By Jean Michel Lemaire

ISBN-10: 0387069682

ISBN-13: 9780387069685

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Gal(Q/Q) → Gal(Q/Q) Geometrically, the above setting can be understood in terms of the cyclotomic tower. This has base Spec Z = V1 . The family is Spec Z[ζn ] = Vn where ζn is a primitive n-th root of unity (n ∈ N∗ ). The set Hom (Vm → Vn ), non-trivial for n|m, corresponds to the map Z[ζn ] → Z[ζm ] of rings. The group Aut(Vn ) = GL1 (Z/nZ) is the Galois group Gal(Q(ζn )/Q). The group of symmetries (14) of the tower is then ˆ G = lim GL1 (Z/nZ) = GL1 (Z), ← − n (15) which is isomorphic to the Galois group Gal(Qab /Q) of the maximal abelian extension of Q.

74) In this case, one obtains an algebra homomorphism fg Ug ∈ C0 (X1 ) Γ1 → (fg ◦ π) Uα(g) ∈ C0 (X2 ) Γ2 . (75) When the continuous map π fails to be proper, the above formula only deﬁnes a homomorphism to the multiplier algebra C0 (X1 ) Γ1 → M (C0 (X2 ) Γ2 ). In the case of the BC and the GL2 systems, we consider the map α : GL1 (Q) → GL2 (Q) of the form α(r) = r0 01 (76) . We also take π to be the determinant map, (ρ, u) ∈ M2 (Af )×GL2 (R) → π(ρ, u) = (det(ρ), det(u)) ∈ Af ×GL1 (R). (77) We then have the following result extending the classical map of Shimura varieties (70).

For 1 < β ≤ ∞, elements of Eβ are indexed by the classes of invertible ˆ hence by the classical points (16) of the ˆ ∗ = GL1 (Z), Q-lattices ρ ∈ Z noncommutative Shimura variety (26), Eβ ∼ = GL1 (Q)\GL1 (A)/R∗+ ∼ = CQ /DQ ∼ = IQ /Q∗+ , (34) with IQ as in (1). In this range of temperatures, the values of states ϕβ,ρ ∈ Eβ on the elements e(r) ∈ A1,Q is given, for 1 < β < ∞ by polylogarithms evaluated at roots of unity, normalized by the Riemann zeta function, ∞ ϕβ,ρ (e(r)) = 1 n−β ρ(ζrk ). ζ(β) n=1 KMS states and complex multiplication (Part II) 31 ˆ acts by automorphisms of the system.

### Aigebres Connexes et Homologie des Espaces de Lacets by Jean Michel Lemaire

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