+. 2. (Q)|αy ∈ Y },. Hy = l. 2. (Γ\Gy),. (πy(f)ξ)(α) := ∑ β∈Γ\Gy f(αβ. −1. ,βy)ξ(β). • The ν-regular ... m=0,m≡a(1). (m1z + m2. ) −k . When k = 1 or k = 2, replace by G. (k) a (z,0),. G. (k) a (z, s) := ∑ ... For any weight k ≥ 1 and level N,. E. (k) x (z) :=.
Noncommutative Geometry on Q-Spaces of Q-Lattices H. Moscovici This is joint work, in progress, with A. Connes on the complex NC-geometry of the quotient space of Q-lattices in C modulo commensurability. It builds on our prior work on modular Hecke algebras and their Hopf symmetry, and on the Connes-Marcolli C ∗-algebraic framework for Q-lattice spaces. The emerging spectral-geometric picture, resembling the transverse geometry of a generic codimension 1 foliation, has notable arithmetic features.
1
Q-lattices [Connes-Marcolli] Λ = Zω1 + Zω2, ω1, ω2 ∈ C, R-linearly indep. (Λ, φ)
with
φ : Q2/Z2 −→ QΛ/Λ
additive.
• (Λ1, φ1) ∼ (Λ2, φ2) iff QΛ1 = QΛ2, and φ1 − φ2 ≡ 0 mod Λ1 + Λ2. φ general case
invertible case
φ
(Λ, φ) is invertible if φ is an isomorphism; two invertible Q-lattices are commensurable iff they are equal, therefore L× = GL2(Q)\ GL2(A) ⊂ L = {(Λ, φ)}/ ∼ 2
Groupoid description ∼ the quotient L is a locally compact groupoid = of the locally compact groupoid + ˆ U = {(α, ρ, g) ∈ GL+ ( Q ) × M ( Z ) × GL 2 2 2 (R ) ; ˆ )} , αρ ∈ M2(Z
r[α, ρ, g] = [αρ, αg]
s[α, ρ, g] = [ρ, g] ,
[α1, ρ1, g1] ◦ [α2, ρ2, g2] = [α1α2, ρ2, g2] , by the action of Γ × Γ, where Γ = SL2(Z) (γ1, γ2) (g, ρ, g) = (γ1αγ2−1, γ2ρ, γ2g) . The above isomorphism is implemented by
[α, ρ, g] 7→ (g −1α−1Λ0, g −1ρ), (g −1Λ0, g −1ρ) Λ0 = Ze1 + Ze2 , with e1 = 1 and e2 = −i.
3
Coordinates • Cc(L) := algebra of Γ × Γ-invariant continuous functions on U with compact support modulo Γ × Γ, with product (f1 ∗ f2)(α, ρ, g) = f1(αβ −1, βρ, βg) f2(β, ρ, g),
X ˆ β∈Γ\ GL+ 2 (Q), βρ∈M2 (Z)
and involution
f ∗(α, ρ, g) = f (α−1, αρ, αg).
• C ∗(L) := C ∗-completion in the regular representation of the groupoid (cf. infra). • B := C∗\L, although not quite a groupoid, ∗ has C ∗(B) := C ∗(L)C as algebra of coordinates. Here ! λ = a + ib ∈ C is identified to a b λ= ∈ GL+ 2 (R). −b a • Similarly, LK := K\L has coordinate algebra C ∗(LK ) := C ∗(L)K , where K := SO(2). 4
ˆ ) × GL+(R)) • The units space L(0) = Γ\(M2(Z 2 is endowed with the invariant measure dν(ρ, g) := d+ρ ⊗ d+g , where d+ρ, d+g are additive measures on M2(). The contravariant change of variables on the finite adeles is compensated by the covariant change on 2 × 2-matrices over R: d+(β −1ρ) = (det β)2 d+ρ d+(β −1g) = (det β)−2 d+g . ˆ ) × GL+(R), let For y = (ρ, g) ∈ Y := M2(Z 2 + Gy = {α ∈ GL2 (Q) | αy ∈ Y }, Hy = `2(Γ\Gy ), (πy (f ) ξ)(α) :=
f (αβ −1, βy) ξ(β) .
X β∈Γ\Gy
• The ν-regular representation of C ∗(L) is π(f ) :=
Z ⊕ Γ\Y
πy (f ) dν(y) ,
||f || := supy∈Y ||πy (f )|| . 5
Modular forms as lattice functions A modular form of weight k ∈ Z+ is a holomorphic function F on H = {z ∈ C; Im(z) > 0}, s.t. F |k γ = F ,
∀ γ ∈ Γ ≡ Γ(1) := SL2(Z) ;
F |k g (z) := det(α)k/2 (cz + d)−k F (g · z) , !
g=
a b ∈ GL+(2, R), c d
g·z =
az + b , cz + d
whose Fourier expansion at ∞, Fˆ(q), q = e2πiz , is holomorphic at q = 0. The corresponding lattice function is ω F˜(Zω1 + Zω2)) := ω2−k F ( 1 ) ω2 ω ω1, ω2 ∈ C∗, Im 1 > 0 , ω2 or equivalently,
∀ g ∈ GL+ 2 (R), k
F˜(g −1Λ0) := (det g) 2 (F |k g) (i) . 6
Eisenstein series • For k > 2 and a = (a1, a2) ∈ (Q/Z)2 (k) Ga (z) :=
(m1z + m2)−k .
X m6=0, m≡a (1)
(k)
When k = 1 or k = 2, replace by Ga (z, 0), X (k) Ga (z, s) := (m1z + m2)−k |m1z + m2|−s , (2)
with Ga (z) only quasi-holomorphic, that is (2) z 7→ Ga (z) +
2πi z − z¯
is holomorphic in z ∈ H. For any weight k ≥ 1 and level N , (k − 1)! (k) Ex (z) := k
X
(2πiN )
a∈
a a ψx 1 , 2 N N
ψx(a) · Ga(z) , 2
1 N Z/Z
:= e2πi(a2x1−a1x2) ,
aj 1 ∈ Z/Z. N N 7
(2)
E0
is quasi-holomorphic, more precisely
1 (2) E0 (z) = − 2πi
d 1 2 (log η) + . dz z − z¯ (k)
• Fourier expansion:
Ex (z) =
X Bk (x1) + − k 0 0,
a2 X =− B2(x1) − B2(0) c2 x·γˇ =0
ρ(γ1, γ2)
1
X d2 X − B2(x1) − B2(x1) c2 x·γˇ γˇ =0 x·γˇ =0 2 1
+2
X
c2 −1 X
x·γˇ1=0 j=0
2
!
B1
x1 + j x +j B1 a2 1 + x2 c2 c2 c2 −1 X
!
!
!
j a2 j −2 B1 B1 ; c2 c2 j=0 1 B1(x) = x − [x] − , 2 1 B2(x) = (x − [x])2 − (x − [x]) + . 6 where x ∈ Q2/Z2,
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Euler class: Petersson cocycle An alternative expression of the 1-cocycle µ is 1 2πi c (0) (0) µα (z) = G (z) − G |α (z) − , 2 2 2 2π cz + d
(0)
where G2
is the quasimodular holomorphic (0)
weight 2 Eisenstein series, related to E2
by
2πi (0) (0) 2 . G2 (z) = 4π E2 (z) + z − z¯ Thus, µ is equivalent (up to a factor) to a b
!
c d
d c = log(cz + d) , z 7→ cz + d dz d d = log(cz + d) − log(c¯ z + d) dz dz d 1 d 1 = log − log |α , dz z − z¯ dz z − z¯
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Therefore, replacing X by XR amounts to replacing µ by the equivalent cocycle 1 c 1 d να (z) = · = log j(α, z). πi cz + d πi dz The 2-cocycle 1 γ 2 z0 (γ1, γ2) 7→ νγ1 (z) dz 2 z0 is clearly equivalent to Z
1 (γ1, γ2) := log j(γ2, z0) + log j(γ1, γ2z0) 2πi − log j(γ1γ2, z0) ∈ Z , where the logarithm is determined by Im log ∈ [−π, π) . This is precisely the integral 2-cocycle introduced by Petersson.
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Splitting formula • Let γ1, γ2 ∈ SL(2, Q), then one has 1 (γ1, γ2)− ρ(γ1, γ2) = Φ(γ1γ2)−Φ(γ1)−Φ(γ2), 2 with Φ the Dedekind-Rademacher function, defined as follows: for γ ∈ SL(2, Z) and c > 0 X j aj 1 a + d c−1 − B1 B1 − ; Φ(γ) = 12 c c c 4 j=0
for γ ∈ SL(2, Q) , with c = 0 and d > 0, !
Φ
a b 0 d
=
1 b . 12 d
• When γ1, γ2 ∈ SL(2, Z), then ρ(γ1, γ2) = 0. One recovers the classical splitting formula for the Petersson cocycle, displaying the triviality of the Euler class of SL(2, Z). 25
Concluding comments: expected outcome • The splitting formula has a K-homological counterpart, consisting of a related transgression formula between the two local cocycles describing the Connes-Chern character of the ∂¯b spectral triple over C ∗(LK ), corresponding to the two connections X and XR. • The above Chern character is obtained as the image via characteristic map of a universal class in the Hopf cyclic cohomology of a † quatum double of the Hopf algebra H1, defined over the ring E(Q) of rational Eisenstein series for GL2(A).
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A. Connes, M. Marcolli, Q-lattices: quantum statistical mechanics and Galois theory. J. Geom. Phys. 56 (2006), no.1, 2–23. A. Connes, H. Moscovici, The local index formula in noncommutative geometry. Geom. Funct. Anal. 5 (1995), 174-243. A. Connes, H. Moscovici, Hopf algebras, cyclic cohomology and the transverse index theorem. Comm. Math. Phys. 198 (1998), no. 1, 199– 246. A. Connes, H. Moscovici, Modular Hecke algebras and their Hopf symmetry; Rankin-Cohen brackets and the Hopf algebra of transverse geometry. Mosc. Math. J. 4 (2004), no. 1, 67–130. A. Connes, H. Moscovici, Transgression of the Godbillon-Vey class and Rademacher functions. arXiv.org/math.QA/0510683. 27
