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 D₄ has a mass formula for certain weights, but we show that D₄ does not have a mass formula when the local Galois representations to D₄ are weighted in the same way as representations to S₄ are weighted in Bhargava’s mass formula.

Keywords

Local Field, Mass Formula, Counting Field Extension

Mathematical Subject Classification

Primary: 11S15

Secondary: 11R45

Authors