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Artin reciprocity law

Contents

Idea

Emil Artin’s reciprocity law is a reciprocity law in class field theory for global fields.

For KK a global field there is a canonical map

𝕀 K⟢Gal(K ab/K) \mathbb{I}_K \longrightarrow Gal(K^{ab}/K)

from its group of ideles to the abelianization of its Galois group, given by

(β‹―,a v,β‹―)β†¦βˆ vFrob v ord v(a v). (\cdots, a_v, \cdots) \mapsto \prod_v Frob_v^{ord_v(a_v)} \,.

For KK a number field, Artin’s reciprocity law says that this map is surjective, that it factors through the idele class group K Γ—\𝕀 KK^\times \backslash \mathbb{I}_K and moreover that further quotienting this by the connected component π’ͺ\mathcal{O} of 1 in the idele class group yields an isomorphism

K Γ—\𝕀 K/π’ͺβŸΆβ‰ƒGal(K ab/K). K^\times \backslash \mathbb{I}_K / \mathcal{O} \stackrel{\simeq}{\longrightarrow} Gal(K^{ab}/K) \,.

For K=β„šK = \mathbb{Q} this is also the statement of the Kronecker-Weber theorem, and together this is a starting point of the Langlands correspondence conjecture, see there for more.

For KK a function field the map is no longer surjective, but yields on the quotient by the restricted product ∏ vπ’ͺ v Γ—\prod_v \mathcal{O}_v^\times an injection with dense image

K Γ—\𝕀 K/∏ vπ’ͺ v Γ—β†ͺGal(K ab/K). K^\times \backslash \mathbb{I}_K / \prod_v \mathcal{O}_v^\times \hookrightarrow Gal(K^{ab}/K) \,.

(e.g. Toth 11, p. 3)

Notice that the double quotients appearing here are by the Weil uniformization theorem analogous to moduli stacks of bundles on the arithmetic curve on which KK is the field of rational functions. Under this function field analogy the analog of Artin’s reciprocity law plays a central role in the geometric Langlands correspondence.

Properties

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[t]\mathbb{F}_q[t] (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(t)\mathbb{F}_q(t) (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
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„‚[[tβˆ’x]]\mathbb{C}[ [t-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((tβˆ’x))\mathbb{F}_q((t-x)) (Laurent series around xx)β„‚((tβˆ’x))\mathbb{C}((t-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„‚((tβˆ’x))\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 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„‚((tβˆ’x)))\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 (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)
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„‚((t x))\mathbb{C}((t_x)) (function algebra on punctured formal disk around xx)
π’ͺ K v\mathcal{O}_{K_v} (ring of integers of formal completion)β„‚[[t x]]\mathbb{C}[ [ t_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles)∏ x∈Σ β€²β„‚((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 Ξ£\Sigma)
π’ͺ\mathcal{O}∏ xβˆˆΞ£β„‚[[t x]]\prod_{x\in \Sigma} \mathbb{C}[ [t_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„‚((t x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((t_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
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface

References

Disucssion with an eye towards geometric class field theory is in

  • Peter Toth, Geometric abelian class field theory, 2011 (web)

Revised on July 27, 2014 20:14:47 by Urs Schreiber (89.204.155.95)