How Noncommutative Geometry Looks at Arithmetic

How Noncommutative Geometry looks at Arithmetic

Matilde Marcolli

2007 Based on:

Alain Connes, Caterina Consani, Matilde Marcolli, • Noncommutative geometry and motives: the thermodynamics 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 number 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 { ∈ A | R − } 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 phenomena)

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)

general case

invertible case

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 = Zˆ n ∈ ←− Up to scaling λ: algebra C(Zˆ)

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 + \ f × { } 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(Zˆ) o N ⇒ Crossed product algebra

f f (r, ρ) = f (rs 1, sρ)f (s, ρ) 1 ∗ 2 1 − 2 s QX,sρ Zˆ ∈ +∗ ∈ 1 f ∗(r, ρ) = f(r− , rρ)

Representations: Rρ = r Q : rρ Zˆ { ∈ +∗ ∈ } 1 (πρ(f)ξ)(r) = f(rs− , sρ)ξ(s)

sXRρ ∈ 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: ϕ : C linear ϕ(1) = 1, ϕ(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 ϕ KMS ∈ 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: ∀ ∈ A ∃ a,b Fa,b(t) = ϕ(aσt(b)) Fa,b(t + iβ) = ϕ(σt(b)a) t R ∀ ∈

9 Example: Gibbs states inverse temperature 0 < β < ∞ 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

ϕ (a) = lim ϕβ(a) ∞ β →∞ Classical points of NC space

Action of symmetries (mod inner): endomorphisms 1 ρ∗(ϕ) = ϕ ρ, for ϕ(e) = 0 ϕ(e) ◦ 6 On warming up/cooling down E∞ β H Tr(πϕ(a) e ) W (ϕ)(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(Zˆ) o N, σt)

Classiﬁcation of KMS states:

β 1 unique KMS state • ≤ ⇒ β β > 1 = Sh(GL1, 1) • ⇒ Eβ 1 βH ϕβ,α(x) = Tr πα(x) e− ζ(β)

= β = Galois action θ : Gal(Qcycl/Q) ∼ Zˆ • ∞ → ∗ γ ϕ(x) = ϕ(θ(γ) x)

11 Generalizations:

System GL1 GL2 K = Q(√ d) −

Z(β) ζ(β) ζ(β)ζ(β 1) ζK(β) −

Symm AQ∗ ,f /Q∗ GL2(AQ,f )/Q∗ AK∗ ,f /K∗

Aut Zˆ GL2(Zˆ) ˆ / ∗ 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∗ E∞ ,f

In 2-dimensions GL2 (Connes-Marcolli) Q(√ 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 ∈ ∞ ∞ × XI

12 The function ﬁeld case (Consani-M.)

2-dim Q-lattices pointed Tate module [CMR] ⇔ T E = H1(E, Zˆ), ξ , ξ T E. 1 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 ∞ ⇒ K = completion at v = ; K¯ = algebraic closure; ∞ ∞ ∞ C = completion; Goss L-function ∞ s Z(s) = I− s S = C∗ Zp ∞ XI ∈ ∞ ×

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: étale correspondences (Xα, S) (X0 , S0) spaces Z such that the G − G α0 right action of (X0 , S0) is étale. 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 ϕ 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) ∈ − 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

ˆ 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 ∈ 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 ϕ : C (from uniform µ) • A → Galois action G Aut( ) • ⊂ A

Enters Thermodynamics: ( , ϕ) σ with ϕ KMS (Tomita–Takesaki) A ⇒ t 1 GNS ϕ with cyclic separating vector ξ H ξ and ξ dense in ϕ ( =von Neumann alg) M M0 H M Sϕ : ξ ξ aξ Sϕ(aξ) = a∗ξ M → M 7→ S∗ : 0ξ 0ξ a0ξ S∗(a0ξ) = a0∗ξ ϕ M → M 7→ ϕ 1/2 closable polar decomposition Sϕ = Jϕ∆ ⇒ ϕ 1 Jϕ conjugate-linear involution Jϕ = Jϕ∗ = Jϕ− 1 ∆ϕ = Sϕ∗Sϕ self-adjoint positive Jϕ∆ϕJϕ = SϕSϕ∗ = ∆ϕ−

it it Jϕ Jϕ = and ∆ ∆ = • M M0 ϕM ϕ− M α (a) = ∆ita∆ it a t ϕ ϕ− ∈ M The state ϕ is a KMS1 state for the modular • ϕ automorphism group σt = α t − 16 Classical points: Ωβ ϕ 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. factor type I ) ∞ M ∞ Ω irreducible representation ∈ β ⇒ π : ( ()) A → B H = () 0 = T 1 : T ( ()) H H ⊗ H M { ⊗ ∈ B H }

ϕ itH itH Gibbs states: σt (π(a)) = e π(a)e− with Tr(e βH) < − ∞ βH Tr(π(a)e ) (a) = − βH Tr(e− ) Notice: H not uniquely determined H H + c ↔ Real line bundle Ω˜ = (ε, H) β { } λ (ε, H) = (ε, H + log λ) λ R ∀ ∈ +∗ R Ω˜ Ω with section Tr(e β H) = 1 +∗ → β → β − Ω˜ Ω R∗ β ' β × + Injections c : Ω Ω for β > β β0,β β → β0 0 17 Dual system: ( ˆ, θ) algebra: ˆ = oσ R A A A

(x?y)(s) = x(t) σt(y(s t)) dt, x, y (R, C) ZR − ∈ S A x(t) Ut dt ˆC R ∈ A Scaling action θ of λ R on ˆ ∈ +∗ A it θ ( x(t) Ut dt) = λ x(t) Ut dt λ Z Z

(ε, H) Ω˜ irred reps of ˆ ∈ β ⇒ A itH π ( x(t) Ut dt) = πε(x(t)) e dt ε,H Z Z Scaling action: π θ = π ε,H ◦ λ λ(ε,H)

Function ﬁeld case: (Consani-M.): Scaling action on dual system = Frobenius action (up to Wick rotation) “Scaling = Frobenius in characteristic zero” ⇒

18 Function ﬁelds: scaling and Frobenius Z v (x) ∞ iθ K∗ = Fq∗d u U x = sign(x)u x like z = re ∞ ∞ × ∞ × ∞ h i s S = C∗ Zp, λ K∗ ∈ ∞ ∞ × ∈ ∞ λs = xdeg(λ) λ y Is = xdeg(I) I y h i ⇒ h i I y time evolution σy(f)(L, L0) = h i f(L, L0) J y h i

Dual system and scaling by λ K∗ ∈ ∞ H = G Zp, G C∗ , ` C(H, ) × ⊂ ∞ ∈ A

X = `(s)Usdµ(s) ZH s scaling θλ(X) = H `(s)λ Usdµ(s) R deg(λ) θλ G(X) = `(s)x Usdµ(s) | ZH

y θλ Zp (X) = `(s) λ Usdµ(s) | ZH h i

Algebra ˆ maps to (A /K ) A A K ∗ injective Artin homomorphism Θ : K Gal(Kab/K) (lo- → cal class ﬁeld theory K = K ) ∞ θ F rZ Frobenius • λ|G 7→ θ Gal(Kab/Kun) inertia • λ|Zp 7→ 18-1 Trace class property:

1 π ( x(t) Ut dt) ( (ε)) ε,H Z ∈ L H for x ˆ (analytic continuation to strip of KMS ∈ Aβ β with rapid decay along boundary)

Restriction map:

π Tr ˆ C(Ω˜ , 1) C(Ω˜ ) Aβ −→ β L −→ β π(x)(ε, H) = π (x) (ε, H) Ω˜ ε,H ∀ ∈ β (no obstruction hypothesis for Tr)

Morphism of cyclic modules π ˆ\ C(Ω˜ , 1)\ Aβ → β L (Tr π)\ ˆ\ ◦ C(Ω˜ )\ equivariant for scaling action of R Aβ −→ β +∗ Abelian category: can take cokernels D( , ϕ) = Coker(δ) A Cyclic cohomology: HC0(D( , ϕ)) with Scaling action: induced R Arepresentation • +∗ If ( , ϕ) from an endomotive: Galois representation• A 19 Quick excursus: cyclic category and NC spaces (Connes) Cyclic category: [n] Obj(Λ) δ : [n 1] [n], σ :∈[n + 1] [n] i − → j → δjδi = δiδj 1 for i < j, σjσi = σiσj+1, i j − ≤

δiσj 1 i < j − σjδi =  1n if i = j or i = j + 1  δi 1σj i > j + 1. − τn : [n] [n]  → τnδi = δi 1τn 1 1 i n, τnδ0 = δn − − ≤ ≤ 2 τnσi = σi 1τn+1 1 i n, τnσ0 = σnτ − ≤ ≤ n+1 n+1 τn = 1n. Category cyclic objects: covariant functors Λ C → C Unital algebra over a ﬁeld K: K(Λ)-module A \ = covariant functor Λ V ect A → K (n+1) [n] ⊗ = ⇒ A A ⊗ A · · · ⊗ A δ (a0 an) (a0 aiai+1 an) i ⇒ ⊗ · · · ⊗ 7→ ⊗ · · · ⊗ ⊗ · · · ⊗ σ (a0 an) (a0 ai 1 ai+1 an) j ⇒ ⊗ · · · ⊗ 7→ ⊗ · · · ⊗ ⊗ ⊗ ⊗ · · · ⊗ 0 n n 0 n 1 τn (a a ) (a a a − ) ⇒ ⊗ · · · ⊗ 7→ ⊗ ⊗ · · · ⊗

20 More morphisms: Morphism of algebras φ : φ\ : \ \ • Traces A → B ⇒ A → B • τ : K τ \ : \ K\ A → ⇒ A → τ \(x0 xn) = τ(x0 xn) ⊗ · · · ⊗ · · · bimodules \ = τ \ ρ\ • A − B E ⇒ E ◦ ρ : End ( ) and τ : End ( ) A → B E B E → B

Abelian category: HCn( ) = Extn( \, K\) A A For non-unital : comp essential ideal (e.g. comp = ˜ =A1-pointA ⊂compactiﬁcation)A Λ-moduleA ( , Acomp)\ A A comp a0 an aj X ⊗ · · · ⊗ ∈ A at least one aj belongs to Trace class operators 1, algebA ra L B ( ˜1)\ \ B ⊗ L → B Tr((x0 t0) . . . (xn tn)) = x0 . . . xn Tr(t0 tn) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ · · ·

21 Back to our chosen example: = C(Zˆ) o N A it ϕ(f) = f(1, ρ)dµ(ρ) σt(f)(r, ρ) = r f(r, ρ) ZZˆ ⇒ Ω˜ = Zˆ R = C = A /Q (for β > 1) β ∗ × +∗ Q Q∗ ∗

Dual system ˆ = C ( ˜) A ∗ G it h(r, ρ, λ) = ft(r, ρ)λ Utdt Z where commensurability of Q-lattices (not up to scale): groupoid

˜ = (r, ρ, λ) Q∗ Zˆ R∗ : rρ Zˆ G { ∈ + × × + ∈ } = C ( ) with = ˜/R A ∗ G G G +∗

Scaling+Galois Zˆ R = C action ⇒ ∗ × +∗ Q χ = character of Zˆ∗

pχ = gχ(g) dg

ZZˆ∗ pχ = idempotent in cat of endomotives and in EndΛD( , ϕ) A

HC (pχ D( , ϕ)) 0 A 22 Adeles class space ˆ Morita equivalence C(Z) o N = (C0(AQ,f ) o Q+∗ )π (π = char function of Zˆ)

AQ,f /Q+∗ dual system AQ· /Q∗

A = AQ R Q· ,f × ∗ Adeles class space XQ := AQ/Q∗ (added point 0 R) ∈ The adeles class space and Riemann’s zeta (Connes)

2 2 2 0 L (AQ/Q∗)0 L (AQ/Q∗) C 0 → δ → δ → → f(0) = 0 and fˆ(0) = 0 E 0 L2(A /Q ) L2(C ) 0 → δ Q ∗ 0 → δ Q → H → E(f)(g) = g 1/2 f(qg), g C | | ∀ ∈ Q qXQ ∈ ∗ compatible with CQ actions

U(h) = h(g) Ug d∗g h (CQ) ZCQ ∈ S (comp support) acts on H = χ χ χ characters of Zˆ∗ H ⊕ H ∈ χ = ξ : Ugξ = χ(g)ξ w/ R -action gen by Dχ H { ∈ H } +∗ 23 Connes’ spectral realization and trace formula 1 Spec(Dχ) = s iR Lχ + s = 0 ∈ | 2 Lχ = L-function with Gr¨ossencharakter χ χ = 1 ζ(s) Riemann zeta ⇒

Trace formula (semi-local): S = ﬁn many places 1 0 h(u− ) Tr(RΛU(h)) = 2h(1) log Λ+ d∗u+o(1) ZQ vXS ∗v 1 u ∈ | − | RΛ = cutoﬀ regularization, 0 = principal value R Weil’s explicit formula (distributional form):

1 0 h(u− ) ˆh(0) + ˆh(1) ˆh(ρ) = d∗u − ZQ 1 u Xρ Xv v∗ | − |

Geometric idea: periodic orbits of the action of CQ on XQ r CQ

24 Guillemin–Sternberg distributional trace formula

Flow on manifold Ft = exp(tv)

(U f)(x) = f(F (x)) f C (M) t t ∈ ∞

(Ft) : Tx/Rvx Tx/Rvx = Nx ∗ → transversality: 1 (Ft) invertible − ∗ h(u) Trdistr( h(t)Ut dt) = d∗u Z Xγ ZIγ 1 (Fu) | − ∗| γ = periodic orbits and Iγ = isotropy group

Schwartz kernel (T f)(x) = k(x, y)f(y) dy R Trdistr(T ) = k(x, x) dx Z For (T f)(x) = f(F (x)) kernel

(T f)(x) = δ(y F (x))f(y)dy Z −

25 Cohomological interpretation: (from joint work w/ Connes and Consani) The restriction morphism δ = (Tr π)\ for the ◦ BC system (C(Zˆ) o N, σt): δ(f) = f(1, nρ, nλ) = f˜(q(ρ, λ)) = E(f˜) nXN qXQ ∈ ∈ ∗ + f˜= ext by zero outside Zˆ R AQ × ⊂ 2 \ Hilbert space L (AQ/Q )0 replaced by Λ-module ˆ δ ∗ Aβ,0 diﬀerent analytic techniques (as in R.Meyer)

Cohomological interpretation of E via ⇒ scaling action θ on HC (D( , ϕ)) 0 A 1 Action of CQ = A /Q on = HC0(D( , ϕ)) Q∗ ∗ H A

ϑ(f) = f(g)ϑg d∗g f S(CQ) ZCQ ∈ Weil’s explicit formula ⇒ 1 0 h(u− ) Tr(ϑ(f) 1 ) = fˆ(0)+fˆ(1) ∆ ∆f(1) d∗u |H − • − Z(K ,e ) 1 u Xv v∗ Kv | − | f S(CK) (strong Schwartz space) ∈ Self inters of diagonal ∆ ∆ = log a = log D • | | − | | (D = discriminant for #-ﬁeld, Euler char χ(C) for Fq(C)) 26 Observation:

Tr(R U(f)): only zeros on critical line • Λ Trace formula (global) RH ⇔

Tr(ϑ(f) 1): all zeros involved • | RH positivitH y ⇔ ] Tr ϑ(f ? f ) 1 0 f S(CQ) |H ≥ ∀ ∈ where 1 (f1 ? f2)(g) = f1(k)f2(k− g)d∗g Z multiplicative Haar measure d∗g and adjoint

] 1 1 f (g) = g − f(g ) | | −

Better for comparing with Weil’s proof for ⇒ function ﬁelds

Explicit formula • Positivity: (correspondences, linear equiv, RR, etc.) • 27 Weil’s proof in a nutshell

K = Fq(C) function ﬁeld, ΣK = places deg nv = # orbit of Fr on ﬁber C(F¯q) ΣK → P (q s) ζ (s) = (1 q nvs) 1 = − K − − s 1 s Y − (1 q )(1 q ) ΣK − − − − 1 P (T ) = (1 λnT ) char polynomial of Fr∗ on H (C¯, Q ) − et ` Q j k j k C(F¯q) Fix(Fr ) = ( 1) Tr(Fr∗ H (C¯, Q )) ⊃ − | et ` Xk 1/2 RH eigenvalues λn with λ = q ⇔ | j|

Correspondences: divisors Z C C; degree, codegree, trace: ⊂ ×

d(Z) = Z (P C) d0(Z) = Z (C P ) • × • × Tr(Z) = d(Z) + d0(Z) Z ∆ − •

RH Weil positivity Tr(Z ? Z ) > 0 ⇔ 0

28 Steps: Frobenius correspondence Adjust degree mod trivial correspondences • C P and P C × × Riemann–Roch: P Z(P ) of deg = g lin • 7→ equiv to eﬀective Using d(Z ? Z ) = d(Z)d (Z) = gd (Z) = d (Z ? Z ) • 0 0 0 0 0 Tr(Z ? Z ) = 2gd (Z) + (2g 2)d (Z) Y ∆ 0 0 − 0 − • (4g 2)d (Z) (4g 4)d (Z) = 2d (Z) 0 ≥ − 0 − − 0 0 ≥

Building a dictionary

Alg Geom/NT NCG C(F¯q) alg points ΞK classical points Weil explicit formula Tr(ϑ(f) 1 ) |H Frobenius correspondence Z(f) = f(g) Zg d g CK ∗ Trivial correspondences = Range(TR r π) Adjusting the degree V Fubini step◦ by trivial correspondences on test functions in Principal divisors ??? V Riemann–Roch Index theorem

29 Classical points of the periodic orbits

CK action on XK = AK/K∗

P = (x, u) XK CK u x = x u = 1 v ΣK: { ∈ × | } 6 ⇒ ∃ ∈

X = x X xv = 0 K,v { ∈ K | }

Isotropy cocompact K = (kw) kw = 1 w = v CK ⊃ v∗ { | ∀ 6 } ⊂ X NC spaces A /K (AK = a AK av = 0 ) K,v K,v ∗ ,v { ∈ | }

= C ( ), = (k, x) xv = 0 AK,v ∗ GK,v GK,v { ∈ GK | } groupoid K = K n AK with C ( K) alg of XK = AK/K G ∗ ∗ G ∗ smooth subalgebra ( K ) S G ,v restricted groupoid for XK,v:

(1) (1) (1) := K∗ n A = (g, a) K a, ga A GK,v K,v { ∈ G ,v | ∈ K,v} (1) (1) (1) A = K with K = interior of x Kw x 1 K,v w w w { ∈ | | | ≤ } Q

30 Quantum Statistical Mechanics on XK,v state ϕ(f) = A(1) f(1, a)da time evolution: K,v ⇒ R σv(f)(k, x) = k it f(k, x) t | |v additive Haar measure scales d(kav) = k vdav KMS1 | | ⇒ (v) ΞK := CK a v [Σ · ∈ K with a(v) = 1 for w = v and a(v) = 0 w 6 v y Ξ positive energy representation of ∈ K AK,v Hamiltonian

(Hy ξ)(k, y) = log k v ξ(k, y) | | low temperature KMS states: classical points of XK,v

Function ﬁeld: ΞK = C(F¯q) Z n Z equiv Fr action: N/Nv = q /q v = Z/nvZ

ΞQ = C(F¯1) over “ﬁeld with one element”? ⇒

31 Correspondences:

Graph of scaling action by g CK ∈ 1 Zg = (x, g x) A /K A /K { − } ⊂ K ∗ × K ∗ Z(f) = f(g) Zg d∗g with f S(CK) CK ∈ R degree and codegree

d(Z(f)) = fˆ(1) = f(u) u d∗u Z | | with d(Zg) = g | | ] d0(Z(f)) = d(Z(f¯ )) = f(u) d∗u = fˆ(0) Z

Adjusting degree d(Z(f)) = fˆ(1) adding h ∈ V h(u, λ) = η(nλ) nXZ ∈ × λ R , u Zˆ , CQ = Zˆ R ∈ +∗ ∈ ∗ ∗ × +∗ Notice: can ﬁnd h with ˆh(1) = 0 since ∈ V 6 η(nλ) dλ = η(nλ)dλ = 0 ZR Xn 6 Xn ZR Fubini thm does not apply 32 Back to the dictionary:

Virtual correspondences bivariant class Γ

Modulo torsion KK(A, B II1) ⊗

Eﬀective correspondences Epimorphism of C∗-modules

Degree of correspondence Pointwise index d(Γ)

deg D(P ) g eﬀective d(Γ) > 0 K, Γ + K onto ≥ ⇒∼ ⇒ ∃

Lefschetz formula bivariant Chern of Z(h) (localization on graph Z(h))