Vol. 2, No. 4, 2008

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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

Bhargava proved a formula for counting, with certain weights, degree n étale extensions of a local field, or equivalently, local Galois representations to Sn. This formula is motivation for his conjectures about the density of discriminants of Sn-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 D4 has a mass formula for certain weights, but we show that D4 does not have a mass formula when the local Galois representations to D4 are weighted in the same way as representations to S4 are weighted in Bhargava’s mass formula.

local field, mass formula, counting field extensions
Mathematical Subject Classification 2000
Primary: 11S15
Secondary: 11R45
Received: 28 November 2007
Revised: 31 March 2008
Accepted: 28 April 2008
Published: 15 June 2008
Melanie Matchett Wood
Princeton University
Department of Mathematics
Fine Hall, Washington Road
Princeton, NJ 08544
United States