Some remarks on the local class field theory of Serre and Hazewinkel

Abstract

We give a refinement of the local class field theory of Serre and Hazewinkel. This refinement allows the theory to treat extensions that are not necessarily totally ramified. Such a refinement was obtained and used in the authors' paper on Fontaine's property (P m ), where the explanation had to be rather brief. In this paper, we give a complete account, from necessary knowledge of an appropriate Grothendieck site to the details of the proof. We start by reviewing the local class field theory of Serre and Hazewinkel.

On the ramified class field theory of relative curves

“…More precisely, Serre considers étale isogenies over L × and the link with Tors ⊗ (L × , G) is provided by 2.4. In [Su13], Suzuki shows that the functor from Theorem 3.8 is a quasi-inverse to Serre's functor when k is algebraically closed, and extends the result to arbitrary perfect residue fields. In particular, the equivalence from Theorem 3.8 is canonical, even though its definition depends on the choice of π. Suzuki's proof of Theorem 3.8 relies on the Albanese property of the morphism ϕ, previously established by Contou-Carrère.…”

“…We denote by ϕ : Spec(L) → L × the corresponding morphism. We follow here Contou-Carrère's convention; in [Su13], the morphism ϕ corresponds to the point (Π − π)Π −1 instead. This is harmless since the inversion is an automorphism of the abelian group L × .…”

Geometric local epsilon factors

“…Since this topic is of independent interest, we choose to present more material than what is strictly necessary in order to prove the main results of this text. We discuss in particular the relations between different formulations of geometric local class field theory, namely those of Serre [Se61], of Contou-Carrère [CC13] and Suzuki [Su13], or of Gaitsgory. We also prove local-global compatibility in geometric class field theory (cf.…”

From the Function-Sheaf Dictionary to Quasicharacters of -Adic Tori

“…It is natural to ask if the vertical surjections can be defined directly, without making use of local class field theory, for which the results of [32,33] may be helpful. The case when T K = G m,K is already very interesting, in which case (14) becomes…”

“…We anticipate that future work on quasicharacter sheaves will make use of [32,33], and will clarify the relation between this project and other attempts to geometrize admissible distributions on p-adic groups, such as [13] (limited to quasicharacters of Z × p ) and [1] (limited to characters of depth-zero representations). We are actively pursuing the question of how to extend the notion of quasicharacter sheaves to provide a geometrization of admissible distributions on connected reductive algebraic groups over p-adic fields, not just commutative ones.…”