$6$-torsion and integral points on quartic threefolds

TL;DR

This paper establishes bounds on the average 6-torsion in class groups of quadratic fields and counts integer solutions on a specific quartic threefold, advancing understanding in algebraic number theory and Diophantine geometry.

Contribution

It provides new bounds for 6-torsion in class groups and counts solutions on a quartic threefold, combining algebraic and geometric methods.

Findings

Matching upper and lower bounds for 6-torsion averages

Quantitative counts of integer solutions on quartic threefolds

Enhanced understanding of class groups and Diophantine equations

Abstract

We prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.