Mass formulas for local Galois representations

2005/11/05 by Kiran S. Kedlaya, Kedlaya, Kiran S., Daniel Gulotta +1 · 1 citation
Mathematics · #11S20 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11S20

paper · pdf · doi:10.48550/arxiv.math/0511135

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

arxiv created 2007/02/14 · arxiv updated 2009/12/01

Abstract

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 G₂.

Cited by

• Malle's Conjecture for Galois octic fields over \mathbb Q

Related

• Counting wildly ramified quartic extensions with prescribed discriminant and Galois closure group
• Motivic versions of mass formulas by Krasner, Serre and Bhargava
• Moduli stacks of two-dimensional Galois representations
• Distinguishing pure representations by normalized traces
• Purity for families of Galois representations