  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.

Number Theory