DENSITY MEASURE OF RATIONAL POINTS ON ABELIAN

27 Pages·2009·0.19 MB·English

M. Waldschmidt
Nagoya Math. J.
Vol. 155 (1999), 27{53
DENSITY MEASURE OF RATIONAL POINTS ON
ABELIAN VARIETIES
MICHEL WALDSCHMIDT1
Abstract. Let A be a simple Abelian variety of dimension g over (cid:0) , and let
‘ be the rank of the Mordell-Weil group A((cid:0) ). Assume ‘(cid:21)1. A conjecture of
MazurassertsthattheclosureofA((cid:0) ) intoA( ) fortherealtopologycontains
theneutralcomponentA( )0 oftheorigin.Th(cid:1)isisknownonlyundertheextra
hypothesis‘(cid:21)g2(cid:0)g+1.(cid:1)Weinvestigatehereaquantitativere(cid:12)nementofthis
question: for each given positive h, the set of points in A((cid:0) ) of N(cid:19)eron-Tate
height (cid:20) h is (cid:12)nite, and we study how these points are distributed into the
connectedcomponentA( )0.MoregenerallyweconsideranAbelianvarietyA
(cid:1)
overanumber(cid:12)eldK embeddedin ,andasubgroup(cid:0)ofA(K)ofsu(cid:14)ciently
(cid:1)
large rank. The e(cid:11)ective result of density we obtain relies on an estimate of
Diophantine approximation, namely a lower bound for linear combinations of
determinants involving Abelian logarithms.
x1. Introduction
Let K be a number (cid:12)eld with a given real embedding. Let V be a
smooth variety over K. Denote by Z the closure, for the real topology, of
V(K) in V(R). In his paper [9] on the topology of rational points, Mazur
assumes that K = Qand thatV(Q) isZariski dense; he askswhetherZ isa
unionofconnected components ofV(R).However an example isgiven in[2]
of a smooth surface V over Q, whose Q-rational points are Zariski-dense,
but such that the closure Z in V(R) of the set of Q-points is not a union
of connected components.
We consider here a quantitative re(cid:12)nement. Assume that V is embed-
ded,asaquasi-projectivevariety,intoaprojectivespaceP overK.Denote
N
by h the absolute logarithmic height (Weil height) on P (K) (see [4], [5,
N
Chap. IV, x1], [6, Chap. III], [7, Chap. II], or [12, Chap. II]):
[K : Q ]
v v
h((cid:13) :(cid:1)(cid:1)(cid:1) :(cid:13) ) = logmax j(cid:13) j ;:::;j(cid:13) j ;
0 N [K : Q] 0 v N v
v2XMK (cid:8) (cid:9)
Received December 16, 1996.
1Work partially supported by the Japan Society for the Promotion of Science: grant
JSPS 96029
27
28 M.WALDSCHMIDT
where M denotes the set of places of K normalized so that the product
K
formula reads
[K :Q ]logj(cid:13)j = 0 for any (cid:13) 2 K(cid:2):
v v v
v2XMK
Further, for (cid:16) = ((cid:16) : (cid:1)(cid:1)(cid:1) : (cid:16) ) and (cid:24) = ((cid:24) :(cid:1)(cid:1)(cid:1) :(cid:24) ) in P (R), write
0 N 0 N N
max j(cid:16) (cid:24) (cid:0)(cid:16) (cid:24) j
i j j i
0(cid:20)i;j(cid:20)N
dist((cid:16);(cid:24)) = :
max j(cid:16) j(cid:1) max j(cid:24) j
i j
0(cid:20)i(cid:20)N 0(cid:20)j(cid:20)N
For each su(cid:14)ciently large real number H, de(cid:12)ne (cid:17) (H) as follows:
V
for any (cid:16) 2 Z, there exists (cid:13) 2 V(K)
(cid:17) (H) = inf (cid:15) > 0 ; :
V
(cid:26) with h((cid:13))(cid:20) logH and dist((cid:16);(cid:13)) (cid:20) (cid:15) (cid:27)
Since V(K) is dense in Z,
lim (cid:17) (H) = 0:
V
H!1
An upper bound for (cid:17) (H) can be considered as a measure of the density
V
in Z of the rational points V(K).
Alowerboundfor(cid:17) (H)iseasilyachievedasfollows:denoteby (H)
V V
the number of points (cid:13) in V(K) of height h((cid:13))(cid:20) logH. Then
liminf (H)(cid:17) (H)dimV > 0:
V V
H!1
Since the estimate (H) (cid:20) CH(cid:20) always holds, with a constant C which
V
depends only on V and K, but not on H, and with (cid:20)= (dimV +1)[K : Q],
we deduce
liminf(cid:17) (H)(cid:1)H(cid:20)0 >0 with (cid:20)0 = (cid:20) :
V
H!1 dimV
In certain cases a sharper lower bound for (cid:17) (H) is available. For instance
V
when V is an Abelian variety A, if ‘ is the rank over Z of the Mordell-Weil
group A(K), then
liminf(cid:17) (H)(cid:1)(logH)‘=2g > 0:
A
H!1
Indeed there exists a positive constant C such that, for all H (cid:21) e,
(H) (cid:20) C(logH)‘=2:
A
DENSITYMEASUREONABELIANVARIETIES 29
ThisfollowseasilyfromthepropertiesoftheN(cid:19)eron-Tateheighth^associated
with the given embedding of A into P , which is de(cid:12)ned for (cid:13) 2 A(K) by
N
1
h^((cid:13)) = lim h(n(cid:13)):
n!1 n2
For elliptic curve, see [5, Chap. IV], or [13, Chap. VIII, x6 and x9]. For
Abelian varieties, see [6, Chap. V], [12, x3.3], [7, Chap. III, x1], or [4, x3].
Since jh(cid:0)h^j is bounded on A(K), it does not make a di(cid:11)erence for us
to replace h by h^. Accordingly we de(cid:12)ne
for any (cid:16) 2 A(R)0, there exists (cid:13) 2 A(K)
(cid:17)^ (h) = inf (cid:15) > 0 ; ;
A (cid:26) with h^((cid:13)) (cid:20) h and dist((cid:16);(cid:13)) (cid:20) (cid:15) (cid:27)
where A(R)0 denotes the connected component of the origin in A(R).
Conjecture 1.1. Let A be a simple Abelian variety of dimension
g over a number (cid:12)eld K embedded in R. Denote by ‘ the rank over Z of
the Mordell-Weil group A(K). For any (cid:15) > 0, there exists h > 0 (which
0
depends only on the Abelian variety A, the real number (cid:12)eld K and (cid:15)) such
that, for any h (cid:21) h ,
0
(cid:17)^ (h) (cid:20) h(cid:0)(‘=2g)+(cid:15):
A
This means that for su(cid:14)ciently large h, any point in A(R)0 should be
at distance (cid:20) h(cid:0)(‘=2g)+(cid:15) of a point of A(K) of N(cid:19)eron-Tate height (cid:20) h. We
shall see that this Conjecture 1.1 can be stated in an equivalent way as an
estimate of Diophantine approximation, for which we shall produce partial
results.
We start with the special case of an elliptic curve. Here is a measure of
density for the set of rational points.
Theorem 1.2. Let E be an elliptic curve over a real number (cid:12)eld K.
There exist two positive constants C and C , which depend only on E and
1 2
K, such that for any h (cid:21) ee,
(cid:17)^ (h) (cid:20) C exp (cid:0)C (logh)(loglogh)(cid:0)1(cid:0)(2=‘) :
E 1 2
(cid:8) (cid:9)
According to Conjecture 1.1, it should be possible to replace the factor
C (loglogh)(cid:0)1(cid:0)(2=‘) by (‘=2)(cid:0)(cid:15) for h> h ((cid:15)):
2 0
30 M.WALDSCHMIDT
The proof of Theorem 1.2 (see x3) rests on an estimate, due to N.
Hirata-Kohno, on the simultaneous approximation of quotients of elliptic
logarithms (namely Theorem 3.2).
As we shall see, the case of an elliptic curve is the only one where the
problem is reduced to a lower bound for a linear form in logarithms.
Let A be a simple Abelian variety of dimension g over a number (cid:12)eld
K (cid:26) R, with Mordell-Weil group A(K) of rank ‘. If ‘ (cid:21) g2(cid:0)g+1, then the
closure of A(K) in A(R) contains A(R)0. Moreover, if ‘ (cid:21) g2, then there
exists(cid:13) 2 A(K)\A(R)0 such thatZ(cid:13) isdense inA(R)0 (seeCorollary4.6).
Our aim is to provide quantitative re(cid:12)nements to these density results.
We assume that A is embedded into a projective space P over K.
N
Theorem 1.3. Assume ‘ (cid:21) 2g2. Then there exist two positive con-
stants C and C such that, for any h(cid:21) e,
1 2
2g2
(cid:17)^ (h) (cid:20) C exp (cid:0)C (logh)(cid:18) where (cid:18) = 1(cid:0) :
A 1 2
‘+1
(cid:8) (cid:9)
The constants C and C depend on A and K, as well as on the em-
1 2
bedding of A into P .
N
Here is the plan of this paper. The purpose of section 2 is twofold:
(cid:12)rstly we state a consequence of Kronecker’s Theorem which reduces the
density problem to a question of irrationality; secondly we give a transfer-
ence lemma which reduces the question of density measure to a problem of
Diophantine approximation on Abelian varieties. Next in x3 we discuss the
case of elliptic curves. Section 4 is devoted to the qualitative problem on
Abelian varieties, and sections 5 and 6 to the quantitative one. The main
results of Diophantine approximation (namely Theorems 5.2 and 6.1) are
deduced from [19].
Acknowledgements. It is a pleasure for the author to thank Noriko
Hirata Kohno for her helpful comments on this paper.
x2. Kronecker and Khinchine
Let E be a vector space of dimension n (cid:21) 1 over the (cid:12)eld R of real
numbers and let Y be a (cid:12)nitely generated subgroup of E. From a result of
Kronecker one deduces (see [18, Chap. II, Prop. 4.3]):
DENSITYMEASUREONABELIANVARIETIES 31
Lemma 2.1. The following conditions are equivalent.
(i) Y is dense in E.
(ii) For any vector subspace V of E distinct from E, we have
rank (Y=Y \V)> dim (E=V):
Z R
(iii) For any hyperplane H of E, we have rank (Y=Y \H) (cid:21) 2.
Z
(iv) For any nonzero linear form ’ 2 Hom (E;R), we have ’(Y) 6(cid:26) Z.
R
(v) For any non-trivial character (cid:31) of E, we have (cid:31)(Y)6= f1g.
(vi) Let y ;:::;y be a set of elements of Y which span a subgroup
1 ‘
of (cid:12)nite index in Y; let (e ;:::;e ) be a basis of E over R; consider the
1 n
coordinates u of y in this basis:
ij j
y = u e +(cid:1)(cid:1)(cid:1)+u e ; (1 (cid:20) j (cid:20) ‘);
j 1j 1 nj n
then for any nonzero s = (s ;:::;s ) in Z‘, the matrix
1 ‘
u (cid:1)(cid:1)(cid:1) u
11 1‘
0 ... ... ... 1
M =
s
u (cid:1)(cid:1)(cid:1) u
B n1 n‘C
B s (cid:1)(cid:1)(cid:1) s C
@ 1 ‘ A
has rank n+1.
Let (cid:10) be a lattice in E (which means a discrete Z-subgroup of maximal
rank n). Denote by E(cid:3) = Hom (E;R) the dual vector space of E, and by
R
(cid:10)(cid:3) the dual lattice of (cid:10):
(cid:10)(cid:3) = ’ 2E(cid:3) ; ’((cid:10)) (cid:26) Z :
(cid:8) (cid:9)
From the equivalence (i) , (iv) in Lemma 2.1, it follows that a (cid:12)nitely
generated subgroup Y of E which contains (cid:10) is dense in E if and only if,
for any nonzero ’ 2 (cid:10)(cid:3), we have ’(Y) 6(cid:26) Z. We now give a quantitative
version of this statement. For x 2 R we denote by kxk the distance to the
nearest integer:
kxk = minjx(cid:0)nj:
n2Z
Lemma 2.2. Let! ;:::;! beabasisofthelattice (cid:10)andlety ;:::;y
1 n 1 m
be elements in E. Let F and G be two monotonically increasing and un-
bounded real valued functions of a real variable which are inverse of each
other:
G(cid:14)F(S) = S for all su(cid:14)ciently large S
32 M.WALDSCHMIDT
and
F (cid:14)G(T) = T for all su(cid:14)ciently large T:
Then the two following conditions are equivalent.
(i) There exist three positive constants c , c and S such that, for any
1 2 0
nonzero ’2 (cid:10)(cid:3), if we set
S = maxfj’(! )j;:::;j’(! )j ;S g;
1 n 0
then we have
max k’(y )k (cid:21) c =F(c S):
i 1 2
1(cid:20)i(cid:20)m
(ii) There exist three positive constants c0, c0 and T such that, for any
1 2 0
x 2 E and any integer T (cid:21) T , there exist ! 2 (cid:10) and (t ;:::;t ) 2 Zm
0 1 m
which satisfy
maxfjt j;:::;jt jg (cid:20) T and x(cid:0)!(cid:0)t y (cid:0)(cid:1)(cid:1)(cid:1)(cid:0)t y (cid:20) c0=G(c0T):
1 m 1 1 m m 1 2
(cid:12) (cid:12)
Proof. The proof is given in(cid:12) [20, Lemme 3.1], and reli(cid:12)es on Khinchine
transference Theorem (see also [18, Chap. V, Lemme 4.1]). More precisely,
one gets the following explicit estimates.
(cid:15) If (i) holds, then (ii) is true with
c0 = (cid:17)c j!j ; c0 = c =(cid:17) and T = (1=c0)F(c S );
1 2 + 2 1 0 2 2 0
where
j!j = j! j+(cid:1)(cid:1)(cid:1)+j! j and (cid:17) = 2(cid:0)n(cid:0)m (n+m)! 2:
+ 1 n
(cid:0) (cid:1)
(cid:15) In the other direction, if (ii) holds, then (i) is true with
c = c0=2(n+m); c = 2(n+m)c0j!j and S =(1=c )G(c0T );
1 2 2 1 (cid:0) 0 2 2 0
where
j!j = sup j! z +(cid:1)(cid:1)(cid:1)+! z j(cid:0)1; jz j= (cid:1)(cid:1)(cid:1) = jz j = 1 :
(cid:0) 1 1 n n 1 n
(cid:8) (cid:9)
Remark. Lemma 2.2 deals with the equivalence (i) , (iv) of Lem-
ma 2.1. One could also state the result in terms of (vi): the problem is then
to produce lower bounds for at least one of the (n+1)(cid:2)(n+1) minors of
M in terms of maxjs j.
s j
DENSITYMEASUREONABELIANVARIETIES 33
x3. Diophantine approximation on elliptic curves
In this section we deal only with elliptic curves.
Let E (cid:26) P be an elliptic curve over a sub(cid:12)eld K of C given with a
2
Weierstra(cid:25) model:
E(K) = (x: y :t) 2 P (K); y2t = 4x3(cid:0)g xt2(cid:0)g t3 ;
2 2 3
(cid:8) (cid:9)
where g and g are elements of K satisfying g3 6=27g2.
2 3 2 3
To begin with, assume K (cid:26) R. Then the exponential map of the Lie
group E(R) is the restriction to R of a Weierstra(cid:25) }-function:
exp : R (cid:0)! E(R)
E;R
z 7(cid:0)! }(z) :}0(z) :1
(cid:0) (cid:1)
The image E(R)0 of this map is the connected component of the origin in
E(R), while itskernel is ofthe formZ! for some nonzeroperiod! 2R. The
induced isomorphism of topological groups between E(R)0 and R=Z shows
that any (cid:12)nitely generated subgroup of E(R)0 of positive rank is dense. In
this case no arithmetic assumption is needed.
We now consider the quantitative density problem. Let (cid:0) be a (cid:12)nitely
generated subgroup of E(R)0 of rank ‘. Let (cid:13) ;:::;(cid:13) be Z-linearly inde-
1 ‘
pendent elements in (cid:0). For 1 (cid:20) j (cid:20) ‘, let y 2 R be an elliptic logarithm
j
of (cid:13) , viz. a real number such that exp (y ) = (cid:13) . From Lemma 2.2, we
j E;R j j
shall deduce:
Lemma 3.1. The following properties are equivalent.
(i) For any (cid:15) > 0, there exists a constant T > 0, with the following
0
property. For any integer T (cid:21) T and any (cid:16) 2 E(R)0, there exist rational
0
integers t ;:::;t satisfying
1 ‘
max jt j(cid:20) T and dist (cid:16);t (cid:13) +(cid:1)(cid:1)(cid:1)+t (cid:13) (cid:20) T(cid:0)‘+(cid:15):
j 1 1 ‘ ‘
1(cid:20)j(cid:20)‘
(cid:0) (cid:1)
(ii) For any (cid:15) > 0 there exists a constant T > 0, with the following
0
property. For anyinteger T (cid:21) T andany(cid:24) 2 R,there existrational integers
0
t ;t ;:::;t satisfying
0 1 ‘
max jt j (cid:20)T and (cid:24) (cid:0)t !(cid:0)t y (cid:0)(cid:1)(cid:1)(cid:1)(cid:0)t y (cid:20) T(cid:0)‘+(cid:15):
j 0 1 1 ‘ ‘
1(cid:20)j(cid:20)‘
(cid:12) (cid:12)
(cid:12) (cid:12)
34 M.WALDSCHMIDT
(iii) For any (cid:15) > 0 there exists a constant Q > 0, with the following
property: for any rational integers q;p ;:::;p with q (cid:21) Q,
1 ‘
y p
max j (cid:0) j > q(cid:0)(1=‘)(cid:0)1(cid:0)(cid:15):
1(cid:20)j(cid:20)‘(cid:12)! q (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Moreover, if the (cid:12)eld K is an(cid:12) algebrai(cid:12)c extension of Q and if (cid:0) is contained
in E(K), then these properties are also equivalent to the next one:
(iv) For any (cid:15) > 0 there exists a constant h > 0 such that, for any
0
h (cid:21) h and any (cid:16) 2 E(R)0, there exists (cid:13) 2 (cid:0) with h^((cid:13)) (cid:20) h and
0
dist((cid:16);(cid:13)) (cid:20) h(cid:0)(‘=2)+(cid:15):
Ineachoftheseproperties,the\constants"aresupposedtodependonly
on (cid:15), g , g ,K andy ;:::;y | notice that thereal number ! dependsonly
2 3 1 ‘
on g and g .
2 3
Proof. There is no loss (apart from the constants, which we do not
compute explicitly here) to replace if necessary (cid:0) by a subgroup of (cid:12)nite
index. Hence one may assume (cid:0) = Z(cid:13) +(cid:1)(cid:1)(cid:1)+Z(cid:13) .
1 ‘
For the proof of (i) , (ii), write any (cid:16) 2 E(R)0 as exp ((cid:24)) for some
E;R
(cid:24) 2 R.Alsowriteany(cid:13) 2 (cid:0)asexp (t y +(cid:1)(cid:1)(cid:1)+t y )forsome(t ;:::;t ) 2
E;R 1 1 ‘ ‘ 1 ‘
Z‘. Up to a (multiplicative) constant, dist((cid:16);(cid:13)) is nothing else than
min (cid:24)(cid:0)t !(cid:0)t y (cid:0)(cid:1)(cid:1)(cid:1)(cid:0)t y :
0 1 1 ‘ ‘
t02Z
(cid:12) (cid:12)
(cid:12) (cid:12)
For the proof of the equivalence between (ii) and (iii), use Lemma 2.2
with
E = R; n= 1; (cid:10) = Z!; F(S) = S(1=‘)+(cid:15)1 and G(T) = T‘(cid:0)(cid:15)2;
where (cid:15) and (cid:15) are related by (cid:15) ‘(cid:0)(cid:15) =‘ = (cid:15) (cid:15) .
1 2 1 2 1 2
Finally, thecorrespondencebetween hin(iv)andT in (i)ish = T2.
It seems reasonable to conjecture that the equivalent properties stated
in Lemma 3.1 always hold when the (cid:12)eld K is a (real) number (cid:12)eld. This
would imply Conjecture 1.1 for g = 1 (just take for (cid:13) ;:::;(cid:13) a basis of the
1 ‘
Mordell-Weil group E(K)). Moreover, as far as assertion (iii) is concerned,
one expects that such an estimate should hold for any number (cid:12)eld K,
not only real ones. In this direction, the best known estimate for the si-
multaneous approximation of the numbers y =! is due to N. Hirata-Kohno
j
(cf. [3]):
DENSITYMEASUREONABELIANVARIETIES 35
Theorem 3.2. (Hirata-Kohno) Let E be an elliptic curve over a num-
ber(cid:12)eldK.Let! 2Cbe anon-zeroperiod ofexp . Lety ;:::;y be complex
E 1 ‘
numbers such that exp (y ) 2 E(K) for 1 (cid:20) j (cid:20) ‘. Assume the numbers
E j
!;y ;:::;y are linearly independent over Q. Then there exists a constant
1 ‘
c, which depends only on g , g , !, K and y ;:::;y , such that, for any
2 3 1 ‘
(q;p ;:::;p )2 Z‘+1 with q > ee,
1 ‘
y p
max j (cid:0) j (cid:21) exp (cid:0)c(logq)(loglogq)1+(2=‘) :
1(cid:20)j(cid:20)‘(cid:12)! q (cid:12)
(cid:12) (cid:12) (cid:8) (cid:9)
(cid:12) (cid:12)
We now dedu(cid:12)ce from (cid:12)Theorem 3.2 a stronger statement than Theo-
rem 1.2 (we do not restrict the discussion to the full Mordell-Weil group
here).
Corollary 3.3. Let E be an elliptic curve over a real number (cid:12)eld K
and let (cid:0) be a subgroup of E(K) of positive rank ‘. There exists a constant
C > 0 such that, for any h (cid:21) ee and any (cid:16) 2E(R)0, there exists (cid:13) 2 (cid:0) with
h^((cid:13)) (cid:20) h and
dist((cid:16);(cid:13)) (cid:20) exp (cid:0)C(logh)(loglogh)(cid:0)1(cid:0)(2=‘) :
(cid:8) (cid:9)
Proof. We use the implication (i) ) (ii) of Lemma 2.2 with
F(S) = exp (logS)(loglogS)1+(2=‘) for S (cid:21) ee:
(cid:8) (cid:9)
Then the inverse function G is bounded from below by
G(T) (cid:21) exp (logT)(loglogT)(cid:0)1(cid:0)(2=‘) for T (cid:21) ee:
(cid:8) (cid:9)
x4. Irrationality, transcendence and density
Let A be an Abelian variety of dimension g de(cid:12)ned over a sub(cid:12)eld K
of C. We denote by T (C) the tangent space at the origin of A, by
A
exp : T (C)(cid:0)! A(C)
A A
the exponential map of A(C), and by (cid:10) its kernel, which is a discrete
A
subgroup of T (C) of rank 2g over Z. If K is a sub(cid:12)eld of R, then the
A
exponential map of the real Lie group A(R) is the restriction
exp : T (R) (cid:0)! A(R)
A;R A
36 M.WALDSCHMIDT
of exp to T (R). Its image is A(R)0 and its kernel
A A
(cid:10) = (cid:10) \T (R)
A;R A A
is a discrete subgroup of T (R) of rank g over Z.
A
We denote by
L (K) = exp(cid:0)1 A(K)
A A
(cid:0) (cid:1)
the Z-module of Abelian logarithms of rational points in A(K). If K is
algebraically closed, then the group A(K) is divisible, hence L (K) is a
A
Q-vector space.
The following result is a consequence of Lemma 2.1.
Lemma 4.1. Let A be an Abelian variety de(cid:12)ned over R. Let (cid:0) be
a (cid:12)nitely generated subgroup of A(R)0. Then the following properties are
equivalent.
(i) The subgroup (cid:0) is dense in A(R)0.
(ii) De(cid:12)ne Y = exp(cid:0)1 ((cid:0)). Then for any nonzero linear form ’ 2
A;R
Hom (T (R);R), we have ’(Y)6(cid:26) Q.
R A
(iii) Let ! ;:::;! be linearly independent elements of (cid:10) .Let y ;:::;
1 g A;R 1
y be elements in T (R) whose exponentials exp (y ), (1 (cid:20) j (cid:20) ‘) span a
‘ A A;R j
subgroup of (cid:12)nite index of (cid:0). Further, consider the coordinates of y in the
j
basis ! ;:::;! of T (R):
1 g A
y = u ! +(cid:1)(cid:1)(cid:1)+u ! ; (1 (cid:20) j (cid:20) ‘):
j 1j 1 gj g
Then for any ((cid:27) ;:::;(cid:27) ;s ;:::;s ) in Zg+‘ with ((cid:27) ;:::;(cid:27) ) 6= (0;:::;0),
1 g 1 ‘ 1 g
the matrix
u (cid:1)(cid:1)(cid:1) u
11 1‘
0 I ... ... ... 1
g
u (cid:1)(cid:1)(cid:1) u
B g1 g‘C
B(cid:27) (cid:1)(cid:1)(cid:1) (cid:27) s (cid:1)(cid:1)(cid:1) s C
@ 1 g 1 ‘ A
has rank g+1.
For instance a subgroup Z(cid:13) of A(R)0 is dense if and only if (cid:13) =
exp (y) with y = # ! +(cid:1)(cid:1)(cid:1)+# ! and 1;# ;:::;# linearly independent
A;R 1 1 g g 1 g
over Q.
It seems reasonable to hope that these properties are true when A
is simple and K is a (real) number (cid:12)eld. This would imply the following