Arithmetic of the BC-system. A. Connes (CDF)

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1 Arithmetic of the BC-system A. Connes (CDF) (a project in collaboration with C. Consani) Oberwolfach September 2011 The three Witt constructions The BC-endomotive and W 0 ( F p ). The completion W( F p ) and p-adic representations of the BCsystem. The KMS theory and division relations of polylogarithms. Iwasawa theory and extension of the KMS theory to a covering of C p. 1

2 The Witt construction and the BC system The initial motivation to seek for this relation came from the discovery that the algebraic relations fulfilled by the operators σ n and ρ n of H Z and the relations of the Frobenius endomorphisms F n : W 0 (A) W 0 (A), n N and Verschiebung additive functorial maps V n : W 0 (A) W 0 (A), n N, in the Witt construction are identical. 2

3 2 fundamental operators act on Z[Q/Z]: 1) ρ n : Z[Q/Z] Z[Q/Z], ρ n (e(γ)) = e(γ ) nγ =γ 2) σ n : Z[Q/Z] Z[Q/Z], σ n (e(γ)) = e(nγ) Compatibility relations σ nm = σ n σ m, ρ mn = ρ m ρ n, m, n N ρ m (σ m (x)y) = x ρ m (y), x, y Z[Q/Z] σ c ρ b (x) = (b, c) ρ b σ c (x), b = b (b,c), c = c (b,c) σ n ρ n (x) = nx, n N σ n ρ m = ρ m σ n, if (m, n) = 1 3

4 The Witt functor W 0 W 0 (A) is functorially defined as the Grothendieck group of the category End A whose objects are pairs (E, f) where E is a finite projective module over A and f End A (E) is an endomorphism of E. (E 1, f 1 ) (E 2, f 2 ) = (E 1 E 2, f 1 f 2 ), (E 1, f 1 ) (E 2, f 2 ) = (E 1 E 2, f 1 f 2 ) turn the Grothendieck group K 0 (End A ) into a (commutative) ring. The pairs of the form (E, f = 0) generate the ideal K 0 (A) K 0 (End A ). W 0 (A) = K 0 (End A )/K 0 (A). 4

5 By construction W 0 is a functor from the category Ring of commutative rings with unit to itself. The key additional structures are given by The Teichmüller lift which is a multiplicative map τ : A W 0 (A). The Frobenius endomorphisms F n for n N. The Verschiebung (shift) additive functorial endomorphisms V n, n N. The ghost components gh n : W 0 (A) A for n N. 5

6 (1) The Teichmüller lift is simply given by the map A f (A, f). (2) For n N, the following operations on End A induce endomorphisms in W 0 (A) which are the Frobenius endomorphisms F n (E, f) = (E, f n ). (3) The Verschiebung maps V n are described by the following operation on matrices: V n (f) = f

7 (4) The ghost components are given by gh n : W 0 (A) A, gh n (E, f) = Trace(f n ).

8 One has (a) F n V n (x) = nx. (b) V n (F n (x)y) = xv n (y). (c) If (m, n) = 1, V m F n = F n V m. (d) For n N, V n (x)v n (y) = nv n (xy). (e) For n N, F n (τ(f)) = τ(f n ). (f) For n, m N, (g) gh n (V m (f)) = gh n (F m (f)) = gh nm (f). m gh n/m (f) if m n 0 otherwise. 7

9 The BC-system and W 0 ( F p ) Theorem To each σ : F p (Q/Z) (p) Q/Z, corresponds a canonical isomorphism σ W 0 ( F p ) σ Z[(Q/Z) (p) ] Z[Q/Z] r=id ϵ Z[(Q/Z) (p) ] The Frobenius F n and Verschiebung maps V n of W 0 ( F p ) are obtained by restriction of the endomorphisms σ n and maps ρ n of Z[Q/Z] by the formulas σ F n = σ n σ, σ V n = r ρ n σ 8

10 This Theorem shows that the integral BC-system with its full structure is, if one drops the p-component, completely described as W 0 ( F p ). As a corollary one gets a representation π σ of the integral BC-system H Z on W 0 ( F p ), π σ (x)ξ = σ 1 (r(x)) ξ, π σ (µ n ) = F n, π σ ( µ n ) = V n for all ξ W 0 ( F p ), x Z[Q/Z] and n N. 9

11 The big Witt functor W(A) Let Λ(A) := 1 + ta[[t]]. The characteristic polynomial defines a map L : W 0 (A) Λ(A), L(E, f) = det(1 E tf) 1, and by a fundamental result of Almkvist, the map L is always injective and its image is the subset of Λ(A) whose elements are rational fractions Range(L) = {(1+a 1 t+...+a n t n )/(1+b 1 t+...+b n t n ) a j, b j A}. 10

12 The addition in W 0 (A) by L(f g) = L(f)L(g), f, g W 0 (A) The Teichmüller lift which gives L(τ(f)) = (1 tf) 1 Λ(A) The shift V n L(Vn(f))(t) = L(f)(t n ) The ghost components 1 gh n (f)t n = t d dt log(l(f(t))). 11

13 This makes it clear how to extend the addition, the Teichmüller lift, the shifts and the ghost components to the completion Λ(A). The corresponding product on Λ(A) is uniquely determined by functoriality and requiring that the ghost components define ring homomorphisms. It is given by explicit polynomials with integral coefficients of the form (1 + a n t n ) (1 + b n t n ) = 1 + a 1 b 1 t + (a 2 1 b2 1 a 2b 2 1 a 2 1 b 2 + 2a 2 b 2 )t 2 + (a 3 1 b3 1 2a 1a 2 b a 3b 3 1 2a3 1 b 1b 2 + 5a 1 a 2 b 1 b 2 3a 3 b 1 b 2 + a 3 1 b 3 3a 1 a 2 b 3 + 3a 3 b 3 )t

14 Every element f(t) Λ(A) can be written uniquely as an infinite product f(t) = (1 z n t n ), z n A, n This shows that the following map is a bijection from the set W(A) of sequences (x n ) of elements of A to Λ(A) φ A : W(A) Λ(A) := 1 + ta[[t]], x = (x n ) n N f x (t) = N (1 x n t n ) 1. 13

15 At the level of the Witt vector components x j the Frobenius F n is given by polynomials with integral coefficients and for instance the first 5 components of F 3 (x) are F 3 (x) 1 = x x 3 F 3 (x) 2 = x 3 2 3x3 1 x 3 3x x 6 F 3 (x) 3 = 3x 6 1 x 3 9x 3 1 x2 3 8x x 9 F 3 (x) 4 = 3x 9 1 x 3 + 3x 3 1 x3 2 x 3 18x 6 1 x x3 2 x2 3 36x3 1 x3 3 24x x3 4 3x3 2 x 6 + 9x 3 1 x 3x 6 + 9x 2 3 x 6 3x x 12 F 3 (x) 5 = 3x 12 1 x 3 18x 9 1 x2 3 54x6 1 x3 3 81x3 1 x4 3 48x5 3 + x x 1 14

16 Artin-Hasse exponential E p (t) = hexp(t) = exp(t + tp p + tp2 p 2 + ) Λ(Z (p) ). (1) E p (t) is an idempotent of Λ(Z (p) ). (2) For n I(p), the series E p (n)(t) := 1 n V n(e p )(t) Λ(Z (p) ) determine an idempotent. As n varies in I(p), the E p (n) form a partition of unity by idempotents. (3) For n / p N, F n (E p )(t) = 1(= 0 Λ ) and F p k(e p )(t) = E p (t), k N. 15

17 (a) The map ψ A : W p (A) Λ(A) Ep, ψ A (x)(t) := h x (t) = N E p (x p nt pn ) is an isomorphism onto the reduced ring Λ(A) Ep = {x Λ(A) x E p = x}. (b) The composite θ A (x) = (θ A (x)) n = ψ 1 A F n(x E p (n)), n I(p), x Λ(A) is a canonical isomorphism θ A : Λ(A) W p (A) I(p) = W(A). (c) The composite isomorphism Θ A := θ A φ A : W(A) W p (A) I(p) is given explicitly on the components by (Θ A (x) n ) p k = F n (x) p k, x W(A), n I(p). 16

18 The completion W( F p ) of W 0 ( F p ) One has the isomorphism W( F p ) = (W p ( F p )) I(p) where W p is the Witt functor using the set of powers of the prime p. Let (Q un p ) C p be the completion of the maximal unramified extension Q un p of p-adic numbers. Then W p ( F p ) is the completion O = O (Q un Qun p ) of the p subring generated by roots of unity. 17

19 The p-adic representations of H Z Theorem Let σ X p and ρ : Q cycl,p C p associated to σ. The representation π σ of H Z extends to W( F p ). For n I(p), the π σ (µ n ) and for x Z[Q/Z], the π σ (x) are O-linear and π σ (µ n )ϵ m = ϵ nm, π σ (e(a/b))ϵ m = ρ(ζ m a/b )ϵ m n N, m, b I(p). Moreover π σ (µ p ) = Fr 1 is the inverse of the Frobenius. 18

20 Ideal J p of H Z Proposition Let J p H Z be the two-sided ideal generated by Then 1 e(p k ), k N J p = Ker(π σ ) and Z[Q/Z] J p is the ideal Jp 0 the 1 e(p k ). of Z[Q/Z] generated by 0 J 0 p Z[Q/Z] r Z[(Q/Z) (p) ] 0 19

21 The algebra H (p) Z Let H (p) Z quotient by J p of subalgebra of H Z generated by Z[Q/Z], µ n, µ n, n I(p). It is generated by Z[(Q/Z) (p) ], µ n, µ n for n I(p), the relations are µ nm = µ n µ m, µ nm = µ n µ m, n, m I(p)) µ n µ n = n, n I(p) µ n µ m = µ m µ n, n, m I(p), (n, m) = 1. µ n xµ n = ρ n(x), µ n x = σ n(x)µ n, x µ n = µ n σ n (x), where ρ n, n I(p) ρ n (e(γ)) = nγ =γ e(γ ), γ µ (p). 20

22 Automorphism Fr Aut(H (p) Z ) Unique Fr Aut(H (p) ) such that Z Fr(e(γ) = e(γ) p, γ µ (p), Fr( µ n ) = µ n, Fr(µ n) = µ n, One has an isomorphism H Z /J p = H (p) Z Fr, p Z n I(p) A θ, p Z { Z a n V n a n A} by the condition: a n p n A, n N Passing from H (p) Z to H Z is given by covariance for the action of the Frobenius Fr of O = O. Qun p 21

23 Theorem (1) The restriction π σ (p) of the representation π σ to H Z H (p) Z is O-linear and indecomposable over O. (2) The representations π σ H (p) are pairwise inequivalent. Z (3) The representation π σ is linear and indecomposable over Z p. (4) Two representations π σ and π σ are equivalent over Z p if and only if there exists α Aut( F p ) such that σ = σ α. 22

24 The KMS theory and division relations of polylogs These representations are the p-adic analogues of the complex, extremal KMS states of the BC-system. We have pursued this analogy much further by implementing the Iwasawa theory of p-adic L-functions to construct, in the p-adic case, the partition function and the KMS β states. In particular, we have shown that the division relations for the p-adic polylogarithms at roots of unity correspond to the KMS condition. 23

25 The time evolution in the p-adic case q = 4, if p = 2, q = p, if p 2. D p = {β C p β p < qp 1/(p 1) > 1}, Let r Z. There exists a unique analytic function (p) such that D p C p, β r (β) r (β) = r β, β = 1 kφ(q). r (β) := r exp((β 1) log p (r)), β D p. is a group homomorphism. Z (p) r r(β) C p 24

26 The automorphisms σ (β) Aut(H (p) C p ) (1) For β D p there exists a unique automorphism σ (β) Aut(H (p) C p ) such that σ (β) ( µ a e(γ)µ b ) = ( b a) (β) µ a e(γ)µ b, a, b I(p), γ (Q/Z)(p). (2) One has σ (β 1) σ (β 2) = σ (β 1+β 2 ) σ (0), β j D p and σ (0) is an automorphism of order φ(q). 25

27 KMS β states One looks for a linear form such that φ : H (p) C p C p, φ(1) = 1, φ(xσ β (y)) = φ(y x), x, y H (p) C p. 26

28 Formal expression in the p-adic case One looks for Z( a b, β) = m I(p) ρ(ζ m a/b ) m β, β D p ρ(ζ a/b m ) only depends on m modulo b. Let f = bp, one decomposes the sum in terms of the residue c, c < f, of m modulo f. Z( a b, β) = 1 c<bp c/ pn ρ(ζa/b c ) (c + fn) β n N = 1 c<bp c/ pn ρ(ζa/b c ) f β ( c n N f + n) β 27

29 For z = c f, one has c / pn and z 1 p < 1. n N (z + n) β = z1 β 1 + β j=0 ( 1 β j ) Bj z j f β (z + n) β = 1 n N f c 1 β β 1 j=0 ( 1 β j ) Bj z j 28

30 Euler-Maclaurin formula b k=a f(k) = b a f(t)dt + f(a) + f(b) 2 + m j=2 B j j! (f (j 1) (b) f (j 1) (a)) R m R m = ( 1)m m! b a f (m) (x)b m (x [x])dx k=0 f(k) 0 f(t)dt f(0) j=2 B j j! f (j 1) (0) 29

31 Hurwitz function One takes f(x) = (z + x) s. f (k) (0) = k! ( s) z s k, k 0 f(x)dx = z1 s 1 + s B j j! f (j 1) (0) = z1 s ( 1 s 1 + s j ) Bj z j For j = 0 one gets j = 1 z1 s 1+s contribution of 0 f(t)dt, for z 1 s 1 + s ( 1 s 1 ) ( 1 2 )z 1 = 1 2 z s 30

32 contribution of 1 2f(0). One gets k=0 (z + k) s z1 s 1 + s 0 ( 1 s j ) Bj z j

33 Construction of Z( a b, β) One defines Z( a b, β) := 1 bq 1 c<bq c/ pn ρ(ζ c a/b ) c 1 β β 1 j=0 ( 1 β j ) ( bp c )j B j Lemma The function Z( a b, β) is meromorphic with a simple pole at β = 1 in D p. 31

34 Solution of KMS β Theorem For any β D p, β 1, and ρ Hom(Q cycl,p, C p ) the linear form φ β,ρ fulfills the KMS β condition: φ β,ρ (xσ (β) (y)) = φ β,ρ (y x), x, y H (p) C p. The partition function is the p-adic L-function for the character χ = 1, Z(β) = L p (β, 1). One has Z(β) 0 for β D p and a pole at β = 1 with residu p 1 p. 32

35 Partition function a β dη = ( 1 (1 + q) 1 β) L p (β, 1) There exists η(t ) O p [[T ]] such that ((1 + q) 1 β 1)Z(β) = η((1 + q) β 1) 1 2 η(t ) O p[[t ]] 33

36 Change of parameter β λ ((1 + q) 1 β 1)Z(β) = η((1 + q) β 1) shows that λ = (1 + q) β is the right parameter. M = D(1, 1 ) = {λ C p λ 1 p < 1} open disk and multiplicative group. One has a homomorphism λ M = D(1, 1 ) β = surjective with kernel µ p. log p λ log p (1 + q) C p 34

37 Extension to the covering Theorem There exists an analytic family of functionals ψ λ,ρ, λ M, on H (p) Z such that ψ λ,ρ (1) = 1. ψ λ,ρ fulfills the KMS condition ψ λ,ρ (xσ[λ](y)) = ψ λ,ρ (y x), x, y H (p) C p. For β D p and λ = (1 + q) β one has ψ λ,ρ = Z(β) 1 φ β,ρ. 35

38 For β D p and r Z (p) one has r (β) = ω(r) λ i p(r), r β = λ i p(r) i p (r) = log p(r) log p (1 + q) Z p, λ = (1 + q) β 36

39 One compares Z ρ ( a, β) := b 1 f 1 c<f c/ pn ρ(ζ c a/b ) c 1 β β 1 with p-adic L-functions j=0 ( 1 β ) ( ) f j B j j c L p (β, χ) := 1 f 1 c<f c/ pn χ(c) c 1 β β 1 j=0 ( 1 β where χ is a primitive Dirichlet character. j ) ( ) f j B j c 37

40 Lemma Let a/b (Q/Z) (p), there exists c(d, χ) C p such that d b Z ρ ( a b, β) = c(d, χ)l p (β, χ)d 1 d 1 β (1 χ(l)l 1 l 1 β ) where d divides b, and χ is a primitive character of conductor f dividing m = b/d and the l are the prime divisors of m/f prime to f. 38

41 L p (β, χ) Let χ be a primitive Dirichlet character with conductor prime to p. If χ is odd (χ( 1) = 1) one has L p (β, χ) = 0. If χ is even there exists h χ O p [[T ]] such that L p (β, χ) = h χ ((1 + q) β 1) 39

42 Case β = 1 when β 1 one has Z(β) 1 Z ρ ( a { 1, if a b, β) b Z 0, otherwise. One obtains the regular representation σ (1) ( µ a e(γ)µ b ) = b a µ ae(γ)µ b 40

43 Lemma: Let λ Ẑ, λ ±1. The graph of multiplication by λ in Q/Z is dense in R/Z R/Z. Let θ Aut(µ (p) ), θ / {±p Z }. The graph of θ is dense in R/Z R/Z. 41

44 Let λ Ẑ G = {(α, λα) α Q/Z} subgroup of T = R/Z R/Z. If Ḡ T there exists (n, m) Z 2 non zero such that G is in the kernel of the character (n, m), nα + mλα Ẑ, α A Q, f (n + mλ p )α Z p, α Q p et n + mλ p = 0 for all p. In the second case Aut(µ (p) ) = Z l l p n + mλ l = 0 for l p and n/m {±p Z }, θ {±p Z } 42

45 Complex case Thus if f : {z C z = 1} C is continuous nonconstant, and ρ j : Q cycl C, an equality f(ρ 1 (ζ a/b )) = f(ρ 2 (ζ a/b )), a/b Q/Z implies ρ 2 = ρ 1 or ρ 2 = ρ 1. This case f( z) = f(z), z, z = 1. cannot happen for f(z) = n=1 n β z n, R(β) > 1. 43

46 p-adic case Let θ Aut(µ (p) ). If θ {±1} one has Z ρ ( a b, β) = Z θ ρ( a b, β), a/b µ(p), β D p. If θ / {±1} and β = 1 m = 1 kφ(q), the functionals Z ρ (, β) and Z θ ρ (, β) are distincts. 1 c b ρ (ζ c a/b ) B m( c b ) = 0 c b 1 ρ(ζa/b c ) B m( b c ). b B m (1 x) = B m (x), m 2N. 44

47 Let Fr Aut( (Q un p )) be the Frobenius, one has (1 p β Fr) 1 Z ρ ( a b, β) = bm 1 m 1 c b and the equality Z ρ = Z ρ implies 1 c b ρ(ζ c a/b ) B m( c b ) = ρ(ζa/b c ) B m( c ) (Q un p b ) 1 c b ρ (ζ c a/b ) B m( c b ) B m (θ 1 ( c b )) = B m( c b ) c/b µ(p). B m (x) = B m (p a x [p a x]), x [0, 1] would imply B m (x) B m (p a x) = 0. 45

On the arithmetic of the BC-system II.

On the arithmetic of the BC-system II. A. Connes (in collaboration with K. Consani) Jami Conference, March 23 2011 Review of the BC-system The BC-system and W 0 ( F p ) The KMS theory at p and Iwasawa