Some works of Furtw'angler and Vandiver revisited and Fermat's last theorem

摘要

From some works of P. Furtw\"angler and H.S. Vandiver, we put the basis of anew cyclotomic approach to Fermat's last theorem for p>3 and to a strongerversion called SFLT, by introducing governing fields of the form Q(exp(2 ipi/q-1)) for prime numbers q. We prove for instance that if there existinfinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) =A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complexconjugation), then Fermat's last theorem holds for p. More generally, the mainpurpose of the paper is to show that the existence of nontrivial solutions forSFLT implies some strong constraints on the arithmetic of the fields Q(exp(2 ipi /q-1)). From there, we give sufficient conditions of nonexistence that wouldrequire further investigations to lead to a proof of SFLT, and we formulatevarious conjectures. This text must be considered as a basic tool for futureresearchs (probably of analytic or geometric nature) - This second versionincludes some corrections in the English language, an in depth study of thecase p=3 (especially Theorem 8), further details on some conjectures, and someminor mathematical improvements.