Mass formulas for local Galois representations ( after Serre , Bhargava )

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 link with integration on p-adic groups. We also check the G2 case, where the analogy seems to break down at residual characteristic 2.