Algebra & Number Theory

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

Melanie Wood

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Local Field Mass Formula Counting Field Extension


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.

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  • M. Bhargava, “The density of discriminants of quartic rings and fields”, Ann. of Math. $(2)$ 162:2 (2005), 1031–1063.
  • M. Bhargava, “Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants”, Int. Math. Res. Not. 2007:17 (2007), rnm052.
  • M. Bhargava, “The density of discriminants of quintic rings and fields”, Ann. of Math.. To appear.
  • H. Cohen, F. Diaz y Diaz, and M. Olivier, “Enumerating quartic dihedral extensions of $\mathbb Q$”, Compositio Math. 133:1 (2002), 65–93.
  • H. Cohen, F. Diaz y Diaz, and M. Olivier, “Counting discriminants of number fields”, J. Théor. Nombres Bordeaux 18:3 (2006), 573–593.
  • J. H. Conway, “personal communication”, 2006.
  • H. Davenport and H. Heilbronn, “On the density of discriminants of cubic fields. II”, Proc. Roy. Soc. London Ser. A 322:1551 (1971), 405–420.
  • J. S. Ellenberg and A. Venkatesh, “Counting extensions of function fields with bounded discriminant and specified Galois group”, pp. 151–168 in Geometric methods in algebra and number theory (Miami, 2003), edited by F. Bogomolov and Y. Tschinkel, Progr. Math. 235, Birkhäuser, Boston, 2005.
  • J. W. Jones and D. P. Roberts, “A database of local fields”, J. Symbolic Comput. 41:1 (2006), 80–97.
  • K. S. Kedlaya, “Mass formulas for local Galois representations”, Int. Math. Res. Not. 2007:17 (2007), rnm021.
  • S. G. Kolesnikov, “On the rationality and strong reality of Sylow 2-subgroups of Weyl and alternating groups”, Algebra Logika 44:1 (2005), 44–53, 127.
  • V. D. Mazurov and E. I. Khukhro (editors), The Kourovka notebook. Unsolved problems in group theory, 14th augmented ed., Russian Academy of Sciences Siberian Division Institute of Mathematics, Novosibirsk, 1999.
  • W. Narkiewicz, Number theory, World Scientific Publishing Co., Singapore, 1983. Translated from the Polish by S. Kanemitsu.
  • PARI/GP, 2.3.2, 2006,
  • G. Pfeiffer, “Character tables of Weyl groups in GAP”, Bayreuth. Math. Schr. 47 (1994), 165–222.
  • D. O. Revin, “The characters of groups of type $X\wr \mathbb Z\sb p$”, Sib. Èlektron. Mat. Izv. 1 (2004), 110–116.
  • J.-P. Serre, “Une “formule de masse” pour les extensions totalement ramifiées de degré donné d'un corps local”, C. R. Acad. Sci. Paris Sér. A-B 286:22 (1978), A1031–A1036.
  • M. M. Wood, “On the probabilities of local behaviors in abelian field extensions”, preprint, 2008.