Quartic rings associated to binary quartic forms

Abstract

We give a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring, relating the rings associated to binary quartic forms with Bhargava's cubic resolvent rings. This gives a parametrization of quartic rings with monogenic cubic resolvents. We also give a geometric interpretation of this parametrization.

to appear in IMRN, section on geometric analogs over an arbitrary base added since version 1

References in corpus (2)

• Gauss composition over an arbitrary base
• Rings and ideals parametrized by binary n-ic forms

Cited by in corpus (5)

• Quartic Index Form Equations and Monogenizations of Quartic Orders
• Counting integral points on symmetric varieties with applications to arithmetic statistics
• The number of quartic $D_4$-fields with monogenic cubic resolvent ordered by conductor
• Binary quartic forms with bounded invariants and small Galois groups
• Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms