On the number of cubic orders of bounded discriminant having automorphism group $C_3$, and related problems

Abstract

For a binary quadratic form $Q$, we consider the action of ${\mathrm{SO}}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed "lattice shape". Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant $D$, the shape is equidistributed in the class group ${\mathrm{Cl}}_D$ of binary quadratic forms of discriminant $D$. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field.

33 pages

References in corpus (2)

• Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves
• Counting Cubic Extensions with given Quadratic Resolvent

Cited by in corpus (8)

• The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields
• The trace form over cyclic number fields
• On $D_{\ell }$-extensions of odd prime degree $\ell$
• The shape of $\mathbb{Z}/\ell \mathbb{Z}$-number fields
• Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent
• Binary quartic forms with vanishing $J$-invariant
• Binary quartic forms with bounded invariants and small Galois groups
• Composed varieties and reflection theorems of Ohno-Nakagawa type