nLab
arithmetic curve

Context

Arithmetic geometry

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Contents

Idea

A curve in arithmetic geometry, hence an arithmetic scheme of suitable dimension 1 etc.

Properties

Function field analogy

(“ of over ”) of over 𝔽 q\mathbb{F}_q ()/
and
\mathbb{Z} ()𝔽 q[z]\mathbb{F}_q[z] (, on 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} ( on )
\mathbb{Q} ()𝔽 q(z)\mathbb{F}_q(z) () on
pp (/non-archimedean )x𝔽 px \in \mathbb{F}_pxx \in \mathbb{C}
\infty ()\infty
Spec()Spec(\mathbb{Z}) ()𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} ()
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} ()
p() p()p\partial_p \coloneqq \frac{(-)^p - (-)}{p} ()z\frac{\partial}{\partial z} ( )
= 0 = 0
p\mathbb{Z}_p ()𝔽 q[[tx]]\mathbb{F}_q[ [ t -x ] ] ( around xx)[[zx]]\mathbb{C}[ [z-x] ] ( on around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-” of XX at pp) in XX
p\mathbb{Q}_p ()𝔽 q((zx))\mathbb{F}_q((z-x)) ( around xx)((zx))\mathbb{C}((z-x)) ( on punctured around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p ()𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((zx))\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x)) ( of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) ()𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ) xGL 1(((zx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))
curves
KK a (K\mathbb{Q} \hookrightarrow K a possibly )KK a of an Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma ( on Σ\Sigma)
𝒪 K\mathcal{O}_K ()𝒪 Σ\mathcal{O}_{\Sigma} ()
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) ( with archimedean )Σ\Sigma ()ΣP 1\Sigma \to \mathbb{C}P^1 ( being )
() pΦ()p\frac{(-)^p - \Phi(-)}{p} (lift of / structure)z\frac{\partial}{\partial z}
vv prime ideal in 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v ( at vv)((z x))\mathbb{C}((z_x)) ( on punctured around xx)
𝒪 K v\mathcal{O}_{K_v} ( of )[[z x]]\mathbb{C}[ [ z_x ] ] ( on around xx)
𝔸 K\mathbb{A}_K () xΣ ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) ( of on all punctured around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) () xΣ GL 1(((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
π 1(Σ)\pi_1(\Sigma)
(“”) on Σ\Sigma
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) ()
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) (, by )
non-abelian class field theory and automorphy
number field function field
GL n(K)\GL n(𝔸 K)//GL n(𝒪)GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O}) ( on this form )Bun GL n()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma, by )
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
of /chiral on Σ\Sigma
/ on Σ\Sigma

Created on July 17, 2014 at 12:04:34. See the history of this page for a list of all contributions to it.