INTERNATIONAL CONFERENCE on ALGEBRA and NUMBER THEORY Hyderabad, December 11 16, 2003 Dependence of logarithms on commutative algebraic groups Michel Waldschmidt miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ http://www.uohyd.ernet.in/ 1
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 http://www.math.jussieu.fr/ miw/ 2
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 http://www.math.jussieu.fr/ miw/ 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. Notice: λ 11 λ 22 λ 12 λ 21 = det λ 11 λ 12 λ 21 λ 22. miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 4
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 http://www.math.jussieu.fr/ miw/ 5
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 http://www.math.jussieu.fr/ miw/ 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. A. Selberg, C.L. Siegel, Th Schneider, S. Lang, K. Ramachandra. miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 7
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 http://www.math.jussieu.fr/ miw/ 8
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 http://www.math.jussieu.fr/ miw/ 9
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 http://www.math.jussieu.fr/ miw/ 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. 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 21 0. λ 12 λ 22 miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 11
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 http://www.math.jussieu.fr/ miw/ 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. 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 http://www.math.jussieu.fr/ miw/ 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. 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 http://www.math.jussieu.fr/ miw/ 14
Remark: λ 11 λ 12 bλ 11 aλ 12 bλ 12 = a b miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 15
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 http://www.math.jussieu.fr/ miw/ 16
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 1 + + r h numbers u ih, (1 i r h, 1 h n) are algebraically independent. miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 17
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 1 + + r h numbers u ih, (1 i r h, 0 h n) are algebraically independent. miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 18
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 http://www.math.jussieu.fr/ miw/ 19
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 http://www.math.jussieu.fr/ miw/ 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. 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 http://www.math.jussieu.fr/ miw/ 21
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 http://www.math.jussieu.fr/ miw/ 22
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 http://www.math.jussieu.fr/ miw/ 23
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 http://www.math.jussieu.fr/ miw/ 24
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 http://www.math.jussieu.fr/ miw/ 25
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 http://www.math.jussieu.fr/ miw/ 26
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 http://www.math.jussieu.fr/ miw/ 27
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 http://www.math.jussieu.fr/ miw/ 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 ( λ12 λ 21 λ 3, 13 λ ) 21 Q Q. λ 11 λ 22 λ 11 λ 23 miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 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 ( 4 λ12 λ λ 22, λ 13 11 λ 21 λ λ ) 23 Q Q. 11 λ 21 miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 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 13 11 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 http://www.math.jussieu.fr/ miw/ 31
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 http://www.math.jussieu.fr/ miw/ 32
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 http://www.math.jussieu.fr/ miw/ 33
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 http://www.math.jussieu.fr/ miw/ 34
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 1 + + 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 http://www.math.jussieu.fr/ miw/ 35
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 1 + + 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 http://www.math.jussieu.fr/ miw/ 36
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 http://www.math.jussieu.fr/ miw/ 37
Remarks. 1. The proofs of results 1 (six exponentials theorem), 2, 3, 4 above are similar: they also rest on Theorem 3. 2. 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 http://www.math.jussieu.fr/ miw/ 38
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 http://www.math.jussieu.fr/ miw/ 39
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 http://www.math.jussieu.fr/ miw/ 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. 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 http://www.math.jussieu.fr/ miw/ 41
Special case: the four exponentials conjecture. 0 λ 11 λ 12 0 λ 11 0 0 λ 21 λ 12 0 0 λ 22 0 λ 21 λ 22 0 miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 42
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 1 + + 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 1 + + M r X r Conjecture. The rank of M is r str (M). miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 43
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 1 + + ZX n. miw@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 44
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@math.jussieu.fr http://www.math.jussieu.fr/ miw/ 45