affine and projective line | | | |
| (integers) | (polynomials, polynomial algebra on affine line ) | (holomorphic functions on complex plane) |
| (rational numbers) | (rational fractions/rational function on affine line ) | meromorphic functions on complex plane |
| (prime number/non-archimedean place) | , where is the irreducible monic polynomial of degree one | , where is the function which subtracts the complex number from the variable |
| (place at infinity) | | |
| (Spec(Z)) | (affine line) | complex plane |
| | (projective line) | Riemann sphere |
| (Fermat quotient) | (coordinate derivation) | “ |
| genus of the rational numbers = 0 | | genus of the Riemann sphere = 0 |
formal neighbourhoods | | | |
| (prime power local ring) | (-th order univariate local Artinian -algebra) | (-th order univariate Weil -algebra) |
| (p-adic integers) | (power series around ) | (holomorphic functions on formal disk around ) |
| (“-arithmetic jet space” of at ) | | formal disks in |
| (p-adic numbers) | (Laurent series around ) | (holomorphic functions on punctured formal disk around ) |
| (ring of adeles) | ( adeles of function field ) | (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks) |
| (group of ideles) | ( ideles of function field ) | |
theta functions | | | |
| Jacobi theta function | | |
zeta functions | | | |
| Riemann zeta function | Goss zeta function | |
| | | |
branched covering curves | | | |
| a number field ( a possibly ramified finite dimensional field extension) | a function field of an algebraic curve over | (sheaf of rational functions on complex curve ) |
| (ring of integers) | | (structure sheaf) |
| (spectrum with archimedean places) | (arithmetic curve) | (complex curve being branched cover of Riemann sphere) |
| (lift of Frobenius morphism/Lambda-ring structure) | | “ |
| genus of a number field | genus of an algebraic curve | genus of a surface |
formal neighbourhoods | | | |
| prime ideal in ring of integers | | |
| (formal completion at ) | | (function algebra on punctured formal disk around ) |
| (ring of integers of formal completion) | | (function algebra on formal disk around ) |
| (ring of adeles) | | (restricted product of function rings on all punctured formal disks around all points in ) |
| | | (function ring on all formal disks around all points in ) |
| (group of ideles) | | |
Galois theory | | | |
| Galois group | “ | fundamental group |
| Galois representation | “ | flat connection (“local system”) on |
class field theory | | | |
| class field theory | “ | geometric class field theory |
| Hilbert reciprocity law | Artin reciprocity law | Weil reciprocity law |
| (idele class group) | “ | |
| | “ | (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 |
| (constant sheaves on this stack form unramified automorphic representations) | “ | (moduli stack of bundles on the curve , 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 |
zeta functions | | | |
| Dedekind zeta function | Weil zeta function | zeta function of a Riemann surface/of the Laplace operator on |
| | | |
higher dimensional spaces | | | |
zeta functions | Hasse-Weil zeta function | | |