A Note on Divisible Points of Curves

0.22 MB·English

by  Martin Bays

A NOTE ON DIVISIBLE POINTS OF CURVES
M. BAYS AND P. HABEGGER
Abstract. Let C be an irreducible algebraic curve defined over a number field and
inside analgebraictorusofdimensionatleast3. We partiallyansweraquestionposed
by Levin on points on C for which a non-trivial power lies again on C. Our results
haveconnectionstoZilber’sConjectureonIntersectionswithToriandyieldtomethods
arising in transcendence theory and the theory of o-minimal structures.
3
1
0 1. Introduction
2
n Let C and C be irreducible algebraic curves in the algebraic torus GN with N ≥ 3.
1 2 m
a
Aaron Levin asked what can be said of points x on C for which there is n ≥ 2 such
J 1
that xn is on C . In this paper we give a partial answer to Levin’s question in the case
3 2
2 C = C .
1 2

The maximal compact subgroup of the algebraic torus is the real torus
]
T
T = {(x ,...,x ) ∈ GN(C); |x | = ··· = |x | = 1}.
N 1 N m 1 N
. It is convenient to call a translate of an algebraic subgroup of GN a coset (of GN).
h m m
t Moreover, a torsion coset (of GN) is a coset containing an element of finite order.
a m
m Torsion cosets are precisely irreducible components of algebraic subgroups of GN. We
m
[ call a coset or torsion coset proper if it is not equal to GN.
m

Say C ⊂ GN is an algebraic curve defined over a number field F. If σ : F → C is
1 m
v a field embedding, then C is the curve defined over C by polynomials obtained from
σ
4 applying σ to polynomials defining C. Let Q be a fixed algebraic closure of Q; we take
7
number fields to be subfields of Q.
6
5
. Theorem 1. Let N ≥ 3 and let C ⊂ GN be an irreducible closed algebraic curve defined
1 m
0 over a number field F. We assume that C is not contained in a proper torsion coset of
3 GN. Let us also assume that C (C) ∩ T is finite for some σ : F → C. Then C(Q)
1 m σ
contains only finitely many points x with xn ∈ C(Q) for some n ≥ 2.
:
v
i The condition N ≥ 3 is natural for this type problem on unlikely intersections, cf.
X
Zannier’s book [18] where Levin’s question appears in print. If x is as in (i) of the
r
a theorem, then (x,xn) is contained in the surface C × C. However, it is also in an
algebraic subgroup of codimension N > dim(C × C). Our result gives new evidence
towards Zilber’s Conjecture [20] on intersections with tori for surfaces in GN. The class
m
of degenerate subvarieties, of which C ×C is a member, has eluded recent progress by
Maurin [11] and the second author [8] towards this conjecture. Degenerate subvarieties
are defined in Maurin’s work [11]; an equivalent definition is (C×C)oa = ∅ in Bombieri,
Masser, and Zannier’s notation, cf. [8].
However, the finiteness condition on C (C) ∩ T does not appear in the theory of
σ
unlikely intersections, and it would follow from Zilber’s conjecture that the condition is
1
2
not necessary. Any curve that is also a torsion coset intersects T in the infinite set of its
points of finite order. There are however also algebraic curves that are not contained in
a proper torsion coset but that intersect T in infinitely many points. An example is
x−2
x, ; x ∈ Cr{1/2} .
(cid:26)(cid:18) 2x−1(cid:19) (cid:27)
Indeed, if |x| = 1 then |x−2| = |2x−1|.
More generally, the birational map C → C;z 7→ z−1 maps the unit circle onto
iz+1
P(R) = R ∪ {∞}, and so sets up a bijective correspondence between curves in GN
m
which have infinite intersection with T and curves in CN whose closure in P(C)N have
infinite intersection with P(R)N. So examples are plentiful.
The intersection of the line x +x = 1 with T, however, consists precisely of the two
1 2
points {(exp(±2πi/6),exp(∓2πi/6)}.
Corvaja, Masser, and Zannier [5] recently proved finiteness results when intersecting
analgebraiccurve withthemaximal compact subgroupofcertain commutative algebraic
groups.
Our proof involves the Theorem of Pila-Wilkie [13] which is playing an increasingly
important role in diophantine problems revolving around unlikely intersections. Zannier
proposed to use this tool to give a new proof of the Manin-Mumford Conjecture for
abelian varieties in joint work with Pila [14]. Maurin’s work [11] relies on a generalized
Vojta inequality due to R´emond. Our work uses an earlier variant of this result also
due to R´emond [15], which almost immediately implies (see Lemma 6) that x as in
Theorem 1 has height bounded only in terms of C. The new ingredient in our work is
the use of Baker’s inequality on linear forms in logarithms. Its effect is to obtain a lower
bound for the degree of x from the height bound obtained from R´emond’s result. We
refer to Baker and Wu¨stholz’s estimate [2], which is completely explicit.
Using a p-adic version of linear forms in logarithms due to Bugeaud and Laurent
we also obtain the following partial result for curves that do not satisfy the finiteness
condition in Theorem 1.
LetS beasetofrationalprimes. Atuple(x ,...,x )ofalgebraicnumbersiscalledS-
1 N
integral if max{|x | ,...,|x | } ≤ 1 for all finite places v of Q(x ,...,x ) with residue
1 v N v 1 N
characteristic outside of S.
Theorem 2. Let N ≥ 3 and let C ⊂ GN be an irreducible closed algebraic curve defined
m
over a number field F. We assume that C is not contained in a proper torsion coset of
GN. There is a constant p with the following property. If S is a finite set of rational
m 0
primes with infS > p , then C(Q) contains only finitely many S-integral points x such
0
that xn ∈ C(Q) for some n ≥ 2.
By convention the infimum of the empty set is +∞. So S = ∅ is allowed and yields a
finiteness statement on points whose coordinates are algebraic integers.
The paper is organized as follows. In the next section we set up common notation
used throughout the article. The third section contains an elementary metric argument
which is used in connection with Baker-type estimates in section 4. In section 5 we
bound from below the size of Galois orbits and section 6 contains the arguments from
o-minimality. Finally, both theorems are proved simultaneously in section 7.
3
Bays was partiallysupported by the Agence NationaledeRecherche [MODIG, Project
ANR-09-BLAN-0047], and both authors thank Zo´e Chatzidakis and the ANR for sup-
porting Habegger’s invitation to Paris in Summer of 2011, where the authors worked
together on this problem. Habegger is also grateful to Yann Bugeaud for answering
questions in connection with linear forms in p-adic logarithms.
2. Notation
By a place v of a number field K we mean an absolute value that is, when restricted
to Q, either the complex absolute value or a p-adic absolute value for some prime p.
A non-Archimedean place is called finite and the others are called infinite. We let K
v
denote a completion of K with respect to v and, by abuse of notation, Q a completion
v
of Q with respect to the restriction of v.
For purposes of bookkeeping it is convenient to work with height functions. We recall
here the absolute logarithmic Weil height used throughout the article. Let x ∈ Q. There
is a unique irreducible polynomial P ∈ Z[X] with P(x) = 0 and positive leading term
a . We define the height x as
0
1
h(x) = loga max{1,|z|}.
0
degP
 Yz∈C 
 P(z)=0 
If K is a number field containing x, then the height h(x) defined above is
1
(1) d logmax{1,|x| }
[K : Q] v v
Xv
where v ranges over places of K, and d = [K : Q ], the degree of the corresponding
v v v
field extension of the completions. Equivalently,
1
(2) h(x) = logmax{1,|σ(x)| }
[K : Q] p
Xp σ:KX֒→Cp
where p ranges over ∞ and the primes, C = C, and C is a completion of an algebraic
∞ p
closure of the field of p-adic numbers if p 6= ∞.
N
The height of a tuple x = (x ,...,x ) ∈ Q is
1 N
h(x) = max{h(x ),...,h(x )}.
1 N
The exponential height of x is H(x) = exp(h(x)).
3. A Metric Argument
The following elementary lemma is well-known and sometimes proved using Puiseux
series. We have decided to include an elementary argument that essentially relies only
on the triangle inequality.
We let h·,·i denote the standard inner product on R2 and k·k the Euclidean norm.
If i ∈ R2 we use i and i to denote its coordinates. A non-zero vector i ∈ Z2 is called
1 2
reduced if i ,i are coprime and if either i ≥ 1 or i = (0,1). We note that two reduced
1 2 1
vectors are linearly dependent if and only if they are equal.
4
If K is a field then K× is its multiplicative group. For this section we suppose that
K is algebraically closed and endowed with an absolute value |·| : K → R.
Lemma 1. Suppose P ∈ K[X±1,Y±1] is a Laurent polynomial with at least 2 non-zero
terms. We set D = max ki−i′k and let σ +1 ≥ 2 denote the number of non-zero
pipi′6=0
terms of P if |·| is Archimedean and σ = 1 otherwise. There exists a finite set Σ ⊂ K×
depending only on P with the following property. Let x ,x ∈ K× with P(x ,x ) = 0
1 2 1 2
satisfy kLk > 16D2(logσ + max log|p /p |) where L = (log|x |,log|x |) ∈ R2.
pipi′6=0 i i′ 1 2
Then there is α ∈ Σ and a reduced j ∈ Z2 with kjk ≤ D such that
1
xj1xj2 = α or log|xj1xj2 −α| ≤ − kLk.
1 2 1 2 16D2
Proof. The set Σ and the value of c > 0 will be determined during the argument. Say
(x ,x ) is as in the hypothesis. We write P = p Xi1Yi2. After possibly dividing P
1 2 i i
by some non-zero term piXi1Yi2 we may suppoPse that the constant term of P is 1 and
|p xi1xi2| ≤ 1 for all i ∈ Z2. By abuse of notation we sometimes also write |· | for the
i 1 2
standard absolute value on R.
Let us fix a non-zero element i of Z2 for which |p xi1xi2| is maximal.
i 1 2
Then hL,ii ≤ −log|p | and we will now bound this quantity from below. In the
i
Archimedean case we estimate
1−|p xi1xi2| ≤ |1+p xi1xi2| = p xi′1xi′2 ≤ |p xi′1xi′2| ≤ (σ −1)|p xi1xi2|
i 1 2 i 1 2 (cid:12) i′ 1 2 (cid:12) i′ 1 2 i 1 2
(cid:12)(cid:12)iX′6=0,i (cid:12)(cid:12) iX′6=0,i
(cid:12) (cid:12)
by the triangle inequality and the(cid:12)choice of i. H(cid:12)ence |p xi1xi2| ≥ σ−1 which is also true
i 1 2
in the non-Archimedean case. Therefore, |hL,ii| ≤ |log|p || + logσ < kLk/(8D2) by
i
hypothesis. We divide by kik and set v = i/kik to get
1
kLk
(3) |hL,v i| < .
1 8D2
We fix v ∈ R2 of norm 1 with hv ,v i = 0. After multiplying by −1 we may suppose
2 1 2
hL,v i ≥ 0. We have hL,v i2 = kLk2 −hL,v i2 and use (3) to estimate
2 2 1
kLk
(4) hL,v i = |hL,v i| ≥ .
2 2
2
Suppose i′ ∈ Z2 is written as λ v + λ v with new real coordinates λ . Then
1 1 2 2 1,2
λ = hi′,v i. But the coordinates of v are up-to sign the coordinates of v = i/kik and
2 2 2 1
so either λ = 0 or |λ | ≥ 1/kik.
2 2
Now suppose also p 6= 0. We claim λ ≤ 0. Indeed, otherwise we would have
i′ 2
λ ≥ 1/kik ≥ 1/D. Now 0 ≥ log|p | + hL,i′i, so 0 ≥ log|p | + λ hL,v i + λ hL,v i.
2 i′ i′ 1 1 2 2
We remark that |λ | = |hi′,v i| ≤ ki′k ≤ D. Using (3) and (4) yields 0 ≥ log|p | −
1 1 i′
kLk/(4D)+kLk/(2D) = log|p |+kLk/(4D). So kLk ≤ 4D|log|p || which contradicts
i′ i′
our hypothesis. Thus λ ≤ 0.
2
We have 0 = P(x ,x ) = A+B with
1 2
i′ i′ i′ i′
A = p x1x2 and B = p x1x2.
i′ 1 2 i′ 1 2
i′=Xλ1v1 i′=λ1v1+Xλ2v2,λ2<0
We first treat the error term B. If it is non-zero, the triangle or ultrametric triangle
inequality yields log|B| ≤ logσ+max {log|p |+λ hL,v i+λ hL,v i}. To treat the
λ2<0 i′ 1 1 2 2
5
terms in the maximum we use the bounds (3), (4), |λ | ≤ D, and λ ≤ −1/D proved
1 2
above. Indeed,
kLk kLk kLk
(5) log|B| ≤ logσ +max{log|p |}+ − ≤ − .
i′
i′ 8D 2D 4D
Now we consider the main term A to which we associated the polynomial F =
i′=λ1v1 pi′Xi′1Yi′2. We fix a primitive generator j of the rank 1 group iQ ∩Z2. Then
Pj = µv with µ ∈ R. We can write F = f Xλj1Yλj2 where λ runs over integers
1 λ λ
and the fλ are certain coefficients of P. WePdefine G = λfλTλ ∈ K[T±1] and with
this new Laurent polynomial we have G(Xj1Yj2) = F. NotPe that 1 is the constant term
of G. Say a is the minimal integer such that TaG is a polynomial. Let us now factor
TaG = p(T − α )···(T − α ) where p 6= 0 is some coefficient of P. We remark that
1 d
α ,...,α do not vanish and come from a finite set depending only on P.
1 d
For brevity we write z = xj1xj2 ∈ K×. Then p(z − α )···(z − α ) = zaG(z) =
1 2 1 d
zaF(x ,x ) = −zaB. Without loss of generality we may assume |z −α | ≤ |z −α | for
1 2 1 k
1 ≤ k ≤ d. So |z −α |d ≤ |p|−1|z|a|B|.
1
If z = α then we are in the first case of the conclusion. Else we take the logarithm
1
and use (5) to estimate
kLk
(6) dlog|z −α | ≤ −log|p|+alog|z|+log|B| ≤ |log|p||+|alog|z||− .
1
4D
We conclude by first bounding |alog|z|| = |aµhL,v i|. The inequality |aµ| = |aµ|kv k =
1 1
kajk ≤ D and (3) imply |alog|z|| ≤ kLk/(8D). Moreover, |log|p|| ≤ kLk/(16D).
Inserting these two inequalities into (6) yields dlog|z−α | ≤ −kLk/(16D). The lemma
1
follows since d ≤ D. (cid:3)
Remark 1. Note that in the case that |·| is non-Archimedean and all coefficients of P
have trivial absolute value, the condition on kLk in Lemma 1 is just that it is non-trivial.
We will use this in our proof of Theorem 2 below.
Let us also remark that a qualitative version of Lemma 1, which does not give this
information needed for Theorem 2 but which suffices for our uses in proving Theorem 1,
admits a proof using the model theory of valued fields. We sketch this here.
Let K be an algebraically closed field with an absolute value |·| : K → R.
Define v : K → R∪{∞}; v(x) := −log|x|. This is a valuation if and only if |·| is
non-Archimedean. Consider the two-sorted structure hhK;+,·i,hR;+,<i;vi consisting
of the field K, the ordered group R, and the map v.
Let (∗K,∗R) be an elementary extension, extending v to a map v : ∗K → ∗R∪{∞}.
Let O := {x ∈ ∗K | ∃n ∈ N. v(x) > −n}. Then O is a local ring, since v(x−1) = −v(x).
Let v′ be the corresponding valuation, and let res be the corresponding residue map. Note
that the restriction of res to K is an embedding.
(In the case that | ·| is non-Archimedean, v′ is the coarsening of v obtained by quo-
tienting the value group v(∗K) by the convex hull of the standard value group v(K). In
the Archimedean case with K = C = R+iR, we can consider res as being induced by
the standard part map ∗R → R)
Now let C ⊆ Gn be a curve defined over K, and let x ∈ C(∗K). Suppose ||v′(x)|| :=
m
max |v′(x )| > 0. By the transcendental valuation inequality [7, Theorem 3.4.3], we have
i i
the following inequality on transcendence degrees of fields and dimensions of Q-vector
6
spaces:
1 = trd(K(x)/K) ≥ dim (v′(K(x))/v′(K))+trd(res(K(x))/res(K))
Q
But v′(K) = 0 and v′(x) 6= 0, so we deduce dim (v′(K(x))) = 1 and res(K(x)) =
Q
res(K). So say θ : Gn → Gn−1 is an algebraic epimorphism such that v′(θ(x)) = 0.
m m
Then res(θ(x)) = res(α) for some α ∈ Gn−1(K).
m
Now let β := θ(x)−α. Then res(β) = 0, so v′(β) 6= 0, so since dim (v′(K(x))) = 1,
Q
if β 6= 0 we have ||v′(β)|| = q||v′(x)|| for some q ∈ Q, q > 0.
Applying the compactness theorem of first-order logic, it follows that there exist finitely
many pairs (θ,α) of algebraic epimorphisms
θ : Gn → Gn−1
m m
and points α ∈ Gn−1(K), and there exist q ∈ Q, q > 0 and B > 0 such that for any
m
x ∈ K if ||v(x)|| := max (|v(x )|) > B then for one of the finitely many pairs (θ,α),
i i
||v(θ(x)−α)|| > q||v(x)||.
4. Baker’s Linear Forms in Logarithms
In order to prove our theorems, we must treat Archimedean and non-Archimedean
places separately. We begin by proving a technical lemma for the latter. Its proof is
elementary.
Lemma 2. For F a number field, α ∈ F×, p a prime, and B ≥ 1 there is a constant
c > 0 with the following property. Say z 6= 0 is algebraic over F and not a root of unity
such that z and α are multiplicatively dependent. We also suppose that h(z) ≤ B and
that |z − 1| < 1 for some finite place v of F(z) with residue characteristic p. Then
v
[F(z) : F]+logn ≥ −clog|zn −α| for all n ≥ 2 for which znα−1 has infinite order.
v
Proof. In this lemma, the constants involved in Vinogradov symbols ≪,≫ depend only
on F,α,p, and B. Let e be the ramification index of v above p.
The setup and the ultrametric triangle inequality imply |z| = 1.
v
Let us suppose first that α has finite order, say m ≪ 1. We fix the integer t ≥ 0
with pt−1 ≤ e/(p−1) < pt. The corollary after lemme 3 [4] yields |zpt −1| < p−1/(p−1).
v
Hence zpt and lies in the image of the domain of convergence of the p-adic exponential
exp(x) = Σ xi/i!; cf. Chapter II.5 [12]. It follows that |zkpt −1| = |k| |zpt −1| for any
i v v v
k ∈ N.
Hence we have
|mn| |zpt −1| = |zmnpt −1| ≤ |zmn −1| ≤ |zn −α|
v v v v v
and note that the very left-hand side is non-zero. Using |mn| ≥ 1/(mn) and taking the
v
ordinary logarithm yields
(7) log|zpt −1| ≤ log(mn)+log|zn −α| .
v v
By the local nature of our height (1), and since d ≥ e, we have −elog|zpt − 1| ≤
v v
[F(z) : Q]h((zpt −1)−1). Basic height properties, h(z) ≤ B, and pt ≤ ep/(p−1) ≤ 2e
now imply
elog|zpt −1| ≥ −[F(z) : Q]h(zpt −1) ≥ −[F(z) : Q](log2+ptB) ≫ −[F(z) : F]e.
v
7
The lemma follows if α has finite order after cancelling e and using (7).
Now we assume that α has infinite order. There is a unique reduced (k,l) ∈ Z2 such
that zkαl = ζ is a root of unity. We note that kl 6= 0 and that any pair (k′,l′) ∈ Z2 for
which zk′αl′ has finite order is an integral multiple of (k,l). To complete the proof we
may assume |zn − α| < p−1/(p−1). This implies |α| = 1 and this time we fix m ∈ N
v v
such that |αm −1| < p−1/(p−1) and m ≪ 1. Thus
v
|ζn−αnl+k| = |(ζα−l)n −αk| = |znk −αk| ≤ |zn −α| < p−1/(p−1)
v v v v
and passing to the m-th power gives
(8) |ζmn −αm(nl+k)| ≤ |zn −α| < p−1/(p−1).
v v
But |αm(nl+k)−1| < p−1/(p−1) holds as well. We use the ultrametric inequality to obtain
v
|ζmn−1| < p−1/(p−1). The only root of unity that is p-adically closer to 1 than p−1/(p−1)
v
is 1 itself. So ζmn = 1 and therefore |αm(nl+k)−1| ≤ |zn−α| by (8). By our choice of m
v v
the element αm is in the image of the domain of convergence of the p-adic exponential.
So as above, we can estimate
|nl+k| |αm −1| = |αm(nl+k) −1| ≤ |zn −α| ,
v v v v
so
|nl+k| ≪ |zn −α|
v v
since α is not a root of unity. We remark that the left-hand side is non-zero. Indeed,
nl +k 6= 0 since zn/α is not a root of unity
Using |nl +k| ≥ |2nlk|−1 we get |nlk| ≫ |zn −α|−1. So log|nlk| ≫ −log|zn −α|
v v v
because |nlk| ≥ n ≥ 2. The heights satisfy kh(z) = |l|h(α) since zk = ζα−l. But α 6= 0
is not a root of unity. So h(α) > 0 by Kronecker’s Theorem and therefore |l| ≪ k. This
leaves us with
(9) log(nk) ≫ −log|zn −α|
v
and to complete the proof we need to control k.
The lemma follows immediately if k = 1. Else k ≥ 2 and Dirichlet’s Theorem from
diophantine approximation provides us with integers k′ and l′ such that 1 ≤ k′ ≤ k/2
and |k′l−l′k| ≤ 2. So h(zk′αl′) = |k′l/k−l′|h(α) ≤ 2h(α)/k. On the other hand, zk′αl′
cannot have finite order since 1 ≤ k′ < k. By a weak version of Dobrowolski’s Theorem
[6] we have h(zk′αl′) ≫ [F(z) : F]−2. So k ≪ [F(z) : F]2. The lemma follows from this
(cid:3)
inequality in combination with (9).
In the following lemma, we apply Baker’s technique of estimating linear forms in
logarithms. IntheArchimedeancaseweusetheexplicitestimatesofBakerandWu¨stholz
[2], and in the non-Archimedean case we use a p-adic version obtained by Bugeaud and
Laurent [4].
It is useful to define L : GN(C) → RN by L(x ,...,x ) = (log|x |,...,log|x |).
m 1 N 1 N
Lemma 3. Let F be a number field and C ⊂ G2 a geometrically irreducible algebraic
m
curve defined over F. We suppose that C is not contained in a proper coset of G2 . Let
m
B ≥ 1. Let x = (x ,x ) ∈ C(Q) with h(x) ≤ B and xn ∈ C(Q) where n ≥ 2 is an
1 2
integer.
8
(i) Say σ : F(x) → C is an embedding and L = L(σ(x)) then
1/6
n
(10) [F(x) : F] ≥ c kLk
(cid:18)logn (cid:19)
where c > 0 depends only on C and B.
(ii) Say v is a finite place of the number field F(x) lying above a rational prime that
is sufficiently large with respect to C. If (log|x | ,log|x | ) 6= 0, then
1 v 2 v
(11) [F(x) : F] ≥ cn1/9
where c > 0 depends only on C,B, and the residue characteristic of v.
Proof. For part (i) the constant c is meant to be sufficiently small with respect to B,C,
and F. It will be fixed throughout the proof. The constants implicit in ≪ and ≫ below
are positive and depend only on B,C, and F. They are independent of x and c. Clearly,
we may assume
nkLk > c−6logn
as the conclusion (10) is immediate otherwise.
The conclusion of Lemma 1 applied to the point (σ(x )n,σ(x )n) is
1 2
(12) log|znβ −1| ≪ −nkLk where z = σ(xj1xj2),
1 2
for algebraic β ∈ C× and j ∈ Z2r{0}, both coming from a finite set depending only on
C. After decreasing c we may assume that the left-hand side of (12) is less than −1.
Let l ,l ∈ C be choices of logarithms of z and β. That is, el1 = z and el2 = β. Some
1 2
elementary calculus yields
log|Λ| ≪ −nkLk with Λ = nl +l +2πik
1 2
where k is an appropriate integer with |k| ≪ n.
In order to apply Baker’s theory to bound |Λ| from below we must first treat the case
Λ = 0, so zn = β−1.
If β is a not a root of unity, then z is not one either. The same weak version of
Dobrowolski’s Theorem as is used in Lemma 2 implies
h(β) 1 1 1
(13) = h(z) ≫ ≫ ≥ .
n [Q(z) : Q]2 [F(z) : F]2 [F(x) : F]2
Hence n ≪ [F(x) : F]2 because β is from a finite set depending only on P. The
definition of the height implies kLk ≪ [Q(x) : Q]h(x). But h(x) ≤ B by hypothesis, so
kLk ≪ [Q(x) : Q] ≪ [F(x) : F]. We may thus bound
n
(14) kLk ≪ [F(x) : F]n ≪ [F(x) : F]3
logn
and the lemma follows in this case.
But what if β is a root of unity? Then some positive power of zn is 1. Hence xn is
contained in a proper algebraic subgroup of G2 . But this is also a point on C. Over 10
m
years ago Bombieri, Masser, and Zannier [3] proved that h(xn) is bounded from above
by a constant depending only on C. Here it is essential that C is not contained in a
9
proper coset. So h(x) ≪ 1/n. By height inequalities similar to those used above we find
kLk ≪ [F(x) : F]h(x) ≪ [F(x) : F]/n and therefore
n
kLk ≪ [F(x) : F].
logn
We have proved (10) in the case Λ = 0.
Let us now assume Λ 6= 0; we apply results on linear forms in logarithms. An explicit
version due to Baker and Wu¨stholz [2] yields
log|Λ| ≫ −[Q(x) : Q]6(logA )(logA )(logA )logn
1 2 3
where A ,A ,A are bounded in terms of the heights of xj1xj2 and β. But h(xj1xj2) ≤
1 2 3 1 2 1 2
|j |h(x )+|j |h(x ) ≪ 1 and so we may streamline the inequality from above to get
1 1 2 2
log|Λ| ≫ −[Q(x) : Q]6logn.
We conclude by comparing this with the upper bound for log|Λ| derived above, since
this gives n kLk ≪ [Q(x) : Q]6 and thus completes the proof of (i).
logn
Say v is as in (ii) and let e be the ramification index of v above the rational prime.
Say C is cut out by a polynomial P. We may suppose that all non-zero coefficients of
P and all α provided by Lemma 1 (which do not depend on v) lie in F and have v-adic
absolute value 1.
In the proof of part (ii) we allow the constants in ≪ and ≫ to also depend on
the residue characteristic of v. In this non-Archimedean case we only assume L =
(log|x | ,log|x | ) 6= 0 which implies the lower bound kLk ≥ 1/e. In order to complete
1 v 2 v
the proof we may assume e < n1/2 since e ≥ n1/2 implies [F(x) : F] ≫ e ≥ n1/2.
By Lemma 1 there are reduced vectors j,j′ ∈ Z2 with
n
(15) log|zn −α| ≪ − ≤ −n1/2
v
e
and |z′ −α′|v < 1 where z = xj11xj22, z′ = x1j1′x2j2′, and α,α′ depend only on P.
Since |α| = |α′| = 1 we find |z| = |z′| = 1. Moreover, L 6= 0 so the two equalities
v v v v
j log|x | +j log|x | = 0 and j′ log|x | +j′ log|x | = 0
1 1 v 2 2 v 1 1 v 2 2 v
imply that j and j′ are linearly dependent. Hence j = j′ because they are reduced. In
particular, z = z′. There is g ≪ 1 with |α′g − 1| < 1 and it satisfies |zg − 1| < 1
v v
because |z −α′| < 1.
v
The exponent g plays an important role in Bugeaud and Laurent’s work [4]. Before
applying their th´eor`eme 3 to z and α′ we must first treat the case where these elements
are multiplicatively dependent.
If this is the case and if both z and zn/α have infinite order we may apply Lemma
2. In this case part (ii) follows with ample margin because of (15). If z has finite order,
then ekLk ≪ [F(x) : F]/n as in the Archimedean case by appealing to the Theorem
of Bombieri, Masser, and Zannier. But ekLk ≥ 1 and hence [F(x) : F] ≫ n, which
is better than the claim. If zn/α has finite order and z has infinite order, then α has
infinite order and nh(z) = h(α) ≪ 1. We can argue as near (13) using a height lower
bound to again obtain [F(x) : F] ≫ n1/2.
10
Finally, suppose z and α′ are multiplicatively independent. Then Bugeaud and Lau-
rent’s result [4, Th´eor`eme 2] implies
−log|zn −α| ≪ [F(z) : F]4(logn)2.
v
(cid:3)
We combine this inequality with (15) to conclude the proof.
5. Galois Orbits
Let N ≥ 2. Throughout this section F is a number field and, if not stated otherwise,
C ⊂ GN is a geometrically irreducible algebraic curve defined over F. The following
m
lemma is a weak equidistribution statement on curves
Lemma 4. Suppose that C is not contained in a proper coset of GN and that there
m
exists an embedding σ : F → C such that the curve C (C) ∩ T is finite. For any
0 σ0
B ≥ 1 there exists ǫ > 0 with the following property. For x ∈ C(Q) with h(x) ≤ B and
[F(x) : F] ≥ (2#C (C)∩T)N there is an embedding σ : F(x) → C extending σ such
σ0 0
that
kL(σ(x))k ≥ ǫ.
Proof. By hypothesis, the intersection C (C)∩T is finite. If the intersection is empty,
σ0
the lemma is immediate. So we define m = #C (C)∩T ≥ 1.
σ0
By symmetry it suffices to prove the lemma for points x = (x ,...,x ) as in the
1 N
assertion for which d = [F(x ) : F] is maximal among [F(x ) : F],...,[F(x ) : F]. So
1 1 N
[F(x) : F] ≤ dN and the hypothesis on [F(x) : F] yields d ≥ 2m.
We fix δ ∈ (0,1) with
(16) −logδ = 2[F : Q]2m2(4B +log2).
We take ǫ > 0 such that if (x′,...,x′ ) ∈ C (C) and kL(x′)k < ǫ then |x′ −t| < δ/2
1 N σ0 1
for some t ∈ C appearing as a first coordinate of a point in C (C)∩T. The number of
σ0
such values t is at most m.
Let us suppose that kL(σ(x))k < ǫ for all σ as in the hypothesis. This will lead to a
contradiction.
By thePigeonhole Principle there ist ∈ C anda set Σ of at least d/m ≥ 2 embeddings
σ : F(x ) → C extending σ such that |σ(x )−t| < δ/2. Then |σ(x )−σ′(x )| < δ for
1 0 1 1 1
two such embeddings.
The absolute value of the discriminant of the minimal polynomial of x is a positive
1
integer. Its logarithm is non-negative and hence
0 ≤ 2[Q(x ) : Q]2h(x )+ log|σ(x )−σ′(x )|
1 1 1 1
σX6=σ′
where σ,σ′ run over all embeddings Q(x ) → C. We note that h(x ) ≤ h(x) ≤ B. Thus
1 1
(17) 0 ≤ 2[F(x ) : Q(x )]2[Q(x ) : Q]2h(x )+D ≤ 2d2[F : Q]2B +D
1 1 1 1
with D = log|σ(x ) − σ′(x )| where the sum is now over all complex em-
σ(x1)6=σ′(x1) 1 1
beddings oPf F(x ).
1