Class numbers of real quadratic function fields of genus one
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 \(N\).
scientific article; zbMATH DE number 1199977