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

  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},
  • M. Bhargava
  • Published 2007
  • Mathematics
  • International Mathematics Research Notices

Configuration spaces, graded spaces, and polysymmetric functions

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


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

Probabilistic Galois Theory over $P$-adic Fields

The asymptotic growth of torsion homology for arithmetic groups

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

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

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

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


Abstract We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ , almost all (in a density sense) realizations of $G$ over

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

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



On the density of discriminants of cubic fields. II

  • H. DavenportH. Heilbronn
  • Mathematics
    Proceedings 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.

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

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


. 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

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

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