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 Ferrero & Washington (1979). A different proof was given by Sinnott (1984).

History

Iwasawa (1959) introduced the μ-invariant of a Z_p-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 Z_p-extension of the rationals for all primes less than 4000. Iwasawa (1971) later conjectured that the μ-invariant vanishes for any Z_p-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 Z_p-extensions.

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 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 ${\displaystyle {\hat {K}}}$, and the maximal unramified abelian p-extension of ${\displaystyle {\hat {K}}}$ by A^(p). Let the Tate module

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

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.^[1]

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

${\displaystyle \lambda m+\mu p^{m}+\kappa \ .}$

The Ferrero–Washington theorem states that μ is zero.^[2]

References