Dependence of logarithms on commutative algebraic groups

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1 INTERNATIONAL CONFERENCE on ALGEBRA and NUMBER THEORY Hyderabad, December 11 16, 2003 Dependence of logarithms on commutative algebraic groups Michel Waldschmidt miw/ 1

2 Transcendence of periods of K3 surfaces Conjecture. Let and be two non isogeneous Weierstraß elliptic functions with algebraic invariants g 2, g 3 and g 2, g 3 respectively. Let {ω 1, ω 2 } and {ω 1, ω 2} be a pair of fundamental periods of and respectively. Then the number is transcendental. ω 1 ω 2 ω 1ω 2 miw@math.jussieu.fr miw/ 2

3 Multiplicative analog Conjecture. Let α 11, α 12, α 21, α 22 be four non zero algebraic numbers. For i = 1, 2 and j = 1, 2, let λ ij C satisfy e λ ij = α ij. Then the number λ 11 λ 22 λ 12 λ 21 is either 0 or else transcendental. miw@math.jussieu.fr miw/ 3

4 Multiplicative analog Conjecture. Let α 11, α 12, α 21, α 22 be four non zero algebraic numbers. For i = 1, 2 and j = 1, 2, let λ ij C satisfy e λ ij = α ij. Then the number λ 11 λ 22 λ 12 λ 21 is either 0 or else transcendental. Notice: λ 11 λ 22 λ 12 λ 21 = det λ 11 λ 12 λ 21 λ 22. miw@math.jussieu.fr miw/ 4

5 Vanishing of λ 11 λ 22 λ 12 λ 21 Denote by L the Q vector space of logarithms of algebraic numbers: L = {λ C ; e λ Q} = {log α ; α Q } = exp 1 (Q ). For r Q, λ L and λ L, det λ λ rλ rλ = 0 and det λ rλ λ rλ = 0. miw@math.jussieu.fr miw/ 5

6 Four Exponentials Conjecture. For i = 1, 2 and j = 1, 2, let α ij be a non zero algebraic number and λ ij a complex number satisfying e λ ij = α ij. Assume λ 11, λ 12 are linearly independent over Q and also λ 11, λ 21 are linearly independent over Q. Then λ 11 λ 22 λ 12 λ 21. miw@math.jussieu.fr miw/ 6

7 Four Exponentials Conjecture. For i = 1, 2 and j = 1, 2, let α ij be a non zero algebraic number and λ ij a complex number satisfying e λ ij = α ij. Assume λ 11, λ 12 are linearly independent over Q and also λ 11, λ 21 are linearly independent over Q. Then λ 11 λ 22 λ 12 λ 21. A. Selberg, C.L. Siegel, Th Schneider, S. Lang, K. Ramachandra. miw@math.jussieu.fr miw/ 7

8 Algebraic independence of logarithms of algebraic numbers Conjecture. Let α 1,..., α n be non zero algebraic numbers. For 1 j n let λ j C satisfy e λ j = α j. Assume λ 1,..., λ n are linearly independent over Q. Then λ 1,..., λ n are algebraically independent. Write λ j = log α j. If log α 1,..., log α n are Q-linearly independent then they are algebraically independent. miw@math.jussieu.fr miw/ 8

9 Recall that L denotes the Q vector space of logarithms of algebraic numbers: L = {λ C ; e λ Q} = {log α ; α Q } = exp 1 (Q ). The conjecture on algebraic independence of logarithms of algebraic numbers can be stated: The injection of L into C extends into an injection of the symmetric algebra Sym Q (L) on L into C. miw@math.jussieu.fr miw/ 9

10 Consequence of the conjecture on algebraic independence of logarithms of algebraic numbers 1 Four Exponentials Conjecture. For i = 1, 2 and j = 1, 2, let α ij be a non zero algebraic number and λ ij a complex number satisfying e λ ij = α ij. Assume λ 11, λ 12 are linearly independent over Q and also λ 11, λ 21 are linearly independent over Q. Then λ 11 λ 22 λ 12 λ 21. miw@math.jussieu.fr miw/ 10

11 Consequence of the conjecture on algebraic independence of logarithms of algebraic numbers 1 Four Exponentials Conjecture. For i = 1, 2 and j = 1, 2, let α ij be a non zero algebraic number and λ ij a complex number satisfying e λ ij = α ij. Assume λ 11, λ 12 are linearly independent over Q and also λ 11, λ 21 are linearly independent over Q. Then λ 11 λ 22 λ 12 λ 21. λ 11 λ 12λ 21 λ 11 λ 22 0, 0, λ 22 λ 12 λ 21 λ 11 λ 21 0, λ 11 λ 22 λ 12 λ λ 12 λ 22 miw@math.jussieu.fr miw/ 11

12 Consequence of the conjecture on algebraic independence of logarithms of algebraic numbers Let λ ij (i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers. 2 Assume Then λ 11 λ 12λ 21 λ 22 Q. λ 11 λ 22 = λ 12 λ 21. miw@math.jussieu.fr miw/ 12

13 Consequence of the conjecture on algebraic independence of logarithms of algebraic numbers Let λ ij (i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers. 3 Assume Then λ 11 λ 22 λ 12 λ 21 Q. λ 11 λ 22 λ 12 λ 21 Q. miw@math.jussieu.fr miw/ 13

14 Consequence of the conjecture on algebraic independence of logarithms of algebraic numbers Let λ ij (i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers. 4 Assume λ 11 λ 21 Q. λ 12 λ 22 Then either λ 11 /λ 12 Q and λ 21 /λ 22 Q or λ 12 /λ 22 Q and λ 11 λ 12 λ 21 λ 22 Q. miw@math.jussieu.fr miw/ 14

15 Remark: λ 11 λ 12 bλ 11 aλ 12 bλ 12 = a b miw@math.jussieu.fr miw/ 15

16 Consequence of the conjecture on algebraic independence of logarithms of algebraic numbers Let λ ij (i = 1, 2, j = 1, 2) be four non zero logarithms of algebraic numbers. 5 Assume Then λ 11 λ 22 λ 12 λ 21 Q. λ 11 λ 22 = λ 12 λ 21. miw@math.jussieu.fr miw/ 16

17 Algebraic independence of elliptic logarithms Conjecture. Let E 1,..., E n be elliptic curves which are pairwise non isogeneous. For 1 h n, let h be the Weierstraß elliptic function associated to E h ; assume its invariants g 2h, g 3h are algebraic. Let k h = End(E h ) Z Q be the field of endomorphisms of E h and u 1h,..., u rh h be complex numbers which are linearly independent over k h such that h (u ih ) Q for 1 i r h. Then the r r h numbers u ih, (1 i r h, 1 h n) are algebraically independent. miw@math.jussieu.fr miw/ 17

18 Mixed conjecture of algebraic independence Conjecture. Let E h, h, u ih be as in the conjecture of algebraic independence of elliptic logarithms. Let u 10,..., u r0 0 be Q-linearly independent of L. Then the r 0 + r r h numbers u ih, (1 i r h, 0 h n) are algebraically independent. miw@math.jussieu.fr miw/ 18

19 General Conjectures Generalized Grothendieck Conjecture of periods (Yves André) Special case of 1-motives [Z r G s m Elliptico-toric Conjecture n h=1 E h ] (Cristiana Bertolin) For n = 0: Schanuel s Conjecture. [Z r G s m] miw@math.jussieu.fr miw/ 19

20 Schanuel s Conjecture. Let x 1,..., x n be Q-linearly independent complex numbers. Then the transcendence degree of the field Q ( x 1,..., x n, e x 1,..., e x n ) is at least n. miw@math.jussieu.fr miw/ 20

21 Schanuel s Conjecture. Let x 1,..., x n be Q-linearly independent complex numbers. Then the transcendence degree of the field Q ( x 1,..., x n, e x 1,..., e x n ) is at least n. Special case: Taking x i L for 1 i n one recovers the conjecture of algebraic independence of logarithms of algebraic numbers. miw@math.jussieu.fr miw/ 21

22 Elliptic conjecture of C. Bertolin. Let E 1,..., E n be pairwise non isogeneous elliptic curves with modular invariants j(e h ). Let ω 1h, ω 2h be a pair of fundamental periods of h and η 1h, η 2h the associated quasi-periods, P ih points on E h (C), p ih and d ih elliptic integrals of the first (resp. second) kind associated to P ih. Define κ h = [k h : Q] and let d h be the dimension of the k h - subspace of C/(k h ω 1h + k h ω 2h ) spanned by p 1h,..., p rh h. Then the transcendence degree of the field ( {j(eh } ) ), ω 1h, ω 2h, η 1h, η 2h, P ih, p ih, d ih Q 1 i r h 1 h n is at least 2 n h=1 d h + 4 n h=1 κ 1 h n + 1. miw@math.jussieu.fr miw/ 22

23 Example of a mixed situation Conjecture. Let be a Weierstraß elliptic function with algebraic invariants g 2, g 3 and let ω 1, ω 2 be a pair of fundamental periods of. Let λ 1, λ 2 be two non zero elements of L. Then the determinant does not vanish. λ 1 λ 2 ω 1 ω 2 miw@math.jussieu.fr miw/ 23

24 λ 1 λ 2 ω 1 ω 2 0. Special case: λ 1 = 2iπ: Define τ = ω 2 /ω 1, q = e 2iπτ, J(q) = j(τ). Theorem (K. Barré-Sirieix, F. Gramain, G. Philibert). If q C satisfies 0 < q < 1, then one at least of the two numbers q, J(q) is transcendental. miw@math.jussieu.fr miw/ 24

25 Transcendence results Six Exponentials Theorem. Let x 1, x 2 be two complex numbers which are Q-linearly independent. Let y 1, y 2, y 3 be three complex numbers which are Q-linearly independent. Then one at least of the six numbers is transcendental. e x iy j (i = 1, 2, j = 1, 2, 3) miw@math.jussieu.fr miw/ 25

26 Transcendence results Six Exponentials Theorem (again). For i = 1, 2 and j = 1, 2, 3, let α ij be a non zero algebraic number and λ ij a complex number satisfying e λ ij = α ij. Assume λ 11, λ 12, λ 13 are linearly independent over Q and also λ 11, λ 21 are linearly independent over Q. Then the matrix has rank 2. ( λ11 λ 12 λ 13 λ 21 λ 22 λ 23 ) miw@math.jussieu.fr miw/ 26

27 Transcendence results 1 (Six exponentials) (λ 11 λ 22 λ 12 λ 21, λ 11 λ 23 λ 13 λ 21 ) (0, 0). miw@math.jussieu.fr miw/ 27

28 Transcendence results 1 (Six exponentials) (λ 11 λ 22 λ 12 λ 21, λ 11 λ 23 λ 13 λ 21 ) (0, 0). ( 2 λ 12 λ 11λ 22, λ 13 λ ) 11λ 23 Q Q. λ 21 λ 21 miw@math.jussieu.fr miw/ 28

29 Transcendence results 1 (Six exponentials) (λ 11 λ 22 λ 12 λ 21, λ 11 λ 23 λ 13 λ 21 ) (0, 0). ( 2 λ 12 λ 11λ 22, λ 13 λ ) 11λ 23 Q Q. λ 21 λ 21 ( λ12 λ 21 λ 3, 13 λ ) 21 Q Q. λ 11 λ 22 λ 11 λ 23 miw@math.jussieu.fr miw/ 29

30 Transcendence results 1 (Six exponentials) (λ 11 λ 22 λ 12 λ 21, λ 11 λ 23 λ 13 λ 21 ) (0, 0). ( 2 λ 12 λ 11λ 22, λ 13 λ ) 11λ 23 Q Q. λ 21 λ 21 ( λ12 λ 21 λ 3, 13 λ ) 21 Q Q. λ 11 λ 22 λ 11 λ 23 ( 4 λ12 λ λ 22, λ λ 21 λ λ ) 23 Q Q. 11 λ 21 miw@math.jussieu.fr miw/ 30

31 Transcendence results 1 (Six exponentials) (λ 11 λ 22 λ 12 λ 21, λ 11 λ 23 λ 13 λ 21 ) (0, 0). ( 2 λ 12 λ 11λ 22, λ 13 λ ) 11λ 23 Q Q. λ 21 λ 21 ( λ12 λ 21 λ 3, 13 λ ) 21 Q Q. λ 11 λ 22 λ 11 λ 23 ( 4 λ12 λ λ 22, λ λ 21 λ λ ) 23 Q Q. 11 λ 21 ( ) 5 λ11 λ 22 λ 12 λ 21, λ 11 λ 23 λ 13 λ 21 Q Q? miw@math.jussieu.fr miw/ 31

32 Theorem 1. Let λ ij (i = 1, 2, j = 1, 2, 3, 4, 5) be ten non zero logarithms of algebraic numbers. Assume λ 11, λ 21 are linearly independent over Q and λ 11,..., λ 15 are linearly independent over Q. Then one at least of the four numbers is transcendental. λ 1j λ 21 λ 2j λ 11, (j = 2, 3, 4, 5) miw@math.jussieu.fr miw/ 32

33 Elliptic Analogue Theorem 2. Let and be two non isogeneous Weierstraß elliptic functions with algebraic invariants g 2, g 3 and g2, g3 respectively. For 1 j 9 let u j (resp. u j ) be a non zero logarithm of an algebraic point of (resp. ). Assume u 1,..., u 9 are Q-linearly independent. Then one at least of the eight numbers u j u 1 u ju 1 (j = 2,..., 9) is transcendental miw@math.jussieu.fr miw/ 33

34 Sketch of Proofs Theorem 3. Let G = G d 0 a G d 1 m G 2 be a commutative algebraic group over Q of dimension d = d 0 + d 1 + d 2, V a hyperplane of the tangent space T e (G), Y a finitely generated subgroup of V of rank l 1 such that exp G (Y ) G(Q) with l 1 > (d 1)(d 1 + 2d 2 ). Then V contains a non zero algebraic Lie subalgebra of T e (G) defined over Q. miw@math.jussieu.fr miw/ 34

35 Sketch of Proof of Theorem 1 Assume λ 1j λ 21 λ 2j λ 11 = γ j for 1 j l 1. Take G = G a G 2 m, d 0 = 1, d 1 = 2, G 2 = {0}, V is the hyperplane z 0 = λ 21 z 1 λ 11 z 2 and Y = Zy Zy l1 with y j = (γ j, λ 1j, λ 2j ), (1 j l 1 ). Since (d 1)(d 1 + 2d 2 ) = 4, we need l 1 5. miw@math.jussieu.fr miw/ 35

36 Assume Sketch of Proof of Theorem 2 u j u 1 u ju 1 = γ j for 1 j l 1. Take G = G a E E, d 0 = 1, d 1 = 0, d 2 = 2, V is the hyperplane z 0 = u 1z 1 u 1 z 2 and Y = Zy Zy l1 with y j = (γ j, u j, u j), (1 j l 1 ). Since (d 1)(d 1 + 2d 2 ) = 8, we need l 1 9. miw@math.jussieu.fr miw/ 36

37 In both cases we need to check that V does not contain a non zero Lie subalgebra of T e (G). For Theorem 1 this follows from the assumption λ 11, λ 21 are Q-linearly independent, while for Theorem 2 this follows from the assumption E, E are not isogeneous. miw@math.jussieu.fr miw/ 37

38 Remarks. 1. The proofs of results 1 (six exponentials theorem), 2, 3, 4 above are similar: they also rest on Theorem Theorem 3 also contains Baker s Theorem on linear independence of logarithms. 3. A refinement of Theorem 3 is available (and useful). If V contains a subspace W of dimension l 0 which is rational over Q, then the condition can be replaced by l 1 > (d 1)(d 1 + 2d 2 ) l 1 > (d l 0 1)(d 1 + 2d 2 ). miw@math.jussieu.fr miw/ 38

39 Theorem 4. Let G = G d 0 a G d 1 m G 2 be a commutative algebraic group over Q of dimension d = d 0 + d 1 + d 2, V a hyperplane of the tangent space T e (G), Y a finitely generated subgroup of V of rank l 1 and W a subspace of T e (G) of dimension l 0 which is rational over Q and contained in V. Assume exp G (Y ) G(Q) and l 1 > (d l 0 1)(d 1 + 2d 2 ). Then V contains a non zero algebraic Lie subalgebra of T e (G) defined over Q. A further extension to subspaces of T e (G) (in place of hyperplanes) yields the Algebraic subgroup theorem. The case V = W is due to G. Wüstholz. miw@math.jussieu.fr miw/ 39

40 Algebraic independence of logarithms (again) Conjecture. Let n be a positive integer, X an affine algebraic variety of C n defined over Q, P a point on X with coordinates on L and V the smallest vector subspace of C n defined over Q which contains P. Then V is contained in X. miw@math.jussieu.fr miw/ 40

41 Algebraic independence of logarithms (again) Conjecture. Let n be a positive integer, X an affine algebraic variety of C n defined over Q, P a point on X with coordinates on L and V the smallest vector subspace of C n defined over Q which contains P. Then V is contained in X. Conjecture. Let M be a 4 4 skew symmetric matrix with entries in L and with Q-linearly independent rows. Assume that the Q-vector space spanned by the columns of M in C 4 does not contain a non zero element of Q 4. Then the rank of M is 3. miw@math.jussieu.fr miw/ 41

42 Special case: the four exponentials conjecture. 0 λ 11 λ 12 0 λ λ 21 λ λ 22 0 λ 21 λ 22 0 miw@math.jussieu.fr miw/ 42

43 Rank of matrices whose entries are logarithms of algebraic numbers Let M be a matrix whose entries are in L. Let λ 1,..., λ r be a basis of the Q-vector space spanned by the entries of M. Write M = M 1 λ M r λ r where M 1,..., M r have rational entries. Denote by r str (M) the rank of the matrix in the field Q(X 1,..., X r ). M = M 1 X M r X r Conjecture. The rank of M is r str (M). miw@math.jussieu.fr miw/ 43

44 Proposition (D. Roy). This conjecture is equivalent to the conjecture of homogeneous algebraic independence of logarithms of algebraic numbers: if λ 1,..., λ n are Q-linearly independent elements of L and if P is a non zero homogeneous polynomial with rational coefficients, then P (λ 1,..., λ n ) 0. Lemma (D. Roy). Any polynomial A Z[X 1,..., X n ] is the determinant of a matrix with entries in the Z-module Z + ZX ZX n. miw@math.jussieu.fr miw/ 44

45 Further developments Quadratic relations among logarithms of algebraic numbers. Abelian varieties in place of product of elliptic curves. Semi abelian varieties. Commutative algebraic groups. Taking periods into account. miw/ 45

The four exponentials conjecture, the six exponentials theorem and related statements.

Number Theory Seminar at NCTS October 29, 2003 The four exponentials conjecture, the six exponentials theorem and related statements. Michel Waldschmidt miw@math.jussieu.fr http://www.math.jussieu.fr/