nLab
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, algebraic geometry over non-algebraically closed fields or fields of positive characteristic is also referred to as β€œarithmetic algebraic geometry”.

Since an affine variety in this context is given by solutions to Diophantine equations, this is alse called Diophantine 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.

and with an eye towards anabelian geometry:

Further resources include

Revised on December 8, 2014 18:28:28 by Urs Schreiber (87.183.144.209)