nLab genus of a number field

Idea

Under the function field analogy it makes sense to regard a number field as the rational functions on an “arithmetic curve over F1”. Accordingly, there is a sensible version of the concept of genus of a curve for number fields.

This genus of a number field was originally introduced in (Weil 39) in a maybe somewhat ad hoc way. It is derived as being that definition which makes the Riemann-Roch theorem come out right in (Neukirch 92, chapter III, def. (3.5)).

Examples

• The genus of the rational numbers themselves is 0.

Properties

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

References

• André Weil, Sur l’analogie entre les corps de nombres algébriques et les corps de fonctions algébriques, Revue Scient. 77, 104-106, 1939

• Jürgen Neukirch, Algebraische Zahlentheorie (1992), English translation Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322, 1999 (pdf)