Vol. 8, No. 1, 2014

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

Manjul Bhargava and Ariel Shnidman

Vol. 8 (2014), No. 1, 53–88

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.

cubic fields, discriminant
Mathematical Subject Classification 2010
Primary: 11R16
Secondary: 11R45
Received: 18 June 2012
Revised: 6 August 2013
Accepted: 19 November 2013
Published: 20 April 2014
Manjul Bhargava
Department of Mathematics
Princeton University
Princeton, NJ 08544
United States
Ariel Shnidman
Department of Mathematics
University of Michigan
530 Church St.
Ann Arbor, 48109
United States