Class numbers of real quadratic function fields of genus one
type: Item
instance of: scholarly article
title: Class numbers of real quadratic function fields of genus one
zbMATH Open document ID: 0919.11070
author: Humio Ichimura
published in: Finite Fields and their Applications
publication date: 1998-10-26T00:00:00Z
review text: The author proves a strengthened analog of an infamous conjecture of Gauss. He formulates the family of questions \(P(g,N)\): Do there exist infinitely many pairs \((p, K/{\mathbb F}_{p}(x))\) of a prime \(p\) and a real quadratic function field extension \(K/{\mathbb F}_{p}(x)\) for which the genus of the field \(K\) is \(g\) and the ideal class number of \({\mathcal O}_{K,x}\), the integral closure of \({\mathbb F}_{p}[x]\) in \(K\), is \(N\)? Previous affirmative results are of \textit{M. L. Madan} [J. Number Theory 2, 279-281 (1970; Zbl 0197.32503)] for \(P(1,1)\) and of the reviewer [ \textit{Th. Schmidt}, J. Number Theory 54, 203-205 (1995; Zbl 0839.11053)] for \(P(2,1)\). The author proves that \(P(1,N)\) has an affirmative answer for all \(N\). The proof is constructive: based upon a number theoretic lemma, one can show that there exist infinitely many \((p,b)\) with elliptic curves over \({\mathbb F}_{p}\) of order \(bN\) having a point of order \(b\). For such a curve, of function field say \(k\), and point \(P\), the divisor class \([P- {\mathcal O}]\) has order \(b\). Let \(y \in k\) have divisor \(A - (P+{\mathcal O})\), with positive divisor \(A\), and \(P, {\mathcal O} \notin A\). Then, \(K = k(y)\) is a quadratic extension over \({\mathbb F}_{p}(y)\); one checks that the class number here is \(N\).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 1199977
zbMATH Keywords: class numbers
zbMATH Keywords: elliptic curves
zbMATH Keywords: real quadratic function fields
reviewed by: Thomas A. Schmidt
MaRDI profile type: Publication
full work available at URL: ffta.1997.0182
OpenAlex ID: W2068783063
cites work: Q4336972
cites work: Hecke polynomials as congruence \(\zeta\) functions in elliptic modular case
cites work: On unique factorization in certain rings of algebraic functions
cites work: Note on a problem of S. Chowla
cites work: Algebraic function fields of class number one
cites work: Algebraic function fields with small class number
cites work: Quadratic function fields with invariant class group
cites work: On the Difference between Consecutive Squarefree Numbers
cites work: \(S\)-units and \(S\)-class group in algebraic function fields
cites work: Analytische Zahlentheorie in Körpern der Charakteristik \(p\)
cites work: Infinitely many real quadratic fields of class number one
cites work: Class numbers of elliptic function fields and the distribution of prime numbers
DOI: 10.1006/FFTA.1997.0182
description: scientific article; zbMATH DE number 1199977