nLab
Hecke theta function

Contents

Idea

A theta function for number fields which generalizes the Jacobi theta function. As the latter is related to the Riemann zeta function, so Hasse theta functions are related to the Dedekind zeta function.

(Kowalski (2.4))

Properties

Relation to Dedekind zeta function

A certain integral over a kernel involving the Hecke theta function gives the Dedekind zeta function. See at Dedekind zeta function – Relation to Hecke theta function.

Function field analogy

function field analogy

number fields (β€œfunction 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
β„€\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})π’ͺ β„‚\mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
β„š\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational functions)meromorphic functions on complex plane
pp (prime number/non-archimedean place)xβˆˆπ”½ px \in \mathbb{F}_pxβˆˆβ„‚x \in \mathbb{C}
∞\infty (place at infinity)∞\infty
Spec(β„€)Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec(β„€)βˆͺplace ∞Spec(\mathbb{Z}) \cup place_{\infty}β„™ 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
βˆ‚ p≔(βˆ’) pβˆ’(βˆ’)p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)βˆ‚βˆ‚z\frac{\partial}{\partial z} (coordinate derivation)β€œ
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
β„€ p\mathbb{Z}_p (p-adic integers)𝔽 q[[tβˆ’x]]\mathbb{F}_q[ [ t -x ] ] (power series around xx)β„‚[[zβˆ’x]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx)
Spf(β„€ p)Γ—Spec(β„€)XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (β€œpp-arithmetic jet space” of XX at pp)formal disks in XX
β„š p\mathbb{Q}_p (p-adic numbers)𝔽 q((zβˆ’x))\mathbb{F}_q((z-x)) (Laurent series around xx)β„‚((zβˆ’x))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 β„š=∏ β€²pplaceβ„š 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 )∏ β€²xβˆˆβ„‚β„‚((zβˆ’x))\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 β„š=GL 1(𝔸 β„š)\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 )∏ β€²xβˆˆβ„‚GL 1(β„‚((zβˆ’x)))\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 (β„šβ†ͺK\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Ξ£\Sigma over 𝔽 p\mathbb{F}_pK Ξ£K_\Sigma (sheaf of rational functions on complex curve Ξ£\Sigma)
π’ͺ K\mathcal{O}_K (ring of integers)π’ͺ Ξ£\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(π’ͺ K)β†’Spec(β„€)Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Ξ£\Sigma (arithmetic curve)Ξ£β†’β„‚P 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
(βˆ’) pβˆ’Ξ¦(βˆ’)p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)βˆ‚βˆ‚z\frac{\partial}{\partial z}β€œ
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers π’ͺ K\mathcal{O}_Kx∈Σx \in \Sigmax∈Σx \in \Sigma
K vK_v (formal completion at vv)β„‚((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx)
π’ͺ K v\mathcal{O}_{K_v} (ring of integers of formal completion)β„‚[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles)∏ x∈Σ β€²β„‚((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 Ξ£\Sigma)
π’ͺ\mathcal{O}∏ xβˆˆΞ£β„‚[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Ξ£\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles)∏ x∈Σ β€²GL 1(β„‚((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
Galois theory
Galois groupβ€œΟ€ 1(Ξ£)\pi_1(\Sigma) fundamental group
Galois representationβ€œflat connection (β€œlocal system”) on Ξ£\Sigma
class field theory
class field theoryβ€œgeometric 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)β€œ
GL 1(K)\GL 1(𝔸 K)/GL 1(π’ͺ)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})β€œBun GL 1(Ξ£)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(π’ͺ)GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O}) (constant sheaves on this stack form unramified automorphic representations)β€œBun GL n(β„‚)(Ξ£)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Ξ£\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 Ξ£\Sigma
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on Ξ£\Sigma
higher dimensional spaces
zeta functionsHasse-Weil zeta function
context/function field analogytheta function ΞΈ\thetazeta function ΞΆ\zeta (= Mellin transform of ΞΈ(0,βˆ’)\theta(0,-))L-function L zL_{\mathbf{z}} (= Mellin transform of ΞΈ(z,βˆ’)\theta(\mathbf{z},-))eta function Ξ·\etaspecial values of L-functions
physics/2d CFTpartition function ΞΈ(z,Ο„)=Tr(exp(βˆ’Ο„β‹…(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of complex structure Ο„\mathbf{\tau} of worldsheet Ξ£\Sigma (hence polarization of phase space) and background gauge field/source z\mathbf{z}analytically continued trace of Feynman propagator ΞΆ(s)=Tr reg(1(D 0) 2) s=∫ 0 βˆžΟ„ sβˆ’1ΞΈ(0,Ο„)dΟ„\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued trace of Feynman propagator in background gauge field z\mathbf{z}: L z(s)≔Tr reg(1(D z) 2) s=∫ 0 βˆžΟ„ sβˆ’1ΞΈ(z,Ο„)dΟ„L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tauanalytically continued trace of Dirac propagator in background gauge field z\mathbf{z} Ξ· z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s regularized 1-loop vacuum amplitude pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / regularized fermionic 1-loop vacuum amplitude pvΞ· z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / vacuum energy βˆ’12L z β€²(0)=Z H=12lndet reg(D z 2)-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)
Riemannian geometry (analysis)zeta function of an elliptic differential operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
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 ℂ g→J(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})zeta function of a Riemann surfaceSelberg zeta functionDedekind eta 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=0\mathbf{z} = 0 the trivial Galois representation)Artin L-function L zL_{\mathbf{z}} of a Galois representation z\mathbf{z}, expressible β€œin 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 β‹…\cdot regulator
arithmetic geometry for β„š\mathbb{Q}Jacobi theta function (z=0\mathbf{z} = 0)/ Dirichlet theta function (z=Ο‡\mathbf{z} = \chi a Dirichlet character)Riemann zeta function (being the Dirichlet L-function L zL_{\mathbf{z}} for Dirichlet character z=0\mathbf{z} = 0)Artin L-function of a Galois representation z\mathbf{z} , expressible β€œin coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

  • E. Kowalski, first part of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)

Last revised on August 21, 2014 at 10:54:22. See the history of this page for a list of all contributions to it.