4 Noncommutative geometry at arithmetic in- finity
4.4 Arakelov geometry and hyperbolic geometry
where the first term counts the contribution from the finite places as what happens over Sp(OK) and the second termis the contribution of the archimedean primes, at the part of the intersection that happens over arithmetic infinity.
While the first term is computed in algebro-geometric terms, from the local equations for the divisorsDi at a point p, the second term is defined as a sum of values of Green functionsgα on the Riemann surfacesXα(C) as
[D1, D2]∞=−
α
α
β,γ
gα(pα1,β, pα2,γ),
at pointspα1,β andpα2,γ ofXα(C) for 1≤β≤[K(D1),K] and 1≤γ≤[K(D2),K]
respectively, for finite extensions K(Dj) of K determined by Dj for j = 1,2, whereα= 1 forαreal embeddings andα= 2 forαcomplex embeddings.
For amore detailed account of these notions of Arakelov geometry,may refer to [65] and [99].
Further evidence for the similarity between the archimedean and the totally degenerate fibers comes from an explicit computation of the Green function at the archimedean places, as derived by Manin [106], in terms of a Schottky uniformization of the Riemann surfaceXα(C). Such a uniformization has an analog at a finite prime, in terms ofp-adic Schottky groups, only in the totally degenerate case. Another source of evidence comes froma cohomological theory of the local factors at archimedean primes, as developed by Consani [57], in which shown is that the resulting description of the local factor as regularized determinant at the archimedean primes resembles closely the case of the totally degenerate reduction at a finite prime.
Presented soon are both results in the light of the noncommutative space as a spectral triple (OA, H, D) introduced in the previous section. As shown by Consani and Marcolli in [58], [59], [60], and [61], the noncommutative geometry of this space is naturally related to both the result of Manin on the Arakelov Green function and the cohomological construction of Consani.
• Singularities: gA−mxlog|z|is locally real analytic, forza local coordinate in a neighbourhood of x.
• Normalization:
XgAdμ= 0 is satisfied.
If B =
y∈Xny(y) is another divisor, such that |A| ∩ |B| = ∅, then the expression defined as gμ(A, B) =
unygA,μ(y) is symmetric and biadditive in A, B. In general, such an expression as gμ depends on μ, where the choice of μ is equivalent to the choice of a real analytic Riemannian metric on X, compatible with the complex structure. However, in the special case of degree zero divisors as degA = degB = 0, the expression gμ(A, B) are conformal invariants, denoted asg(A, B), named as theArakelov Green function.
In the case of the Riemann sphere P1(C) ≈ S2 ≈ C∪ {∞}, if wA is a meromorphic function with Div(wA) =A, then we have
g(A, B) = log Πy∈|B||wA(y)|ny = Re
γB
dwA
wA
, whereγB is a 1-chain with boundaryB.
In the case of degree zero divisorsA, Bon a Riemann surface of higher genus, the formula as above forg(A, B) can be generalized, replacing the logarithmic differentialdwA/wA withwA a differential of the third kind, as ameromoprhic differential with non-vanishing residues, with purely imaginary and residues as mx atx, as given as
g(A, B) = Re
γB
wA.
Thus, g(A, B) can be explicitly computed from a basis of differentials of the third kind with purely imaginary periods.
Remark. Recall from[116] the following. Adifferentialon a Riemann surface Ris a 1-formasw=udx+vdywithz=x+iylocally, withw∗=−vdx+udy conjugate differential, with (w∗)∗=−w. Ifdw= 0, thewis said to beclosed.
Then, for α∈H1(R) the homology, the integral
αwis defined uniquely to be theperiodalong a 1-cycle representing the classα. If there is a functionF on R such that dF =w, thenw is said to be exact. ifw∗ =−iw, thenwis said to bepure. Thenw=f(z)dz locally. Iff is holomorphic, thenwis said to be holomorphicoranalytic, which is equivalent to thatwis closed and pure. Iff ismeromorphic (as holomorphic or with only poles), thenwismeromorphicor Abeliandifferential. An Abelian differential on a closed (or compact) Riemann surface have the sumof residues equal to zero. Abelian differentials are divided into being of thefirst,second, andthirdkind, respectively, iffis holomorphic, f has zero residues for poles, and otherwise. The indefinite integralp
p0wis said to beAbelian integral, wherep0 is not a pole. In particular, an Abel integral on a closed Riemann surface of genus 1 is said to beellipticintegral.
As well, a divisoron a Riemann surface R is defined to be a formal sum with coefficients inZas D=
p∈Rnpp, where thesupport ofD is a discrete set ofp∈Rwith nonzeronp∈Z. Define thedegreeofDas degD=
p∈Rnp.
A divisor onDonRis said to be adivisorforf orw=f(z)dzif the support of Dconsists of zeros and poles off, and ifpis a zero,npis its degree (i.e.,f(z) = (z−p)nph(z) locally, withhholomorphic withh(p)= 0) and ifpis a pole, then
−np ≥1 is its degree (i.e.,f(z) = (z−h(z)p)−np locally, similarly as above). On a closed (or compact) Riemann surface of genusg≥1, a divisor of ameromorphic function is said to beprincipal, and a divisor of an Abelian differential is said to becanonical. ForDprincipal, degD= 0. ForDcanonical, degD= 2g−2.
Cross ratio and geodesics. The basic step leading to the result of Manin [106], expressing the Arakelov Green function in terms of geodesics in the hy- perbolic handle-bodyYΓ =H3/Γ is a simple classical fact in hyperbolic geom- etry. Namely, it is the fact that the (logarithmic) cross ratio of four points in P1(C)≈S2 ≈C∪ {∞}can be expressed in terms of geodesics in the interior H3as
log|[a, b, c, d]|=−ord(a∗[c, d], b∗[c, d]),
where [a, b, c, d] =−(a−b)(c−d)(a−d)−1(b−c)−1is thecross ratioof points a, b, c, d∈P1(C), and ord(·,·)means theoriented distanceinH3, anda∗[c, d]
indicates the on the geodesic [c, d] inH3with end pointsc, d∈P1(C), obtained as the intersection of [c, d] with the unique geodesic fromathat intersects with [c, d] at a right angle. (No original figure).
Differentials and Schottky uniformization. The next important step in the result of Manin [106] is to show that ifX(C) = Γ\ΩΓ is a Riemann surface with a Schottky uniformization, then obtained is a basis of differentials of the third kind with purely imaginary periods, by taking suitable averages over the group Γ of expressions involving the cross ratio of points ofP1(C).
Denote by C(γ) a set of representatives of Γ/γZ, and by C(ρ, γ) a set of representativesρZ\Γ/γZ, and byS(γ) the conjugacy class of γin Γ.
LetwAbe ameromorphic function onP1(C) with divisorA= (a)−(b),such that the support|A|is contained in the complement of an open neighbourhood of ΛΓ.
For a fixed choice of a base pointz0∈ΩΓ, the series ν(a)−(b)=
γ∈Γ
dlog[a, b, γz, γz0], with [a, b, γz, γz0] = (a−b)(γz−γz0)
−(a−γz0)(b−γz), gives the lift to ΩΓ of a differential of the third kind on the Riemann surface X(C), endowed with the choice of Schottky uniformization. These differentials have residues ±1 at the image of a, b in X(C), and they have vanishing ak
periods, whereak, bk for 1≤k≤gare the generators of the (first) homology of X(C).
Similarly, obtained are the lifts of differentials of the first kind onX(C), by
considering the series ωγ =
h∈C(γ)
dlog[hz+(γ), hz−(γ), z, z0],
with [hz+(γ), hz−(γ), z, z0] = (hz+(γ)−hz−(γ))(z−z0)
−(hz+(γ)−z0)(hz−(γ)−z),
wherez±(γ)∈ΛΓ denote the pair of the attractive and repelling fixed points of γ∈Γ.
The series asν(a)−(b)andωγ converge absolutely on compact subsets of ΩΓ, whenever dimHΛΓ<1. Moreover, they do not depend on the choice of the base point asz0∈ΩΓ.
In particular, given {γk}gk=1 a choice of generators of the Schottky group Γ, obtained is a basis of holomorphic differentials ωγk, satisfying
akωγl = 2π√
−1δkl.
Then use a linear combination of the holomorphic differentials ωγk to cor- rect the meromorphic differentials ν(a)−(b) in such a way that the resulting meromorphic differentials have purely imaginarybk-periods. Letxl(a, b) denote coefficients such that the differential of the third kind defined as
ω(a)−(b)=ν(a)−(b)− g l=1
xl(a, b)ωγl
(corrected) have purely imaginarybk-periods. The coefficients xl(a, b) satisfy the systemof equations
g l=1
xl(a, b)Re
bk
ωγl= Re
bk
ν(a)−(b)=
h∈S(γk)
log|[a, b, z+(h), z−(h)]|
=−
h∈S(γk)
ord(a∗[z+(h), z−(h)], b∗[z+(h), z−(h)])
(specified). Thus, it is obtained ([106], cf. also [157]) the Arakelov Green functiong(A, B) forX(C) with Schottky uniformization can be computed as
g((a)−(b),(c)−(d)) = Re
γ(c)−(d)
ω(a)−(b)
= Re
γ(c)−(d)
[ν(a)−(b)− g l=1
xl(a, b)ωγl]
= Re
γ(c)−(d)
ν(a)−(b)− g
l=1
xl(a, b)Re
γ(c)−(d)
ωγl
=
h∈Γ
log|[a, b, hc, hd]| − g l=1
xl(a, b)
h∈S(γl)
log|[z+(h), z,−(h), c, d]|.
Note that
[a, b, c, d] = (a−b)(c−d)
−(a−d)(b−c)= (c−d)(a−b)
−(c−b)(d−a) = [c, d, a, b].
It is noticed that the result above seems to indicate that there is a choice of Schottky uniformization involved as an additional datum for Arakerov ge- ometry at arithmetic infinity. However, as remarked previously, at least in the case of archimedean primes that are real embeddings as the case of arithmetic infinity for Q, the Schottky uniformization is detemined by the real structure, by splittingX(C) along the real locusX(R), when the latter is non-trivial.
Green function and geodesics. By combining the basic formula by the oriented distance with the formula calculated above for the Arakelov Green function on a Riemann surface with Shottky uniformization, we can replace each termappearing in the formula with the corresponding termwhich computes the oriented geodesic length of a certain arc of geodesic inYΓ as
g((a)−(b),(c)−(d)) =−
h∈Γ
ord(a∗[hc, hd], b∗[hc, hd])
+ g
l=1
xl(a, b)
h∈S(γl)
ord(z+(h)∗[c, d], z−(h)∗[c, d]).
The coefficients xl(a, b) can also be expressed in terms of geodesics, using the system equations given above.