A DYNAMICAL VERSION OF THE MORDELL-LANG
CONJECTURE FOR THE ADDITIVE GROUP
D. GHIOCA AND T. J. TUCKER
Abstract. We prove a dynamical version of the Mordell-Lang conjecture in
the context of Drinfeld modules. We use analytic methods similar to the
ones employed by Skolem, Chabauty and Coleman for studying diophantine
equations.
1. Introduction
Faltings proved the Mordell-Lang conjecture in the following form (see [Fal94]).
Theorem 1.1 (Faltings). Let G be an abelian variety defined over the field of
complex numbers C. Let X ⊂ G be a closed subvariety and Γ ⊂ G(C) a finitely
generated subgroup of G(C). Then X(C)∩Γ is a finite union of cosets of subgroups
of Γ.
In particular, Theorem 1.1 says that an irreducible subvariety X of an abelian
variety G has a Zariski dense intersection with a finitely generated subgroup of
G(C) only if X is a translate of an algebraic subgroup of G. We also note that
Faltings result was generalized to semiabelian varieties G by Vojta (see [Voj96]),
and then to finite rank subgroups Γ of G by McQuillan (see [McQ95]).
If we try to formulate the Mordell-Lang conjecture in the context of algebraic
subvarieties contained in a power of the additive group scheme Ga, the conclusion
is either false (in the characteristic 0 case, as shown by the curve y = x2 which
has an infinite intersection with the finitely generated subgroup Z × Z, without
being itself a translate of an algebraic subgroup of G2a) or it is trivially true (in
the characteristic p > 0 case, because every finitely generated subgroup of a power
of Ga is finite). Denis [Den92a] formulated a Mordell-Lang conjecture for powers
of Ga in characteristic p in the context of Drinfeld modules. Denis replaced the
finitely generated subgroup from the usual Mordell-Lang statement with a finitely
generated φ-submodule, where φ is a Drinfeld module. He also strengthened the
conclusion of the Mordell-Lang statement by asking that the subgroups whose cosets
are contained in the intersection of the algebraic variety with the finitely generated
φ-submodule be actually φ-submodules. The first author proved several cases of the
Denis-Mordell-Lang conjecture in [Ghi05] and [Ghi06b].
In the present paper we investigate other cases of the Denis-Mordell-Lang con-
jecture through methods different from the ones employed in [Ghi05]. In partic-
ular, we prove the Denis-Mordell-Lang conjecture in the case where the finitely
generated φ-module is cyclic and the Drinfeld modules are defined over a field of
transcendence degree equal to one (this is our Theorem 2.5). Note that [Ghi05] and
Key words and phrases. Drinfeld module, Polynomial Dynamics.
The second author was partially supported by National Security Agency Grant 06G-067.
http://arxiv.org/abs/0704.1333v1
2 D. GHIOCA AND T. J. TUCKER
[Ghi06b] treat only the case where the transcendence degree of the field of definition
is greater than one. One of the methods employed in [Ghi05] (and whose outcome
was later used in [Ghi06b]) was specializations; hence the necessity of dealing with
fields of transcendence degree greater than one. By contrast, the techniques used
in this paper are more akin to those used in treating diophantine problems over
number fields (see [Cha41], [Col85], or [BS66, Chapter 4.6], for example), where
such specialization arguments are also not available. So, making a parallel between
the classical Mordell-Lang conjecture and the Denis-Mordell-Lang conjecture, we
might say that the papers [Ghi05] and [Ghi06b] deal with the “function field case”,
while our present paper deals with the “number field case” of the Denis conjecture.
Moreover, using specializations (as in [Hru98] and [Ghi05]), our Theorem 2.5 can be
extended to Drinfeld modules defined over fields of arbitrary finite transcendence
degree.
We also note that recently there has been significant progress on establishing ad-
ditional links between classical diophantine results over number fields and similar
statements for Drinfeld modules. The first author proved in [Ghi06a] an equidis-
tribution statement for torsion points of a Drinfeld module, which 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 conjec-
ture). 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 [GT06a] to establish certain equidistribu-
tion and integrality statements for Drinfeld modules. Moreover, Bosser’s result is
quite possibly true also for linear forms in logarithms at finite places for a Drinfeld
module. Assuming this last statement, the authors proved in [GT06b] the analog of
Siegel’s theorem for finitely generated φ-submodules. We believe that our present
paper provides an additional proof of the fact that the Drinfeld modules represent
the right arithmetic analog in characteristic p for abelian varieties in characteristic
The idea behind the proof of our Theorem 2.5 can be explained quite simply.
Assuming that an affine variety V ⊂ Gga has infinitely many points in common with
a cyclic φ-submodule Γ, we can find then a suitable submodule Γ0 ⊂ Γ whose coset
lies in V . Indeed, applying the logarithmic map (associated to a suitable place
v) to Γ0 yields a line in the vector space C
v. Each polynomial f that vanishes
on V , then gives rise to an analytic function F on this line (by composing with
the exponential function). Because we assumed there are infinitely many points in
V ∩ Γ, the zeros of F must have an accumulation point on this line, which means
that F vanishes identically on the line. This means that there is an entire translate
of Γ0 contained in the zero locus of f . The inspiration for this idea comes from
the method employed by Chabauty in [Cha41] (and later refined by Coleman in
[Col85]) to study the intersection of a curve C of genus g, embedded in its Jacobian
J , with a finitely generated subgroup of J of rank less than g. Our technique also
bears a resemblance to Skolem’s method for treating diophantine equations (see
[BS66, Chapter 4.6]).
Alternatively, our results can be interpreted purely from the point of view of
polynomial dynamics, as we describe the intersection of affine varieties with the
iterates of a point in the affine space under polynomial actions on each coordinate.
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP3
In this paper we will treat the case of affine varieties embedded in Gga, while the
polynomial action (on each coordinate of Gga) will always be given by Drinfeld
modules. The more general problem of studying intersections of affine varieties with
the iterates of a point in affine space under polynomial actions over number fields or
function fields appears to be quite difficult. To our knowledge, very little about this
question has been proven except in the case of multiplication maps on semiabelian
varieties (see [Voj96] and [McQ95]). We refer the reader to Section 4 of Zhang’s
notes [Zha06] for a number of algebraic dynamical conjectures that would generalize
well-known arithmetic theorems for semiabelian varieties. Although these notes do
not contain a dynamical analog of the Mordell-Lang conjecture, Zhang has indicated
to us that it might be reasonable to conjecture that if ψ : Y −→ Y is a suitable
morphism of a projective variety Y (one that is “polarized”, to use the terminology
of [Zha06]), then the intersection of the ψ-orbit of a point β with a subvariety V
must be finite if V does not contain a positive dimensional preperiodic subvariety.
We briefly sketch the plan of our paper. In Section 2 we set the notation, describe
the Denis-Mordell-Lang conjecture, and then state our main result. In Section 3
we prove this main result (Theorem 2.5), while in Section 4 we prove a couple of
extensions of it (Theorems 4.1 and 4.2).
2. Notation and statement of our main result
All subvarieties appearing in this paper are closed.
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. Let K
sep be the separable closure of
K inside 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
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 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
dimFq(t) Γ⊗A Fq(t).
If φ1 : A→ K{τ}, . . . , φg : A→ K{τ} are Drinfeld modules, then (φ1, . . . , φg) acts
on Gga coordinate-wise (i.e. φi acts on the i-th coordinate). We define as above the
notion of a (φ1, . . . , φg)-submodule of G
a; same for its rank.
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 set of all torsion points is denoted by φtor.
4 D. GHIOCA AND T. J. TUCKER
2.2. Valuations. 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 let MK be the set of valuations on K. Then MK 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;
• for each nonzero x ∈ K, we have
|x|v = 1.
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.
By abuse of notation, we let ∞ ∈MK denote any place extending the place v∞.
For v ∈MK we let Kv be the completion of K with respect to v. Let Cv be the
completion of an algebraic closure of Kv. Then | · |v extends to a unique absolute
value 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.
2.3. Logarithms and exponentials associated to a Drinfeld module. Let
v ∈ MK . According to Proposition 4.6.7 from [Gos96], there exists an unique
formal power series expφ,v ∈ Cv{τ} such that for every a ∈ A, we have
(2.1.1) φa = expφ,v a exp
In addition, the coefficient of the linear term in expφ,v(X) equals 1. We let logφ,v
be the formal power series exp−1
, which is the inverse of expφ,v.
If v = ∞ is an infinite place, then expφ,∞(x) is convergent for all x ∈ C∞ (see
Theorem 4.6.9 of [Gos96]). There exists a sufficiently small ball B∞ centered at
the origin such that expφ,∞ is an isometry on B∞ (see Lemma 3.6 of [GT06a]).
Hence, logφ,∞ is convergent on B∞. Moreover, the restriction of logφ,∞ on B∞ is
an analytic isometry (see also Proposition 4.14.2 of [Gos96]).
If v is a finite place, then expφ,v is convergent on a sufficiently small ball Bv ⊂ Cv
(this follows identically as the proof of the analyticity of expφ,∞ from Theorem 4.6.9
of [Gos96]). Similarly as in the above paragraph, at the expense of replacing Bv by
a smaller ball, we may assume expφ,v is an isometry on Bv. Hence, also logφ,v is
an analytic isometry on Bv.
For every place v ∈ MK , for every x ∈ Bv and for every polynomial a ∈ A, we
have (see (2.1.1))
(2.1.2) a logφ,v(x) = logφ,v(φa(x)) and expφ,v(ax) = φa(expφ,v(x)).
By abuse of language, expφ,∞ and expφ,v will be called exponentials, while logφ,∞
and logφ,v will be called logarithms.
2.4. Integrality and reduction.
Definition 2.2. A 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. All
infinite places of K are places of bad reduction for φ.
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP5
2.5. The Denis-Mordell-Lang conjecture. Let g be a positive integer.
Definition 2.3. Let φ1 : A→ K{τ}, . . . , φg : A→ K{τ} be Drinfeld modules. An
algebraic (φ1, . . . , φg)-submodule of G
a is an irreducible algebraic subgroup of G
invariant under the action of (φ1, . . . , φg).
Denis proposed in Conjecture 2 of [Den92a] the following problem, which we call
the full Denis-Mordell-Lang conjecture because it asks for the description of the
intersection of an affine variety with a finite rank φ-module (as opposed to only
a finitely generated φ-module). Recall that a φ-module M is said to be a finite
rank φ-module if it contains a finitely generated φ-submodule such that M/M ′ is
a torsion φ-module.
Conjecture 2.4 (The full Denis-Mordell-Lang conjecture). Let φ1 : A→ K{τ}, . . . , φg :
A → K{τ} be Drinfeld modules. Let V ⊂ Gga be an affine variety defined over K.
Let Γ be a finite rank (φ1, . . . , φg)-submodule of G
a(K). Then there exist algebraic
(φ1, . . . , φg)-submodules B1, . . . , Bl of G
a and there exist γ1, . . . , γl ∈ Γ such that
V (K) ∩ Γ =
(γi +Bi(K)) ∩ Γ.
In [Den92a], Denis showed that under certain natural Galois theoretical assump-
tions, Conjecture 2.4 would follow from the weaker conjecture which would describe
the intersection of an affine variety with a finitely generated φ-module.
Since then, the case Γ is the product of the torsion submodules of each φi was
proved by Scanlon in [Sca02], while various other instances of Conjecture 2.4 were
worked out in [Ghi05] and [Ghi06b]. We note that Denis asked his conjecture also
for t-modules, which includes the case of products of distinct Drinfeld modules
acting on Gga.
For the sake of simplifying the notation, we denote by φ the action of (φ1, . . . , φg)
on Gga. We also note that if V is an irreducible affine subvariety of G
a which has a
Zariski dense intersection with a finite rank φ-submodule Γ of Gga, then the Denis-
Mordell-Lang conjecture predicts that V is a translate of an algebraic φ-submodule
of Gga by a point in Γ. In particular, if V is an irreducible affine curve, which is not
a translate of an algebraic φ-submodule, then its intersection with any finite rank
φ-submodule of Gga should be finite.
In [Ghi05], the first author studied the Denis-Mordell-Lang conjecture for Drin-
feld modules whose field of definition (for their coefficients) is of transcendence
degree at least equal to 2. The methods employed in [Ghi05] involve specializa-
tions, and so, it was crucial for the φ there not to be isomorphic with a Drinfeld
module defined over Fq(t). In the present paper we will study precisely this case
left out in [Ghi05] and [Ghi06b]. Our methods depend crucially on the hypothesis
that the transcendence degree of the field generated by the coefficients of φi is one,
since we use the fact that at each place v, the number of residue classes in the ring
of integers at v is finite.
The main result of our paper is describing the intersection of an affine subvariety
V ⊂ Gga with a cyclic φ-submodule Γ of G
Theorem 2.5. Let K be a function field of transcendence degree equal to one. Let
φ1 : A→ K{τ}, . . . , φg : A→ K{τ} be Drinfeld modules. Let (x1, . . . , xg) ∈ G
and let Γ ⊂ Gga(K) be the cyclic (φ1, . . . , φg)-submodule generated by (x1, . . . , xg).
6 D. GHIOCA AND T. J. TUCKER
Let V ⊂ Gga be an affine subvariety defined over K. Then V (K) ∩ Γ is a finite
union of cosets of (φ1, . . . , φg)-submodules of Γ.
Using an idea from [Ghi06b], we are able to extend the above result to (φ1, . . . , φg)-
submodules of rank 1 (see our Theorem 4.2) in the special case where V is a curve.
3. Proofs of our main results
We continue with the notation from Section 2. Hence φ1, . . . , φg are Drinfeld
modules. We denote by φ the action of (φ1, . . . , φg) on G
a. Also, let (x1, . . . , xg) ∈
a(K) and let Γ be the cyclic φ-submodule of G
a(K) generated by (x1, . . . , xg).
Unless otherwise stated, V ⊂ Gga is an affine subvariety defined over K.
We first prove an easy combinatorial result which we will use in the proof of
Theorem 2.5.
Lemma 3.1. Let Γ be a cyclic φ-submodule of Gga(K). Let Γ0 be a nontrivial φ-
submodule of Γ, and let S ⊂ Γ be an infinite set. Suppose that for every infinite
subset S0 ⊂ S, there exists a coset C0 of Γ0 such that C0 ∩ S0 6= ∅ and C0 ⊂ S.
Then S is a finite union of cosets of φ-submodules of Γ.
Proof. Since S is infinite, Γ is infinite, and thus Γ is torsion-free. Therefore, Γ is an
infinite cyclic φ-module, which is isomorphic to A (as a module over itself). Hence,
via this isomorphism, Γ0 is isomorphic to a nontrivial ideal I of A. Since A/I is
finite (recall that A = Fq[t]), there are finitely many cosets of Γ0 in Γ. Thus, S
contains at most finitely many cosets of Γ0.
Now, let {yi+Γ0}
i=1 be all of the cosets of Γ0 that are contained in S. Suppose
(3.1.1) S0 := S \
(yi + Γ0) is infinite.
Then using the hypotheses of this Lemma for S0, we see that there is a coset of
Γ0 that is contained in S but is not one of the cosets (yi + Γ0) (because it has a
nonempty intersection with S0). This contradicts the fact that {yi + Γ0}
i=1 are
all the cosets of Γ0 that are contained in S. Therefore S0 must be finite. Since
any finite subset of Γ is a finite union of cosets of the trivial submodule of Γ, this
completes our proof. �
We will also use the following Lemma in the proof of Theorem 2.5.
Lemma 3.2. Let θ : A → K{τ} and ψ : A → K{τ} be Drinfeld modules. Let
v be a place of good reduction for θ and ψ. Let x, y ∈ Cv. Let 0 < rv < 1,
and let Bv := {z ∈ Cv | |z|v < rv} be a sufficiently small ball centered at the
origin with the property that both logθ,v and logψ,v are analytic isometries on Bv.
Then for every polynomials P,Q ∈ A such that (θP (x), ψP (y)) ∈ Bv × Bv and
(θQ(x), ψQ(y)) ∈ Bv ×Bv, we have
logθ,v(θP (x)) · logψ,v(ψQ(y)) = logθ,v(θQ(x)) · logψ,v(ψP (y)).
Proof. Since v is a place of good reduction for θ, all the coefficients of θQ are v-adic
integers and thus, |θQ(θP (x))|v ≤ |θP (x)|v < rv (we use the fact that |θP (x)|v <
rv < 1, and so, each term of θQ(θP (x)) has its absolute value at most equal to
|θP (x)|v). Using (2.1.2), we conclude that
Q · logθ,v(θP (x)) = logθ,v(θQP (x)) = logθ,v(θPQ(x)) = P · logθ,v(θQ(x)).
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP7
Similarly we obtain that Q · logψ,v(ψP (x)) = P · logψ,v(ψQ(x)). This concludes the
proof of Lemma 3.2. �
The following result is an immediate corollary of Lemma 3.2.
Corollary 3.3. With the notation as in Theorem 2.5, assume in addition that
x1 /∈ (φ1)tor. Let v be a place of good reduction for each φi. Suppose Bv is a small
ball (of radius less than 1) centered at the origin such that each logφi,v is an analytic
isometry on Bv. Then for each i ∈ {2, . . . , g}, the fractions
λi :=
logφi,v ((φi)P (xi))
logφ1,v ((φ1)P (x1))
are independent of the choice of the nonzero polynomial P ∈ A for which φP (x1, . . . , xg) ∈
Bgv .
The following simple result on zeros of analytic functions can be found in [Gos96,
Proposition 2.1, p. 42]. We include a short proof for the sake of completeness.
Lemma 3.4. Let F (z) =
i=0 aiz
i be a power series with coefficients in Cv that
is convergent in an open disc B of positive radius around the point z = 0. Suppose
that F is not the zero function. Then the zeros of F in B are isolated.
Proof. Let w be a zero of F in B. We may rewrite F in terms of (z−w) as a power
series F (z) =
i=1 bi(z−w)
i that converges in a disc Bw of positive radius around
w. Let m be the smallest index n such that bn 6= 0.
Because F is convergent in Bw, then there exists a positive real number r such
that for all n > m, we have
∣∣∣ bnbm
< rn−m. Then, for any u ∈ Bw such that
0 < |u − w|v <
, we have |bm(u − w)
m|v > |bn(u − w)
n|v for all n > m. Hence
|F (u)|v = |bm(u − w)
m|v 6= 0. Thus F (u) 6= 0, and so, F has no zeros other than
w in a nonempty open disc around w. �
We are ready to prove Theorem 2.5.
Proof of Theorem 2.5. We may assume V (K) ∩ Γ is infinite (otherwise the conclu-
sion of Theorem 2.5 is obvisouly satisfied). Assuming V (K) ∩ Γ is infinite, we will
show that there exists a nontrivial φ-submodule Γ0 ⊂ Γ such that each infinite sub-
set of points S0 in V (K)∩Γ has a nonempty intersection with a coset C0 of Γ0, and
moreover, C0 ⊂ V (K) ∩ Γ. Then Lemma 3.1 will finish the proof of Theorem 2.5.
First we observe that Γ is not a torsion φ-submodule. Otherwise Γ is finite,
contradicting our assumption that V (K) ∩ Γ is infinite. Hence, from now on, we
assume (without loss of generality) that x1 is not a torsion point for φ1.
We fix a finite set of polynomials {fj}
j=1 ⊂ K[X1, . . . , Xg] which generate the
vanishing ideal of V .
Let v ∈MK be a place of K which is of good reduction for all φi (for 1 ≤ i ≤ g).
In addition, we assume each xi is integral at v (for 1 ≤ i ≤ g). Then for each
P ∈ A, we have
φP (x1, . . . , xg) ∈ G
a(ov),
where ov is the ring of v-adic integers in Kv (the completion of K at v). Because
ov is a compact space (we use the fact that K is a function field of transcendence
degree 1 and thus has a finite residue field at v), we conclude that every infinite
sequence of points φP (x1, . . . , xg) ∈ V (K)∩Γ contains a convergent subsequence in
8 D. GHIOCA AND T. J. TUCKER
v. Using Lemma 3.1, it suffices to show that there exists a nontrivial φ-submodule
Γ0 ⊂ Γ such that every convergent sequence of points in V (K)∩Γ has a nonempty
intersection with a coset C0 of Γ0, and moreover, C0 ⊂ V (K) ∩ Γ.
Now, let S0 be an infinite subsequence of distinct points in V (K) ∩ Γ which
converges v-adically to (x0,1, . . . , x0,g) ∈ o
v, let 0 < rv < 1, and let Bv := {z ∈ Cv |
|z|v < rv} be a small ball centered at the origin on which each of the logarithmic
functions logφi,v is an analytic isometry (for 1 ≤ i ≤ g). Since (x0,1, . . . , x0,g) is the
limit point for S0, there exists a d ∈ A and an infinite subsequence {φd+Pn}n≥0 ⊂ S0
(with Pn = 0 if and only if n = 0), such that for each n ≥ 0, we have
(3.4.1)
∣∣(φi)d+Pn (xi)− x0,i
for each 1 ≤ i ≤ g.
We will show that there exists an algebraic group Y0, independent of S0 and in-
variant under φ, such that φd(x1, . . . , xg) + Y0 is a subvariety of V containing
φd+Pn(x1, . . . , xg) for all Pn. Thus the submodule Γ0 := Y0(K) ∩ Γ will satisfy the
hypothesis of Lemma 3.1 for the infinite subset V (K) ∩ Γ ⊂ Γ; this will yield the
conclusion of Theorem 2.5.
Using (3.4.1) for n = 0 (we recall that P0 = 0), and then for arbitrary n, we see
(3.4.2)
∣∣(φi)Pn (xi)
for each 1 ≤ i ≤ g.
Hence logφi,v is well-defined at (φi)Pn (xi) for each i ∈ {1, . . . , g} and for each n ≥ 1.
Moreover, the fact that
(φi)Pn+d (xi)
converges to a point in ov means that(
(φi)Pn (xi)
converges to a point which is contained in Bv (see (3.4.2)).
Without loss of generality, we may assume
(3.4.3) | logφ1,v
(φ1)P1 (x1)
| logφi,v
(φi)P1 (xi)
Using the result of Corollary 3.3, we conclude that for each i ∈ {2, . . . , g}, the
following fraction is independent of n and of the sequence {Pn}n:
(3.4.4) λi :=
logφi,v
(φi)Pn (xi)
logφ1,v
(φ1)Pn (x1)
Note that since x1 is not a torsion point for φ1, the denominator of λi (3.4.4) is
nonzero. Because of equation (3.4.3), we may conclude that |λi|v ≤ 1 for each i.
The fact that λi is independent of the sequence {Pn}n will be used later to show
that the φ-submodule Γ0 that we construct is independent of the sequence {Pn}n.
For each n ≥ 1 and each 2 ≤ i ≤ g, we have
(3.4.5) logφi,v
(φi)Pn (xi)
= λi · logφ1,v
(φ1)Pn (x1)
For each i, applying the exponential function expφi,v to both sides of (3.4.5) yields
(3.4.6) (φi)Pn (xi) = expφi,v
λi · logφ1,v
(φ1)Pn (x1)
Since φd+Pn (x1, . . . , xg) ∈ V (K), for each j ∈ {1, . . . , ℓ} we have
(3.4.7) fj (φd+Pn(x1, . . . , xg)) = 0 for each n.
For each j ∈ {1, . . . , ℓ} we let fd,j ∈ K[X1, . . . , Xg] be defined by
(3.4.8) fd,j (X1, . . . , Xg) := fj (φd(x1, . . . , xg) + (X1, . . . , Xg)) .
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP9
We let Vd ⊂ G
a be the affine subvariety defined by the equations
fd,j(X1, . . . , Xg) = 0 for each j ∈ {1, . . . , ℓ}.
Using (3.4.7) and (3.4.8), we see that for each j ∈ {1, . . . , ℓ} we have
(3.4.9) fd,j (φPn(x1, . . . , xg)) = 0
for each n, and so,
(3.4.10) φPn(x1, . . . , xg) ∈ Vd(K).
For each j ∈ {1, . . . , ℓ}, we let Fd,j(u) be the analytic function defined on Bv by
Fd,j(u) := fd,j
u, expφ2,v
λ2 logφ1,v(u)
, . . . , expφg ,v
λg logφ1,v(u)
We note, because of (3.4.3), and the fact that logφ1,v is an analytic isometry on
Bv, that for each u ∈ Bv, we have
(3.4.11) |λi · logφ1,v(u)|v = |λi|v · | logφ1,v(u)|v ≤ |u|v < rv.
Equation (3.4.11) shows that λi · logφ1,v(u) ∈ Bv, and so, expφi,v
λi · logφ1,v(u)
well-defined.
Using (3.4.6) and (3.4.9) we obtain that for every n ≥ 1, we have
(3.4.12) Fd,j
(φ1)Pn (x1)
(φ1)Pn (x1)
is a sequence of zeros for the analytic function Fd,j which
has an accumulation point in Bv. Lemma 3.4 then implies that Fd,j = 0, and so,
for each j ∈ {1, . . . , ℓ}, we have
(3.4.13) fd,j
u, expφ2,v
λ2 logφ1,v(u)
, . . . , expφg,v
λg logφ1,v(u)
For each u ∈ Bv, we let
Zu :=
u, expφ2,v
λ2 logφ1,v(u)
, . . . , expφg,v
λg logφ1,v(u)
∈ Gga(Cv).
Then (3.4.13) implies that
(3.4.14) Zu ∈ Vd for each u ∈ Bv.
Let Y0 be the Zariski closure of {Zu}u∈Bv . Then Y0 ⊂ Vd. Note that Y0 is inde-
pendent of the sequence {Pn}n (because the λi are independent of the sequence
{Pn}n, according to Corollary 3.3).
We claim that for each u ∈ Bv and for each P ∈ A, we have
(3.4.15) φP (Zu) = Z(φ1)P (u).
Note that for each u ∈ Bv, then also (φ1)P (u) ∈ Bv for each P ∈ A, because each
coefficient of φ1 is a v-adic integer. To see that (3.4.15) holds, we use (2.1.2), which
implies that for each i ∈ {2, . . . , g} we have
expφi,v
λi logφ1,v ((φ1)P (u))
= expφi,v
λi · P · logφ1,v(u)
= expφi,v
P · λi logφ1,v(u)
= (φi)P
expφi,v
λi logφ1,v(u)
Hence, (3.4.15) holds, and so, Y0 is invariant under φ. Furthermore, since all of the
expφi,v and logφi,v are additive functions, we have Zu1+u2 = Zu1 + Zu2 for every
u1, u2 ∈ Bv. Hence Y0 is an algebraic group, which is also a φ-submodule of G
Moreover, Y0 is defined independently of Γ.
10 D. GHIOCA AND T. J. TUCKER
Let Γ0 := Y0(K)∩Γ. Because Y0 is invariant under φ, then Γ0 is a submodule of
Γ. Because Y0 ⊂ Vd, it follows that the translate φd(x1, . . . , xg)+Y0 is a subvariety
of V which contains all {φd+Pn(x1, . . . , xg)}n. In particular, the (infinite) translate
C0 of Γ0 by φd(x1, . . . , xg) is contained in V (K)∩Γ. Hence, every infinite sequence
of points in V (K)∩Γ has a nontrivial intersection with a coset C0 of (the nontrivial
φ-submodule) Γ0, and moreover, C0 ⊂ V (K)∩Γ. Applying Lemma 3.1 thus finishes
the proof of Theorem 2.5. �
In the course of our proof of Theorem 2.5 we also proved the following statement.
Theorem 3.5. Let Γ be an infinite cyclic φ-submodule of Gga. Then there exists
an infinite φ-submodule Γ0 ⊂ Γ such that for every affine subvariety V ⊂ G
a, if
V (K) ∩ Γ is infinite, then V (K) ∩ Γ contains a coset of Γ0.
Proof. Let v be a place of good reduction for φ; in addition, we assume the points
in Γ are v-adic integers. Suppose that V (K) ∩ Γ is infinite. As shown in the proof
of Theorem 2.5, there exists a positive dimensional algebraic group Y0, invariant
under φ, and depending only on Γ and v (but not on V ), such that a translate of Y0
by a point in Γ lies in V . Moreover, Γ0 := Y0(K) ∩ Γ is infinite. Hence Γ0 satisfies
the conclusion of Theorem 3.5. �
4. Further extensions
We continue with the notation from Section 3: φ1, . . . , φg are Drinfeld modules.
As usual, we denote by φ the action of (φ1, . . . , φg) on G
a. First we prove the
following consequence of Theorem 2.5.
Theorem 4.1. Let V ⊂ Gga be an affine subvariety defined over K. Let Γ ⊂ G
be a finitely generated φ-submodule of rank 1. Then V (K) ∩ Γ is a finite union of
cosets of φ-submodules of Γ. In particular, if V is an irreducible curve which is not
a translate of an algebraic φ-submodule, then V (K) ∩ Γ is finite.
Proof. Since A = Fq[t] is a principal ideal domain, Γ is the direct sum of its finite
torsion submodule Γtor and a free submodule Γ1, which is cyclic because Γ has rank
1. Therefore
γ∈Γtor
γ + Γ1,
and so,
V (K) ∩ Γ =
γ∈Γtor
V (K) ∩ (γ + Γ1) =
γ∈Γtor
(γ + (−γ + V (K)) ∩ Γ1) .
Using the fact Γtor is finite and applying Theorem 2.5 to each intersection (−γ + V (K))∩
Γ1 thus completes our proof. �
We use the ideas from [Ghi06b] to describe the intersection of a curve C with a
φ-module of rank 1. So, let (x1, . . . , xg) ∈ G
a(K), let Γ be the cyclic φ-submodule of
a(K) generated by (x1, . . . , xg), and let Γ be the φ-submodule of rank 1, containing
all (z1, . . . , zg) ∈ G
a(K) for which there exists a nonzero polynomial P such that
φP (z1, . . . , zg) ∈ Γ.
Since all polynomials φP (for P ∈ A) are separable, we have Γ ⊂ G
sep).
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP11
With the notation above, we prove the following result; this may be viewed
as a Drinfeld module analog of McQuillan’s result on semiabelian varieties (see
[McQ95]), which had been conjectured by Lang.
Theorem 4.2. Let C ⊂ Gga be an affine curve defined over K. Then C(K) ∩ Γ is
a finite union of cosets of φ-submodules of Γ.
Before proceeding to the proof of Theorem 4.2 we first prove two facts which will
be used later. The first fact is an immediate consequence of Theorem 1 of [Sca02]
(the Denis-Manin-Mumford conjecture for Drinfeld modules), which we state below.
Theorem 4.3 (Scanlon). Let V ⊂ Gga be an affine variety defined over K. Then
there exist algebraic φ-submodules B1, . . . , Bℓ of G
a and elements γ1, . . . , γℓ of φtor
such that
V (K) ∩ φtor =
γi +Bi(K)
∩ φtor.
Moreover, in Remark 19 from [Sca02], Scanlon notes that his proof of the Denis-
Manin-Mumford conjecture yields a uniform bound on the degree of the Zariski
closure of V (K)∩φtor, depending only on φ, g, and the degree of V . In particular,
one obtains the following uniform statement for translates of curves.
Fact 4.4. Let C ⊂ Gga be an irreducible curve which is not a translate of an
algebraic φ-module of Gga. Then there exists a positive integer N such that for
every y ∈ Gga(K), the set
y + C(K)
∩ φtor has at most N elements.
Proof. The curve C contains no translate of a positive dimensional algebraic φ-
submodule of Gga, so for every y ∈ G
a(K), the algebraic φ-modules Bi appear-
ing in the intersection
y + C(K)
∩ φtor are all trivial. In particular, the set(
y + C(K)
∩φtor is finite. Thus, using the uniformity obtained by Scanlon for his
Manin-Mumford theorem, we conclude that the cardinality of
y + C(K)
∩ φtor is
uniformly bounded above by some positive integer N . �
We will also use the following fact in the proof of our Theorem 4.2.
Fact 4.5. Let φ : A→ K{τ} be a Drinfeld module. Then for every positive integer
D, there exist finitely many torsion points y of φ such that [K(y) : K] ≤ D.
Proof. If y ∈ φtor, then the canonical height ĥ(y) of y (as defined in [Den92b])
equals 0. Also, as shown in [Den92b], the difference between the canonical height
and the usual Weil height is uniformly bounded on K. Then Fact 4.5 follows by
noting that there are finitely many points of bounded Weil height and bounded
degree over the field K (using Northcott’s theorem applied to the global function
field K). �
We are now ready to prove Theorem 4.2.
Proof of Theorem 4.2. Arguing as in the proof of Theorem 2.5, it suffices to show
that if C is an irreducible affine curve (embedded in Gga), then C(K)∩ Γ is infinite
only if C is a translate of an algebraic φ-submodule (because any translate of an
algebraic φ-module intersects Γ in a coset of some φ-submodule of Γ). Therefore,
from now on, we assume C is irreducible, that C(K) ∩ Γ is infinite, and that C is
not a translate of an algebraic φ-submodule. We will derive a contradiction.
12 D. GHIOCA AND T. J. TUCKER
Let z ∈ C(K) ∩ Γ. For each field automorphism σ : Ksep → Ksep that restricts
to the identity on K, we have zσ ∈ C (Ksep) (because C is defined over K). By
the definition of Γ, there exists a nonzero polynomial P ∈ A such that φP (z) ∈ Γ.
Since φP has coefficients in K, we obtain
φP (z
σ) = (φP (z))
= φP (z).
The last equality follows from the fact that φP (z) ∈ Γ ⊂ G
a(K). We conclude that
φP (z
σ − z) = 0, and, thus, we have
Tz,σ := z
σ − z ∈ φtor.
Moreover, Tz,σ ∈ (−z+C(K))∩φtor (because z
σ ∈ C). Using Fact 4.4 we conclude
that for each fixed z ∈ C(K)∩Γ, the set {Tz,σ}σ has cardinality bounded above by
some number N (independent of z). In particular, this implies that z has finitely
many Galois conjugates, so [K(z) : K] ≤ N . Similarly we have [K(zσ) : K] ≤ N ;
thus, we may conclude that
(4.5.1) [K (Tz,σ) : K] ≤ [K(z, z
σ) : K] ≤ N2.
As shown by Fact 4.5, there exists a finite set of torsion points w for which [K(w) :
K] ≤ N2. Hence, recalling that N is independent of z, we see that the set
(4.5.2) H := {Tz,σ} z∈C(K)∩Γ
σ:Ksep→Ksep
is finite.
Now, since H is a finite set of torsion points, there must exist a nonzero poly-
nomial Q ∈ A such that φQ(H) = {0}. Therefore, φQ(z
σ − z) = 0 for each
z ∈ C(K)∩Γ and each automorphism σ. Hence φQ(z)
σ = φQ(z) for each σ. Thus,
we have
(4.5.3) φQ(z) ∈ G
a(K) for every z ∈ C(K) ∩ Γ.
Let Γ1 := Γ ∩ G
a(K). Since Γ is a finite rank φ-module and G
a(K) is a tame
module (i.e. every finite rank submodule is finitely generated; see [Poo95] for a
proof of this result), it follows that Γ1 is finitely generated. Let Γ2 be the finitely
generated φ-submodule of Γ generated by all points z ∈ Γ such that φQ(z) ∈
Γ1. More precisely, if w1, . . . , wℓ generate the φ-submodule Γ1, then for each i ∈
{1, . . . , ℓ}, we find all the finitely many zi such that φQ(zi) = wi. Then this finite set
of all zi generate the φ-submodule Γ2. Thus Γ2 is a finitely generated φ-submodule,
and moreover, using equation (4.5.3), we obtain C(K) ∩ Γ = C(K) ∩ Γ2. Since Γ2
is a finitely generated φ-submodule of rank 1 (because Γ2 ⊂ Γ and Γ has rank 1),
Theorem 4.1 finishes the proof of Theorem 4.2. �
References
[Bak75] A. Baker, Transcendental number theory, Cambridge University Press, Cambridge, 1975.
[Bos99] V. Bosser, Minorations de formes linéaires de logarithmes pour les modules de Drinfeld,
J. Number Theory 75 (1999), no. 2, 279–323.
[BS66] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by
Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New
York, 1966.
[Cha41] C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur á
l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885.
[Col85] R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770.
[Dav95] S. David, Minorations de formes linéaire de logarithmes elliptiques, Mem. Soc. Math.
France 62 (1995), 143 pp.
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP13
[Den92a] L. Denis, Géométrie diophantienne sur les modules de Drinfel′d, The arithmetic of
function fields (Columbus, OH, 1991), Ohio State Univ. Math. Res. Inst. Publ., vol. 2,
de Gruyter, Berlin, 1992, pp. 285–302.
[Den92b] , Hauteurs canoniques et modules de Drinfel′d, Math. Ann. 294 (1992), no. 2,
213–223.
[Fal94] G. Faltings, The general case of S. Lang’s conjecture, Barsotti Symposium in Algebraic
Geometry (Abano Terme, 1991), Perspect. Math., no. 15, Academic Press, San Diego,
CA, 1994, pp. 175–182.
[Ghi05] D. Ghioca, The Mordell-Lang theorem for Drinfeld modules, Int. Math. Res. Not.
(2005), no. 53, 3273–3307.
[Ghi06a] D. Ghioca, Equidistribution for torsion points of a Drinfeld module, Math. Ann. 336
(2006), no. 4, 841–865.
[Ghi06b] D. Ghioca, Towards the full Mordell-Lang conjecture for Drinfeld modules, submitted
for publication, 6 pages, 2006.
[Gos96] D. Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und
ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-
Verlag, Berlin, 1996.
[GT06a] D. Ghioca and T. J. Tucker, Equidistribution and integral points for Drinfeld modules,
submitted for publication, 29 pages, 2006.
[GT06b] , Siegel’s Theorem for Drinfeld modules, submitted for publication, 25 pages,
2006.
[Hru98] E. Hrushovski, Proof of Manin’s theorem by reduction to positive characteristic, Model
theory and algebraic geometry, Lecture Notes in Math., no. 1696, Springer, Berlin, 1998,
pp. 197–205.
[McQ95] M. McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995),
no. 1, 143–159.
[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. Number
Theory 97 (2002), no. 1, 10–25.
[Ser97] J.-P. Serre, Lectures on the Mordell-Weil theorem, third ed., Aspects of Mathematics,
Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by
Martin Brown from notes by Michel Waldschmidt, with a foreword by Brown and Serre.
[SUZ97] L. Szpiro, E. Ullmo, and S. Zhang, Equirépartition des petits points, Invent. Math. 127
(1997), 337–347.
[Voj96] P. Vojta, Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126
(1996), no. 1, 133–181.
[Zha98] S. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147
(1998), no. 1, 159–165.
[Zha06] S. Zhang, Distributions and heights in algebraic dynamics, prepint, Available at
www.math.columbia.edu/~szhang/papers/dynamics.pdf, 66 pages, 2006.
E-mail address: dghioca@math.mcmaster.ca
Dragos Ghioca, Department of Mathematics, McMaster University, 1280 Main Street
West, Hamilton, Ontario, Canada L8S 4K1,
E-mail address: ttucker@math.rochester.edu
Thomas Tucker, Department of Mathematics, Hylan Building, University of Rochester,
Rochester, NY 14627
	1. Introduction
	2. Notation and statement of our main result
	2.1. Drinfeld modules
	2.2. Valuations
	2.3. Logarithms and exponentials associated to a Drinfeld module
	2.4. Integrality and reduction
	2.5. The Denis-Mordell-Lang conjecture
	3. Proofs of our main results
	4. Further extensions
	References

  We prove a dynamical version of the Mordell-Lang conjecture in the context of
Drinfeld modules. We use analytic methods similar to the ones employed by
Skolem, Chabauty, and Coleman for studying diophantine equations.

Introduction
Faltings proved the Mordell-Lang conjecture in the following form (see [Fal94]).
Theorem 1.1 (Faltings). Let G be an abelian variety defined over the field of
complex numbers C. Let X ⊂ G be a closed subvariety and Γ ⊂ G(C) a finitely
generated subgroup of G(C). Then X(C)∩Γ is a finite union of cosets of subgroups
of Γ.
In particular, Theorem 1.1 says that an irreducible subvariety X of an abelian
variety G has a Zariski dense intersection with a finitely generated subgroup of
G(C) only if X is a translate of an algebraic subgroup of G. We also note that
Faltings result was generalized to semiabelian varieties G by Vojta (see [Voj96]),
and then to finite rank subgroups Γ of G by McQuillan (see [McQ95]).
If we try to formulate the Mordell-Lang conjecture in the context of algebraic
subvarieties contained in a power of the additive group scheme Ga, the conclusion
is either false (in the characteristic 0 case, as shown by the curve y = x2 which
has an infinite intersection with the finitely generated subgroup Z × Z, without
being itself a translate of an algebraic subgroup of G2a) or it is trivially true (in
the characteristic p > 0 case, because every finitely generated subgroup of a power
of Ga is finite). Denis [Den92a] formulated a Mordell-Lang conjecture for powers
of Ga in characteristic p in the context of Drinfeld modules. Denis replaced the
finitely generated subgroup from the usual Mordell-Lang statement with a finitely
generated φ-submodule, where φ is a Drinfeld module. He also strengthened the
conclusion of the Mordell-Lang statement by asking that the subgroups whose cosets
are contained in the intersection of the algebraic variety with the finitely generated
φ-submodule be actually φ-submodules. The first author proved several cases of the
Denis-Mordell-Lang conjecture in [Ghi05] and [Ghi06b].
In the present paper we investigate other cases of the Denis-Mordell-Lang con-
jecture through methods different from the ones employed in [Ghi05]. In partic-
ular, we prove the Denis-Mordell-Lang conjecture in the case where the finitely
generated φ-module is cyclic and the Drinfeld modules are defined over a field of
transcendence degree equal to one (this is our Theorem 2.5). Note that [Ghi05] and
Key words and phrases. Drinfeld module, Polynomial Dynamics.
The second author was partially supported by National Security Agency Grant 06G-067.
http://arxiv.org/abs/0704.1333v1
2 D. GHIOCA AND T. J. TUCKER
[Ghi06b] treat only the case where the transcendence degree of the field of definition
is greater than one. One of the methods employed in [Ghi05] (and whose outcome
was later used in [Ghi06b]) was specializations; hence the necessity of dealing with
fields of transcendence degree greater than one. By contrast, the techniques used
in this paper are more akin to those used in treating diophantine problems over
number fields (see [Cha41], [Col85], or [BS66, Chapter 4.6], for example), where
such specialization arguments are also not available. So, making a parallel between
the classical Mordell-Lang conjecture and the Denis-Mordell-Lang conjecture, we
might say that the papers [Ghi05] and [Ghi06b] deal with the “function field case”,
while our present paper deals with the “number field case” of the Denis conjecture.
Moreover, using specializations (as in [Hru98] and [Ghi05]), our Theorem 2.5 can be
extended to Drinfeld modules defined over fields of arbitrary finite transcendence
degree.
We also note that recently there has been significant progress on establishing ad-
ditional links between classical diophantine results over number fields and similar
statements for Drinfeld modules. The first author proved in [Ghi06a] an equidis-
tribution statement for torsion points of a Drinfeld module, which 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 conjec-
ture). 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 [GT06a] to establish certain equidistribu-
tion and integrality statements for Drinfeld modules. Moreover, Bosser’s result is
quite possibly true also for linear forms in logarithms at finite places for a Drinfeld
module. Assuming this last statement, the authors proved in [GT06b] the analog of
Siegel’s theorem for finitely generated φ-submodules. We believe that our present
paper provides an additional proof of the fact that the Drinfeld modules represent
the right arithmetic analog in characteristic p for abelian varieties in characteristic
The idea behind the proof of our Theorem 2.5 can be explained quite simply.
Assuming that an affine variety V ⊂ Gga has infinitely many points in common with
a cyclic φ-submodule Γ, we can find then a suitable submodule Γ0 ⊂ Γ whose coset
lies in V . Indeed, applying the logarithmic map (associated to a suitable place
v) to Γ0 yields a line in the vector space C
v. Each polynomial f that vanishes
on V , then gives rise to an analytic function F on this line (by composing with
the exponential function). Because we assumed there are infinitely many points in
V ∩ Γ, the zeros of F must have an accumulation point on this line, which means
that F vanishes identically on the line. This means that there is an entire translate
of Γ0 contained in the zero locus of f . The inspiration for this idea comes from
the method employed by Chabauty in [Cha41] (and later refined by Coleman in
[Col85]) to study the intersection of a curve C of genus g, embedded in its Jacobian
J , with a finitely generated subgroup of J of rank less than g. Our technique also
bears a resemblance to Skolem’s method for treating diophantine equations (see
[BS66, Chapter 4.6]).
Alternatively, our results can be interpreted purely from the point of view of
polynomial dynamics, as we describe the intersection of affine varieties with the
iterates of a point in the affine space under polynomial actions on each coordinate.
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP3
In this paper we will treat the case of affine varieties embedded in Gga, while the
polynomial action (on each coordinate of Gga) will always be given by Drinfeld
modules. The more general problem of studying intersections of affine varieties with
the iterates of a point in affine space under polynomial actions over number fields or
function fields appears to be quite difficult. To our knowledge, very little about this
question has been proven except in the case of multiplication maps on semiabelian
varieties (see [Voj96] and [McQ95]). We refer the reader to Section 4 of Zhang’s
notes [Zha06] for a number of algebraic dynamical conjectures that would generalize
well-known arithmetic theorems for semiabelian varieties. Although these notes do
not contain a dynamical analog of the Mordell-Lang conjecture, Zhang has indicated
to us that it might be reasonable to conjecture that if ψ : Y −→ Y is a suitable
morphism of a projective variety Y (one that is “polarized”, to use the terminology
of [Zha06]), then the intersection of the ψ-orbit of a point β with a subvariety V
must be finite if V does not contain a positive dimensional preperiodic subvariety.
We briefly sketch the plan of our paper. In Section 2 we set the notation, describe
the Denis-Mordell-Lang conjecture, and then state our main result. In Section 3
we prove this main result (Theorem 2.5), while in Section 4 we prove a couple of
extensions of it (Theorems 4.1 and 4.2).
2. Notation and statement of our main result
All subvarieties appearing in this paper are closed.
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. Let K
sep be the separable closure of
K inside 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
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 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
dimFq(t) Γ⊗A Fq(t).
If φ1 : A→ K{τ}, . . . , φg : A→ K{τ} are Drinfeld modules, then (φ1, . . . , φg) acts
on Gga coordinate-wise (i.e. φi acts on the i-th coordinate). We define as above the
notion of a (φ1, . . . , φg)-submodule of G
a; same for its rank.
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 set of all torsion points is denoted by φtor.
4 D. GHIOCA AND T. J. TUCKER
2.2. Valuations. 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 let MK be the set of valuations on K. Then MK 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;
• for each nonzero x ∈ K, we have
|x|v = 1.
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.
By abuse of notation, we let ∞ ∈MK denote any place extending the place v∞.
For v ∈MK we let Kv be the completion of K with respect to v. Let Cv be the
completion of an algebraic closure of Kv. Then | · |v extends to a unique absolute
value 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.
2.3. Logarithms and exponentials associated to a Drinfeld module. Let
v ∈ MK . According to Proposition 4.6.7 from [Gos96], there exists an unique
formal power series expφ,v ∈ Cv{τ} such that for every a ∈ A, we have
(2.1.1) φa = expφ,v a exp
In addition, the coefficient of the linear term in expφ,v(X) equals 1. We let logφ,v
be the formal power series exp−1
, which is the inverse of expφ,v.
If v = ∞ is an infinite place, then expφ,∞(x) is convergent for all x ∈ C∞ (see
Theorem 4.6.9 of [Gos96]). There exists a sufficiently small ball B∞ centered at
the origin such that expφ,∞ is an isometry on B∞ (see Lemma 3.6 of [GT06a]).
Hence, logφ,∞ is convergent on B∞. Moreover, the restriction of logφ,∞ on B∞ is
an analytic isometry (see also Proposition 4.14.2 of [Gos96]).
If v is a finite place, then expφ,v is convergent on a sufficiently small ball Bv ⊂ Cv
(this follows identically as the proof of the analyticity of expφ,∞ from Theorem 4.6.9
of [Gos96]). Similarly as in the above paragraph, at the expense of replacing Bv by
a smaller ball, we may assume expφ,v is an isometry on Bv. Hence, also logφ,v is
an analytic isometry on Bv.
For every place v ∈ MK , for every x ∈ Bv and for every polynomial a ∈ A, we
have (see (2.1.1))
(2.1.2) a logφ,v(x) = logφ,v(φa(x)) and expφ,v(ax) = φa(expφ,v(x)).
By abuse of language, expφ,∞ and expφ,v will be called exponentials, while logφ,∞
and logφ,v will be called logarithms.
2.4. Integrality and reduction.
Definition 2.2. A 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. All
infinite places of K are places of bad reduction for φ.
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP5
2.5. The Denis-Mordell-Lang conjecture. Let g be a positive integer.
Definition 2.3. Let φ1 : A→ K{τ}, . . . , φg : A→ K{τ} be Drinfeld modules. An
algebraic (φ1, . . . , φg)-submodule of G
a is an irreducible algebraic subgroup of G
invariant under the action of (φ1, . . . , φg).
Denis proposed in Conjecture 2 of [Den92a] the following problem, which we call
the full Denis-Mordell-Lang conjecture because it asks for the description of the
intersection of an affine variety with a finite rank φ-module (as opposed to only
a finitely generated φ-module). Recall that a φ-module M is said to be a finite
rank φ-module if it contains a finitely generated φ-submodule such that M/M ′ is
a torsion φ-module.
Conjecture 2.4 (The full Denis-Mordell-Lang conjecture). Let φ1 : A→ K{τ}, . . . , φg :
A → K{τ} be Drinfeld modules. Let V ⊂ Gga be an affine variety defined over K.
Let Γ be a finite rank (φ1, . . . , φg)-submodule of G
a(K). Then there exist algebraic
(φ1, . . . , φg)-submodules B1, . . . , Bl of G
a and there exist γ1, . . . , γl ∈ Γ such that
V (K) ∩ Γ =
(γi +Bi(K)) ∩ Γ.
In [Den92a], Denis showed that under certain natural Galois theoretical assump-
tions, Conjecture 2.4 would follow from the weaker conjecture which would describe
the intersection of an affine variety with a finitely generated φ-module.
Since then, the case Γ is the product of the torsion submodules of each φi was
proved by Scanlon in [Sca02], while various other instances of Conjecture 2.4 were
worked out in [Ghi05] and [Ghi06b]. We note that Denis asked his conjecture also
for t-modules, which includes the case of products of distinct Drinfeld modules
acting on Gga.
For the sake of simplifying the notation, we denote by φ the action of (φ1, . . . , φg)
on Gga. We also note that if V is an irreducible affine subvariety of G
a which has a
Zariski dense intersection with a finite rank φ-submodule Γ of Gga, then the Denis-
Mordell-Lang conjecture predicts that V is a translate of an algebraic φ-submodule
of Gga by a point in Γ. In particular, if V is an irreducible affine curve, which is not
a translate of an algebraic φ-submodule, then its intersection with any finite rank
φ-submodule of Gga should be finite.
In [Ghi05], the first author studied the Denis-Mordell-Lang conjecture for Drin-
feld modules whose field of definition (for their coefficients) is of transcendence
degree at least equal to 2. The methods employed in [Ghi05] involve specializa-
tions, and so, it was crucial for the φ there not to be isomorphic with a Drinfeld
module defined over Fq(t). In the present paper we will study precisely this case
left out in [Ghi05] and [Ghi06b]. Our methods depend crucially on the hypothesis
that the transcendence degree of the field generated by the coefficients of φi is one,
since we use the fact that at each place v, the number of residue classes in the ring
of integers at v is finite.
The main result of our paper is describing the intersection of an affine subvariety
V ⊂ Gga with a cyclic φ-submodule Γ of G
Theorem 2.5. Let K be a function field of transcendence degree equal to one. Let
φ1 : A→ K{τ}, . . . , φg : A→ K{τ} be Drinfeld modules. Let (x1, . . . , xg) ∈ G
and let Γ ⊂ Gga(K) be the cyclic (φ1, . . . , φg)-submodule generated by (x1, . . . , xg).
6 D. GHIOCA AND T. J. TUCKER
Let V ⊂ Gga be an affine subvariety defined over K. Then V (K) ∩ Γ is a finite
union of cosets of (φ1, . . . , φg)-submodules of Γ.
Using an idea from [Ghi06b], we are able to extend the above result to (φ1, . . . , φg)-
submodules of rank 1 (see our Theorem 4.2) in the special case where V is a curve.
3. Proofs of our main results
We continue with the notation from Section 2. Hence φ1, . . . , φg are Drinfeld
modules. We denote by φ the action of (φ1, . . . , φg) on G
a. Also, let (x1, . . . , xg) ∈
a(K) and let Γ be the cyclic φ-submodule of G
a(K) generated by (x1, . . . , xg).
Unless otherwise stated, V ⊂ Gga is an affine subvariety defined over K.
We first prove an easy combinatorial result which we will use in the proof of
Theorem 2.5.
Lemma 3.1. Let Γ be a cyclic φ-submodule of Gga(K). Let Γ0 be a nontrivial φ-
submodule of Γ, and let S ⊂ Γ be an infinite set. Suppose that for every infinite
subset S0 ⊂ S, there exists a coset C0 of Γ0 such that C0 ∩ S0 6= ∅ and C0 ⊂ S.
Then S is a finite union of cosets of φ-submodules of Γ.
Proof. Since S is infinite, Γ is infinite, and thus Γ is torsion-free. Therefore, Γ is an
infinite cyclic φ-module, which is isomorphic to A (as a module over itself). Hence,
via this isomorphism, Γ0 is isomorphic to a nontrivial ideal I of A. Since A/I is
finite (recall that A = Fq[t]), there are finitely many cosets of Γ0 in Γ. Thus, S
contains at most finitely many cosets of Γ0.
Now, let {yi+Γ0}
i=1 be all of the cosets of Γ0 that are contained in S. Suppose
(3.1.1) S0 := S \
(yi + Γ0) is infinite.
Then using the hypotheses of this Lemma for S0, we see that there is a coset of
Γ0 that is contained in S but is not one of the cosets (yi + Γ0) (because it has a
nonempty intersection with S0). This contradicts the fact that {yi + Γ0}
i=1 are
all the cosets of Γ0 that are contained in S. Therefore S0 must be finite. Since
any finite subset of Γ is a finite union of cosets of the trivial submodule of Γ, this
completes our proof. �
We will also use the following Lemma in the proof of Theorem 2.5.
Lemma 3.2. Let θ : A → K{τ} and ψ : A → K{τ} be Drinfeld modules. Let
v be a place of good reduction for θ and ψ. Let x, y ∈ Cv. Let 0 < rv < 1,
and let Bv := {z ∈ Cv | |z|v < rv} be a sufficiently small ball centered at the
origin with the property that both logθ,v and logψ,v are analytic isometries on Bv.
Then for every polynomials P,Q ∈ A such that (θP (x), ψP (y)) ∈ Bv × Bv and
(θQ(x), ψQ(y)) ∈ Bv ×Bv, we have
logθ,v(θP (x)) · logψ,v(ψQ(y)) = logθ,v(θQ(x)) · logψ,v(ψP (y)).
Proof. Since v is a place of good reduction for θ, all the coefficients of θQ are v-adic
integers and thus, |θQ(θP (x))|v ≤ |θP (x)|v < rv (we use the fact that |θP (x)|v <
rv < 1, and so, each term of θQ(θP (x)) has its absolute value at most equal to
|θP (x)|v). Using (2.1.2), we conclude that
Q · logθ,v(θP (x)) = logθ,v(θQP (x)) = logθ,v(θPQ(x)) = P · logθ,v(θQ(x)).
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP7
Similarly we obtain that Q · logψ,v(ψP (x)) = P · logψ,v(ψQ(x)). This concludes the
proof of Lemma 3.2. �
The following result is an immediate corollary of Lemma 3.2.
Corollary 3.3. With the notation as in Theorem 2.5, assume in addition that
x1 /∈ (φ1)tor. Let v be a place of good reduction for each φi. Suppose Bv is a small
ball (of radius less than 1) centered at the origin such that each logφi,v is an analytic
isometry on Bv. Then for each i ∈ {2, . . . , g}, the fractions
λi :=
logφi,v ((φi)P (xi))
logφ1,v ((φ1)P (x1))
are independent of the choice of the nonzero polynomial P ∈ A for which φP (x1, . . . , xg) ∈
Bgv .
The following simple result on zeros of analytic functions can be found in [Gos96,
Proposition 2.1, p. 42]. We include a short proof for the sake of completeness.
Lemma 3.4. Let F (z) =
i=0 aiz
i be a power series with coefficients in Cv that
is convergent in an open disc B of positive radius around the point z = 0. Suppose
that F is not the zero function. Then the zeros of F in B are isolated.
Proof. Let w be a zero of F in B. We may rewrite F in terms of (z−w) as a power
series F (z) =
i=1 bi(z−w)
i that converges in a disc Bw of positive radius around
w. Let m be the smallest index n such that bn 6= 0.
Because F is convergent in Bw, then there exists a positive real number r such
that for all n > m, we have
∣∣∣ bnbm
< rn−m. Then, for any u ∈ Bw such that
0 < |u − w|v <
, we have |bm(u − w)
m|v > |bn(u − w)
n|v for all n > m. Hence
|F (u)|v = |bm(u − w)
m|v 6= 0. Thus F (u) 6= 0, and so, F has no zeros other than
w in a nonempty open disc around w. �
We are ready to prove Theorem 2.5.
Proof of Theorem 2.5. We may assume V (K) ∩ Γ is infinite (otherwise the conclu-
sion of Theorem 2.5 is obvisouly satisfied). Assuming V (K) ∩ Γ is infinite, we will
show that there exists a nontrivial φ-submodule Γ0 ⊂ Γ such that each infinite sub-
set of points S0 in V (K)∩Γ has a nonempty intersection with a coset C0 of Γ0, and
moreover, C0 ⊂ V (K) ∩ Γ. Then Lemma 3.1 will finish the proof of Theorem 2.5.
First we observe that Γ is not a torsion φ-submodule. Otherwise Γ is finite,
contradicting our assumption that V (K) ∩ Γ is infinite. Hence, from now on, we
assume (without loss of generality) that x1 is not a torsion point for φ1.
We fix a finite set of polynomials {fj}
j=1 ⊂ K[X1, . . . , Xg] which generate the
vanishing ideal of V .
Let v ∈MK be a place of K which is of good reduction for all φi (for 1 ≤ i ≤ g).
In addition, we assume each xi is integral at v (for 1 ≤ i ≤ g). Then for each
P ∈ A, we have
φP (x1, . . . , xg) ∈ G
a(ov),
where ov is the ring of v-adic integers in Kv (the completion of K at v). Because
ov is a compact space (we use the fact that K is a function field of transcendence
degree 1 and thus has a finite residue field at v), we conclude that every infinite
sequence of points φP (x1, . . . , xg) ∈ V (K)∩Γ contains a convergent subsequence in
8 D. GHIOCA AND T. J. TUCKER
v. Using Lemma 3.1, it suffices to show that there exists a nontrivial φ-submodule
Γ0 ⊂ Γ such that every convergent sequence of points in V (K)∩Γ has a nonempty
intersection with a coset C0 of Γ0, and moreover, C0 ⊂ V (K) ∩ Γ.
Now, let S0 be an infinite subsequence of distinct points in V (K) ∩ Γ which
converges v-adically to (x0,1, . . . , x0,g) ∈ o
v, let 0 < rv < 1, and let Bv := {z ∈ Cv |
|z|v < rv} be a small ball centered at the origin on which each of the logarithmic
functions logφi,v is an analytic isometry (for 1 ≤ i ≤ g). Since (x0,1, . . . , x0,g) is the
limit point for S0, there exists a d ∈ A and an infinite subsequence {φd+Pn}n≥0 ⊂ S0
(with Pn = 0 if and only if n = 0), such that for each n ≥ 0, we have
(3.4.1)
∣∣(φi)d+Pn (xi)− x0,i
for each 1 ≤ i ≤ g.
We will show that there exists an algebraic group Y0, independent of S0 and in-
variant under φ, such that φd(x1, . . . , xg) + Y0 is a subvariety of V containing
φd+Pn(x1, . . . , xg) for all Pn. Thus the submodule Γ0 := Y0(K) ∩ Γ will satisfy the
hypothesis of Lemma 3.1 for the infinite subset V (K) ∩ Γ ⊂ Γ; this will yield the
conclusion of Theorem 2.5.
Using (3.4.1) for n = 0 (we recall that P0 = 0), and then for arbitrary n, we see
(3.4.2)
∣∣(φi)Pn (xi)
for each 1 ≤ i ≤ g.
Hence logφi,v is well-defined at (φi)Pn (xi) for each i ∈ {1, . . . , g} and for each n ≥ 1.
Moreover, the fact that
(φi)Pn+d (xi)
converges to a point in ov means that(
(φi)Pn (xi)
converges to a point which is contained in Bv (see (3.4.2)).
Without loss of generality, we may assume
(3.4.3) | logφ1,v
(φ1)P1 (x1)
| logφi,v
(φi)P1 (xi)
Using the result of Corollary 3.3, we conclude that for each i ∈ {2, . . . , g}, the
following fraction is independent of n and of the sequence {Pn}n:
(3.4.4) λi :=
logφi,v
(φi)Pn (xi)
logφ1,v
(φ1)Pn (x1)
Note that since x1 is not a torsion point for φ1, the denominator of λi (3.4.4) is
nonzero. Because of equation (3.4.3), we may conclude that |λi|v ≤ 1 for each i.
The fact that λi is independent of the sequence {Pn}n will be used later to show
that the φ-submodule Γ0 that we construct is independent of the sequence {Pn}n.
For each n ≥ 1 and each 2 ≤ i ≤ g, we have
(3.4.5) logφi,v
(φi)Pn (xi)
= λi · logφ1,v
(φ1)Pn (x1)
For each i, applying the exponential function expφi,v to both sides of (3.4.5) yields
(3.4.6) (φi)Pn (xi) = expφi,v
λi · logφ1,v
(φ1)Pn (x1)
Since φd+Pn (x1, . . . , xg) ∈ V (K), for each j ∈ {1, . . . , ℓ} we have
(3.4.7) fj (φd+Pn(x1, . . . , xg)) = 0 for each n.
For each j ∈ {1, . . . , ℓ} we let fd,j ∈ K[X1, . . . , Xg] be defined by
(3.4.8) fd,j (X1, . . . , Xg) := fj (φd(x1, . . . , xg) + (X1, . . . , Xg)) .
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP9
We let Vd ⊂ G
a be the affine subvariety defined by the equations
fd,j(X1, . . . , Xg) = 0 for each j ∈ {1, . . . , ℓ}.
Using (3.4.7) and (3.4.8), we see that for each j ∈ {1, . . . , ℓ} we have
(3.4.9) fd,j (φPn(x1, . . . , xg)) = 0
for each n, and so,
(3.4.10) φPn(x1, . . . , xg) ∈ Vd(K).
For each j ∈ {1, . . . , ℓ}, we let Fd,j(u) be the analytic function defined on Bv by
Fd,j(u) := fd,j
u, expφ2,v
λ2 logφ1,v(u)
, . . . , expφg ,v
λg logφ1,v(u)
We note, because of (3.4.3), and the fact that logφ1,v is an analytic isometry on
Bv, that for each u ∈ Bv, we have
(3.4.11) |λi · logφ1,v(u)|v = |λi|v · | logφ1,v(u)|v ≤ |u|v < rv.
Equation (3.4.11) shows that λi · logφ1,v(u) ∈ Bv, and so, expφi,v
λi · logφ1,v(u)
well-defined.
Using (3.4.6) and (3.4.9) we obtain that for every n ≥ 1, we have
(3.4.12) Fd,j
(φ1)Pn (x1)
(φ1)Pn (x1)
is a sequence of zeros for the analytic function Fd,j which
has an accumulation point in Bv. Lemma 3.4 then implies that Fd,j = 0, and so,
for each j ∈ {1, . . . , ℓ}, we have
(3.4.13) fd,j
u, expφ2,v
λ2 logφ1,v(u)
, . . . , expφg,v
λg logφ1,v(u)
For each u ∈ Bv, we let
Zu :=
u, expφ2,v
λ2 logφ1,v(u)
, . . . , expφg,v
λg logφ1,v(u)
∈ Gga(Cv).
Then (3.4.13) implies that
(3.4.14) Zu ∈ Vd for each u ∈ Bv.
Let Y0 be the Zariski closure of {Zu}u∈Bv . Then Y0 ⊂ Vd. Note that Y0 is inde-
pendent of the sequence {Pn}n (because the λi are independent of the sequence
{Pn}n, according to Corollary 3.3).
We claim that for each u ∈ Bv and for each P ∈ A, we have
(3.4.15) φP (Zu) = Z(φ1)P (u).
Note that for each u ∈ Bv, then also (φ1)P (u) ∈ Bv for each P ∈ A, because each
coefficient of φ1 is a v-adic integer. To see that (3.4.15) holds, we use (2.1.2), which
implies that for each i ∈ {2, . . . , g} we have
expφi,v
λi logφ1,v ((φ1)P (u))
= expφi,v
λi · P · logφ1,v(u)
= expφi,v
P · λi logφ1,v(u)
= (φi)P
expφi,v
λi logφ1,v(u)
Hence, (3.4.15) holds, and so, Y0 is invariant under φ. Furthermore, since all of the
expφi,v and logφi,v are additive functions, we have Zu1+u2 = Zu1 + Zu2 for every
u1, u2 ∈ Bv. Hence Y0 is an algebraic group, which is also a φ-submodule of G
Moreover, Y0 is defined independently of Γ.
10 D. GHIOCA AND T. J. TUCKER
Let Γ0 := Y0(K)∩Γ. Because Y0 is invariant under φ, then Γ0 is a submodule of
Γ. Because Y0 ⊂ Vd, it follows that the translate φd(x1, . . . , xg)+Y0 is a subvariety
of V which contains all {φd+Pn(x1, . . . , xg)}n. In particular, the (infinite) translate
C0 of Γ0 by φd(x1, . . . , xg) is contained in V (K)∩Γ. Hence, every infinite sequence
of points in V (K)∩Γ has a nontrivial intersection with a coset C0 of (the nontrivial
φ-submodule) Γ0, and moreover, C0 ⊂ V (K)∩Γ. Applying Lemma 3.1 thus finishes
the proof of Theorem 2.5. �
In the course of our proof of Theorem 2.5 we also proved the following statement.
Theorem 3.5. Let Γ be an infinite cyclic φ-submodule of Gga. Then there exists
an infinite φ-submodule Γ0 ⊂ Γ such that for every affine subvariety V ⊂ G
a, if
V (K) ∩ Γ is infinite, then V (K) ∩ Γ contains a coset of Γ0.
Proof. Let v be a place of good reduction for φ; in addition, we assume the points
in Γ are v-adic integers. Suppose that V (K) ∩ Γ is infinite. As shown in the proof
of Theorem 2.5, there exists a positive dimensional algebraic group Y0, invariant
under φ, and depending only on Γ and v (but not on V ), such that a translate of Y0
by a point in Γ lies in V . Moreover, Γ0 := Y0(K) ∩ Γ is infinite. Hence Γ0 satisfies
the conclusion of Theorem 3.5. �
4. Further extensions
We continue with the notation from Section 3: φ1, . . . , φg are Drinfeld modules.
As usual, we denote by φ the action of (φ1, . . . , φg) on G
a. First we prove the
following consequence of Theorem 2.5.
Theorem 4.1. Let V ⊂ Gga be an affine subvariety defined over K. Let Γ ⊂ G
be a finitely generated φ-submodule of rank 1. Then V (K) ∩ Γ is a finite union of
cosets of φ-submodules of Γ. In particular, if V is an irreducible curve which is not
a translate of an algebraic φ-submodule, then V (K) ∩ Γ is finite.
Proof. Since A = Fq[t] is a principal ideal domain, Γ is the direct sum of its finite
torsion submodule Γtor and a free submodule Γ1, which is cyclic because Γ has rank
1. Therefore
γ∈Γtor
γ + Γ1,
and so,
V (K) ∩ Γ =
γ∈Γtor
V (K) ∩ (γ + Γ1) =
γ∈Γtor
(γ + (−γ + V (K)) ∩ Γ1) .
Using the fact Γtor is finite and applying Theorem 2.5 to each intersection (−γ + V (K))∩
Γ1 thus completes our proof. �
We use the ideas from [Ghi06b] to describe the intersection of a curve C with a
φ-module of rank 1. So, let (x1, . . . , xg) ∈ G
a(K), let Γ be the cyclic φ-submodule of
a(K) generated by (x1, . . . , xg), and let Γ be the φ-submodule of rank 1, containing
all (z1, . . . , zg) ∈ G
a(K) for which there exists a nonzero polynomial P such that
φP (z1, . . . , zg) ∈ Γ.
Since all polynomials φP (for P ∈ A) are separable, we have Γ ⊂ G
sep).
A DYNAMICAL VERSION OF THE MORDELL-LANG CONJECTURE FOR THE ADDITIVE GROUP11
With the notation above, we prove the following result; this may be viewed
as a Drinfeld module analog of McQuillan’s result on semiabelian varieties (see
[McQ95]), which had been conjectured by Lang.
Theorem 4.2. Let C ⊂ Gga be an affine curve defined over K. Then C(K) ∩ Γ is
a finite union of cosets of φ-submodules of Γ.
Before proceeding to the proof of Theorem 4.2 we first prove two facts which will
be used later. The first fact is an immediate consequence of Theorem 1 of [Sca02]
(the Denis-Manin-Mumford conjecture for Drinfeld modules), which we state below.
Theorem 4.3 (Scanlon). Let V ⊂ Gga be an affine variety defined over K. Then
there exist algebraic φ-submodules B1, . . . , Bℓ of G
a and elements γ1, . . . , γℓ of φtor
such that
V (K) ∩ φtor =
γi +Bi(K)
∩ φtor.
Moreover, in Remark 19 from [Sca02], Scanlon notes that his proof of the Denis-
Manin-Mumford conjecture yields a uniform bound on the degree of the Zariski
closure of V (K)∩φtor, depending only on φ, g, and the degree of V . In particular,
one obtains the following uniform statement for translates of curves.
Fact 4.4. Let C ⊂ Gga be an irreducible curve which is not a translate of an
algebraic φ-module of Gga. Then there exists a positive integer N such that for
every y ∈ Gga(K), the set
y + C(K)
∩ φtor has at most N elements.
Proof. The curve C contains no translate of a positive dimensional algebraic φ-
submodule of Gga, so for every y ∈ G
a(K), the algebraic φ-modules Bi appear-
ing in the intersection
y + C(K)
∩ φtor are all trivial. In particular, the set(
y + C(K)
∩φtor is finite. Thus, using the uniformity obtained by Scanlon for his
Manin-Mumford theorem, we conclude that the cardinality of
y + C(K)
∩ φtor is
uniformly bounded above by some positive integer N . �
We will also use the following fact in the proof of our Theorem 4.2.
Fact 4.5. Let φ : A→ K{τ} be a Drinfeld module. Then for every positive integer
D, there exist finitely many torsion points y of φ such that [K(y) : K] ≤ D.
Proof. If y ∈ φtor, then the canonical height ĥ(y) of y (as defined in [Den92b])
equals 0. Also, as shown in [Den92b], the difference between the canonical height
and the usual Weil height is uniformly bounded on K. Then Fact 4.5 follows by
noting that there are finitely many points of bounded Weil height and bounded
degree over the field K (using Northcott’s theorem applied to the global function
field K). �
We are now ready to prove Theorem 4.2.
Proof of Theorem 4.2. Arguing as in the proof of Theorem 2.5, it suffices to show
that if C is an irreducible affine curve (embedded in Gga), then C(K)∩ Γ is infinite
only if C is a translate of an algebraic φ-submodule (because any translate of an
algebraic φ-module intersects Γ in a coset of some φ-submodule of Γ). Therefore,
from now on, we assume C is irreducible, that C(K) ∩ Γ is infinite, and that C is
not a translate of an algebraic φ-submodule. We will derive a contradiction.
12 D. GHIOCA AND T. J. TUCKER
Let z ∈ C(K) ∩ Γ. For each field automorphism σ : Ksep → Ksep that restricts
to the identity on K, we have zσ ∈ C (Ksep) (because C is defined over K). By
the definition of Γ, there exists a nonzero polynomial P ∈ A such that φP (z) ∈ Γ.
Since φP has coefficients in K, we obtain
φP (z
σ) = (φP (z))
= φP (z).
The last equality follows from the fact that φP (z) ∈ Γ ⊂ G
a(K). We conclude that
φP (z
σ − z) = 0, and, thus, we have
Tz,σ := z
σ − z ∈ φtor.
Moreover, Tz,σ ∈ (−z+C(K))∩φtor (because z
σ ∈ C). Using Fact 4.4 we conclude
that for each fixed z ∈ C(K)∩Γ, the set {Tz,σ}σ has cardinality bounded above by
some number N (independent of z). In particular, this implies that z has finitely
many Galois conjugates, so [K(z) : K] ≤ N . Similarly we have [K(zσ) : K] ≤ N ;
thus, we may conclude that
(4.5.1) [K (Tz,σ) : K] ≤ [K(z, z
σ) : K] ≤ N2.
As shown by Fact 4.5, there exists a finite set of torsion points w for which [K(w) :
K] ≤ N2. Hence, recalling that N is independent of z, we see that the set
(4.5.2) H := {Tz,σ} z∈C(K)∩Γ
σ:Ksep→Ksep
is finite.
Now, since H is a finite set of torsion points, there must exist a nonzero poly-
nomial Q ∈ A such that φQ(H) = {0}. Therefore, φQ(z
σ − z) = 0 for each
z ∈ C(K)∩Γ and each automorphism σ. Hence φQ(z)
σ = φQ(z) for each σ. Thus,
we have
(4.5.3) φQ(z) ∈ G
a(K) for every z ∈ C(K) ∩ Γ.
Let Γ1 := Γ ∩ G
a(K). Since Γ is a finite rank φ-module and G
a(K) is a tame
module (i.e. every finite rank submodule is finitely generated; see [Poo95] for a
proof of this result), it follows that Γ1 is finitely generated. Let Γ2 be the finitely
generated φ-submodule of Γ generated by all points z ∈ Γ such that φQ(z) ∈
Γ1. More precisely, if w1, . . . , wℓ generate the φ-submodule Γ1, then for each i ∈
{1, . . . , ℓ}, we find all the finitely many zi such that φQ(zi) = wi. Then this finite set
of all zi generate the φ-submodule Γ2. Thus Γ2 is a finitely generated φ-submodule,
and moreover, using equation (4.5.3), we obtain C(K) ∩ Γ = C(K) ∩ Γ2. Since Γ2
is a finitely generated φ-submodule of rank 1 (because Γ2 ⊂ Γ and Γ has rank 1),
Theorem 4.1 finishes the proof of Theorem 4.2. �
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E-mail address: dghioca@math.mcmaster.ca
Dragos Ghioca, Department of Mathematics, McMaster University, 1280 Main Street
West, Hamilton, Ontario, Canada L8S 4K1,
E-mail address: ttucker@math.rochester.edu
Thomas Tucker, Department of Mathematics, Hylan Building, University of Rochester,
Rochester, NY 14627
	1. Introduction
	2. Notation and statement of our main result
	2.1. Drinfeld modules
	2.2. Valuations
	2.3. Logarithms and exponentials associated to a Drinfeld module
	2.4. Integrality and reduction
	2.5. The Denis-Mordell-Lang conjecture
	3. Proofs of our main results
	4. Further extensions
	References
