The Mordell-Lang Theorem for finitely generated subgroups of a semiabelian variety defined over a finite field

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by  Dragos Ghioca

THE MORDELL-LANG THEOREM FOR FINITELY GENERATED
SUBGROUPS OF A SEMIABELIAN VARIETY DEFINED OVER A
FINITE FIELD
DRAGOS GHIOCA
6
0
0 Abstract. We determine the structureofthe intersectionofafinitely generatedsubgroup
2
of a semiabelian variety G defined over a finite field with a closed subvariety X ⊂G.

n
a
J

4
1 1. Introduction

] Let G be a semiabelian variety defined over a finite field Fq. Let K be a regular field
T
extension of F . Let F be the corresponding Frobenius for F . Then F ∈ End(G).
N q q
Let X be a subvariety of G defined over K (in this paper, all subvarieties will be closed).
.
h In [3] and [4], Moosa and Scanlon discussed the intersection of the K-points of X with a
t
a finitely generated Z[F]-submodule Γ of G(K). They proved that the intersection is a finite
m
union of F-sets in Γ (see Definition 2.4). Our goal is to extend their result to the case when
[
Γ is a finitely generated subgroup of G(K) (not necessarily invariant under F).

1 In Section 2 we will state our main results, which include, besides the Mordell-Lang
v
statement for subgroups of semiabelian varieties defined over finite fields, also a similar
0
6 Mordell-Lang statement for Drinfeld modules defined over finite fields. The Mordell-Lang
3 Theorem for Drinfeld modules was also studied by the author in [1]. In Section 3 we will
1
prove our main theorem for semiabelian varieties, while in Section 4 we will show how the
0
6 Mordell-Lang statement for Drinfeld modules defined over finite fields can be deduced from
0
the results in [3]. We will conclude Section 4 with two counterexamples for two possible
/
h
extensions of our statement for Drinfeld modules towards results similar with the ones true
t
a for semiabelian varieties.
m
v: 2. Statement of our main results
i
X
Everywhere in this paper, Y represents the Zariski closure of the set Y.
r A central notion for the present paper is the notion of a Frobenius ring. This notion was
a
first introduced by Moosa and Scanlon (see Definition 2.1 in [4]). We extend their definition
to include also rings of finite characteristic.
Definition 2.1. Let R be a Dedekind domain with the property that for every nonzero
prime ideal p ⊂ R, R/p is a finite field. We call R[F] a Frobenius ring if the following
properties are satisfied:
(i) R[F] is a simple extension of R generated by a distinguished element F.
(ii) R[F] is a finite integral extension of R.
(iii) F is not a zero divisor in R[F].
(iv) The ideal F∞R[F] := FnR[F] is trivial.
n≥0
1
2000 AMS Subject ClassificatioTn: Primary 11G10; Secondary 11G09, 11G25
1
The classical example of a Frobenius ring associated to a semiabelian variety G defined
over the finitefield F is Z[F], where F isthe corresponding FrobeniusforF . This Frobenius
q q
ring is discussed in [3] and [4]. We will show later in this section thatA[F] is also a Frobenius
ring when F is the Frobenius on F and φ : A → F [F] is a Drinfeld module (in this case, A
q q
is a Dedekind domain of finite characteristic).
We define the notion of groupless F-sets contained in a module over a Frobenius ring.
Definition 2.2. Let R[F] be a Frobenius ring and let M be an R[F]-module. For a ∈ M
and δ ∈ N∗, we denote the Fδ-orbit of a by S(a;δ) := {Fδna | n ∈ N}. If a ,...,a ∈ M
1 k
and δ ,...,δ ∈ N∗, then we denote the sum of the Fδi-orbits of a by
1 k i
k
S(a ,...,a ;δ ,...,δ ) = { Fδinia | (n ,...,n ) ∈ Nk}.
1 k 1 k i 1 k
i=1
X
A set of the form b + S(a ,...,a ;δ ,...,δ ) with b,a ,...,a ∈ M is called a groupless
1 k 1 k 1 k
F-set based in M. We do allow in our definition of groupless F-sets k = 0, in which case,
the groupless F-set consists of the single point b. We denote by GF the set of all groupless
M
F-sets based in M. For every subgroup Γ ⊂ M, we denote by GF (Γ) the collection of
M
groupless F-sets contained in Γ and based in M. When M is clear from the context, we will
drop the index M from our notation.
Remark 2.3. Each groupless F-set O is based in a finitely generated Z[F]-module.
Definition 2.4. Let M be a module over a Frobenius ring R[F]. Let Γ ⊂ M be a subgroup.
A set of the form (C + H), where C ∈ GF (Γ) and H is a subgroup of Γ invariant under
M
F is called an F-set in Γ based in M. The collection of all such F-sets in Γ is denoted by
F (Γ). When M is clear from the context, we will drop the index M from our notation.
M
Let G be a semiabelian variety defined over F . Let F be the corresponding Frobenius
q
for F . Let K be a finitely generated regular extension of F . We fix an algebraic closure
q q
Kalg of K. Let Γ be a finitely generated subgroup of G(K). We denote by F(Γ) and GF(Γ)
the collection of F-sets and respectively, the collection of groupless F-sets in Γ based in
G(Kalg) (which is obviously a Z[F]-module). When we do not mention the Z[F]-submodule
containing the base points for the F-sets contained in Γ, then we will always understand
that the corresponding submodule is G(Kalg). The following theorem is our main result for
semiabelian varieties.
Theorem 2.5. Let G, K and Γ be as in the above paragraph. Let X be a K-subvariety of
G. Then X(K)∩Γ = r (C +H ), where (C +∆ ) ∈ F(Γ). Moreover, the subgroups ∆
i=1 i i i i i
from X(K)∩Γ are of the form G (K)∩Γ, where G is an algebraic subgroup of G defined
i i
S
over F .
q
Asmentioned inSection1,theresult ofourTheorem 2.5was establised in[3](see Theorem
7.8) and in [4] (see Theorem 2.1) for finitely generated Z[F]-modules Γ ⊂ G(K). Because
Z[F] isa finite extension ofZ, each finitely generated Z[F]-moduleis also a finitely generated
group (but not every finitely generated group is invariant under F).
We describe now the setting for our Drinfeld modules statements. We start by defining
Drinfeld modules over finite fields.
Let p bea prime number andlet q bea power ofp. Let C bea projective nonsingular curve
defined over F . We fix a closed point ∞ on C. Let A be the ring of F -valued functions on
q q
2
C, regular away from ∞. Then A is a Dedekind domain. Moreover, A is a finite extension
of F [t]. Hence, for every nonzero prime ideal p ⊂ A, A/p is a finite field.
q
Let F be the corresponding Frobenius on F . We call a Drinfeld module defined over F a
q q
ringhomomorphismφ : A → F [F]suchthatthereexistsa ∈ Aforwhichφ := φ(a) ∈/ F ·F0
q a q
(i.e. the degree of φ as a polynomial in F is positive). In general, for every a ∈ A, we write
a
φ to denote φ(a) ∈ F [F]. We note that this is not the most general definition for Drinfeld
a q
modules defined over finite fields (see Example 4.8).
For each field extension L of F , φ induces an action on G (L) by a∗x := φ (x) for every
q a a
x ∈ L and for every a ∈ A. For each g ≥ 1, we extend the action of A diagonally on Gg.
a
Clearly, for every a ∈ A, Fφ = φ F. This means F is an endomorphism of φ (see Section
a a
4 of Chapter 2 in [2]). We let A[F] ∈ End(φ) be the finite extension of A generated by F,
where we identified A with its image in F [F] through φ. Actually, A[F] is isomorphic to
q
F [F]. However, we keep the notation A[F] instead of F [F], when we talk about modules
q q
over this ring only to emphasize the Drinfeld module action given by A.
Lemma 2.6. The ring A[F] defined in the above paragraph is a Frobenius ring.
Proof. Because for some a ∈ A, φ is a polynomial in F of positive degree, we conclude
a
F is integral over A. Because F [F] is a domain, we conclude F is not a zero divisor.
q
Also, no nonzero element of A[F] is infinitely divisible by F because all elements of A[F]
are polynomials in F and so, no nonzero polynomial can be infinitely divisible by some
polynomial of positive degree. Therefore A[F] is a Frobenius ring. (cid:3)
Let K be a regular field extension of F . We fix an algebraic closure Kalg of K. Let Γ be
q
a finitely generated A[F]-submodule of Gg(K). We denote by F(Γ) and GF(Γ) the F-sets
a
and respectively, the groupless F-sets in Γ based in Gg(Kalg). When we do not mention the
a
A[F]-submodule containing the base points for the F-sets contained in Γ, we will always
understand that the corresponding submodule is Gg(Kalg). We will explain in Section 4 that
a
the following Mordell-Lang statement for Drinfeld modules defined over finite fields follows
along the same lines as Theorem 7.8 in [3].
Theorem 2.7. Let φ : A → F [F] be a Drinfeld module. Let K be a regular extension of F .
q q
Let g be a positive integer. Let Γ be a finitely generated A[F]-submodule of Gg(K) and let
a
X be an affine subvariety of Gg defined over K. Then X(K)∩Γ is a finite union of F-sets
a
in Γ.
3. The Mordell-Lang Theorem for semiabelian varieties defined over
finite fields
Proof of Theorem 2.5. We first observe that the subgroups ∆ from the intersection of X
i
with Γ are indeed of the form G (K)∩Γ for algebraic groups G defined over F . Otherwise,
i i q
we can always replace a subgroup ∆ appearing in the intersection X(K)∩Γ with its Zariski
i
closure G and then intersect with Γ (see also the proof of Lemma 7.4 in [3]). Because G
i i
is the Zariski closure of a subset of G(K), then G is defined over K. Because G is an
i i
algebraic subgroup of G, then G is defined over Falg. Because K is a regular extension of
i q
F , we conclude that G is defined over F = K ∩Falg.
q i q q
We will prove the main statement of Theorem 2.5 by induction on dim(X). Clearly, when
dim(X) = 0 the statement holds (the intersection is a finite collection of points in that case).
Assume the statement holds for dim(X) < d and we prove that it holds also for dim(X) = d.
3
We will use in our proof a number of reduction steps.
Step 1. Because X(K) ∩ Γ = X(K)∩Γ ∩ Γ we may assume that X(K) ∩ Γ is Zariski
dense in X.
Step 2. At the expense of replacing X by one of its irreducible components, we may
assume X is irreducible. Each irreducible component of X has Zariski dense intersection
with Γ. If our Theorem 2.5 holds for each irreducible component of X, then it also holds for
X.
Step 3. We may assume the stabilizer Stab (X) of X in G is finite. Indeed, let H :=
G
Stab (X). Then H is defined over K (because X is defined over K) and also, H is defined
G
over Falg (because it is an algebraic subgroup of G). Thus H is defined over F . Let
q q
π : G → G/H be the natural projection. Let Gˆ, Xˆ and Γˆ be the images of G, X and Γ
through π. Clearly Γˆ is a finitely generated subgroup of Gˆ(K) and also, Xˆ is defined over
K.
Ifdim(H) > 0,thendim(Xˆ) < dim(X) = d. Hence, bytheinductive hypothesis, Xˆ(K)∩Γˆ
is a finite union of F-sets in Γˆ. Using the fact that the kernel of π| stabilizes X(K)∩Γ, we
Γ
conclude
X(K)∩Γ = π|−1 Xˆ(K)∩Γˆ ,
Γ
(cid:16) (cid:17)
which shows that X(K)∩Γ is also a finite union of F-sets, because ker(π| ) is a subgroup
Γ
of Γ invariant under F (we recall that ker(π) = H is invariant under F).
Therefore, we work from now on under the assumptions that
(i) X(K)∩Γ = X;
(ii) X is irreducible;
(iii) Stab (X) is finite.
G
Let Γ˜ be the Z[F]-module generated by Γ. Because Γ is finitely generated and F is integral
over Z, then also Γ˜ is finitely generated. By Theorem 7.8 of [3], X(K)∩Γ˜ is a finite union
of F-sets in Γ˜. So, there are finitely many groupless F-sets C and Z[F]-submodules H ⊂ Γ˜
i i
such that
X(K)∩Γ˜ = ∪ (C +H ).
i i i
We want to show (C +H )∩Γ is a finite union of F-sets in Γ. It suffices to show that for
i i i
each i, there exists a finite union B of F-sets in Γ such that (C +H )∩Γ ⊂ B ⊂ X(K).
i i i i
S
Indeed, the existence of such B yields
i
X(K)∩Γ = ∪ B ,
i i
as desired.
Case 1. dimC +H < d.
i i
Let X := C +H . Then X is defined over K (because (C +H ) ⊂ G(K)) and dim(X ) <
i i i i i i i
d. So, by the induction hypothesis, B := X (K)∩Γ is a finite union of F-sets in Γ. Clearly,
i i
(C +H )∩Γ ⊂ B ⊂ X(K) (because X ⊂ X).
i i i i
Case 2. dimC +H = d.
i i
˜
Because X = X(K)∩Γ, then X = X(K)∩Γ. Moreover, X is irreducible and so, because
dimC +H = dim(X), then X = C +H . Hence H ⊂ Stab (X) because
i i i i i G
C +H +H = C +H and so, C +H +H ⊂ C +H .
i i i i i i i i i i
4
Because Stab (X) is finite, we conclude H is finite. Thus (C + H ) is a finite union of
G i i i
groupless F-sets because it can be written as a finite union ∪ (h+C ). We let B :=
h∈Hi i i
(C +H )∩Γ. We will show that for each (of the finitely many elements) h ∈ H ,
i i i
(1) (h+C )∩Γ is a finite union of groupless F-sets in Γ.
i
The following lemma will prove (1) and so, it will conclude the proof of Theorem 2.5.
Lemma 3.1. Let M be a finitely generated Z[F]-submodule of G(Kalg) and let O ∈ GF . If
M
Γ is a finitely generated subgroup of G(Kalg), then O∩Γ is a finite union of groupless F-sets
based in M.
Proof. If O∩Γ is finite, then we are done. So, from now on, we may assume O∩Γ is infinite.
Also, we may and do assume Γ ⊂ M (otherwise we replace Γ with Γ∩M).
Let O := Q+S(P ,...,P ;δ ,...,δ ), where Q,P ,...,P ∈ M and δ ∈ N∗ for every i ∈
1 k 1 k 1 k i
{1,...,k}. We may assume that δ = ··· = δ = 1, in which case S(P ,...,P ;δ ,...,δ ) :=
1 k 1 k 1 k
S(P ,...,P ;1). Indeed, if we show that
1 k
(Q+S(P ,...,P ;1))∩Γ is a union of groupless F-sets,
1 k
then also its subsequent intersection with (Q+S(P ,...,P ;δ ,...,δ )) is a finite union of
1 k 1 k
groupless F-sets, as shown in part (a) of Lemma 3.7 in [3].
Because M is a finitely generated Z-module, M is isomorphic with a direct sum of its
finite torsion M and a free Z-submodule M .
tor 1
Let
g−1
(2) f(X) := Xg − α Xi
i
i=0
X
be the minimal polynomial for F over Z (i.e. f(F) = 0 in End(G)). Let r ,...,r be all
1 g
the roots in C of f(X). Clearly, each r 6= 0 because F is not a zero-divisor in End(G).
i
Also, each r has absolute value larger than 1 (actually, their absolute values equal q or
i
q1, according to the Riemann hypothesis for semiabelian varieties defined over F ). Finally,
2 q
all r are distinct. At most one of the r is real and it equals q (and it corresponds to the
i i
1
multiplicative part of G), while all of the other r have absolute value equal to q (and they
i 2
correspond to the abelian part of G). If
0 → T → G → A → 0
is a short exact sequence of group varieties, with T being a torus and A an abelian variety,
1
then the roots r of absolute value q correspond to roots of the minimal polynomial over
i 2
Z for the Frobenius morphism on A. The abelian variety A is isogenuous with a product of
simple abelian varieties A , all defined over a finite field. If f is the minimal polynomial of
i i
the corresponding Frobenius on A , then the minimal polynomial f of the Frobenius on A
i 0
is the least common multiple of all f . For each i, End(A ) is a domain and so, f has simple
i i i
roots. Therefore f (and so, f) has simple roots.
0
The definition of f shows that for every point P ∈ G(Kalg),
g−1
(3) FgP = α FjP.
j
j=0
X
5
We conclude that for all n ≥ g,
g−1
(4) FnP = α Fn−g+jP.
j
j=0
X
For each j we define the sequence {z } as follows
j,n n≥0
(5) z = 0 if 0 ≤ n ≤ g −1 and n 6= j;
j,n
(6) z = 1 and
j,j
g−1
(7) z = α z for all n ≥ g.
j,n l j,n−g+l
l=0
X
Using (5) and (6) we obtain that
g−1
(8) FnP = z FjP, for every 0 ≤ n ≤ g −1.
j,n
j=0
X
We prove by induction on n that
g−1
(9) FnP = z FjP, for every n ≥ 0.
j,n
j=0
X
We already know (9) is valid for all n ≤ g −1 due to (8). Thus we assume (9) holds for all
n < N, where N ≥ g and we prove that (9) also holds for n = N. Using (4), we get
g−1
(10) FNP = α FN−g+j.
j
j=0
X
We apply the induction hypothesis to all FN−g+j for 0 ≤ j ≤ g −1 and conclude
g−1 g−1 g−1 g−1 g−1
(11) α FN−g+j = α z FiP = α z FiP.
j j i,N−g+j j i,N−g+j
!
j=0 j=0 i=0 i=0 j=0
X X X X X
We use (7) in (11) and conclude
g−1 g−1
(12) α FN−g+j = z FiP.
j i,N
j=0 i=0
X X
Combining (10) and (12) we obtain the statement of (9) for n = N. This concludes the
inductive proof of (9).
Because {z } is a recursive defined sequence, then for each j ∈ {0,...,g−1} there exist
j,n n
{γ } ⊂ Qalg such that for every n ∈ N,
j,l 1≤l≤g
(13) z = γ rn.
j,n j,l l
1≤l≤g
X
To derive (13) we also use the fact that all r are distinct, nonzero numbers.
i
6
Equations (9) and (13) show that for every n and for every P ∈ G(Kalg),
(14) FnP = γ rn FjP.
j,l l
!
0≤j≤g−1 1≤l≤g
X X
For each i ∈ {1,...,k} and for each j ∈ {0,...,g − 1}, let FjP := T(j) + Q(j), with
i i i
(j) (j)
T ∈ M and Q ∈ M . Also, let Q := T +Q , where T ∈ M and Q ∈ M .
i tor i 1 0 0 0 tor 0 1
Let R ,...,R be a basis for the Z-module M . For each j ∈ {0,...,g−1} and for each
1 m 1
i ∈ {1,...,k}, let
m
(j) (l)
(15) Q := a R ,
i i,j l
l=1
X
where a(l) ∈ Z. Finally, let a(1),...,a(m) ∈ Z such that Q = m a(j)R .
i,j 0 0 0 j=1 0 j
For every n ∈ N and for every i ∈ {1,...,k}, (9) and the definitions of Q(j) and T(j) yield
P i i
(16) FnP = z T(j) +Q(j) = z T(j) + z Q(j).
i j,n i i j,n i j,n i
0≤Xj≤g−1 (cid:16) (cid:17) 0≤Xj≤g−1 0≤Xj≤g−1
Because T(j) ∈ M , then for each (n ,...,n ) ∈ Nk,
i tor 1 k
k g−1
(j)
z T ∈ M .
j,ni i tor
i=1 j=0
XX
Also, because Q andallQ(j) areinM andbecause z ∈ Z, thenfor each (n ,...,n ) ∈ Nk,
0 i 1 j,n 1 k
k g−1
(j)
Q + z Q ∈ M .
0 j,ni i 1
i=1 j=0
XX
Moreover,
k
(17) Q+ FniP = T + z T(j) + Q + z Q(j) .
i  0 j,ni i   0 j,ni i 
i=1 1≤i≤k 1≤i≤k
X X X
 0≤j≤g−1   0≤j≤g−1 
   
For each h ∈ M , if (h+M )∩Γ is not empty, we fix (h+U ) ∈ Γ for some U ∈ M .
tor 1 h h 1
Let Γ := Γ∩M . Then
1 1
(18) (h+M )∩Γ = h+U +Γ .
1 h 1
For each h ∈ M , we let O := {P ∈ O | P = h+P′ with P′ ∈ M }. Then using (18), we
tor h 1
get
(19) O ∩Γ = O ∩(h+U +Γ ) = (h+((−h+O )∩(U +Γ ))).
h h 1 h h 1
h∈[Mtor h∈[Mtor
Clearly, (−h+O ) ∈ M . Therefore (19) and (17) yield
h 1
(j)
(20) O ∩Γ = h+ Q + z Q (U +Γ ) .
0 j,ni i h 1
! !!
h∈[Mtor Xi,j \
T0+ i,jzj,niTi(j)=h
7
P
In (20), the union is over the finitely many torsion points of M (M is finitely generated)
tor
and it might be that not for each h ∈ M there is a corresponding nonempty intersection
tor
in (20).
Fix h ∈ M . We show that the set of tuples (n ,...,n ) ∈ Nk for which
tor 1 k
(j)
(21) h = T + z T
0 j,ni i
i,j
X
is a finite union of cosets of semigroups of Nk (a semigroup of Nk is the intersection of a
subgroup of Zk with Nk). Indeed, let N ∈ N∗ such that M ⊂ G[N]. Because for each
tor
j ∈ {0,...,g−1},z isarecursivelydefinedsequence(asshownby(5),(6)and(7)),thenthe
j,n
sequence {z } iseventually periodicmoduloN (arecursively definedsequence iseventually
j,n n
(j)
periodic modulo any integral modulus). Thus each value taken by T + z T is
0 i,j j,ni i
attained for tuples (n ,...,n ) which belong to a finite union of cosets of semigroups of Nk.
1 k P
We will prove next that for each fixed h ∈ M , the tuples (n ,...,n ) for which
tor 1 k
(j)
(22) Q + z Q ∈ (U +Γ )
0 j,ni i h 1
!
i,j
X
form a finite union of cosets of semigroups of Nk. This will finish the proof of our theorem
because this result, combined with the one from the previous paragraph and combined with
(20), will show that the tuples (n ,...,n ) for which
1 k
k
Q+ FniP ∈ Γ
i
i=1
X
form a finite union of cosets of semigroups of Nk (we are also using the fact that the intersec-
tion of two finite unions of cosets of semigroups is also a finite union of cosets of semigroups).
Lemma 3.4 of [3] shows that the set of points in O corresponding to a finite union of cosets
of semigroups containing the tuples of exponents (n ,...,n ) is a finite union of groupless
1 k
F-sets.
Because Γ ⊂ M and M is a free Z-module with basis {R ,...,R }, we can find (after a
1 1 1 1 m
possible relabelling of R ,...,R ) a Z-basis V ,...,V (n ≤ m) of Γ of the following form:
1 m 1 n 1
V = β(i1)R +···+β(m)R ;
1 1 i1 1 m
V = β(i2)R +···+β(m)R ;
2 2 i2 2 m
and in general, V = β(ij)R +···+β(m)R , where
j j ij j m
1 ≤ i < i < ··· < i ≤ m
1 2 n
and all β(i) ∈ Z. We also assume β(ij) 6= 0 for every j ∈ {1,...,n} (n ≥ 1 because we
j j
assumed the intersection O ∩ Γ is infinite, which means Γ is infinite, because otherwise
1
|O∩Γ| ≤ |M |).
tor
Let b(1),...,b(m) ∈ Z such that U = m b(j)R . Then a point
0 0 h j=1 0 j
m
P
P := c(j)R ∈ (U +Γ )
j h 1
j=1
X
8
if and only if there exist integers k ,...,k such that
1 n
n
(23) P = U + k V .
h i i
i=1
X
Using the expressions of the V , U and P in terms of the Z-basis {R ,...,R } of M , we
i h 1 m 1
obtain the following relations:
(24) c(j) = b(j) for every 1 ≤ j < i ;
0 1
(25) c(j) = b(j) +k β(j) for every i ≤ j < i ;
0 1 1 1 2
(26) c(j) = b(j) +k β(j) +k β(j) for every i ≤ j < i
0 1 1 2 2 2 3
and so on, until
n
(27) c(m) = b(m) + k β(m).
0 i i
i=1
X
We express equation (25) for j = i as a linear congruence modulo β(i1) and obtain
1 1
(28) c(i1) ≡ b(i1) mod β(i1) .
0 1
(cid:16) (cid:17)
Also from (25) for j = i , we get k = c(i1)−b(0i1). Then we substitute this formula for k in
1 1 β(i1) 1
1
(25) for all i < j < i and obtain
1 2
c(i1) −b(i1)
(29) c(j) = b(j) + 0 β(j) for every i < j < i .
0 β(i1) 1 1 2
1
Then we express (26) forj = i as a linear congruence moduloβ(i2) (also using the expression
2 2
for k computed above). We obtain
1
c(i1) −b(i1)
(30) c(i2) ≡ b(i2) + 0 β(i2) mod β(i2) .
0 β(i1) 1 2
1 (cid:16) (cid:17)
Next we equate k from (26) for j = i (also using the formula for k ) and obtain
2 2 1
c(i2) −b(i2) − c(i1)−b(0i1)β(i2)
0 β(i1) 1
k = 1
2 β(i2)
2
Then we substitute this formula for k in (26) for i < j < i and obtain
2 2 3
(31) c(j) = b(j) + c(i1) −b(0i1) ·β(j) + c(i2) −b(0i2) − c(i1β)1(−i1b)(0i1)β1(i2) ·β(j).
0 β(i1) 1 β(i2) 2
1 2
We go on as above until we express c(m) in terms of
c(i1),...,c(in)
9
(m) (l)
and b and the β . We observe that all congruences can be written as linear congruences
0 j
over Z. For example, the above congruence equation (30) modulo β(i2) can be written as the
2
following linear congruence over Z:
β(i1) ·c(i2) ≡ c(i1) −b(i1) β(i2) +β(i1)b(i2) mod β(i1) ·β(i2) .
1 0 1 1 0 1 2
(cid:16) (cid:17) (cid:16) (cid:17)
Hence all the above conditions are either linear congruences or linear equations for the c(j).
A typical intersection point from the inner intersection in (20) corresponding to a tuple
(n ,...,n ) ∈ Nk is
1 k
(j)
Q + z Q ∩(U +Γ )
0 j,ni i h 1
!
i,j
X
and it can be written in the following form (see also (13)):
g g
a(l) + a(l) γ rni R .
 0 i,j j,e e  l
l=1 1≤i≤k e=1
X X X
 0≤j≤g−1 
 
Such a point lies in (U +Γ ) if and only if its coefficients
h 1
g
a(l) + a(l) γ rni
0 i,j j,e e
1≤i≤k e=1
X X
0≤j≤g−1
with respect to the Z-basis {R ,...,R } of M satisfy the linear congruences and linear
1 m 1
equations such as (24), (28), (29), (30) and (31), associated to (U +Γ ). A linear equation
h 1
as above yields an equation of the following form (after collecting the coefficients of rni for
e
each 1 ≤ e ≤ g and each 1 ≤ i ≤ k):
g k
(32) d rni = D.
e,i e
e=1 i=1
XX
All d and D are algebraic numbers. A tuple (n ,...,n ) ∈ Nk satisfying (32) corresponds
e,i 1 k
to an intersection point of the linear variety L in (Gg )k(Qalg) given by the equation
m
g k
(33) d X = D
e,i e,i
e=1 i=1
XX
and the finitely generated subgroup G of (Gg )k(Qalg) spanned by
0 m
(34) (r ,...,r ,1,...,1); (1,...,1,r ,...,r ,1,...,1); ...,(1,...,1,r ,...,r ).
1 g 1 g 1 g
Each vector in (34) has gk components. There are k multiplicatively independent generators
above for G (we are using the fact that |r | > 1, for each i). Hence G ≃ Zk. By Lang
0 i 0
Theorem for Ggk, we conclude the intersection of L(Qalg) and G is a finite union of cosets
m 0
of subgroups of G . The subgroups of G correspond to subgroups of Zk. Hence the tuples
0 0
(n ,...,n ) ∈ Nk which satisfy (32) belong to a finite union of cosets of semigroups of Nk.
1 k
A congruence equation as (28) or (30), corresponding to conditions for a point to lie in
(U +Γ ) yields a congruence relation between the coefficients (with respect to the Z-basis
h 1
10