The prime at infinity and the rank of the class group in global function fields
Let \({\mathbb F}\) be the finite field of \(q\) elements and \(k={\mathbb F}(T)\). For any finite extension \(K\) of \(k\) let \(Cl_ K\) be the ideal class group of the integral closure \({\mathfrak O} _ K\) of \({\mathbb F}[T]\) in \(K\). The author proved in [J. Number Theory 106, No. 1, 26--49 (2004; Zbl 1100.11039)] that for any integers \(m\) and \(n\) not divisible by the characteristic there are infinitely many function fields of fixed degree \(m\) with class number divisible by \(n\). \textit{Y. Lee} and the author [Proc. Am. Math. Soc. 133, No. 10, 2883--2889 (2005; Zbl 1100.11037)] proved that for any relatively prime integers \(m\) and \(n\), not divisible by the characteristic and with some other restrictions, there exist infinitely many function fields \(K\) of degree \(m\) over \(k\) such that the infinite prime in \(k\) is inert in \(K\) and \(Cl_ K\) contains a subgroup isomorphic to \(({\mathbb Z}/n {\mathbb Z})^ {m-1}\). The objective of the paper under review is to generalize the above results. The main theorem is that if \(m\) and \(n\) are any positive integers larger than one, not both even, and not divisible by the characteristic of \(k\), and if \(h\) is an integer with \(2 \leq h \leq m-1\), then there exist infinitely many function fields \(K\) of degree \(m\) over \(k\) such that the infinite prime in \(k\) splits into exactly \(h\) prime divisors of \(K\), one with ramification index \(m-h+1\), the rest unramified, all with relative degree one and \(Cl_ K\) contains a subgroup isomorphic to \(({\mathbb Z}/ n{\mathbb Z}) ^ {m-h}\). The main tools used in this paper are the same as the ones used in the papers cited above and an application of Newton's polygon.
scientific article; zbMATH DE number 5015054
scientific article