Michel Waldschmidt
Abstract. According to the Four Exponentials Conjecture, a 2× 2 matrix whose entries λij (i = 1,2, j = 1,2) are logarithms of algebraic numbers is regular, as soon as the two rows as well as the two columns are linearly independent over the field Q of rational numbers. The question we address is as follows: are the numbers
λ12−(λ11λ22/λ21), (λ11λ22)/(λ12λ21), (λ12/λ11)−(λ22/λ21) and
λ11λ22−λ21λ12
transcendental?
Denote byLethe set of linear combination, with algebraic coef- ficients, of 1 and logarithms of algebraic numbers. A strong form of the Four Exponentials Conjecture states that a 2×2 matrix whose entries are in Leis regular, as soon as the two rows as well as the two columns are linearly independent over the field Qof algebraic numbers. From this conjecture follows a positive answer (apart from trivial cases) to the previous question for the first three num- bers: not only they are transcendental, but, even more, they are not in the setL. This Strong Four Exponentials Conjecture doese not seem sufficient to settle the question for the last number, which amounts to prove that a 3×3 matrix is regular; the Conjecture of algebraic independence of logarithms of algebraic numbers provides the answer.
2000Mathematics Subject Classification. 11J81 11J86 11J89.
Key words and phrases. Transcendental numbers, logarithms of algebraic num- bers, four exponentials Conjecture, six exponentials Theorem, algebraic indepen- dence, elliptic functions, periods ofK3-surfaces.
The author is thankful to the organizing committee of the International Confer- ence on Algebra and Number Theory, especially its secretary Rajat Tandon, for their invitation, to the faculty, staff and students of Hyderabad University for the excellent job they did to welcome the participants, to Stephane Fischler for his comments on a preliminary version of this paper, to Guy Diaz whose suggestions enabled the author to improve the results and to Hironori Shiga for contributing with his appendix.
337
The first goal of this paper is to give the state of the art on these questions: we replace the Strong Four Exponen- tials Conjecture by the Strong Six Exponentials Theorem of D. Roy; we deduce that in a set of 2 numbers, at least one element is not inLe(and therefore is transcendental). The second goal is to replace the Conjecture of algebraic inde- pendence of logarithms by the Linear Subgroup Theorem;
we obtain partial results on the non existence of quadratic relations among logarithms of algebraic numbers. The third and last goal is to consider elliptic analogs of these state- ments.
An appendix by Hironori Shiga provides a link with periods ofK3-surfaces.
1. Conjectures — Exponential Case
Denote by
Qthe field of algebraic numbers (algebraic closure of
Qin
C) and byL the
Q-vector space of logarithms of algebraic numbers:L = {λ ∈
C; e
λ∈
Q×} = {log α ; α ∈
Q×} = exp
−1(Q
×).
Here is the main Conjecture (see for instance [5], Historical Note of Chapter III, [4], Chap. 6 p. 259 and [12], Conjecture 1.15):
Conjecture 1.1. (Algebraic Independence of Logarithms of Alge- braic Numbers). Let λ
1, . . . , λ
nbe
Q-linearly independent elements ofL.
Then λ
1, . . . , λ
nare algebraically independent.
The following special case of Conjecture 1.1 was already investigated by Th. Schneider ([9], end of Chap. 5), S. Lang ([5], Chap. II § 1) and K. Ramachandra ([6] II § 4) in the 1960’s (see [12], Conjecture 1.13):
Conjecture 1.2. (Four Exponentials Conjecture). Let M be a 2 ×2 matrix with entries in L. Assume that the two rows of M are linearly independent over
Qand also that the two columns of M are linearly independent over
Q. ThenM has rank 2.
It is plain that a 2 × 2 matrix
(1.3) M =
µ
λ
11λ
12λ
21λ
22¶
has rank < 2 if and only if there exist x
1, x
2, y
1, y
2such that λ
ij= x
iy
j(i = 1, 2, j = 1, 2). Hence Conjecture 1.2 is equivalent to the following statement: if x
1, x
2are two complex numbers which are
Q-linearly independent and if y
1, y
2are two complex numbers which are also
Q-linearly independent, then one at least of the four numberse
x1y1, e
x1y2, e
x2y1, e
x2y2is transcendental.
Let λ
ij(i = 1, 2, j = 1, 2) be four non-zero elements of L. Assume that the two rows of the matrix (1.3) are linearly independent over
Qand also that the two columns are linearly independent over
Q. TheFour Exponentials Conjecture 1.2 states that each of the following four numbers is not zero:
(1.4) λ
12− λ
11λ
22λ
21, 1 − λ
11λ
22λ
21λ
12, λ
12λ
11− λ
22λ
21, λ
11λ
22− λ
21λ
12. We investigate the transcendence of these numbers. From Conjecture 1.1, it follows that each of them is algebraic only in trivial cases (where in fact it is rational). More precisely, we show that the diophantine nature (rational, algebraic irrational or transcendental) of each of the three first numbers in (1.4) is settled by the following special case of Conjecture 1.1, suggested by D. Roy (see for instance [12] Conjecture 11.17).
Denote by L
ethe
Q-vector space spanned by 1 andL in
C. HenceL
eis the set of linear combinations of logarithms of algebraic numbers with algebraic coefficients:
L
e=
nβ
0+ β
1log α
1+ · · · + β
nlog α
n;
n ≥ 0, (α
1, . . . , α
n) ∈ (Q
×)
n, (β
0, β
1, . . . , β
n) ∈
Qn+1 o. Conjecture 1.5. (Strong Four Exponentials Conjecture). Let M be a 2 × 2 matrix with entries in L. Assume that the two rows of
eM are linearly independent over
Qand also that the two columns of M are linearly independent over
Q. ThenM has rank 2.
Equivalently, if x
1, x
2are two complex numbers which are
Q-linearlyindependent and if y
1, y
2are two complex numbers which are also
Q-linearly independent, then one at least of the four numbers
x
1y
1, x
1y
2, x
2y
1, x
2y
2does not belong to L.
eWe derive several consequences of the Strong Four Exponentials Con- jecture 1.5.
Consequence 1.6. Let Λ be a transcendental element of L. Then
e1/Λ is not in L.
eConsequence 1.7. Let Λ
1, Λ
2be two elements of L. Assume that
eΛ
1and Λ
2/Λ
1are transcendental. Then this quotient Λ
2/Λ
1is not in
L.
eConsequence 1.8. Let Λ
1, Λ
2be two transcendental elements of L.
eThen the product Λ
1Λ
2is not in L.
eConsequence 1.9. Let Λ
1, Λ
2, Λ
3be three elements of L
ewith Λ
26=
0. Assume that the two numbers Λ
1/Λ
2and Λ
3/Λ
2are transcendental.
Then Λ
1Λ
3Λ
26∈ L.
eExamples where Consequences 1.7, 1.8 and 1.9 hold uncondition- ally (i.e. without assuming Conjecture 1.5) are given in § 2 below (see Corollaries 2.7, 2.8 and 2.9).
Proof of Consequences 1.6, 1.7, 1.8 and 1.9 assuming Conjecture 1.5.
Let Λ
0be an element in L. We apply the Strong Four Exponentials Con-
ejecture 1.5 to the matrices
µ
Λ 1 1 Λ
0¶
,
µ
Λ
1Λ
21 Λ
0¶
,
µ
1 Λ
2Λ
1Λ
0¶
and
µ
Λ
2Λ
3Λ
1Λ
0¶
.
¤
Since 1 ∈ L
eone may also deduce Consequences 1.7 and 1.8 from 1.9 and then 1.6 from 1.7.
We recall Baker’s Theorem on linear independence of logarithms of algebraic numbers: if λ
1, . . . , λ
nare
Q-linearly independent elements ofL, then the numbers 1, λ
1, . . . , λ
nare
Q-linearly independent.Consequence 1.9 applies to the three first numbers in (1.4). Let λ
ij(i = 1, 2, j = 1, 2) be four non-zero elements of L. Assuming Conjecture 1.5, we deduce
λ
12− λ
11λ
22λ
216∈
Q\ {0}, λ
11λ
22λ
12λ
216∈
Q\
Qand λ
12λ
11− λ
22λ
216∈
Q\
Q.Moreover, again under Conjecture 1.5, if the number λ
12λ
11− λ
22λ
21is rational, then
• either λ
12/λ
11∈
Qand λ
22/λ
21∈
Q• or λ
21/λ
11∈
Q.Notice that if λ
11and λ
12are two elements of L and a, b two rational numbers with bλ
116= 0, then
λ
12λ
11− bλ
12− aλ
11bλ
11= a
b ·
Consequence 1.8 shows that, if Conjecture 1.5 holds, and if λ
11, λ
12, λ
22are three elements of L such that λ
11λ
226= 0, then both numbers
λ
11λ
22− λ
12and λ
12λ
11− λ
22are transcendental.
The transcendence of the last number λ
11λ
22− λ
21λ
12in (1.4) does not seem to follow from the Strong Four Exponentials Conjecture. How- ever Conjecture 1.1 claims that any algebraic relation among logarithms of algebraic numbers is homogeneous, hence:
Consequence 1.10. (of Conjecture 1.1). Let λ
ij(i = 1, 2, j = 1, 2) be four transcendental elements of L. Assume
λ
11λ
226= λ
21λ
12. Then the number
λ
11λ
22− λ
21λ
12is not in L.
eConsequence 1.10 amounts to study certain quadratic relations among logarithms of algebraic numbers. One may deduce it directly from the next conjecture of D. Roy on the rank of matrices whose entries are in L
e(see [8], remarks (i) and (ii) p. 54, as well as section 12.1.4 in [12]).
Let M be a d × ` matrix with entries in L. Let Λ
e 1, . . . , Λ
sbe a basis over
Qof the vector space spanned by these entries. Write
M = M
1Λ
1+ · · · + M
sΛ
s,
where M
1, . . . , M
sare d × ` matrices with algebraic entries. The struc- tural rank of M with respect to
Qis defined as the rank of the matrix
M
1X
1+ · · · + M
sX
swhose entries are in the field
Q(X1, . . . , X
s). This definition is indepen- dent of the chosen basis Λ
1, . . . , Λ
s.
As noted by D. Roy (see [12] Prop. 12.13), Conjecture 1.1 is equiv- alent to the next statement:
Conjecture 1.11. The rank of a matrix with entries in L
eis equal to its structural rank with respect to
Q.By homogeneity, for any λ
ij(i = 1, 2, j = 1, 2) in L \ {0} with
λ
11λ
226= λ
12λ
21and for any Λ ∈ L \ {0}
ethe structural rank with respect to
Qof
1 0 λ
110 1 λ
12λ
22−λ
21Λ
is 3. Hence Conjecture 1.11 implies Consequence 1.10.
The two next statements, which are consequences of Conjecture 1.11, involve cubic relations among logarithms of algebraic numbers.
Consequence 1.12. Let λ
ij(i = 1, 2, j = 1, 2) be four non-zero elements of L. Then
λ
12λ
11− λ
22λ
21is not in L \
e Q.Consequence 1.13. Let λ
ij(i = 1, 2, j = 1, 2) be four non-zero elements of L. Then
λ
11λ
22λ
21λ
12is not in L \
e Q.One deduces Consequences 1.12 and 1.13 from Conjecture 1.11 by introducing the matrices
λ
11λ
120 λ
21λ
22Λ 0 λ
211
and
λ
11λ
120 λ
210 1 0 λ
22Λ
with Λ ∈ L.
e2. Theorems — Exponential Case
The sharpest known result in direction of the Strong Four Exponen- tials Conjecture 1.5 is the following one, due to D. Roy ([7] Corollary 2
§4 p. 38; see also [12] Corollary 11.16).
Theorem 2.1. (Strong Six Exponentials Theorem). Let M be a 2 × 3 matrix with entries in L. Assume that the two rows of
eM are linearly independent over
Qand also that the three columns of M are linearly independent over
Q. ThenM has rank 2.
Equivalently, if x
1, x
2are two complex numbers which are
Q-linearlyindependent and if y
1, y
2, y
3are three complex numbers which are also
Q-linearly independent, then one at least of the six numbersx
1y
1, x
1y
2, x
1y
3, x
2y
1, x
2y
2, x
2y
3does not belong to L.
eWe select a sample of consequences of Theorem 2.1 related to the conjectural statements of § 1.
The first one involves the inverse of an element in L, like in Conse-
equence 1.6, and also a quotient of two elements in L, like in Consequence
e1.7.
Corollary 2.2. Let Λ
1, Λ
2be two elements of L. Assume that the
ethree numbers 1, Λ
1, Λ
2are
Q-linearly independent. Then one at leastof the two numbers 1/Λ
1, Λ
2/Λ
1is not in L.
eThe second corollary deals with two quotients of elements in L
eand provides a partial answer to Consequence 1.7.
Corollary 2.3. Let Λ
1, Λ
2, Λ
3be three elements of L
eAssume that Λ
1is transcendental and that the three numbers Λ
1, Λ
2, Λ
3are
Q-linearly independent. Then one at least of the two numbersΛ
2/Λ
1, Λ
3/Λ
1is not in L.
eThe third one involves two products of elements in L
eand therefore is related to Consequence 1.8.
Corollary 2.4. Let Λ
1, Λ
2, Λ
3be three transcendental elements of L
eAssume that the three numbers 1, Λ
2, Λ
3are
Q-linearly independent.Then one at least of the two numbers Λ
1Λ
2, Λ
1Λ
3is not in L.
eThe next corollary combines Consequence 1.7 and Consequence 1.9.
Corollary 2.5. Let Λ
1, Λ
2, Λ
3be three transcendental elements of L
eAssume that Λ
1/Λ
2is transcendental and that the three numbers 1, Λ
2, Λ
3are
Q-linearly independent. Then one at least of the two numbersΛ
1/Λ
2, Λ
1Λ
3/Λ
2is not in L.
eThe last one is a weak but unconditional version of Consequence 1.9.
Corollary 2.6. Let Λ
1, Λ
2, Λ
3, Λ
4be four elements of L. Assume
ethat Λ
1/Λ
2is transcendental and that the three numbers Λ
2, Λ
3, Λ
4are linearly independent over
Q. Then one at least of the two numbersΛ
1Λ
3Λ
2, Λ
1Λ
4Λ
2is not in L.
eProof of Corollaries 2.2, 2.3, 2.4, 2.5 and 2.6. We apply The- orem 2.1 to the matrices
µΛ
11 Λ
21 Λ Λ
0¶
,
µ
Λ
1Λ
2Λ
31 Λ Λ
0¶
,
µ
1 Λ
2Λ
3Λ
1Λ Λ
0¶
,
µ
Λ
2Λ
31 Λ
1Λ Λ
0¶
,
µ
Λ
2Λ
3Λ
4Λ
1Λ Λ
0¶
, with Λ and Λ
0in L.
e¤
From Corollary 2.4 we deduce an example due to G. Diaz [3] where the statement 1.8 holds. See also [2]. The basic idea occurs initially in [6] I p. 68: a complex number is algebraic if and only if both its real part and its imaginary part are algebraic.
Corollary 2.7. (G. Diaz). Let Λ
1∈ L∩(R∪iR)\Q
eand let Λ
2∈ L.
eAssume that the three numbers 1, Λ
2and Λ
2are linearly independent over
Q. Then the productΛ
1Λ
2is not in L.
eIndeed Corollary 2.4 with Λ
3= Λ
2shows that one at least of the two numbers Λ
1Λ
2, Λ
1Λ
2is not in L. From the assumption Λ
e 1= ±Λ
1the result follows.
Using Hermite-Lindemann’s Theorem L ∩
Q= {0},
we deduce that for Λ
1∈ L ∩
e(R ∪ iR) \
Qand λ
2∈ L \ (R ∪ iR), the product Λ
1λ
2is not in L.
eIn the same vein we may produce examples where Consequences 1.7 and 1.9 hold. We give only two such examples.
Corollary 2.8. Let Λ
1∈ L ∩
e(R ∪ iR) \
Qand Λ
2∈ L. Assume
ethat the three numbers Λ
1, Λ
2and Λ
2are linearly independent over
Q.Then the quotient Λ
2/Λ
1is not in L.
eCorollary 2.9. Let Λ
1, Λ
2, Λ
3be three elements in L
ewith Λ
1and Λ
2in
R∪ iR and Λ
26= 0. Assume that Λ
1/Λ
2is transcendental and that the three numbers Λ
2, Λ
3and Λ
3are
Q-linearly independent. ThenΛ
1Λ
3/Λ
2is not in L.
eOne deduces from Corollary 2.6 the following results. Let M be a 2 × 3 matrix with entries in L:
(2.10) M =
µ
λ
11λ
12λ
13λ
21λ
22λ
23¶
.
Assume that the two numbers λ
11, λ
21are
Q-linearly independent, andalso that the three numbers λ
21, λ
22, λ
23are
Q-linearly independent.Then
(i) one at least of the two numbers λ
12− λ
11λ
22λ
21, λ
13− λ
11λ
23λ
21is transcendental,
(ii) if λ
12λ
136= 0, then one at least of the two numbers λ
11λ
22λ
21λ
12, λ
11λ
23λ
21λ
13is transcendental,
(iii) one at least of the two numbers λ
12λ
11− λ
22λ
21, λ
13λ
11− λ
23λ
21is transcendental
and
(iv) one at least of the two numbers λ
11λ
21− λ
12λ
22, λ
11λ
21− λ
13λ
23is transcendental.
From Corollary 2.4 it follows that, if λ
11, λ
12, λ
13are three ele- ments of L with λ
116= 0 and if the two numbersλ
22, λ
23are
Q-linearlyindependent elements of L, then
(i) one at least of the two numbers
λ
11λ
22− λ
12, λ
11λ
23− λ
13is transcendental
and
(ii) one at least of the two numbers λ
12λ
11− λ
22, λ
13λ
11− λ
23is transcendental.
For a 2 × 3 matrix (2.10) with entries in L, assuming that on each row and on each column the entries are linearly independent over
Q, onewould like to prove that one at least of the two numbers
λ
11λ
22− λ
21λ
12, λ
11λ
23− λ
21λ
13is transcendental (see Corollary 2.13 for a special case). In a forthcoming paper we shall prove that one at least of the three 2 × 2 determinants
λ
11λ
22− λ
21λ
12, λ
11λ
23− λ
21λ
13, λ
12λ
23− λ
22λ
13is not in L.
eA partial answer will follow from the next result, whose proof is
given in § 3.
Theorem 2.11. Let M = (Λ
ij)
1≤i≤m;1≤j≤`be a m × ` matrix with entries in L. Denote by
eI
mthe identity m × m matrix and assume that the m + ` column vectors of the matrix (I
m, M ) are linearly independent over
Q. LetΛ
1, . . . , Λ
mbe elements of L. Assume that the numbers
e1, Λ
1, . . . , Λ
mare
Q-linearly independent. Assume further` > m
2. Then one at least of the ` numbers
Λ
1Λ
1j+ · · · + Λ
mΛ
mj(j = 1, . . . , `) is not in L.
eTaking m = 1, ` = 2 in Theorem 2.11 we recover Corollary 2.4. In case m = 2, ` = 5 we deduce the next two corollaries.
Corollary 2.12. Let M be a 2 × 5 matrix with entries in L:
e µΛ
11Λ
12Λ
13Λ
14Λ
15Λ
21Λ
22Λ
23Λ
24Λ
25¶
.
Assume that Λ
216= 0, that the number Λ
11/Λ
21is transcendental and that the seven columns of the matrix (I
2, M ) are linearly independent over
Q. Then one at least of the four numbersΛ
11Λ
2j− Λ
21Λ
1j(j = 2, 3, 4, 5) is not in L.
eProof. In Theorem 2.11 we set
m = 2, ` = 5, Λ
1= −Λ
21and Λ
2= Λ
11.
¤
By adding further hypotheses and by using the complex conjugation
`a la Diaz we can reduce from 4 to 2 the number of elements in the set which we show not to be included in L:
eCorollary 2.13. Let M be a 2 × 3 matrix with entries in L:
e µΛ
11Λ
12Λ
13Λ
21Λ
22Λ
23¶
.
Assume that Λ
11, Λ
12, Λ
13and Λ
21are in
R∪ iR and that Λ
216= 0. As- sume further that the number Λ
11/Λ
21is transcendental and furthermore that the six numbers 1, Λ
21, Λ
22, Λ
23, Λ
22, Λ
23are linearly independent over
Q. Then one at least of the two numbersΛ
11Λ
22− Λ
21Λ
12, Λ
11Λ
23− Λ
21Λ
13is not in L.
eProof. For j = 2 and j = 3 define
Λ
j= Λ
11Λ
2j− Λ
21Λ
1j. We apply Corollary 2.12 to the 2 × 5 matrix
µ
Λ
11Λ
12Λ
13Λ
12Λ
13Λ
21Λ
22Λ
23Λ
22Λ
23¶
with
Λ
14= Λ
12, Λ
15= Λ
13, Λ
24= Λ
22, Λ
25= Λ
23.
We deduce that one at least of the four numbers Λ
2, Λ
2, Λ
3, Λ
3is not in L, hence one at least of the two numbers Λ
e 2, Λ
3is not in L.
e¤
We conclude this section with the following consequence of Theorem 2.11, whose conclusion goes further than [12] Exercise 11.10 (which is [11] Corollary 2.5).
Corollary 2.14. Let M = (λ
ij)
1≤i≤m;1≤j≤`be a m × ` matrix with entries in L whose columns are linearly independent over
Qand let λ
1, . . . , λ
mbe
Q-linearly independent elements inL. Assume ` > m
2. Then one at least of the ` numbers
λ
1λ
1j+ · · · + λ
mλ
mj(j = 1, . . . , `) is not in L.
eProof. According to part (iii) of Exercise 11.5 in [12], Baker’s ho- mogeneous Theorem implies that
Q-linearly independent elements inL
mare linearly independent over
Q. In the same way, using the full force ofBaker’s Theorem, we deduce that the columns of the matrix (I
m, M ) are linearly independent over
Q. We use again Baker’s Theorem to deducefrom the assumptions of Corollary 2.14 that the numbers 1, λ
1, . . . , λ
mare
Q-linearly independent. Hence the hypotheses of Theorem 2.11 arefulfilled for Λ
i= λ
iand Λ
ij= λ
ij.
¤
3. Proof of Theorem 2.11 and further results
Theorem 5 of [7] (see also Corollary 11.15 in [12]) deals with the intersection V ∩ L
edwhen V is a vector subspace of
Cd. For the proof of Theorem 2.11 we shall need only the special case where V is a hyperplane of
Cd.
Theorem 3.1. Let V be a hyperplane in
Cdsuch that V ∩
Qd= {0}.
Then V ∩ L
edis a finite dimensional vector space over
Qof dimension
≤ d(d − 1).
Proof of Theorem 2.11. Let d = m + 1. The hyperplane V of
Cdof equation
Λ
1z
1+ · · · + Λ
mz
m= z
m+1contains the m + ` following points:
γ
i= (²
i, Λ
i) (1 ≤ i ≤ m) and
γ
m+j= (Λ
1j, . . . , Λ
mj, Λ
0j) (1 ≤ j ≤ `), where (²
1, . . . , ²
m) is the canonical basis of
Cmand where
Λ
0j= Λ
1Λ
1j+ · · · + Λ
mΛ
mj(j = 1, . . . , `).
The definition of Λ
01, . . . , Λ
0`means that the (m + 1) × (m + `) matrix M
f=
µ
I
mM
Λ
1· · · Λ
mΛ
01· · · Λ
0`¶
. has rank m.
Since 1, Λ
1, . . . , Λ
mare
Q-linearly independent, we haveV ∩
Qd= {0}. Since the columns of M
fare
Q-linearly independent, them + ` points γ
1, . . . , γ
m+`are linearly independent over
Q. The assumption` > m
2yields m + ` > d(d − 1), hence Theorem 3.1 shows that one at least of the ` numbers Λ
01, . . . , Λ
0`is not in L.
e¤
Another corollary of Theorem 3.1 is the following.
Corollary 3.2. Let M = (Λ
ij)
1≤i≤m;1≤j≤`be a m × ` matrix with entries in L
ewhose columns are linearly independent over
Qand let x
1, . . . , x
mbe
Q-linearly independent complex numbers. Assume` >
m(m − 1). Then one at least of the ` numbers
x
1Λ
1j+ · · · + x
mΛ
mj(j = 1, . . . , `) is not zero.
Proof. Apply Theorem 3.1 with d = m to the hyperplane V of equation
x
1z
1+ · · · + x
mz
m= 0
in
Cm.
¤Further related results, which deserve to be compared with Theorem
2.11, follow from Corollary 11.6 in [12] which reads as follows.
Theorem 3.3. Let d
0≥ 0 and d
1≥ 1 be two positive integers. Set d = d
0+ d
1. Le V be a vector subspace of
Cdof dimension n which satisfies
V ∩ (Q
d0× {0}) = {0} and V ∩ ({0} ×
Qd1) = {0}.
Denote by `
0the dimension of the
Q-vector spaceV ∩
Qd. Then the
Q-vector spaceV ∩ (Q
d0× L
d1) has finite dimension bounded by
dim
Q¡V ∩ (Q
d0× L
d1)
¢≤ d
1(n − `
0).
A consequence of Theorem 3.3 is Corollary 1.6 in [11]. However there are two misprints in this last statement which we take the oppor- tunity to correct here: firstly the assumption in Corollary 1.6 of [11]
that t
1, . . . , t
mare
Q-linearly independent should be replaced by the as-sumption that 1, t
1, . . . , t
mare
Q-linearly independent, and secondly inthe condition ` > (r − 1)(n + 1) one should read m in place of n. Here is an equivalent formulation of the corrected statement.
Corollary 3.4. Let M = (λ
ij)
1≤i≤m;1≤j≤`be a m × ` matrix with entries in L whose columns are linearly independent over
Qand let x
1, . . . , x
mbe
Q-linearly independent complex numbers. Denote byr the dimension of the
Q-vector space spanned byx
1, . . . , x
mand assume
` > m(r − 1). Then one at least of the ` numbers x
1λ
1j+ · · · + x
mλ
mj(j = 1, . . . , `) is not zero.
If we use only the upper bound r ≤ m in Corollary 3.4, the result we obtain is a consequence of Corollary 3.2 (compare with part a) of Exercise 11.9 in [12]).
Here is another consequence of Theorem 3.3 (with d
0= 1, d
1= n = m, `
0= m − r).
Corollary 3.5. Let M = (λ
ij)
1≤i≤m;1≤j≤`be a m × ` matrix with entries in L whose columns are linearly independent over
Qand let x
1, . . . , x
mbe
Q-linearly independent complex numbers. Denote byr + 1 the dimension of the
Q-vector space spanned by1, x
1, . . . , x
mand assume
` > mr. Then one at least of the ` numbers
x
1λ
1j+ · · · + x
mλ
mj(j = 1, . . . , `) is transcendental.
When 1, x
1, . . . , x
mare linearly independent over
Q, the condition` > mr in Corollary 3.5 for getting a transcendental number is the same
as in Theorem 2.11 for getting an element outside of L. However the
eformer result holds with arbitrary complex numbers x
ibut restricts λ
ijto be in L, while the later one, namely Theorem 2.11, requires Λ
i∈ L
eand allows Λ
ijin L.
e4. Elliptic Case
The previous study involves the multiplicative group
Gm. Similar statements hold for an elliptic curve E which is defined over
Q: wereplace the field
Qby the field
kE= EndE ⊗
ZQof endomorphisms of E, the
Q-vector spaceL by the
kE-vector space L
Eof elliptic logarithms of algebraic points on E, namely
L
E= exp
−1EE (Q)
and the
Q-vector spaceL
eby the
Q-vector spaceL
eEspanned by 1 and L
Ein
C.Recall that the elliptic curve E has complex multiplication or is a CM curve if
kE6=
Q, in which casekEis an imaginary quadratic field.
The Weierstrass elliptic function ℘ associated to E satisfies a differ- ential equation
℘
02= 4℘
3− g
2℘ − g
3with algebraic invariants g
2, g
3; from the definition it follows that L
Eis the set of elliptic logarithm of algebraic points of ℘, namely the set of complex numbers u such that either u is a period of ℘ or else ℘(u) is algebraic.
A special case of the elliptico-toric Conjecture of C. Bertolin in [1]
is the following elliptic analog of Conjecture 1.1.
Conjecture 4.1. (Algebraic Independence of Elliptic Logarithms of Algebraic Numbers). Let u
1, . . . , u
nbe
kE-linearly independent elements of L
E. Then u
1, . . . , u
nare algebraically independent.
From this conjecture one readily deduces elliptic analogs to all the statements in § 1: one replaces the
Q-vector spaceL
eby the
Q-vectorspace L
eEconsisting of linear combinations
β
0+ β
1u
1+ · · · + β
nu
nof elliptic logarithms u
i∈ L
Ewith algebraic coefficients β
j.
Elliptic analogs of the results in § 2 deserve to be considered – we plan to do it elsewhere. Here we content ourselves with two transcendence statements (compare with Corollary 2.12) whose proofs are given in § 5.
Theorem 4.2. Let ℘ be a Weierstrass elliptic functions with alge-
braic invariants g
2, g
3. For 1 ≤ j ≤ 7, let λ
j∈ L and u
j∈ L
E. Assume
that the seven points (λ
1, u
1), . . . , (λ
7, u
7) are
Q-linearly independent inC2
. Assume also λ
16= 0 and u
16= 0. Then one at least of the six numbers
λ
1u
j− λ
ju
1(j = 2, . . . , 7) is transcendental.
Furthermore, if u
1is a period ω of ℘ and λ
1= 2iπ, then one at least of the four numbers
2iπu
j− λ
jω (j = 2, . . . , 5) is transcendental.
Theorem 4.3. Let ℘ and ℘
∗be two Weierstrass elliptic functions with algebraic invariants g
2, g
3and g
2∗, g
3∗respectively. For 1 ≤ j ≤ 9, let u
j(resp. u
∗j) be an elliptic logarithm of an algebraic point of ℘ (resp.
℘
∗). Assume that the nine points (u
1, u
∗1), . . . , (u
9, u
∗9) are
Q-linearlyindependent in
C2. Assume further either that the elliptic curves E and E
∗are non isogeneous and u
1u
∗16= 0, or else that E = E
∗and that the two numbers u
1, u
∗1are linearly independent over
kE. Then one at least of the eight numbers
u
ju
∗1− u
∗ju
1(j = 2, . . . , 9) is transcendental.
Furthermore, if u
1is a period ω of ℘ and u
∗1a period ω
∗of ℘
∗, then one at least of the six numbers
u
jω
∗− u
∗jω (j = 2, . . . , 7) is transcendental.
Remark. From the elliptico-toric Conjecture of C. Bertolin in [1]
one deduces that, under the assumptions of Theorem 4.3, the number u
2u
∗1− u
∗2u
1is transcendental. More precisely, in the case where E and E
∗are non iso- geneous, her conjecture implies that the number u
1u
∗1is transcendental, and that the transcendence degree t over
Qof the field
Q(u1, u
2, u
∗1, u
∗2) satisfies
t =
4 if u
1, u
2are linearly independent over
kEand u
∗1, u
∗2are linearly independent over
kE∗, 2 if u
1, u
2are linearly dependent over
kEand u
∗1, u
∗2are linearly dependent over
kE∗,
3 otherwise.
In particular if we denote by (ω
1, ω
2) (resp. (ω
∗1, ω
2∗)) a pair of fundamen- tal periods of ℘ (resp. of ℘
∗), then, according to Bertolin’s Conjecture, the number
ω
2ω
∗1− ω
∗2ω
1is transcendental. I wish to thank H. Shiga who pointed out to me that such numbers occur as periods of K3 surfaces (we refer to the appendix).
His remark was the initial motivation for this paper.
5. Proofs of Theorems 4.2 and 4.3
The proofs of Theorems 4.2 and 4.3 rely on the following special case of the Algebraic Subgroup Theorem ([10] Th. 1.1 and [11] Th. 4.1) – the full statement deals with a vector subspace V, here we need only to consider a hyperplane.
Theorem 5.1. Let d
0, d
1, d
2be three non negative integers with d = d
0+ d
1+ d
2> 0. Let G
2be a commutative algebraic group over
Qof dimension d
2and set G =
Gda0×
Gdm1× G
2. Let V be a hyperplane of the tangent space T
e(G) and Y a finitely generated subgroup of V of rank ` such that exp
G(Y ) ⊂ G(Q). Let κ be the
Z-rank ofV ∩ ker exp
G. Assume
` > (d − 1)(d
1+ 2d
2− κ).
Then V contains a non-zero algebraic Lie subalgebra of T
e(G) defined over
Q.Proof of Theorem 4.2. Consider the matrix
µλ
1λ
2. . . λ
`u
1u
2. . . u
`¶