A note on tamely ramified towers of global function fields
If \(N_q(g)\) denotes the maximum number of \(\mathbb F_q\)-rational points that a smooth irreducible curve with genus \(g\) can have, then \(A(q)=\lim\sup N_q (g)/g\) is an important asymptotic measure of \(\mathbb F_q\)-rational points. The Drinfel'd-Vlădut bound \(A(q)\leq q^{\frac{1}{2}}-1\) is attained when \(q\) is a square but the value of \(A(q)\) is not known for non-square \(q\). The authors extend previous work by \textit{R. Schoof} [J. Number Theory 41, 6-14 (1992; Zbl 0762.11026)] and \textit{H. Niederreiter} and \textit{C. Xing} [Math. Nachr. 195, 171-186 (1998; Zbl 0920.11039)] to construct \(T\)-tamely ramified and \(S\)-decomposed Hilbert class field towers \(F_{\infty}(S,T)/F\) (where \(S\) and \(T\) are disjoint finite sets of places of \(F\) and \(S\neq\emptyset\)) and obtain a lower bound for \(A(q)\) when the tower is infinite. The key result is Theorem 7, which gives a sufficient condition on the \(l\)-rank of the \((S,T)\)-divisor class group for the tower to be infinite. Simple examples are then constructed which give \(A(3)\geq\frac{8}{17}\) and \(A(5)\geq\frac{8}{11}\), which improve the lower bounds obtained by Niederreiter and Xing using unramified Hilbert class field towers with decomposition. Reviewer's remark: The paper contains a few misprints, for example: on p. 208, line 3, change ``lower'' to ``upper''; on p. 213 third line from bottom, change \(\geq\) to \(\leq\).
scientific article; zbMATH DE number 1781828