Ferrero–Washington theorem

Short description: Iwasawa's μ-invariant is 0 for cyclotomic extensions of abelian algebraic number fields

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 in	1973
First proof by	Bruce Ferrero Lawrence C. Washington
First proof in	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 (Ferrero Washington). 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) 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) 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 [math]\displaystyle{ \hat K }[/math], and the maximal unramified abelian p-extension of [math]\displaystyle{ \hat K }[/math] by A^(p). Let the Tate module

[math]\displaystyle{ T_p(K) = \mathrm{Gal}(A^{(p)}/\hat K) \ . }[/math]

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[[T]] and this implies a formula for the exponent of p in the order of the class groups C_m of the form

[math]\displaystyle{ \lambda m + \mu p^m + \kappa \ . }[/math]

The Ferrero–Washington theorem states that μ is zero.

References

Sources

• Ferrero, Bruce; Washington, Lawrence C. (1979), "The Iwasawa invariant μ_p vanishes for abelian number fields", Annals of Mathematics, Second Series 109 (2): 377–395, doi:10.2307/1971116, ISSN 0003-486X 
• Iwasawa, Kenkichi (1958), "On some invariants of cyclotomic fields", American Journal of Mathematics 81 (3): 773–783, doi:10.2307/2372857, ISSN 0002-9327  (And correction JSTOR 2372857)
• Iwasawa, Kenkichi (1959), "On Γ-extensions of algebraic number fields", Bulletin of the American Mathematical Society 65 (4): 183–226, doi:10.1090/S0002-9904-1959-10317-7, ISSN 0002-9904 
• Iwasawa, Kenkichi (1971), "On some infinite Abelian extensions of algebraic number fields", Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, pp. 391–394, http://ada00.math.uni-bielefeld.de/ICM/ICM1970.1/ 
• Iwasawa, Kenkichi (1973), "On the μ-invariants of Z1-extensions", Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Tokyo: Kinokuniya, pp. 1–11, https://books.google.com/books?id=4_buAAAAMAAJ 
• Iwasawa, Kenkichi; Sims, Charles C. (1966), "Computation of invariants in the theory of cyclotomic fields", Journal of the Mathematical Society of Japan 18: 86–96, doi:10.2969/jmsj/01810086, ISSN 0025-5645 
• Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, 49 (Second ed.), ISBN 978-3-540-20364-3, ISSN 0938-0396 
• Sinnott, W. (1984), "On the μ-invariant of the Γ-transform of a rational function", Inventiones Mathematicae 75 (2): 273–282, doi:10.1007/BF01388565, ISSN 0020-9910