Asymptotically Good Towers of Global Fields

Abstract

The study of the maximal p -extension of a global field k unram­ified everywhere and totally split at a finite set of places of k has at least two important applications: it gives information on the asymptotic behav­ior of discriminants versus degree in the number field case (as measured by the Martinet constant a(t)), and on the relationship between genus and the number of places of degree one (for large genus) in the function field case (as measured by the Ihara constant A(q)). We survey recent work on class-field­theoretical constructions of towers of global fields which are optimal for the study of these phenomena, including best known examples in both settings; these contain, among others, an infinite unramified tower of totally complex number fields with small root discriminant improving Martinet’s record. We show that allowing wild ramification to limited depth leads to asymptoti­cally good towers. However, we demonstrate also that the investigation of the infinitude of these towers involves difficulties absent in the tame case.