# The shapes of pure cubic fields

@article{Harron2015TheSO, title={The shapes of pure cubic fields}, author={Robert Harron}, journal={arXiv: Number Theory}, year={2015} }

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 rectangular and that they are equidistributed, in a regularized sense, when ordered by discriminant, in the one-dimensional space of all rectangular lattices. We do the same for Type II fields, which are however no longer rectangular. We obtain as a corollary of…

## 11 Citations

The shapes of Galois quartic fields

- Mathematics
- 2019

We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the…

Equidistribution of shapes of complex cubic fields of fixed quadratic resolvent

- Mathematics
- 2019

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…

SHAPES OF MULTIQUADRATIC EXTENSIONS A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI‘I AT MĀNOA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

- Mathematics
- 2019

For a positive integer n, we compute the shape of a totally real multiquadratic extension of degree 2 in which the prime 2 does not ramify. From this calculation, we see that the shape of such a…

Characterization of number fields by their integral trace form

- Mathematics
- 2020

We prove that the integral trace form is a complete invariant for totally real number fields of fundamental discriminant. We also study the relations of this invariant with the trace-zero form and…

A proof of a conjecture on trace-zero forms and shapes of number fields

- MathematicsResearch in Number Theory
- 2020

In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based…

Heights on stacks and a generalized Batyrev-Manin-Malle conjecture

- Mathematics
- 2021

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational…

An introduction to Casimir pairings and some arithmetic applications

- Mathematics
- 2018

Inspired by the Casimir invariant of a semisimple Lie algebra we introduce the notion of Casimir pairing; the classical Casimir invariant is just a value taken by the pairing in a very specific case.…

Maass form twisted Shintani $\mathscr{L}$-functions

- Mathematics
- 2016

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…

On a question of Mordell

- MathematicsProceedings of the National Academy of Sciences
- 2021

This paper concludes a 65-y search with an affirmative answer to Mordell’s question and strongly supports a related conjecture of Heath-Brown and makes several improvements to methods for finding integer solutions to x3+y3+z3=k for small values of k.

The Shape of cyclic number fields.

- Mathematics
- 2019

Let $K$ be a cyclic number field without wild ramification. We show that the shape of $K$ is completely determined by the discriminant and the degree of $K$. Furthermore, given integers $m>1$ and…

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