Main conjecture of Iwasawa theory
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles (1984). The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.
Contents
Motivation
Iwasawa (1969) was partly motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian. In this analogy,
- The action of the Frobenius corresponds to the action of the group Γ.
- The Jacobian of a curve corresponds to a module X over Γ defined in terms of ideal class groups
- The zeta function of a curve over a finite field corresponds to a p-adic L-function.
- Weil's theorem relating the eigenvalues of Frobenius to the zeros of the zeta function of the curve corresponds to Iwasawa's main conjecture relating the action of the Iwasawa algebra on X to zeros of the p-adic zeta function.
History
The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by Mazur & Wiles (1984) for Q, and for all totally real number fields by Wiles (1990). These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the Herbrand–Ribet theorem).
Karl Rubin found a more elementary proof of the Mazur–Wiles theorem by using Thaine's method and Kolyvagin's Euler systems, described in Lang (1990) and Washington (1997), and later proved other generalizations of the main conjecture for imaginary quadratic fields.^{[1]}
Statement
- p is a prime number.
- F_{n} is the field Q(ζ) where ζ is a root of unity of order p^{n+1}.
- Γ is the largest subgroup of the absolute Galois group of F_{∞} isomorphic to the p-adic integers.
- γ is a topological generator of Γ
- L_{n} is the p-Hilbert class field of F_{n}.
- H_{n} is the Galois group Gal(L_{n}/F_{n}), isomorphic to the subgroup of elements of the ideal class group of F_{n} whose order is a power of p.
- H_{∞} is the inverse limit of the Galois groups H_{n}.
- V is the vector space H_{∞}⊗_{Zp}Q_{p}.
- ω is the Teichmüller character.
- V^{i} is the ω^{i} eigenspace of V.
- h_{p}(ω^{i},T) is the characteristic polynomial of γ acting on the vector space V^{i}
- L_{p} is the p-adic L function with L_{p}(ω^{i},1–k) = –B_{k}(ω^{i–k})/k, where B is a generalized Bernoulli number.
- u is the unique p-adic number satisfying γ(ζ) = ζ^{u} for all p-power roots of unity ζ
- G_{p} is the power series with G_{p}(ω^{i},u^{s}–1) = L_{p}(ω^{i},s)
The main conjecture of Iwasawa theory proved by Mazur and Wiles states that if i is an odd integer not congruent to 1 mod p–1 then the ideals of Z_{p}[[T]] generated by h_{p}(ω^{i},T) and G_{p}(ω^{1–i},T) are equal.
The proof of Iwasawa's main conjecture by Mazur-Wiles was later improved by Rubin using Kolyvagin's Euler Systems, see entry for Iwasawa theory.
References
- ^ Manin & Panchishkin (2007) p.246
- Coates, John; Sujatha, R. (2006), Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics, Springer-Verlag, ISBN 3-540-33068-2, Zbl 1100.11002
- Iwasawa, Kenkichi (1964), "On some modules in the theory of cyclotomic fields", Journal of the Mathematical Society of Japan, 16: 42–82, doi:10.4099/jmath.16.42, ISSN 0025-5645, MR 0215811
- Iwasawa, Kenkichi (1969), "Analogies between number fields and function fields", Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Belfer Graduate School of Science, Yeshiva Univ., New York, pp. 203–208, MR 0255510
- Iwasawa, Kenkichi (1969b), "On p-adic L-functions", Annals of Mathematics, Second Series, 89: 198–205, doi:10.2307/1970817, ISSN 0003-486X, JSTOR 1970817, MR 0269627
- 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, Zbl 1079.11002
- Mazur, Barry; Wiles, Andrew (1984), "Class fields of abelian extensions of Q", Inventiones Mathematicae, 76 (2): 179–330, doi:10.1007/BF01388599, ISSN 0020-9910, MR 0742853
- Wiles, Andrew (1990), "The Iwasawa conjecture for totally real fields", Annals of Mathematics, Second Series, 131 (3): 493–540, doi:10.2307/1971468, ISSN 0003-486X, MR 1053488