Ferrero–Washington theorem explained

Ferrero–Washington theorem
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 FerreroLawrence C. Washington
First Proof Date:	1979

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 . A different proof was given by .

History

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

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 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

T _{p(K)} = 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.

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

λ m + \mu p ^{m} + \kappa .

The Ferrero–Washington theorem states that μ is zero.

References

Sources

• • (And correction)