Vol. 302, No. 1, 2019

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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
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(x,y) of nonzero discriminant. Then, we shall enumerate the GL2()-equivalence classes of all such forms associated to a fixed f(x,y).

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
Authors
Cindy (Sin Yi) Tsang
Yau Mathematical Sciences Center
Tsinghua University
Beijing
China
School of Mathematics (Zhuhai)
Sun Yat-Sen University
Tangjiawan, Zhuhai
Guangdong
China
Stanley Yao Xiao
Mathematical Institute
University of Oxford
United Kingdom
Department of Mathematics
University of Toronto
Canada