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Backward Orbit Conjecture for Lattès Maps

Vijay Sookdeo

Department of Mathematics, The Catholic University of America, Washington, DC

Abstract: For a Lattès map defined over a number field K, we prove a conjecture on the integrality of points in the backward orbit of under

Keywords: backward orbit conjecture, Lattès maps

Received March 14, 2015; Revised May 15, 2015; Accepted July 12, 2015

Citation: Vijay Sookdeo, Backward Orbit Conjecture for Lattès Maps. Turkish Journal of Analysis and Number Theory, 3 (3). 75-77. January 2015. doi:10.12691/tjant-3-3-1

1. Introduction

Let be a rational map of degree ≥ 2 defined over a number field and write for the nth iterate of For a point let be the forward orbit of under and let

be the backward orbit of under We say P is -preperiodic if and only if is finite.

Viewing the projective line as and taking a theorem of Silverman [4] states that if is not a fixed point for then contains at most finitely many points in the ring of algebraic integers in If is the set of all archimedean places for then is the set of points in which are S-integral relative to (see section 2). Replacing with any point and S with any finite set of places containing all the archimedean places, Silverman's Theorem can be stated as: If is not a fixed point for then contains at most finitely many points which are S-integral relative to

A conjecture for finiteness of integral points in backward orbits was stated in [[6], Conj. 1.2].

Conjecture 1.1. If is not S-preperiodic, then contains at most finitely many points in which are S-integral relative to

In [6], Conjecture 1.1 was shown true for the powering map with degree and consequently for Chebyschev polynomials. A gener-alized version of this conjecture, which is stated over a dynamical family of maps is given in [[1], Sec. 4]. Along those lines, our goal is to prove a general form of Conjecture 1.1 where is the family of Lattès maps associate to a fixed elliptic curve E defined over K (see Section 3).

2. The Chordal Metric and Integrality

2.1. The Chordal Metric on Let be the set of places on K normalized so that the product formula holds: for all

For points and in define the v-adic chordal metric as

Note that is independent of choice of projective coordinates for P and Q, and (see [2]).

2.2. Integrality on Projective Curves. Let C be an irreducible curve in defined over K and S a finite subset of which includes all the archimedean places. A divisor on C defined over is a finite formal sum with and The divisor is effective if for each i, and its support is the set Supp(D) =

Let and when This makes an arithmetic distance function on C (see [3]) and as with any arithmetic distance function, we may use it to classify the integral points on C.

For an effective divisor on C defined over , we say is S-integral relative to D, or P is a (D, S)-integral point, if and only if for all embeddings and for all places Furthermore, we say the set is S-integral relative to D if and only if each point in is S-integral relative to D.

As an example, let C be the projective line S be the Archimedean place of and For with x and y are relatively prime in we have for each prime v. Therefore, P is S-integral relative to D if and only if that is, P is S-integral relative to D is and only if

From the definition we find that if are finite subsets of which contains all the archimedean places, then P is a -integral point implies that P is a -integral point. Similarly, if Supp ⊂ Supp, then P is a -integral point implies that P is also a -integral point. Therefore enlarging S or Supp(D) only enlarges the set of -integrals points on

For a finite morphism between projective curves and write

where is the ramification index of at Q. Furthermore, if is a divisor on C, then we define

Theorem 2.1 (Distribution Relation). Let be a finite mor-phism between irreducibly smooth curves in Then for there is a finite set of places S, depending only on and containing all the archimedean places, such that for all

Proof. See [[3], Prop. 6.2b] and note that for projective varieties the term is not required, and that the big-O constant is an -bounded constant not depending on P and Q.

Corollary 2.2. Let be a finite morphism between irreducibly smooth curves in let and let D be an effective divisor on C₂ defined over K. Then there is a finite set of places S, depending only on and containing all the archimedean places, such that is S-integral relative to D if and only P is S-integral relative to

Proof. Extend S so that the conclusion of Theorem 2.1 holds. Then for with each and we have that.

So if and only if

3. Main Result

Let E be an elliptic curve, a morphism, and be a finite covering. A Lattès map is a rational map making the following diagram commute:

For instance, if E is defined by the Weierstrass equation is the multiplication-by-2 endomorphism on E, and then

Fix an elliptic curve E defined over a number field K, and for define:

A point Q is -preperiodic if and only if Q is -preperiodic for some We write for the set of -preperiodic points in

Theorem 3.1. If is not -periodic, then contains at most finitely many points in which are S-integral relative to Q.

Proof. Let be the End(E)-submodule of that is finitely generated by the points in and let

Then Indeed, if is not -preperiodic, then is non torsion and for some Lattès map . So for some morphism and this gives for some Lattès map . Therefore for some morphism Since any morphism is of the form where and (see [[5], 6.19]), we find that there is a such that is in , the End(E)-submodule generated by Otherwise, if is -preperiodic, then ([[5], Prop. 6.44]) gives that may be a torsion point; again since Hence

Let D be an effective divisor whose support lies entirely in let be the set of points in which are S-integral relative to Q, and let be the set of points in which are S-integral relative to D. Extending S so that Theorem 2.1 holds for the map , and since Supp(D) ⊂ Supp, we have: if is S-integral relative to Q, then is S-integral relative to D. Therefore . Now is a finite map and so to complete the proof, it suffices to show that D can be chosen so that is finite.

From [[5], Prop. 6.37], we find that if is a nontrivial subgroup of Aut(E), then and the map can be determine explicitly. The four possibilities for , which are or correspond respectively to the four possibilities for , which are or which in turn depends only on the j-invariant of E. (Here, denotes the Nth roots of unity in .)

First assume that . Since Q is not [']-preperiodic, take to be non torsion. Then since and is non-torsion. Taking [[1], Thm. 3.9(i)] gives that is finite.

Suppose that Then where and is non-torsion since Q is not -preperiodic. Assuming that both and are torsion give that is torsion, and this contradicts the fact that is torsion. Therefore, we may assume that is non-torsion. Now taking [[1], Thm. 3.9(i)] again gives that is finite. Hence RQ, the set of points in which are S-integral relative to Q, is finite.

References

[1]   | David Grant and Su-Ion Ih, Integral division points on curves, Compositio Math-ematica 149 (2013), no. 12, 2011-2035.
[2]   | Shu Kawaguchi and J. H. Silverman, Nonarchimedean green functions and dynam-ics on projective space, Mathematische Zeitschrift 262 (2009), no. 1, 173-197.
[3]   | J. H. Silverman, Arithmetic distance functions and height functions in Diophantine geometry, Mathematische Annalen 279 (1987), no. 2, 193-216.
[4]   | Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J. 71 (1993), no. 3, 793-829.
[5]   | The arithmetic of dynamical systems, Graduate Text in Mathematics 241, Springer, New York, 2007.
[6]   | V. A. Sookdeo, Integer points in backward orbits, J. Number Theory 131 (2011), no. 7, 1229-1239.