Main conjecture of Iwasawa theory

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 totally real fields,[1] CM fields, elliptic curves, and so on.

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).

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.[2]

In 2014, Christopher Skinner and Eric Urban proved several cases of the main conjectures for a large class of modular forms.[3] As a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil L-function L(E, s) of E at s = 1 implies that the p-adic Selmer group of E is infinite. Combined with theorems of Gross-Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has infinitely many rational points if and only if L(E, 1) = 0, a (weak) form of the Birch–Swinnerton-Dyer conjecture. These results were used by Manjul Bhargava, Skinner, and Wei Zhang to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.[4][5]

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_∞⊗_ZpQ_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 ${\displaystyle \mathbf {Z} _{p}[[T]]}$ generated by h_p(ω^i,T) and G_p(ω^1–i,T) are equal.

Notes

1. Wiles 1990, Kakde 2013
2. Manin & Panchishkin 2007, p. 246.
3. Skinner & Urban 2014, pp. 1–277.
4. Bhargava, Skinner & Zhang 2014.
5. 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", arXiv:1407.1826 [math.NT]
• Coates, John; Sujatha, R. (2006), Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics, Springer-Verlag, ISBN 978-3-540-33068-4, 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:10.2969/jmsj/01610042, ISSN 0025-5645, 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", Annals of Mathematics, Second Series, 89 (1): 198–205, doi:10.2307/1970817, ISSN 0003-486X, JSTOR 1970817, MR 0269627
• Kakde, Mahesh (2013), "The main conjecture of Iwasawa theory for totally real fields", Inventiones Mathematicae, 193 (3): 539–626, arXiv:1008.0142, Bibcode:2013InMat.193..539K, doi:10.1007/s00222-012-0436-x, MR 3091976
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• Manin, Yu I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 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, Bibcode:1984InMat..76..179M, doi:10.1007/BF01388599, ISSN 0020-9910, MR 0742853, S2CID 122576427
• Skinner, Christopher; Urban, Eric (2014), "The Iwasawa main conjectures for GL₂", Inventiones Mathematicae, 195 (1): 1–277, Bibcode:2014InMat.195....1S, CiteSeerX 10.1.1.363.2008, doi:10.1007/s00222-013-0448-1, MR 3148103, S2CID 120848645
• Washington, Lawrence C. (1997), Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4
• 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, JSTOR 1971468, MR 1053488