transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
analytic geometry (complex, rigid, global)
geometry+analysis/analytic number theory
analytic space, analytic variety, Berkovich space
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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. (Also the entire holomorphic functions on the complex plane are, while not quite an principal ideal domain still a BΓ©zout domain. )
But the analogy ranges much deeper than this similarity alone might suggest. For instance (Weil 39) defined an invariant of a number field β the genus of a number fieldβ 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 βarithmetic curves over the would-be field with one element $\mathbb{F}_1$β.
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 theory of number fields and of function fields both looks much like a global analytic geometry-version of the theory complex curves.
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
geometry βover F1β.
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 Borgerβs absolute geometry β 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 (βbigβ) toposes
factors through an alternative base topos $\widetilde Spec(\mathbb{F}_q)$
which, while different from $Spec(\mathbb{F}_q)$ is βcloseβ 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
where the notation $\widetilde Spec(\mathbb{F}_1)$ here stands for Borgerβs the topos over Lambda-rings, see at Borger's absolute geometry for the actual details.
number fields (βfunction fields of curves over F1β) | 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, function 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 functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (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) | β | |
genus of the rational numbers = 0 | genus of the Riemann sphere = 0 | ||
formal neighbourhoods | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (power series around $x$) | $\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$) | |
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (β$p$-arithmetic jet spaceβ 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) | $\frac{\partial}{\partial z}$ | β | |
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 | β | $\pi_1(\Sigma)$ fundamental group | |
Galois representation | β | flat connection (βlocal systemβ) on $\Sigma$ | |
class field theory | |||
class field theory | β | 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) | β | ||
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$ | β | $Bun_{GL_1}(\Sigma)$ (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) | β | $Bun_{GL_n(\mathbb{C})}(\Sigma)$ (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 $\;$ β | moduli stack of vector bundles on $\Sigma$ |
automorphic representation | Hitchin connection D-module on bundle of conformal blocks over the moduli stack |
Original articles includes
AndrΓ© Weil, Sur lβanalogie entre les corps de nombres algΓ©brique et les corps de fonctions algΓ©brique, 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. 203β208, MR 0255510
for more on this see: Wikipedia, Main conjecture of Iwasawa theory
Jean-Marc Fontaine, Jean-Pierre Wintenberger, Extensions algΓ©brique et corps des normes des extensions APF des corps locaux, C. R. Acad. Sci. Paris SΓ©r. AβB 288(8) (1979), A441βA444
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
JΓΌrgen 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
A collection of more recent developments is in
Discussion including also the complex-analytic side includes
and a comparison of the number theory to that of foliations is in
An actual formalization of the analogy between geometry over number fields and function fields is in