Binary quartic forms with bounded invariants and small Galois groups

Cindy (Sin Yi) Tsang and Stanley Yao Xiao

Vol. 302 (2019), No. 1, 249–291

DOI: 10.2140/pjm.2019.302.249

Abstract

We consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f\left(x,y\right)$ of nonzero discriminant. Then, we shall enumerate the ${GL}_2\left(\mathbb{Z}\right)$-equivalence classes of all such forms associated to a fixed $f\left(x,y\right)$.

Keywords

binary quartic forms, coregular spaces, arithmetic statistics

Mathematical Subject Classification 2010

Primary: 11E76, 11R45

Milestones

Received: 27 February 2018

Revised: 21 October 2018

Accepted: 26 January 2019

Published: 5 November 2019