Ferrero–Washington theorem

History

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

showed that the vanishing of the μ-invariant for cyclotomic Zp-extensions of the rationals is equivalent to certain congruences between Bernoulli numbers, and 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 Zp-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.

Iwasawa exhibited T**p(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 C**m of the form : \lambda m + \mu p^m + \kappa \ .

The Ferrero–Washington theorem states that μ is zero.

References

Sources

• (And correction )

References

1. {{harvnb. Manin. Panchishkin. 2007
2. {{harvnb. Manin. Panchishkin. 2007