nLab arithmetic geometry

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 “arithmetic algebraic geometry”.

Since an affine variety in this context is given by solutions to Diophantine equations, this is also 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 “at 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 $\mathbb{F}_1$

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

$$Et(Spec(\mathbb{Z})) \longrightarrow Et(Spec(\mathbb{F}_1)) \,.$$

Function field analogy

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Related concepts

• arithmetic variety, arithmetic scheme

  • arithmetic curve

• arithmetic Chow group

• analytic geometry

• Weil-étale topology for arithmetic schemes

• absolute cohomology

• function field analogy

• Weil conjecture on Tamagawa numbers

• Borger's absolute geometry

• Mordell conjecture

• Iwasawa-Tate theory

• arithmetic jet space

• arithmetic differential geometry

• adelic integration

• shtuka

• higher arithmetic geometry

• E-∞ arithmetic geometry

• arithmetic topology

• Frobenioid

• p-adic physics

  • p-adic string theory

  • p-adic AdS/CFT correspondence

References

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

An almost entirely self-contained introduction and (according to Werner Kleinert) “the most comprehensive and detailed elaboration of the theory of algebraic schemes available in (text-)book form (after Grothendieck’s EGA)”:

• Qing Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics 6, 2002/2006, 600pp.

Lecture notes include

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

• C. Soulé, 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:

• Minhyong Kim, Galois Theory and Diophantine geometry, 2009 (pdf)

Further resources:

• Wikipedia: Glossary of arithmetic and Diophantine geometry,

• Wikipedi, Arakelov geometry

• Arakelov geometry preprint arxiv, list of links

• conferences in arithmetic geometry, at Kiran Kedlaya‘s wiki

Some relation to modular tensor categories:

• Orit Davidovich, Tobias Hagge, Zhenghan Wang, On Arithmetic Modular Categories, arXiv preprit, 2013 [arXiv:1305.2229]