- Herbrand–Ribet theorem
In
mathematics , the Herbrand–Ribet theorem is a result on the class number of certain number fields. It is a strengthening ofKummer 's theorem to the effect that the prime "p" divides the class number of the cyclotomic field of "p"-th roots of unity if and only if "p" divides the numerator of the nthBernoulli number "B""n" for some "n", 0 < "n" < "p" − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when "p" divides such an "B""n".The
Galois group Σ of thecyclotomic field of "p"th roots of unity for an odd prime "p", Q(ζ) with ζ"p" = 1, consists of the "p" − 1 group elements σ"a", where σ"a" is defined by the fact that sigma_a(zeta) = zeta^a. As a consequence of thelittle Fermat theorem , in the ring of "p"-adic integers Bbb{Z}_p we have "p" − 1 roots of unity, each of which is congruent mod "p" to some number in the range 1 to "p" − 1; we can therefore define aDirichlet character ω (the Teichmüller character) with values in Bbb{Z}_p by requiring that for "n" relatively prime to "p", ω("n") be congruent to "n" modulo "p". The "p" part of the class group is a Bbb{Z}_p-module, and we can apply elements in thegroup ring Bbb{Z}_p [Sigma] to it and obtain elements of the class group. We now may define an idempotent element of the group ring for each "n" from 1 to "p" − 1, as:epsilon_n = frac{1}{p-1}sum_{a=1}^{p-1} omega(a)^n sigma_a^{-1}.
We now can break up the "p" part of the ideal class group "G" of Q(ζ) by means of the idempotents; if "G" is the ideal class group, then "G""n" = ε"n"("G").
Then we have the theorem of Herbrand–Ribet [Ribet, Ken, A modular construction of unramified p-extensions of Bbb{Q}(μp), Inv. Math. 34 (3), 1976, pp. 151-162.] : "G""n" is nontrivial if and only if "p" divides the Bernoulli number "B""p"−"n". The part saying p divides "B""p"−"n" if "G""n" is not trivial is due to
Herbrand . The converse, that if "p" divides "B""p"−"n" then "G""n" is not trivial is due toRibet , and is considerably more difficult. Byclass field theory , this can only be true if there is an unramified extension of the field of "p"th roots of unity by a cyclic extension of degree "p" which behaves in the specified way under the action of Σ; Ribet proves this by actually constructing such an extension using methods in the theory ofmodular forms . A more elementary proof of Ribet's converse to Herbrand's theorem can be found in Washington's book. [Washington, Lawrence C., Introduction to Cyclotomic Fields, Second Edition, Springer-Verlag, 1997.]Ribet's methods were pushed further by
Barry Mazur andAndrew Wiles in order to prove the Main Conjecture of Iwasawa Theory, [Mazur, Barry, and Wiles, Andrew, Class Fields of Abelian Extension of Bbb{Q}, Inv. Math. 76 (2), 1984, pp. 179-330.] a corollary of which is a strengthening of the Herbrand-Ribet theorem: the power of "p" dividing "B""p"−"n" is exactly the power of "p" dividing the order of "G""n".ee also
References
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