578 research outputs found
Analysis of the classical cyclotomic approach to fermat's last theorem
We give again the proof of several classical results concerning the
cyclotomic approach to Fermat's last theorem using exclusively class field
theory (essentially the reflection theorems), without any calculations. The
fact that this is possible suggests a part of the logical inefficiency of the
historical investigations. We analyze the significance of the numerous
computations of the literature, to show how they are probably too local to get
any proof of the theorem. However we use the derivation method of Eichler as a
prerequisite for our purpose, a method which is also local but more effective.
Then we propose some modest ways of study in a more diophantine context using
radicals; this point of view would require further nonalgebraic investigations.Comment: Publications Math\'ematiques UFR Sciences Techniques Besan\c{c}on
2010 (2010) 85-11
A Cup Product in the Galois Cohomology of Number Fields
Let K be a number field containing the group of n-th roots of unity and S a
set of primes of K including all those dividing n and all real archimedean
places. We consider the cup product on the first Galois cohomology group of the
maximal S-ramified extension of K with coefficients in n-th roots of unity,
which yields a pairing on a subgroup of the multiplicative group of K
containing the S-units. In this general situation, we determine a formula for
the cup product of two elements which pair trivially at all local places.
Our primary focus is the case that K is the cyclotomic field of p-th roots of
unity for n = p an odd prime and S consists of the unique prime above p in K.
We describe a formula for this cup product in the case that one element is a
p-th root of unity. We explain a conjectural calculation of the restriction of
the cup product to p-units for all p < 10,000 and conjecture its surjectivity
for all p satisfying Vandiver's conjecture. We prove this for the smallest
irregular prime p = 37, via a computation related to the Galois module
structure of p-units in the unramified extension of K of degree p.
We describe a number of applications: to a product map in K-theory, to the
structure of S-class groups in Kummer extensions of K, to relations in the
Galois group of the maximal pro-p extension of K unramified outside p, to
relations in the graded Z_p-Lie algebra associated to the representation of the
absolute Galois group of Q in the outer automorphism group of the pro-p
fundamental group of P^1 minus three points, and to Greenberg's pseudo-nullity
conjecture.Comment: final versio
Ideal class groups of cyclotomic number fields II
We first study some families of maximal real subfields of cyclotomic fields
with even class number, and then explore the implications of large plus class
numbers of cyclotomic fields. We also discuss capitulation of the minus part
and the behaviour of p-class groups in cyclic ramified p-extensions
Vanishing of eigenspaces and cyclotomic fields
We use a result of Thaine to give an alternative proof of the fact that, for
a prime p>3 congruent to 3 modulo 4, the component e_{(p+1)/2} of the p-Sylow
subgroup of the ideal class group of \mathbb Q(\zeta_{p}) is trivial.Comment: 6 pages, minor corrections made, to appear in the International
Mathematics Research Notice
On the failure of pseudo-nullity of Iwasawa modules
We consider the family of CM-fields which are pro-p p-adic Lie extensions of
number fields of dimension at least two, which contain the cyclotomic
Z_p-extension, and which are ramified at only finitely many primes. We show
that the Galois groups of the maximal unramified abelian pro-p extensions of
these fields are not always pseudo-null as Iwasawa modules for the Iwasawa
algebras of the given p-adic Lie groups. The proof uses Kida's formula for the
growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact,
we give a new proof of Kida's formula which includes a slight weakening of the
usual assumption that mu is trivial. This proof uses certain exact sequences
involving Iwasawa modules in procyclic extensions. These sequences are derived
in an appendix by the second author.Comment: 26 page
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