[Paper Review] Tame ramification and group cohomology

Chandan Singh Dalawat, Jung-Jo Lee|arXiv (Cornell University)|May 12, 2013

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper provides an intrinsic parametrization of tamely ramified Galois extensions of a local field with finite residue field using group cohomology, showing that two natural definitions of the cohomology class of such extensions coincide. It gives an elementary proof of Serre’s mass formula in the tame case and classifies all Galois extensions of degree $ l^3 $ for prime $ l \neq p $, including dihedral and quaternionic cases.

ABSTRACT

We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class of a tamely ramified finite galoisian extension coincide, and can be recovered from the parameter. We also give an elementary proof of Serre's mass formula in the tame case and in the simplest wild case, and we classify tame galoisian extensions of degree the cube of a prime.

Motivation & Objective

• To provide an intrinsic parametrization of tamely ramified extensions of a local field with finite residue field using group cohomology.
• To show that two natural definitions of the cohomology class of a tamely ramified Galois extension coincide.
• To recover key invariants—such as Galois closure, splitness of extension towers, and Galois group structure—from the cohomological parameter.
• To give an elementary proof of Serre’s mass formula in the tame case and in the case where degree is divisible by $ p $ but not $ p^2 $.
• To classify all Galois extensions of degree $ l^3 $ for prime $ l \neq p $, including dihedral and quaternionic types.

Proposed method

• Use group cohomology to parametrize the set $ \mathcal{T}_{e,f}(K) $ of tamely ramified extensions of a local field $ K $ with ramification index $ e $ and residual degree $ f $.
• Define 'ramified lines' as images of sections of the normalized valuation map $ \bar{w}_f: K_f^\times / K_f^{\times e} \to \mathbb{Z}/e\mathbb{Z} $, which classify extensions in $ \mathcal{T}_{e,f}(K) $.
• Establish a canonical bijection between $ \mathcal{T}_{e,f}(K) $ and the set of $ G_f $-orbits on the set of ramified lines, where $ G_f = \mathrm{Gal}(K_f/K) $.
• Apply the Kummer pairing and its equivariance to relate cohomological invariants to ramification data.
• Use the action of $ G_f $ on $ k_f^\times / k_f^{\times e} $ to compute the number of Galois extensions and determine their Galois groups.
• Leverage the vanishing of $ H^2(G_f, k_f^\times / k_f^{\times e}) $ in certain cases to show that extension towers split.

Experimental results

Research questions

• RQ1 How can tamely ramified extensions of a local field be intrinsically parametrized using group cohomology?
• RQ2 Do the two standard definitions of the cohomology class of a tamely ramified Galois extension coincide?
• RQ3 Can Serre’s mass formula be proven elementarily in the tame case and in the case where the degree is divisible by $ p $ but not $ p^2 $?
• RQ4 What are the possible Galois groups of extensions of degree $ l^3 $ over a local field of residual characteristic $ p \neq l $?
• RQ5 Under what conditions do non-abelian Galois extensions of degree $ l^3 $ arise, and how can they be classified?

Key findings

• The set $ \mathcal{T}_{e,f}(K) $ of tamely ramified extensions of $ K $ with ramification index $ e $ and residual degree $ f $ is in canonical bijection with the set of $ G_f $-orbits on the set of ramified lines in $ K_f^\times / K_f^{\times e} $.
• Two natural definitions of the cohomology class of a tamely ramified Galois extension coincide and can be recovered from the ramified line parameter.
• An elementary proof of Serre’s mass formula is given for the tame case and for extensions of degree divisible by $ p $ but not $ p^2 $, analogous to the proof in prime degrees.
• For $ l \neq p $, there are $ l $ Galois extensions of degree $ l^3 $ over $ K $ that are non-abelian and isomorphic to the dihedral group $ D_{l^2,l} $, provided $ v_l(q-1) = 1 $.
• The Heisenberg group $ H_{l^3} $ does not occur as the Galois group of any tamely ramified extension of degree $ l^3 $ over a local field of residual characteristic $ p \neq l $.
• When $ v_2(q-1) = m-1 $, there are $ 2^{m-1} $ Galois extensions in $ \mathcal{T}_{2^m,2}(K) $ that are non-abelian, and the corresponding extension tower splits due to the vanishing of the relevant cohomology group.