Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1325

The Norm Index Theorem (An Analytic Proof)

Rainer Weissauer
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give an analytic proof of the norm index theorem $[I_:K^* N(I_L)] =[L:K]$ for cyclic extensions of number fields using spectral theory of the idele class group.

Explore related subjects

math.NTmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rainer Weissauer. 2007-07-09. The Norm Index Theorem (An Analytic Proof). https://arxiv.org/abs/0707.1325

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An explicit type number formula for quaternion orders of level $(N_1,N_2)$

We give an explicit type number formula for the quaternion orders of level $(N_1,N_2)$ specified in this paper, where $N_1=p_1^{2u_1+1}\cdots p_w^{2u_w+1}$, the primes $p_i$ are distinct, $u_i\ge0$, $w$ is odd, and $\gcd(N_1,N_2)=1$. The formula includes Eichler orders and the orders with odd prime-power ramified level considered by Boyd, and allows nonmaximal local orders at several ramified primes, including $2$. We express the answer in terms of a generalized modified Hurwitz class number and explicit local correction factors. The proof combines the correspondence between quaternion orders and ternary quadratic forms with the Siegel--Weil formula and local representation densities. We tabulate class and type numbers for $N_1N_2\le100$. As an application, a mass bound and a finite exact calculation determine the $27$ pairs in this family with type number one.

math.NT↗
arXiv2026

Nombres de Pisot, nombres de Salem et la conjecture de Lehmer

We investigate the relationship between the set S of Pisot numbers and the set T of Salem numbers. Salem first established that: " every Pisot number is an accumulation point of the set T ". Building on Boyd's method, we show that every accumulation point of T belongs to S. Together, these results imply that the union S U T forms a closed subset of the real half-line ]1,+infinity[. Consequently, this settles Boyd's conjecture while disproving Lehmer's conjecture.

math.NT↗
arXiv2026

Multiplicative Subgroups of $\mathbb{Z}_p^*$ that are Generalized Arithmetic Progressions

We prove that a multiplicative subgroup $A_k$ of $\mathbb{Z}_p^*$ is a generalized arithmetic progression if and only if $|A_k| = 2,\ 4,$ or $p-1$. Much of the argument builds upon recent work studying additive decompositions of subgroups, and we generalize a result of Hanson and Petridis to show that any additive $n$-decomposition of a subgroup must be a direct sum. We also show how this classification quickly follows from Kalmynin's recent work resolving Sárközy's conjecture for quadratic residues.

math.NT↗