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zeta function

Contents

Idea

The concept of zeta function originates in number theory, but to get an idea of what they “really are” it is helpful to proceed anachronistically:

ζ\zeta-functions are meromorphic functions sζ(s)s \mapsto \zeta(s) on the complex plane, which behave like regularized Feynman propagators, namely which behave like analytic continuations of traces of powers

sTr(1H) s s \mapsto Tr \left(\frac{1}{H}\right)^s

of suitable elliptic differential operators HH, which means that for sufficiently nice such HH these are analytic continuations in ss of sums of the form

sλλ s, s \mapsto \underset{\lambda}{\sum} \lambda^{-s} \,,

where the summation is over the eigenvalues λ\lambda of HH.

Indeed, such zeta functions of elliptic differential operators constitutes one class of examples of zeta functions. Of particular interest is the case where HH is a Laplace operator of a hyperbolic manifold and in particular on a hyperbolic Riemann surface, for that case one obtains the zeta function of a Riemann surface, in particular the Selberg zeta function.

In modern language one also speaks of L-functions. Where a zeta function of some space is like the Feynman propagator of the canonical Laplace operator of that space, an L-function is defined from an extra “twisting” information such as that of a flat bundle/local system of coefficients on the space and is like the Feynman propagator of the corresponding twisted/coupled Laplace operator.

Proceeding from this setup in complex analytic geometry one may wonder if there are analogs also in arithmetic geometry. Indeed, by the function field analogy there are. All the way down “on Spec(Z)” the analog of the Selberg zeta function is the Riemann zeta function, which historically is the first of all zeta functions, defined by analytic continuation of the series

sn=1n s. s \mapsto \underoverset{n = 1}{\infty}{\sum} n^{-s} \,.

The Riemann hypothesis conjectures a characterization of the roots of this zeta function and is regarded as one of the outstanding problems in mathematics. It has evident analogs for all other zeta functions (for some of which it has been proven).

More generally, over arithmetic curves which are spectra of rings of integers of more general number fields, the Riemann zeta function has generalization to the Artin L-functions defined intrinsically in terms of characteristic polynomials of Galois representations. When the Galois representation is 1-dimensional, then the Artin L-function may be expressed (by “Artin reciprocity”) in terms of “more arithmetic” data by Dirichlet L-functions and Hecke L-functions. When the Galois representation is higher dimensional, then the Langlands correspondence conjecture asserts that the Artin L-function may be expressed “arithmetically” as the automorphic L-function of an automorphic form.

Similarly on arithmetic curves given by function fields there is the Goss zeta function and in higher dimensional arithmetic geometry the Weil zeta function, famous from the Weil conjectures. The

When interpreting the Frobenius morphisms that appear in the Artin L-functions geometrically as flows (as discussed at Borger’s arithmetic geometry – Motivation) then this induces an evident analog of zeta function of a dynamical system. This in turn has strong analogies with Alexander polynomials in knot theory (see at arithmetic topology).

context/function field analogytheta function θ\thetazeta function ζ\zeta (= Mellin transform of θ(0,)\theta(0,-))eta function η\eta and L-function L zL_{\mathbf{z}} of Galois representation/flat connection z\mathbf{z}special values of L-functions
physics/2d CFTpartition function θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of complex structure τ\mathbf{\tau} of worldsheet Σ\Sigma (hence polarization of phase space) and background gauge field/source z\mathbf{z}analytically continued Feynman propagator ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued Dirac propagator η z(s)=Tr reg(sgn(D z)D z) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ D_{\mathbf{z}} }\right)^s regularized Feynman propagator pvζ(1)=Tr reg(1(D z) 2)pv\, \zeta(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / regularized Dirac propagator pvη(1)=Tr reg(D z(D z) 2)pv\, \eta(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / path integral ζ (0)=lndet reg(D z 2)\zeta^\prime(0) = - \ln\;det_{reg}(D_{\mathbf{z}}^2)
analysiszeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant
complex analytic geometrysection θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of line bundle over Jacobian variety J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})zeta function of a Riemann surfaceSelberg zeta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function L zL_{\mathbf{z}} for z\mathbf{z} induced from the trivial representation)Artin L-function L zL_{\mathbf{z}} of a Galois representation z\mathbf{z}, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)class number \cdot regulator
arithmetic geometry for \mathbb{Q}Jacobi theta functionRiemann zeta functionArtin L-function of a Galois representation z\mathbf{z} , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

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[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

General

A useful survey of the zoo of zeta functions is in

Further general review includes

Discussion in the more general context of higher dimensional arithmetic geometry is in

  • Ivan Fesenko, Adelic approch to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273–317 (pdf)

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 August 29, 2014 10:40:01 by Urs Schreiber (185.37.147.12)