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arithmetic geometry

Contents

Idea

Arithmetic geometry is a branch of algebraic geometry studying schemes (usually of finite type) over the spectrum Spec(Z) of the commutative ring of integers. More generally often algebraic geometry over non-algebraically closed fields or fields of positive characteristic is also referred to as β€œarithmetic algebraic geometry”.

An archetypical application of arithmetic geometry is the study of elliptic curves over the integers and the rational numbers.

For number theoretic purposes, i.e. in actual arithmetic; usually one complements this with some data β€œat the prime at infinity” leading to a more modern notion of an arithmetic scheme (cf. Arakelov geometry).

The refinement to higher geometry is E-infinity geometry (spectral geometry).

Properties

Base over 𝔽 1\mathbb{F}_1

Arithmetic geometry naturally has as base topos the topos over F1 in the sense of Borger's absolute geometry, which gives an essential geometric morphism of etale toposes

Et(Spec(β„€))⟢Et(Spec(𝔽 1)). Et(Spec(\mathbb{Z})) \longrightarrow Et(Spec(\mathbb{F}_1)) \,.

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

References

Textbook accounts include

  • Dino Lorenzini, An Invitation to Arithmetic Geometry (Graduate Studies in Mathematics, Vol 9) GSM/9

Lecture notes include

  • Andrew Sutherland, Introduction to Arithmetic Geometry, 2013 (web)

  • C. SoulΓ©, D. Abramovich, J. F. Burnol, J. K. Kramer, Lectures on Arakelov Geometry, Cambridge Studies in Advanced Mathematics 33, 188 pp.

Further resources include

Revised on August 24, 2014 23:24:04 by Urs Schreiber (89.204.138.16)