Related papers: The shape of cubic fields

We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant. Our…

We demonstrate equidistribution of the lattice shape of cubic fields when ordered by discriminant, giving an estimate in the Eisenstein series spectrum with a lower order main term. The analysis gives a separate discussion of the…

We show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field's trace-zero form. We also prove a general such statement for all orders in \'etale Q-algebras. Applying a method of Manjul…

We determine the shapes of pure cubic fields and show that they fall into two families based on whether the field is wildly or tamely ramified (of Type I or Type II in the sense of Dedekind). We show that the shapes of Type I fields are…

For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.

We construct some families of quadratic fields whose class numbers are divisible by $3.$ The main tools used are a trinomial introduced by Kishi and a parametrization of Kishi and Miyake of a family of quadratic fields whose class numbers…

The question of the distribution of shapes of unit lattices in number fields, pioneered by Margulis and Gromov, has lately attracted considerable interest, not least because of the lack of available results. Here we prove that the set of…

We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a…

We present a method for tabulating all cubic function fields over $\mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $\mathbb{F}_{q}^*$, up to a given bound…

We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues.…

The Maass-form twisted Shintani $\mathscr{L}$-functions are introduced, and some of their analytic properties are studied. These functions contain data regarding the distribution of shapes of cubic rings.

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. The shape is a…

In this paper we prove an incidence bound for points and cubic curves over prime fields. The methods generalise those used by Mohammadi, Pham, and Warren (2021).

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…

This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of…

In previous work, the authors introduced "soft" methods to prove the effective (i.e. with power savings error) equidistribution of "shears" in cusped hyperbolic surfaces. In this paper, we study the same problem but now allow full use of…

Let k be an imaginary quadratic number field (with class number 1). We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant…

A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successfully applied to several hard special cases…

We prove the existence of secondary terms of order $X^{5/6}$ in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error…

We give an asymptotic formula for class numbers of orders in cubic number fields.