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zeta function

Contents

Idea

ζ\zeta-functions are functions on parts of the complex plane, which look like generating functions for the counting of points in spaces in algebraic geometry. The archetypical and historically first example is the Riemann zeta function, which is

ζ:sn=1 1n s. \zeta \colon s \mapsto \underset{n = 1}{\sum}^\infty \frac{1}{n^s} \,.

Under the function field analogy one may understand this function in terms of the counting of points in Spec()Spec(\mathbb{Z}). This induces a plethora of generalizations: the Dedekind zeta function generalizes this from integers/rational numbers to number fields/their rings of integers, the Goss zeta function to function fields, the Weil zeta function (which is the topic of the famous Weil conjectures) to more general varieties over finite fields, and the Selberg zeta function to complex curves(Riemann surfaces). How all this hangs together is surveyed below in function field analogy.

This way many fields of mathematics have their own versions of zeta functions: arithmetic geometry and algebraic geometry, dynamical systems, graphs, operator algebra. There are several articles trying to attach zeta functions attached to triangulated categories (and stable model categories…) and study them. Zeta functions express some motivic information, hence the categorical framework must be natural as it is for motives.

(…)

Properties

Function field analogy

One way to understand the plethora of different zeta functions is to see them as the incarnation of the same general concept in different flavors of geometry. This is expressed at least in parts by the

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[t]\mathbb{F}_q[t] (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(t)\mathbb{F}_q(t) (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
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)[[tx]]\mathbb{C}[ [t-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((tx))\mathbb{F}_q((t-x)) (Laurent series around xx)((tx))\mathbb{C}((t-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((tx))\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((t-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(((tx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((t-x)))
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)
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)((t x))\mathbb{C}((t_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[t x]]\mathbb{C}[ [ t_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((t x))\prod^\prime_{x\in \Sigma} \mathbb{C}((t_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[t x]]\prod_{x\in \Sigma} \mathbb{C}[ [t_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(((t x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((t_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
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface

References

General

A general review is in

In algebraic geometry

  • Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228:4 (1995) 121–163, and preprint MPIM1992-50 pdf

  • Nobushige Kurokawa, Zeta functions over F 1F_1, Proc. Japan Acad. Ser. A Math. Sci. 81:10 (2005) 180-184 euclid

Categorical approaches

Revised on July 23, 2014 00:22:47 by David Corfield (129.12.18.232)