# 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 $\mathbb{F}_q$ ()/
and
$\mathbb{Z}$ ()$\mathbb{F}_q[z]$ (, on $\mathbb{A}^1_{\mathbb{F}_q}$)$\mathcal{O}_{\mathbb{C}}$ ( on )
$\mathbb{Q}$ ()$\mathbb{F}_q(z)$ () on
$p$ (/non-archimedean )$x \in \mathbb{F}_p$$x \in \mathbb{C}$
$\infty$ ()$\infty$
$Spec(\mathbb{Z})$ ()$\mathbb{A}^1_{\mathbb{F}_q}$ ()
$Spec(\mathbb{Z}) \cup place_{\infty}$$\mathbb{P}_{\mathbb{F}_q}$ ()
$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ ()$\frac{\partial}{\partial z}$ ( )
= 0 = 0
$\mathbb{Z}_p$ ()$\mathbb{F}_q[ [ t -x ] ]$ ( around $x$)$\mathbb{C}[ [z-x] ]$ ( on around $x$)
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-” of $X$ at $p$) in $X$
$\mathbb{Q}_p$ ()$\mathbb{F}_q((z-x))$ ( around $x$)$\mathbb{C}((z-x))$ ( on punctured around $x$)
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ ()$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ ( of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ ()$\mathbb{I}_{\mathbb{F}_q((t))}$ ( )$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
curves
$K$ a ($\mathbb{Q} \hookrightarrow K$ a possibly )$K$ a of an $\Sigma$ over $\mathbb{F}_p$$K_\Sigma$ ( on $\Sigma$)
$\mathcal{O}_K$ ()$\mathcal{O}_{\Sigma}$ ()
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ ( with archimedean )$\Sigma$ ()$\Sigma \to \mathbb{C}P^1$ ( being )
$\frac{(-)^p - \Phi(-)}{p}$ (lift of / structure)$\frac{\partial}{\partial z}$
$v$ prime ideal in $\mathcal{O}_K$$x \in \Sigma$$x \in \Sigma$
$K_v$ ( at $v$)$\mathbb{C}((z_x))$ ( on punctured around $x$)
$\mathcal{O}_{K_v}$ ( of )$\mathbb{C}[ [ z_x ] ]$ ( on around $x$)
$\mathbb{A}_K$ ()$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ ( of on all punctured around all points in $\Sigma$)
$\mathcal{O}$$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all around all points in $\Sigma$)
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ ()$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
$\pi_1(\Sigma)$
(“”) on $\Sigma$
$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ ()
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$$Bun_{GL_1}(\Sigma)$ (, by )
non-abelian class field theory and automorphy
number field function field
$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ ( on this form )$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by )
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
of /chiral on $\Sigma$
/ on $\Sigma$

Created on July 17, 2014 at 12:01:55. See the history of this page for a list of all contributions to it.