Kato's conductor and generic residual perfection

Conductors and the moduli of residual perfection

Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two general results: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any information about a Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space.

Brauer $p$-维数与 Kato 的 Swan 导子

Brauer $p$-dimension and Kato's Swan Conductor

本文利用 Kato 的 Swan 导子及 Gersten 型序列，研究特征 $p>0$ 的 henselian 离散赋值域与半整体域的 Brauer $p$-维数，并得到好约化曲线函数域的部分结果。

Arithmetic and Differential Swan Conductors of rank one representations with finite local monodromy

We consider a complete discrete valuation field of characteristic p, with possibly non perfect residue field. Let V be a rank one continuous representation with finite local monodromy of its absolute Galois group. We will prove that the Arithmetic Swan conductor of V (defined after Kato in [Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides with the Differential Swan conductor of the associated differential module $D^†(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaine's formalism as presented in [Tsu…

On the ramification of non-abelian Galois coverings of degree $p^3$

The refined Swan conductor is defined by K.\ Kato \cite{KK2}, and generalized by T.\ Saito \cite{wild}. In this part, we consider some smooth $l$-adic étale sheaves of rank $p$ such that we can be define the $rsw$ following T.\ Saito, on some smooth dense open subscheme $U$ of a smooth separated scheme X of finite type over a perfect fields $κ$ of characteristic $p>0$. We give an explicit expression of $rsw(\mathcal{F})$ in some situation. As a consequence, we show that it is integral.

Iwasawa Main Conjecture for Non-Ordinary Modular Forms

Let $p>2$ be a prime. Under mild assumptions, we prove the Iwasawa main conjecture of Kato, for modular forms with general weight and conductor prime to $p$.

特征 p>0 上亨塞尔离散赋值域的 Brauer p-维数

Brauer $p$-dimension of henselian discretely valued fields over characteristic $p>0$

本文利用 Kato 的 Swan 导体研究剩余特征 p>0 的亨塞尔离散赋值域的 Brauer p-维数，并将其周期-指数问题转化为与剩余域 Kähler 微分相关的阿贝尔群的符号长度问题。