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Spec(Z)

Contents

Contents

Idea

Spec()Spec(\mathbb{Z}) denotes the spectrum of the commutative ring \mathbb{Z} of integers. Its closed points are the maximal ideals (p)(p), for each prime number pp in \mathbb{Z}, which are closed, and the non-maximal prime ideal (0)(0), whose closure is the whole of Spec()Spec(\mathbb{Z}). For details see at Zariski topology this example.

Since \mathbb{Z} is the initial object in the category CRing of commutative rings, Spec()Spec(\mathbb{Z}) is the terminal object in the category of affine schemes.

The gros etale topos over Spec()Spec(\mathbb{Z}) is the context for arithmetic geometry. By the discussion at Borger's absolute geometry it sits via an essential geometric morphism over the F1-topos:

Et(Spec())Et(Spec(𝔽 1)) Et(Spec(\mathbb{Z}))\longrightarrow Et(Spec(\mathbb{F}_1))

Properties

As a 3-dimensional space

There are some phenomena that may be interpreted as Spec()Spec(\mathbb{Z}) behaving like a 3-manifold in some ways.

As a 3-sphere containing knots

Several properties of Spec()Spec(\mathbb{Z}) make it behave as if of dimension 3. For instance Spec(){}Spec(\mathbb{Z}) \cup \{\infty\} has étale cohomological dimension equal to 3, up to 2-torsion (Mazur 73). Moreover the étale fundamental group π^ 1(Spec(){})\hat \pi_1(Spec(\mathbb{Z}) \cup \{\infty\}) is trivial, and hence Mazur suggested that Spec(){}Spec(\mathbb{Z}) \cup \{\infty\} is in fact analogous to the 3-sphere.

Similarly, the spectra Spec(𝔽 q)Spec(\mathbb{F}_q) of finite fields look like compact 1-dimensional spaces – circles – in that their étale cohomology with l\mathbb{Z}_l-coefficients for ll coprime to qq is l\mathbb{Z}_l in degrees 0 and 1 and vanishes in all higher degrees.

From this it is folklore (going back to Mazur and Manin, review includes Deninger 05, section 8, Kohno-Morishita 06) that the spectra of prime fields with their canonical embedding into Spec()Spec(\mathbb{Z})

Spec(𝔽 p)Spec() Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})

(formally dual to the canonical mod-pp projection 𝔽 p\mathbb{Z}\to \mathbb{F}_p) are analogous to knots inside this 3-dimensional space (a good exposition is in LeBruyn).

Observations like this give rise to the field of arithmetic topology.

As a hyperbolic 3-manifold containing prime geodesics

However, in view of the analogy between the Selberg zeta function and the Artin L-function it might be more appropriate to think of Spec()Spec(\mathbb{Z}) as analogous to a hyperbolic manifold of dimension 3 (see also Fujiwara 07, slide 7) and then to think of finite field spectra as analogous to the prime geodesics in the manifold. This does not change the fact that every single Spec(𝔽 p)Spec()Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z}) is like an embedded circle, hence like a knot, but it affects the perspective on which role these play. For instance there does not seem to be a differential geometric analog situation where one considers infinite products over all knots in a 3-space, but there are such situations where one considers infinite products over all prime geodesics in a space, namely the Selberg zeta function analogous to the Artin L-function with its product over prime ideals.

Function field analogy

function field analogy

number fields (“function fields of curves over F1”)function fields of curves over finite fields 𝔽 q\mathbb{F}_q (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
\mathbb{Z} (integers)𝔽 q[z]\mathbb{F}_q[z] (polynomials, function algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational functions)meromorphic functions on complex plane
pp (prime number/non-archimedean place)x𝔽 px \in \mathbb{F}_pxx \in \mathbb{C}
\infty (place at infinity)\infty
Spec()Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
p() p()p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)z\frac{\partial}{\partial z} (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
p\mathbb{Z}_p (p-adic integers)𝔽 q[[tx]]\mathbb{F}_q[ [ t -x ] ] (power series around xx)[[zx]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX
p\mathbb{Q}_p (p-adic numbers)𝔽 q((zx))\mathbb{F}_q((z-x)) (Laurent series around xx)((zx))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((zx))\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x)) (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field ) xGL 1(((zx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
KK a number field (K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma)
𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)ΣP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
() pΦ()p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)z\frac{\partial}{\partial z}
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v (formal completion at vv)((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles) xΣ GL 1(((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
Galois theory
Galois groupπ 1(Σ)\pi_1(\Sigma) fundamental group
Galois representationflat connection (“local system”) on Σ\Sigma
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)
GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})Bun GL 1(Σ)Bun_{GL_1}(\Sigma) (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence
GL n(K)\GL n(𝔸 K)//GL n(𝒪)GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O}) (constant sheaves on this stack form unramified automorphic representations)Bun GL n()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma, by Weil uniformization theorem)
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
theta functions
Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on Σ\Sigma
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on Σ\Sigma
higher dimensional spaces
zeta functionsHasse-Weil zeta function

References

  • Barry Mazur, Notes on étale cohomology of number fields, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 6 no. 4 (1973), p. 521-552 (NUMDAM)

  • Barry Mazur, Remarks on the Alexander polynomial, unpublished note

  • Christopher Deninger, Arithmetic Geometry and Analysis on Foliated Spaces (arXiv:math/0505354)

  • Toshitake Kohno, Masanori Morishita (eds.), Primes and Knots, Contemporary Mathematics, AMS 2006 (web)

  • Lieven LeBruyn, talk 2010 (pdf slides (35 mb), MO comment (with more details))

  • K. Fujiwara, pp-adic gauge theory in number theory, 2007 (pdf slides)

Last revised on October 3, 2018 at 04:20:43. See the history of this page for a list of all contributions to it.