nLab function field analogy

Idea

There is a noticeable analogy between phenomena (theorems) in the theory of number fields and those in the theory of function fields over finite fields (Weil 39, Iwasawa 69, Mazur-Wiles 83), hence between the theories of the two kinds of global fields. When regarding number theory dually as arithmetic geometry, then one may see that this analogy extends further to include complex analytic geometry, the theory of complex curves (e.g. Frenkel 05).

At a very basic level the analogy may be plausible from the fact that both the integers $\mathbb{Z}$ as well as the polynomial rings $\mathbb{F}_q[x]$ (over finite fields $\mathbb{F}_q$) are principal ideal domains with finite group of units, all quotients being finite rings and with infinitely many prime ideals, which already implies that a lot of arithmetic over these rings is similar. Since number fields are the finite dimensional field extensions of the field of fractions of $\mathbb{Z}$, namely the rational numbers $\mathbb{Q}$, and since function fields are just the finite-dimensional field extensions of the fields of fractions $\mathbb{F}_q(x)$ of $\mathbb{F}_q[x]$, this similarity plausibly extends to these extensions.

The entire holomorphic functions on the complex plane are, while not quite an principal ideal domain, still a Bézout domain, but in constructive mathematics the integers and the polynomial rings over finite fields are only Bézout domains as well.

But the analogy ranges much deeper than this similarity alone might suggest. For instance (Weil 39) defined an invariant of a number field – the genus of a number field– which is analogous to the genus of the algebraic curve on which a given function field is the rational functions. This is such as to make the statement of the Riemann-Roch theorem for algebraic curves extend to arithmetic geometry (Neukirch 92, chapter II, prop.(3.6)).

Another notable part of the analogy is the fact that there are natural analogs of the Riemann zeta function in all three columns of the analogy. This aspect has found attention notably through the lens of regarding number fields as rational functions on “arithmetic curves over the would-be field with one element $\mathbb{F}_1$”.

The analogy between p-adic numbers and Laurent series over $\mathbb{F}_p$ is strengthened by (Fontaine-Winterberger 79), which shows that the absolute Galois groups of the perfection of $\mathbb{F}_p((t))$ and of $\mathbb{Q}_p[p^{\frac{1}{p^\infty}}]$ are isomorphic. For more review of this see also (Hartl 06). (The generalization of this to higher dimensions is the topic of perfectoid spaces.)

It is also the function field analogy which induces the conjecture of the geometric Langlands correspondence by analogy from the number-theoretic Langlands correspondence. Here one finds that the moduli stack of bundles over a complex curve is analogous in absolute arithmetic geometry to the coset space of the general linear group with coefficients in the ring of adeles of a number field, on which unramified automorphic representations are functions. Under this analogy the Weil conjecture on Tamagawa numbers may be regarded as giving the groupoid cardinality of the moduli stack of bundles in arithmetic geometry.

In summary then the analogy says that the theories of number fields and of function fields both look much like a global analytic geometry-version of the theory of complex curves.

Formalizations

To date the function field analogy remains just that, an analogy, though various research programs may be thought of as trying to provide a context in which the analogy would become a consequence of a systematic theory (see e.g. the introduction of v.d. Geer et al 05). This includes

• Arakelov geometry;

• global analytic geometry.

• geometry “over F1”.

Regarding the last point, in particular Borger's absolute geometry (Borger 09) makes precise the analogy between Spec(Z) and the polynomial ring $k[z]$/entire holomorphic function-ring $\mathcal{O}_{\mathbb{C}}$ by interpreting the analog of the canonical derivation $\frac{\partial}{\partial z}$ on the latter two as the Fermat quotient operation, and more generally by interpreting the lift of this to arithmetic spaces over ${Spec}(\mathbb{Z})$ as lifts of Frobenius homomorphisms as given by Lambda-ring structures. See at Borger’s absolute geometry – Motivation for more on this.

In this context the analogy between geometry over number fields and over function fields is made precise by showing (Borger 09, section 7) that for any smooth connected curve $S/\mathbb{F}_q$ over a finite field $\mathbb{F}_q$ the standard geometric morphism of (“big”) toposes

$$Spec(S/\mathbb{F}_q)\longrightarrow Spec(\mathbb{F}_q)$$

factors through an alternative base topos $\widetilde Spec(\mathbb{F}_q)$

$$Spec(S/\mathbb{F}_q)\longrightarrow \widetilde Spec(\mathbb{F}_q) \longrightarrow Spec(\mathbb{F}_q)$$

which, while different from $Spec(\mathbb{F}_q)$ is “close” to $Spec(\mathbb{F}_q)$ in some precise sense, but which has the advantage that its construction does exist for $q = 1$ in that there is directly analogous

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

where the notation $\widetilde Spec(\mathbb{F}_1)$ here stands for Borger’s the topos over Lambda-rings, see at Borger's absolute geometry for the actual details.

Overview

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

$\,$

analogies in the Langlands program:

arithmetic Langlands correspondence	geometric Langlands correspondence
ring of integers of global field	structure sheaf on complex curve $\Sigma$
Galois group	fundamental group of $\Sigma$
Galois representation	flat connection/local system on $\Sigma$
idele class group mod integral adeles	moduli stack of line bundles on $\Sigma$
nonabelian $\;$ “	moduli stack of vector bundles on $\Sigma$
automorphic representation	Hitchin connection D-module on bundle of conformal blocks over the moduli stack

Related concepts

• MKR analogy in arithmetic topology

References

Original articles includes

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

• Kenkichi Iwasawa, Analogies between number fields and function fields, in Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Belfer Graduate School of Science, Yeshiva Univ., New York, pp. 203–208, MR 0255510

  for more on this see: Wikipedia, Main conjecture of Iwasawa theory

• Jean-Marc Fontaine, Jean-Pierre Wintenberger, Extensions algébrique et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris Sér. A–B 288(8) (1979), A441–A444

• Barry Mazur, Andrew Wiles, Analogies between function fields and number fields, American Journal of Mathematics Vol. 105, No. 2 (Apr., 1983), pp. 507-521 (JStor)

Textbook accounts include

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

• Michael Rosen, Number theory in function fields, Graduate texts in mathematics, 2002

Tables showing the parallels between number fields and function fields are in

• David Goss, Dictionary, in David Goss, David R. Hayes, Michael Rosen (eds.) The Arithmetic of Function Fields, Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin, 1992, pp. 475-482,

• Bjorn Poonen, section 2.6 of Lectures on rational points on curves, 2006 (pdf)

• Urs Hartl, A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic (arXiv:math/0607182)

See also

• M. Blickle,Hélène Esnault, K. Rülling, Characteristic $0$ and $p$ analogies, and somemotivic cohomology (pdf)

A collection of more recent developments is in

• van der Geer et al (eds.) Number Fields and Function Fields – Two Parallel Worlds, Birkhäuser 2005 (publisher page)

Discussion including also the complex-analytic side includes

• Edward Frenkel, section 2 of Lectures on the Langlands Program and Conformal Field Theory (arXiv:hep-th/0512172).

and a comparison of the number theory to that of foliations is in

• Christopher Deninger, Analogies between analysis on foliated spaces and arithmetic geometry (arXiv:0709.2801)

An actual formalization of the analogy between geometry over number fields and function fields is in

• James Borger, section 7 of Lambda-rings and the field with one element (arXiv/0906.3146)