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

TL;DR

This paper extends Bhargava's mass formula for local Galois representations to symmetric groups to a broader class of groups formed via wreath and cross products, and explores their character tables and mass formulas.

Contribution

It proves new mass formulas for local Galois representations to groups built from symmetric groups, and shows these groups have rational character tables.

Findings

Groups formed from symmetric groups via wreath and cross products have mass formulas.

Certain groups like D_4 have mass formulas for specific weights.

D_4 does not have a mass formula under weights similar to S_4 in Bhargava's formula.

Abstract

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 conjectures about the density of discriminants of S_n-number fields. We prove there are analogous ``mass formulas'' that count local Galois representations to any group that can be formed from symmetric groups by wreath products and cross products, corresponding to counting towers and direct sums of etale extensions. We obtain as a corollary that the above mentioned groups have rational character tables. Our result implies that D_4 has a mass formula for certain weights, but we show that D_4 does not have a mass formula when the local Galois representations to D_4 are weighted in the same way as representations to S_4 are weighted in Bhargava's mass formula.