Nicolas Potthast
Paderborn University
Title
Asymptotics of elementary-abelian extensions of function fields
Abstract
In 2004, Gunter Malle published his famous conjecture on the asymptotics of extensions of number fields with a fixed Galois group, counted by discriminant. His conjecture can be generalised directly to global function fields if the order of the Galois group is coprime to the characteristic of the field (the so-called tamely ramified case). For the wildly ramified case, in which the characteristic of the global field divides the order of the Galois group, Thorsten Lagemann showed that the naive generalisation of Malle’s conjecture (as in the tamely ramified case) does not apply. In this talk, we consider local and global function fields and determine the precise asymptotics of wildly ramified elementary-abelian extensions of these fields, counted by discriminant. We mainly focus on local and rational function fields for which a local-global principle in counting holds. To determine the asymptotics, we follow the classical approach of analysing the corresponding Dirichlet series of the counting function.