Central extensions and Hasse norm principle over functions fields (Q5946894)

Let \(K/k\) be a finite extension of global fields. The extension \(K/k\) satisfies the Hasse norm principle (HNP) if an element of \(k\) is a norm from \(K\) iff it is a local norm for every prime divisor of \(K\). The aim of this paper is to study the HNP and the \(\ell\)-rank of the class group for cyclotomic function fields. Let \(k={\mathbb F}_q(T)\) be the rational function field over the finite field of \(q\) elements, and let \(M \in {\mathbb F}_q[T]\) be a monic polynomial. Let \(k(\Lambda _M)\) and \(k(\Lambda _M)^+\) be the \(M\)-th cyclotomic function field and its maximal real subfield respectively. Let \(h\) be the number of distinct irreducible factors of \(M\). The validity of HNP for \(k(\Lambda _M)\) and \(k(\Lambda _M)^+\) depends on \(h\). If \(h=1\) NHP holds for \(k(\Lambda _M)\) and \(k(\Lambda _M)^+\). If \(h\geq 4\) HNP fails for \(k(\Lambda _M)\). For \(h=2\) and \(3\) HNP holds for \(k(\Lambda _M)\) iff some congruence conditions hold. Similar results are proved for \(k(\Lambda _M) ^+\). In the last section, using results of \textit{G. Cornell} and \textit{M. Rosen} [Compos. Math. 53, 133--141 (1984; Zbl 0551.12006)], the authors find lower bounds for the \(\ell\)-rank of the ideal class groups of \(k(\Lambda _M)\), \(k(\Lambda _M)^+\), \(K\) and \(K^+ = k \cap k(\Lambda _M)^+\) where \(K\) is the maximal abelian \(\ell\)-extension of \(k\) with conductor \(M\).