# Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants

@article{Bhargava2007MassFF, title={Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants}, author={Manjul Bhargava}, journal={International Mathematics Research Notices}, year={2007}, volume={2007} }

## 78 Citations

Configuration spaces, graded spaces, and polysymmetric functions

- Mathematics
- 2022

This paper presents techniques for computing motivic measures of configuration spaces when points of the base space are weighted. This generality allows for a more flexible notion of configuration…

On unit signatures and narrow class groups of odd abelian number fields: Galois structure and heuristics

- Mathematics
- 2021

This paper is an extension of the work of Dummit and Voight on modeling the 2-Selmer group of number fields. We extend their model to Sn-number fields of even degree and develop heuristics on the…

SATO–TATE EQUIDISTRIBUTION OF CERTAIN FAMILIES OF ARTIN $L$ -FUNCTIONS

- MathematicsForum of Mathematics, Sigma
- 2019

We study various families of Artin $L$ -functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of…

The asymptotic growth of torsion homology for arithmetic groups

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2012

Abstract When does the amount of torsion in the homology of an arithmetic group grow exponentially with the covolume? We give many examples where this is the case, and conjecture precise conditions.

On the asymptotics of cubic fields ordered by general invariants

- Mathematics
- 2022

In this article, we introduce a class of invariants of cubic ﬁelds termed “generalized discriminants”. We then obtain asymptotics for the families of cubic ﬁelds ordered by these invariants. In…

Low degree Hurwitz stacks in the Grothendieck ring

- Mathematics
- 2022

For $2 \leq d \leq 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of $\mathbb P^1$ stabilizes in the Grothendieck ring of stacks as $g \to \infty$. We will…

Heuristics for the asymptotics of the number of $S_n$-number fields

- Mathematics
- 2020

We give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of $S_n$-number fields having bounded discriminant. We then make our arguments rigorous in the case…

The number of quartic $D_4$-fields ordered by conductor

- Mathematics
- 2017

We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate…

Low-lying zeros of certain families of Artin L-functions

- Mathematics
- 2015

We study various families of Artin $L$-functions attached to geometric parametrizations of number fields. In each case we find the Sato-Tate measure of the family and determine the symmetry type of…

## References

SHOWING 1-9 OF 9 REFERENCES

On the density of discriminants of cubic fields. II

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1971

An asymptotic formula is proved for the number of cubic fields of discriminant δ in 0 < δ < X; and in - X < δ < 0.

Étude heuristique des groupes de classes des corps de nombres.

- Mathematics
- 1990

Le probleme de la repartition des nombres de classes et des groupes de classes d'ideaux de corps de nombres se pose depuis Gauss. Bien que ait fait de remarquables decouvertes dans ce domaine (par…

Enumerating Quartic Dihedral Extensions of ℚ

- MathematicsCompositio Mathematica
- 2002

We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of ℚ. From this series we deduce an asymptotic formula for the number of…

THE DENSITY OF DISCRIMINANTS OF S3-SEXTIC NUMBER FIELDS

- Mathematics
- 2007

. We prove an asymptotic formula for the number of sextic number fields with Galois group S 3 and absolute discriminant < X. In addition, we give an interpretation of the constant in the formula in…

The density of discriminants of quartic rings and fields

- Mathematics
- 2005

We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a…

Mass formulas for local Galois representations to wreath products and cross products

- Mathematics
- 2008

Bhargava proved a formula for counting, with certain weights, degree n etale extensions of a local field, or equivalently, local Galois representations to S_n. This formula is motivation for his…