The main object of the paper under review is the real positive number \(A(q)=\lim \sup _{g\to\infty }N_ q(g)/g\), where \(N_ q(g)\) is the maximal possible number of rational points on a projective smooth absolutely irreducible algebraic curve over \(\mathbb F_ q\). If \(q\) is a square, it is well known that \(A(q)=\sqrt{q}-1\); for all \(q\) one has \(A(q)\leq\sqrt{q}-1\). The authors’ goal is to improve known lower bounds for \(A(q)\). This is done in Corollaries 4.5 and 4.6, and the estimates obtained are usually better than those obtained in their earlier paper [Math. Nachr. 195, 171–186 (1998; Zbl 0920.11039)].
The authors’ method is based on the use of infinite class field towers (as in most earlier papers on this subject). Their main technical innovation consists in using towers whose ground floor is a quadratic extension of a field of higher genus (rather than of a rational function field). To be more precise, they use narrow ray class fields which are constructed explicitly via Drinfeld modules of rank 1.
For the entire collection see [Zbl 0930.00039].