Integral Points and Relative Sizes of Coordinates of Orbits in
We give a generalization to higher dimensions of Silverman’s result on finiteness of integer points in orbits. Assuming Vojta’s conjecture, we prove a sufficient condition for morphisms on so that -integral points in each orbit are Zariski-non-dense. This condition is geometric, and for dimension it corresponds precisely to Silverman’s hypothesis that the second iterate of the map is not a polynomial. In fact, we will prove a more precise formulation comparing local heights outside to the global height. For hyperplanes, this amounts to comparing logarithmic sizes of the coordinates, generalizing Silverman’s precise version in dimension . We also discuss a variant where we can conclude that integral points in orbits are finite, rather than just Zariski-non-dense. Further, we show unconditional results and examples, using Schmidt’s subspace theorem and known cases of Lang–Vojta conjecture. We end with some extensions to the case of rational maps and to the case when the arithmetic of the orbit under one map is controlled by the geometric properties of another. We include many explicit examples to illustrate different behaviors of integral points in orbits in higher dimensions.
Key words: Integral points, orbits, higher-dimensional dynamics, Vojta’s conjecture
Mathematics Subject Classification: 37P55 11J97 37P15
Dynamics is a field studying asymptotic behavior of iterations of a self-map. We will denote the -fold iterations by ; we will not use higher derivatives so this should not cause any confusion. Dynamics was classically studied over , but more recently arithmetic dynamics, studying number-theoretic behaviors of self-maps defined over number fields, has now also become a very active field. For example, one would like to know how frequently “integral points” occur on the orbit of a rational point . Silverman  answered this question for rational functions in one variable, as follows:
Theorem 1 (Silverman).
Let be a rational function of degree .
(i) If is not a polynomial (i.e. not in ), then is a finite set for any .
(ii) Assume that neither nor is a polynomial. If we write in a reduced form, then for any with ,
Viewing a rational function as a morphism on , the hypothesis of being a polynomial is equivalent to having a totally ramified fixed point at . So the upshot is that unless satisfies a special ramification property, only finitely many points in any orbit are integral and in fact orbit points asymptotically become further and further away from being integral. The proof of this theorem uses the powerful Diophantine approximation theorem of Roth, together with a combinatorial description of ramification on iterations based on Riemann–Hurwitz formula.
In this article, we will prove several results that generalize Silverman’s theorem to higher-dimensions. Dynamics in higher-dimension is in general very difficult. Some results are known for powers of : for example, the dynamical Mordell–Lang conjecture can be proved for on under some conditions on , the dynamical Manin–Mumford conjecture is known for lines in , and integral points in orbits of on are known to be finite . These may appear to be highly special at the first glance, but they are already quite difficult. Dynamics on projective spaces in which multiple variables intermingle seems to be even more challenging.
There are also obstacles specific to generalizing Silverman’s result to higher-dimensions. Since the notion of integrality is defined with respect to a divisor, it is natural to look at the pullbacks of this divisor by iterates, just as Silverman analyzes poles of iterates. However, here is one difficulty: irreducible divisors in are simply points, but irreducible divisors in higher-dimensions can be highly singular and can even have self-intersections. Moreover, while there has been great recent progress in Diophantine approximation such as , unfortunately they are not strong enough for our purposes. Thus, we will resort to a very deep Diophantine conjecture of Vojta [15, Conjecture 3.4.3] to treat general situations, while giving many illuminating examples where the use of this conjecture can be avoided. We note that a stronger version of Vojta’s conjecture has been applied to arithmetic dynamics in  to analyze dynamical Zsigmondy sets for .
We now discuss our main results. We will always use the convention that the degree of a rational map is the polarization degree. Given an effective divisor on defined over a number field, we consider all the normal-crossings subdivisors of over and the one with the highest degree will be called a normal-crossings part of , denoted by . We denote the pullback by and its normal-crossings part by when the map is clear. We are now ready to state the main results.
Theorem 2 (cf. Theorem 4).
Let be a morphism defined over of degree , and let be a divisor on . If for some , then Vojta’s conjecture on for the divisor implies that for any , is Zariski-non-dense.
If is the hyperplane , then by writing with with common divisor , is in if and only if . So the theorem says that the Zariski-closure of integer-coordinate points in orbits is not all of . Note that Theorem 2 for is exactly Theorem 1 (i). Indeed, using Riemann–Hurwitz, Silverman proves that is not a polynomial if and only if some iterate has at least three distinct poles. Since Zariski-non-denseness is equivalent to finiteness on , Theorem 2 for completely agrees with Silverman’s result. This is an upgrade from a prior work , as the hypothesis in the main theorem there was strictly stronger for than Silverman’s theorem.
Just as in dimension , we can also obtain a more precise version involving the number of digits of the coordinates, albeit assuming Vojta’s conjecture for more divisors:
Theorem 3 (cf. Corollary 3).
Let be the hyperplane , and let . Assuming Vojta’s conjecture on , for any and , is Zariski-non-dense.
To make up for having to use a deep conjecture to treat general maps, we include many explicit examples throughout the paper which do not require Vojta’s conjecture. In fact, the examples are important components of this paper. In particular, Examples 1, 3, and 4 provide different circumstances where assuming this conjecture can be circumvented, and Examples 6, 7, and 8 analyze interesting rational maps without assuming any conjecture. They should shed some light to the spectrum of behaviors for integral points in orbits in higher-dimensions.
We now describe the results and examples in this paper in slightly more detail. In Section 2, we recall definitions of global and local heights and of normal-crossings divisors, and then discuss Vojta’s conjecture. In Section 3, we will prove the number-field versions of Theorems 2 and 3, involving several places. These theorems and their several variants are all consequences of Theorem 4. One of the variants (Corollary 2) allows us to conclude “finiteness” rather than “Zariski-non-denseness” under the assumption that the orbit of is generic, that is, any infinite subset of is Zariski-dense. It is one of the far-reaching and difficult conjectures of Zhang  that the orbit of most points is generic, but we will show some examples where finiteness can be concluded unconditionally.
In Section 4, we discuss results and examples for which Vojta’s conjecture is not necessary. For example, if is a union of hyperplanes, then Schmidt’s subspace theorem gives us the same results unconditionally (Proposition 1). This situation has a bonus that if is defined over , then the exceptions to the subspace theorem are also hyperplanes defined over . This observation is exploited in Example 1, and we discuss its connection with dynamical (rank-one) Mordell–Lang conjecture. Another situation where the results become unconditional comes from a weaker form of Vojta’s conjecture called Lang–Vojta conjecture for integral points. We take advantage of known cases of this conjecture in Proposition 2 and Example 3. We also mention an example (Example 5) which demonstrates that our criterion given in Theorem 2 for Zariski-non-denseness of integral points in orbits is not yet satisfactory.
In Section 5, we discuss some extensions. The first extension is removing the assumption that is a morphism. For this, we will use the notion of -ratio, introduced by Lee . Using his height inequality for rational maps, we prove an extension to rational maps (Theorem 5). We also give an explicit example of this theorem for which Vojta’s conjecture is unnecessary (Example 8). The second extension is a case when the arithmetic of the orbit under one map is controlled by the geometry of another (Theorem 6). We mention several open questions at the very end.
2. Background on Vojta’s Conjecture
In this section, we briefly introduce Vojta’s conjecture . It is an extremely deep inequality of heights, and its consequences are vast, including Mordell’s conjecture (Faltings ), Schmidt’s subspace theorem, Bombieri–Lang conjecture and the conjecture (Vojta ). There are multiple versions of the conjecture, including the so-called Lang–Vojta conjecture [15, Proposition 4.1.2] which specializes to integral points and a uniform one over [15, Conjecture 5.2.6], but the version we will use in this article is an inequality for rational points over a fixed number field [15, Conjecture 3.4.3].
We first recall important basic properties of heights, and set some notations. Let be a number field, and let be the set of absolute values up to equivalence. For each , let be the absolute value in the class of which is the -th power of the extension of the normalized absolute value on , so that the product formula is simply for . We sometimes use the additive notation . When is a finite subset of containing all of the archimedean ones, the ring of -integers is the set of all such that for all . We define Weil height on by
which is well-defined. When is a rational (algebraic) map, we define the degree of to be the common degree of the homogeneous polynomials defining coordinates of . This can also be viewed as the degree of the map on the Picard group. We have the obvious height inequality , and whenever is a morphism (i.e. it is defined everywhere without indeterminancy), we also have an important inequality in the opposite direction:
Let be a projective variety over , assumed to be irreducible unless otherwise stated. For any Cartier divisor on , we can define a Weil height by writing as a difference of ample divisors and using the heights on the projective spaces. We can also define local height for and a divisor , using an -bounded metric on the line bundle . In essence, is , where is a local equation for , but one needs to glue this together in a coherent way using -bounded functions. As is big when is -adically close to , this is the number-theoretic analog of the proximity function in Nevanlinna theory. For details, see , . With our normalization, we have
Local height functions also satisfy functoriality with respect to pullbacks: given ,
and in fact the inequality with holds even for rational maps .
On , if a divisor is defined (globally) by the homogeneous polynomial of degree , then a local height is simply
In particular, choosing to have coefficients in the ring of integers, we see that for any non-archimedean . We also note that , but a similar relation does not hold with a single .
For , let us write as , where with common divisor . Then letting ,
for each place corresponding to the prime . If a finite subset including the archimedean place , then we denote the prime-to- part of an integer by . By above,
Thus, is precisely the set of points with -integer coordinates, i.e. with . In general, we say a set is -integral if it is of the form
where almost all are . We will abuse the notation and write for an -integral set.
We say that a divisor on a smooth variety is normal-crossings if near each point the divisor is defined by , where is a part of a local (analytic) coordinate system. Note that by definition, the multiplicity of each irreducible component in a normal-crossings divisor is . We are now ready to state Vojta’s conjecture [15, Conjecture 3.4.3].
Conjecture 1 (Vojta’s Conjecture).
Let be a smooth projective variety over , a canonical divisor of , an ample divisor and a normal-crossings divisor. Fix height functions , , and . Let be a finite set of places. Then given , there exists a Zariski-closed such that
for all not on .
Since local heights have logarithmic poles along , this conjecture states that a point cannot get too close -adically to for , and how close a rational point can approximate is controlled by the geometry of the variety, namely how negative the canonical divisor is. We note that the normal-crossings assumption on the divisor is absolutely essential, and this condition will be important in the rest of the paper. Since we mostly work with projective spaces, we also state the following version.
Conjecture 2 (Vojta’s Conjecture for ).
Let be a normal-crossings divisor on defined over , and be a finite set of places. Then given , there exists a finite union of hypersurfaces and a constant such that for any ,
This is precisely Roth’s theorem when and consists of a single archimedean absolute value. In fact, if is a union of hyperplanes in in general position, this conjecture can be shown to be equivalent to Schmidt’s subspace theorem. Thus, one can view Vojta’s conjecture as a higher-degree extension of the subspace theorem.
3. Proofs of the Theorems
Let be a morphism defined over of degree . Let be a divisor on defined over , and let be the normal-crossings part of . Let , and let be a number field that contains the fields of definition of , , and . Assume Vojta’s conjecture for the divisor . Then for any , for any finite set , and for any , the set
is Zariski-non-dense. In particular, is Zariski-non-dense if for some .
By applying Vojta’s conjecture to the divisor on , there exists a constant such that
holds for all except for points on a finite union of hypersurfaces. Since is a morphism, the degree of is , and thus we also have
Then for ,
|(8) and (9)|
Hence, for any , we let and conclude that
as long as . Note that if is finite, then this theorem is trivial. Otherwise, as by Northcott’s theorem. By dividing both sides of the inequality by , can be incorporated into a change in for large enough ’s. Moreover, if , then , which is a Zariski-closed set not equal to the whole of . Therefore, the result follows, as the given set is contained in the union of with a finitely many of the orbit points. The last sentence of the theorem is immediate from the discussion of -integral sets in the previous section. ∎
What we actually prove is the following: there exists a constant such that is Zariski-non-dense. The constant comes partially from (7), so it is not effective. In the following, we will prove similar results in various settings, all of which can be stated as differences of heights, although we state them with ratios for simplicity.
Let and . Then assuming Vojta’s conjecture for , for all ,
In truth, if is a monotone increasing sequence whose limit is , then we only need to assume Vojta’s conjecture for divisors with .
Note that is completely determined geometrically, and it does not depend on the choice of . On the other hand, the (possibly reduced) proper subvariety that contains (10) will depend on and (conjecturally, the hypersurface part does not depend on , but the additional higher-codimensional part will certainly depend on ).
We next discuss another variation of the main results. An infinite set of rational points is called generic if any infinite subset is Zariski-dense. Zhang  has conjectured that any polarized dynamical system has a point whose orbit is generic. If we assume genericity of the orbit, then we can conclude finiteness, rather than just Zariski-non-denseness, as follows.
Let . Let us assume that the orbit of is generic (in particular, is infinite). Assuming Vojta’s conjecture for , for all , then
holds for sufficiently large . In particular,
(i) If there exists such that , then is a finite set.
(ii) If , then
exists and equal to .
By the definition of genericity, the only Zariski-non-dense subset of the orbits is a finite set, so the first statement follows immediately from Corollary 1. (i) then follows as before from the fact that , and (ii) follows from the fact that the numerator inside the limit is , together with the squeeze theorem. ∎
Let , and let be a finite subset containing .
(i) If there exists such that , then assuming Vojta’s conjecture for , for any , the -integral points in the orbit of is Zariski-non-dense.
(ii) Assuming Vojta’s conjecture for , for all ,
(iii) If the orbit of is generic, then assuming Vojta’s conjecture for , for all ,
holds for all sufficiently large . In particular, if there exists such that , is a finite set, and if ,
4. Unconditional Results and Examples
In this section, we discuss some cases for which we obtain results such as Theorem 4 unconditionally without assuming any conjecture. One major case comes from Schmidt’s subspace theorem (Proposition 1 and Example 1). There are other special cases for which Vojta’s conjecture can be proved, and we discuss these examples as well (Proposition 2 and Examples 3 and 4).
First, we discuss cases for which Schmidt’s subspace theorem applies. Since this theorem is equivalent to Vojta’s conjecture for and a union of hyperplanes in general position, whenever normal-crossings divisors are linear, we get results in the previous section unconditionally. We now make this precise. Similar to the definition of , we define a linear normal-crossings part of to be a highest-degree normal-crossings subdivisor over of whose support is a union of hyperplanes. In other words, among all the different general-position unions of -hyperplanes contained in , has the most number of components.
Let be a morphism defined over , let be an effective divisor on defined over , and . Suppose there exists such that is positive. Let contain the fields of definition of , , , and , and let be a finite subset of . Then for all ,
is Zariski-non-dense. Further, if no hyperplane defined over contains infinitely many points of , then (11) is a finite set.
As before, we also have a version for -integral points, a version with , and a version with .
Much progress has been made on Schmidt’s subspace theorem and its exceptional hyperplanes. For example, we have some upper bounds for the number of exceptional hyperplanes. However, in general we still do not have a bound for the heights of the coefficients of the exceptional linear subspaces, and so we do not have an effective bound of which lies in (11).
All of these follow directly from the corresponding statements involving instead of . One notable observation is the fact that Schmidt’s subspace theorem actually lets us conclude that the exceptions to the inequality of Vojta’s conjecture are contained in a finite union of hyperplanes defined over . For general normal-crossings divisors, we only know that the exception is a union of hypersurfaces, so this is much stronger. To conclude finiteness of (11), we only need to show that each hyperplane over contains only finitely many points of . ∎
Let be the morphism on . Letting , , and , satisfying the hypothesis of the proposition. Hence, -integral points in the orbit are Zariski-non-dense, and more precisely,
Now, let , and we will show that no line defined over contains infinitely many points of . Unlike the ratio of the number of digits of coordinates, the ratio of the coordinates is unaffected even when there is a common factor. Since the -coordinate of is much smaller than the first two coordinates and the first two coordinates of are quartic in and while the last coordinate is only linear, the ratio of the last coordinate of to either of the first two coordinates becomes smaller and smaller in absolute value as . Therefore, a hyperplane with a nonzero coefficient for cannot contain infinitely many points of . The orbit points of clearly will not lie on or , so we are left to to check for . Any orbit point lying on this line has its previous iterate a rational point on . When this is a smooth curve, it has genus , so this immediately gives finiteness. Using the Jacobian criterion, the derivatives with respect to and give . So and letting be some cube root of the rational number , we have and . Then the derivative with respect to yields . There are two possibilities. When , then , so . This line does not contain any orbit points of , as their coordinates are all positive. When , then so . For even, the -coordinate of is bigger than the -coordinate, so it will not be on this line. The -coordinate of is about times as big as the -coordinate, and for points with a much larger -coordinate than the -coordinate, the first coordinate of is dominated by while dominates the second coordinate. Hence, the ratio of the -coordinate to the -coordinate becomes bigger and bigger upon every . Again, we observe that a common factor will not affect the ratio of the coordinates. Therefore, the first two coordinates of points in will never have a ratio of .
Therefore, we conclude that (12) is a finite set for , unconditionally without assuming any conjectures. Note that to conclude finiteness, it was very useful to know that the exceptions are lines (so that we can make arguments involving ratios of coordinates) and that the coefficients are in .
The argument above only utilizes standard methods, but it is somewhat adhoc and it is difficult to generalize to arbitrary of similar shape and arbitrary . On the other hand, given a specific situation, one can usually come up with a similar argument to show that a line defined over does not contain infinitely many of its orbit points.
Example 1 can be generalized to higher-dimensions. For example, let on , where are linear forms in general position such that is a morphism. Then is the vanishing locus of , so Proposition 1 applies. On the other hand, it is more difficult in general to conclude that (11) is a finite set, as exceptions are no longer 1-dimensional.
As is evident in Example 1, even in cases for which Schmidt’s subspace theorem applies, it would be beneficial to have an affirmative answer to dynamical Mordell–Lang question in order to conclude finiteness rather than Zariski-non-denseness. This question asks whether an infinite intersection of the orbit with a subvariety forces the subvariety to be preperiodic, and it has been proved affirmatively in various settings using Skolem–Mahler–Lech method (see for example ). The higher-rank case, involving several maps which commute with each other, was introduced in . They prove an affirmative answer in low-dimensional cases and they also demonstrated a couple of counterexamples. The higher-rank case has now been proved to completely fail in general , though no counterexample has yet been found for orbits of a single map. The case relevant to Example 1, namely the case of self-maps on with respect to a line, is not known.
As a next situation when the results become unconditional, we take advantage of the known cases of the “integral point” version of Vojta’s conjecture. This version is sometimes called Lang–Vojta conjecture, and it is a special case [15, Propsoition 4.1.2] of Conjecture 1 in Section 2. This conjectures that when is of log general type, the -integral points are Zariski-non-dense. We will now use a known case of this to obtain an unconditional result:
Let be a morphism defined over of degree . Let be a divisor on defined over , and let . Let be a number field that contains the fields of definition of , of irreducible components over of , and of , and let be a finite subset. Suppose there exists with such that
contains distinct geometrically-irreducible components,
Then is Zariski-non-dense.
The proof is similar in spirit to the proof of Theorem 4. Let be such that is in the set in question and let . Note that
On the other hand,
Putting these two together, we see that is bounded by . For a fixed number field, (4) shows that there are only finitely many non-archimedean places for which the minimum positive value of is below . Therefore, so must belong to a set of -integral points. But Lang–Vojta conjecture is known for and a divisor with distinct geometrically-irreducible components (originally [15, Corollary 2.4.3] and a special case of [16, Corollary 0.3]), so the conclusion follows by taking image by . ∎
Of course, once contains distinct geometrically-irreducible components, a set of -integral points in all of is Zariski-non-dense. Hence, the set of orbit points which are -integral is contained in , which is also Zariski-non-dense. Thus, integral-point statements such as Theorem 2 are trivial, while the number-of-digits comparison statements such as Proposition 2 are less immediate.
Let on , where is a -linear form and is a geometrically irreducible -quadratic form such that neither goes through or . This is a morphism. Let be such that the -coordinate is much larger than the other two coordinates. , so has four distinct components. Then , and as contains a nonzero -term while the other two coordinates of do not, the -coordinate is always the largest in . Therefore, , and hence we have
Therefore, the proposition unconditionally tells us that
is contained in a finite union of algebraic curves. Note that since does not contain four distinct lines, Proposition 1 does not apply.
We next discuss one other situation where our results become unconditional. Sometimes, Vojta’s inequality can be verified even for non-normal-crossings divisors, and our next example takes advantage of this.
because whichever coordinate has the maximum absolute value, it is canceled by the denominator. Hence, Vojta’s inequality is trivially satisfied for , and so if contains , one can replace with and obtain Theorem 4 and its consequences unconditionally. For example, let and let . Since the common intersections of the last two coordinates of are and the first coordinate of is nonzero at these points, is a morphism. Now, , which contains as a subdivisor. Thus, we conclude unconditionally from Theorem 4 that
is contained in a finite union of algebraic curves for any and .
We close this section with an example which demonstrates that our theorems are not yet satisfactory for determining Zariski-non-density of integral points in orbits.
Let . This is a morphism, and since the last two coordinates of only involve and , this property continues to hold for all . On the other hand, any homogeneous polynomial in and factors into linear terms, all of which go through the point . Thus, for , is at most two lines for every , so this map does not satisfy the hypothesis of Proposition 1.
However, we can actually show that an orbit of