Mass formulas for local Galois representations

Details Mass formulas for local Galois representations Mass formulas for local Galois representations

2005-11-05

arxiv.org/abs/math/0511135v4

Mathematics

Number Theory

23 pages; v4: refereed version; title changed, other minor edits; numbering changes in section 7, appendix by Daniel Gulotta

Scientific paper

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 B_n series, which suggests a 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 D_n series (in the appendix) and the exceptional group G_2.

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