genus of a surface in nLab

affine and projective line ℤ\mathbb{Z} (integers)𝔽 q[z]\mathbb{F}_q[z] (polynomials, polynomial algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 ℂ\mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane) ℚ\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational fractions/rational function on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})meromorphic functions on complex plane pp (prime number/non-archimedean place)x∈𝔽 px \in \mathbb{F}_p, where z−x∈𝔽 q[z]z - x \in \mathbb{F}_q[z] is the irreducible monic polynomial of degree onex∈ℂx \in \mathbb{C}, where z−x∈𝒪 ℂz - x \in \mathcal{O}_{\mathbb{C}} is the function which subtracts the complex number xx from the variable zz ∞\infty (place at infinity)∞\infty Spec(ℤ)Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane Spec(ℤ)∪place ∞Spec(\mathbb{Z}) \cup place_{\infty}ℙ 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere ∂ p≔(−) p−(−)p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)∂∂z\frac{\partial}{\partial z} (coordinate derivation)“ genus of the rational numbers = 0genus of the Riemann sphere = 0 formal neighbourhoods ℤ/(p nℤ)\mathbb{Z}/(p^n \mathbb{Z}) (prime power local ring)𝔽 q[z]/((z−x) n𝔽 q[z])\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z]) (nn-th order univariate local Artinian 𝔽 q \mathbb{F}_q -algebra)ℂ[z]/((z−x) nℂ[z])\mathbb{C}[z]/((z-x)^n \mathbb{C}[z]) (nn-th order univariate Weil ℂ \mathbb{C} -algebra) ℤ p\mathbb{Z}_p (p-adic integers)𝔽 q[[z−x]]\mathbb{F}_q[ [ z -x ] ] (power series around xx)ℂ[[z−x]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx) Spf(ℤ p)×Spec(ℤ)XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX ℚ p\mathbb{Q}_p (p-adic numbers)𝔽 q((z−x))\mathbb{F}_q((z-x)) (Laurent series around xx)ℂ((z−x))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx) 𝔸 ℚ=∏ ′pplaceℚ p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field )∏ ′x∈ℂℂ((z−x))\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) 𝕀 ℚ=GL 1(𝔸 ℚ)\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field )∏ ′x∈ℂGL 1(ℂ((z−x)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x))) theta functions Jacobi theta function zeta functions Riemann zeta functionGoss zeta function branched covering curves KK a number field (ℚ↪K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma) 𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf) Spec an(𝒪 K)→Spec(ℤ)Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)Σ→ℂP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere) (−) p−Φ(−)p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)∂∂z\frac{\partial}{\partial z}“ genus of a number fieldgenus of an algebraic curvegenus of a surface formal neighbourhoods vv prime ideal in ring of integers 𝒪 K\mathcal{O}_Kx∈Σx \in \Sigmax∈Σx \in \Sigma K vK_v (formal completion at vv)ℂ((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx) 𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)ℂ[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx) 𝔸 K\mathbb{A}_K (ring of adeles)∏ x∈Σ ′ℂ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma) 𝒪\mathcal{O}∏ x∈Σℂ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Σ\Sigma) 𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles)∏ x∈Σ ′GL 1(ℂ((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x))) Galois theory Galois group“π 1(Σ)\pi_1(\Sigma) fundamental group Galois representation“flat connection (“local system”) on Σ\Sigma class field theory class field theory“geometric class field theory Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)“ GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})“Bun GL 1(Σ)Bun_{GL_1}(\Sigma) (moduli stack of line bundles, by Weil uniformization theorem) non-abelian class field theory and automorphy number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence GL n(K)\GL n(𝔸 K)//GL n(𝒪)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(ℂ)(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma, by Weil uniformization theorem) Tamagawa-Weil for number fieldsTamagawa-Weil for function fields theta functions Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on Σ\Sigma zeta functions Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on Σ\Sigma higher dimensional spaces zeta functionsHasse-Weil zeta function