Heights and multiplicative relations on algebraic varieties

Points on a subvariety X of a semi-abelian variety A that are contained in a subgroup,
let the subgroup be of finite rank or algebraic, are subject to severe restrictions
arithmetical nature.
Finiteness results for intersections of X with subgroups of finite rank have been studied
by Faltings, Hindry, Laurent, McQuillan, Raynaud, Vojta and others. More recently
several authors ([CZ00], [BMZ99], [BMZ03], [BMZ06a], [BMZ06b], [BMZ04],
[Via03], [RV03], [R´em05b], [R´em07], [Pin05b], [Zan00], [Zil02], [Mau06]) have
considered the intersection of X with A[r], the set of complex points in A contained in
an algebraic subgroup of codimension greater or equal to r. If H is a fixed algebraic
subgroup of A with codimension strictly less than dimX, then a dimension counting
argument shows that X\H is either empty or contains a curve. As we are allowing H to
vary with fixed codimension, the intersection X \ A[r] may be quite large if r < dimX.
In this thesis we are only interested in the case r � dimX.
If not stated otherwise we will also assume throughout the introduction that all
varieties are defined over Q, the field of algebraic numbers. One can define a height
function on the set of algebraic points of A. Throughout this thesis we work only in
the algebraic torus Gn
m or an abelian variety. So we can take the Weil height or the
N´eron-Tate height associated to an ample line bundle.
We will pursue two types of questions. First, for which r does the set X0(Q) \ A[r]
have bounded height and how do these bounds depend on X? Second, for which r is the
set X00(Q) \ A[r] finite? Here X0 and X00 are obtained from removing from X certain
subvarieties in order to to eliminate trivial counterexamples. For example if X is a
proper algebraic subgroup of Gn
m with positive dimension, then there is no hope for a
boundedness of height or finiteness result for U(Q) \ (Gn
m)[r] if r � dimX and if U is
Zariski open and dense in X. In this case X0 and X00 are both empty.
The simplest non-trivial example seems to be the curve defined by x + y = 1 in
G2
m. Here we can take X0 and X00 to equal our curve. Algebraic subgroups of G2
m
can be described by at most two monomial relations x�y� = 1 with integer exponents
� and �. For subgroups of dimension 1, one non-trivial relation suffices. If (x, y) is
contained in such a subgroup then x and y are called multiplicatively dependent. Hence
the intersection of our curve with the union of all proper algebraic subgroups of G2
m can
be described by the solutions of
(0.0.1) x�(1 − x)� = 1.
This is an equation in three unknowns x, �, and �, so one should not expect finitely
many solutions. Indeed, taking x 6= 1 a root of unity gives infinitely many solutions.
In [CZ00] Cohen and Zannier showed that if H denotes the absolute non-logarithmic
Weil height then (0.0.1) implies the sharp inequality max{H(x),H(1−x)} � 2. In chapter
2 we start off by giving an alternative proof of Cohen and Zannier’s Theorem. We
even show that the possibly larger height H(x, 1−x) is at most 2. In their paper, Cohen
and Zannier also proved that 2 is an isolated point in the range of max{H(x),H(1−x)}.
We make this result explicit in Theorem 2.2, working instead with H(x, 1 − x). The
proof applies Smyth’s Theorem on lower bounds for heights of non-reciprocal algebraic
numbers and a Theorem of Mignotte.
As was already noticed in [CZ00], solutions of (0.0.1) are closely linked to roots of
certain trinomials whose coefficients are roots of unity. In chapter 3 Theorem 3.2 we
follow this avenue by factoring such trinomials over cyclotomic fields. Having essentially
a minimal polynomial in our hands, we obtain a new proof for the boundedness of
H(x, 1 − x) with x as in (0.0.1). More importantly, in Theorem 3.1 we show that not
only is 2 isolated in the range of the height function, but also that H(x, 1−x) converges
to an absolute constant if [Q(x) : Q] goes to infinity. The proof determines the value
of this limit: it is the Mahler measure of the two-variable polynomial X + Y − 1. In a
certain sense this Mahler measure is the height of the curve in our problem.
In Theorem 3.3 we prove a conjecture of Masser stated in [Mas07]: the number
of solutions of (0.0.1) with [Q(x) : Q] � D is asymptotically equal to c0D3 with c0 =
2.06126 . . . as D ! 1. The constant c0 is defined properly in chapter 3 as a converging
series. This counting result is a further application of Theorem 3.2.
In chapter 4 we generalize the method from chapter 2 to bound the height of multiplicatively
dependent solutions of
(0.0.2) x + y = �.
Here � is now any non-zero algebraic number. In [BMZ99] Bombieri, Masser, and
Zannier prove a more general result which also implies boundedness of height in this
case. Their Proposition A leads to an explicit upper bound for the height; the bound
is polynomial in H(�). We are mainly interested in upper bounds for H(x, y) which
have good dependency in H(�). The value H(�) can be regarded as the height of
the defining equation (0.0.2). In Theorems 4.1 and 4.2 we get the bound H(x, y) � 2H(�) min{H(�), 7 log(3H(�))}. By Theorem 4.3 the exponent of the logarithm cannot
be less than 1. But in some special cases, e.g. if � is a rational integer, we improve the
upper bound to 2H(�), see Theorem 4.4. In this theorem we also show that if � is a
rational integer then 2H(�) is attained as a height if and only if � is a power of two.
Thus if � is a power of two, then our bound is sharp. For such � and if also � � 2 we
prove in Theorem 4.5 that 2H(�) is isolated in the range of the height.
Starting from chapter 6 we work in an algebraic torus of arbitrary dimension. Algebraic
subgroups can still be described by a finite set of monomial equations. For example
(x1, . . . , xn) 2 Gn
m(C) is contained in a proper algebraic subgroup if and only if the xi
satisfy a non-trivial multiplicative relation. In [BMZ99] Bombieri, Masser, and Zannier
proved that if X is an irreducible curve which is not contained in the translate of a
proper algebraic subgroup, then points on X that lie in a proper algebraic subgroup have
bounded height. Moreover, they showed that this statement is false if X is contained in
the translate of a proper algebraic subgroup. The authors also showed that there are
only finitely many points on X that lie in an algebraic subgroup of codimension at least
2. This finiteness result was generalized by the same authors in [BMZ03] to algebraic
curves defined over the field of complex numbers. Hence for curves it makes sense to
take X0 = X if X is not contained in the translate of a proper algebraic subgroup and
X0 = ; else wise. But X00 is more subtle: we take X00 = X if X is not contained in a
proper algebraic subgroup and X00 = ; else wise. The point in making this distinction is
that in [BMZ06a] the authors conjectured that X00 contains only finitely many points
in an algebraic subgroup of codimension at least 2. They proved this conjecture for
n � 5. Recently, in [Mau06] Maurin gave a proof for all n.
Let X � Gn
m be an irreducible subvariety, not necessarily a curve. In the higher
dimensional case we finally need a definition of X0: we get X0 by removing from X
all positive dimension subvarieties that show up in an improper component of the intersection
of X with the translate of an algebraic subgroup. The definition of X00 is
similar but we require the translates of algebraic subgroups to be algebraic subgroups.
In [BMZ06b] Bombieri, Masser, and Zannier showed that X0 is Zariski open in X.
Let h be the absolute logarithmic Weil height. Our contribution in chapter 6 is
Theorem 6.1 where we give an explicit bound for the height of algebraic points p in X0
that lie “uniformly close” to an algebraic subgroup of codimension strictly greater than
n − n/ dimX. By uniformly close we mean that there exist an � > 0, independent of p,
and an a in an algebraic subgroup of said codimension with h(pa−1) � �. Actually, in
Theorem 6.1 we will use a weaker notion of uniformly close. The terminology comes from
the fact that the map (p, a) 7! h(pa−1) has similar properties as a distance function.
For example it satisfies the triangle inequality. This notion of distance was considered
by several authors ([Eve02], [Poo99], [R´em03]) in connection with subgroups of finite
rank.
Theorem 6.1 generalizes the Bounded Height Theorem for curves by Bombieri,
Masser, and Zannier. We state our theorem such that it also gives an explicit version
of a Theorem of Bombieri and Zannier in [Zan00] on the intersection of varieties with
one dimensional subgroups. To do this we will need a slightly more general definition
of X0 which is provided in chapter 6.
The height upper bound in Theorem 6.1 involves, along with n, the degree and
height of the variety X. We define these two notions in chapter 5. In simple terms, the
height of X controls the heights of the coefficients of a certain set of defining equations
for X whereas the degree of X controls their degrees. Just as in the second proof for
height bounds on curves given in [BMZ99], our proof of Theorem 6.1 uses ideas from
the geometry of numbers. Given p 2 X(Q) uniformly close to an algebraic subgroup we
construct a new algebraic subgroup H of codimension dimX and controlled degree, such
that pH has normalized height small compared to the height of p. We then intersect
pH with X. The Arithmetic B´ezout Theorem bounds the height of isolated points in
this intersection leading to an explicit height bound for p.
Lehmer-type lower bounds for heights in spirit of Dobrowolski’s Theorem and its
generalization to higher dimension provide a method for deducing finiteness results from
height bounds as given in chapter 6. This method was used together with algebraic number
theory in Bombieri, Masser, and Zannier’s article [BMZ99] to prove the finiteness
of the set of points on X0 in an algebraic subgroup of codimension at least 2 if X is a
curve. Meanwhile, their intricate argument has been simplified in [BMZ04] by applying
a more advanced height lower bound due to Amoroso and David [AD04]. In this
lower bound the degree over Q of a point is essentially replaced by its degree over the
maximal abelian extension of Q. Using this approach we show in Corollary 6.2 that if
X is a surface in G5
m, then there are only finitely many points on X0 contained in an
algebraic subgroup of codimension at least 3. Thus we have finiteness for the correct
subgroup size at least in an isolated case.
Even in presence of a uniform height bound as in Theorem 6.1, the approaches in
[BMZ99] and [BMZ04] cannot be used to prove the finiteness of the set of p 2 X0(Q)
with h(pa−1) small and a contained in an algebraic subgroups of appropriate dimension:
although pa−1 has small height, its degree cannot be controlled. In chapter 7 we pursue
a new approach using a Bogomolov-type height lower bound. This bound was proved by
Amoroso and David in [AD03]; it bounds from below the height of a generic point on a
variety not equal to the translate of an algebraic subgroup. The main result of chapter
7 is Theorem 7.1: we show that for B 2 R there exists an � = �(X,B) > 0 with the
following property: there are only finitely many p 2 X0(Q) with h(pa−1) � � where a is
contained in an algebraic subgroup of dimension strictly less than m(dimX, n). In other
words, there are only finitely many algebraic points on X0 of bounded height which are
uniformly close to an algebraic subgroup of dimension less than m(dimX, n). Just as
was the case in Theorem 6.1 we actually use a relaxed version of uniformly close in
Theorem 7.1. The somewhat unnatural function m(·, ·) is defined in (7.1.1). At least in
the case of curves we have n − 2 < m(1, n) and so we can take the subgroups to have
the best possible dimension n−2. Unfortunately this is the only interesting case where
m(r, n) > n − r − 1.
With the height upper bound from chapter 6 we can deduce a corollary to Theorem
7.1 which proves finiteness independently of B and where the subgroup dimension is
strictly less than min{n/ dimX,m(dimX, n)}. Let X be a curve, then this result is
optimal with respect to the subgroup dimension. Let us assume that X is not contained
in the translate of a proper algebraic subgroup, hence X0 = X. Then our corollary says
that there are only finitely many algebraic points on X that are close to an algebraic
subgroup of codimension at least 2. Moreover, in Corollary 7.2 we use Dobrowolski’s
Theorem to show that if � in the definition of uniformly close is small enough, then
all points on X close to an algebraic subgroup of codimension at least 2 are actually
contained in such a subgroup.
We now shift our focus from the algebraic torus to abelian varieties: we want to study
the intersection X0(Q)\A[r] where A is an abelian variety and X is an irreducible closed
subvariety of A. The definitions of X0 and X00 make sense in the abelian setting and
are completely analog to the multiplicative case.
Let X be a curve, then in [Via03] Viada proved that X0(Q)\A[1] has bounded height
if A is a power of an elliptic curve. If the elliptic curve has complex multiplication she
also proved that X0(Q)\A[2] is finite. R´emond in [R´em05b] generalized Viada’s height
bound to any abelian variety. In [R´em07] R´emond applied a generalization of Vojta’s
inequality which he proved in [R´em05a] and in Theorem 1.2 showed boundedness of
height of (X(Q)\Z(r)
X )\A[r]. Here X\Z(r)
X � X is a new deprived subset which depends
on r. In fact his result holds for a set larger than A[r] involving also the division closure
of finitely generated group. If A is isogenous to a product of elliptic curves and if X is a
sufficiently general surface which is not contained in the translate of a proper algebraic
subgroup then X\Z(r)
X is non-empty and Zariski open in X for r � (dimA + 3)/2.
In [RV03], R´emond and Viada proved that if X is a curve then X00(Q) \ A[2] is
finite if A is a power of an elliptic curve E with complex multiplication. In a recent
preprint, Viada [Via07] announced the finiteness of X00(Q) \ A[3] for unrestricted E,
the optimal subgroup codimension 2 is thus just missed.
We announce the following result called the Bounded Height Theorem: if A = Eg
is a power of an elliptic curve E and X is an irreducible closed subvariety of arbitrary
dimension, then X0(Q) \ A[dimX] has bounded N´eron-Tate height. Also, using a result
from Kirby’s Thesis [Kir06] and ideas from Bombieri, Masser, and Zannier’s [BMZ06b]
one can show that X0 is Zariski open and give a criterion on X to decide when X0 is
non-empty. Using height lower bounds on abelian varieties with complex multiplication
due to Ratazzi in [Rat07] we can use the Bounded Height Theorem to show that
X0(Q)\A[dimX+1] is finite if E has complex multiplication. For an elliptic curve without
complex multiplication, finiteness of X0(Q)\A[r] can also be obtained, using for example
R´emond’s Theorem 2.1 from [R´em05b]. But r is in general sub-optimal for such elliptic
curves.
The essential difference between the Bounded Height Theorem and Theorem 6.1 is
that the subgroups are now allowed to have the best-possible codimension dimX for all
X.
In the future we plan to publish these results.
Pink has stated a general conjecture on mixed Shimura varieties, see [Pin05a] and
[Pin05b]. One special implication is his Conjecture 5.1 from [Pin05b]: if A is a semiabelian
variety defined over C and if X � A is a subvariety also defined over C which is
not contained in a proper algebraic subgroup of A, then X(C)\A[dimX+1] is not Zariski
dense in X. Zilber’s stronger Conjecture 2 in [Zil02] implies the same conclusion. With
the Bounded Height Theorem we can prove this assertion under the following stronger
hypothesis on A and X: A is a power of an elliptic curve E with complex multiplication
and if ' : Eg ! EdimX is a surjective homomorphism of algebraic groups, then the
restriction '|X : X ! EdimX is dominant.
The proof of the Bounded Height Theorem uses the completeness of A (and X)
in an essential way as it relies on intersection theory. Nevertheless, a proof for the
boundedness of height of X0(Q)\A[dimX] for the non-complete X � A = Gn
m along the
lines of the proof of the Bounded Height Theorem must not be ruled out. For instance
one could compactify Gn
m ,! Pn and work in Pn. Still, there seems to be no suitable
Theorem of the Cube for Gn
m. Future research could consist in finding a proof of the
Bounded Height Theorem in the multiplicative case or in abelian varieties other than a
power of an elliptic curve.
In the two appendices we leave the main path of the thesis. Let P be an irreducible
polynomial in two variables with algebraic coefficients. Say x and y are algebraic with
P(x, y) = 0. In appendix A, motivated by Proposition B of [BMZ99], we consider the
problem of bounding | degX(P)h(x)−degY (P)h(y)| explicitly and with good dependency
in h(x), h(y), and P.
For simple examples such as P = Xp − Y q with p and q coprime integers, the
absolute value is zero. But for general and fixed P it may even be unbounded as (x, y)
runs over all algebraic solutions of P. In Theorem A.1 we prove an upper bound which is
of the form c max{1, hp(P)}1/2 max{1, h(x), h(y)}1/2 where the constant c is completely
explicit and depends only on the partial degrees of P. Here hp(P) is the projective
logarithmic Weil height of the coefficient vector of P. This type of height inequality is
often referred to as quasi-equivalence of heights.
In appendix B we demonstrate four known results using the Quasi-equivalence Theorem
from appendix A. The first application is the Theorem of Bombieri, Masser, and
Zannier, already discussed above, in the case of curves in G2
m. We then prove a version
of Runge’s Theorem on the finiteness of the number of solutions of certain diophantine
equations. Next we show a result of Skolem from 1929: we first generalize the greatest
common divisor of pairs of integers to pairs of algebraic numbers. We then show that
if x and y are coprime algebraic numbers and P(x, y) = 0 where P is an irreducible
polynomial in Q[X, Y ] without constant term, then x and y have uniformly bounded
height. This result has been proved independently by Abouzaid in [Abo06] who used it
to prove a variant of the Quasi-equivalence Theorem. The fourth and final application
is an explicit version of Sprindzhuk’s Theorem: let P have rational coefficients, again
without constant term and such that not both partial derivatives of P vanish at (0, 0).
Then for a sufficiently large prime l, the polynomial P(l, Y ) 2 Q[Y ] is irreducible. Since
the Quasi-equivalence Theorem gives explicit bounds, so do its four applications.
Chapters 1 and 5 contain no new results but serve as reference for certain theorems
which we apply in the rest of the thesis. Chapter 1 introduces the Weil height and
related subjects. It is used throughout the thesis. Chapter 5 contains some results from
algebraic geometry and gives a definition for the height of a positive dimensional variety.
These definitions and results will be used in the second part of the thesis, chapters 6
and 7.