NONCOMMUTATIVE COMPLEX GEOMETRY ON THE MODULI SPACE OF Q-LATTICES IN C

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1 NONCOMMUTATIVE COMPLEX GEOMETRY ON THE MODULI SPACE OF Q-LATTICES IN C Joint work in progress with A. Connes on the hypoelliptic metric structure of the noncommutative complex moduli space of Q-lattices in C modulo commensurability. 1

2 Q-lattices [Connes-Marcolli] φ : Q 2 /Z 2 QΛ/Λ group homomorphism (Λ 1, φ 1 ) (Λ 2, φ 2 ) iff φ 1 = φ 2 mod Λ 1 + Λ 2 (Λ, φ) is invertible if φ is an isomorphism. Two invertible Q-lattices are commensurable iff they are equal.! general case invertible case! The ordinary space of invertible Q-lattices sits inside the quotient nc-space L 2 = GL 2(Q)\ GL 2 (A) L 2 = {(Λ, φ)}/ 2

3 (Λ, φ) = (λ(z + Zτ), λρ), λ C, τ H, and ρ M 2 (Ẑ) = Hom(Q 2 /Z 2, Q 2 /Z 2 ) L 2 the locally compact groupoid obtained as the quotient of U 2 = {(α, ρ, g) GL + 2 (Q) M 2(Ẑ) GL + 2 (R) ; αρ M 2 (Ẑ)}, by the action of Γ Γ, where Γ = SL 2 (Z), (γ 1, γ 2 ) (g, ρ, g) = (γ 1 αγ 1 2, γ 2ρ, γ 2 g); r[α, ρ, g] = [αρ, αg] s[α, ρ, g] = [ρ, g], [α 1, ρ 1, g 1 ] [α 2, ρ 2, g 2 ] = [α 1 α 2, ρ 2, g 2 ]. The quotient by scaling B 2 := L 2 /C is no longer a groupoid but has convolution algebra. 3

4 Coordinate algebras C c (L 2 ) := algebra of Γ Γ-invariant continuous functions on U 2 with compact support modulo Γ Γ, and product (f 1 f 2 )(α, ρ, g) = β Γ\G + (Q), βρ M 2 (Ẑ) f 1 (αβ 1, βρ, βg) f 2 (β, ρ, g) and -involution f (α, ρ, g) = f(α 1, αρ, αg). C (L 2 ) := C -completion in the regular representation w.r.t. invariant measure. 4

5 The space of units L (0) 2 = Γ\(M 2 (Ẑ) GL + 2 (R)) is endowed with the invariant measure dν(ρ, g) := d + ρ d + g ; the contravariant change of variables on the finite adèles is compensated by the covariant one on matrices over R: d + (β 1 ρ) = (det β) 2 d + ρ d + (β 1 g) = (det β) 2 d + g. For y = (ρ, g) Y := M 2 (Ẑ) GL + 2 (R), let G y = {α GL + 2 (Q) αy Y }, H y = l 2 (Γ\G y ), (π y (f) ξ y )(α) := Then π(f) := β Γ\G y f(αβ 1, βy) ξ y (β). Γ\Y π y(f) dν(y), f := sup y Y π y (f) gives the ν-regular representation of C (L 2 ). 5

6 Holomorphic flat connection Both the base and the fibre of C L (0) 2 B (0) 2 = Γ\(M 2 (Ẑ) H) have canonical complex structures. For any u < 1, view the Q-lattice (Λ, φ) as lattice in C u, w.r.t. to complex structure d u z := dz + ud z; one obtains germ of map v (Λ(v), φ(v)), where Λ(v) := {ψ v (ω) = i v ω + ω; ω Λ} 2 covol(λ) φ(v) = ψ v φ. The normalization by covolume scales as (λ Λ)(v) = λ Λ( v λ 2), λ C, giving a canonical identification Jet 1 C (B(0) 2 ) (L(0) 2 ) 2. 6

7 Canonical invariant vertical vector field of the principal C -bundle L 2 : Y := 1 2 λ 1 λ. Canonical invariant horizontal vector field X R (F )(Λ) := 1 d 2πidv v=0f (Λ(v)). However, if F is a modular form of weight 2k, in the coordinates Λ = Zω 1 + Zω 2, X R (F ) = 1 ( df 2πi dz + 2kF ), z = ω 1. z z ω 2 Holomorphic affine connection has horizontal vector field: X = X R + 2 E (2) 0 Y, where E (2) 0 = the quasi-holomorphic Eisenstein series of weight 2. Holomorphic affine framing: T (1,0) = CY + CX, [Y, X ] = X. 7

8 Eisenstein series For k > 2 and a = (a 1, a 2 ) (Q/Z) 2 G (k) a (z) := m 0, m a (1) (m 1 z + m 2 ) k. When k = 1 or k = 2, replace by G (k) a (z, 0), G (k) a (z, s) := (m 1 z + m 2 ) k m 1 z + m 2 s, with G (0) a (z) only quasi-holomorphic, that is z G (0) a (z) + 2πi z z is holomorphic in z H. For any weight k 1 and level N, E (k) x (z) := (k 1)! (2πiN) k ψ x (a) G a (z), ) 2 a ( 1N Z/Z ( a1 ψ x N, a ) 2 N := e 2πi(a 2x 1 a 1 x 2 ), a j N 1 N Z/Z. 8

9 E (2) 0 is quasi-holomorphic, more precisely E (2) 0 (z) = 1 ( 2 d ) 1 (log η) + 2πi dz z z. Fourier expansion: B k(x 1 ) k + 0<r x 1 (1) + 0<r x 1 (1) n=1 E (k) x (z) = n=1 r k 1 e 2πin(x 2+rz) r k 1 e 2πin( x 2+rz), where B k is the periodized Bernoulli function B k (x) = B k (x [x]), x R. Distribution Law: E (k) x = y α=x E (k) y α, α M + 2 (Z). 9

10 Modular forms as lattice functions Let F be a modular form of weight k 2Z + and let F be the corresponding lattice function Then F (Λ(ω 1, ω 2 )) = ω k 2 F (ω 1 ω 2 ) ω 1, ω 2 C, Im ω 1 ω 2 > 0. χ (F )(g) = F (ǧλ 0 ), g GL + 2 (R), where ǧ = det g g 1, Λ 0 := Z 1 + Z ( i), ( ) χ (F )(g) := (ci + d) k a b F (g i), g =. c d Thus the passage to F involves the operation F k g (z) := (cz + d) k F (g z). 10

11 Arithmetic multipliers 1 0. Modular Hecke algebra A (Γ): consists of finitely supported maps F : Γ\G + (Q) M, Γα F α, with values in modular forms, satisfying F αγ = F α γ, α G + (Q), γ Γ, and with product defined by the rule (F 1 F 2 ) α := Γβ Γ\G + (Q) F 1 αβ 1 β F 2 β. Proposition. The elements of A (Γ) supported in Γ\M 2 + (Z) form a subalgebra A (M 2 + (Z), Γ), and the assignment F f F, F A (M 2 + (Z), Γ), f F (α, ρ, g) := χ (F α ) (F α g) (i), maps to the multiplier algebra of C (L 2 ). 11

12 2 0. Lifting by χ the arithmetic algebra A Q of [Connes-Marcolli] gives ρ-dependent arithmetic multipliers of weight Eisenstein multipliers: x = (x 1, x 2 ) Q2 xe (k) (α, ρ, g) := ρ = ( a b c d ) χ ( E (k) ) x ˇρ (g) if α Γ, 0 if α / Γ, ( ) M 2 + d b (Z), ˇρ = c a ; Z 2 Proposition. xe (k) satisfies the cyclotomic condition, hence is an arithmetic multiplier of weight k. 12

13 Hopf algebra symmetry The Ramanujan operator acting on the modular Hecke algebra A (Γ) satisfies X (F α) = det α X (F ) α + µ α Y(F ) α where µ α = 2 ( E (2) 0 E (2) 0 α). This gives rise to an action of H 1 := H 1 C[σ, σ 1 ] on A (Γ), where H 1 is the algebra generated by {X, Y, δ n ; n 1} modulo the relations [Y, X] = X, [Y, δ k ] = kδ k [X, δ k ] = δ k+1, [δ j, δ k ] = 0. H 1 is in fact a Hopf algebra, with coproduct (Y ) = Y Y, (X) = X 1 + σ X + δ 1 Y, (δ 1 ) = δ σ δ 1, (σ) = σ σ, (σ 1 ) = σ 1 σ 1, 13

14 counit ɛ(x) = ɛ(y ) = ɛ(δ k ) = 0, ɛ(σ) = 1, and antipode S(σ) = σ 1, S(X) = σ 1 ( X + δ 1 Y ) S(Y ) = Y, S(δ 1 ) = σ 1 δ 1. The generators of H 1 act on A (Γ) op as follows: F A (Γ) and α GL + 2 (Q), Y(F ) α = Y(F α ), X (F ) α = X (F α ), δ n (F ) α = X n 1 (µ α ) F α, σ(f ) α = det(α) F α. Proposition. A (Γ) op is a Hopf module-algebra, i.e. h H 1 h(a op b) = h (1) (a) op h (2) (b), a, b A (Γ) in Sweedler s notation (h) = h (1) h (2). 14

15 In the coordinates on GL + 2 (R) ( y g = 1/2 xy 1/2 ) ( ) ( r 0 cos θ sin θ 0 y 1/2 0 r sin θ cos θ the holomorphic flat connection becomes Y(f)(α, ρ, g) = 1 ( r 4 r + 1 ) f (α, ρ, g), i θ X (f) = 1 4π r 2 R(E)(f) + 2E 0 (g)y(f) where E 0 = χ (E (2) 0 ), and ( R(E)(f)(α, ρ, g) = e 2iθ iy f x + y f y + 1 ) f 2i θ ), Theorem. There is a unique Hopf action of H 1 on C (L 2 ) op, extending via χ the action on A (Γ) op, defined on generators as follows: Y (f) = Y(f), X(f) = X (f), δ n (f)(α, ρ, g) = X n 1 (µ α )(g) f(α, ρ, g), σ(f)(α, ρ, g) = det(α) f(α, ρ, g), f C (L 2 ). 15

16 Holomorphic Rankin-Cohen deformation Theorem. 1 0 With O(L 2 )= subalgebra of holomorphic multipliers of C (L 2 ), the following formulae define bilinear operations RC n : O(L 2 ) O(L 2 ) O(L 2 ), n 0: RC n(a, b) = n k=0 σ n k A k k! (2Y + k) n k (a) B n k (n k)! (2Y + n k) k(b); where (Z) k := (Z + 1)...(Z + k 1) and A 1 = 0, A 0 = 1, A n+1 = ( X + δ 1 Y )A n nr(ω) ( Y n 1 B 0 = 1, B 1 = X, B n+1 = XB n nl(ω) ( Y n 1 ) 2 Bn 1 ; 2 ) An 1, Ω = χ (E 4 ) is Eisenstein series of weight The product formula a b := n RC n(b, a) gives an associative deformation of O(L 2 ). 16

17 Quasi-regular representation If one ignores the complex affine structure, one can reduce the above action to the quasiregular representation of GL + 2 (R) on the Hilbert space H := Γ\(M 2 (Ẑ) GL + 2 (R)) H (ρ,g) d+ ρ d + g of the ν-regular representation of C (L 2 ), by conjugation with a 1-cocycle u Z 1 (H 1, C (L 2 ) op ). Basis of gl(2, C) = sl(2, C) CI: ( ) 1 i ( 1 i ) E = 1 2 i 1 H = 1 ( 0 i 2 i 0 ), Ē = 1 2, I = with nonzero bracket relations i ( ), 1 [H, E] = 2E, [H, Ē] = 2Ē, [E, Ē] = H. 17

18 Then the non-affine conjugate action is Y = Y = 1 4( R(I) R(H) ), X = 1 4πr 2 R(E), δ n 0, n 1, where R(E) = e 2iθ ( R(Ē) = e 2iθ ( R(H) = 1 i iy x + y y + 1 2i θ iy x + y y 1 2i θ, R(I) = r r. ), θ ), Since d + g = r 3 dr dxdy y 2 dθ = r4 dg, R(E) = R(Ē), R(H) = R(H), while R(I) = R(I) 4 Id. 18

19 Hypoelliptic -operator Complex structure on Γ\(M 2 (Ẑ) GL + 2 (R)) : Y = 1 ( r 4 r + 1 ), i θ X = 1 4π r 2 R(E) + 2E 0 Y, Ȳ = 1 ( r 4 r + 1 ), i θ X = 1 4π r 2 R(Ē) + 2Ē 0 Ȳ. After taking the product by P 1 (C), one can form the -analogue of the hypoelliptic signature operator: Q = ( V V V V ) γ V ( H + H ). Q = D D. 19

20 Local Index Formula ϕ n (a 0,..., a n ) = k N n c n,k a 0 [Q, a 1 ] (k 1)... [Q, a n ] (k n) Q n 2 k, where (T ) = [Q 2, a], T (k) = k (T ), k = k k n, c n,k = 2i( 1) k (k 1!... k n!) 1 Γ ( k + n ) 2 ( (k1 + 1)... (k k n + n) ) 1, defines an (odd) cocycle φ D = {ϕ n } n=1,3,... in the (b, B)-bicomplex of A whose cyclic cohomology class gives the Chern character of (A, H, D). 20

21 Quasi-Hopf symmetry Lie brackets : [Y, Ȳ] = 0, [Y, X ] = 0, [X, X ] = ς 2 H, where H = 2(Y Ȳ) and ς(ρ, g) := 1 4π det(g); Since Y(ς 2 ) = ς 2, X (ς 2 ) = 2E 0 ς 2, Ȳ(ς 2 ) = ς 2, X (ς 2 ) = 2Ē 0 ς 2. X (E 0 ) = c 1 (E 0 )2 + c 2 χ (E 4 ), c 1, c 2 C, these relations bring in the base ring E generated by χ -lifts of Eisenstein series and their complex conjugates. The resulting symmetry structure is a quantum double of H 1 over the ring E. 21

22 Connes, Alain; Marcolli, Matilde Q-lattices: quantum statistical mechanics and Galois theory. J. Geom. Phys. 56 (2006), no.1, Connes, A.; Moscovici, H. Hopf algebras, cyclic cohomology and the transverse index theorem. Comm. Math. Phys. 198 (1998), no. 1, Connes, Alain; Moscovici, Henri Modular Hecke algebras and their Hopf symmetry. Mosc. Math. J. 4 (2004), no. 1, , 310. Connes, Alain; Moscovici, Henri Rankin-Cohen brackets and the Hopf algebra of transverse geometry. Mosc. Math. J. 4 (2004), no. 1, ,