On Furtwängler's theorems and second case of Fermat's Last Theorem

Abstract

This article, complement to the article [Que], deals with some generalizations of Futwängler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $\zeta$ a $p$th primitive root of unity, $K:=\Q(ζ)$ and $C{\ell }_K$ the class group of $K$. A prime $q$ is said $p$-principal if the class $c\ell_K (\mk q_K)\in C\ell_K$ of any prime ideal $\mk q_K$ of ${\mathbb{Z}}_K$ over $q$ is the $p$th power of a class. Assume that FLT2 fails for $(p,x,y,z)$ where $x,y,z$ are mutually coprime integers, $p$ divides $y$ and $x^p+y^p+z^p=0$. Let $q$ be a prime dividing $\frac{(x^p+y^p)(y^p+z^p)(z^p+x^p)}{(x+y)(y+z)(z+x)}$ and $\mk q_K$ be any prime ideal of $K$ over $q$. We obtain the $p$-power residue symbols relations: $$(\frac{p}{\mk q_K})_K=(\frac{1-ζ^j}{\mk q_K})_K for j=1,\dots,p-1.$$ As an application, we prove that: if Vandiver's conjecture holds for $p$ then $q$ is a $p$-principal prime. Similarly, let $q$ be a prime dividing $\frac{(x^p-y^p)(y^p-z^p)(z^p-x^p)}{(x-y)(y-z)(z-x)}$ and $\mk q_K$ be the prime ideal of $K$ over $q$ dividing $(x\zeta -y)(z\zeta -y)(x\zeta -z)$. We give an explicit formula for the $p$-power residue symbols $(\frac{ε_{k}}{\mk q_K})_K$ for all $k$ with $1<k\le \frac{p-1}{2},$ where ${\epsilon }_k$ is the cyclotomic unit given by ${\epsilon }_k=:{\zeta }^{(1-k)/2}\cdot \frac{1+{\zeta }^k}{1+\zeta }.$ The principle of proofs rely on the $p$-Hilbert class field theory.

13 pages; this article is a part of the restructuration of the article : Complements on Furtwängler's second theorem and Vandiver's cyclotomic units, arXiv 1109.0956 (2011)