The shape of quartic fields

TL;DR

This paper proves the joint equidistribution of the shapes of quartic fields and their cubic resolvents using advanced analytic methods, improving previous estimates in the distribution of these algebraic objects.

Contribution

It introduces a refined application of Shintani's method to establish joint distribution results for quartic fields and their resolvents, with improved bounds on Weyl sums.

Findings

Joint equidistribution of quartic and cubic resolvent shapes

Enhanced bounds on Weyl sums in the context of algebraic number fields

Application of Shintani's method to new distribution problems

Abstract

We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant. Our estimate saves a small power in the corresponding Weyl sums.