The norm on the unit filtration, with the Herbrand shift #
Let L/K be a finite Galois extension of nonarchimedean local fields, and let
ψℕ_{L/K} : ℕ → ℕ be its integral inverse Herbrand function. This file proves that the norm
carries the unit filtration of L into that of K after the Herbrand shift:
N_{L/K}(U(L, ψℕ_{L/K}(n))) ⊆ U(K, n) and N_{L/K}(U(L, ψℕ_{L/K}(n) + 1)) ⊆ U(K, n + 1)
for every n : ℕ. The shift cannot be dropped: N_{L/K}(U(L, n)) ⊆ U(K, n) fails in general
for ramified extensions. These inclusions are the input to the comparison of the norm group of
an abelian extension with its upper ramification filtration, through which its conductor is
computed.
The proof is Serre's. The trace bound in
TauCeti.NumberTheory.LocalField.Different.Herbrand comes first: Hilbert's formula
d(L/K) = ∑_{i ≥ 0} (#G_i - 1) and the defining identity #G_1 + ⋯ + #G_m = n · #G_0 of
m = ψℕ_{L/K}(n) give e(L/K) (n + 1) ≤ ψℕ_{L/K}(n) + 1 + d(L/K), so the trace carries
𝓂[L] ^ (ψℕ_{L/K}(n) + 1) into 𝓂[K] ^ (n + 1). For a Galois extension of prime degree, the
expansion N(1 + x) = 1 + Tr(x) + Tr(y) + N(x) with y ∈ 𝓂[L] ^ (2m) reduces the inclusion to
this bound and to N(𝓂[L] ^ m) ⊆ 𝓂[K] ^ m. In general the Galois group is solvable, so it has a
normal subgroup of prime index, whose fixed field F is Galois of prime degree over K; the norm
is transitive, N_{L/K} = N_{F/K} ∘ N_{L/F}, and so is the inverse Herbrand function,
ψℕ_{L/K} = ψℕ_{L/F} ∘ ψℕ_{F/K}, so induction on the degree concludes.
Main results #
TauCeti.map_normUnits_unitFiltration_psiNat_add_one_le:N_{L/K}(U(L, ψℕ_{L/K}(n) + 1)) ⊆ U(K, n + 1).TauCeti.map_normUnits_unitFiltration_psiNat_le:N_{L/K}(U(L, ψℕ_{L/K}(n))) ⊆ U(K, n).
References #
- J.-P. Serre, Corps Locaux, Chapter V, §3 (Lemmas 4 and 5) and §6 (Proposition 8).
The norm on the unit filtration, with the Herbrand shift. For a finite Galois extension
L/K of nonarchimedean local fields, N_{L/K}(U(L, ψℕ_{L/K}(n) + 1)) ⊆ U(K, n + 1).
The norm on the unit filtration, with the Herbrand shift. For a finite Galois extension
L/K of nonarchimedean local fields, N_{L/K}(U(L, ψℕ_{L/K}(n))) ⊆ U(K, n). Without the shift
the inclusion N_{L/K}(U(L, n)) ⊆ U(K, n) fails in general for ramified extensions.