Arithmetic infinity as archimedean primes

It then follows that

[D0⊕L², Sj] =− ∞ n=1

n 2h k=1

ajk[T_j^∗Pk,Πn−Πn−1]

A possible estimate of the norm of this commutator is given by the upper bounded as 2h_∞

n=1n, but which diverges. Thus, it is necessary to have that the norm of the commutators as the direct summands converges to zero at in- finity, of order less thanO(_n¹2) as n→ ∞. Possibly, it is necessary to consider elements ofOA^∞which are smooth, rapidly decreasing, or compactly supported, but with respect to (Π^∧_n) as a sort of basis.

The dimension of the (i=√

−1)n-th (corrected) eigenspace ofDis 2g(2g− 1)ⁿ⁻¹(2g−2) for n≥1, 2g forn= 0, and 2g(2g−1)⁻ⁿ⁻¹(2g−2) forn≤ −1, so that the spectral triple is not finitely summable, since |D|^z is not of trace class. But it is θ-summable, since the operator exp(−tD²) is of trace class, for allt >0.

For instance, suppose that D(ξ ⊕η) = 0 ⊕0 on H = ⊕²L². Then _∞

n=1nΠ^∧_nη = 0 and _∞

n=1nΠ^∧_nξ = 0. If we have the dimension of the ker- nel ofD equal to 2g, the summations in the definition of D should start with n= 1. Moreover, note that for ξn∈Π^∧_nL²,

0 −nΠ^∧_n nΠ^∧_n 0

ξn

−iξn

= inξn

nξn

=in ξn

−iξn

, and 0 −nΠ^∧n

nΠ^∧_n 0

ξn

iξn

=

−inξn

nξn

=−in ξn

iξn

.

Using the description of the noncommutative spaceO^Aas the crossed prod- uctC^∗-algebraFA^TZof an AFFAby the action of the shiftT, itmay be able to find a 1-summable spectral triple, where the dense subalgebra involved in it should not contain any of the group elements. In fact, by the result of Connes [26], the group Γ is a free group, and hence its growth is too fast to support finitely summable spectral triples on its group ring.

For instance, [Q(√ 2,√

3),Q] = 4 =r1withr2= 0. Indeed, as vector spaces, Q(√

2,√

3)∼=Q1⊕Q√ 2⊕Q√

3⊕Q√ 6

∼=Q(√

2)1⊕Q(√ 2)√

3 with [Q(√

2,√

3),Q] = [Q(√

2),Q][Q(√ 2,√

3),Q(√

2)] = 2². There is the identity embedding as Q(√

2,√

3) → R. As well, the others are obtained by sending respectively (√

2,√

3) → (−√ 2,√

3), (√ 2,−√

3), or (−√

2,−√ 3).

As well, [Q(i),Q] = 2 withr2= 1. Indeed, Q(i)∼=Q1⊕Q√

−1, and there is the identity embedding as Q(i)→ C, and the other is obtained by sending i→ −i(cf. [129]).

A general strategy in arithmetic geometry is to adapt the tools of classi- cal algebraic geometry to the arithmetic setting. In particular, the (spectrum) Sp(Z) of primes overQ is the analog of the affine line in arithmetic geometry.

It then becomes clear that some compactification is necessary, at least in or- der to have a well behaved form of intersection theory in arithmetic geometry.

Namely, we need to pass from the affine (Spec) Sp(Z) to the projective case.

The compactification is obtained by adding the infinite prime to the set of finite primes. Then a goal of arithmetic geometry becomes developing a setting that treats the infinite prime and the finite primes in the equal footing.

More generally, for a number fieldK, withO_K as its ring of integers, the set Sp(O_K) of primes is compactified by adding the set of archimedean primes, as

Sp(O_K) = Sp(O_K)∪ {α:K→C}.

Arithmetic surfaces. Let X be a smooth projective algebraic curve defined overQ. Then, obtained by clearing denominators is an equation with coefficients in Z. This determins aschemeX_Z over Sp(Z), as

X_Z⊗_ZSp(Q) =X,

where the closed fiber of X_Z at a prime p ∈ Sp(Z) is the reduction of X mod p. Thus, an algebraic curve viewed as an arithmetic variety becomes a 2-dimensional fibration over the affine line Sp(Z).

Also consider reductions of X defined over Z modulo pⁿ for some prime p∈Sp(Z). The corresponding limit asn→ ∞ defines ap-adic completion of X_Z. This can be thought of as an infinitesimal neighbourhood of the fiber atp.

The corresponding picture ismore complicated at arithmetic infinity, since there is no suitable notion of reductionmod ∞, available to define the closed fiber. On the other hand, there is the analog of thep-adic completion at hand.

This is given by the Riemann surface X(C) as a smooth projective algebraic curve over C, determined by the equation of the algebraic curve X over C, under the embedding ofQ⊂C, as

X(C) =X⊗_QSp(C),

with the absolute value|·|at the infinite prime, replacing thep-adic valuations.

Similarly, for Ka number field with [K:Q] =nandO_K its ring of integers, a smooth proper algebraic curveX overKdetermines a smoothminimalmodel XO_K, which defines an arithmetic surface X^OK over Sp(O_K). The closed fiber X_p of XO_K over a primep∈O_Kis given by the reductionmodp.

When Sp(O_K) is compactified by adding the archimedean primes, obtained aren=r1+2r2Riemann surfacesXα(C), which are obtained fromthe equation definingX overKunder then embeddings α:K→C. The correspondingr1

Riemann surfaces of nhave the real involution.

Then, the picture of an arithmetic surfaceX_Sp(O

K)over Sp(O_K) with inclu- sions is as follows:

X_p −−−−→ X^⊂ O_K =XSp(O_K) −−−−→⊂ X_Sp(O

K)

←−−−−⊃ ?

⏐⏐

⏐⏐ ⏐⏐ ⏐⏐ p −−−−→^⊂ Sp(O_K) −−−−→^⊂ Sp(O_K) ←−−−− {α^⊃ :K→C}

where there is no explicit geometric description of the closed fibers as (?) over the archimedean primesα.

An arithmetic surface over Sp(Z) of primes looks like a bundle of curves or crossed ones over primes, whose limit at infinity is what as (?). (No figure).

It is able to formally enlarge the group of divisors on the arithmetic surface by adding formal real linear combinations as

αλαFα of irreducible closed vertical fibers Fα at infinity, where the fibers Fα are only treated as formal symbols, and no geometric model of such fibers is provided. A remarkable fact says that Hermitian geometry on the Riemann surfaces Xα(C) is sufficient to specify the contribution of such divisors to intersection theory on the arithmetic surface, even without an explicit knowledge of the closed fiber.

The main ideal of Arakelov geometry is that it is sufficient to work with the infinitesimal negihbourhood Xα(C) of the fibers Fα, to have well defined intersection indices.

By analogy, in the case of the classical geometry of a degeneration of al- gebraic curves over a disk with a special fiber at 0, the analogous statement would be that the geometry of the special fiber is completely determined by the generic fibers. This is a strong statement as the form of the degeneration. For instance, blowing up points at the special fibermay be not seen by just looking at the generic fibers. Investigating this analogy leads to expect that the fiber at infinity should behave like the totally degenerate case. This is the case as themaximal degeneration, where all the components of the closed fiber are the same asP¹and the geometry of the degeneration is completely encoded by the dual graph, which describes in a purely combinatorial way how the components asP¹are joined. The dual graph has a vertex for each components of the closed fiber and an edge for each double point.

The local intersectionmultiplicities of two finite, horizontal, irreducible di- visorsD1,D2 onXO_K is given by

[D1, D2] = [D1, D2]f+ [D1, D2]_∞,

where the first term counts the contribution from the finite places as what happens over Sp(O_K) 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(p^α_1,β, p^α_2,γ),

at pointsp^α_1,β andp^α_2,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.