Some examples toward a Manin-Mumford conjecture for T-modules

0.2 MB·English

by  Luca Demangos

Some examples toward a Manin-Mumford conjecture
T
for abelian uniformizable modules
−
Luca Demangos
5 Former address: Laboratoire Paul Painlev´e, USTL,
1
Batiment M2 Cit´e Scientifique,
0
2 59655 Villeneuve d’Ascq C´edex

n Current address: Instituto de Matem´aticas – Unidad Cuernavaca,
a
Universidad Nacional Autonoma de M´exico,
J
Av. Universidad S/N, C.P. 62210
2
2 Cuernavaca, Morelos, ME´XICO

e-mail: [email protected]
]
T
N January 23, 2015
.
h
t
a The aim of this work is to present a possible adaptation of the Manin-Mumford
m
conjecture to the T modules, a mathematical object which has been introduced in the
[ 1980’s by G. Ander−son as the natural analogue of the abelian varieties in the context

1 of modules over rings which are contained in positive characteristic function fields. We
v proposethenageneralisationofsuchanadaptedconjecturetoamodifiedgeneralversion
8 ofMordell-LangconjectureforT moduleswhichmightcorrecttheoneproposedforthe
0 −
first time by L. Denis in [Den2] but no longer compatible with the present results.
4
5
0 We will remind in our first preliminary section the formulation of the Mordell-Lang and
. the Manin-Mumford conjectures and the definition of T modules and sub-T modules,
1 − −
listing nothingmorethanthe basicdefinitions andpropertieswhicharestrictlyessential
0
5 to us in order to state and prove our theorems. All the detailed informations the reader
1 may be interested in might be found in [Goss], chapter 5. We then trace a brief history
: of the recent studies, mainly due to the work of L. Denis, D. Ghioca and T. Scanlon,
v
i about an adaptation of such conjectures to a very special case of T module: the power
X −
of a Drinfeld module. In the second section we will present some examples which prove
r thatanaifadaptationofthe Manin-Mumfordconjectureto the T modulesingeneralis
a −
no longer true, even extending the notion of sub-T module in a way which encode the
−
deeper structure of the involved rings. In one of these examples we will take the case
of a product of different Drinfeld modules, showing that such an intuitive adaptation
of Manin-Mumford conjecture is no longer true even in this appearently simple case of
study. We will then propose a reasonablycorrectedformulation,whose proofis the final
aim of our research.
1 Preliminaries
G. Faltings proved the Mordell-Lang conjecture (see [F]) in the following version.
1
Theorem 1. Let be an abelian variety defined over a number field. Let X be a closed
A
subvariety of and Γ a finitely generated subgroup of the group of C points on .
A ⊂A − A
Then X Γ is a finite union of cosets of subgroups of Γ.
∩
Such a result contains clear similarities with the Manin-Mumford conjecture. The
classic version of this conjecture1, which treats the situation of an elliptic curve embed-
ded in its Jacobianvariety has been proved(see [R1]) and later generalized(see [R2]) to
the following statement.
We call torsion subvariety of an abelian variety the translate of some abelian sub-
A
variety of by a torsion point.
A
Theorem 2. Let X be an algebraic subvariety of an abelian variety defined over a
A
number field. If X contains a Zariski-dense set of torsion points, then X is a torsion
subvariety of .
A
An even stronger evidence of the link existing between these two results is given by
the following extended formulation of G. Faltings’ result, proved by M. Hindry and G.
Faltings (see [H]):
Theorem 3. Let X be a closed subvariety of an abelian variety defined over a number
A
field. Let Γ be a finitely generated subgroup of (Q) and let:
A
Γ:= x (Q), m>0,[m]x Γ .
{ ∈A ∃ ∈ }
Therefore, X(Q) Γ is a finite union of cosets of subgroups of Γ.
∩
In fact, the Manin-Mumford conjecture (proved before such a result was known)
follows if one chooses Γ=0 in Theorem 3. On the other hand, the study of the Manin-
Mumford conjecture still produces relevant ideas, which might be applied to different
aspects of this kind of problems, as we hope to do.
The weakversionofthe M. Raynaud’stheoremprovedin[PZ]usingnew ideas involving
Diophantine Geometry is for example a consequence of Theorem 2:
Theorem 4. Let X be an algebraic subvariety of an abelian variety defined over a
A
number field. If X does not contain any torsion subvariety of of dimension >0, then
A
X contains at most finitely many torsion points.
The techniques involved in proving Theorem 4 appear to particularly adapt to the
studyofaspecificclassofobjects,calledT modules,whichareaffinealgebraicvarieties
−
over function fields provided with a module structure, which we briefly introduce now.
Most of definitions and properties we list below are taken from [Goss], chapter 5. We
send the reader to such a reference for more details.
We call A:=F [T]the ring of polynomials with coefficient in the finite field F , where q
q q
is a power of the prime number p, k :=Frac(A) and is the completion of an algebraic
C
closureofthecompletionofkwithrespecttotheplaceatinfinity. IfK isafieldandn,m
aretwopositiveintegers,thenotationKn,m willindicatetheringofmatriceswithentries
in K, having n lines and m columns. We use τ to indicate the Frobenius automorphism
in the following form:
τ :z zq, z .
7→ ∀ ∈C
1Historically,thefirstversionoftheManin-MumfordconjecturehasbeenprovedbyM.Laurentover
Gnm,see[Lau]
2
Definition 1. A T-module = (Gm,Φ) of degree d and dimension m defined on the
A a
field k is the algebraic group Gm having the structure of A-module given by the
F ⊂ a e
F -algebras homomorphism:
q
Φ:A m,m τ
→F { }
e
d
T a (T)τi;
i
7→
Xi=0
where a (also called dΦ(T), the differentialof Φ(T), which can be seen as a linear map
0
acting on m) is of the form:
C
a =TI +N;
0 m
where N is a nilpotent matrix, and ae= 0. This shows moreover that Φ is injective, as
d 6
in the case of a Drinfeld module (which is just a T-module having dimension 1).
Definition2. AT moduleofdimension 1is calledaDrinfeld module. Aparticularly
−
interesting example of a Drinfeld module is the typical degree 1 case, called the Carlitz
module:
C =(G ,Φ);
a
where:
Φ(T)(τ):=T +τ.
Definition 3. The set of torsion points of the T module is:
− A
:= x , a(T) A 0 ,Φ(a(T))(x)=0 .
tors.
A { ∈A ∃ ∈ \{ } }
Definition 4. A sub-T module of a T module is a reduced connected algebraic
− B − A
subgroup of ( ,+) such that Φ(T)( ) .
A B ⊂B
Weremarkthatasub-T moduleofaT moduleisnotingeneralaT module. This
− − −
can be seen in the following example.
Proposition 1. Let us consider q =2. Let D =(G ,Φ ) and D =(G ,Φ ) be rank-1
1 a 1 2 a 2
Drinfeld modules such that:
Φ (T)(τ)=T +Tτ and Φ (T)(τ)=T +T2τ.
1 2
Therefore, the algebraic subgroup:
:= (x,y) G2,y =x+x2 ;
B { ∈ a }
is a sub-T module of D D .
1 2
− ×
Proof. We remark that is reduced and connected. Now, for each (x,y) we have
B ∈ B
that:
Φ (T)(y)=Ty+T2y2 =T(x+x2)+T2(x2+x4)=Φ (T)(x)+Φ (T)(x)2;
2 1 1
so that:
(Φ (T)(x),Φ (T)(y)) .
1 2
∈B
This proves Proposition 1. As is not a power of G it constitutes an example of a
a
B
sub-T module which is not a T module.
− −
3
Definition 5. Let =(Gm,Φ) be a T moduleof dimension m. We say that it is sim-
A a −
ple if it does not admit any non-trivial sub-T module (in other words, a sub-T module
− −
different from and 0).
A
Definition 6. Let a(T) A F . We call a reduced, connected algebraic sub-group of
q
∈ \ B
the T module =(G ,Φ) a sub-a(T) module if Φ(a(T))( ) .
a
− A − B ⊂B
We call from now the dimension of a sub-T module of , the dimension of
− B A B
as an algebraic variety over . We remark that the dimension of any non-trivial sub-
C
T module < is strictly less than the dimension of .
− B A A
Let be any non-trivial sub-T module of . We call for the moment torsion set
B − A
a subset of under the form:
A
x+ , x .
tors.
B ∈A
We will extend later such a definition to the more general one of torsion subvariety
(see Definition 11) in order to adapt the statement of Theorem 2 and Theorem 4 to the
T module context.
−
Definition7. By calling k thefield generatedover k by theentries of thecoefficient
F ⊂
matrices of Φ(T) (which are in k), the rank of a T module is the rank over [T]
− A F
of the [T] module Hom ( ,G ) (the F additive group homomorphisms from to
F a q
F − A − A
G ). We say that a T module is abelian if Hom ( ,G ) has finite rank.
a F a
− A A
Aneasy example ofanabelianT moduleis givenby a product ofDrinfeld modules.
−
ItisimmediatetoseeindeedthattherankofaDrinfeldmodulecoincideswithitsdegree.
By taking D ,...,D finitely many Drinfeld modules of rank, respectively, d ,...,d and
1 r 1 r
defined over the field k one sees that:
F ⊂
r
Hom (D D ,G ) Hom (D ,G );
F 1 r a F i a
×···× ≃
Mi=1
so that the rank of such a product of Drinfeld modules is r d . This example shows
i=1 i
also that the rank of a T module is not in general the dePgree of this one. Indeed, the
−
degree of D D is max d .
1 r i=1,...,r i
×···× { }
We provide an example of a nonabelian T module in Proposition 7.
−
One can prove (cfr. [Goss], Theorem 5.4.10) that if is abelian, Hom ( ,G ) is also
F a
A A
free as a [T] module.
F −
Let be a T module of dimension m. We call Lie( ) m the tangent space of
A − A ≃ C
at 0.
A
Definition 8. Let be an abelian T module. The exponential function of is the
A − A
unique2 morphism:
e:Lie( ) ;
A →A
such that, for each z Lie( ), we have that:
∈ A
e(dΦ(T)z)=Φ(T)(e(z));
as described in [Goss], Definition 5.9.7.
2See[Goss],Proposition5.9.2.
4
It is known (see [Goss], section 5) that such a morphism is F linear, a local home-
q
−
omorphism and (see Definition 9) entire too.
F−
If is abelian and we write:
A
Λ:=Ker(e);
this kernel is an A lattice inside Lie( ) and its A rank is less or equal the rank of ,
− A − A
cfr. [Goss], Lemma 5.9.12.
The exponential map associated to =(Gm,Φ) projects therefore m in Gm.
A a C a
Proposition 2. Let be a sub-a(T) module of the abelian T module for a given
B − − A
a(T) A F , and Lie( ) its tangent space, contained as a subspace into Lie( ).
q
∈ \ B C− A
Therefore:
e(Lie( )) .
B ⊆B
Proof. By calling:
t:=a(T);
andseeing as a sub-t module of the t module , the statementfollows directly from
B − − A
[Goss] Remark 5.9.8.
As e(Lie( )) it follows that:
B ⊆B
Lie( ) e−1( ).
B ⊆ B
Lemma 1. Let ρ(Λ ) be the A rank of the lattice associated to as the kernel of the
A
− A
exponential function e : Lie( ) , and let ρ( ) be the rank of . The following
A → A A A
properties are equivalent for an abelian T module =(Gm,Φ).
− A a
1. ρ(Λ )=ρ( );
A
A
2. The exponential function e:Lie( ) is surjective.
A →A
Proof. See [Goss], Theorem 5.9.14.
Definition9. Let =(Gm,Φ)beanabelian T module. Ifitrespectsthetwoequivalent
A a −
conditions of Lemma 1 it is called uniformizable.
We remark that the proof of Lemma 1 in [Goss], Theorem 5.9.14 also applies to sub-
T modules of a given abelian T module by the slightly extended notion of T module
− − −
consideredinsuchawork. WecanthereforeeasilyextendDefinition9tosub-T modules
−
too. More specifically, we say that a sub-T module of a given abelian T module
− B − A
is uniformizable if the induced exponential map:
e:Lie( ) ;
B →B
is surjective.
Remark 1. The sub-T modulesof an abelian, uniformizable T moduleare abelian and
− −
uniformizable.
Proof. See [D], Remarque 2.1.8 and Remarque 2.1.19.
5
We also remark (see [D], Lemme 2.1.37) that if is an abelian, uniformizable
A
T moduleofrankd,thenLie( )canbewrittenasadirectsumofak vectorspaceof
∞
− A −
dimension d, into which the associated lattice Λ=Ker(e ) is cocompact (this is called
A
the torsion part of Lie( )), and a k vector space of infinite dimension (the free
∞
A −
part of Lie( )). In other words, by calling ω ,...,ω a generating set of periods of:
1 d
A { }
Λ=<ω ,...,ω > ;
1 d A
we have that:
d
Lie( ) ( k ω ) Free ;
A ≃ ∞ i ⊕ k∞
Mi=1
where Free is an infinite-dimensional k vector space. Up to the automorphism φ
k∞ ∞−
of Lie( ) which sends ω ,...,ω to the canonical basis of kd and leaves unchanged
A { 1 d} ∞
the free part of Lie( ), so that it moves Λ to Ad, we have then that:
A
Lie( ) (k /A)d Free ;
A ≃ ∞ ⊕ k∞
sothateputinbijectiontheset ofthetorsionpointsof withtheset(k/A)d 0.
tors.
A A ×
By the discussion above it is now easy to see that the torsion points of with respect
B
to its structure of sub-a(T) module of (in other words,the a(T) torsionpoints of
− A − A
which belong to ) shall correspond via the exponential map e associated to to the
B A
F (a(T)) rational points of φ(Lie( )) ((k /A)d 0).
q ∞
− B ∩ ×
As one can rapidly check a T module which is abelian and uniformizable can be in-
−
terpretedthereforeby analyzingits associatedfinite-ranklattice in ananalogousfashion
as for an abelian variety. The new techniques recently introduced by U. Zannier and
J. Pila in [PZ] which provided an elegant alternative proof of Theorem 4 involving Dio-
phantineGeometry,arespecificallybasedonsuchaninterpretationandthisisthereason
why we focus on this particular class of T modules.
−
2 A new conjecture
In this section we will study the possibility to adapt the Manin-Mumford conjecture to
the T modules as they have been previously introduced.
−
Results about a connection between the Mordell-Lang and the Manin-Mumford conjec-
turesforT moduleshavebeenworkedoutfirstlybyL.Denis(see[Den2]),whoproposed
−
the following unified conjecture.
Statement 1. Let =(Gm,Φ) be a T module of rank d and dimension m>1. Let Γ
A a −
be a finitely generated submodule of (k) and let X be a closed subvariety of Gm. Let:
A a
Γ:= x (k), a(T) A 0 ,Φ(a(T))(x) Γ .
{ ∈A ∃ ∈ \{ } ∈ }
Therefore, there exist finitely many translates of sub-T modules of in the form γ +
− A 1
,...,γ + such that:
B1 s Bs
X Γ= (γ + Γ).
∩ i Bi∩
1≤[i≤s
6
As we have seen in the previous section, such a statement would imply an intuitive
adaptationofMordell-LangandManin-MumfordconjecturestotheT modulescontext.
−
We will see anyway in the present section that Statement 1 is unfortunately false.
L. Denis proved on the other hand in the same paper, under some technical restric-
tion on the hypotheses, an adapted formulation of the Manin-Mumford conjecture for
finite powers of Drinfeld modules (see [Den2], Th´eor`eme 1). Such a result has been
subsequently extended into a completely analogous formulation of Theorem 2 for finite
powers of Drinfeld modules by T. Scanlon in [Sc1], removing so the restriction on the
hypotheses in L. Denis’ result summarized above. T. Scanlon’s result is the following
one.
Theorem 5. Let = Dm := (Gm,Φ ) be a power of some given Drinfeld module
A a m
D = (G ,Φ) defined over a field , so that Φ (T) = Φ(T)I (in other words, Φ acts
a m m
F
diagonally on Gm). Let X be an irreducible algebraic sub-variety of . If X( ) is
a A F tors.
Zariski-dense in X, then X is the translate of some sub-T module of by a torsion
− A
point.
T. Scanlon provedalso an adapted versionof the Mordell-Lang conjecture to powers
of Drinfeld modules of finite characteristic3 (see [Sc2]), analogous to Theorem 1:
Theorem6. LetDbeaDrinfeldmoduleoffinitecharacteristicandmodulartrascendence
degree4 at least 1. Let Γ be a finitely generated submodule of Dm(k) for some m>1 and
let X be a closed k subvariety of Gm. Then X(k) Γ is a finite union of translates of
− a ∩
subgroups of Γ.
As we will highlight better later such a specific case of T module (a finite power of
−
a Drinfeld module) presents particularlystrong properties whichmake it one ofthe best
casesofstudy. AT moduleinsucha formis moreovereasilyabeliananduniformizable
−
and this would appear to be a first encouraging step for our project to prove a similar
result, using the new techniques introduced in [PZ] for an abelian and uniformizable
T module . We would like therefore to prove a weaker result, analogous to Theorem
− A
4,butforageneralabeliananduniformizableT module,asinthefollowingformulation.
−
Formulation 1. LetX bean algebraic subvarietyof anabelian uniformizable T module
−
defined over k. If X does not contain any torsion set of , then X contains at most
A A
finitely many torsion points.
Such a statement turns out to be unfortunately false. This is an immediate conse-
quence of the following proposition.
Proposition 3. There exists an abelian uniformizable T module and a non-trivial
− A
algebraic subvariety of which contains infinitely many torsion points but no torsion
A
sets of .
A
Proof. We consider the T module of dimension 2 defined by the tensor power C⊗2 =
−
(G2,Φ)oftheCarlitzmoduleC,introducedbyG.AndersonandD.Thakurin[AT]. We
a
suppose that q =2. Such a T module is then under the following form:
−
T 1 0 0
Φ(T)(τ)= + τ.
(cid:18) 0 T (cid:19) (cid:18) 1 0 (cid:19)
3See[Goss],chapter 4.
4Seethereferredpaperforthecompletedefinition.
7
X
We can then show that for each C⊗2:
(cid:18) Y (cid:19)∈
X T2X +X2
Φ(T2) = .
(cid:18) Y (cid:19) (cid:18) T2Y +(T +T2)X2+Y2 (cid:19)
The algebraic sub-group 0 G of C⊗2 is then easily a sub-T2 module, but not a sub-
a
× −
T module. Indeed, any tensor power C⊗m, for any power m N 0 , of the Carlitz
− ∈ \{ }
module C is always a simple, abelian (see [Goss], Corollary 5.9.38) and uniformizable
T module(see[Yu],Proposition1.2),butone canprovethatitpossessessometimes(as
−
in the present case) non-trivial sub-Tj modules for some j depending on m and q. By
−
choosing 0 G as an algebraicsubvariety of C⊗2, we now see that it contains infinitely
a
×
many torsion points, which correspond, by Proposition 2 and discussion subsequent to
Remark 1, to the F (T2) rational points of the torsion part of:
2
−
Lie(0×Ga)/(Ker(eC⊗2)∩Lie(0×Ga)).
As C⊗2 is simple as a T module, 0 G cannot contain on the other hand any torsion
a
− ×
set of positive dimension, which finally prove our statement.
Proposition 3 put in light how the structure of the generic polynomial ring F [T],
q
whichcontainsinfinitely manysubringsisomorphictoF [T]itself, determinesthefailure
q
of any attempt to adapt the Manin-Mumford conjecture (even in its weaker formulation
stated in Theorem 4) to abelian uniformizable T modules, while the same conjecture
−
is on the contrary true for abelian varieties over number fields because their algebraic
structure is that of a group, which is a Z module and Z do not contain as a ring non-
−
trivial subrings.
A similar phenomenon has been observed already studying Mordell-Lang conjecture for
powers of Drinfeld modules of finite characteristic. More precisely, given a finitely gen-
erated submodule Γ of Dm for some Drinfeld module D = (G ,Φ) and m > 1, if X is
a
an algebraic subvariety of Dm then X(k) Γ might not be stabilized by the action of
∩
Φ(T) but it can be invariant anyway under the action of Φ(Tn) for a suitable n>1. A
detailed example can be found in [Ghioca], Remark 4.11. This leads to a formulation of
Mordell-Lang conjecture for powers of some convenient cases of Drinfeld modules which
extends in some sense the notion of submodule of Dm to that of sub-Φ(Tn) module for
−
n N 0 depending on the chosen Φ, Γ and X (see [Ghioca], Theorem 4.6).
∈ \{ }
We extend therefore the class of algebraic sub-modules of in order to avoid counter-
A
examples produced, as we showed in the proof of Proposition 3, by the abundance of
subrings of F [T].
q
We start by giving the following definition.
Definition 10. Let =(Gm,Φ) be a general T module. Let B be a sub-ring of A. We
A a −
call sub-B module of any reduced connected algebraic sub-group of Gm such that:
− A B a
Φ(a(T))( ) , a(T) B.
B ⊆B ∀ ∈
We say that a sub-set of is a torsion subvariety if it is under the following form:
A
x+ ;
B
where x and there exists B a sub-ring of A such that is a sub-B module of
tors.
∈ A B −
.
A
8
The following result will allow us to ease the study of the sub-B modules of for
− A
each B subring of A, reducing all of them to sub-Tj modules for a convenient index
−
j N 0 only depending on .
∈ \{ } A
Theorem 7. Let =(Gm,Φ) be an abelian uniformizable T module of dimension m.
A a −
There exists then a number j( ) N 0 only depending on such that for each B
A ∈ \{ } A
sub-ring of A, every sub-B module of is a sub-Tj(A) module).
− A −
Proof. LetN bethenilpotentmatrixassociatedtothedifferentialdΦ(T)of introduced
A
inDefinition1. Letn( ) N 0 beitsorder. Let beasub-B moduleof forsome
A ∈ \{ } B − A
B subringofA. Itis thenasub-a(T) moduleforeacha(T) B F andinparticulara
q
− ∈ \
reducedconnectedalgebraicsub-groupofGm. In orderto provethe statementit will be
a
sufficienttoshowthatLie( )isstabilizedbytheactionofdΦ(Tj(A))forsomeconvenient
B
j( ) N 0 . ThereasonofthiseasilycomesfromProposition2. Wechoosetherefore:
A ∈ \{ }
j( )=pr(A);
A
by calling pr(A) the smallest power of p to be greater or equal to n( ) (in other words,
A
r( )=[log (n( ))]+1, where we meanby [log (n( ))] the integer partof log (n( ))).
A p A p A p A
It is easy to see then that:
dΦ(Tj(A))=Tj(A)I ;
m
which stabilizes every vector sub-space of Lie( ) over . So in particular, it stabilizes
A C
Lie( ) too.
B
We remark that we just proved the existence of such a number j( ) N 0 , but
A ∈ \{ }
wedidnotfoundactuallyitsminimalpossiblevalueinprinciple. Wecananywayremark
that such a value is 1 in the case where is a power of a Drinfeld module, as we will
A
show here below. This shows that for each subring B of A every sub-B module of is
− A
actually a sub-T module. This is precisely the reason why examples like the one pre-
−
sented in the proof of Proposition 3 would not work for some finite power of a Drinfeld
module, being so another confirmation of Scanlon’s result (Theorem 5).
We show now this property using the following Thiery’s Theorem (see [T]).
Theorem8. LetDm =(Gm,Φm)them thpowerofaDrinfeldmoduleD=(G ,Φ)and
a − a
let beitscoefficients field. Thereexiststherefore abijective correspondence betweenthe
F
family of all sub-T modules of Dm and the family of the vector sub-spaces of Lie(Dm)
−
which are End (Φ) rational. This correspondence is given by the exponential function:
F
−
e:Lie(Dm) Dm;
→
V e(V).
7→
Moreover,wehavethatthedimensionofanysub-T moduleofDm andoftheEnd (Φ) rational
F
− −
vector sub-space of Lie(Dm) corresponding to it are the same.
Proof. See [T], Theorem at page 33.
Proposition 4. Let Dm be as in Theorem 5. Then:
j(Dm)=1.
9
Proof. Let be a sub-B module of Dm for a non-trivial subring B of A. It is therefore
B −
a sub-a(T) module of for any a(T) B F . We assume that the Drinfeld module D
q
− A ∈ \
has rank d. We define:
T′ :=a(T);
and:
Ψ(T′):=Φ(a(T));
is therefore a T′ module which is the m th power of some F [T′] Drinfeld module.
q
B − − −
ByTheorem8 isthe zerolocusofs=m dim( )linearequationsunderthefollowing
B − B
form:
m
P (τ)X =0;
ij j
Xj=1
where P (τ) End (Ψ) for each i = 1,...,s and each j = 1,...,m. Now, it is well
ij F
∈
known that Φ(T) End (Ψ). Moreover, End (Ψ) is a commutative ring because D
F F
∈
has characteristic 0 (see [Goss], Definition 4.4.1). Therefore, one sees that is actually
B
a sub-T module of Dm as it is stabilized by the action of Φ(T).
−
If is absolutely simple (this means that it does not contain any non-trivial sub-
A
Tj module for each j N 0 ) we fix j( )=1.
− ∈ \{ } A
We also remark that this study of powers of some Drinfeld module provides an example
ofa class ofT modules suchthatfor eachi N 0 we havethat j( i)=j( ) and
− A ∈ \{ } A A
one may wonder if it is a general phenomenon.
This would lead us to a more general formulation of Manin-Mumford conjecture on
abelian and uniformizable T modules, where we propose to show that, up to a finite
−
number, the torsion points of could be shared in finitely many sub-Tj(A) modules.
A −
We would like then to state an analogue of Theorem 4 as in the following formulation.
Formulation 2. LetX bean algebraic subvarietyof anabelian uniformizable T module
−
. If X does not contain any torsion subvariety of of dimension >0, then X contains
A A
at most finitely many torsion points.
Such a new formulation is however false again and not yet sufficient to completely
exclude other counter-examples. The nice one which follows has been suggested by
Laurent Denis.
Proposition 5. There exists an algebraic subvariety X of an abelian uniformizable
T module which does not contain non-trivial torsion subvarieties of but it con-
− A A
tains infinitely many torsion points.
Proof. We consider the following case of a product of two non-isogeneousDrinfeld mod-
ules. We assume that q = 2. Let D = (G ,C) be the Carlitz module, so that
1 a
C(T)(τ)=T +τ, being τ the Frobenius automorphism over F . We define the Drinfeld
2
module D =(G ,C ) as follows:
2 a (2)
C (T)(τ):=T +(T1/2+T)τ +τ2.
(2)
One can see then that for each z we have that:
∈C
C (T)(√z)= C(T2)(z).
(2)
p
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