Counting $G$-extensions by discriminant

The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg–Venkatesh, Bhargava, Bhargava–Shankar–Wang, and others. In this paper, we use the geometry of numbers and invariant theory of finite groups, in a manner similar to Ellenberg and Venkatesh, to give an upper bound on the number of extensions $L/K$ with fixed degree, bounded relative discriminant, and specified Galois closure.

Keywords

discriminants, number field counting, $G$-extensions, discriminant counting, polynomial invariant theory, geometry of numbers

2010 Mathematics Subject Classification

Primary 11R21. Secondary 11H06, 11R29, 13A50.

Full Text (PDF format)

Received 29 October 2016

Accepted 3 April 2017

Published 16 November 2018