[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] | | Elements of unique factorization domains| |-|---------| | Base unique factorization domain | $R$ | | Field at irreducible element | $R/p$ | | Weil algebra at power of irreducuble element | $R/p^n$ | | Topological completion | $R_{p}$ | Shapes as types, to define extension types... **[[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]] | arbitrary [[unique factorization domain]] | |----|-------------------|----------------------|----|----| | **[[affine line|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]]) | $R$ ([[unique factorization domain]]) | | | $\mathbb{Q}$ ([[rational numbers]]) | $\mathbb{F}_q(z)$ ([[rational fractions]]/[[rational function on an affine variety|rational function on affine line]] $\mathbb{A}^1_{\mathbb{F}_q}$) | [[meromorphic functions]] on [[complex plane]] | $R[(R^\times)^{-1}]$ [[field of fractions]] | | | $p$ ([[prime number]]/non-archimedean [[place]]) | $x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the [[irreducible element|irreducible]] [[monic polynomial]] of [[degree of a polynomial|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]] $z$ | $p$ ([[irreducible element]]) | | | $\infty$ ([[place at infinity]]) | | $\infty$ | | | | $Spec(\mathbb{Z})$ ([[Spec(Z)]]) | $\mathbb{A}^1_{\mathbb{F}_q}$ ([[affine line]]) | [[complex plane]] | $Spec(R)$ ([[spectrum]] of $R$) | | | $Spec(\mathbb{Z}) \cup place_{\infty}$ | $\mathbb{P}_{\mathbb{F}_q}$ ([[projective line]]) | [[Riemann sphere]] | | | | $\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ ([[Fermat quotient]]) | $\frac{\partial}{\partial z}$ ([[coordinate]] [[derivation]]) | " | | | | [[genus of a number field|genus of the rational numbers]] = 0 | | [[genus of a surface|genus of the Riemann sphere]] = 0 | | | **[[formal neighbourhoods]]** | | | | | | | $\mathbb{Z}/(p^n \mathbb{Z})$ ([[prime power local ring]]) | $\mathbb{F}_q [t]/((t-x)^n \mathbb{F}_q [t])$ ($n$-th order univariate [[local Artinian ring|local Artinian $\mathbb{F}_q$-algebra]]) | $\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate [[Weil algebra|Weil $\mathbb{C}$-algebra]]) | $R/(p^n R)$ | | | $\mathbb{Z}_p$ ([[p-adic integers]]) | $\mathbb{F}_q[ [ t -x ] ]$ ([[power series]] around $x$) | $\mathbb{C}[ [z-x] ]$ ([[holomorphic functions]] on [[formal disk]] around $x$) | $R_p \coloneqq \underset{n \to \infty}\mathrm{colim} R/(p^n R)$ | | | $Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ ("$p$-[[arithmetic jet space]]" of $X$ at $p$) | | [[formal disks]] in $X$ | $Spf(R_p)\underset{Spec(R)}{\times} X$ | | | $\mathbb{Q}_p$ ([[p-adic numbers]]) | $\mathbb{F}_q((z-x))$ ([[Laurent series]] around $x$) | $\mathbb{C}((z-x))$ ([[holomorphic functions]] on punctured [[formal disk]] around $x$) | $R_p[(R_p^\times)^{-1}]$ ([[field of fractions]] of $R_p$) | | | $\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ ([[ring of adeles]]) | $\mathbb{A}_{\mathbb{F}_q((t))}$ ( [adeles of function field](ring%20of%20adeles#ForAGlobalField) ) | $\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ ([[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]]) | $\mathbb{I}_{\mathbb{F}_q((t))}$ ( [[group of ideles|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 set|finite]] [[dimension|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 of a commutative ring|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 bundles|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](moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence) $\Sigma$, by [[Weil uniformization theorem]]) | | | | [Tamagawa-Weil for number fields](Weil+conjecture+on+Tamagawa+numbers#NumberFieldCase) | [Tamagawa-Weil for function fields](Weil+conjecture+on+Tamagawa+numbers#FunctionFieldCase) | | | | **[[theta functions]]** | | | | | | | [[Hecke theta function]] | | [[functional determinant]] [[determinant line bundle|line bundle]] of [[Dirac operator]]/chiral [[Laplace operator]] on $\Sigma$ | | | **[[zeta functions]]** | | | | | | [[Dedekind zeta function]] | [[Weil zeta function]] | [[zeta function of a Riemann surface]]/[[zeta function of an elliptic differential operator|of the Laplace operator]] on $\Sigma$ | | | | | | | | | | **[[higher dimensional arithmetic geometry|higher dimensional spaces]]** | | | | | | | **[[zeta functions]]** | [[Hasse-Weil zeta function]] | | | | | category: redirected to nlab