Self-dual integral normal bases and Galois module structure

Abstract

Let N/F be an odd degree Galois extension of number fields with Galois group G and rings of integers ON and OF = O respectively. Let A be the unique fractional ON -ideal with square equal to the inverse different of N/F . Erez has shown that A is a locally free O[G]-module if and only if N/F is a so called weakly ramified extension. There have been a number of results regarding the freeness of A as a Z[G]-module, however this question remains open. In this paper we prove that A is free as a Z[G]-module assuming that N/F is weakly ramified and under the hypothesis that for every prime ℘ of O which ramifies wildly in N/F , the decomposition group is abelian, the ramification group is cyclic and ℘ is unramified in F/Q.We make crucial use of a construction due to the first named author which uses Dwork's exponential power series to describe self-dual integral normal bases in Lubin-Tate extensions of local fields. This yields a new and striking relationship between the local norm-resolvent and Galois Gauss sum involved. Our results generalise work of the second named author concerning the case of base field Q.

The equivariant local $\varepsilon $-constant conjecture for unramified twists of $\mathbb Z_p(1)$

“…Then Proposition 5.2.1 of loc.cit. implies that∂ Zp[G],B dR [G] (θ) + ∂ Zp[G],B dR [G] (ε D (N/K, V )) is represented by θ χ u −mm χφ τ K (χφ) = d 1/2 K N K/Qp (θ 2 |φ)p 2m if χ = χ 0 d 1/2 K N K/Qp (θ 2 |φ)p m χ(4)φ(b −2 )u −2m if χ = χ 0 .We want to point out that this is a crucial step in the proof and relies on one of the main results of[PV13] which was basic for the proof of [BC, Prop. 5.1.5].…”

Equivariant epsilon constant conjectures for weakly ramified extensions

“…3.9]. Furthermore, τ K (χ) = p mχ (c −1 χ )ψ K (c −1 χ ), by the last displayed formula in the proof of [15,Prop. 3.9].…”

“…By Lemma 3.2.5, if v N (λ 1 θ) = 1, then v N ((a − 1) j λ 1 θ) = j + 1 (recall that j < p), and this contradicts (15). Hence v N (λ 1 θ) > 1, so that Lemma 3.2.6 implies λ 1 θ ∈ p 2 N = (T a , a − 1)θ.…”

“…If K/Q p is unramified, then s = 0 and we obtain the first equality. The second equality is the last displayed equality in the proof of [15,Prop. 3.8] with s = 0 and π K = p.…”

“…The assumptions of the theorem imply that the ramification group is cyclic of order p (cf. [15,Cor. 3.4]).…”