smallest prime of good reduction. 7 Another important study in dimension bigger than one is the papers by J. Bell, D. Ghioca, and T. Tucker [BGT15], [BGT16]. In these papers we can find an example of infinitely many fixed curves for an endomorphism of P2 . Indeed if f is a homogeneous two-variable polynomial of degree n, then the morphism P2 → P2 given by [x : y : z] → [f (x, z) : f (y, z) : k k z n ] has infinitely many f -invariant curves of the form [xz n −1 : f k (x, z) : z n ], where f k is the homogenized kth iterate of the dehomogenized one-variable polynomial x → f (x, 1). Motivated by the example of J. Bell, D. Ghioca, and T. Tucker and the SilvermanMorton Conjecture I study the set of K-rational preperiodic hypersurfaces of PN under an endomorphism of PN . Let φ be an endomorphism of PN , defined over K, of degree d and H an irreducible K-rational hypersurface of PN of degree e. We say that H is periodic under φ if there is an integer n > 0 such that φn (H) = H. It is called preperiodic under φ if there is an integer m ≥ 0 such that φm (H) is periodic. If H is preperiodic but not periodic it is called a tail hypersurface. Let HTail(φ, K, e), HPer(φ, K, e) and HPrePer(φ, K, e) be the sets of K-rational tail, periodic and preperiodic hypersurfaces of degree e of φ, respectively. It is important to notice that the degree of the preperiodic hypersurface will be involved in our study. This new parameter does not come up for points because the degree of a (geometric) point is always 1. However, this extra parameter is a natural condition because similar examples to the one given by J. Bell, D. Ghioca, and T. Tucker could be given if we consider subschemes in place of subvarieties. For instance, if instead of subvarieties we consider more generally integral closed K-subschemes, then a curve does have infinitely many periodic K-integral closed subschemes, because we can just take K-components of the subscheme of periodic points of period n. However, if we bound the degree of the Ksubschemes, then once again we get finiteness by Northcott’s theorem. The main idea of my results on P1 [Tro] lies in an arithmetic relation between K-rational 8 tail points and K-rational periodic points. Using a generalization of the p-adic logarithmic distance in P1 , I was able to generalize the relation between K-rational tail points and Krational periodic points to a relation between K-rational tail hypersurfaces and K-rational periodic points. Theorem 1.0.7. Let φ be an endomorphism of Pn , defined over K. Suppose φ has good reduction outside S. Let H be a K-rational tail hypersurface, m the period of the periodic part of the orbit of H and H the periodic hypersurface such that H = φm0 m (H) for some / supp{H }. Then δv (P ; H) = 0 m0 > 0. Let P ∈ Pn (K) be any periodic point such that P ∈ for every v ∈ / S. In [GTZ11] Bell, Ghioca and Tucker also propose the following question Question: Is there a constant C = C(N, K, d) such that for any periodic K-rational subvariety V of PN , we have PerΦ (V ) ≤ C? Using the previous arithmetic relation together with a result from Ru and Wong [RW91] we give a result that implies a partial answer to the previous question for curves on the projective plane. In fact, we provide a bound for the number of K-rational tail hypersurfaces of degree e in the backwards orbit of a given periodic K-rational hypersurface of Pn . Theorem 1.0.8. Let φ be an endomorphism of Pn , defined over K and suppose φ has good 2N +1 reduction outside S. Consider N = e+n e −1 and let {Pi }i=1 be a set of K-rational periodic points of Pn such that no N + 1 of them lie in a curve of degree e. Consider B = {H ∈ HPer(φ, K) : ∀1 ≤ i ≤ 2N + 1, B and l ≥ 0, φlnq (q) = H Pi ∈ / supp H } and A = {q ∈ HTail(φ, K, e) : there is H ∈ where nq is the period of the periodic part of q}. Then |A| ≤ 233 · (2N + 1)2 9 (N +1)3 (s+2N +1) In 2016 B. Hutz [Hut16] proved that the set of K-rational preperiodic subvarieties of Pn is finite. His proof is based on the theory of canonical height functions. In the special case of K-rational preperiodic curves of P2 we were able to give an alternative proof than the one given by Hutz. This alternative proof is based in a strong result of dynamical systems ([Fak03], Corollary 5.2) which states that if φ is an endomorphism of Pn then the ¯ for an endomorphism φ is Zariski dense in Pn . set Per(φ, K) Theorem 1.0.9. Let K be a number field and φ be an endomorphism of P2 , defined over K. Then for every e ∈ N the set HTail(φ, K, e) is finite. We end this introduction with a brief outline of the rest of the thesis. Chapter 2 introduces some classical notations and definitions from arithmetic dynamics, arithmetic geometry and number theory. We also prove some propositions needed for the main theorems of this manuscript. Chapter 3 presents the proof of our results on P1 . This chapter has three sections: the first section gives all the propositions and lemmas needed for the next two sections, the second section uses S-unit equations to get bounds for the set of K-rational preperiodic points and the third section uses Thue-Mahler equations to gives different bounds for the set of K-rational preperiodic points. Finally, Chapter 4 presents definitions and results on PN . This chapter has four sections: the first one gives definitions and propositions on PN . The second section give effective results for a large subset of the set of K-rational tail hypersurfaces of PN of a given degree. The third section prove finiteness of the set of K-rational tail curves of degree e of P2 . The last section gives examples of K-rational tail and periodic hypersurfaces of PN . 10 Chapter 2 Background Material This chapter contains the background information which will be used throughout the thesis. We’ll discuss, define and prove many of the basics of arithmetic dynamics. ¯ an algebraic For the rest of the chapter/thesis we assume that K is a number field, K closure of K, O its integer ring, p a non-zero prime ideal of O and vp a p-adic valuation on K corresponding to the prime ideal p (we always assume vp to be normalized so that vp (K ∗ ) = Z), and S a fixed finite set of places of K including all archimedean places. We also adopt the multi-index notation, which means that X is (X0 , . . . , Xn ), i is i (i0 , . . . , in ), Xi = X00 · · · Xnin and |i| = i0 + . . . + in . We say that P ∈ Pn (K), or P is defined over K, if we can find a representation of P such that every coordinate lies in K. We write P = [x0 : · · · : xn ] ∈ Pn (K) meaning that every xi ∈ K. We say that φ is an endomorphism of Pn , defined over K if φ = [F0 : · · · : Fn ] where F0 , . . . , Fn ∈ K[X] are homogeneous polynomials of the same degree with no common factors and F0 , . . . , Fn does not have a common zero on Pn . 2.1 Basics of arithmetic dynamics We start this section with the idea of normalized forms with respect to p. 11 Definition 2.1.1. 1. We say that a point P = [x0 : · · · : xn ] ∈ Pn (K) is in normalized form with respect to p if min{vp (x0 ), . . . , vp (xn )} = 0. 2. Let φ be an endomorphism of Pn , defined over K. Assume φ is given by φ = [F0 : · · · : Fn ] where F0 , . . . , Fn ∈ K[X] are homogeneous polynomials with no common factors of degree d. We say that φ = [F0 : · · · : Fn ] is normalized with respect to p or that φ is in normalized form with respect to p if Fi ∈ Op [X ] for every i ∈ {0, . . . , n} and at least one coefficient of Fi is not in the maximal ideal of Op for some i ∈ {0, . . . , n}. 3. Let H be a hypersurface of Pn defined over K of degree d. Suppose that H is defined by ai Xi ∈ K[X]. We say that H is in normalized an homogeneous polynomial f = |i|=d form with respect to p if min {vp (ai )} = 0. |i|=d Let P ∈ Pn (K) and φ be an endomorphism of Pn , defined over K. Since Op is a discrete valuation ring, we can always find a representation of P and φ in normalized form with respect to p. However, it is not always true that the same representation is normalized for every p. For this reason we need a more global definition of normalized forms. Definition 2.1.2. 1. We say that P = [x0 : · · · : xn ] ∈ Pn (K) is normalized with respect to S if [x0 : · · · : xn ] is in normalized form with respect to p for every p ∈ / S. 12 2. Let φ = [F0 : · · · : Fn ] be an endomorphism of Pn , defined over K. We say that φ is in normalized form with respect to S if [F0 : · · · : Fn ] is normalized with respect to p for every p ∈ / S. 3. Let H be a hypersurface of Pn defined over K of degree d. Suppose that H is defined by ai Xi ∈ K[X]. We say that H is in normalized an homogeneous polynomial f = |i|=d form with respect to S if min {vp (ai )} = 0 for every |i|=d p∈ /S The following remark is a characterization of the previous definition and it will be a useful idea to keep in mind in the next chapters. Remark 2.1.3. A point P = [x0 : · · · : xn ] ∈ Pn (K) admits a normalized form with respect to S if and only if the OS -fractional ideal (x0 , . . . , xn ) is principal. Similarly, an endomorphism φ or a hypersurface H admit a normalized form with respect to S if and only if the coefficients of the polynomials defined by φ or H generate an OS principal ideal. Now we define the important concept of good reduction with respect to a prime p. Definition 2.1.4. Let φ be an endomorphism of Pn , defined over K and write φ = [F0 : · · · : Fn ] in normalized form with respect to p. We say that φ has good reduction at p if ¯ where F˜0 , . . . , F˜n are the reductions of F˜0 (X) = . . . = F˜n (X) = 0 has no solutions in Pn (k), F0 , . . . , Fn modulo p respectively and k is the residue field of Op . We say that φ has good reduction outside S if φ has good reduction at p for every p ∈ / S. 13 We define P˜ as the reduction of P modulo p after P is written in normalized form with respect to p. Similarly, we define φ˜ as the reduction of φ modulo p after φ is written in normalized form with respect to p Proposition 2.1.5. Let φ, ϕ be two endomorphism of Pn defined over K. In addition, assume that φ and ϕ have good reduction at p. If ∗˜ is the reduction modulo p then ˜ P˜ ) = φ(P ) for all P ∈ Pn (K). 1. φ( 2. The composition φ ◦ ϕ has good reduction at p and φ ◦ ϕ = φ˜ ◦ ϕ˜ Proof. The proof for the one dimentional case can be found in [[Sil07], p.59.]. The higher dimensional case follows exactly as in P1 . From the previous proposition we emphasize two properties that we will use in future chapters. Remark 2.1.6. Let φ be an endomorphism of Pn , defined over K and assume φ has good reduction at p. We write φ = [F0 : · · · : Fn ] in normalized form with respect to p and let P = [x0 : · · · : xn ] ∈ P1 (K) be in normalized form with respect to p. 1. φk has good reduction at p for every k ≥ 2. Even more, φk = [G0,k : · · · : Gn,k ] is in normalized form with respect to p, where Gi,k (X0 , . . . , Xn ) = Fi (G0,k−1 (X0 , . . . , Xn ), . . . , Gn,k−1 (X0 , . . . , Xn )), Gi,1 (X0 , . . . , Xn ) = Fi (X0 , . . . , Xn ) and i ∈ {0, . . . , n}. 14 2. φ(P ) = [F0 (x0 : . . . : xn ) : · · · : Fn (x0 , . . . , xn )] is in normalized form with respect to p. Let us define the most important concept of this work: K-rational tail, periodic and preperiodic subvarieties of an endomorphism of Pn . Definition 2.1.7. Let V be subvariety of Pn (K) and φ be an endomorphism of Pn defined over K. 1. The orbit of V is the set {φi (V )}i≥0 . 2. We say that a subvariety W Pn is in the backward orbit of V if φm (W ) = V for some m > 0. The backward orbit of V is denoted by O−1 (V ). 3. We say that V is a K-rational periodic subvariety if φm (V ) = V for some m ≥ 1. 4. We say that V is a K-rational preperiodic subvariety if there is n0 ≥ 0 such that φm (V ) = φn0 (V ) for some m > n0 . Equivalent, V is a preperiodic subvariety if its orbit is finite. 5. Given V a K-rational periodic subvariety we say that V is in the tail of V if V is a preperiodic but not a periodic subvariety and V is in the orbit of P . 6. We say that V is a K-rational tail subvariety if V is in the tail of some periodic subvariety. Notation 2.1.8. Let φ be an endomorphism of Pn defined over K and e ∈ N then • We denote by Per(φ, K) the set of K-rational periodic points, Tail(φ, K) the set of K-rational tail points and PrePer(φ, K) the set of K-rational preperiodic points. 15 • We denote by Per(φ, K, e) the set of K-rational periodic hypersurfaces of degree e, Tail(φ, K, e) the set of K-rational tail hypersurfaces of degree e and PrePer(φ, K, e) the set of K-rational preperiodic hypersurfaces of degree e. • We denote by Per(φ, K) = Per(φ, K, e) the set of K-rational periodic hypersurfaces, e>0 Tail(φ, K) = Tail(φ, K, e) the set of K-rational tail hypersurfaces and PrePer(φ, K) = e>0 PrePer(φ, K, e) the set of K-rational preperiodic hypersurfaces. e>0 Definition 2.1.9. Let V be subvariety of Pn (K) and φ an endomorphism of Pn defined over K. 1. We say that V has period n if φn (V ) = V . 2. We say that V has exact (or primitive) period n if φn (V ) = V and φi (V ) = V for all 1 ≤ i < n. A point of period one is called a fixed point, an irreducible hypersurface of degree 1 is called a hyperplane and an irreducible hypersurface of degree 2 is called a quadratic hypersurface. Theorem 2.1.10 (Bezout’s Theorem). Suppose that X and Y are two plane projective curves defined over K that do not have a common component. Then the total number of ¯ counted with their multiplicities, is intersection points of X and Y with coordinates in K, equal to the product of the degrees of X and Y . Now we define the standard absolute values of Q. There is an archimedean absolute value on Q defined by |x|∞ = max{x, −x}. 16 This is just the restriction to Q is the usual absolute value on R. Further, for each prime number p there is a nonarchimedean (or p-adic) absolute value defined as follows. For any nonzero rational number x ∈ Q, let ordp (x) be the unique integer such that x can be written in the form x = pordp (x) · a b with a, b ∈ Z and p ab. (If x = 0, we set ordp (x) = ∞ by convention.) Then the p-adic absolute value of x ∈ Q is the quantity |x|p = p−ordp (x) . The set of standard absolute values on Q is the set MQ consisting of the archimedean absolute value | · |∞ and the p-adic absolute values | · |p for every prime p. We denote MK the set of all absolute values on K whose restriction to Q is one of the standard absolute values on Q. We define the local degree of v ∈ MK by nv = [Kv : Qv ] where Kv and Qv are the completions of K and Q at v, respectively. To simplify notation, we write the absolute value corresponding to v ∈ MK as | · |v . Definition 2.1.11. Let P ∈ PN (K) be a point with homogeneous coordinates P = [x0 : · · · : xN ] x0 , . . . , xN ∈ K. The height of P (relative to K) is the quantity max{|x0 |v , . . . , |xn |v }nv . HK (P ) = v∈MK 17 The (absolute) height of P is the quantity H(P ) = HK (P )1/[K:Q] . Note that HK (P ) and H(P ) are independent of the choice of homogeneous coordinates. Theorem 2.1.12 (Northcott’s Theorem). Let K be a number field and let B be any constant. Then the set of points {P ∈ PN (K) : HK (P ) ≤ B} is finite. More generally, for any constants B and D, the set of points ¯ : H(P ) ≤ B and [Q(P ) : Q] ≤ D} {P ∈ PN (Q) 2.2 is finite. Definitions and results for P1 We start by defining the p-adic logarithmic distance between two points in P1 . Definition 2.2.1. Let P1 = [x1 : y1 ] and P2 = [x2 : y2 ] be points in P1 (K). We will denote by δp (P1 , P2 ) = vp (x1 y2 − x2 y1 ) − min{vp (x1 ), vp (y1 )} − min{vp (x2 ), vp (y2 )} the p-adic logarithmic distance between the points P1 and P2 . Note that δp (P1 , P2 ) is independent of the choice of homogeneous coordinates. We use the convention that vp (0) = ∞. 18 Remark 2.2.2. Note that if P = [x1 : x2 ] and Q = [y1 : y2 ] are in normalized form with respect to p then δp (P1 , P2 ) = vp (x1 y2 − x2 y1 ). Next we will give the definition and some results on the nth dynatomic polynomial associated to an endomorphism φ of P1 defined over K. Definition 2.2.3. Let φ be an endomorphism of P1 defined over K of degree d. For any n ≥ 0 write φn (X, Y ) = [Fn (X, Y ) : Gn (X, Y )] with homogeneous polynomials Fn , Gn ∈ K[X, Y ] of degree dn . The n-period polynomial of φ is the polynomial Φφ,n (X, Y ) = Y Fn (X, Y ) − XGn (X, Y ). Φφ,n is well defined up to a constant. Notice that Φφ,n (P ) = 0 if and only if φn (P ) = P . The nth dynatomic polynomial of φ is the polynomial Φ∗φ,n (X, Y ) = Φφ,k (X, Y )µ(n/k) (Y Fk (X, Y ) − XGk (X, Y ))µ(n/k) = k|n k|n where µ is the M¨obius function. If φ is fixed, we write Φn and Φ∗n for Φφ,n and Φ∗φ,n respectively. The following remark will give us the degree of the dynatomic polynomial which will be useful in the end of the next chapter. Remark 2.2.4. The degree of the nth dynatomic polynomial is given by deg(Φ∗φ,n ) = µ k|n 19 n (dk + 1). k In particular, if n = 1 the degree of Φ∗φ,n is d + 1 and if n is a prime number then the degree of Φ∗φ,n is dn − d. Definition 2.2.5. Let φ be an endomorphism of P1 defined over K of degree d ≥ 2 and let ¯ be a periodic point for φ. P ∈ P1 (K) 1. We say that P has formal period n if Φ∗n (P ) = 0. 2. Suppose that P has primitive period, we say that P a (primitive) n-periodic point. The relation between the three previous definition is primitive period n =⇒ formal period n =⇒ period n. ¯ Definition 2.2.6. Consider P1 = A1 ∪ {∞}. Let φ ∈ K(z), and α ∈ P1 a periodic point of exact period n. We define the multiplier of φ at α by λ(α) =     (φn ) (α) if α = ∞  α−2 φ (α−1 )    lim α→0 φ(α−1 )2 if α = ∞ (2.1) where φ is the derivative of φ with respect to the variable z. The following theorem will guarantee that Φ∗φ,n is a polynomial. Also, the theorem will give us some useful tools for the next chapter. Theorem 2.2.7 ([Sil07], p.151). Let φ be an endomorphism of P1 defined over K of degree ¯ let d ≥ 2. For each P ∈ P1 (K), aP (n) = ordP (Φφ,n (X, Y )) and 20 a∗P (n) = ordP (Φ∗φ,n (X, Y )) where ordP (Φφ,n (X, Y )) and ordP (Φ∗φ,n (X, Y )) are the order of zero or pole at P of Φφ,n (X, Y ) and Φ∗φ,n (X, Y ), respectively. Then (a) Φ∗φ,n ∈ K[X, Y ], or equivalently, a∗P (n) ≥ 0 for all n ≥ 1 and all P ∈ P1 . (b) Let P be a point of primitive period m and let λ(P ) = (φm ) (P ) be the multiplier of P . Then P has formal period n, i.e., a∗P (n) > 0, if and only if one of the following is true: (i) n = m (ii) n = mr and λ(P ) is a primitive rth root of unity. In particular, a∗P (n) is nonzero for at most two values of n. Now we state a weak version of the Riemann-Hurwitz formula. Theorem 2.2.8 (Weak Riemann-Hurwitz Formula [Sil07], p.15). Let φ be an endomorphism of P1 defined over K of degree d. Then d − |φ−1 (α)| 2d − 2 = α∈P1 Using the Weak Riemann-Hurwitz Formula we deduce that an endomorphism of P1 of degree greater than one has at most two totally ramified points. The following result proves the existence of n-periodic points for an endomorphism of P1 . 21 Theorem 2.2.9 (Baker [Bak64]). Let φ be an endomorphism of P1 defined over K of degree d ≥ 2. Suppose that φ has no primitive n-periodic points. Then (n, d) is one of the pairs (2, 2), (2, 3), (3, 2), (4, 2). ¯ Baker’s theorem is extremely strong because it guarantees K-rational n-periodic points for every (n, d) but the four exceptions described in the theorem. However, these four exceptions are fully known as it can be seen in the next remark. Remark 2.2.10 (Kisaka [Kis95]). Kisaka completely classifies all the endomorphisms associated to the exceptional pairs (n, d) mentioned in Baker’s Theorem. Each of these exceptional ¯ endomorphisms have at least two distinct fixed points in K. 2.3 General results In this section we will cite strong theorems that can be found in the literature. Each of these results will be useful in one way or another in future chapters. The next two theorems will give bounds for the number of solutions of two important equations, the S-unit equation and the Thue-Mahler equation. First we start with the S-unit equation. We can consider the S-unit equation ax + by = 1 where a, b ∈ K ∗ and x, y are S-units. Bounds on the number of solutions of this equation give powerful consequences in different areas of mathematics. Among many studies on the S-unit equation, one of the best bounds is the following: Theorem 2.3.1 (Beukers and Schlickewei [BS96]). Let Γ be a subgroup of (K ∗ )2 = K ∗ ×K ∗ 22 of rank r. Then the equation x+y =1 in (x, y) ∈ Γ has at most 28(r+1) solutions. Corollary 2.3.2. Let Γ0 be a subgroup of K ∗ of rank r. Consider Γ = Γ0 × Γ0 and assume a, b ∈ K ∗ . Then the equation ax + by = 1 in (x, y) ∈ Γ has at most 28(2r+2) solutions. Now we gives bound for the Thue-Mahler equation. Let F (X, Y ) be a binary form of degree r ≥ 3 with coefficients in OS . An OS∗ -coset of solutions of F (x, y) ∈ OS∗ in (x, y) ∈ OS2 (2.2) is a set { (x, y) : ∈ OS∗ }, where (x, y) is a fixed solution of (2.2). Theorem 2.3.3 (Evertse [Eve97]). Let F (X, Y ) be a binary form of degree r ≥ 3 with coefficients in OS which is irreducible over K. Then the set of solutions of F (x, y) ∈ OS∗ in is the union of at most (5 · 106 r)s 23 (x, y) ∈ OS2 OS∗ -cosets of solutions. Now we will give some results to bound equations in more than two variables. First we say that a point x = (x1 , . . . , xn ) ∈ K n is said to be an S-integral point if xi ∈ OS for all 1 ≤ i ≤ n. l Let (Di )li=1 be distinct K-rational hyperplanes of Pn . Let D = Di . We call a set i=1 R ⊂ Pn (K)\D a set of (D, S)-integral points if there exists an affine embedding Pn \D ⊂ An K such that every P ∈ R has S-integral coordinates. Before stating the theorem we will recall the definition of general position for hyperplanes. Definition 2.3.4. A set H of hyperplanes of Pn is in general position if, for any 1 ≤ k ≤ n, the intersection of j hyperplanes in H is of dimension n − j, and the intersection of any n + 1 hyperplanes in H is empty. Now we state an important result from Ru and Wong. Theorem 2.3.5 (Ru and Wong [RW91] ). Let (Di )li=1 be distinct K-rational hyperplanes l of Pn that are in general position. Let D = points of Pn (K) is finite. Di . If l > 2n then any set of (D, S)-integral i=1 A generalization of the previous theorem can be found in [NW02]. Next we will state a result coming from decomposable form equations. Let F (X) = l1 (X) . . . lr (X) ∈ OS [X] be a decomposable form of degree r, where X = (X1 , . . . , Xn ) and l1 , . . . , lr are linear form with coefficients in some extension of K such that {x ∈ K n : l1 (x) = 0, . . . , lr (x) = 0} = {0}. 24 Theorem 2.3.6 ( [Eve95]). Assume that the number of OS∗ -cosets of solutions of F (x) ∈ OS∗ in x ∈ OSn (2.3) 3 is finite. Then this number is at most (233 r2 )n s . Finally we end this subsection with a strong consequence of Dirichlet’s Theorem on primes in arithmetic progression. Theorem 2.3.7 ([Rib01], p.527). If I is a fractional ideal of OS , then there is a prime ideal P0 of OS such that [I] = [P0 ] as OS -ideal classes i.e. there is a λ ∈ K such that I = (λ)P0 . The next proposition shows that after slightly enlarging any given set S, we can always write a map (or a point) in normalized form with respect to S. Proposition 2.3.8. Let φ = [F0 : . . . : Fn ] be an endomorphism of Pn , defined over K of degree d with ai,j Xi Fj (X) = where 0 ≤ j ≤ n. |i|=d Then there exists a prime ideal p0 of K and an element α ∈ K such that φ = [α−1 F0 : . . . : α−1 Fn ] is in normalized form with respect to S = S ∪ {p0 }. Proof. Consider the fractional ideal I = (ai,j )i,j OS . Then by Theorem 2.3.7 there is a prime pI of K and αI ∈ K such that I = (αI )pI OS . Consider the representation of φ given by φ = [αI−1 F0 : . . . : αI−1 Fn ] and let S = S∪{pI }. Then vp ((αI−1 ai,j )i,j ) = 0 for every p ∈ / S . In other words, [αI−1 F0 : . . . αI−1 Fn ] is normalized with respect to S . 25 Proposition 2.3.9. For every P = [x0 : . . . : xn ] ∈ Pn (K) there exists a prime ideal p0 of K and an element α ∈ K such that P = [α−1 x0 : . . . α−1 x0 ] is in normalized form with respect to S = S ∪ {p0 }. Proof. The proof follows the proof of the previous proposition. 26 Chapter 3 Arithmetic Dynamics on P1 This chapter will be divided in three sections. The first section will state and prove a fundamental arithmetic relation between K-rational tail points and K-rational periodic points. We will also prove some results needed for the rest of the chapter. The last two sections will give two different approaches to bound the cardinality of PrePer(φ, K) for an endomorphism φ of P1 defined over K. The first approach is using S-unit equations and the second one is using Thue-Mahler equations. 3.1 Main propositions on P1 First we will prove the key arithmetic relation between K-rational tail and periodic points, which will used throughout the chapter. Proposition 3.1.1. Let φ be an endomorphism of P1 , defined over K. Suppose φ has good reduction outside S. Let P ∈ P1 (K) be a periodic point, Q ∈ P1 (K) a fixed point with P = Q and R ∈ P1 (K) a tail point of Q. Then δp (P, R) = 0 for every p ∈ / S. Proof. Let p ∈ / S be a prime of good reduction. Consider P = [p1 : p2 ], Q = [q1 : q2 ], R = [r1 : r2 ] and φ = [F (x, y) : G(x, y)] all in normalized form with respect to p. Let n be the period of P and LQ (x, y) = q2 x − q1 y a linear form defining Q . Given N > 1 consider φN = [FN (x, y) : GN (x, y)] where FN (x, y) = FN −1 (F (x, y), G(x, y)), 27 GN (x, y) = GN −1 (F (x, y), G(x, y)), F1 (x, y) = F (x, y) and G1 (x, y) = G(x, y). By Remark 2.1.6, φN = (FN , GN ) is in normalized form with respect to p and [FN (p1 , p2 ) : GN (p1 , p2 )] is in normalized form with respect to p. Therefore for every m > 0 we can find λ ∈ Op∗ such that Fnm (p1 , p2 ) = λp1 and Gnm (P ) = λp2 . We conclude vp (LQ (Fnm (p1 , p2 ), Gnm (p1 , p2 ))) = vp (LQ (p1 , p2 )) + vp (λ) = vp (LQ (p1 , p2 )). (3.1) Pick m big enough so that φmn (R) = Q. Then LQ (Fnm (r1 , r2 ), Gnm (r1 , r2 )) = 0. Let LR (x, y) = r2 x − r1 y be a linear form defining R, and notice that LQ (x, y), LR (x, y) are factors of LQ (Fnm (x, y), Gnm (x, y)). By Gauss’s lemma, we can find a polynomial H(x, y) ∈ (OS )p [x, y] such that LQ (Fnm (x, y), Gnm (x, y)) = LR (x, y)LQ (x, y)H(x, y). Hence vp (LQ (Fnm (p1 , p2 ), Gnm (p1 , p2 ))) = vp (LR (p1 , p2 )) + vp (LQ (p1 , p2 )) + vp (H(p1 , p2 )). So by (3.1) 0 = vp (LR (p1 , p2 )) + vp (H(p1 , p2 )). Since vp (LR (p1 , p2 )) ≥ 0 and vp (H(p1 , p2 )) ≥ 0 we get vp (LR (p1 , p2 )) = 0. Finally, since R and P are in normalized form with respect to p, we have vp (LR (p1 , p2 )) = δp (P, R) = 0 by Remark 2.2.2. 28 Theorem 3.1.2. Let φ be an endomorphism of P1 , defined over K. Suppose φ has good reduction outside S. Let R ∈ P1 (K) be a tail point and let n be the period of the periodic part of the orbit of R. Let P ∈ P1 (K) be any periodic point that is not φmn (R) for some m. Then δp (P, R) = 0 for every p ∈ / S. Proof. Take the minimum m > 0 such that φmn (R) is a periodic point. By Remark 2.1.6, φn also has good reduction outside S. Now apply the previous proposition using φn for φ, φmn (R) for the fixed point and P as the periodic point different from φmn (R). The last theorem tells us that R is an S-integral point with respect to P (and vice versa). For instance, if P = [x1 : y1 ] and R = [x2 : y2 ] are written with coprime S-integral coordinates, then x1 y2 − x2 y1 is an S-unit. 3.2 S-unit equation approach The goal of this section is to prove two statements. The first gives a bound for | Per(φ, K)| and | Tail(φ, K)| depending only on the amount of places of bad reduction, provided that | Tail(φ, K)| ≥ 3 and | Per(φ, K)| ≥ 4, respectively. Theorem 3.2.1. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. 29 (a) If there are at least three K-rational tail points of φ then | Per(φ, K)| ≤ 216s + 3. (b) If there are at least four K-rational periodic points of φ then | Tail(φ, K)| ≤ 4(216s ). The second theorem gives bounds for | Tail(φ, K)|, | Per(φ, K)| and | PrePer(φ, K)| in terms of |S| and the degree of φ for any endomorphism of P1 . Theorem 3.2.2. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. Then 3 (a) | Per(φ, K)| ≤ 216sd + 3. 3 (b) | Tail(φ, K)| ≤ 4(216sd ). 3 (c) | PrePer(φ, K)| ≤ 5(216sd ) + 3. In the last part of this section we will provide examples to show the necessity of the hypotheses of Theorem 3.2.1. 3.2.1 Proof of Theorem 3.2.1 For this subsection, we will state a notation presented in [Can07]. 30 Let a1 , ..., ah be a full system of integral representatives for the ideal classes of OS . Hence, for each i ∈ {1, .., h} there is an S-integer αi ∈ OS such that ai h = αi OS . Let L be the extension of K given by √ √ L = K(ζ, h α1 , ..., h αh ) where ζ is a primitive h-th root of unity. Consider the following subgroups of L∗ : √ K ∗ := {a ∈ L∗ : ∃m ∈ Z>0 with am ∈ K ∗ } and OS∗ := {a ∈ L∗ : ∃m ∈ Z>0 with am ∈ OS∗ }. Denote by S the set of places of L lying above the places in S and by OS and OS∗ the ring √ of S-integers and the group of S-units, respectively in L. By definition OS∗ ∩ K ∗ = OS∗ and OS∗ is a subgroup of L∗ of free rank s − 1 by Dirichlet’s unit theorem. Lemma 3.2.3. Assume the notation above. There exist fixed representations [xP : yP ] ∈ P1 (L) for every rational point P ∈ P1 (K) satisfying the following two conditions. (a) For every P ∈ P1 (K), we have xP , yP ∈ √ K ∗ and xP OS + yP OS = OS . 31 (b) If P, Q ∈ P1 (K) then xP yQ − yP xQ ∈ √ K ∗. Proof. Let P = [x : y] be a representation of P in P1 (K) and consider b ∈ {a1 , ..., ah } a representative of xOS + yOS . We can find β ∈ K ∗ such that bh = βOS . Then there is λ ∈ K ∗ such that (xOS + yOS )h = λh βOS . (3.2) We define in L x = λ x √ h y = β and with this definition, it is clear that x , y ∈ y √ h λ β √ K ∗ such that x OS + y OS = OS . Furthermore, let P = [x1 , y1 ] and Q = [x2 : y2 ] where xi = λi xi √ h yi = βi λi yi √ h βi and λi , βi are as the ones described in equation (3.2) for i ∈ {1, 2}. Then (x1 y2 − y1 x2 )h = (x1 y2 − y1 x2 )h ∈ K ∗. h h λ1 λ2 β1 β2 The previous representation is given the name S-radical coprime coordinates in recent paper of J. Canci and L. Paladino [CP16] and J. Canci and S. Vishkautsan [CV]. Proof of Theorem 3.2.1 part (a). Let P1 , P2 , P3 be three different K-rational tail points and let ni be the period of the periodic part of the orbit of Pi with i ∈ {1, 2, 3}. Let P be a 32 K-rational periodic point such that φmni (Pi ) = P for every m ∈ Z≥0 and i ∈ {1, 2, 3} (if such a P does not exist then | Per(φ, K)| ≤ 3 and the proof will be complete). By Lemma 3.2.3, for every i ∈ {1, 2, 3} there exist P = [x : y], Pi = [xi : yi ] with x, y, xi , yi ∈ L such that (a) xi OS + yi OS = OS , (b) xOS + yOS = OS , (c) xi y − yi x ∈ √ K ∗. By (a) and (b) we have δp (P, Pi ) = vp (xi y − yi x) for every p ∈ / S and every i ∈ {1, 2, 3}. Using Corollary 3.1.2 we can find S-units u1 , u2 , u3 ∈ OS∗ such that Notice that by (c), ui ∈ x1 y − y1 x = u1 , (3.3) x2 y − y2 x = u2 , (3.4) x3 y − y3 x = u3 . (3.5) √ K ∗ ∩ OS∗ = OS∗ for each i ∈ {1, 2, 3}. Using equations (3.3) and (3.4) we get x= u 2 x1 u 1 x2 − y2 x1 − y1 x2 y2 x1 − y1 x2 and y= u1 y 2 u2 y 1 − . y2 x1 − y1 x2 y2 x1 − y1 x2 Then by (3.5) we get (x3 y2 − y3 x2 )u1 + (y3 x1 − x3 y1 )u2 = (y2 x1 − y1 x2 )u3 . 33 Thus Au + Bv = 1 (x y −y x ) (y x −x y ) −1 where A = (y 3x2 −y3 x2 ) , B = (y3 x1 −y 3x1 ) , u = u1 u−1 3 and v = u2 u3 . 2 1 1 2 2 1 1 2 Notice that A, B = 0 since P2 = P3 , P1 = P3 and the denominator is not 0 since P1 = P2 . Hence by Corollary 2.3.2 with Γ0 = OS∗ , the total number of solutions (u, v) ∈ OS∗ × OS∗ of Au + Bv = 1 is bounded by 28(2s) . From equations (3.3) and (3.5), we can solve for x/y in terms of x1 , y1 , x3 , y3 , u. Therefore there are 28(2s) possible [x : y]. Finally notice that there are at most three periodic points P such that φmni (Pi ) = P for some m ∈ Z≥0 and some i ∈ {1, 2, 3}. Therefore | Per(φ, K)| ≤ 216s + 3. The proof of Theorem 3.2.1 part (b) is similar and requires only minor changes at the start and conclusion of the proof. Proof of Theorem 3.2.1 part (b). Let P1 , P2 , P3 , P4 be 4 different K-rational periodic points and let ni be the period of Pi with i ∈ {1, 2, 3, 4}. Let P be a K-rational tail point such that φmni (P ) = Pi for every m ∈ Z≥0 and i ∈ {1, 2, 3}. By Lemma 3.2.3, for every i ∈ {1, 2, 3} we can take P = [x : y], Pi = [xi : yi ] with x, y, xi , yi ∈ L such that (a) xi OS + yi OS = OS , (b) xOS + yOS = OS , 34 (c) xi y − yi x ∈ √ K ∗. Using the same argument of proof of Theorem 3.2.1 part (a), we get that there are at most 28(2s) possible [x : y]. Now for the K-rational tail points given by φmn1 ([x : y]) = P1 , φmn2 ([x : y]) = P2 , φmn3 ([x : y]) = P3 we use the same argument with the triples (P2 , P3 , P4 ), (P1 , P3 , P4 ) and (P1 , P2 , P4 ), respectively. In each case we get the same bound 216s . Therefore, | Tail(φ, K)| ≤ 4(216s ). 3.2.2 Proof of Theorem 3.2.2 Proof of Theorem 3.2.2. We claim that we can find a field extension of K to a field E such that φ has at least three E-rational tail points (resp. four E-rational periodic points) and [E : K] ≤ d3 . Suppose the claim is true and let S be the set of places of E lying above the places of S. Then Theorem 3.2.2 follows by applying Theorem 3.2.1 to get 3 | Per(φ, K)| ≤ | Per(φ, E)| ≤ 216|S | + 3 = 216|S|d + 3 and 3 | Tail(φ, K)| ≤ | Tail(φ, E)| ≤ 4(216|S | ) = 4(216|S|d ) respectively. Now we just have to prove our claim. Part (a). Assume φ has at least three periodic points; otherwise the bound trivially holds. 35 By the weak Riemann-Hurwitz formula a rational function has at most two totally ramified points. Therefore at least one of our periodic points admits a non-periodic preimage. Let P1 be one possible preimage of such a point and consider E1 the field of definition of P1 over K. Notice that [E1 : K] ≤ d. ¯ a preimage of P1 and P2 , respectively. Let E2 be the field of Consider P2 , P3 ∈ P1 (K) definition of P2 over E1 and E the field of definition of P3 over E2 . Notice that [E2 : E1 ] ≤ d, [E : E2 ] ≤ d, P2 ∈ P1 (E2 ) and P3 ∈ P1 (E). So [E : K] ≤ d3 and φ has at least three E-rational tail points. Part (b). If | Per(φ, K)| > 4 then we can apply Theorem 3.2.1 to get the desired bound. Now assume 1 ≤ | Per(φ, K)| ≤ 3. Case 1: Suppose there exists a point P ∈ P1 (K) of period 3 under φ. Considering the field extension E = K(Q) of K where Q is a fixed point of φ. Notice that [E : K] ≤ d + 1 ≤ d3 by Remark 2.2.4 and φ has at least four E-rational periodic points. Case 2: Suppose there exists no periodic point of period 3 in P1 (K) but there is a point ¯ − P1 (K) of period 3 under φ. Considering the field extension E = K(P ) of K P ∈ P1 (K) we have that φ has a 3-periodic point in P1 (E). Notice that [E : K] ≤ d3 − d ≤ d3 by Remark 2.2.4 and φ has at least four E-rational periodic points since 1 ≤ | Per(φ, K)|. ¯ of period 3 under φ. Then by TheCase 3: Suppose there exists no point P ∈ P1 (K) ¯ of period 2 and two distinct orem 2.2.9 and Remark 2.2.10, φ admits a point P1 ∈ P1 (K) ¯ Since 1 ≤ | Per(φ, K)| ≤ 3 we can assume that at least one of fixed points P2 , P3 ∈ P1 (K). P1 , P2 , P3 is K-rational. Let E = K(P1 , P2 , P3 ). Notice that [E : K] ≤ d3 by Remark 2.2.4 and | Per(φ, E)| ≥ 4. 36 3.2.3 Examples In this subsection we will present two examples that show the sharpness of the hypotheses of Theorem 3.2.1 part (a) and (b). The first example gives a family of rational functions with exactly two Q-rational tail points and a fixed set of places of bad reduction. However the size of the set of Q-rational periodic points grows with the degree of the rational functions in the family. This proves that the hypothesis of Theorem 3.2.1 part (a) is necessary. Example 3.2.4. Consider fd (x) = 1 (x − 2−d )(x − 2−d+1 )...(x − 1)...(x − 2d−1 )(x − 2d ) + x x2d+1 ∈ Q(x). If we take S = {∞, 2} then fd (x) has good reduction outside S. Now notice that 0 and ∞ are tail points and 1 is a fixed point with orbit 0 → ∞ → 1 → 1. Also for every i ∈ {−d, .., −1, 1, .., d} the points 2i are Q-rational periodic points of period 2. Finally by Theorem 3.2.1 if d > 231 + 1, then Tail(fd , Q) = {0, ∞}. Thus, this gives an example of a family of rational functions fd such that each rational function fd has exactly two Q-rational tail points, good reduction outside of a fixed finite set of places S, and the number of Q-rational periodic points grows with the degree of fd (x). The second example gives a family of rational functions with exactly three Q-rational periodic points and a fixed set of places of bad reduction. However the size of the set of Q-rational tail points grows with the degree of the rational functions in the family. This proves that the hypothesis of Theorem 3.2.1 part (b) is necessary. 37 Example 3.2.5. Consider fd (x) = (x − 1)(x − 2)(x − 22 )...(x − 2d−1 ) xd ∈ Q(x). If we take S = {∞, 2} then fd (x) has good reduction outside S. Now we notice that 0 is a periodic point with orbit 0 → ∞ → 1 → 0 and that 2, .., 2d−1 are in the tail of 0. Finally by Theorem 3.2.1 if d > 234 + 1, then Per(fd , Q) = {0, 1, ∞}. Thus, this gives an example of a family of rational functions fd such that each rational function fd has exactly three Q-rational periodic points, good reduction outside of a fixed finite set of places S, and the number of Q-rational tail points grows with the degree of fd (x). 3.3 Thue-Mahler approach The goal of this section is to improve Theorem 3.2.1 and Theorem 3.2.2. In order to do so we will use Thue-Mahler equations instead of S-unit equations. The first result will improve the bound for | Tail(φ, K)| given in the previous section. Theorem 3.3.1. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. Then | Tail(φ, K)| ≤ d max (5 · 106 (d3 + 1))s+4 , 4(264(s+3) ) . Similarly, the second result improves the bound for | Per(φ, K)| given in the previous section. This improvement needs the extra assumption that φ has at least one K-rational 38 tail point. Theorem 3.3.2. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K, and d ≥ 2 the degree of φ. Assume φ has good reduction outside S. If φ has at least one K-rational tail point then | Per(φ, K)| ≤ max (5 · 106 (d − 1))s+3 , 4(2128(s+2) ) + 1. 3.3.1 Proof of Theorem 3.3.1 In this subsection, we assume all the hypotheses of Theorem 3.3.1. Notice that if φ has at least four K-rational periodic points, then by Theorem 3.2.1 | Tail(φ, K)| ≤ 4(216s ). Therefore until the end of the proof of Theorem 3.3.1 we assume | Per(φ, K)| ≤ 3. If | Per(φ, K)| = 0 then | Tail(φ, K)| = 0. So there is nothing to prove in this case. The remaining possibilities can be divided into two cases: when | Per(φ, K)| = 2 or 3 and when | Per(φ, K)| = 1. Before we start analyzing these two cases, we will prove a proposition that will be useful in both. Proposition 3.3.3. Let K be a number field and S a finite set of places of K containing all the archimedean ones. Let φ be an endomorphism of P1 , defined over K and d ≥ 2 the degree of φ. Assume φ has good reduction outside S and φ admits a normalized form with respect to S. Let A ⊂ Tail(φ, K) be such that every point in A admits a normalized form 39 with respect to S. Then |A | ≤ max (5 · 106 (d3 + 1))s+1 , 4(264s ) . ¯ Proof. Suppose that there exists a K-rational periodic point P∗ of period 1, 2 or 3 such that [E : K] ≥ 3 where E = K(P∗ ). Notice that [E : K] ≤ d3 . Let SE be the set of places of E lying above places in S. Applying Proposition 2.3.9 to P∗ and SE , we can find a prime pE in E such that P∗ can be written in normalized form with respect to SE ∪ {pE }. Consider S = S ∪ {pK } where pK is the prime of K lying below pE and let SE be the set of places in E lying above places in S . Let P = [x : y] ∈ A be in normalized form with respect to S and P∗ = [a : b] ∈ P1 (E) in normalized form with respect to SE . Notice that P∗ is not in the orbit of P since it is not K-rational. / SE For every prime pE ∈ / SE , δp (P, P∗ ) = 0. Then for every pE ∈ E vp (ay − bx) = 0. (3.6) E Denote by NE/K the norm from E to K and consider F (X, Y ) = NE/K (aY − bX) ∈ K[X, Y ] where the embedding of E over K act trivially on X and Y . Since P∗ is in normalized form with respect to SE , we have that a, b ∈ OE,S . Hence F (X, Y ) ∈ OK,S [X, Y ]. Notice E that the degree of F is [E : K]. Since P∗ is a root of F (X, Y ) and E is the field of definition of P∗ we have that F (X, Y ) is irreducible over K. Finally using that every P = [x : y] ∈ A is in normalized form with respect to S and equation (3.6) we have F (x, y) ∈ O∗ K,S 40 . Now we have all the hypotheses to apply Theorem 2.3.3. Therefore in this case we get |A | ≤ (5 · 106 [E : K])s+1 ≤ (5 · 106 d3 )s+1 . ¯ Now suppose that for every K-periodic point P of period 1, 2 or 3, we have [K(P ) : K] ≤ 2. We claim that in this case we can find a field E of degree [E : K] ≤ 4 such that φ has at least 4 distinct E-rational periodic points. To prove the claim we just need to use Theorem 2.2.9 and Remark 2.2.10 as follows. ¯ of period 3 under φ. Let Q ∈ P1 (K) ¯ be a fixed point Case 1: There exists a point P ∈ P1 (K) of φ and E = K(P, Q). Then by assumption [E : K] ≤ 4 and we have | Per(φ, E)| ≥ 4. ¯ of period 3 under φ. By Theorem 2.2.9 Case 2: There does not exist a point P ∈ P1 (K) ¯ of period 2 and two distinct fixed and Remark 2.2.10, φ admits a point P1 ∈ P1 (K) ¯ Since 1 ≤ | Per(φ, K)| ≤ 3, we can assume that at least one of points P2 , P3 ∈ P1 (K). P1 , P2 , P3 is K-rational. Let E = K(P1 , P2 , P3 ). Then again we have [E : K] ≤ 4 and | Per(φ, E)| ≥ 4. Then by Theorem 3.2.1 |A | ≤ | Tail(φ, K)| ≤ | Tail(φ, E)| ≤ 4(216(4(s)) ) = 4(264s ). In any case |A | ≤ max (5 · 106 (d3 + 1))s+1 , 4(264s ) . Notice that if OS is a PID then Theorem 3.3.1 follows immediately from Proposition 3.3.3. 41 Proof of Theorem 3.3.1. Case 1: | Per(φ, K)| ∈ {2, 3} By Proposition 2.3.8 we can assume φ is in normalized form with respect to S1 , for some S1 with |S1 | = |S| + 1 and S ⊂ S1 . Let P1 = [x1 : y1 ], P2 = [x2 : y2 ] be two different K-rational periodic points. For every P = [xP : yP ] ∈ Tail(φ, K) there is iP ∈ {1, 2} such that δp (P, PiP ) = 0 for every p ∈ / S1 . Then (xP yiP − yP xiP )OK,S1 = (xP , yP )(xiP , yiP )OK,S1 for every P ∈ Tail(φ, K). Applying Proposition 2.3.9 on P1 , P2 and S1 , we can find a representation of P1 and P2 such that P1 = [x1 : y1 ] and P2 = [x2 : y2 ] are in normalized form with respect to S2 , for some S2 with S1 ⊂ S2 and |S2 | = |S1 | + 2. Hence, for every P ∈ Tail(φ, K) (xP yi − yP xi )OK,S2 = (xP , yP )OK,S2 P P and xP and yP generate a principal OK,S2 -ideal. Therefore, for every P ∈ Tail(φ, K) we −1 x : can find a representation of P that is normalized with respect to S2 , namely P = [αP P −1 y ], where α = x y − y x αP P P P i P i P P (Remark 2.1.3). Every point P ∈ Tail(φ, K) admits a normalized form with respect to S2 and φ is in normalized form with respect to S2 with good reduction outside S2 . Applying Proposition 3.3.3 gives | Tail(φ, K)| ≤ max (5 · 106 (d3 + 1))s+4 , 4(264(s+3) ) . 42 Case 2: | Per(φ, K)| = 1 By Proposition 2.3.8 we can assume φ is in normalized form with respect to S1 , for some S1 with |S1 | = |S| + 1 and S ⊂ S1 . Let Q ∈ P1 (K) be the only K-rational periodic point. Applying Proposition 2.3.9 on Q and S1 , we can find a representation of Q such that Q = [q1 : q2 ] is in normalized form with respect to S2 , for some S2 with S1 ⊂ S2 and |S2 | = |S1 | + 1. Let P = [xP : yP ] ∈ Tail(φ, K). Since φ = [F, G] is in normalized form with respect to S2 and φ has good reduction outside S2 . Thus vp ((F (xP , yP ), G(xP , yP ))) = vp ((xP , yP )d ) for every p ∈ / S2 . Therefore, (F (xP , yP ), G(xP , yP )) = (xP , yP )d for every OK,S2 -ideals. (3.7) n Applying the last equality repeatedly we get that the OK,S2 -ideal class [(xP , yP )d ] = [(q1 , q2 )] = [1] is trivial for some n > 0 depending on P . Assume the notation of Theorem 2.2.7. By Theorem 2.2.7 there are at most two values of n such that a∗Q (n) = ordQ (Φ∗φ,n (X, Y )) = 0. Since a∗Q (1) = 0 we get that either a∗Q (2) = 0 or a∗Q (3) = 0. Set l = min{i : a∗Q (i) = 0}. Consider Φ∗φ,l (X, Y ) and notice that every root of Φ∗φ,l is a periodic point of period 1 or l, different from Q. Let 43 Φ∗φ,l (X, Y ) = cf1 (X, Y )α1 · · · fi (X, Y )αi · · · fr (X, Y )αr be the irreducible factorization of Φ∗φ,l (X, Y ) over K and c ∈ K ∗ . Let ei = deg fi for i = 1, ..., r. Note that the degree of Φ∗φ,l is dl − d. ¯ be a root of fi (X, Y ). Consider Ei = K(Qi ) Fix i ∈ {1, ..., r}. Let Qi = [ai : bi ] ∈ P1 (K) the field of definition of Qi and ei = [Ei : K]. Let SEi be the set of places of Ei lying above places of S2 . Denote by NE /K the norm from Ei to K and notice that fi (X, Y ) = NE /K (ai Y −bi X) ∈ i i K[X, Y ] up to a constant. For every P ∈ Tail(φ, K) and for every pEi ∈ / SEi we have δpE (P, Qi ) = 0. Then i (xP bi − yP ai ) = (ai , bi )(xP , yP ) as OEi ,S -ideals. E i (3.8) Applying NE /K to (3.8) we get i (fi (xP , yP ))OK,S2 = Ii (xP , yP )ei OK,S2 (3.9) where Ii = NE /K ((ai , bi )) is an OK,S2 -ideal. Taking appropriate powers and multiplying i over all i gives (Φ∗φ,l (xP , yP ))OK,S2 = I(xP , yP ) i αi ei OK,S2 (3.10) α where I = Πi Ii i is an OK,S2 -ideal. By Theorem 2.3.7 applied to the OK,S2 -ideal I, there is a prime ideal p0 in K and βI ∈ K such that (βI )I = p0 OK,S2 . Consider S2 = S2 ∪ {p0 }. Then multiplying (3.10) by βI we get 44 l l βI (Φ∗φ,l (xP , yP ))OK,S2 = βI I(xP , yP )d −d OK,S2 = p0 OK,S2 (xP , yP )d −d OK,S2 . Notice that p0 OK,S2 is the trivial ideal in OK,S . Therefore 2 l βI (Φ∗φ,l (xP , yP ))OK,S = (xP , yP )d −d OK,S . (3.11) 2 2 n l Thus, the ideal class of (xP , yP )d −d in OK,S is trivial. Then the ideal class of (xP , yP )d 2 n in OK,S is trivial since the ideal class of (xP , yP )d in OK,S2 is trivial . Taking the g.c.d. 2 of dl − d and dn we get that the ideal class of (xP , yP )d in OK,S is trivial. 2 Let A be the set of all K-rational tail points excluding the initial point in each maximal orbit. Using equation (3.7) and Remark 2.1.3 every point P ∈ A admits a normalized form with respect to S2 . Now applying Proposition 3.3.3 to A and S2 , we get |A | ≤ max (5 · 106 (d3 + 1))s+4 , 4(264(s+3) ) . This gives us | Tail(φ, K)| ≤ d|A | ≤ d max (5 · 106 (d3 + 1))s+4 , 4(264(s+3) ) . 45 3.3.2 Proof of Theorem 3.3.2 In this subsection, we assume the hypotheses in Theorem 3.3.2. Hence | Tail(φ, K)| ≥ 1. Notice that if φ has at least three K-rational tail points, then by Theorem 3.2.1 we have that | Per(φ, K)| < 216s + 3. Therefore in the rest of the section we assume | Tail(φ, K)| ∈ {1, 2}. As before, we will need to prove a proposition to use in the proof of Theorem 3.3.2. Proposition 3.3.4. Let φ be an endomorphism of P1 , defined over K. Let d ≥ 2 be the degree of φ. Assume φ has good reduction outside S and φ is in normalized form with respect to S. Let A ⊂ Per(φ, K) such that every point in A admits a normalized form with respect to S. Then |A | ≤ max (5 · 106 (d − 1))s+1 , 4(2128s ) . 1 (K) such that φ(P ) is a K-rational ¯ Proof. Suppose that for every tail point P∗ ∈ P1 (K)−P ∗ periodic point, [K(P∗ ) : K] ≥ 3 where E = K(P∗ ) is the field of definition of P∗ . Then the same proof as the first part of the proof of Proposition 3.3.3 yields the desired result ( notice that [E : K] ≤ d − 1). ¯ − P1 (K) such that φ(P∗ ) is a KNow suppose that for every tail point P∗ ∈ P1 (K) rational periodic point, [K(P∗ ) : K] < 3. In this case, assume we can find three different K-rational periodic points Q1 , Q2 , Q3 which are not totally ramified (otherwise Per(φ, K) is ¯ − P1 (K) such that trivially bounded ). We can find three different tail points Pi ∈ P1 (K) φ(Pi ) = Qi and 1 ≤ [K(Pi ) : K] ≤ 2 where 1 ≤ i ≤ 3. Applying Theorem 3.2.1 gives |A | ≤ 4(2128s ). 46 Therefore we get |A | ≤ max (5 · 106 (d − 1))s+1 , 4(2128s ) . Notice that if OS is a PID then every point in Per(φ, K) admits a normalized form with respect to S. Thus, in this case Proposition 3.3.4 gives a bound for Per(φ, K) and the hypothesis on the existence of a K-rational tail point will not be required in Theorem 3.3.2. Proof of Theorem 3.3.2. This proof follows the proof of Case I of Theorem 3.3.1 except that we use Proposition 3.3.4 instead of Proposition 3.3.3. After this change we obtain | Per(φ, K)| ≤ max (5 · 106 (d − 1))s+3 , 4(2128(s+2) ) + 1. 47 Chapter 4 Arithmetic Dynamics on Pn This chapter consists of four sections. The first one gives a generalization of the logarithmic p-adic chordal distance and we prove the main tools needed for our study of dynamics in Pn . In the second section we prove some effective results for K-rational tail hypersurfaces. In the third section we prove finiteness of K-rational tail curves on P2 . Finally, in the last section we provide examples of preperiodic hypersurfaces. 4.1 Main definitions and propositions We start this section by giving a generalization of the logarithmic p-adic chordal distance between two points in P1 to a logarithmic v-adic distance between a K-rational hypersurface and a point in Pn with respect to a nonarchimedean place v. Definition 4.1.1. Let H be a hypersurface of Pn defined over K of degree e. Further, ai Xi ∈ K[X] of degree e. suppose H is defined by an homogeneous polynomial f = |i|=e Let v be a nonarchimedean place of K and P = [x0 : · · · : xn ] a point in Pn (K) such that P ∈ / supp(H). We define the logarithmic v-adic distance between P and H with respect to v by δv (P ; H) = v(f (x0 , . . . , xn )) − d min {v(xi )} − min {v(ai )}. 0≤i≤n 48 |i|=e (4.1) Note that δv (P ; H) is independent of the choice of homogeneous coordinates for P and the choice of defining polynomial for H. Now we will prove some important properties of our logarithmic distance. We would like to emphasize that in the following two propositions there is no assumption of irreducibility or preperiodicity on the hypersurface. Proposition 4.1.2. Let H, H be two hypersurfaces of Pn defined over K and v be a nonarchimedean place of K. Consider x, y ∈ Pn (K) such that x ∈ / supp{H} and y ∈ / supp{H +H } then δv (x; H) ≥ 0 (4.2) δv (y; H + H ) = δv (y; H) + δv (y; H ). (4.3) and ai Xi a homogeneous polynomial of degree e and Proof. Suppose H is defined by f (X) = |i|=e bi Xi a homogeneous polynomial of degree e . Without loss of H is defined by g(X) = |i|=e generality assume that min {v(ai )} = 0 and min {v(bi )} = 0. Assume that x = [x0 : · · · : xn ] |i|=e |i|=e and y = [y: · · · : yn ] are in normalized form with respect to v. Proof of (4.2):After our assumptions (4.2) is just v(f (x)) ≥ 0 which is trivially true. ci Xi , where Proof of (4.3): Notice that H + H is defined by f (X)g(X) = |i|=e+e min {v(ci )} = 0. |i|=e+e Now (4.3) is just v(f (y)g(y)) = v(f (y)) + v(g(y)) which is trivially true. The definition of distance above can also be defined for archimedean places; we will just have to change the notation of valuation to absolute values. For our study we will be working 49 with nonarchimedean places so the valuation notation is more convenient in our case. Proposition 4.1.3. Let φ be an endomorphism of Pn , defined over K and S a set finite set of places, including the archimedean ones. Suppose φ has good reduction outside S. Let H be a hypersurface of Pn defined over K and v ∈ / S. If x ∈ Pn (K) such that φ(x) ∈ / supp{H} then δv (z; φ∗ H) = δv (φ(z); H). (4.4) ai Xi a homogeneous polynomial of degree Proof. Suppose H is defined by f (X) = |i|=e e. Without loss of generality assume that min {v(ai )} = 0 and x = [x0 : · · · : xn ] is in |i|=e normalized form with respect to v. Since φ is in normalized form with respect to v and it has good reduction at v we have that φ(z) is in normalized form with respect to v. Notice that φ∗ H is defined by h(X) = f (φ(X)) = cj Xj . Further, since f, φ are in normalized form with respect to v and φ has good reduction outside S, we have that min{v(cj )} = 0. Now (4.4) is just v(h(x)) = v(f (φ(x))) which is trivially true. The following result is a key arithmetic relation between a tail hypersurface and a periodic point. We will be using this property for the rest of the chapter. Proposition 4.1.4. Let φ be an endomorphism of Pn , defined over K. Suppose φ has good reduction outside S. Let P ∈ Pn (K) be a periodic point of φ, and consider H a K-rational fixed hypersurface and H a K-rational tail hypersurface in the tail of H . Suppose that P ∈ / supp{H }. Then δv (P ; H) = 0 for every v ∈ / S. 50 Proof. Let m be the exact period of P , v ∈ / S and M = ml be the smallest natural number such that φM (H) = H . Since P is periodic of period m we have φM (P ) = P . Suppose H is defined by f an irreducible homogeneous polynomial with coefficients in K. Let φ∗M (H ) be the pullback of H by φM and notice that φ∗M (H ) is a hypersurface defined by F = f (φM ). Take the irreducible decomposition of F (over K) given by and let Ci be the prime divisor defined by fi . Then φ∗M (H ) = k fi i ki Ci . Even more, since H is a fixed hypersurface and φM (H) = H we have that φ∗M (H ) = H + H + D for some K-rational hypersurface D (not necessarily irreducible). Using the properties of the logarithmic v-adic distance we get δv (P ; H ) = δv (φM (P ); H ) = δv (P ; φ∗M H ) = δv (P ; H +H+D) = δv (P ; H )+δv (P ; H)+δv (P ; D). Then 0 = δv (P ; H) + δv (P ; D) Since δv (P ; D) ≥ 0 we conclude that δv (P ; H) = 0. Theorem 4.1.5. Let φ be an endomorphism of Pn , defined over K. Suppose φ has good reduction outside S. Let H be a K-rational tail hypersurface, m the period of the periodic part of the orbit of H and H the periodic hypersurface such that H = φm0 m (H) for some m0 > 0. Let P ∈ Pn (K) be any periodic point such that P ∈ / supp{H }. Then δv (P ; H) = 0 for every v ∈ / S. Proof. By Remark 2.1.6, φm also has good reduction outside S. Now apply the previous proposition using φm for φ, H for the fixed hypersurface and P as the periodic point. 51 The following two lemmas will normally be used together with Proposition 2.3.9. We also point out that H is not necessarily an irreducible hypersurface and neither P nor H are assumed to be preperiodic. Lemma 4.1.6. Let H be a K-rational hypersurface of Pn and P a point of Pn (K) that does not lie on H. Assume that P admits a normalized form with respect to S and δv (P ; H) = 0 for every v ∈ / S. Then H admits a normalized form with respect to S. ai Xi . Take P = [x0 : Proof. Suppose H is defined by an homogeneous polynomial f = |i|=e · · · : xn ] in normalized form with respect to S. By definition, for every v ∈ / S, v(f (x0 , . . . , xn )) = min {v(ai )} since δv (P ; H) = 0. Then |i|=e (f (x0 , . . . , xn ))OS = (ai )i OS . Therefore, the ideal generated by (ai )i is a principal OS -ideal which implies that H admits a normalized form with respect to S by Remark 2.1.3. More explicitly, H given by g= (f (x0 , . . . , xn ))−1 ai Xi is in normalized form with respect to S. |i|=d To have a similar result in the other direction we have to use that H is a hyperplane. Lemma 4.1.7. Let H be a K-rational hyperplane of Pn and P a point of Pn (K) that does not lie on H. Assume that H admits a normalized form with respect to S and δv (P ; H) = 0 for every v ∈ / S. Then P admits a normalized form with respect to S. Proof. The proof is analogous to the previous one. 52 4.2 Effective results For the rest of the section we fix e ∈ N and N (e) = N = e+n − 1. e Consider the e-th veronese embedding Ψe : Pn → PN x → xi |i|=e = (wi )|i|=e . With this embedding in mind we can associate a point of Pn to a hyperplane of PN as follows: xi w i . P = x → HP : |i|=e Similarly, we associate a degree e hypersurface of Pn to a point of PN as follows: ci Xi → PC = (ci )|i|=e . C: |i|=e We notice that HP (PC ) = C(P ). (4.5) Let V be the subvariety of (Pn )N +1 defined by the determinant of the square matrix M, where the i-th row of M consists of all monomials of degree e in the coordinate variables of the i-th copy of Pn for 0 ≤ i ≤ N . For the rest of the section we assume the above notation for M, V and for the association of points and hypersurfaces of degree e to hyperplanes and points, respectively. Our objective now is to use this association together with (4.5) to study | HTail(φ, K, e)| for an endomorphism φ of Pn , defined over K. More precisely, we will give an effective bound for 53 the cardinality of a large subset of HTail(φ, K, e). First, we need to establish some condition on a set of points of Pn , in order to guarantee general position for the hyperplanes of PN associated to those points. Proposition 4.2.1. Let A = {Pi }li=1 be a set of points of Pn with l ≥ N + 1. Then the associated set of hyperplanes B = is a point (Q0 , . . . , QN ) ∈ AN +1 ⊂ HPi l is not in general position if and only if there i=1 (Pn )N +1 with different coordinates that lies in V if and only if there are N + 1 points of A that lie in hypersurface of degree e. Proof. To simplify notation we write Hi for HPi for 1 ≤ i ≤ l. Notice that: B is not in general position if and only if there are N + 1 hyperplanes Hi0 , . . . , HiN of B such that    Ψe (Pi0 )       Ψe (Pi )   1  det  =0   ..   .     Ψe (PiN ) if and only if the point (Pi0 , . . . , PiN ) lies in V if and only if there is a non-zero vector v = (vi )|i|=e such that    Ψe (Pi0 )       Ψe (Pi )   1   v = 0   ..   .     Ψe (PiN ) 54 vi xi , where x consists of the if and only if the points Pi0 , . . . , PiN are zeros of |i|=e coordinate variables of Pn . Just for the next theorem we assume the following notation: Let φ be an endomorphism of Pn , defined over K. For T ∈ HTail(φ, K, e) we denote by nT the the exact period of the periodic part of the orbit of T . Theorem 4.2.2. Let φ be an endomorphism of Pn , defined over K and suppose φ has good +1 n reduction outside S. Let {Pi }2N i=1 be a set of K-rational periodic points of P . Assume that no N + 1 of them lie in a hypersurface of degree e. Consider B = {H ∈ HPer(φ, K) : ∀1 ≤ i ≤ 2N + 1, Pi ∈ / supp H } and A = {T ∈ HTail(φ, K, e) : there is l ≥ 0, φlnT (T ) ∈ B}. Then |A| ≤ 233 · (2N + 1)2 (N +1)3 (s+2N +1) +1 is in normalized Proof. We can assume by Proposition 2.3.9 that every point of {Pi }2N i=1 +1 form with respect to S1 , for some S1 with |S1 | = |S| + 2N + 1 and S ⊂ S1 . Since {Pi }2N i=1 +1 are are in normalized form with respect to S1 then the associated hyperplanes {HPi }2N i=1 in normalized form with respect to S1 . To simplify notation, for the rest of the proof we write Hi for HPi . By Lemma 4.1.6 every T ∈ A admits a normalized form with respect to S1 because P1 is in normalized form with respect to S1 and δv (P1 ; T ) = 0 for every v ∈ / S1 (by Theorem 4.1.5). Take T ∈ A in normalized form with respect to S1 and PT be the associated point in PN . Then by (4.5) and Theorem 4.1.5 we have that Hi (PT ) ∈ OS∗ for every 1 ≤ i ≤ 2N + 1. 1 55 Hence PT is a solution of the system      H1 (x0 , . . . , xN ) ∈ OS∗   1    .. .         H (x0 , . . . , x ) ∈ O∗ 2N +1 N with (x0 , . . . , xN ) ∈ OSN +1 (4.6) 1 S1 Notice that the solutions of the set of OS∗ -cosets of solution of the system (4.6) are 1 2N +1 Hi , S1 )-integral point. Then by Theorem 2.3.5 the set of OS∗ -cosets of solution of the ( 1 1 system (4.6) is finite. Finally, by Theorem 2.3.6 the number of OS∗ -cosets of solutions of 1 F (x0 , . . . , xN ) = H1 (x0 , . . . , xN ) · · · H2N +1 (x0 , . . . , xN ) ∈ OS∗ with 1 is bounded by 233 · (2N + 1)2 (N +1)3 (s+2N +1) |A| ≤ 233 · (2N + 1)2 4.3 (x0 , . . . , xN ) ∈ OSN +1 . This implies that (N +1)3 (s+2N +1) . Finiteness Finiteness for the set of K-rational preperiodic subvarieties of Pn has been proven by B. Hutz [Hut16] using the theory of canonical height functions. Our goal in this section is to give an alternative proof to the one given by Hutz. We divide this section in two subsections. In the first one we give a result on finiteness for the set of K-rational tail hypersurfaces on 56 1 the backward orbit of a K-rational periodic hypersurface. In the second subsection we prove finiteness of the set of K-rational preperiodic curves for an endomorphism of P2 . 4.3.1 Finiteness on Pn For this subsection we assume that e is a fixed natural number and N (e) = N = e+n e − 1. We start recalling a strong result from dynamical systems. Theorem 4.3.1 ([Fak03], Corollary 5.2). ¯ of degree d ≥ 2. Then the set Per(φ, K) ¯ is Zariski Let φ be an endomorphism of Pn over K dense in Pn . Now we can prove a lemma which will have many application in this section. Lemma 4.3.2. Let φ be an endomorphism of Pn , defined over K and Y Pn a closed ¯ \ Y such that no N + 1 lie subset of Pn . For any l ∈ N, there are l periodic points in Pn (K) on a curve of degree e. Proof. First notice that it is enough to prove the result for l > N , so Without loss of generality assume l > N . Let I be a subset of {1, . . . , l} of size N + 1. Let φI : (Pn )l → (Pn )N +1 be the projection onto (Pn )N +1 corresponding to the elements of I. φ−1 I (V) where V is the Consider the proper algebraic subset of (Pn )l given by X = I subvariety of (Pn )N +1 defined at the beginning of the section. Take a point P = (P1 , . . . , Pl ) ∈ (Pn )l and notice that by Proposition 4.2.1 P lies on X if and only if no subset of N + 1 distinct points in {P1 , . . . , Pl } lie on a curve of degree e. For every 1 ≤ i ≤ l take πi : (Pn )l → Pn the projection of the i-coordinate onto Pn . πi−1 (Y ). Since Consider the proper algebraic subset of (Pn )l given by X = X ∪ 1≤i≤l 57 ¯ is a Zariski dense subset of Pn then Per(φ, K ¯ l is Zariski dense in (Pn )l . Hence, Per(φ, K) ¯ l \ X . So {Q1 , . . . , Ql } is the desired set of we can find a point (Q1 , . . . , Ql ) ∈ Per(φ, K) points. Theorem 4.3.3. Let φ be an endomorphism of Pn , defined over K and H be a K-rational periodic hypersurface. If A = {T ∈ HTail(φ, K, e) : ∃m > 0 φm (T ) = H} then |A| is finite. In other words, the set HTail(φ, K, e) ∩ O−1 (H) is finite. Proof. First notice that it is enough to prove the theorem for H a fixed hypersurface. Without loss of generality assume that H is a fixed hypersurface. Let S be the set of places of bad reduction of φ (including the archimedean places). By Lemma 4.3.2 we can find 2N + 1 periodic points such that none lie on H and no N + 1 of them lie on a curve of degree e. Let E be the field of definition of our 2N + 1 points and SE the set of places of E lying above the places of S. Now applying Theorem 4.2.2 we get that the cardinality of B = {T ∈ HTail(φ, E, e) : ∃m > 0 φm (T ) = H} is bounded. Since A ⊂ B we get that A is finite. 4.3.2 Finiteness on P2 Now we have all of the tools to prove finiteness of K-rational tail curves of a given degree. Theorem 4.3.4. Let K be a number field and φ be an endomorphism of P2 , defined over K. Then for every e ∈ N the set HTail(φ, K, e) is finite. 58 Proof. Let e be a fixed natural number, S be the set of places of bad reduction of φ (including the archimedean places) and N = e+2 2 − 1. We split this proof in two main cases. CASE 1: Suppose that all K-rational periodic curves have degree at most e. For this case we assume the following notation: If T ∈ HTail(φ, K, e) we denote by nT the exact period of the periodic part of the orbit of T . By Lemma 4.3.2 we can find 2N + 1 + e2 periodic points such that no N + 1 of them lie on a curve of degree e. We denote these points by P = {Pi }l1 where l = 2N + 1 + e2 . Let E be the field of definition of the points P and SE the set of places of E lying above the places of S. Let I be a subset of {1, . . . , l} of size 2N + 1. Consider BI = {H ∈ HPer(φ, E) : ∀i ∈ I Pi ∈ / supp H } and AI = {T ∈ HTail(φ, E, e) : there is l ≥ 0 φlnT (T ) ∈ BI }. Notice that AI is finite by Theorem 4.2.2. Consider D = {H ∈ HPer(φ, E) : at least e2 + 1 points of P are in supp H } and C = {T ∈ HTail(φ, E, e) : there is l ≥ 0 φlnT (T ) ∈ D}. By Bezout’s Theorem |D| ≤ l e2 +1 and applying Theorem 4.3.3 we have that the set C is finite. Finally we notice that HTail(φ, K, e) ⊂ C ∪ AI . So | HTail(φ, K, e)| < ∞. I CASE 2: Suppose there is a K-rational periodic curve H that has degree f > e. Let n be the exact period of H, ψ = φn and B = {T ∈ HTail(φ, K, e) : ∃m > 0 φmn (T ) = H}. Let T ∈ HTail(φ, K, e) \ B, nT the period of the periodic part of T and HT = φmnT (T ) ∈ HPer(φ, K) for some m > 0. Then we claim that every point in T ∩ H is preperiodic with respect to ψ. Indeed, notice that if P ∈ T ∩ H then all but finitely 59 many points of OψnT (P ) will lie on H ∩ HT and by Bezout’s theorem H and HT intersect in finitely many points. Therefore OψnT (P ) is finite which implies that Oψ (P ) is finite, i.e. every point of T ∩ H is preperiodic. We also notice that every point in T ∩ H has degree bounded by ef over K. Let R be the set of preperiodic points of ψ restricted to H of degree at most ef over K. This set is finite by Northcott’s Theorem. We claim that the set of curves T of degree e such that T ∩ H ⊂ R is finite. First notice that up to multiplication by constants, there are finitely many functions on H of degree at most ef that are regular on H \ R and their multiplicative inverses are also regular on H \ R (i.e. no zeros or poles). Fix a curve T0 of degree e such that T0 ∩ H ⊂ R and let t0 be an homogeneous polynomial of degree e defining T0 . If T is a curve of degree e such that T ∩ H ⊂ R, t then let t be a homogeneous polynomial of degree e defining T . Then the function t0 on H will be a function of degree at most ef with no zeros or poles on H \ R. Now consider T1 and T2 two distinct curves of degree e such that T1 ∩ H ⊂ R and T2 ∩ H ⊂ R. Let t1 and t2 be two homogeneous polynomials of degree e defining T1 t t and T2 respectively. Then the functions 1 and 2 on H do not differ by multiplication t0 t0 t t by a constant because if there were a constant c such that 1 = c 2 then t1 − ct2 will t0 t0 be divisible by the polynomial defining H and this is impossible since f > e. This finishes our claim. Therefore, the set HTail(φ, K, e) \ B is finite and by Theorem 4.3.3 on H and ψ we have that B is finite. In other words, 60 | HTail(φ, K, e)| < ∞. 4.4 Examples We will give three examples of preperiodic hypersurfaces of an endomorphism of Pn . Example 4.4.1. Let Sn+1 be the symmetric group on n + 1 elements, σ ∈ Sn+1 and φσ,d = [xdσ(0) , ..., xdσ(n) ] a degree d endomorphism of Pn . Then all the canonical hyperplanes Hi defined by xi are periodic. Further, their orbits are given by the decomposition into disjoint cycles of σ. Example 4.4.2. Consider the morphism ψd : P2 → P2 (x : y : z) → (xd : y d : z d ). i i For any i ∈ N take the curve Ci ⊂ P2 defined by z d −1 y = xd . Notice that Ci is an irreducible curve of degree di . Further Ci is fixed under ψd . i i If ρ is a root of unity then the curve Cρ defined by z d −1 y = ρxd is preperiodic under ψd . Further, if ρ is a dk -root of unity for some k ∈ N then Cρ is in the tail of Ci . Otherwise, Cρ is periodic. Therefore, ψd has infinitely many Q-rational periodic curves. Notice that the degree of the fixed curves go to infinity. 61 It is important to notice that a number field contains only finitely many roots of unity. Therefore, in this way we can construct only finitely many tail curves that are rational over a given number field. This example illustrates the importance of including the degree of a hypersurface as a parameter in order to obtain finiteness results for K-rational preperiodic hypersurfaces. Example 4.4.3 (Bell, Ghioca and Tucker [BGT15]). Let f be a homogeneous two-variable polynomial of degree d and consider the morphism ψ : P2 → P2 (x : y : z) → [f (x, z) : f (y, z) : z d ]. k Then ψ has infinetely many fixed curves. Indeed, all curves of the form [xz d −1 : f k (x, z) : k z d ] are fixed under ψ, where f k is the homogenized kth iterate of the dehomogenized one- variable polynomial x → f (x, 1). Like the previous example, we have infinitely many fixed curves. However just like before, the degree of the the fixed curves go to infinity. 62 BIBLIOGRAPHY 63 BIBLIOGRAPHY [Bak64] I. Baker. Fixpoints of polynomials and rational functions. J. London Math. Soc., 39:615–622, 1964. [BCH+ 14] R. Benedetto, R. Chen, T. Hyde, Y. Kovacheva, and C. White. Small dynamical heights for quadratic polynomials and rational functions. Exp. Math., 23(4):433– 447, 2014. [Ben07] R. Benedetto. Preperiodic points of polynomials over global fields. J. Reine Angew. Math., 608:123–153, 2007. [BGT15] J. Bell, D. Ghioca, and T. Tucker. Applications of p-adic analysis for bounding periods for subvarieties under ´etale maps. Int. Math. Res. Not. IMRN, (11):3576– 3597, 2015. [BGT16] J. Bell, D. Ghioca, and T. Tucker. The dynamical Mordell-Lang conjecture, volume 210 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2016. [BS96] F. Beukers and H. Schlickewei. The equation x + y = 1 in finitely generated groups. Acta Arith., 78(2):189–199, 1996. [Can07] J. Canci. Finite orbits for rational functions. Indag. Math. (N.S.), 18(2):203–214, 2007. [Can10] J. Canci. Rational periodic points for quadratic maps. Ann. Inst. Fourier (Grenoble), 60(3):953–985, 2010. [CP16] J. Canci and L. Paladino. Preperiodic points for rational functions defined over a global field in terms of good reduction. Proc. Amer. Math. Soc., 144(12):5141– 5158, 2016. [CV] J. Canci and S. Vishkautsan. Scarcity of cycles for rational functions over a number field. Trans. Amer. Math. Soc. Ser. B, to appear. [Eve95] J. Evertse. The number of solutions of decomposable form equations. Invent. Math., 122(3):559–601, 1995. [Eve97] J. Evertse. The number of solutions of the Thue-Mahler equation. J. Reine Angew. Math., 482:121–149, 1997. 64 [Fak01] N. Fakhruddin. Boundedness results for periodic points on algebraic varieties. Proc. Indian Acad. Sci. Math. Sci., 111(2):173–178, 2001. [Fak03] N. Fakhruddin. Questions on self maps of algebraic varieties. J. Ramanujan Math. Soc., 18(2):109–122, 2003. [FHI+ 09] X. Faber, B. Hutz, P. Ingram, R. Jones, M. Manes, T. Tucker, and M. Zieve. Uniform bounds on pre-images under quadratic dynamical systems. Math. Res. Lett., 16(1):87–101, 2009. [Ghi07] D. Ghioca. The Lehmer inequality and the Mordell-Weil theorem for Drinfeld modules. J. Number Theory, 122(1):37–68, 2007. [GTZ11] D. Ghioca, T. Tucker, and S. Zhang. Towards a dynamical Manin-Mumford conjecture. Int. Math. Res. Not. IMRN, (22):5109–5122, 2011. [HI13] B. Hutz and P. Ingram. On Poonen’s conjecture concerning rational preperiodic points of quadratic maps. Rocky Mountain J. Math., 43(1):193–204, 2013. [Hut15] B. Hutz. Determination of all rational preperiodic points for morphisms of PN. Math. Comp., 84(291):289–308, 2015. [Hut16] B. Hutz. Good reduction and canonical heights of subvarieties. ArXiv e-prints, March 2016. [Kis95] M. Kisaka. On some exceptional rational maps. Proc. Japan Acad. Ser. A Math. Sci., 71(2):35–38, 1995. [Man07] M. Manes. Arithmetic dynamics of rational maps. ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–Brown University. [Maz77] ´ B. Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Etudes Sci. Publ. Math., (47):33–186 (1978), 1977. [Mer96] L. Merel. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math., 124(1-3):437–449, 1996. [MN06] R. Marszalek and W. Narkiewicz. Finite polynomial orbits in quadratic rings. Ramanujan J., 12(1):91–130, 2006. [MS94] P. Morton and J. Silverman. Rational periodic points of rational functions. Internat. Math. Res. Notices, (2):97–110, 1994. [Nor50] D. Northcott. Periodic points on an algebraic variety. Ann. of Math. (2), 51:167– 177, 1950. 65 [NW02] J. Noguchi and J. Winkelmann. Holomorphic curves and integral points off divisors. Math. Z., 239(3):593–610, 2002. [Poo98] B. Poonen. The classification of rational preperiodic points of quadratic polynomials over Q: a refined conjecture. Math. Z., 228(1):11–29, 1998. [Rib01] P. Ribenboim. Classical theory of algebraic numbers. Universitext. SpringerVerlag, New York, 2001. [RW91] M. Ru and P. Wong. Integral points of Pn − {2n + 1 hyperplanes in general position}. Invent. Math., 106(1):195–216, 1991. [Sil07] J. Silverman. The arithmetic of dynamical systems, volume 241 of Graduate Texts in Mathematics. Springer, New York, 2007. [Tro] S. Troncoso. Bound for preperiodic points for maps with good reduction. J. Number Theory, to appear. [Voj87] P. Vojta. Diophantine approximations and value distribution theory, volume 1239 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1987. 66