Sharp exponents for a tame frame #
Let L/K be an extension of fields with finite automorphism group G = Aut_K(L), and suppose
that L has finitely many p-power roots of unity, forming the cyclic group μ_{p^∞}(L) of
order q(L) = p^k. Let σ, τ generate G, with τ of order prime to p (a tame frame, as
supplied by a Frobenius lift and a generator of tame inertia on a tamely ramified layer).
This file proves the existence of sharp exponents (the exact sequence (∗) in the proof of
NSW (7.4.1)): natural numbers a, b through which σ, τ act on μ_{p^∞}(L), such that the
left ideal J = ℤ_p[G]·(σ - a) + ℤ_p[G]·(τ - b) is exactly the annihilator of μ_{p^∞}(L). In
other words the left ℤ_p[G]-module ℤ_p[G] ⧸ J has order q(L), and it is isomorphic to the
p-power torsion of the p-adic completion A(L) of Lˣ, which is μ_{p^∞}(L) with its Galois
action.
The exponents are sharp in that b is chosen so that b ^ orderOf τ - 1 has p-adic valuation
exactly that of q(L); this bounds ℤ_p[G] ⧸ J, a cyclic ℤ_p-module, by q(L). Not every lift
works: for a = b = 1 the quotient is ℤ_p[G] ⧸ I_G ≅ ℤ_p, which is infinite.
Main statements #
TauCeti.nonempty_quotient_tameFrameExponents_linearEquiv: supplied sharp exponents identify their quotient with the torsion ofA(L).TauCeti.exists_tameFrame_exponents: the existence of sharp exponents for a tame frame.TauCeti.finite_and_natCard_le_of_pow_orderOf_sub_one:ℤ_p[G] ⧸ Jhas at mostp ^ kelements whenb ^ orderOf τ - 1isp ^ ktimes ap-adic unit.TauCeti.not_exists_tameFrame_exponents_one_one: whenq(L) = 2and the automorphism group is nontrivial, the identity pair cannot have sharp exponents.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, proof of (7.4.1).
If σ, τ generate the monoid G and b ^ orderOf τ - 1 = p ^ k * c with c a unit of
ℤ_p, then ℤ_p[G] ⧸ ℤ_p[G]·(σ - a, τ - b) is finite with at most p ^ k elements: it is a
cyclic ℤ_p-module killed by p ^ k.
Supplied sharp tame-frame exponents identify their quotient with the p-power torsion of the
p-adic completion A(L).
Sharp exponents for a tame frame (the exact sequence (∗) in the proof of NSW (7.4.1)).
Let σ, τ generate the finite automorphism group G of L/K, with τ of order prime to p, and
suppose L has finitely many p-power roots of unity. Then there are natural numbers a, b
through which σ, τ act on μ_{p^∞}(L) such that the left ideal J = (σ - a, τ - b) of
ℤ_p[G] is the annihilator of μ_{p^∞}(L): the quotient ℤ_p[G] ⧸ J has order
q(L) = #μ_{p^∞}(L), and it is isomorphic as a left ℤ_p[G]-module to the p-power torsion of
the p-adic completion A(L) of Lˣ, which is μ_{p^∞}(L) with its Galois action.
Generation is needed: when σ, τ generate a proper subgroup H of G and q(L) > 1, the
quotient has order at least q(L) ^ [G : H].
The generating hypothesis is necessary #
The identity pair does not have sharp tame-frame exponents. Suppose the finite group of
2-power roots of unity of L has order two and Aut_K(L) is finite and nontrivial. There are
no exponents through which the identity pair acts on those roots for which the corresponding
two-generator quotient of ℤ_2[Aut_K(L)] also has order two.
Indeed, the action equations on -1 force both exponents to be odd. Reduction modulo two then
makes both relations vanish, so the quotient surjects onto 𝔽₂[Aut_K(L)], which has at least
four elements. This shows that the generating hypothesis of exists_tameFrame_exponents cannot
be omitted.
The identity pair on the unramified quadratic extension ℚ₂(ζ₃)/ℚ₂ does not admit sharp
tame-frame exponents. This is the concrete rejection test showing that the generating hypothesis
of exists_tameFrame_exponents cannot be removed.