How Noncommutative Geometry looks at Arithmetic Matilde Marcolli 2007 Based on: Alain Connes, Caterina Consani, Matilde Marcolli, • Noncommutative geometry and motives: the ther- modynamics of endomotives, Advances in Mathe- matics, Vol.214 (2007) N.2, 761-831. Caterina Consani, Matilde Marcolli, Quantum sta- • tistical mechanics over function fields Journal of Number Theory 123 (2007) 487-528. Alain Connes, Matilde Marcolli, Niranjan Ramachan- • dran, KMS states and complex multiplication, Se- lecta Mathematica, Vol.11 (2005) N.3-4, 325{347. Alain Connes, Matilde Marcolli, Quantum statisti- • cal mechanics of Q-lattices in \Frontiers in num- ber theory, physics, and geometry. I", 269{347, Springer, Berlin, 2006. Alain Connes, Caterina Consani, Matilde Marcolli, • The Weil proof and the geometry of the adeles class space, to appear in Manin's Festschrift. 1 Some Goals: Moduli spaces in arithmetic geometry: Enrich the boundary structure with \invisible" degenerations: Modular curves • (elliptic curves and noncommutative tori) Shimura varieties • Moduli spaces of Drinfeld modules, etc. • L-functions: Modular forms and modular symbols • extensions on the boundary Spectral realization of L-functions • and cohomological interpretation L-functions of motives • (geometry of archimedean factors) L-functions for real quadratic fields • (Stark's numbers) 2 Noncommutative geometry as a tool Equivalence relation on X: R quotient Y = X= . R Even for very good X X= pathological! ) R Classical: functions on the quotient (Y ) := f (X) f is invariant A f 2 A j R − g often too few functions ) (Y ) = C only constants A NCG: (Y ) noncommutative algebra A (Y ) := (Γ ) A A R functions on the graph Γ X X of the equivalence relation R ⊂ × (compact support or rapid decay) Convolution product (f1 f2)(x; y) = f1(x; u)f2(u; y) ∗ x Xu y ∼ ∼ involution f ∗(x; y) = f(y; x). 3 (Γ ) noncommutative algebra Y = X= A ) R noncommutativeR space Recall: C0(X) X Gelfand{Naimark equiv of categories , abelian C∗-algebras, loc comp Hausdorff spaces Result of NCG: Y = X= noncommutative space with NC al- R gebra of functions (Y ) := (Γ ) is A A R as good as X to do geometry • (deRham forms, cohomology, vector bundles, con- nections, curvatures, integration, points and subva- rieties) but with new phenomena • (time evolution, thermodynamics, quantum phe- nomena) 4 An example: Q-lattices (from joint work with Alain Connes) Definition: (Λ; φ) Q-lattice in Rn lattice Λ Rn + labels of torsion points ⊂ φ : Qn=Zn QΛ=Λ −! group homomorphism (invertible Q-lat if isom) φ general case invertible case φ Commensurability (Λ ; φ ) (Λ ; φ ) 1 1 ∼ 2 2 iff QΛ1 = QΛ2 and φ1 = φ2 mod Λ1 + Λ2 Q-lattices / Commensurability NC space ) 5 More concretely: 1-dimension (Λ; φ) = (λ Z; λ ρ) λ > 0 ρ Hom(Q=Z; Q=Z) = lim Z=nZ = Z^ n 2 − Up to scaling λ: algebra C(Z^) Commensurability Action of N = Z>0 1 αn(f)(ρ) = f(n− ρ) (partially defined action of Q+∗ ) Invertible: A∗ =Q∗ = GL1(Q) (GL1(A ) 1 ) = Sh(GL1; 1) f + n f × { g simplest (zero-dim) example of Shimura variety Non-invertible: nc C0(A ) o Q∗ Sh (GL1; 1) f + , \non-commutative" Shimura variety 6 1-dimensional Q-lattices up to scale / Commens. NC space C(Z^) o N ) Crossed product algebra f f (r; ρ) = f (rs 1; sρ)f (s; ρ) 1 ∗ 2 1 − 2 s QX,sρ Z^ 2 +∗ 2 1 f ∗(r; ρ) = f(r− ; rρ) Representations: Rρ = r Q : rρ Z^ f 2 +∗ 2 g 1 (πρ(f)ξ)(r) = f(rs− ; sρ)ξ(s) sXRρ 2 2 on ` (Rρ). Completion: f = sup πρ(f) k k ρ k k Time evolution (ratio of covolumes of commensurable pairs) it covol(Λ0) (σt f)((Λ; φ); (Λ0; φ0)) = f((Λ; φ); (Λ0; φ0)) covol(Λ) it (σt f)(r; ρ) = r f(r; ρ) 7 Quantum statistical mechanics ( ; σ ) C -algebra and time evolution A t ∗ State: ' : C linear '(1) = 1, '(a a) 0 A ! ∗ ≥ Representation π : ( ): Hamiltonian A ! B H itH itH π(σt(a)) = e π(a)e− Symmetries: Automorphisms: G Aut( ), gσ = σ g • ⊂ A t t Inner: u = unitary, σ (u) = u, a uau t 7! ∗ Endomorphisms: ρσ = σ ρ, e = ρ(1) • t t it Inner: u = isometry σt(u) = λ u 8 KMS states ' KMS 2 β Im z = β iβ β ϕ σ F(t + i ) = ( t(b)a) ϕ σ 0 F(t) = (a t(b)) Im z = 0 a; b holom function F (z) on strip: 8 2 A 9 a;b Fa;b(t) = '(aσt(b)) Fa;b(t + iβ) = '(σt(b)a) t R 8 2 9 Example: Gibbs states inverse temperature 0 < β < 1 1 βH βH Tr a e− Z(β) = Tr e− Z(β) At T > 0 simplex KMSβ ; extremal β At T = 0 (ground states) weak limits E ' (a) = lim 'β(a) 1 β !1 Classical points of NC space Action of symmetries (mod inner): endomor- phisms 1 ρ∗(') = ' ρ, for '(e) = 0 '(e) ◦ 6 On warming up/cooling down E1 β H Tr(π'(a) e ) W (')(a) = − β β H Tr( e− ) 10 Main idea: Recover classical moduli spaces as set of low temperature KMS states. Extra structure: high temperature regime, phase transitions, algebra of quantum mechanical observables Bost{Connes: For (C(Z^) o N; σt) Classification of KMS states: β 1 unique KMS state • ≤ ) β β > 1 = Sh(GL1; 1) • ) Eβ 1 βH 'β,α(x) = Tr πα(x) e− ζ(β) = β = Galois action θ : Gal(Qcycl=Q) ∼ Z^ • 1 ! ∗ γ '(x) = '(θ(γ) x) 11 Generalizations: System GL1 GL2 K = Q(p d) − Z(β) ζ(β) ζ(β)ζ(β 1) ζK(β) − Symm AQ∗ ;f =Q∗ GL2(AQ;f )=Q∗ AK∗ ;f =K∗ Aut Z^ GL2(Z^) ^ = ∗ O∗ O∗ End GL+(Q) Cl( ) 2 O Gal Gal(Qab=Q) Aut(F ) Gal(Kab=K) Sh(GL1; 1) Sh(GL2; H) AK∗ =K∗ E1 ;f In 2-dimensions GL2 (Connes-Marcolli) Q(p d) (Connes-Marcolli-Ramachandran) − Shimura varieties (Ha-Paugam) Function fields (Consani-Marcolli) Function fields case K = Fq(C) requires char p valued functions; time evolution σ : Zp Aut( ) and Goss L-function ! A s Z(s) = I− s S = C∗ Zp 2 1 1 × XI 12 The function field case (Consani-M.) 2-dim Q-lattices pointed Tate module [CMR] , T E = H1(E; Z^); ξ ; ξ T E: 1 2 2 Commensurability isogeny , Function fields K = Fq(C): Elliptic curves/lattices Drinfeld modules ) 1-dim K-lattices mod commens. A =K LK;1 K;f ∗ (choice of v = on curve C) convolution algebra 1 ) K = completion at v = ; K¯ = algebraic closure; 1 1 1 C = completion; Goss L-function 1 s Z(s) = I− s S = C∗ Zp 1 XI 2 1 × Quantum statistical mechanics: p-adic time evolution σ : Zp Aut( ) ! A Partition function Z(s), KMS states A =K ) K∗ ∗ 12-1 General procedure: Endomotives (from joint work with Connes and Consani) Algebraic category of endomotives: Objects: = A o S AK A = lim Aα Xα = Spec(Aα) −!α Xα = Artin motives over K S = unital abelian semigroup of endomorphisms with ρ : A ' eAe with e = ρ(1) ! Morphisms: ´etale correspondences (Xα; S) (X0 ; S0) spaces Z such that the G − G α0 right action of (X0 ; S0) is ´etale. G α0 (i.e. Z = Spec(M) right K-module: M finite projective) A Q-linear space (( ) ( )) formal linear combi- M Xα; S ; Xα0 0 ; S0 nations U = i aiZi CompP osition: Z W = Z W ◦ ×G0 fibered product over groupoid of the action of S0 on X0 13 Analytic category of endomotives: X(K¯)=S Objects: C -algebras C( ) o S ∗ X C algebra = C(X(K¯)) o S = C ( ) ∗ − A ∗ G Uniform condition: µ = lim µα counting on Xα − dρ µ ∗ loc constant on = X(K¯) dµ X state ' on ) A Morphisms: ´etale correspondences Z g : discrete fiber and 1 =comp operator in Z ! X MZ right module over C ( ) from Cc( )-valued inn prod ∗ X G ξ; η (x; s) := ξ¯(z)η(z s) h i ◦ z Xg 1(x) 2 − For { spaces Z Z(K¯) = G G0 7! Z Cc( ) right mod over Cc( ) Z G Morphisms in KK or cyclic category ) 14 Galois action: G = Gal(K¯=K) on characters X(K¯) = Hom(A; K¯) χ χ g An K¯ An K¯ K¯ ! 7! ! ! automorphisms of = C( ) o S A X ^ Revisit the example: C(Z) =∼ C∗(Q=Z) (Fourier transform) Xn = Spec(An); An = Q[Z=nZ] A = lim An = Q[Q=Z] −!n S = N action on canonical basis er, r Q=Z 2 1 ρn(er) = es n nsX=r Galois action ζn = χ(e ) cyclotomic action of 1=n ) Gab = Gal(Qcycl=Q) Examples from self-maps of algebraic varieties 15 Data: (X; S) endomotive =K: C -algebra = C( ) o S • ∗ A X arithmetic subalgebra = A o S • AK state ' : C (from uniform µ) • A ! Galois action G Aut( ) • ⊂ A Enters Thermodynamics: ( ; ') σ with ' KMS (Tomita{Takesaki) A ) t 1 GNS ' with cyclic separating vector ξ H ξ and ξ dense in ' ( =von Neumann alg) M M0 H M S' : ξ ξ aξ S'(aξ) = a∗ξ M ! M 7! S∗ : 0ξ 0ξ a0ξ S∗(a0ξ) = a0∗ξ ' M ! M 7! ' 1=2 closable polar decomposition S' = J'∆ ) ' 1 J' conjugate-linear involution J' = J'∗ = J'− 1 ∆' = S'∗S' self-adjoint positive J'∆'J' = S'S'∗ = ∆'− it it J' J' = and ∆ ∆ = • M M0 'M '− M α (a) = ∆ita∆ it a t ' '− 2 M The state ' is a KMS1 state for the modular • ' automorphism group σt = α t − 16 Classical points: Ωβ ' if σ preserves K o C KMS1 state on t A ) A Ω regular extremal KMS states β ⊂ Eβ β (low temparature: type I , i.e.