Main conjecture of Iwasawa theory
Field | Algebraic number theory Iwasawa theory |
---|---|
Conjectured by | Kenkichi Iwasawa |
Conjectured in | 1969 |
First proof by | Barry Mazur Andrew Wiles |
First proof in | 1984 |
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
Motivation
Iwasawa (1969a) 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 variety. 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).
In 2014,
Statement
- p is a prime number.
- Fn is the field Q(ζ) where ζ is a root of unity of order pn+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
- Ln is the p-Hilbert class field of Fn.
- Hn is the Galois group Gal(Ln/Fn), isomorphic to the subgroup of elements of the ideal class group of Fn whose order is a power of p.
- H∞ is the inverse limit of the Galois groups Hn.
- V is the vector space H∞⊗ZpQp.
- ω is the Teichmüller character.
- Vi is the ωi eigenspace of V.
- hp(ωi,T) is the characteristic polynomial of γ acting on the vector space Vi
- Lp is the generalized Bernoulli number.
- u is the unique p-adic number satisfying γ(ζ) = ζu for all p-power roots of unity ζ
- Gp is the power series with Gp(ωi,us–1) = Lp(ωi,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 generated by hp(ωi,T) and Gp(ω1–i,T) are equal.
Notes
- ^ Wiles 1990, Kakde 2013
- ^ Manin & Panchishkin 2007, p. 246.
- ^ Skinner & Urban 2014, pp. 1–277.
- ^ Bhargava, Skinner & Zhang 2014.
- ^ Baker 2014.
Sources
- Baker, Matt (2014-03-10), "The BSD conjecture is true for most elliptic curves", Matt Baker's Math Blog, retrieved 2019-02-24
- Bhargava, Manjul; Skinner, Christopher; Zhang, Wei (2014-07-07), "A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture", ]
- 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, MR 0215811
- Iwasawa, Kenkichi (1969a), "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", MR 0269627
- MR 3091976
- Zbl 0704.11038
- Zbl 1079.11002
- S2CID 122576427
- Skinner, Christopher; Urban, Eric (2014), "The Iwasawa main conjectures for GL2", S2CID 120848645
- ISBN 978-0-387-94762-4
- MR 1053488