

Backward Orbit Conjecture for Lattès Maps
Department of Mathematics, The Catholic University of America, Washington, DCAbstract
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
Copyright © 2015 Science and Education Publishing. All Rights Reserved.Cite this article:
- Vijay Sookdeo. Backward Orbit Conjecture for Lattès Maps. Turkish Journal of Analysis and Number Theory. Vol. 3, No. 3, 2015, pp 75-77. http://pubs.sciepub.com/tjant/3/3/1
- Sookdeo, Vijay. "Backward Orbit Conjecture for Lattès Maps." Turkish Journal of Analysis and Number Theory 3.3 (2015): 75-77.
- Sookdeo, V. (2015). Backward Orbit Conjecture for Lattès Maps. Turkish Journal of Analysis and Number Theory, 3(3), 75-77.
- Sookdeo, Vijay. "Backward Orbit Conjecture for Lattès Maps." Turkish Journal of Analysis and Number Theory 3, no. 3 (2015): 75-77.
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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
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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
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For points and
in
define the v-adic chordal metric as
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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
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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.
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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:
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For instance, if E is defined by the Weierstrass equation
is the multiplication-by-2 endomorphism on E, and
then
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Fix an elliptic curve E defined over a number field K, and for define:
![]() |
![]() |
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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
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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. | ||
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[2] | Shu Kawaguchi and J. H. Silverman, Nonarchimedean green functions and dynam-ics on projective space, Mathematische Zeitschrift 262 (2009), no. 1, 173-197. | ||
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[3] | J. H. Silverman, Arithmetic distance functions and height functions in Diophantine geometry, Mathematische Annalen 279 (1987), no. 2, 193-216. | ||
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[4] | Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J. 71 (1993), no. 3, 793-829. | ||
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[5] | The arithmetic of dynamical systems, Graduate Text in Mathematics 241, Springer, New York, 2007. | ||
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[6] | V. A. Sookdeo, Integer points in backward orbits, J. Number Theory 131 (2011), no. 7, 1229-1239. | ||
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