Algebra & Number Theory

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

Manjul Bhargava and Ariel Shnidman

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Abstract

For a binary quadratic form Q, we consider the action of SOQ on a 2-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 ClD 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.

Article information

Source
Algebra Number Theory, Volume 8, Number 1 (2014), 53-88.

Dates
Received: 18 June 2012
Revised: 6 August 2013
Accepted: 19 November 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730135

Digital Object Identifier
doi:10.2140/ant.2014.8.53

Mathematical Reviews number (MathSciNet)
MR3207579

Zentralblatt MATH identifier
1298.11102

Subjects
Primary: 11R16: Cubic and quartic extensions
Secondary: 11R45: Density theorems

Keywords
cubic fields discriminant

Citation

Bhargava, Manjul; Shnidman, Ariel. On the number of cubic orders of bounded discriminant having automorphism group $C_3$, and related problems. Algebra Number Theory 8 (2014), no. 1, 53--88. doi:10.2140/ant.2014.8.53. https://projecteuclid.org/euclid.ant/1513730135


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