SIEGEL’S THEOREM FOR DRINFELD MODULES
D. GHIOCA AND T. J. TUCKER
Abstract. We prove a Siegel type statement for finitely generated φ-
submodules of Ga under the action of a Drinfeld module φ. This provides
a positive answer to a question we asked in a previous paper. We also
prove an analog for Drinfeld modules of a theorem of Silverman for
nonconstant rational maps of P1 over a number field.
1. Introduction
In 1929, Siegel ([Sie29]) proved that if C is an irreducible affine curve
defined over a number field K and C has at least three points at infinity,
then there are at most finitely many K-rational points on C that have
integral coordinates. The proof of this famous theorem uses diophantine
approximation along with the fact that certain groups of rational points are
finitely generated; when C has genus greater than 0, the group in question
is the Mordell-Weil group of the Jacobian of C, while when C has genus 0,
the group in question is the group of S-units in a finite extension of K.
Motivated by the analogy between rank 2 Drinfeld modules and elliptic
curves, the authors conjectured in [GT06] a Siegel type statement for finitely
generated φ-submodules Γ of Ga (where φ is a Drinfeld module of arbitrary
rank). For a finite set of places S of a function field K, we defined a notion of
S-integrality and asked whether or not it is possible that there are infinitely
many γ ∈ Γ which are S-integral with respect to a fixed point α ∈ K. We
also proved in [GT06] a first instance of our conjecture in the case where Γ
is a cyclic submodule and α is a torsion point for φ. Our goal in this paper
is to prove our Siegel conjecture for every finitely generated φ-submodule
of Ga(K), where φ is a Drinfeld module defined over the field K (see our
Theorem 2.4). We will also establish an analog (also in the context of
Drinfeld modules) of a theorem of Silverman for nonconstant morphisms of
P1 of degree greater than 1 over a number field (see our Theorem 2.5).
We note that recently there has been significant progress on establish-
ing additional links between classical diophantine results over number fields
and similar statements for Drinfeld modules. Denis [Den92a] formulated
analogs for Drinfeld modules of the Manin-Mumford and the Mordell-Lang
2000 Mathematics Subject Classification. Primary 11G50, Secondary 11J68, 37F10.
Key words and phrases. Drinfeld module, Heights, Diophantine approximation.
1dghioca@math.mcmaster.ca; ttucker@math.rochester.edu
http://arxiv.org/abs/0704.1331v1
2 D. GHIOCA AND T. J. TUCKER
conjectures. The Denis-Manin-Mumford conjecture was proved by Scan-
lon in [Sca02], while a first instance of the Denis-Mordell-Lang conjecture
was established in [Ghi05] by the first author (see also [Ghi06b] for an ex-
tension of the result from [Ghi05]). The authors proved in [GT07] several
other cases of the Denis-Mordell-Lang conjecture. In addition, the first au-
thor proved in [Ghi06a] an equidistribution statement for torsion points of a
Drinfeld module that is similar to the equidistribution statement established
by Szpiro-Ullmo-Zhang [SUZ97] (which was later extended by Zhang [Zha98]
to a full proof of the famous Bogomolov conjecture). Breuer [Bre05] proved a
special case of the André-Oort conjecture for Drinfeld modules, while special
cases of this conjecture in the classical case of a number field were proved
by Edixhoven-Yafaev [EY03] and Yafaev [Yaf06]. Bosser [Bos99] proved a
lower bound for linear forms in logarithms at an infinite place associated to
a Drinfeld module (similar to the classical result obtained by Baker [Bak75]
for usual logarithms, or by David [Dav95] for elliptic logarithms). Bosser’s
result was used by the authors in [GT06] to establish certain equidistribution
and integrality statements for Drinfeld modules. Moreover, Bosser’s result
is believed to be true also for linear forms in logarithms at finite places for a
Drinfeld module (as was communicated to us by Bosser). Assuming this last
statement, we prove in this paper the natural analog of Siegel’s theorem for
finitely generated φ-submodules. We believe that our present paper provides
additional evidence that the Drinfeld modules represent a good arithmetic
analog in characteristic p for abelian varieties in characteristic 0.
The basic outline of this paper can be summarized quite briefly. In Sec-
tion 2 we give the basic definitions and notation, and then state our main
results. In Section 3 we prove these main results: Theorems 2.4 and 2.5.
2. Notation
Notation. N stands for the non-negative integers: {0, 1, . . . }, while N∗ :=
N \ {0} stands for the positive integers.
2.1. Drinfeld modules. We begin by defining a Drinfeld module. Let p
be a prime and let q be a power of p. Let A := Fq[t], let K be a finite
field extension of Fq(t), and let K be an algebraic closure of K. We let τ
be the Frobenius on Fq, and we extend its action on K. Let K{τ} be the
ring of polynomials in τ with coefficients from K (the addition is the usual
addition, while the multiplication is the composition of functions).
A Drinfeld module is a morphism φ : A→ K{τ} for which the coefficient
of τ0 in φ(a) =: φa is a for every a ∈ A, and there exists a ∈ A such that
φa 6= aτ
0. The definition given here represents what Goss [Gos96] calls a
Drinfeld module of “generic characteristic”.
We note that usually, in the definition of a Drinfeld module, A is the ring
of functions defined on a projective nonsingular curve C, regular away from
a closed point η ∈ C. For our definition of a Drinfeld module, C = P1
and η
is the usual point at infinity on P1. On the other hand, every ring of regular
SIEGEL’S THEOREM FOR DRINFELD MODULES 3
functions A as above contains Fq[t] as a subring, where t is a nonconstant
function in A.
For every field extension K ⊂ L, the Drinfeld module φ induces an action
on Ga(L) by a∗x := φa(x), for each a ∈ A. We call φ-submodules subgroups
of Ga(K) which are invariant under the action of φ. We define the rank of
a φ-submodule Γ be
dimFrac(A) Γ⊗A Frac(A).
As shown in [Poo95], Ga(K) is a direct sum of a finite torsion φ-submodule
with a free φ-submodule of rank ℵ0.
A point α is torsion for the Drinfeld module action if and only if there
exists Q ∈ A \ {0} such that φQ(α) = 0. The monic polynomial Q of
minimal degree which satisfies φQ(α) = 0 is called the order of α. Since each
polynomial φQ is separable, the torsion submodule φtor lies in the separable
closure Ksep of K.
2.2. Valuations and Weil heights. Let MFq(t) be the set of places on
Fq(t). We denote by v∞ the place in MFq(t) such that v∞(
) = deg(g) −
deg(f) for every nonzero f, g ∈ A = Fq[t]. We letMK be the set of valuations
on K. ThenMK is a set of valuations which satisfies a product formula (see
[Ser97, Chapter 2]). Thus
• for each nonzero x ∈ K, there are finitely many v ∈ MK such that
|x|v 6= 1; and
• for each nonzero x ∈ K, we have
|x|v = 1.
We may use these valuations to define a Weil height for each x ∈ K as
(2.0.1) h(x) =
max log(|x|v , 1).
Convention. Without loss of generality we may assume that the nor-
malization for all the valuations of K is made so that for each v ∈ MK , we
have log |x|v ∈ Z.
Definition 2.1. Each place in MK which lies over v∞ is called an infinite
place. Each place in MK which does not lie over v∞ is called a finite place.
2.3. Canonical heights. Let φ : A → K{τ} be a Drinfeld module of rank
d (i.e. the degree of φt as a polynomial in τ equals d). The canonical height
of β ∈ K relative to φ (see [Den92b]) is defined as
ĥ(β) = lim
h(φtn(β))
Denis [Den92b] showed that a point is torsion if and only if its canonical
height equals 0.
For every v ∈MK , we let the local canonical height of β ∈ K at v be
(2.1.1) ĥv(β) = lim
log max(|φtn(β)|v , 1)
4 D. GHIOCA AND T. J. TUCKER
Furthermore, for every a ∈ Fq[t], we have ĥv(φa(x)) = deg(φa) · ĥv(x) (see
[Poo95]). It is clear that ĥv satisfies the triangle inequality, and also that∑
ĥv(β) = ĥ(β).
2.4. Completions and filled Julia sets. By abuse of notation, we let
∞ ∈ MK denote any place extending the place v∞. We let K∞ be the
completion of K with respect to | · |∞. We let K∞ be an algebraic closure
of K∞. We let C∞ be the completion of K∞. Then C∞ is a complete,
algebraically closed field. Note that C∞ depends on our choice for ∞ ∈MK
extending v∞. However, each time we will work with only one such place
∞, and so, there will be no possibility of confusion.
Next, we define the v-adic filled Julia set Jφ,v corresponding to the Drin-
feld module φ and to each place v of MK . Let Cv be the completion of an
algebraic closure of Kv. Then | · |v extends to a unique absolute value on
all of Cv. The set Jφ,v consists of all x ∈ Cv for which {|φQ(x)|v}Q∈A is
bounded. It is immediate to see that x ∈ Jφ,v if and only if {|φtn(x)|v}n≥1
is bounded.
One final note on absolute values: as noted above, the place v ∈ MK
extends to a unique absolute value | · |v on all of Cv. We fix an embedding
of i : K −→ Cv. For x ∈ K, we denote |i(x)|v simply as |x|v , by abuse of
notation.
2.5. The coefficients of φt. Each Drinfeld module is isomorphic to a Drin-
feld module for which all the coefficients of φt are integral at all the places
inMK which do not lie over v∞. Indeed, we let B ∈ Fq[t] be a product of all
(the finitely many) irreducible polynomials P ∈ Fq[t] with the property that
there exists a place v ∈MK which lies over the place (P ) ∈MFq(t), and there
exists a coefficient of φt which is not integral at v. Let γ be a sufficiently
large power of B. Then ψ : A → K{τ} defined by ψQ := γ
−1φQγ (for each
Q ∈ A) is a Drinfeld module isomorphic to φ, and all the coefficients of ψt
are integral away from the places lying above v∞. Hence, from now on, we
assume that all the coefficients of φt are integral away from the places lying
over v∞. It follows that for every Q ∈ A, all coefficients of φQ are integral
away from the places lying over v∞.
2.6. Integrality and reduction.
Definition 2.2. For a finite set of places S ⊂MK and α ∈ K, we say that
β ∈ K is S-integral with respect to α if for every place v /∈ S, and for every
morphisms σ, τ : K → K (which restrict to the identity on K) the following
are true:
• if |ατ |v ≤ 1, then |α
τ − βσ|v ≥ 1.
• if |ατ |v > 1, then |β
σ|v ≤ 1.
We note that if β is S-integral with respect to α, then it is also S′-integral
with respect to α, where S′ is a finite set of places containing S. Moreover,
the fact that β is S-integral with respect to α, is preserved if we replace
SIEGEL’S THEOREM FOR DRINFELD MODULES 5
K by a finite extension. Therefore, in our results we will always assume
α, β ∈ K. For more details about the definition of S-integrality, we refer the
reader to [BIR05].
Definition 2.3. The Drinfeld module φ has good reduction at a place v if
for each nonzero a ∈ A, all coefficients of φa are v-adic integers and the
leading coefficient of φa is a v-adic unit. If φ does not have good reduction
at v, then we say that φ has bad reduction at v.
It is immediate to see that φ has good reduction at v if and only if all
coefficients of φt are v-adic integers, while the leading coefficient of φt is a
v-adic unit.
We can now state our Siegel type result for Drinfeld modules.
Theorem 2.4. With the above notation, assume in addition K has only
one infinite place. Let Γ be a finitely generated φ-submodule of Ga(K), let
α ∈ K, and let S be a finite set of places in MK . Then there are finitely
many γ ∈ Γ such that γ is S-integral with respect to α.
As mentioned in Section 1, we proved in [GT06] that Theorem 2.4 holds
when Γ is a cyclic φ-module generated by a nontorsion point β ∈ K and
α ∈ φtor(K) (see Theorem 1.1 and Proposition 5.6 of [GT06]). Moreover,
in [GT06] we did not have in our results the extra hypothesis from Theo-
rem 2.4 that there exists only one infinite place in MK . Even though we
believe Theorem 2.4 is true without this hypothesis, our method for proving
Theorem 2.4 requires this technical hypothesis. On the other hand, we are
able to prove the following analog for Drinfeld modules of a theorem of Sil-
verman (see [Sil93]) for nonconstant morphisms of P1 of degree greater than
1 over a number field, without the hypothesis of having only one infinite
place in MK .
Theorem 2.5. With the above notation, let β ∈ K be a nontorsion point,
and let α ∈ K be an arbitrary point. Then there are finitely many Q ∈ A
such that φQ(β) is S-integral for α.
As explained before, in [GT06] we proved Theorem 2.5 in the case α is a
torsion point in K.
3. Proofs of our main results
We continue with the notation from Section 2. In our argument, we will
be using the following key fact.
Fact 3.1. Assume ∞ ∈ MK is an infinite place. Let γ1, . . . , γr, α ∈ K.
Then there exist (negative) constants C0 and C1 (depending only on φ,
γ1, . . . , γr, α) such that for any polynomials P1, . . . , Pr ∈ A (not all con-
stants), either φP1(γ1) + · · ·+ φPr(γr) = α or
log |φP1(γ1) + · · · + φPr(γr)− α|∞ ≥ C0 +C1 max
1≤i≤r
(deg(Pi) log deg(Pi)).
6 D. GHIOCA AND T. J. TUCKER
Fact 3.1 follows easily from the lower bounds for linear forms in logarithms
established by Bosser (see Théorème 1.1 in [Bos99]). Essentially, it is the
same proof as our proof of Proposition 3.7 of [GT06] (see in particular the
derivation of the inequality (3.7.2) in [GT06]). For the sake of completeness,
we will provide below a sketch of a proof of Fact 3.1.
Proof of Fact 3.1. We denote by exp∞ the exponential map associated to
the place ∞ (see [Gos96]). We also let L be the corresponding lattice for
exp∞, i.e. L := ker(exp∞). Finally, let ω1, . . . , ωd be an A-basis for L of
“successive minima” (see Lemma (4.2) of [Tag93]). This means that for
every Q1, . . . , Qd ∈ A, we have
(3.1.1) |Q1ω1 + · · ·+Qdωd|∞ =
|Qiωi|∞.
Let u0 ∈ C∞ such that exp∞(u0) = α. We also let u1, . . . , ur ∈ C∞
such that for each i, we have exp∞(ui) = γi. We will find constants C0
and C1 satisfying the inequality from Fact 3.1, which depend only on φ and
u0, u1, . . . , ur.
There exists a positive constant C2 such that exp∞ induces an isomor-
phism from the ball B := {z ∈ C∞ : |z|∞ < C2} to itself (see Lemma 3.6 of
[GT06]). If we assume there exist no constants C0 and C1 as in the conclu-
sion of Fact 3.1, then there exist polynomials P1, . . . , Pr, not all constants,
such that
(3.1.2)
φPi(γi) 6= α
and |
i=1 φPi(γi) − α|∞ < C2. Thus we can find y ∈ B such that |y|∞ =
i=1 φPi(γi)− α|∞ and
(3.1.3) exp∞(y) =
φPi(γi)− α.
Moreover, because exp∞ is an isomorphism on the metric space B, then for
every y′ ∈ C∞ such that exp∞(y
i=1 φPi(γi)−α, we have |y
′|∞ ≥ |y|∞.
But we know that
(3.1.4) exp∞
Piui − u0
φPi(γi)− α.
Therefore |
i=1 Piui − u0|∞ ≥ |y|∞. On the other hand, using (3.1.3) and
(3.1.4), we conclude that there exist polynomials Q1, . . . , Qd such that
Piui − u0 = y +
Qiωi.
SIEGEL’S THEOREM FOR DRINFELD MODULES 7
Hence |
i=1Qiωi|∞ ≤ |
i=1 Piui − u0|∞. Using (3.1.1), we obtain
∣∣∣∣∣
∣∣∣∣∣
|Qiωi|∞ ≤
∣∣∣∣∣
Piui − u0
∣∣∣∣∣
≤ max
|u0|∞,
|Piui|∞
≤ C3 ·
|Pi|∞,
(3.1.5)
where C3 is a constant depending only on u0, u1, . . . , ur. We take logarithms
of both sides in (3.1.5) and obtain
degQi ≤
degPi + logC3 −
log |ωi|∞
degPi + C4,
(3.1.6)
where C4 depends only on φ and u0, u1, . . . , ur (the dependence on the ωi is
actually a dependence on φ, because the ωi are a fixed basis of “successive
minima” for φ at ∞). Using (3.1.6) and Proposition 3.2 of [GT06] (which is a
translation of the bounds for linear forms in logarithms for Drinfeld modules
established in [Bos99]), we conclude that there exist (negative) constants C0,
C1, C5 and C6 (depending only on φ, γ1, . . . , γr and α) such that
∣∣∣∣∣
φPi(γi)− α
∣∣∣∣∣
= log |y|∞
= log
∣∣∣∣∣
Piui − u0 −
∣∣∣∣∣
≥ C5 + C6
degPi + C4
(degPi + C4)
≥ C0 + C1
degPi
(degPi) ,
(3.1.7)
as desired. �
In our proofs for Theorems 2.5 and 2.4 we will also use the following state-
ment, which is believed to be true, based on communication with V. Bosser.
Therefore we assume its validity without proof.
Statement 3.2. Assume v does not lie above v∞. Let γ1, . . . , γr, α ∈
K. Then there exist positive constants C1, C2, C3 (depending only on v,
φ, γ1, . . . , γr and α) such that for any P1, . . . , Pr ∈ Fq[t], either φP1(γ1) +
· · ·+ φPr(γr) = α or
log |φP1(γ1) + · · ·+ φPr(γr)− α|v ≥ −C1 − C2 max
1≤i≤r
(deg(Pi))
8 D. GHIOCA AND T. J. TUCKER
Statement 3.2 follows after one establishes a lower bound for linear forms
in logarithms at finite places v. In a private communication, V. Bosser told
us that it is clear to him that his proof ([Bos99]) can be adapted to work
also at finite places with minor modifications.
We sketch here how Statement 3.2 would follow from a lower bound for
linear forms in logarithms at finite places. Let v be a finite place and let
expv be the formal exponential map associated to v. The existence of expv
and its convergence on a sufficiently small ball Bv := {x ∈ Cv : |x|v < Cv}
is proved along the same lines as the existence and the convergence of the
usual exponential map at infinite places for φ (see Section 4.6 of [Gos96]).
In addition,
(3.2.1) | expv(x)|v = |x|v
for every x ∈ Bv. Moreover, at the expense of replacing Cv with a smaller
positive constant, we may assume that for each F ∈ A, and for each x ∈ Bv,
we have (see Lemma 4.2 in [GT06])
(3.2.2) |φF (x)|v = |Fx|v .
Assume we know the existence of the following lower bound for (nonzero)
linear forms in logarithms at a finite place v.
Statement 3.3. Let u1, . . . , ur ∈ Bv such that for each i, expv(ui) ∈ K.
Then there exist positive constants C4, C5, and C6 (depending on u1, . . . , ur)
such that for every F1, . . . , Fr ∈ A, either
i=1 Fiui = 0, or
∣∣∣∣∣
∣∣∣∣∣
≥ −C4 − C5
degFi
As mentioned before, Bosser proved Statement 3.3 in the case v is an
infinite place (in his result, C6 = 1 + ǫ and C4 = Cǫ for every ǫ > 0).
We will now derive Statement 3.2 assuming Statement 3.3 holds.
Proof. (That Statement 3.3 implies Statement 3.2.) Clearly, it suffices to
prove Statement 3.2 in the case α = 0. So, let γ1, . . . , γr ∈ K, and assume
by contradiction that there exists an infinite sequence {Fn,i} n∈N∗
1≤i≤r
such that
for each n, we have
(3.3.1) −∞ < log
∣∣∣∣∣
φFn,i(γi)
∣∣∣∣∣
< logCv.
For each n ≥ 1, we let Fn := (Fn,1, . . . , Fn,r) ∈ A
r. We view Ar as
an r-dimensional A-lattice inside the r-dimensional Frac(A)-vector space
Frac(A)r. In addition, we may assume that for n 6= m, we have Fn 6= Fm.
Using basic linear algebra, because the sequence {Fn,i} n∈N∗
1≤i≤r
is infinite, we
can find n0 ≥ 1 such that for every n > n0, there exist Hn, Gn,1, . . . , Gn,n0 ∈
SIEGEL’S THEOREM FOR DRINFELD MODULES 9
A (not all equal to 0) such that
(3.3.2) Hn · Fn =
Gn,j · Fj.
Essentially, (3.3.2) says that F1, . . . ,Fn0 span the linear subspace of Frac(A)
generated by all Fn. Moreover, we can choose theHn in (3.3.2) in such a way
that degHn is bounded independently of n (e.g. by a suitable determinant
of some linearly independent subset of the first n0 of the Fj). Furthemore,
there exists a constant C7 such that for all n > n0, we have
(3.3.3)
degGn,j < C7 +
degFn,i.
Because
i=1 φFn,i(γi)
< Cv, equation (3.2.2) yields
(3.3.4)
∣∣∣∣∣φHn
φFn,i(γi)
)∣∣∣∣∣
= |Hn|v ·
∣∣∣∣∣
φFn,i(γi)
∣∣∣∣∣
Using (3.3.2), (3.3.4), and the fact that |Hn|v ≤ 1, we obtain
∣∣∣∣∣
φFn,i(γi)
∣∣∣∣∣
∣∣∣∣∣φHn
φFn,i(γi)
)∣∣∣∣∣
∣∣∣∣∣∣
φGn,j
φFj,i(γi)
)∣∣∣∣∣∣
(3.3.5)
Since
i=1 φFj,i(γi)
< Cv for all 1 ≤ j ≤ n0, there exist u1, . . . , un0 ∈ Bv
such that for every 1 ≤ j ≤ n0, we have
expv(uj) =
φFj,i(γi).
Then Statement 3.3 implies that there exist constants C4, C5, C6, C8, C9 (de-
pending on u1, . . . , un0), such that
∣∣∣∣∣∣
φGn,j
φFj,i(γi)
)∣∣∣∣∣∣
= log
∣∣∣∣∣∣
Gn,juj
∣∣∣∣∣∣
≥ −C4 − C5
degGn,j
≥ −C8 − C9
degFn,i
(3.3.6)
where in the first equality we used (3.2.1), while in the last inequality we
used (3.3.3). Equations (3.3.5) and (3.3.6) show that Statement 3.2 follows
from Statement 3.3, as desired. �
10 D. GHIOCA AND T. J. TUCKER
Next we prove Theorem 2.5 which will be a warm-up for our proof of
Theorem 2.4. For its proof, we will only need the following weaker (but also
still conjectural) form of Statement 3.2 (i.e., we only need Statement 3.3 be
true for non-homogeneous 1-forms of logarithms).
Statement 3.4. Assume v does not lie over v∞. Let γ, α ∈ K. Then there
exist positive constants C1, C2 and C3 (depending only on v, φ, γ and α)
such that for each polynomial P ∈ Fq[t], either φP (γ) = α or
log |φP (γ)− α|v ≥ −C1 − C2 deg(P )
Proof of Theorem 2.5. The following Lemma is the key to our proof.
Lemma 3.5. For each v ∈MK , we have ĥv(β) = limdegQ→∞
log |φQ(β)−α|v
qd degQ
Proof of Lemma 3.5. Let v ∈ MK . If β /∈ Jφ,v, then |φQ(β)|v → ∞, as
degQ → ∞. Hence, if degQ is sufficiently large, then |φQ(β) − α|v =
|φQ(β)|v = max{|φQ(β)|v , 1}, which yields the conclusion of Lemma 3.5.
Thus, from now on, we assume β ∈ Jφ,v. Hence ĥv(β) = 0, and we need
to show that
(3.5.1) lim
degQ→∞
log |φQ(β)− α|v
qddegQ
Also note that since β ∈ Jφ,v, then |φQ(β) − α|v is bounded, and so,
lim supdegQ→∞
log |φQ(β)−α|v
qd degQ
≤ 0. Thus, in order to prove (3.5.1), it suf-
fices to show that
(3.5.2) lim inf
degQ→∞
log |φQ(β)− α|v
qddegQ
If v is an infinite place, then Fact 3.1 implies that for every polynomial Q
such that φQ(β) 6= α, we have log |φQ(β)−α|∞ ≥ C0+C1 deg(Q) log deg(Q)
(for some constants C0, C1 < 0). Then taking the limit as degQ → ∞, we
obtain (3.5.2), as desired.
Similarly, if v is a finite place, then (3.5.2) follows from Statement 3.4. �
Theorem 2.5 follows easily using the result of Lemma 3.5. We assume
there exist infinitely many polynomials Qn such that φQn(β) is S-integral
with respect to α. We consider the sum
log |φQn(β) − α|v
qddegQn
Using Lemma 3.5, we obtain that Σ = ĥ(β) > 0 (because β /∈ φtor).
Let T be a finite set of places consisting of all the places in S along with
all places v ∈MK which satisfy at least one of the following conditions:
1. |β|v > 1.
2. |α|v > 1.
3. v is a place of bad reduction for φ.
SIEGEL’S THEOREM FOR DRINFELD MODULES 11
Therefore by our choice for T (see 1. and 3.), for every v /∈ T , we have
|φQn(β)|v ≤ 1. Thus, using also 2., we have |φQn(β) − α|v ≤ 1. On the
other hand, φQn(β) is also T -integral with respect to α. Hence, because of
2., then for all v /∈ T , we have |φQn(β) − α|v ≥ 1. We conclude that for
every v /∈ T , and for every n, we have |φQn(β) − α|v = 1. This allows us
to interchange the summation and the limit in the definition of Σ (because
then Σ is a finite sum over all v ∈ T ). We obtain
Σ = lim
qddegQn
log |φQn(β)− α|v = 0,
by the product formula applied to each φQn(β)−α. On the other hand, we
already showed that Σ = ĥ(β) > 0. This contradicts our assumption that
there are infinitely many polynomials Q such that φQ(β) is S-integral with
respect to α, and concludes the proof of Theorem 2.5. �
Before proceeding to the proof of Theorem 2.4, we prove several facts
about local heights. In Lemma 3.10 we will use the technical assumption of
having only one infinite place in K.
From now on, let φt =
i=0 aiτ
i. As explained in Section 2, we may
assume each ai is integral away from v∞. Also, from now on, we work under
the assumption that there exists a unique place ∞ ∈MK lying above v∞.
Fact 3.6. For every place v of K, there exists Mv > 0 such that for each
x ∈ K, if |x|v > Mv, then for every nonzero Q ∈ A, we have |φQ(x)|v > Mv.
Moreover, if |x|v > Mv, then ĥv(x) = log |x|v +
log |ad|v
Fact 3.6 is proved in Lemma 4.4 of [GT06]. In particular, Fact 3.6 shows
that for each v ∈ MK and for each x ∈ K, we have ĥv(x) ∈ Q. Indeed, for
every x ∈ K of positive local canonical height at v, there exists a polynomial
P such that |φP (x)|v > Mv. Then ĥv(x) =
bhv(φP (x))
qd degP
. By Fact 3.6, we already
know that ĥv(φP (x)) ∈ Q. Thus also ĥv(x) ∈ Q.
Fact 3.7. Let v ∈MK \{∞}. There exists a positive constant Nv, and there
exists a nonzero polynomial Qv, such that for each x ∈ K, the following
statements are true
(i) if |x|v ≤ Nv, then for each Q ∈ A, we have |φQ(x)|v ≤ |x|v ≤ Nv.
(ii) either |φQv(x)|v ≤ Nv, or |φQv(x)|v > Mv.
Proof of Fact 3.7. This was proved in [Ghi07b]. It is easy to see that
Nv := min
1≤i≤d
satisfies condition (i), but the proof of (ii) is much more complicated. In
[Ghi07b], the first author proved that there exists a positive integer dv such
that for every x ∈ K, there exists a polynomial Q of degree at most dv such
that either |φQ(x)|v > Mv, or |φQ(x)|v ≤ Nv (see Remark 5.12 which is
12 D. GHIOCA AND T. J. TUCKER
valid for every place which does not lie over v∞). Using Fact 3.6 and (i), we
conclude that the polynomial Qv :=
deg P≤dv
P satisfies property (ii). �
Using Facts 3.6 and 3.7 we prove the following important result valid for
finite places.
Lemma 3.8. Let v ∈ MK \ {∞}. Then there exists a positive integer Dv
such that for every x ∈ K, we have Dv · ĥv(x) ∈ N. If in addition we assume
v is a place of good reduction for φ, then we may take Dv = 1.
Proof of Lemma 3.8. Let x ∈ K. If ĥv(x) = 0, then we have nothing to
show. Therefore, assume from now on that ĥv(x) > 0. Using (ii) of Fact 3.7,
there exists a polynomial Qv (depending only on v, and not on x) such
that |φQv(x)|v > Mv (clearly, the other option from (ii) of Lemma 3.7 is
not available because we assumed that ĥv(x) > 0). Moreover, using the
definition of the local height, and also Fact 3.6,
(3.8.1) ĥv(x) =
ĥv(φQv(x))
qddegQv
log |φQv(x)|v +
log |ad|v
qddegQv
Because both log |φQv(x)|v and log |ad|v are integer numbers, (3.8.1) yields
the conclusion of Lemma 3.8 (we may take Dv = q
ddegQv(qd − 1)).
The second part of Lemma 3.8 follows immediately from Lemma 4.13 of
[Ghi07b]. Indeed, if v is a place of good reduction for φ, then |x|v > 1
(because we assumed ĥv(x) > 0). But then, ĥv(x) = log |x|v (here we use
the fact that v is a place of good reduction, and thus ad is a v-adic unit).
Hence ĥv(x) ∈ N, and we may take Dv = 1. �
The following result is an immediate corollary of Fact 3.8.
Corollary 3.9. There exists a positive integer D such that for every v ∈
MK \ {∞}, and for every x ∈ K, we have D · ĥv(x) ∈ N.
Next we prove a similar result as in Lemma 3.8 which is valid for the only
infinite place of K.
Lemma 3.10. There exists a positive integer D∞ such that for every x ∈ K,
either ĥv(x) > 0 for some v ∈MK \ {∞}, or D∞ · ĥ∞(x) ∈ N.
Before proceeding to its proof, we observe that we cannot remove the
assumption that ĥv(x) = 0 for every finite place v, in order to obtain the
existence of D∞ in the statement of Lemma 3.10. Indeed, we know that
in K there are points of arbitrarily small (but positive) local height at ∞
(see Example 6.1 from [Ghi07b]). Therefore, there exists no positive integer
D∞ which would clear all the possible denominators for the local heights at
∞ of those points. However, it turns out (as we will show in the proof of
Lemma 3.10) that for such points x of very small local height at ∞, there
exists some other place v for which ĥv(x) > 0.
SIEGEL’S THEOREM FOR DRINFELD MODULES 13
Proof of Lemma 3.10. Let x ∈ K. If x ∈ φtor, then we have nothing to
prove (every positive integer D∞ would work because ĥ∞(x) = 0). Thus,
we assume x is a nontorsion point. If ĥv(x) > 0 for some place v which
does not lie over v∞, then again we are done. So, assume from now on that
ĥv(x) = 0 for every finite place v.
By proceeding as in the proof of Lemma 3.8, it suffices to show that there
exists a polynomial Q∞ of degree bounded independently of x such that
|φQ∞(x)|∞ > M∞ (with the notation as in Fact 3.6). This is proved in
Theorem 4.4 of [Ghi07a]. The first author showed in [Ghi07a] that there
exists a positive integer L (depending only on the number of places of bad
reduction of φ) such that for every nontorsion point x, there exists a place
v ∈ MK , and there exists a polynomial Q of degree less than L such that
|φQ(x)|v > Mv . Because we assumed that ĥv(x) = 0 for every v 6= ∞, then
the above statement yields the existence of D∞. �
We will prove Theorem 2.4 by showing that a certain lim sup is positive.
This will contradict the existence of infinitely many S-integral points in a
finitely generated φ-submodule. Our first step will be a result about the
lim inf of the sequences which will appear in the proof of Theorem 2.4.
Lemma 3.11. Suppose that Γ is a torsion-free φ-submodule of Ga(K) gener-
ated by elements γ1, . . . , γr. For each i ∈ {1, . . . , r} let (Pn,i)n∈N∗ ⊂ Fq[t] be
a sequence of polynomials such that for each m 6= n, the r-tuples (Pn,i)1≤i≤r
and (Pm,i)1≤i≤r are distinct. Then for every v ∈MK , we have
(3.11.1) lim inf
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
Proof. Suppose that for some ǫ > 0, there exists a sequence (nk)k≥1 ⊂ N
such that
i=1 φPnk,i
(γi) 6= α and
(3.11.2)
log |
i=1 φPnk,i
(γi)− α|v
i=1 q
ddeg Pnk,i
< −ǫ,
for every k ≥ 1. But taking the lower bound from Fact 3.1 or Statement 3.2
(depending on whether v is the infinite place or not) and dividing through
i=1 q
ddeg Pnk,i , we see that this is impossible. �
The following proposition is the key technical result required to prove
Theorem 2.4. This proposition plays the same role that Lemma 3.5 plays
in the proof of Theorem 2.5, or that Corollary 3.13 plays in the proof of
Theorem 1.1 from [GT06]. Note that is does not provide an exact formula
for the canonical height of a point, however; it merely shows that a certain
limit is positive. This will suffice for our purposes since we only need that a
certain sum of limits be positive in order to prove Theorem 2.4.
Proposition 3.12. Let Γ be a torsion-free φ-submodule of Ga(K) generated
by elements γ1, . . . , γr. For each i ∈ {1, . . . , r} let (Pn,i)n∈N∗ ⊂ Fq[t] be a
14 D. GHIOCA AND T. J. TUCKER
sequence of polynomials such that for each m 6= n, the r-tuples (Pn,i)1≤i≤r
and (Pm,i)1≤i≤r are distinct. Then there exists a place v ∈MK such that
(3.12.1) lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
Proof. Using the triangle inequality for the v-adic norm, and the fact that
qddeg Pn,i = +∞,
we conclude that proving that (3.12.1) holds is equivalent to proving that
for some place v, we have
(3.12.2) lim sup
log |
i=1 φPn,i(γi)|v∑r
i=1 q
ddeg Pn,i
We also observe that it suffices to prove Proposition 3.12 for a subsequence
(nk)k≥1 ⊂ N
We prove (3.12.2) by induction on r. If r = 1, then by Corollary 3.13 of
[GT06] (see also our Lemma 3.5),
(3.12.3) lim sup
deg P→∞
log |φP (γ1)|v
qddegP
= ĥv(γ1)
and because γ1 /∈ φtor, there exists a place v such that ĥv(γ1) > 0, thus
proving (3.12.2) for r = 1. Therefore, we assume (3.12.2) is established for
all φ-submodules Γ of rank less than r and we will prove it for φ-submodules
of rank r.
In the course of our argument for proving (3.12.2), we will replace several
times a given sequence with a subsequence of itself (note that passing to a
subsequence can only make the lim sup smaller). For the sake of not cluster-
ing the notation, we will drop the extra indices which would be introduced
by dealing with the subsequence.
Let S0 be the set of places v ∈MK for which there exists some γ ∈ Γ such
that ĥv(γ) > 0. The following easy fact will be used later in our argument.
Fact 3.13. The set S0 is finite.
Proof of Fact 3.13. We claim that S0 equals the finite set S
0 of places v ∈
MK for which there exists i ∈ {1, . . . , r} such that ĥv(γi) > 0. Indeed, let
v ∈ MK \ S
0. Then for each i ∈ {1, . . . , r} we have ĥv(γi) = 0. Moreover,
for each i ∈ {1, . . . , r} and for each Qi ∈ Fq[t], we have
(3.13.1) ĥv(φQi(γi)) = deg(φQi) · ĥv(γi) = 0.
Using (3.13.1) and the triangle inequality for the local canonical height, we
obtain that
φQi(γi)
SIEGEL’S THEOREM FOR DRINFELD MODULES 15
This shows that indeed S0 = S
0, and concludes the proof of Fact 3.13. �
If the sequence (nk)k≥1 ⊂ N
∗ has the property that for some j ∈ {1, . . . , r},
we have
(3.13.2) lim
qddeg Pnk,j
i=1 q
ddeg Pnk,i
then the inductive hypothesis will yield the desired conclusion. Indeed, by
the induction hypothesis, and also using (3.13.2), there exists v ∈ S0 such
(3.13.3) lim sup
log |
i 6=j φPnk,i
(γi)|v
i=1 q
ddeg Pnk,i
If ĥv(γj) = 0, then
∣∣∣φPnk,j(γj)
is bounded as k → ∞. Thus, for large
enough k, ∣∣∣∣∣
φPnk,i
∣∣∣∣∣
∣∣∣∣∣∣
i 6=j
φPnk,i
∣∣∣∣∣∣
and so, (3.13.3) shows that (3.12.2) holds.
Now, if ĥv(γj) > 0, then we proved in Lemma 4.4 of [GT06] that
(3.13.4) log |φP (γj)|v − q
ddeg P ĥv(γj)
is uniformly bounded as degP → ∞ (note that this follows easily from
simple arguments involving geometric series and coefficients of polynomials).
Therefore, using (3.13.2), we obtain
(3.13.5) lim
∣∣∣φPnk,j(γj)
i=1 q
ddeg Pnk,i
Using (3.13.3) and (3.13.5), we conclude that for large enough k,
∣∣∣∣∣
φPnk,i
∣∣∣∣∣
∣∣∣∣∣∣
i 6=j
φPnk,i
∣∣∣∣∣∣
and so,
(3.13.6) lim sup
i=1 φPnk,i
i=1 q
ddeg Pnk,i
as desired. Therefore, we may assume from now on that there exists B ≥ 1
such that for every n,
(3.13.7)
max1≤i≤r q
ddegPn,i
min1≤i≤r q
ddegPn,i
≤ B or equivalently,
(3.13.8) max
1≤i≤r
degPn,i − min
1≤i≤r
degPn,i ≤
logq B
16 D. GHIOCA AND T. J. TUCKER
We will prove that (3.12.2) holds for some place v by doing an analysis
at each place v ∈ S0. We know that |S0| ≥ 1 because all γi are nontorsion.
Our strategy is to show that in case (3.12.2) does not hold, then we can
find δ1, . . . , δr ∈ Γ, and we can find a sequence (nk)k≥1 ⊂ N
∗, and a sequence
of polynomials (Rk,i) k∈N∗
1≤i≤r
such that
(3.13.9)
φPnk,i
(γi) =
φRk,i(δi) and
(3.13.10)
ĥv(δi) <
ĥv(γi) and
(3.13.11) 0 < lim inf
i=1 q
ddegPnk,i
i=1 q
ddegRk,i
≤ lim sup
i=1 q
ddegPnk,i
i=1 q
ddegRk,i
< +∞.
Equation (3.13.9) will enable us to replace the γi by the δi and proceed with
our analysis of the latter. Inequality (3.13.10) combined with Corollary 3.9
and Lemma 3.10 will show that for each such v, in a finite number of steps we
either construct a sequence δi as above for which all ĥv(δi) = 0, or (3.12.2)
holds for δ1, . . . , δr and the corresponding polynomials Rk,i, i.e.
(3.13.12) lim sup
log |
i=1 φRk,i(δi)|v∑r
i=1 q
ddegRk,i
Equation (3.13.11) shows that (3.12.2) is equivalent to (3.13.12) (see also
(3.13.9)).
We start with v ∈ S0 \ {∞}. As proved in Lemma 4.4 of [GT06], for each
i ∈ {1, . . . , r} such that ĥv(γi) > 0, there exists a positive integer di such
that for every polynomial Qi of degree at least di, we have
(3.13.13) log |φQi(γi)|v = q
ddegQiĥv(γi)−
log |ad|v
qd − 1
We know that for each i, we have limn→∞ degPn,i = +∞ because of (3.13.8).
Hence, for each n sufficiently large, and for each i ∈ {1, . . . , r} such that
ĥv(γi) > 0, we have
(3.13.14) log |φPn,i(γi)|v = q
ddeg Pn,iĥv(γi)−
log |ad|v
qd − 1
We now split the problem into two cases.
Case 1. There exists an infinite subsequence (nk)k≥1 such that for every
k, we have
(3.13.15)
∣∣∣∣∣
φPnk,i
∣∣∣∣∣
= max
1≤i≤r
∣∣∣φPnk,i(γi)
For the sake of not clustering the notation, we drop the index k from
(3.13.15) (note that we need to prove (3.12.2) only for a subsequence). At
SIEGEL’S THEOREM FOR DRINFELD MODULES 17
the expense of replacing again N∗ by a subsequence, we may also assume
that for some fixed j ∈ {1, . . . , r}, we have
(3.13.16)
∣∣∣∣∣
φPn,i(γi)
∣∣∣∣∣
∣∣φPn,i(γi)
∣∣φPn,j (γj)
for all n ∈ N∗. Because we know that there exists i ∈ {1, . . . , r} such that
ĥv(γi) > 0, then for such i, we know |φPn,i(γi)|v is unbounded (as n → ∞).
Therefore, using (3.13.16), we conclude that also |φPn,j (γj)|v is unbounded
(as n→ ∞). In particular, this means that ĥv(γj) > 0.
Then using (3.13.14) for γj, we obtain that
lim sup
log |
i=1 φPn,i(γi)|v∑r
i=1 q
ddeg Pn,i
= lim sup
log |φPn,j (γj)|v∑r
i=1 q
ddeg Pn,i
= lim sup
qddegPn,j ĥv(γj)−
log |ad|v
qd−1∑r
i=1 q
ddeg Pn,i
= lim
qddeg Pn,j ĥv(γj)−
log |ad|v
qddeg Pn,j
· lim sup
qddegPn,j∑r
i=1 q
ddeg Pn,i
(3.13.17)
since
qddeg Pn,j ĥv(γj)−
log |ad|v
qddegPn,j
= ĥv(γj) > 0 and
lim sup
qddegPn,j∑r
i=1 q
ddeg Pn,i
> 0 because of (3.13.8).
Case 2. For all but finitely many n, we have
(3.13.18)
∣∣∣∣∣
φPn,i(γi)
∣∣∣∣∣
< max
1≤i≤r
∣∣φPn,i(γi)
Using the pigeonhole principle, there exists an infinite sequence (nk)k≥1 ⊂
N∗, and there exist j1, . . . , js ∈ {1, . . . , r} (where s ≥ 2) such that for each
k, we have
(3.13.19)
|φPnk,j1
(γj1)|v = · · · = |φPnk,js
(γjs)|v > max
i∈{1,...,r}\{j1,...,js}
|φPnk,i
(γi)|v.
Again, as we did before, we drop the index k from the above subsequence of
N∗. Using (3.13.19) and the fact that there exists i ∈ {1, . . . , r} such that
ĥv(γi) > 0, we conclude that for all 1 ≤ i ≤ s, we have ĥv(γji) > 0. Hence,
using (3.13.14) in (3.13.19), we obtain that for sufficiently large n, we have
(3.13.20) qddeg Pn,j1 ĥv(γj1) = · · · = q
ddeg Pn,js ĥv(γjs).
18 D. GHIOCA AND T. J. TUCKER
Without loss of generality, we may assume ĥv(γj1) ≥ ĥv(γji) for all i ∈
{2, . . . , s}. Then (3.13.20) yields that degPn,ji ≥ degPn,j1 for i > 1. We
divide (with quotient and remainder) each Pn,ji (for i > 1) by Pn,j1 and for
each such ji, we obtain
(3.13.21) Pn,ji = Pn,j1 · Cn,ji +Rn,ji ,
where degRn,ji < degPn,j1 ≤ degPn,ji . Using (3.13.8), we conclude that
degCn,ji is uniformly bounded as n→ ∞. This means that, at the expense
of passing to another subsequence of N∗, we may assume that there exist
polynomials Cji such that
Cn,ji = Cji for all n.
We let Rn,i := Pn,i for each n and for each i ∈ {1, . . . , r} \ {j2, . . . , js}.
Let δi for i ∈ {1, . . . , r} be defined as follows:
δi := γi if i 6= j1; and
δj1 := γj1 +
φCji (γji).
Then for each n, using (3.13.21) and the definition of the δi and Rn,i, we
obtain
(3.13.22)
φPn,i(γi) =
φRn,i(δi).
Using (3.13.8) and the definition of the Rn,i (in particular, the fact that
Rn,j1 = Pn,j1 and degRn,ji < degPn,j1 for 2 ≤ i ≤ s), it is immediate to see
(3.13.23) 0 < lim inf
i=1 q
ddegPn,i
i=1 q
ddegRn,i
≤ lim sup
i=1 q
ddeg Pn,i
i=1 q
ddegRn,i
< +∞.
Moreover, because of (3.13.22) and (3.13.23), we get that
(3.13.24) lim sup
log |
i=1 φPn,i(γi)|v∑r
i=1 q
ddeg Pn,i
if and only if
(3.13.25) lim sup
log |
i=1 φRn,i(δi)|v∑r
i=1 q
ddegRn,i
SIEGEL’S THEOREM FOR DRINFELD MODULES 19
We claim that if ĥv(δj1) ≥ ĥv(γj1), then (3.13.25) holds (and so, also
(3.13.24) holds). Indeed, in that case, for large enough n, we have
log |φRn,j1 (δj1)|v = q
ddegRn,j1 ĥv(δj1)−
log |ad|v
qd − 1
≥ qddeg Pn,j1 ĥv(γj1)−
log |ad|v
qd − 1
= log |φPn,j1 (γj1)|v
log |φRn,ji (γji)|v,
(3.13.26)
where in the last inequality from (3.13.26) we used (3.13.20) and (3.13.14),
and that for each i ∈ {2, . . . , s} we have degRn,ji < degPn,ji . Moreover,
using (3.13.26) and (3.13.19), together with the definition of the Rn,i and
the δi, we conclude that for large enough n, we have
∣∣∣∣∣
φRn,i(δi)
∣∣∣∣∣
= log
∣∣∣φRn,j1 (δj1)
= qddeg Pn,j1 ĥv(γj1)−
log |ad|v
qd − 1
(3.13.27)
Because Rn,j1 = Pn,j1 , equations (3.13.8) and (3.13.23) show that
(3.13.28) lim sup
qddegRn,j1∑r
i=1 q
ddegRn,i
Equations (3.13.27) and (3.13.28) show that we are now in Case 1 for the
sequence (Rn,i) n∈N∗
1≤i≤r
. Hence
(3.13.29) lim sup
log |
i=1 φRn,i(δi)|v∑r
i=1 q
ddegRn,i
as desired.
Assume from now on that ĥv(δj1) < ĥv(γj1). Because v ∈ MK \ {∞},
using Corollary 3.9 and also using that if i 6= j1, then δi = γi, we conclude
ĥv(γi)−
ĥv(δi) ≥
Our goal is to prove (3.13.24) by proving (3.13.25). Because we replace
some of the polynomials Pn,i with other polynomials Rn,i, it may very well
be that (3.13.8) is no longer satisfied for the polynomials Rn,i. Note that
in this case, using induction and arguing as in equations (3.13.2) through
(3.13.6), we see that
lim sup
log |
j=1 φRn,j (δj)|w∑r
j=1 q
ddegRn,j
20 D. GHIOCA AND T. J. TUCKER
for some place w. This would yield that (see (3.13.22) and (3.13.23))
lim sup
log |
j=1 φPn,j (γj)|w∑r
j=1 q
ddeg Pn,j
as desired. Hence, we may assume again that (3.13.8) holds.
We continue the above analysis this time with the γi replaced by δi.
Either we prove (3.13.25) (and so, implicitly, (3.13.24)), or we replace the
δi by other elements in Γ, say βi and we decrease even further the sum of
their local heights at v:
ĥv(δi)−
ĥv(βi) ≥
The above process cannot go on infinitely often because the sum of the local
heights
i=1 ĥv(γi) is decreased each time by at least
. Our process ends
when we cannot replace anymore the eventual ζi by new βi. Thus, at the
final step, we have ζ1, . . . , ζr for which we cannot further decrease their sum
of local canonical heights at v. This happens either because all ζi have
local canonical height equal to 0, or because we already found a sequence of
polynomials Tn,i for which
(3.13.30) lim sup
log |
i=1 φTn,i(ζi)|v∑r
i=1 q
ddeg Tn,i
Since
(3.13.31)
φPn,i(γi) =
φTn,i(ζi),
this would imply that (3.12.2) holds, which would complete the proof. Hence,
we may assume that we have found a sequence (ζi)1≤i≤r with canonical local
heights equal to 0. As before, we let the (Tn,i) n∈N∗
1≤i≤r
be the corresponding
sequence of polynomials for the ζi, which replace the polynomials Pn,i.
Next we apply the above process to another w ∈ S0 \{∞} for which there
exists at least one ζi such that ĥw(ζi) > 0. Note that when we apply the
above process to the ζ1, . . . , ζr at the place w, we might replace (at certain
steps of our process) the ζi by
(3.13.32)
φCj (ζj) ∈ Γ.
Because for the places v ∈ S0 for which we already completed the above
process, ĥv(ζi) = 0 for all i, then by the triangle inequality for the local
height, we also have
φCj (ζj)
 = 0.
SIEGEL’S THEOREM FOR DRINFELD MODULES 21
If we went through all v ∈ S0 \ {∞}, and if the above process did not
yield that (3.13.24) holds for some v ∈ S \ {∞}, then we are left with
ζ1, . . . , ζr ∈ Γ such that for all i and all v 6= ∞, we have ĥv(ζi) = 0. Note
that since ĥv(ζi) = 0 for each v 6= ∞ and each i ∈ {1, . . . , r}, then by the
triangle inequality for local heights, for all polynomials Q1, . . . , Qr, we have
(3.13.33) ĥv
φQi(ζi)
= 0 for v 6= ∞.
Lemma 3.10 and (3.13.33) show that for all polynomials Qi,
(3.13.34) D∞ · ĥ∞
φQi(ζi)
We repeat the above process, this time for v = ∞. As before, we conclude
that either
(3.13.35) lim sup
log |
i=1 φTn,i(ζi)|∞∑r
i=1 q
ddeg Tn,i
or we are able to replace the ζi by some other elements βi (which are of the
form (3.13.32)) such that
ĥ∞(βi) <
ĥ∞(ζi).
Using (3.13.34), we conclude that
(3.13.36)
ĥ∞(ζi)−
ĥ∞(βi) ≥
Therefore, after a finite number of steps this process of replacing the ζi must
end, and it cannot end with all the new βi having local canonical height 0,
because this would mean that all βi are torsion (we already knew that for
v 6= ∞, we have ĥv(ζi) = 0, and so, by (3.13.33), ĥv(βi) = 0). Because the
βi are nontrivial “linear” combinations (in the φ-module Γ) of the γi which
span a torsion-free φ-module, we conclude that indeed, the βi cannot be
torsion points. Hence, our process ends with proving (3.13.35) which proves
(3.13.24), and so, it concludes the proof of our Proposition 3.12. �
Remark 3.14. If there is more than one infinite place in K, then we cannot
derive Lemma 3.10, and in particular, we cannot derive (3.13.36). The idea is
that in this case, for each nontorsion ζ which has its local canonical height
equal to 0 at finite places, we only know that there exists some infinite
place where its local canonical height has bounded denominator. However,
we do not know if that place is the one which we analyze at that particular
moment in our process from the proof of Proposition 3.12. Hence, we would
not necessarily be able to derive (3.13.36).
Now we are ready to prove Theorem 2.4.
22 D. GHIOCA AND T. J. TUCKER
Proof of Theorem 2.4. Let (γi)i be a finite set of generators of Γ as a module
over A = Fq[t]. At the expense of replacing S with a larger finite set of places
of K, we may assume S contains all the places v ∈MK which satisfy at least
one of the following properties:
1. ĥv(γi) > 0 for some 1 ≤ i ≤ r.
2. |γi|v > 1 for some 1 ≤ i ≤ r.
3. |α|v > 1.
4. φ has bad reduction at v.
Expanding the set S leads only to (possible) extension of the set of S-integral
points in Γ with respect to α. Clearly, for every γ ∈ Γ, and for every v /∈ S
we have |γ|v ≤ 1. Therefore, using 3., we obtain
γ ∈ Γ is S-integral with respect to α⇐⇒ |γ − α|v = 1 for all v ∈MK \ S.
(3.14.1)
Moreover, using 1. from above, we conclude that for every γ ∈ Γ, and for
every v /∈ S, we have ĥv(γ) = 0 (see the proof of Fact 3.13).
Next we observe that it suffices to prove Theorem 2.4 under the assump-
tion that Γ is a free φ-submodule. Indeed, because A = Fq[t] is a principal
ideal domain, Γ is a direct sum of its finite torsion submodule Γtor and a
free φ-submodule Γ1 of rank r, say. Therefore,
γ∈Γtor
γ + Γ1.
If we show that for every γ0 ∈ Γtor there are finitely many γ1 ∈ Γ1 such
that γ1 is S-integral with respect to α − γ0, then we obtain the conclusion
of Theorem 2.4 for Γ and α (see (3.14.1)).
Thus from now on, we assume Γ is a free φ-submodule of rank r. Let
γ1, . . . , γr be a basis for Γ as an Fq[t]-module. We reason by contradiction.
φPn,i(γi) ∈ Γ
be an infinite sequence of elements S-integral with respect to α. Because of
the S-integrality assumption (along with the assumptions on S), we conclude
that for every v /∈ S, and for every n we have
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
Thus, using the product formula, we see that
lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
= lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
SIEGEL’S THEOREM FOR DRINFELD MODULES 23
On the other hand, by Proposition 3.12, there is some place w ∈ S and
some number δ > 0 such that
lim sup
log |
i=1 φPn,i(γi)− α|w∑r
i=1 q
ddeg Pn,i
= δ > 0.
So, using Lemma 3.11, we see that
lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
v 6=w
lim inf
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
+ lim sup
log |
i=1 φPn,i(γi)− α|w∑r
i=1 q
ddegPn,i
≥ 0 + δ
Thus, we have a contradiction which shows that there cannot be infinitely
many elements of Γ which are S-integral for α. �
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[Poo95] B. Poonen, Local height functions and the Mordell-Weil theorem for Drinfel′d
modules, Compositio Math. 97 (1995), no. 3, 349–368.
[Sca02] T. Scanlon, Diophantine geometry of the torsion of a Drinfeld module, J. Num-
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Dragos Ghioca, Department of Mathematics, McMaster University, 1280
Main Street West, Hamilton, Ontario, Canada L8S 4K1,
E-mail address: dghioca@math.mcmaster.ca
Thomas Tucker, Department of Mathematics, Hylan Building, University
of Rochester, Rochester, NY 14627,
E-mail address: ttucker@math.rochester.edu
	1. Introduction
	2. Notation
	2.1. Drinfeld modules
	2.2. Valuations and Weil heights
	2.3. Canonical heights
	2.4. Completions and filled Julia sets
	2.5. The coefficients of t
	2.6. Integrality and reduction
	3. Proofs of our main results
	References

  We prove an analog of Siegel's theorem for integral points in the context of
Drinfeld modules. The result holds for finitely generated submodules of the
additive group over a function field of transcendence dimension 1.

Introduction
In 1929, Siegel ([Sie29]) proved that if C is an irreducible affine curve
defined over a number field K and C has at least three points at infinity,
then there are at most finitely many K-rational points on C that have
integral coordinates. The proof of this famous theorem uses diophantine
approximation along with the fact that certain groups of rational points are
finitely generated; when C has genus greater than 0, the group in question
is the Mordell-Weil group of the Jacobian of C, while when C has genus 0,
the group in question is the group of S-units in a finite extension of K.
Motivated by the analogy between rank 2 Drinfeld modules and elliptic
curves, the authors conjectured in [GT06] a Siegel type statement for finitely
generated φ-submodules Γ of Ga (where φ is a Drinfeld module of arbitrary
rank). For a finite set of places S of a function field K, we defined a notion of
S-integrality and asked whether or not it is possible that there are infinitely
many γ ∈ Γ which are S-integral with respect to a fixed point α ∈ K. We
also proved in [GT06] a first instance of our conjecture in the case where Γ
is a cyclic submodule and α is a torsion point for φ. Our goal in this paper
is to prove our Siegel conjecture for every finitely generated φ-submodule
of Ga(K), where φ is a Drinfeld module defined over the field K (see our
Theorem 2.4). We will also establish an analog (also in the context of
Drinfeld modules) of a theorem of Silverman for nonconstant morphisms of
P1 of degree greater than 1 over a number field (see our Theorem 2.5).
We note that recently there has been significant progress on establish-
ing additional links between classical diophantine results over number fields
and similar statements for Drinfeld modules. Denis [Den92a] formulated
analogs for Drinfeld modules of the Manin-Mumford and the Mordell-Lang
2000 Mathematics Subject Classification. Primary 11G50, Secondary 11J68, 37F10.
Key words and phrases. Drinfeld module, Heights, Diophantine approximation.
1dghioca@math.mcmaster.ca; ttucker@math.rochester.edu
http://arxiv.org/abs/0704.1331v1
2 D. GHIOCA AND T. J. TUCKER
conjectures. The Denis-Manin-Mumford conjecture was proved by Scan-
lon in [Sca02], while a first instance of the Denis-Mordell-Lang conjecture
was established in [Ghi05] by the first author (see also [Ghi06b] for an ex-
tension of the result from [Ghi05]). The authors proved in [GT07] several
other cases of the Denis-Mordell-Lang conjecture. In addition, the first au-
thor proved in [Ghi06a] an equidistribution statement for torsion points of a
Drinfeld module that is similar to the equidistribution statement established
by Szpiro-Ullmo-Zhang [SUZ97] (which was later extended by Zhang [Zha98]
to a full proof of the famous Bogomolov conjecture). Breuer [Bre05] proved a
special case of the André-Oort conjecture for Drinfeld modules, while special
cases of this conjecture in the classical case of a number field were proved
by Edixhoven-Yafaev [EY03] and Yafaev [Yaf06]. Bosser [Bos99] proved a
lower bound for linear forms in logarithms at an infinite place associated to
a Drinfeld module (similar to the classical result obtained by Baker [Bak75]
for usual logarithms, or by David [Dav95] for elliptic logarithms). Bosser’s
result was used by the authors in [GT06] to establish certain equidistribution
and integrality statements for Drinfeld modules. Moreover, Bosser’s result
is believed to be true also for linear forms in logarithms at finite places for a
Drinfeld module (as was communicated to us by Bosser). Assuming this last
statement, we prove in this paper the natural analog of Siegel’s theorem for
finitely generated φ-submodules. We believe that our present paper provides
additional evidence that the Drinfeld modules represent a good arithmetic
analog in characteristic p for abelian varieties in characteristic 0.
The basic outline of this paper can be summarized quite briefly. In Sec-
tion 2 we give the basic definitions and notation, and then state our main
results. In Section 3 we prove these main results: Theorems 2.4 and 2.5.
2. Notation
Notation. N stands for the non-negative integers: {0, 1, . . . }, while N∗ :=
N \ {0} stands for the positive integers.
2.1. Drinfeld modules. We begin by defining a Drinfeld module. Let p
be a prime and let q be a power of p. Let A := Fq[t], let K be a finite
field extension of Fq(t), and let K be an algebraic closure of K. We let τ
be the Frobenius on Fq, and we extend its action on K. Let K{τ} be the
ring of polynomials in τ with coefficients from K (the addition is the usual
addition, while the multiplication is the composition of functions).
A Drinfeld module is a morphism φ : A→ K{τ} for which the coefficient
of τ0 in φ(a) =: φa is a for every a ∈ A, and there exists a ∈ A such that
φa 6= aτ
0. The definition given here represents what Goss [Gos96] calls a
Drinfeld module of “generic characteristic”.
We note that usually, in the definition of a Drinfeld module, A is the ring
of functions defined on a projective nonsingular curve C, regular away from
a closed point η ∈ C. For our definition of a Drinfeld module, C = P1
and η
is the usual point at infinity on P1. On the other hand, every ring of regular
SIEGEL’S THEOREM FOR DRINFELD MODULES 3
functions A as above contains Fq[t] as a subring, where t is a nonconstant
function in A.
For every field extension K ⊂ L, the Drinfeld module φ induces an action
on Ga(L) by a∗x := φa(x), for each a ∈ A. We call φ-submodules subgroups
of Ga(K) which are invariant under the action of φ. We define the rank of
a φ-submodule Γ be
dimFrac(A) Γ⊗A Frac(A).
As shown in [Poo95], Ga(K) is a direct sum of a finite torsion φ-submodule
with a free φ-submodule of rank ℵ0.
A point α is torsion for the Drinfeld module action if and only if there
exists Q ∈ A \ {0} such that φQ(α) = 0. The monic polynomial Q of
minimal degree which satisfies φQ(α) = 0 is called the order of α. Since each
polynomial φQ is separable, the torsion submodule φtor lies in the separable
closure Ksep of K.
2.2. Valuations and Weil heights. Let MFq(t) be the set of places on
Fq(t). We denote by v∞ the place in MFq(t) such that v∞(
) = deg(g) −
deg(f) for every nonzero f, g ∈ A = Fq[t]. We letMK be the set of valuations
on K. ThenMK is a set of valuations which satisfies a product formula (see
[Ser97, Chapter 2]). Thus
• for each nonzero x ∈ K, there are finitely many v ∈ MK such that
|x|v 6= 1; and
• for each nonzero x ∈ K, we have
|x|v = 1.
We may use these valuations to define a Weil height for each x ∈ K as
(2.0.1) h(x) =
max log(|x|v , 1).
Convention. Without loss of generality we may assume that the nor-
malization for all the valuations of K is made so that for each v ∈ MK , we
have log |x|v ∈ Z.
Definition 2.1. Each place in MK which lies over v∞ is called an infinite
place. Each place in MK which does not lie over v∞ is called a finite place.
2.3. Canonical heights. Let φ : A → K{τ} be a Drinfeld module of rank
d (i.e. the degree of φt as a polynomial in τ equals d). The canonical height
of β ∈ K relative to φ (see [Den92b]) is defined as
ĥ(β) = lim
h(φtn(β))
Denis [Den92b] showed that a point is torsion if and only if its canonical
height equals 0.
For every v ∈MK , we let the local canonical height of β ∈ K at v be
(2.1.1) ĥv(β) = lim
log max(|φtn(β)|v , 1)
4 D. GHIOCA AND T. J. TUCKER
Furthermore, for every a ∈ Fq[t], we have ĥv(φa(x)) = deg(φa) · ĥv(x) (see
[Poo95]). It is clear that ĥv satisfies the triangle inequality, and also that∑
ĥv(β) = ĥ(β).
2.4. Completions and filled Julia sets. By abuse of notation, we let
∞ ∈ MK denote any place extending the place v∞. We let K∞ be the
completion of K with respect to | · |∞. We let K∞ be an algebraic closure
of K∞. We let C∞ be the completion of K∞. Then C∞ is a complete,
algebraically closed field. Note that C∞ depends on our choice for ∞ ∈MK
extending v∞. However, each time we will work with only one such place
∞, and so, there will be no possibility of confusion.
Next, we define the v-adic filled Julia set Jφ,v corresponding to the Drin-
feld module φ and to each place v of MK . Let Cv be the completion of an
algebraic closure of Kv. Then | · |v extends to a unique absolute value on
all of Cv. The set Jφ,v consists of all x ∈ Cv for which {|φQ(x)|v}Q∈A is
bounded. It is immediate to see that x ∈ Jφ,v if and only if {|φtn(x)|v}n≥1
is bounded.
One final note on absolute values: as noted above, the place v ∈ MK
extends to a unique absolute value | · |v on all of Cv. We fix an embedding
of i : K −→ Cv. For x ∈ K, we denote |i(x)|v simply as |x|v , by abuse of
notation.
2.5. The coefficients of φt. Each Drinfeld module is isomorphic to a Drin-
feld module for which all the coefficients of φt are integral at all the places
inMK which do not lie over v∞. Indeed, we let B ∈ Fq[t] be a product of all
(the finitely many) irreducible polynomials P ∈ Fq[t] with the property that
there exists a place v ∈MK which lies over the place (P ) ∈MFq(t), and there
exists a coefficient of φt which is not integral at v. Let γ be a sufficiently
large power of B. Then ψ : A → K{τ} defined by ψQ := γ
−1φQγ (for each
Q ∈ A) is a Drinfeld module isomorphic to φ, and all the coefficients of ψt
are integral away from the places lying above v∞. Hence, from now on, we
assume that all the coefficients of φt are integral away from the places lying
over v∞. It follows that for every Q ∈ A, all coefficients of φQ are integral
away from the places lying over v∞.
2.6. Integrality and reduction.
Definition 2.2. For a finite set of places S ⊂MK and α ∈ K, we say that
β ∈ K is S-integral with respect to α if for every place v /∈ S, and for every
morphisms σ, τ : K → K (which restrict to the identity on K) the following
are true:
• if |ατ |v ≤ 1, then |α
τ − βσ|v ≥ 1.
• if |ατ |v > 1, then |β
σ|v ≤ 1.
We note that if β is S-integral with respect to α, then it is also S′-integral
with respect to α, where S′ is a finite set of places containing S. Moreover,
the fact that β is S-integral with respect to α, is preserved if we replace
SIEGEL’S THEOREM FOR DRINFELD MODULES 5
K by a finite extension. Therefore, in our results we will always assume
α, β ∈ K. For more details about the definition of S-integrality, we refer the
reader to [BIR05].
Definition 2.3. The Drinfeld module φ has good reduction at a place v if
for each nonzero a ∈ A, all coefficients of φa are v-adic integers and the
leading coefficient of φa is a v-adic unit. If φ does not have good reduction
at v, then we say that φ has bad reduction at v.
It is immediate to see that φ has good reduction at v if and only if all
coefficients of φt are v-adic integers, while the leading coefficient of φt is a
v-adic unit.
We can now state our Siegel type result for Drinfeld modules.
Theorem 2.4. With the above notation, assume in addition K has only
one infinite place. Let Γ be a finitely generated φ-submodule of Ga(K), let
α ∈ K, and let S be a finite set of places in MK . Then there are finitely
many γ ∈ Γ such that γ is S-integral with respect to α.
As mentioned in Section 1, we proved in [GT06] that Theorem 2.4 holds
when Γ is a cyclic φ-module generated by a nontorsion point β ∈ K and
α ∈ φtor(K) (see Theorem 1.1 and Proposition 5.6 of [GT06]). Moreover,
in [GT06] we did not have in our results the extra hypothesis from Theo-
rem 2.4 that there exists only one infinite place in MK . Even though we
believe Theorem 2.4 is true without this hypothesis, our method for proving
Theorem 2.4 requires this technical hypothesis. On the other hand, we are
able to prove the following analog for Drinfeld modules of a theorem of Sil-
verman (see [Sil93]) for nonconstant morphisms of P1 of degree greater than
1 over a number field, without the hypothesis of having only one infinite
place in MK .
Theorem 2.5. With the above notation, let β ∈ K be a nontorsion point,
and let α ∈ K be an arbitrary point. Then there are finitely many Q ∈ A
such that φQ(β) is S-integral for α.
As explained before, in [GT06] we proved Theorem 2.5 in the case α is a
torsion point in K.
3. Proofs of our main results
We continue with the notation from Section 2. In our argument, we will
be using the following key fact.
Fact 3.1. Assume ∞ ∈ MK is an infinite place. Let γ1, . . . , γr, α ∈ K.
Then there exist (negative) constants C0 and C1 (depending only on φ,
γ1, . . . , γr, α) such that for any polynomials P1, . . . , Pr ∈ A (not all con-
stants), either φP1(γ1) + · · ·+ φPr(γr) = α or
log |φP1(γ1) + · · · + φPr(γr)− α|∞ ≥ C0 +C1 max
1≤i≤r
(deg(Pi) log deg(Pi)).
6 D. GHIOCA AND T. J. TUCKER
Fact 3.1 follows easily from the lower bounds for linear forms in logarithms
established by Bosser (see Théorème 1.1 in [Bos99]). Essentially, it is the
same proof as our proof of Proposition 3.7 of [GT06] (see in particular the
derivation of the inequality (3.7.2) in [GT06]). For the sake of completeness,
we will provide below a sketch of a proof of Fact 3.1.
Proof of Fact 3.1. We denote by exp∞ the exponential map associated to
the place ∞ (see [Gos96]). We also let L be the corresponding lattice for
exp∞, i.e. L := ker(exp∞). Finally, let ω1, . . . , ωd be an A-basis for L of
“successive minima” (see Lemma (4.2) of [Tag93]). This means that for
every Q1, . . . , Qd ∈ A, we have
(3.1.1) |Q1ω1 + · · ·+Qdωd|∞ =
|Qiωi|∞.
Let u0 ∈ C∞ such that exp∞(u0) = α. We also let u1, . . . , ur ∈ C∞
such that for each i, we have exp∞(ui) = γi. We will find constants C0
and C1 satisfying the inequality from Fact 3.1, which depend only on φ and
u0, u1, . . . , ur.
There exists a positive constant C2 such that exp∞ induces an isomor-
phism from the ball B := {z ∈ C∞ : |z|∞ < C2} to itself (see Lemma 3.6 of
[GT06]). If we assume there exist no constants C0 and C1 as in the conclu-
sion of Fact 3.1, then there exist polynomials P1, . . . , Pr, not all constants,
such that
(3.1.2)
φPi(γi) 6= α
and |
i=1 φPi(γi) − α|∞ < C2. Thus we can find y ∈ B such that |y|∞ =
i=1 φPi(γi)− α|∞ and
(3.1.3) exp∞(y) =
φPi(γi)− α.
Moreover, because exp∞ is an isomorphism on the metric space B, then for
every y′ ∈ C∞ such that exp∞(y
i=1 φPi(γi)−α, we have |y
′|∞ ≥ |y|∞.
But we know that
(3.1.4) exp∞
Piui − u0
φPi(γi)− α.
Therefore |
i=1 Piui − u0|∞ ≥ |y|∞. On the other hand, using (3.1.3) and
(3.1.4), we conclude that there exist polynomials Q1, . . . , Qd such that
Piui − u0 = y +
Qiωi.
SIEGEL’S THEOREM FOR DRINFELD MODULES 7
Hence |
i=1Qiωi|∞ ≤ |
i=1 Piui − u0|∞. Using (3.1.1), we obtain
∣∣∣∣∣
∣∣∣∣∣
|Qiωi|∞ ≤
∣∣∣∣∣
Piui − u0
∣∣∣∣∣
≤ max
|u0|∞,
|Piui|∞
≤ C3 ·
|Pi|∞,
(3.1.5)
where C3 is a constant depending only on u0, u1, . . . , ur. We take logarithms
of both sides in (3.1.5) and obtain
degQi ≤
degPi + logC3 −
log |ωi|∞
degPi + C4,
(3.1.6)
where C4 depends only on φ and u0, u1, . . . , ur (the dependence on the ωi is
actually a dependence on φ, because the ωi are a fixed basis of “successive
minima” for φ at ∞). Using (3.1.6) and Proposition 3.2 of [GT06] (which is a
translation of the bounds for linear forms in logarithms for Drinfeld modules
established in [Bos99]), we conclude that there exist (negative) constants C0,
C1, C5 and C6 (depending only on φ, γ1, . . . , γr and α) such that
∣∣∣∣∣
φPi(γi)− α
∣∣∣∣∣
= log |y|∞
= log
∣∣∣∣∣
Piui − u0 −
∣∣∣∣∣
≥ C5 + C6
degPi + C4
(degPi + C4)
≥ C0 + C1
degPi
(degPi) ,
(3.1.7)
as desired. �
In our proofs for Theorems 2.5 and 2.4 we will also use the following state-
ment, which is believed to be true, based on communication with V. Bosser.
Therefore we assume its validity without proof.
Statement 3.2. Assume v does not lie above v∞. Let γ1, . . . , γr, α ∈
K. Then there exist positive constants C1, C2, C3 (depending only on v,
φ, γ1, . . . , γr and α) such that for any P1, . . . , Pr ∈ Fq[t], either φP1(γ1) +
· · ·+ φPr(γr) = α or
log |φP1(γ1) + · · ·+ φPr(γr)− α|v ≥ −C1 − C2 max
1≤i≤r
(deg(Pi))
8 D. GHIOCA AND T. J. TUCKER
Statement 3.2 follows after one establishes a lower bound for linear forms
in logarithms at finite places v. In a private communication, V. Bosser told
us that it is clear to him that his proof ([Bos99]) can be adapted to work
also at finite places with minor modifications.
We sketch here how Statement 3.2 would follow from a lower bound for
linear forms in logarithms at finite places. Let v be a finite place and let
expv be the formal exponential map associated to v. The existence of expv
and its convergence on a sufficiently small ball Bv := {x ∈ Cv : |x|v < Cv}
is proved along the same lines as the existence and the convergence of the
usual exponential map at infinite places for φ (see Section 4.6 of [Gos96]).
In addition,
(3.2.1) | expv(x)|v = |x|v
for every x ∈ Bv. Moreover, at the expense of replacing Cv with a smaller
positive constant, we may assume that for each F ∈ A, and for each x ∈ Bv,
we have (see Lemma 4.2 in [GT06])
(3.2.2) |φF (x)|v = |Fx|v .
Assume we know the existence of the following lower bound for (nonzero)
linear forms in logarithms at a finite place v.
Statement 3.3. Let u1, . . . , ur ∈ Bv such that for each i, expv(ui) ∈ K.
Then there exist positive constants C4, C5, and C6 (depending on u1, . . . , ur)
such that for every F1, . . . , Fr ∈ A, either
i=1 Fiui = 0, or
∣∣∣∣∣
∣∣∣∣∣
≥ −C4 − C5
degFi
As mentioned before, Bosser proved Statement 3.3 in the case v is an
infinite place (in his result, C6 = 1 + ǫ and C4 = Cǫ for every ǫ > 0).
We will now derive Statement 3.2 assuming Statement 3.3 holds.
Proof. (That Statement 3.3 implies Statement 3.2.) Clearly, it suffices to
prove Statement 3.2 in the case α = 0. So, let γ1, . . . , γr ∈ K, and assume
by contradiction that there exists an infinite sequence {Fn,i} n∈N∗
1≤i≤r
such that
for each n, we have
(3.3.1) −∞ < log
∣∣∣∣∣
φFn,i(γi)
∣∣∣∣∣
< logCv.
For each n ≥ 1, we let Fn := (Fn,1, . . . , Fn,r) ∈ A
r. We view Ar as
an r-dimensional A-lattice inside the r-dimensional Frac(A)-vector space
Frac(A)r. In addition, we may assume that for n 6= m, we have Fn 6= Fm.
Using basic linear algebra, because the sequence {Fn,i} n∈N∗
1≤i≤r
is infinite, we
can find n0 ≥ 1 such that for every n > n0, there exist Hn, Gn,1, . . . , Gn,n0 ∈
SIEGEL’S THEOREM FOR DRINFELD MODULES 9
A (not all equal to 0) such that
(3.3.2) Hn · Fn =
Gn,j · Fj.
Essentially, (3.3.2) says that F1, . . . ,Fn0 span the linear subspace of Frac(A)
generated by all Fn. Moreover, we can choose theHn in (3.3.2) in such a way
that degHn is bounded independently of n (e.g. by a suitable determinant
of some linearly independent subset of the first n0 of the Fj). Furthemore,
there exists a constant C7 such that for all n > n0, we have
(3.3.3)
degGn,j < C7 +
degFn,i.
Because
i=1 φFn,i(γi)
< Cv, equation (3.2.2) yields
(3.3.4)
∣∣∣∣∣φHn
φFn,i(γi)
)∣∣∣∣∣
= |Hn|v ·
∣∣∣∣∣
φFn,i(γi)
∣∣∣∣∣
Using (3.3.2), (3.3.4), and the fact that |Hn|v ≤ 1, we obtain
∣∣∣∣∣
φFn,i(γi)
∣∣∣∣∣
∣∣∣∣∣φHn
φFn,i(γi)
)∣∣∣∣∣
∣∣∣∣∣∣
φGn,j
φFj,i(γi)
)∣∣∣∣∣∣
(3.3.5)
Since
i=1 φFj,i(γi)
< Cv for all 1 ≤ j ≤ n0, there exist u1, . . . , un0 ∈ Bv
such that for every 1 ≤ j ≤ n0, we have
expv(uj) =
φFj,i(γi).
Then Statement 3.3 implies that there exist constants C4, C5, C6, C8, C9 (de-
pending on u1, . . . , un0), such that
∣∣∣∣∣∣
φGn,j
φFj,i(γi)
)∣∣∣∣∣∣
= log
∣∣∣∣∣∣
Gn,juj
∣∣∣∣∣∣
≥ −C4 − C5
degGn,j
≥ −C8 − C9
degFn,i
(3.3.6)
where in the first equality we used (3.2.1), while in the last inequality we
used (3.3.3). Equations (3.3.5) and (3.3.6) show that Statement 3.2 follows
from Statement 3.3, as desired. �
10 D. GHIOCA AND T. J. TUCKER
Next we prove Theorem 2.5 which will be a warm-up for our proof of
Theorem 2.4. For its proof, we will only need the following weaker (but also
still conjectural) form of Statement 3.2 (i.e., we only need Statement 3.3 be
true for non-homogeneous 1-forms of logarithms).
Statement 3.4. Assume v does not lie over v∞. Let γ, α ∈ K. Then there
exist positive constants C1, C2 and C3 (depending only on v, φ, γ and α)
such that for each polynomial P ∈ Fq[t], either φP (γ) = α or
log |φP (γ)− α|v ≥ −C1 − C2 deg(P )
Proof of Theorem 2.5. The following Lemma is the key to our proof.
Lemma 3.5. For each v ∈MK , we have ĥv(β) = limdegQ→∞
log |φQ(β)−α|v
qd degQ
Proof of Lemma 3.5. Let v ∈ MK . If β /∈ Jφ,v, then |φQ(β)|v → ∞, as
degQ → ∞. Hence, if degQ is sufficiently large, then |φQ(β) − α|v =
|φQ(β)|v = max{|φQ(β)|v , 1}, which yields the conclusion of Lemma 3.5.
Thus, from now on, we assume β ∈ Jφ,v. Hence ĥv(β) = 0, and we need
to show that
(3.5.1) lim
degQ→∞
log |φQ(β)− α|v
qddegQ
Also note that since β ∈ Jφ,v, then |φQ(β) − α|v is bounded, and so,
lim supdegQ→∞
log |φQ(β)−α|v
qd degQ
≤ 0. Thus, in order to prove (3.5.1), it suf-
fices to show that
(3.5.2) lim inf
degQ→∞
log |φQ(β)− α|v
qddegQ
If v is an infinite place, then Fact 3.1 implies that for every polynomial Q
such that φQ(β) 6= α, we have log |φQ(β)−α|∞ ≥ C0+C1 deg(Q) log deg(Q)
(for some constants C0, C1 < 0). Then taking the limit as degQ → ∞, we
obtain (3.5.2), as desired.
Similarly, if v is a finite place, then (3.5.2) follows from Statement 3.4. �
Theorem 2.5 follows easily using the result of Lemma 3.5. We assume
there exist infinitely many polynomials Qn such that φQn(β) is S-integral
with respect to α. We consider the sum
log |φQn(β) − α|v
qddegQn
Using Lemma 3.5, we obtain that Σ = ĥ(β) > 0 (because β /∈ φtor).
Let T be a finite set of places consisting of all the places in S along with
all places v ∈MK which satisfy at least one of the following conditions:
1. |β|v > 1.
2. |α|v > 1.
3. v is a place of bad reduction for φ.
SIEGEL’S THEOREM FOR DRINFELD MODULES 11
Therefore by our choice for T (see 1. and 3.), for every v /∈ T , we have
|φQn(β)|v ≤ 1. Thus, using also 2., we have |φQn(β) − α|v ≤ 1. On the
other hand, φQn(β) is also T -integral with respect to α. Hence, because of
2., then for all v /∈ T , we have |φQn(β) − α|v ≥ 1. We conclude that for
every v /∈ T , and for every n, we have |φQn(β) − α|v = 1. This allows us
to interchange the summation and the limit in the definition of Σ (because
then Σ is a finite sum over all v ∈ T ). We obtain
Σ = lim
qddegQn
log |φQn(β)− α|v = 0,
by the product formula applied to each φQn(β)−α. On the other hand, we
already showed that Σ = ĥ(β) > 0. This contradicts our assumption that
there are infinitely many polynomials Q such that φQ(β) is S-integral with
respect to α, and concludes the proof of Theorem 2.5. �
Before proceeding to the proof of Theorem 2.4, we prove several facts
about local heights. In Lemma 3.10 we will use the technical assumption of
having only one infinite place in K.
From now on, let φt =
i=0 aiτ
i. As explained in Section 2, we may
assume each ai is integral away from v∞. Also, from now on, we work under
the assumption that there exists a unique place ∞ ∈MK lying above v∞.
Fact 3.6. For every place v of K, there exists Mv > 0 such that for each
x ∈ K, if |x|v > Mv, then for every nonzero Q ∈ A, we have |φQ(x)|v > Mv.
Moreover, if |x|v > Mv, then ĥv(x) = log |x|v +
log |ad|v
Fact 3.6 is proved in Lemma 4.4 of [GT06]. In particular, Fact 3.6 shows
that for each v ∈ MK and for each x ∈ K, we have ĥv(x) ∈ Q. Indeed, for
every x ∈ K of positive local canonical height at v, there exists a polynomial
P such that |φP (x)|v > Mv. Then ĥv(x) =
bhv(φP (x))
qd degP
. By Fact 3.6, we already
know that ĥv(φP (x)) ∈ Q. Thus also ĥv(x) ∈ Q.
Fact 3.7. Let v ∈MK \{∞}. There exists a positive constant Nv, and there
exists a nonzero polynomial Qv, such that for each x ∈ K, the following
statements are true
(i) if |x|v ≤ Nv, then for each Q ∈ A, we have |φQ(x)|v ≤ |x|v ≤ Nv.
(ii) either |φQv(x)|v ≤ Nv, or |φQv(x)|v > Mv.
Proof of Fact 3.7. This was proved in [Ghi07b]. It is easy to see that
Nv := min
1≤i≤d
satisfies condition (i), but the proof of (ii) is much more complicated. In
[Ghi07b], the first author proved that there exists a positive integer dv such
that for every x ∈ K, there exists a polynomial Q of degree at most dv such
that either |φQ(x)|v > Mv, or |φQ(x)|v ≤ Nv (see Remark 5.12 which is
12 D. GHIOCA AND T. J. TUCKER
valid for every place which does not lie over v∞). Using Fact 3.6 and (i), we
conclude that the polynomial Qv :=
deg P≤dv
P satisfies property (ii). �
Using Facts 3.6 and 3.7 we prove the following important result valid for
finite places.
Lemma 3.8. Let v ∈ MK \ {∞}. Then there exists a positive integer Dv
such that for every x ∈ K, we have Dv · ĥv(x) ∈ N. If in addition we assume
v is a place of good reduction for φ, then we may take Dv = 1.
Proof of Lemma 3.8. Let x ∈ K. If ĥv(x) = 0, then we have nothing to
show. Therefore, assume from now on that ĥv(x) > 0. Using (ii) of Fact 3.7,
there exists a polynomial Qv (depending only on v, and not on x) such
that |φQv(x)|v > Mv (clearly, the other option from (ii) of Lemma 3.7 is
not available because we assumed that ĥv(x) > 0). Moreover, using the
definition of the local height, and also Fact 3.6,
(3.8.1) ĥv(x) =
ĥv(φQv(x))
qddegQv
log |φQv(x)|v +
log |ad|v
qddegQv
Because both log |φQv(x)|v and log |ad|v are integer numbers, (3.8.1) yields
the conclusion of Lemma 3.8 (we may take Dv = q
ddegQv(qd − 1)).
The second part of Lemma 3.8 follows immediately from Lemma 4.13 of
[Ghi07b]. Indeed, if v is a place of good reduction for φ, then |x|v > 1
(because we assumed ĥv(x) > 0). But then, ĥv(x) = log |x|v (here we use
the fact that v is a place of good reduction, and thus ad is a v-adic unit).
Hence ĥv(x) ∈ N, and we may take Dv = 1. �
The following result is an immediate corollary of Fact 3.8.
Corollary 3.9. There exists a positive integer D such that for every v ∈
MK \ {∞}, and for every x ∈ K, we have D · ĥv(x) ∈ N.
Next we prove a similar result as in Lemma 3.8 which is valid for the only
infinite place of K.
Lemma 3.10. There exists a positive integer D∞ such that for every x ∈ K,
either ĥv(x) > 0 for some v ∈MK \ {∞}, or D∞ · ĥ∞(x) ∈ N.
Before proceeding to its proof, we observe that we cannot remove the
assumption that ĥv(x) = 0 for every finite place v, in order to obtain the
existence of D∞ in the statement of Lemma 3.10. Indeed, we know that
in K there are points of arbitrarily small (but positive) local height at ∞
(see Example 6.1 from [Ghi07b]). Therefore, there exists no positive integer
D∞ which would clear all the possible denominators for the local heights at
∞ of those points. However, it turns out (as we will show in the proof of
Lemma 3.10) that for such points x of very small local height at ∞, there
exists some other place v for which ĥv(x) > 0.
SIEGEL’S THEOREM FOR DRINFELD MODULES 13
Proof of Lemma 3.10. Let x ∈ K. If x ∈ φtor, then we have nothing to
prove (every positive integer D∞ would work because ĥ∞(x) = 0). Thus,
we assume x is a nontorsion point. If ĥv(x) > 0 for some place v which
does not lie over v∞, then again we are done. So, assume from now on that
ĥv(x) = 0 for every finite place v.
By proceeding as in the proof of Lemma 3.8, it suffices to show that there
exists a polynomial Q∞ of degree bounded independently of x such that
|φQ∞(x)|∞ > M∞ (with the notation as in Fact 3.6). This is proved in
Theorem 4.4 of [Ghi07a]. The first author showed in [Ghi07a] that there
exists a positive integer L (depending only on the number of places of bad
reduction of φ) such that for every nontorsion point x, there exists a place
v ∈ MK , and there exists a polynomial Q of degree less than L such that
|φQ(x)|v > Mv . Because we assumed that ĥv(x) = 0 for every v 6= ∞, then
the above statement yields the existence of D∞. �
We will prove Theorem 2.4 by showing that a certain lim sup is positive.
This will contradict the existence of infinitely many S-integral points in a
finitely generated φ-submodule. Our first step will be a result about the
lim inf of the sequences which will appear in the proof of Theorem 2.4.
Lemma 3.11. Suppose that Γ is a torsion-free φ-submodule of Ga(K) gener-
ated by elements γ1, . . . , γr. For each i ∈ {1, . . . , r} let (Pn,i)n∈N∗ ⊂ Fq[t] be
a sequence of polynomials such that for each m 6= n, the r-tuples (Pn,i)1≤i≤r
and (Pm,i)1≤i≤r are distinct. Then for every v ∈MK , we have
(3.11.1) lim inf
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
Proof. Suppose that for some ǫ > 0, there exists a sequence (nk)k≥1 ⊂ N
such that
i=1 φPnk,i
(γi) 6= α and
(3.11.2)
log |
i=1 φPnk,i
(γi)− α|v
i=1 q
ddeg Pnk,i
< −ǫ,
for every k ≥ 1. But taking the lower bound from Fact 3.1 or Statement 3.2
(depending on whether v is the infinite place or not) and dividing through
i=1 q
ddeg Pnk,i , we see that this is impossible. �
The following proposition is the key technical result required to prove
Theorem 2.4. This proposition plays the same role that Lemma 3.5 plays
in the proof of Theorem 2.5, or that Corollary 3.13 plays in the proof of
Theorem 1.1 from [GT06]. Note that is does not provide an exact formula
for the canonical height of a point, however; it merely shows that a certain
limit is positive. This will suffice for our purposes since we only need that a
certain sum of limits be positive in order to prove Theorem 2.4.
Proposition 3.12. Let Γ be a torsion-free φ-submodule of Ga(K) generated
by elements γ1, . . . , γr. For each i ∈ {1, . . . , r} let (Pn,i)n∈N∗ ⊂ Fq[t] be a
14 D. GHIOCA AND T. J. TUCKER
sequence of polynomials such that for each m 6= n, the r-tuples (Pn,i)1≤i≤r
and (Pm,i)1≤i≤r are distinct. Then there exists a place v ∈MK such that
(3.12.1) lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
Proof. Using the triangle inequality for the v-adic norm, and the fact that
qddeg Pn,i = +∞,
we conclude that proving that (3.12.1) holds is equivalent to proving that
for some place v, we have
(3.12.2) lim sup
log |
i=1 φPn,i(γi)|v∑r
i=1 q
ddeg Pn,i
We also observe that it suffices to prove Proposition 3.12 for a subsequence
(nk)k≥1 ⊂ N
We prove (3.12.2) by induction on r. If r = 1, then by Corollary 3.13 of
[GT06] (see also our Lemma 3.5),
(3.12.3) lim sup
deg P→∞
log |φP (γ1)|v
qddegP
= ĥv(γ1)
and because γ1 /∈ φtor, there exists a place v such that ĥv(γ1) > 0, thus
proving (3.12.2) for r = 1. Therefore, we assume (3.12.2) is established for
all φ-submodules Γ of rank less than r and we will prove it for φ-submodules
of rank r.
In the course of our argument for proving (3.12.2), we will replace several
times a given sequence with a subsequence of itself (note that passing to a
subsequence can only make the lim sup smaller). For the sake of not cluster-
ing the notation, we will drop the extra indices which would be introduced
by dealing with the subsequence.
Let S0 be the set of places v ∈MK for which there exists some γ ∈ Γ such
that ĥv(γ) > 0. The following easy fact will be used later in our argument.
Fact 3.13. The set S0 is finite.
Proof of Fact 3.13. We claim that S0 equals the finite set S
0 of places v ∈
MK for which there exists i ∈ {1, . . . , r} such that ĥv(γi) > 0. Indeed, let
v ∈ MK \ S
0. Then for each i ∈ {1, . . . , r} we have ĥv(γi) = 0. Moreover,
for each i ∈ {1, . . . , r} and for each Qi ∈ Fq[t], we have
(3.13.1) ĥv(φQi(γi)) = deg(φQi) · ĥv(γi) = 0.
Using (3.13.1) and the triangle inequality for the local canonical height, we
obtain that
φQi(γi)
SIEGEL’S THEOREM FOR DRINFELD MODULES 15
This shows that indeed S0 = S
0, and concludes the proof of Fact 3.13. �
If the sequence (nk)k≥1 ⊂ N
∗ has the property that for some j ∈ {1, . . . , r},
we have
(3.13.2) lim
qddeg Pnk,j
i=1 q
ddeg Pnk,i
then the inductive hypothesis will yield the desired conclusion. Indeed, by
the induction hypothesis, and also using (3.13.2), there exists v ∈ S0 such
(3.13.3) lim sup
log |
i 6=j φPnk,i
(γi)|v
i=1 q
ddeg Pnk,i
If ĥv(γj) = 0, then
∣∣∣φPnk,j(γj)
is bounded as k → ∞. Thus, for large
enough k, ∣∣∣∣∣
φPnk,i
∣∣∣∣∣
∣∣∣∣∣∣
i 6=j
φPnk,i
∣∣∣∣∣∣
and so, (3.13.3) shows that (3.12.2) holds.
Now, if ĥv(γj) > 0, then we proved in Lemma 4.4 of [GT06] that
(3.13.4) log |φP (γj)|v − q
ddeg P ĥv(γj)
is uniformly bounded as degP → ∞ (note that this follows easily from
simple arguments involving geometric series and coefficients of polynomials).
Therefore, using (3.13.2), we obtain
(3.13.5) lim
∣∣∣φPnk,j(γj)
i=1 q
ddeg Pnk,i
Using (3.13.3) and (3.13.5), we conclude that for large enough k,
∣∣∣∣∣
φPnk,i
∣∣∣∣∣
∣∣∣∣∣∣
i 6=j
φPnk,i
∣∣∣∣∣∣
and so,
(3.13.6) lim sup
i=1 φPnk,i
i=1 q
ddeg Pnk,i
as desired. Therefore, we may assume from now on that there exists B ≥ 1
such that for every n,
(3.13.7)
max1≤i≤r q
ddegPn,i
min1≤i≤r q
ddegPn,i
≤ B or equivalently,
(3.13.8) max
1≤i≤r
degPn,i − min
1≤i≤r
degPn,i ≤
logq B
16 D. GHIOCA AND T. J. TUCKER
We will prove that (3.12.2) holds for some place v by doing an analysis
at each place v ∈ S0. We know that |S0| ≥ 1 because all γi are nontorsion.
Our strategy is to show that in case (3.12.2) does not hold, then we can
find δ1, . . . , δr ∈ Γ, and we can find a sequence (nk)k≥1 ⊂ N
∗, and a sequence
of polynomials (Rk,i) k∈N∗
1≤i≤r
such that
(3.13.9)
φPnk,i
(γi) =
φRk,i(δi) and
(3.13.10)
ĥv(δi) <
ĥv(γi) and
(3.13.11) 0 < lim inf
i=1 q
ddegPnk,i
i=1 q
ddegRk,i
≤ lim sup
i=1 q
ddegPnk,i
i=1 q
ddegRk,i
< +∞.
Equation (3.13.9) will enable us to replace the γi by the δi and proceed with
our analysis of the latter. Inequality (3.13.10) combined with Corollary 3.9
and Lemma 3.10 will show that for each such v, in a finite number of steps we
either construct a sequence δi as above for which all ĥv(δi) = 0, or (3.12.2)
holds for δ1, . . . , δr and the corresponding polynomials Rk,i, i.e.
(3.13.12) lim sup
log |
i=1 φRk,i(δi)|v∑r
i=1 q
ddegRk,i
Equation (3.13.11) shows that (3.12.2) is equivalent to (3.13.12) (see also
(3.13.9)).
We start with v ∈ S0 \ {∞}. As proved in Lemma 4.4 of [GT06], for each
i ∈ {1, . . . , r} such that ĥv(γi) > 0, there exists a positive integer di such
that for every polynomial Qi of degree at least di, we have
(3.13.13) log |φQi(γi)|v = q
ddegQiĥv(γi)−
log |ad|v
qd − 1
We know that for each i, we have limn→∞ degPn,i = +∞ because of (3.13.8).
Hence, for each n sufficiently large, and for each i ∈ {1, . . . , r} such that
ĥv(γi) > 0, we have
(3.13.14) log |φPn,i(γi)|v = q
ddeg Pn,iĥv(γi)−
log |ad|v
qd − 1
We now split the problem into two cases.
Case 1. There exists an infinite subsequence (nk)k≥1 such that for every
k, we have
(3.13.15)
∣∣∣∣∣
φPnk,i
∣∣∣∣∣
= max
1≤i≤r
∣∣∣φPnk,i(γi)
For the sake of not clustering the notation, we drop the index k from
(3.13.15) (note that we need to prove (3.12.2) only for a subsequence). At
SIEGEL’S THEOREM FOR DRINFELD MODULES 17
the expense of replacing again N∗ by a subsequence, we may also assume
that for some fixed j ∈ {1, . . . , r}, we have
(3.13.16)
∣∣∣∣∣
φPn,i(γi)
∣∣∣∣∣
∣∣φPn,i(γi)
∣∣φPn,j (γj)
for all n ∈ N∗. Because we know that there exists i ∈ {1, . . . , r} such that
ĥv(γi) > 0, then for such i, we know |φPn,i(γi)|v is unbounded (as n → ∞).
Therefore, using (3.13.16), we conclude that also |φPn,j (γj)|v is unbounded
(as n→ ∞). In particular, this means that ĥv(γj) > 0.
Then using (3.13.14) for γj, we obtain that
lim sup
log |
i=1 φPn,i(γi)|v∑r
i=1 q
ddeg Pn,i
= lim sup
log |φPn,j (γj)|v∑r
i=1 q
ddeg Pn,i
= lim sup
qddegPn,j ĥv(γj)−
log |ad|v
qd−1∑r
i=1 q
ddeg Pn,i
= lim
qddeg Pn,j ĥv(γj)−
log |ad|v
qddeg Pn,j
· lim sup
qddegPn,j∑r
i=1 q
ddeg Pn,i
(3.13.17)
since
qddeg Pn,j ĥv(γj)−
log |ad|v
qddegPn,j
= ĥv(γj) > 0 and
lim sup
qddegPn,j∑r
i=1 q
ddeg Pn,i
> 0 because of (3.13.8).
Case 2. For all but finitely many n, we have
(3.13.18)
∣∣∣∣∣
φPn,i(γi)
∣∣∣∣∣
< max
1≤i≤r
∣∣φPn,i(γi)
Using the pigeonhole principle, there exists an infinite sequence (nk)k≥1 ⊂
N∗, and there exist j1, . . . , js ∈ {1, . . . , r} (where s ≥ 2) such that for each
k, we have
(3.13.19)
|φPnk,j1
(γj1)|v = · · · = |φPnk,js
(γjs)|v > max
i∈{1,...,r}\{j1,...,js}
|φPnk,i
(γi)|v.
Again, as we did before, we drop the index k from the above subsequence of
N∗. Using (3.13.19) and the fact that there exists i ∈ {1, . . . , r} such that
ĥv(γi) > 0, we conclude that for all 1 ≤ i ≤ s, we have ĥv(γji) > 0. Hence,
using (3.13.14) in (3.13.19), we obtain that for sufficiently large n, we have
(3.13.20) qddeg Pn,j1 ĥv(γj1) = · · · = q
ddeg Pn,js ĥv(γjs).
18 D. GHIOCA AND T. J. TUCKER
Without loss of generality, we may assume ĥv(γj1) ≥ ĥv(γji) for all i ∈
{2, . . . , s}. Then (3.13.20) yields that degPn,ji ≥ degPn,j1 for i > 1. We
divide (with quotient and remainder) each Pn,ji (for i > 1) by Pn,j1 and for
each such ji, we obtain
(3.13.21) Pn,ji = Pn,j1 · Cn,ji +Rn,ji ,
where degRn,ji < degPn,j1 ≤ degPn,ji . Using (3.13.8), we conclude that
degCn,ji is uniformly bounded as n→ ∞. This means that, at the expense
of passing to another subsequence of N∗, we may assume that there exist
polynomials Cji such that
Cn,ji = Cji for all n.
We let Rn,i := Pn,i for each n and for each i ∈ {1, . . . , r} \ {j2, . . . , js}.
Let δi for i ∈ {1, . . . , r} be defined as follows:
δi := γi if i 6= j1; and
δj1 := γj1 +
φCji (γji).
Then for each n, using (3.13.21) and the definition of the δi and Rn,i, we
obtain
(3.13.22)
φPn,i(γi) =
φRn,i(δi).
Using (3.13.8) and the definition of the Rn,i (in particular, the fact that
Rn,j1 = Pn,j1 and degRn,ji < degPn,j1 for 2 ≤ i ≤ s), it is immediate to see
(3.13.23) 0 < lim inf
i=1 q
ddegPn,i
i=1 q
ddegRn,i
≤ lim sup
i=1 q
ddeg Pn,i
i=1 q
ddegRn,i
< +∞.
Moreover, because of (3.13.22) and (3.13.23), we get that
(3.13.24) lim sup
log |
i=1 φPn,i(γi)|v∑r
i=1 q
ddeg Pn,i
if and only if
(3.13.25) lim sup
log |
i=1 φRn,i(δi)|v∑r
i=1 q
ddegRn,i
SIEGEL’S THEOREM FOR DRINFELD MODULES 19
We claim that if ĥv(δj1) ≥ ĥv(γj1), then (3.13.25) holds (and so, also
(3.13.24) holds). Indeed, in that case, for large enough n, we have
log |φRn,j1 (δj1)|v = q
ddegRn,j1 ĥv(δj1)−
log |ad|v
qd − 1
≥ qddeg Pn,j1 ĥv(γj1)−
log |ad|v
qd − 1
= log |φPn,j1 (γj1)|v
log |φRn,ji (γji)|v,
(3.13.26)
where in the last inequality from (3.13.26) we used (3.13.20) and (3.13.14),
and that for each i ∈ {2, . . . , s} we have degRn,ji < degPn,ji . Moreover,
using (3.13.26) and (3.13.19), together with the definition of the Rn,i and
the δi, we conclude that for large enough n, we have
∣∣∣∣∣
φRn,i(δi)
∣∣∣∣∣
= log
∣∣∣φRn,j1 (δj1)
= qddeg Pn,j1 ĥv(γj1)−
log |ad|v
qd − 1
(3.13.27)
Because Rn,j1 = Pn,j1 , equations (3.13.8) and (3.13.23) show that
(3.13.28) lim sup
qddegRn,j1∑r
i=1 q
ddegRn,i
Equations (3.13.27) and (3.13.28) show that we are now in Case 1 for the
sequence (Rn,i) n∈N∗
1≤i≤r
. Hence
(3.13.29) lim sup
log |
i=1 φRn,i(δi)|v∑r
i=1 q
ddegRn,i
as desired.
Assume from now on that ĥv(δj1) < ĥv(γj1). Because v ∈ MK \ {∞},
using Corollary 3.9 and also using that if i 6= j1, then δi = γi, we conclude
ĥv(γi)−
ĥv(δi) ≥
Our goal is to prove (3.13.24) by proving (3.13.25). Because we replace
some of the polynomials Pn,i with other polynomials Rn,i, it may very well
be that (3.13.8) is no longer satisfied for the polynomials Rn,i. Note that
in this case, using induction and arguing as in equations (3.13.2) through
(3.13.6), we see that
lim sup
log |
j=1 φRn,j (δj)|w∑r
j=1 q
ddegRn,j
20 D. GHIOCA AND T. J. TUCKER
for some place w. This would yield that (see (3.13.22) and (3.13.23))
lim sup
log |
j=1 φPn,j (γj)|w∑r
j=1 q
ddeg Pn,j
as desired. Hence, we may assume again that (3.13.8) holds.
We continue the above analysis this time with the γi replaced by δi.
Either we prove (3.13.25) (and so, implicitly, (3.13.24)), or we replace the
δi by other elements in Γ, say βi and we decrease even further the sum of
their local heights at v:
ĥv(δi)−
ĥv(βi) ≥
The above process cannot go on infinitely often because the sum of the local
heights
i=1 ĥv(γi) is decreased each time by at least
. Our process ends
when we cannot replace anymore the eventual ζi by new βi. Thus, at the
final step, we have ζ1, . . . , ζr for which we cannot further decrease their sum
of local canonical heights at v. This happens either because all ζi have
local canonical height equal to 0, or because we already found a sequence of
polynomials Tn,i for which
(3.13.30) lim sup
log |
i=1 φTn,i(ζi)|v∑r
i=1 q
ddeg Tn,i
Since
(3.13.31)
φPn,i(γi) =
φTn,i(ζi),
this would imply that (3.12.2) holds, which would complete the proof. Hence,
we may assume that we have found a sequence (ζi)1≤i≤r with canonical local
heights equal to 0. As before, we let the (Tn,i) n∈N∗
1≤i≤r
be the corresponding
sequence of polynomials for the ζi, which replace the polynomials Pn,i.
Next we apply the above process to another w ∈ S0 \{∞} for which there
exists at least one ζi such that ĥw(ζi) > 0. Note that when we apply the
above process to the ζ1, . . . , ζr at the place w, we might replace (at certain
steps of our process) the ζi by
(3.13.32)
φCj (ζj) ∈ Γ.
Because for the places v ∈ S0 for which we already completed the above
process, ĥv(ζi) = 0 for all i, then by the triangle inequality for the local
height, we also have
φCj (ζj)
 = 0.
SIEGEL’S THEOREM FOR DRINFELD MODULES 21
If we went through all v ∈ S0 \ {∞}, and if the above process did not
yield that (3.13.24) holds for some v ∈ S \ {∞}, then we are left with
ζ1, . . . , ζr ∈ Γ such that for all i and all v 6= ∞, we have ĥv(ζi) = 0. Note
that since ĥv(ζi) = 0 for each v 6= ∞ and each i ∈ {1, . . . , r}, then by the
triangle inequality for local heights, for all polynomials Q1, . . . , Qr, we have
(3.13.33) ĥv
φQi(ζi)
= 0 for v 6= ∞.
Lemma 3.10 and (3.13.33) show that for all polynomials Qi,
(3.13.34) D∞ · ĥ∞
φQi(ζi)
We repeat the above process, this time for v = ∞. As before, we conclude
that either
(3.13.35) lim sup
log |
i=1 φTn,i(ζi)|∞∑r
i=1 q
ddeg Tn,i
or we are able to replace the ζi by some other elements βi (which are of the
form (3.13.32)) such that
ĥ∞(βi) <
ĥ∞(ζi).
Using (3.13.34), we conclude that
(3.13.36)
ĥ∞(ζi)−
ĥ∞(βi) ≥
Therefore, after a finite number of steps this process of replacing the ζi must
end, and it cannot end with all the new βi having local canonical height 0,
because this would mean that all βi are torsion (we already knew that for
v 6= ∞, we have ĥv(ζi) = 0, and so, by (3.13.33), ĥv(βi) = 0). Because the
βi are nontrivial “linear” combinations (in the φ-module Γ) of the γi which
span a torsion-free φ-module, we conclude that indeed, the βi cannot be
torsion points. Hence, our process ends with proving (3.13.35) which proves
(3.13.24), and so, it concludes the proof of our Proposition 3.12. �
Remark 3.14. If there is more than one infinite place in K, then we cannot
derive Lemma 3.10, and in particular, we cannot derive (3.13.36). The idea is
that in this case, for each nontorsion ζ which has its local canonical height
equal to 0 at finite places, we only know that there exists some infinite
place where its local canonical height has bounded denominator. However,
we do not know if that place is the one which we analyze at that particular
moment in our process from the proof of Proposition 3.12. Hence, we would
not necessarily be able to derive (3.13.36).
Now we are ready to prove Theorem 2.4.
22 D. GHIOCA AND T. J. TUCKER
Proof of Theorem 2.4. Let (γi)i be a finite set of generators of Γ as a module
over A = Fq[t]. At the expense of replacing S with a larger finite set of places
of K, we may assume S contains all the places v ∈MK which satisfy at least
one of the following properties:
1. ĥv(γi) > 0 for some 1 ≤ i ≤ r.
2. |γi|v > 1 for some 1 ≤ i ≤ r.
3. |α|v > 1.
4. φ has bad reduction at v.
Expanding the set S leads only to (possible) extension of the set of S-integral
points in Γ with respect to α. Clearly, for every γ ∈ Γ, and for every v /∈ S
we have |γ|v ≤ 1. Therefore, using 3., we obtain
γ ∈ Γ is S-integral with respect to α⇐⇒ |γ − α|v = 1 for all v ∈MK \ S.
(3.14.1)
Moreover, using 1. from above, we conclude that for every γ ∈ Γ, and for
every v /∈ S, we have ĥv(γ) = 0 (see the proof of Fact 3.13).
Next we observe that it suffices to prove Theorem 2.4 under the assump-
tion that Γ is a free φ-submodule. Indeed, because A = Fq[t] is a principal
ideal domain, Γ is a direct sum of its finite torsion submodule Γtor and a
free φ-submodule Γ1 of rank r, say. Therefore,
γ∈Γtor
γ + Γ1.
If we show that for every γ0 ∈ Γtor there are finitely many γ1 ∈ Γ1 such
that γ1 is S-integral with respect to α − γ0, then we obtain the conclusion
of Theorem 2.4 for Γ and α (see (3.14.1)).
Thus from now on, we assume Γ is a free φ-submodule of rank r. Let
γ1, . . . , γr be a basis for Γ as an Fq[t]-module. We reason by contradiction.
φPn,i(γi) ∈ Γ
be an infinite sequence of elements S-integral with respect to α. Because of
the S-integrality assumption (along with the assumptions on S), we conclude
that for every v /∈ S, and for every n we have
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
Thus, using the product formula, we see that
lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
= lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
SIEGEL’S THEOREM FOR DRINFELD MODULES 23
On the other hand, by Proposition 3.12, there is some place w ∈ S and
some number δ > 0 such that
lim sup
log |
i=1 φPn,i(γi)− α|w∑r
i=1 q
ddeg Pn,i
= δ > 0.
So, using Lemma 3.11, we see that
lim sup
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
v 6=w
lim inf
log |
i=1 φPn,i(γi)− α|v∑r
i=1 q
ddeg Pn,i
+ lim sup
log |
i=1 φPn,i(γi)− α|w∑r
i=1 q
ddegPn,i
≥ 0 + δ
Thus, we have a contradiction which shows that there cannot be infinitely
many elements of Γ which are S-integral for α. �
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Dragos Ghioca, Department of Mathematics, McMaster University, 1280
Main Street West, Hamilton, Ontario, Canada L8S 4K1,
E-mail address: dghioca@math.mcmaster.ca
Thomas Tucker, Department of Mathematics, Hylan Building, University
of Rochester, Rochester, NY 14627,
E-mail address: ttucker@math.rochester.edu
	1. Introduction
	2. Notation
	2.1. Drinfeld modules
	2.2. Valuations and Weil heights
	2.3. Canonical heights
	2.4. Completions and filled Julia sets
	2.5. The coefficients of t
	2.6. Integrality and reduction
	3. Proofs of our main results
	References
