## Algebra & Number Theory

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

Melanie Wood

#### Abstract

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.

#### Article information

Source
Algebra Number Theory, Volume 2, Number 4 (2008), 391-405.

Dates
Received: 28 November 2007
Revised: 31 March 2008
Accepted: 28 April 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513797267

Digital Object Identifier
doi:10.2140/ant.2008.2.391

Mathematical Reviews number (MathSciNet)
MR2411405

Zentralblatt MATH identifier
1176.11063

Subjects
Primary: 11S15: Ramification and extension theory
Secondary: 11R45: Density theorems

#### Citation

Wood, Melanie. Mass formulas for local Galois representations to wreath products and cross products. Algebra Number Theory 2 (2008), no. 4, 391--405. doi:10.2140/ant.2008.2.391. https://projecteuclid.org/euclid.ant/1513797267

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