Ferrero–Washington theorem

{{short description|Iwasawa's μ-invariant is 0 for cyclotomic extensions of abelian algebraic number fields}}

{{Infobox mathematical statement

| name = Ferrero–Washington theorem

| image =

| caption =

| field = Algebraic number theory

| statement = Iwasawa's μ-invariant is zero for cyclotomic p-adic extensions of abelian number fields.

| first stated by = Kenkichi Iwasawa

| first stated date = 1973

| first proof by = Bruce Ferrero{{br}}Lawrence C. Washington

| first proof date = 1979

| implied by =

| generalizations =

}}

In algebraic number theory, the Ferrero–Washington theorem states that Iwasawa's μ-invariant vanishes for cyclotomic Z_p-extensions of abelian algebraic number fields. It was first proved by {{harvtxt|Ferrero|Washington|1979}}. A different proof was given by {{harvtxt|Sinnott|1984}}.

History

{{harvtxt|Iwasawa|1959}} introduced the μ-invariant of a Z_p-extension and observed that it was zero in all cases he calculated. {{harvtxt|Iwasawa|Sims|1966}} used a computer to check that it vanishes for the cyclotomic Z_p-extension of the rationals for all primes less than 4000.

{{harvtxt|Iwasawa|1971}} later conjectured that the μ-invariant vanishes for any Z_p-extension, but shortly after {{harvtxt|Iwasawa|1973}} discovered examples of non-cyclotomic extensions of number fields with non-vanishing μ-invariant showing that his original conjecture was wrong. He suggested, however, that the conjecture might still hold for cyclotomic Z_p-extensions.

{{harvtxt|Iwasawa|1958}} showed that the vanishing of the μ-invariant for cyclotomic Z_p-extensions of the rationals is equivalent to certain congruences between Bernoulli numbers, and {{harvtxt|Ferrero|Washington|1979}} showed that the μ-invariant vanishes in these cases by proving that these congruences hold.

Statement

For a number field K, denote the extension of K by p^m-power roots of unity by K_m, the union of the K_m as m ranges over all positive integers by $\hat K$ , and the maximal unramified abelian p-extension of $\hat K$ by A^(p). Let the Tate module

: $T_p(K) = \mathrm{Gal}(A^{(p)}/\hat K) \ .$

Then T_p(K) is a pro-p-group and so a Z_p-module. Using class field theory one can describe T_p(K) as isomorphic to the inverse limit of the class groups C_m of the K_m under norm.{{harvnb|Manin|Panchishkin|2007|p=245}}

Iwasawa exhibited T_p(K) as a module over the completion Z_p{{brackets|T}} and this implies a formula for the exponent of p in the order of the class groups C_m of the form

: $\lambda m + \mu p^m + \kappa \ .$

The Ferrero–Washington theorem states that μ is zero.{{harvnb|Manin|Panchishkin|2007|p=246}}

References

=Sources=

{{Citation | author1-link=Bruce Ferrero | author2-link=Larry Washington | last1=Ferrero | first1=Bruce | last2=Washington | first2=Lawrence C. | title=The Iwasawa invariant μ_p vanishes for abelian number fields | doi=10.2307/1971116 | year=1979 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=109 | issue=2 | pages=377–395 | mr=528968 | zbl=0443.12001 | jstor=1971116 }}
{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=On some invariants of cyclotomic fields | jstor=2372782 |mr=0124317 | year=1958 | journal=American Journal of Mathematics | issn=0002-9327 | volume=81 | issue=3 | pages=773–783 | doi=10.2307/2372857}} (And correction {{JSTOR|2372857}})
{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=On Γ-extensions of algebraic number fields | doi=10.1090/S0002-9904-1959-10317-7 |mr=0124316 | year=1959 | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=65 | issue=4 | pages=183–226| doi-access=free }}
{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1 | chapter-url=http://ada00.math.uni-bielefeld.de/ICM/ICM1970.1/ | publisher=Gauthier-Villars |mr=0422205 | year=1971 | chapter=On some infinite Abelian extensions of algebraic number fields | pages=391–394}}
{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | title=Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki | chapter-url=https://books.google.com/books?id=4_buAAAAMAAJ | publisher=Kinokuniya | location=Tokyo |mr=0357371 | year=1973 | chapter=On the μ-invariants of Z1-extensions | pages=1–11}}
{{Citation | last1=Iwasawa | first1=Kenkichi | author1-link=Kenkichi Iwasawa | last2=Sims | first2=Charles C. | title=Computation of invariants in the theory of cyclotomic fields | doi= 10.2969/jmsj/01810086|mr=0202700 | year=1966 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=18 | pages=86–96| doi-access=free }}
{{citation | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 }}
{{Citation | last1=Sinnott | first1=W. | title=On the μ-invariant of the Γ-transform of a rational function | doi=10.1007/BF01388565 | year=1984 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=75 | issue=2 | pages=273–282 | mr=732547 | zbl=0531.12004 }}

Category:Theorems in algebraic number theory