非 CM 形式的类群与局部不可分解性

关于二进制局部域上经典n-通用二次型

On classic $n$-universal quadratic forms over dyadic local fields

本文研究了二进制局部域上经典n-通用二次型的等价条件和最小测试集，确定了其数学性质和表示能力。

Igusa local zeta functions of a class of hybrid polynomials

In this paper, we study the Igusa's local zeta functions of a class of hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. Such class of hybrid polynomial was first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We prove the rationality of these local zeta functions and describe explicitly their poles. The proof is based on Igusa's stationary phase formula.

Two-Dimensional Local-Global Class Field Theory in Positive Characteristic

Using the higher tame symbol and Kawada and Satake's Witt vector method, A. N. Parshin developed class field theory for higher local fields, defining reciprocity maps separately for the tamely ramified and wildly ramified cases. We extend this technique to the case of local-global fields associated to points and curves on an algebraic surface over a finite field.

Class field theory for open curves over local fields

Theory for open curves over a local field. After introducing the reciprocity map, we determine the kernel and the cokernel of this map. In addition to this, the Pontrjagin dual of the reciprocity map is also investigated. This gives the one to one correspondence between the set of finite abelian étale coverings and the set of finite index open subgroups of the idèle class group as in the classical class field theory under some assumptions.

Invitation to higher local fields, Part I, section 5: Kato's higher local class field theory

This is a presentation of main ingredients of Kato's higher local class field theory.

Invitation to higher local fields, Part I, section 7: Parshin's higher local class field theory in characteristic p

This is a review of Parshin's higher local class field theory in characteristic p.

Invitation to higher local fields, Part I, section 10: Explicit higher local class field theory

This is a presentation of explicit methods to construct higher local class field theory by using topological K-groups, explicit symbols and a generalization of Neukirch-Hazewinkel's axiomatic approaches. The existence theorem is discussed as well.

Invitation to higher local fields, Part I, section 11: Generalized class formations and higher class field theory

Koya's and author's approach to the higher local reciprocity map as a generalization of the classical class formations approach to the level of complexes of Galois modules.

\ell-adic class field theory for regular local rings

In this paper, we prove the $\ell$-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is a $\ell$-adic variant of a result of Matsumi [13].

Invitation to higher local fields, Interlude: Existence theorem for higher local class field theory

Viewing higher local fields as ring objects in the category of iterated pro-ind-objects, a definition of open subgroups in Milnor K-groups of the fields is given. The self-duality of the additive group of a higher local field is proved. By studying norm groups of cohomological objects and using cohomological approach to higher local class field theory the existence theorem is proved.

Class Field Theory and Arithmetic of Abelian Varieties over Local Fields

We use knowledge of local fields to adapt Jonathan Lubin and Michael Rosen's proof of Mazur's Proposition 4.39. This changes the result about abelian varieties from only working over local fields with a finite residue field to working with local fields with an arbitrary perfect residue field of positive characteristic. We then briefly discuss the Local Class Field Theory implications of such information

On the cycle class map for zero-cycles over local fields

We study the Chow group of zero-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic different from l. The same statement holds for semistable K3 surfaces defined over C((t)), but does not hold in general for surfaces over strictly local fields.

Classification of irreducible representations of metaplectic covers of the general linear group over a non-archimedean local field

Let $F$ be a non-archimedean local field. The classification of the irreducible representations of $GL_n(F)$, $n\ge0$ in terms of supercuspidal representations is one of the highlights of the Bernstein--Zelevinsky theory. We give an analogous classification for metaplectic coverings of $GL_n(F)$, $n\ge0$.

Serre-Hazewinkel Local Class Field Theory and a Geometric Proof of the Local Langlands Correspondence for $\operatorname{GL}(1)$

In this expository paper we provide a geometric proof of the local Langlands Correspondence for the groups $\operatorname{GL}_{1}$ defined over $p$-adic fields $K$. We do this by redeveloping the theory of proalgebraic groups and use this to derive local class field theory in the style of Serre and Hazewinkel. In particular, we show that the local class field theory of Serre and Hazewinkel is valid for both equal characteristic and mixed characteristic ultrametric local fields. Finally, we use this to prove an equivalence of the categories of smooth representations of $K^{\ast}$ with continuou…