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Ferrero–Washington theorem

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Iwasawa's μ-invariant is 0 for cyclotomic extensions of abelian algebraic number fields
Ferrero–Washington theorem
FieldAlgebraic number theory
StatementIwasawa's μ-invariant is zero for cyclotomic p-adic extensions of abelian number fields.
First stated byKenkichi Iwasawa
First stated in1973
First proof byBruce Ferrero
Lawrence C. Washington
First proof in1979

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

History

Iwasawa (1959) introduced the μ-invariant of a Zp-extension and observed that it was zero in all cases he calculated. Iwasawa & Sims (1966) used a computer to check that it vanishes for the cyclotomic Zp-extension of the rationals for all primes less than 4000. Iwasawa (1971) later conjectured that the μ-invariant vanishes for any Zp-extension, but shortly after 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 Zp-extensions.

Iwasawa (1958) showed that the vanishing of the μ-invariant for cyclotomic Zp-extensions of the rationals is equivalent to certain congruences between Bernoulli numbers, and 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-power roots of unity by Km, the union of the Km as m ranges over all positive integers by K ^ {\displaystyle {\hat {K}}} , and the maximal unramified abelian p-extension of K ^ {\displaystyle {\hat {K}}} by A. Let the Tate module

T p ( K ) = G a l ( A ( p ) / K ^ )   . {\displaystyle T_{p}(K)=\mathrm {Gal} (A^{(p)}/{\hat {K}})\ .}

Then Tp(K) is a pro-p-group and so a Zp-module. Using class field theory one can describe Tp(K) as isomorphic to the inverse limit of the class groups Cm of the Km under norm.

Iwasawa exhibited Tp(K) as a module over the completion Zp] and this implies a formula for the exponent of p in the order of the class groups Cm of the form

λ m + μ p m + κ   . {\displaystyle \lambda m+\mu p^{m}+\kappa \ .}

The Ferrero–Washington theorem states that μ is zero.

References

  1. Manin & Panchishkin 2007, p. 245
  2. Manin & Panchishkin 2007, p. 246

Sources

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