Arithmetic varieties and L -factors

while the C^∗-algebra crossed productC(S)T Z as a noncommutative space, corresponding to the quotient ΛΓ×ΓΛΓ, gives the set of edges of the dual graph.

Moreover, given by using noncommutative geometry is a notion of reduction mod ∞, analogous to the reduction maps mod p^m defined by the subgraphs ΔK,n of the BT tree ΔK in the case of Mumford curves. In fact, the reduc- tion map corresponds to the paths connecting ends of the graph ΔK,n/Γ to the corresponding vertices of the graph Δ_Γ/Γ = ΔK,0/Γ. Then the analog at arithmetic infinity consists of geodesics inYΓ which are the images of geodesics inH³ starting at some pointx0∈H³∪ΩΓ and having the other end as a point of ΛΓ. These are parameterized by the set ΛΓ×Γ(H³∪ΩΓ). Thus, in terms of NC geometry, the reductionmod∞corresponds to a compactification of the homotopy quotient ΛΓ ×Γ H³ with H³ =EΓ and BΓ = H³/Γ = YΓ. Hence, we can view ΛΓ/Γ as the quotient of a foliation on the homotopy quotient with contractible leaves asH³. Then the reductionmod∞is given by the assembly mapμas

μ:K^∗+1(ΛΓ×ΓH³)→K_∗(C(ΛΓ)Γ), ∗= 0,1.

This shows that the spectral triple (O^A, H, D) as a noncommutative (Riemann) space is closely related to the geometry of the fiber at arithmetic infinity of an algebraic variety. Then,may ask a question that what arithmetic information is captured by the Dirac operator D of the spectral triple. Let us see in the next section that as proved in [61], the Dirac operator gives another important arithmetic invariant, namely the localL-factor at the Archimedean prime.

withγ theEulerconstant andck the generalizedEulerconstants.

However, to have the functional equation as below, we needs to consider the product

ζ(s)Γ(s 2) 1

π^s2 =ζ(1−s)Γ(1−s 2 ) 1

π^1−s2

included with a contribution of the archimedean prime, expressed in terms of theGamma function, with factorials

Γ(s) = _∞

0

t^s

te^tdt= (s−1)Γ(s−1), Γ(n) = (n−1)!, Γ(1) = 1.

An analogy with ordinary geometry suggests to think of the functional equa- tion as a sort of Poincar´e duality, which holds for a compactmanifold, hence for which, we need to compactify arithmetic varieties by adding the archimedean primes and the corresponding archimedean fibers.

Looking at an arithmetic variety over a finite prime p ∈ Sp(O_K), the fact that the reduction lives over a residue field of positive characteristic implies that there is a special operator as the geometric Frobenius F r_p^∗, acting on a suitable cohomology theory as ´etale cohomology, induced by the Frobenius automorphismϕ_p of Gal(Fp/Fp) (as apⁿ-powermap).

Note that Fp ∼=∪1≤n<∞Fpⁿ as an algebraic closure of the finite field F^p with order a primep, with [Fpⁿ,Fp] =n(cf. [123]).

The local L-factors of the L-function first given as the product at finite primesp encode the action of the geometric Frobenius F r_p^∗ as in the form, as in [144]

L_p(H^m(X), s) = det(1−F r^∗_pN(p)⁻¹|H^m(X,Ql)^Ip)⁻¹,

where we consider the action of the geometric Frobenius F r^∗_p on the inertia invariantsH^m(X,Ql)^Ip of the ´etale cohomology, as explained in the following.

We use the notationN for the norm map in the localL-factors (converted).

An introduction to ´etale cohomology as well as a precise definition of these arithmetic structures are beyond the scope of this survey. In fact, our primary concern would only be the contribution of the archimedean primes to the L- function, where the construction is based on the ordinary de Rhamcohomology.

Thus, we only give a quick and somewhat heuristic explanation of the local L-factors. May refer to [144], [155] for a detailed and rigorous account and for the precise hypotheses under which the following holds.

LetX =X(Q) be a smooth projective algebraic variety of any dimension defined over Q. Set X = X ⊗Sp(Q) ≡X(Q), where Q denotes an algebraic closure. Forla prime, the (´etale) cohomologyH^∗(X,Ql) is a finite-dimensional Ql-vector space, satisfying

H^j(X(C),C)∼=H^j(X,Ql)⊗C.

The (countable!) algebraic closureQof Qmay be given as the set of all algebraic numbers overQ(such as√

2 and√

3), up to aQ-isomorphism. Then

Q⊂QbutQ C, because there are transendental numbers inCsuch aseand π(andγ?) (cf. [116], [129]).

TheabsoluteGalois group Gal(Q/Q) = Aut(Q/Q) acts onH^∗(X,Q^l).

Similarly, we can consider the (´etale) cohomology H^∗(X,Q^l) forX =X(K) defined over a number field K ⊃ Q, with X = X ⊗Sp(K) ≡ X(K). For p ∈ Sp(O_K) of maximal ideals of O_K the ring of integers in K and l a prime such that (l, q) = 1, where q is the cardinality of the residue field O_K/p at p, the (physical law of)inertiainvariants are defined to be

H^∗(X,Ql)^Ip⊂H^∗(X,Ql)

as the part of the l-adic cohomology, on which the inertia group at p acts trivially, and the inertia group is defined as the kernel of the short exact sequence of groups withr_p as the quotientmap:

1→I_p= ker(r_p)→D_p={σ∈Gal(Q/Q)|σ(p) =p} −−−−→^rp Gal(Fp/Fp)→1.

The Frobenius automorphism ϕ_p of Gal(Fp/Fp) (as a pⁿ-power map) lifts to ϕ_p∈D_p/I_pby the same symbol, which induces the geometricFrobeniusF r^∗_p= (ϕ⁻_p¹)^∗ acting on H^∗(X,Ql)^Ip. Thus, the local L-factor given above can be written equivalently as

L_p(H^m(X), s) = Πλ∈sp(F r^∗_p)

1

(1−λq^−s)^{dimHm(XΓ)Iλp} ,

whereH^m(XΓ)^I_λ^p denotes the eigen-space of the Frobenius with eigenvalueλ∈ sp(F r^∗_p) the (operator) (point) spectrum.

The notationH^m(XΓ)^I_λ^p may better be replaced withH^m(X,Q^l)^I_λ^p. For our purpose, what ismost important to retain fromthe discussion above is that the local L-factors depend on the data (H^∗(X,Ql)^Ip, F r_p^∗) of a vector space and a linear operator on it, which have a cohomological interpretation.

ArchimedeanL-factors. Since the ´etale cohomology satisfies the compatibil- ity as

H^j(X,Ql)⊗C∼=H^j(X(C),C),

if we again resort to the general philosophy, according to which we can work with the smooth complex manifold X(C) and gain information on the closed fiber at arithmetic infinity, then we are led to expect that the contribution of the archimedean primes to the L-function may be expressed in terms of the cohomology H^∗(X(C),C), or equivalently in terms of de Rhamcohomology.

In fact, it is shown by Serre [144] that the expected contribution of the archimedean primes depends on theHodgestructure as

H^m(X(C)) =⊕p+q=mH^p,q(X(C))

and is again expressed in terms of Gamma functions, as in the case of the func- tional equation forζ(s)Γ(^s₂)π^−s2. Namely, given is the product of Gamma func- tions according to theHodgenumbersh^p,q (as the dimension ofH^p,q(X(C)))

for the (another) local L-factor:

L(H^m(X(C)), s) =

Πp<qΓ_C(s−p)^hp,qΠpΓ_R(s−p)^hp,+Γ_R(s−p+ 1)^hp,−, for the real embedding Πp,qΓ_C(s−min{p, q})^hp,q, for the complex one whereh^p,± is the dimension of the±(−1)^p-eigenspace of the involution onH^p,p induced by the real structure, and

Γ_C(s)≡(2π)^−sΓ(s) and Γ_R(s)≡√ 2⁻¹√

π^−sΓ(2⁻¹s).

One of the general ideas in arithmetic geometry is the always seeking a uni- fied picture of what happens at the finite and at the infinite primes. In particu- lar, it should be a suitable reformulation of the localL-factorsL_p(H^m(X(K)), s) and L(H^m(X(C)), s) as the determinant or the product, where both formulae can be expressed in the similar way.

By seeking a unified description of local L-factors at finite and infinite primes, both L-factors as above are expressed as infinite determinants, by Deninger in [68], [69], [70].

Recall that theRay-Singerdeterminant of an operatorT with pure (imag- inary) point spectrum λ∈sp(T) with finitemultiplicitiesmλ is defined to be

det∞(s−T)≡exp

−d

dzζT(s, z)|z=0

, ζT(s, z) =

λ∈sp(T)

mλ

(s−λ)^z as the zeta function of the operatorT. Described in [107] are suitable conditions for the convergence of these expressions in the case of the localL-factors.

It is shown by Deninger that the localL-factorL_p(H^m(X), s) as the product can be written equivalently in the form

L_p(H^m(X), s)⁻¹= det

∞(s−Θq), for an operators−Θq with spectrum given as

sp(s−Θq) ={s−αλ+2πin

logq|n∈Z, λ∈sp(F r^∗_p)} withmultiplicitiesdλ andq^αλ =λ.

Moreover, the (another) local L-factor L(H^m(X(C)), s) at infinity can be written similarly in the form

L(H^m(X(C)), s) = det

∞

((2π)⁻¹(s−Φ)|H^m)₋1

,

whereH^mis an infinite dimensional vector space and Φ is a linear operator with spectrum sp(Φ) = Z with finite multiplicities. This operator is regarded as a logarithmof Frobenius at arithmetic infinity.

Given the two formulae of Deninger for the local L-factors by det_∞, it is natural to ask for a cohomological interpretation of the data (H^m,Φ), somewhat analogous to the data (H^∗(X,Q^l)^Ip, F r_p^∗).

Arithmetic surfaces: L-factor and Dirac operator. As a return, let us now consider the special case of arithmetic surfaces, in the case of genusg≥2.

At an archimedean prime, consider the Riemann surfaceXα(C) with a Schottky uniformization X(C) = ΩΓ/Γ. In the case of a real embedding α:K→C, we can assume that the choice of a Schottky uniformization corresponds to the real structure, obtained by cuttingX(C) along the real locus X(R).

Now consider the spectral triple (O^A, H, D) associated to the Schottky group Γ acting on its limit set ΛΓ, where H =⊕²L²(ΛΓ) andO^A∼=C(ΛΓ)Γ and

D= ∞ n=1

nΠ^∧_n%

− ∞ n=1

nΠ^∧_n

.

Since the spectral triple is not finitely summable (butθ-summable), we can not define zeta functions of the spectral triple as in the form tr(a|D|^z). However, we can consider the restriction of D to a suitable subspace of H (if any), on which the trace is finite.

In particular, consider the zeta function with respect to the (Dirac) operator D (restricted)

ζπ_V,D(s, z) =

λ∈sp(D)i

tr(π_VΠ(λ, D)) 1 (s−λ)^z

(iinserted), where Π(λ, D)(=p(λ, D)) is the projection on the eigenspace for an eigenvalueλofD, andπ_Vis the orthogonal projection ofHon the the subspace V of 0⊕L²⊂H defined by

Πnπ_VΠn= g j=1

sⁿ_j(s^∗_j)ⁿ onL²,

where Πn are the projections ontoPn,Cin L² and are also the spectral projec- tions of the Dirac operatorD, with Π(−n, D) = Πn forn≥0.

In the case of an arithmetic surface, as the interesting local factor. the first cohomology L(H¹(X), s) is computed ([59], [61]) as in the following, from the zeta functionζπ_V,D of the spectral triple (OA, H, D).

Theorem 4.5. The localL-factor given asL(H^m(X(C)), s)above is computed as

L(H¹(X(C)), s) = exp

−d

dzζπ_V,(2π)⁻¹D((2π)⁻¹s, z)|z=0

₋1

. In the case of a real embedding, the same holds, with the projectionπ_V,F

∞ onto the +1eigenspace of the involution F_∞ induced by the real structure onV.

In particular, this result shows that, for the special case of arithmetic surfaces with X(C) of genusg ≥2, the pair (V ⊂L²(ΛΓ), D|V) is a possible geometric construction of the pair (H¹,Φ) of (H^m,Φ). Moreover, the Dirac operator of the spectral triple has an arithmeticmeaning, as that it recovers the logarithm of Frobenius asD|_V = Φ.

Looking more closely at the subspace V of the Hilbert space H, we see that it has a simple geometric interpretation in terms of the geodesics in the handle-bodyYΓ. Recall that the filtered subspaceP_C=C(ΛΓ,Z)⊗Cof L²= L²(ΛΓ, dμ) describes 1-cochains on the mapping torus ST, with P_C/δ_∗P_C = H¹(ST).

Themapping torusST is viewed as a copy of the tangle of bounded geodesics inside YΓ. Among these geodesics there are g fundamental closed geodesics that correspond to the generators of Γ, which correspond to geodesics in H³ connecting the fixed points{z^±(γj)} forj = 1,· · ·, g. Topologically, these are thegcore handles of the handle-bodyYΓ, which generate the homologyH1(YΓ).

Consider the cohomology H¹(ST). Suppose that there are elements of H¹(ST) supported on those fundamental closed geodesics. But this is impos- sible, because 1-cochains on ST are defined by functions in C(S,Z), that are supported on some clopen subsets covering the totally disconnected setS, which contains no isolated points. However, it is possible to choose a sequence of 1- cochains onST whose supports aremuch smaller clopen subsets containing the infinite word in S corresponding to one of the g fundamental geodesics. The complex finite dimensional subspaceVn=V ∩ Pn,C⊂ Pn,C with dim_CVn= 2g gives representatives of such cohomology classes in H¹(ST), where we get 2g instead ofgbecause we take into account the two possible choices of orientation.

Thus, it gives us the cohomological interpretation of the space H¹ =V of the pair (H¹,Φ), in the case of arithmetic surfaces. Moreover, the Schottky uniformization also provides us with a way of expressing the cohomology as H¹=Vin terms of the de RhamcohomologyH¹(X(C)). In fact, already seen in the calculation of the Green function, under the hypothesis that dimH(ΛΓ)<1, to each generator γj of the Schottky group Γ, we can associate a holomorphic differentialωγ_j on the Riemann surface X(C). Namely, themap

γj→ωγ_j =

h∈C(γ_j)

dlog[hz⁺(γj), hz⁻(γj), z, z0] induces an identification as

V ∼=⊕n∈Z,n≥0H¹(X(C)).

See in the next section that, in fact, the right hand side above is a particular case of a more general construction that works for arithmetic varieties in any dimension and that gives a cohomological interpretation of (H^m,Φ).

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