The canonical fractional Galois ideal at s=0

Finite quotients of Galois pro-$p$ groups and rigid fields

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.

The Galois structure of ambiguous ideals in cyclic extensions of degree 8

In cyclic, degree 8 extensions of algebraic number fields $N/K$, ambiguous ideals in N are canonical $\mathbb{Z}[C_8]$-modules. Their $\mathbb{Z}[C_8]$-structure is determined here. It is described in terms of indecomposable modules and determined by ramification invariants. Although infinitely many indecomposable $\mathbb{Z}[C_8]$-modules are available (classification by Yakovlev), only 23 appear.

关于分式理想的标准根

On canonical roots of fractional ideals

研究如何在多项式时间内计算序\(R\)的分式理想的根，因不假定\(R\)是戴德金的，输出不唯一，故要求算法具函子性，通过推广相关结果来满足这两个约束。

奇数$n$下$A_{n}$、$D_{n}$、$E_{n}$型的伽罗瓦迹形式

Galois trace forms of type $A_{n}, D_{n}, E_{n}$ for odd $n$

该研究证明奇数次伽罗瓦扩张不存在对应$A_n$、$D_n$、$E_n$型迹形式的分式理想，同时发现循环三次域含无穷多$A_3$型子格，与二次域的相关结论形成对比。

The Chevalley-Weil Formula for Orbifold Curves

In the 1930s Chevalley and Weil gave a formula for decomposing the canonical representation on the space of differential forms of the Galois group of a ramified Galois cover of Riemann surfaces. In this article we prove an analogous Chevalley-Weil formula for ramified Galois covers of orbifold curves. We then specialize the formula to the case when the base orbifold curve is the (reduced) modular orbifold. As an application of this latter formula we decompose the canonical representations of modular curves of full, prime level and of Fermat curves of arbitrary exponent.

Scaffolds and Generalized Integral Galois Module Structure

Let $L/K$ be a finite, totally ramified $p$-extension of complete local fields with residue fields of characteristic $p > 0$, and let $A$ be a $K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where $L/K$ was Galois and $A=K[G]$ for $G=\mathrm{Gal}(L/K)$. When a suitable $A$-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, f…

Minoration de la hauteur canonique pour les modules de Drinfeld à multiplications complexes

Lower Bound for the Canonical Height for Drinfeld Modules with Complex Multiplication. Let K be a fi nite extension of Fq(T), let L=K be a Galois extension with Galois group G and let E be the sub eld of L fixed by the center of G. Assume that there exists a finite place v of K such that the local degrees of E=K above v are bounded. Let $ϕ$ be a Drinfeld module with complex multiplication. We give an e fective lower bound for the canonical height of $ϕ$ on L outside the torsion points of $ϕ$ . In the number field case, this problem was solved by F. Amoroso, S. David and U. Zannier.

Self-Dual Integral Normal Bases and Galois Module Structure

Let $N/F$ be an odd degree Galois extension of number fields with Galois group $G$ and rings of integers ${\mathfrak O}_N$ and ${\mathfrak O}_F=\bo$ respectively. Let $\mathcal{A}$ be the unique fractional ${\mathfrak O}_N$-ideal with square equal to the inverse different of $N/F$. Erez has shown that $\mathcal{A}$ is a locally free ${\mathfrak O}[G]$-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 $\mathcal{A}$ as a $\Z[G]$-module, however this question remains open. In this paper we prove that $\mathcal{A}$ i…

高余维数中的同余模与伽罗瓦上同调中的 zeta 线

Congruence modules in higher codimension and zeta lines in Galois cohomology

本文基于高余维数同余模概念，提出了检测局部环正则性的准则与形变下同余模变化的精细结果，并应用于由 Hilbert 模形式伴随动机导出的 Galois 上同调群中构造典范线。

Relationships between conjectures on the structure of pro-p Galois groups unramified outside p

We consider the canonical representation of the absolute Galois group of the rational numbers in the outer automorphism group of the pro-p completion of the fundamental group of the projective line minus 0,1, and infinity. Deligne has conjectured that a certain graded Z_p-Lie algebra arising from this representation becomes a free p-adic Lie algebra on one element in each odd degree starting with 3 when tensored with the Q_p. We construct good choices of these elements and use them to examine the structure of the Z_p-Lie algebra. In particular, we consider how its structure depends upon the re…

On infinite extensions of Dedekind domains, upper semicontinuous functions and the ideal class semigroups

In this article, we study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain O obtained by an infinite integral extension of a Dedekind domain. We introduce a notion of "upper semicontinuous functions" whose domain is the maximal spectrum of O equipped with a certain topology, and whose codomain is a certain totally ordered monoid containing the set of real numbers. We construct an isomorphism between a monoid consisting of such upper semicontinuous functions satisfying certain conditions and the monoid of fractional ideals of O. T…

Generalized Steinberg Relations

We consider a field $F$ and positive integers $n$, $m$, such that $m$ is not divisible by $\mathrm{Char}(F)$ and is prime to $n!$. The absolute Galois group $G_F$ acts on the group $\mathbb{U}_n(\mathbb{Z}/m)$ of all $(n+1)\times(n+1)$ unipotent upper-triangular matrices over $\mathbb{Z}/m$ cyclotomically. Given $0,1\neq z\in F$ and an arbitrary list $w$ of $n$ Kummer elements $(z)_F$, $(1-z)_F$ in $H^1(G_F,μ_m)$, we construct in a canonical way a quotient $\mathbb{U}_w$ of $\mathbb{U}_n(\mathbb{Z}/m)$ and a cohomology element $ρ^z$ in $H^1(G_F,\mathbb{U}_w)$ whose projection to the superdiago…