Some works of Furtwängler and Vandiver revisited and Fermat's last theorem

Abstract

From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat ′ s last theorem for p > 3 and to a stronger version called SFLT, by introducing governing fields of the form Q(µ q−1 ) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2 ), such that for q | q in Q(µ q−1 ), we have q 1−c = a p (α) with α ≡ 1 (mod p 2 ) (where c is the complex conjugation), then Fermat ′ s last theorem holds for p. More generally, the main purpose of the paper is to show that the existence of nontrivial solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(µ q−1 ). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for future researches (probably of analytic or geometric nature).Résumé. Reprenant des travaux de P. Furtwängler et H.S. Vandiver, nous posons les bases d'une nouvelle approche cyclotomique du dernier théorème de Fermat pour p > 3 et d'une version plus forte appelée SFLT, en introduisant des corps gouvernants de la forme Q(µ q−1 ) pour q premier. Nous prouvons par exemple que s'il existe une infinité de nombres premiers q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2 ), tels que pour q | q dans Q(µ q−1 ), on ait q 1−c = a p (α) avec α ≡ 1 (mod p 2 ) (où c est la conjugaison complexe), alors le théorème de Fermat est vrai pour p. Plus généralement, le but principal de l'article est de montrer que l'existence de solutions non triviales pour SFLT implique de fortes contraintes sur l'arithmétique des corps Q(µ q−1 ). A partir de là, nous donnons des conditions suffisantes de non existence qui nécessiteraient des investigations supplémentaires pour conduire à une preuve de SFLT, et nous formulons diverses conjectures. Ce texte doit être considéré comme un outil de base pour de futures recherches (probablement analytiques ou géométriques). ******* This second version includes some corrections in the English language, an in depth study of the case p = 3 (especially Theorem 8), further details on some conjectures, and some minor mathematical improvements. 1 Equation (u + v ζ) Z[ζ] = w p 1 or p w p 1 , in integers u, v with g.c.d. (u, v) = 1, equivalent to N K/Q (u + v ζ) = w p 1 or p w p 1 , where ζ := e 2iπ/p , K := Q(ζ), p := (ζ − 1) Z[ζ] (see Conjecture 1). Remark that the important condition g.c.d. (u, v) = 1 implies w 1 prime to p.Note that if u v = 0, the condition g.c.d. (u, v) = 1 implies (u, v) = (±1, 0) or (0, ±1).2 If ν ≥ 1, then α := a+c ζ a+c ζ −1 is a pseudo-unit (i.e., the pth power of an ideal), congruent to 1 modulo p; so, from [Gr1, Theorem 2.2, Remark 2.3, (ii)], α is locally a pth power in K giving easily α ≡ 1 (mod p p+1 ), then c (ζ−ζ −1 ) a+c ζ −1 ≡ 0 (mod p p+1 ), hence c ≡ 0 (mod p 2 ).