On class number of quadratic extensions over function fields
type: Item
instance of: scholarly article
title: On class number of quadratic extensions over function fields
zbMATH Open document ID: 0911.11053
author: Iwao Kimura
published in: Manuscripta Mathematica
publication date: 1998-11-29T00:00:00Z
review text: Let \({\mathbb{F}} _p (T)\) be a rational function field over the field of \(p\) elements, \(p \geq 5\) and let \(k\) be a finite extension of \({\mathbb{F}} _p (T)\). Let \(n\) be a natural number and let \(\alpha : M _\infty \rightarrow \prod _{v \in M _\infty} Q _v\) be any map, where \(M _\infty\) denotes the set of infinite prime divisors of \(k\) and \(Q _v\) denotes the finite set of extensions over \(k _v\), the completion of \(k\) at \(v\), whose degrees over \(k _v\) are bounded by 2. Let \({\mathcal Q} _\alpha ^\ast (n)\) denote the set of quadratic extensions \(F\) over \(k\) such that for any place \(v \in M _\infty\), \(v\) is ramified, splits or is inert in \(F\) according to whether \(\alpha(v)\) is a ramified quadratic extension over \(k _v\), \(k _v\) itself or an unramified quadratic extension over \(k _v\), and \(D _{F/k} = q ^{2n}\). Here \(D _{F/k}\) denotes the norm in \(k\) of the discriminant of \(F/k\). Let \(S _0\), \(S _1\), \(S _2\) be mutually disjoint finite sets of finite places of \(k\). The main results of the paper are the following density theorems: \[ \lim _{n \rightarrow \infty} {{1} \over {| {\mathcal Q} _\alpha ^\ast (n)| }} \sum _{F \in {\mathcal Q} _\alpha ^\ast (n)} h _{F, 3} = 1 + {{1} \over 3 ^{r(\alpha)}} \tag{I} \] and \[ \liminf _{n \rightarrow \infty} {{| \left\{F \in {\mathcal Q} _\alpha ^\ast (n) \mid 3 \nmid h _{F/k}\right\}| } \over {| {\mathcal Q} _\alpha ^\ast (n)| }} \geq \left(1 - {{1}\over{2 \cdot 3 ^{r(\alpha)}}}\right) \left(\prod _{v \in S _0} {{1} \over {q _v + 1}} \right) \left(\prod _{v \in S _1 \cup S _2} {{q _v} \over {2 (q _v + 1)}} \right), \tag{II} \] where \(r(\alpha)\) denotes the number of places \(v \in M _\infty\) with \(\alpha(v) = k _v\) and \(h _{F,3}\) denotes the order of the maximal elementary abelian 3-group in the kernel of the norm map from the class group of \(F\) to that of \(k\). As consequences of the above results the author shows that there exist infinitely many quadratic extens…
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1215552
zbMATH Keywords: class numbers
zbMATH Keywords: quadratic extensions
zbMATH Keywords: congruence function fields
zbMATH Keywords: density theorems
zbMATH Keywords: Iwasawa invariants
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s002290050087
OpenAlex ID: W1999769169
DOI: 10.1007/S002290050087
description: scientific article; zbMATH DE number 1215552