Mass formulas for local Galois representations to wreath products and cross products

Wood, Melanie

doi:10.2140/ant.2008.2.391

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Mass formulas for local Galois representations to wreath products and cross products

Melanie Matchett Wood

Vol. 2 (2008), No. 4, 391–405

DOI: 10.2140/ant.2008.2.391

Abstract

Bhargava proved a formula for counting, with certain weights, degree $n$ étale 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 étale 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.

Keywords

local field, mass formula, counting field extensions

Mathematical Subject Classification 2000

Primary: 11S15

Secondary: 11R45

Milestones

Received: 28 November 2007

Revised: 31 March 2008

Accepted: 28 April 2008

Published: 15 June 2008

Authors

Melanie Matchett Wood
Princeton University Department of Mathematics Fine Hall, Washington Road Princeton, NJ 08544 United States

Vol. 2, No. 4, 2008