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

Abstract

Bhargava proved a formula for counting, with certain weights, degree $n$ étale 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 étale 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.