Rationality problems for relation modules of dihedral groups

二面体扩张的范数一环面的合理性问题

Rationality problem for norm one tori for dihedral extensions

本文利用Endo与Miyata及Endo的定理，完整解决了Galois闭包为二面体扩张的范数一环面的合理性（稳定等价）问题，并在附录中细化了稳定有理情形的证明。

A p-adic family of dihedral (phi,Gamma)-modules

The goal of this article is to construct explicitly a p-adic family of representations (which are dihedral representations), to construct their associated (phi,Gamma)-modules by writing down explicit matrices for phi and for the action of Gamma, and finally to determine which of these are trianguline.

Hopf Galois module structure of dihedral degree $2p$ extensions of $\mathbb{Q}_p$

Let $p$ be an odd prime. For field extensions $L/\mathbb{Q}_p$ with Galois group isomorphic to the dihedral group $D_{2p}$ of order $2p$, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and the module structure of the ring of integers $\mathcal{O}_L$. We solve the case in which $L/\mathbb{Q}_p$ is not totally ramified and present a practical method which provides a complete answer for the cases $p=3$ and $p=5$. We see that within this family of dihedral extensions, the ring of integers is always free over the associated orders in the different…

The ring of integers of Hopf-Galois degree p extensions of p-adic fields with dihedral normal closure

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mat…

Universal deformation rings and dihedral defect groups

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal 2-modular block of a finite simple group. We determine the universal deformation ring R(G,V) for every kG-module V which belongs to B and has stable endomorphism ring k. It follows that R(G,V) is always isomorphic to a subquotient ring of WD. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersection…

On Brauer-Kuroda type relations of S-class numbers in dihedral extensions

Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D_{2q} and for arbitrary Galois-stable sets of primes S, containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D_{2q}-extension from class numbers and S-class numbers. The techniques we use …