Mass Formulas for Local Galois Representations (with an Appendix by Daniel Gulotta)

Bhargava has given a formula, derived from a formula of Serre, computing a certain count of extensions of a local field, weighted by conductor and by number of automorphisms. We interpret this result as a counting formula for permutation representations of the absolute Galois group of the local field, then speculate on variants of this formula in which the role of the symmetric group is played by other groups. We prove an analogue of Bhargava's formula for representations into a Weyl group in the Bn series, which suggests a possible link with integration on p-adic groups. We also obtain analogous positive results in odd residual characteristic, and negative results in residual characteristic 2, for the Dn series (in the appendix) and the exceptional group G2.