Main conjecture of Iwasawa theory

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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 Barry Mazur and Andrew Wiles.Template:Sfnp 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,Template:Sfnp Template:Sfnp CM fields, elliptic curves, and so on.

Motivation

Iwasawa 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.Template:Sfnp 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 the two methods of defining p-adic L-functions (by Galois modules or by interpolation of special values of L-functions) should coincide, as far as that was well-defined. This was proved by Mazur and Wiles for Q.Template:Sfnp Wiles later proved it for all totally real number fields.Template:Sfnp 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,Template:Sfnp Template:Sfnp and later proved other generalizations of the main conjecture for imaginary quadratic fields.Template:Sfnp

In 2014, Christopher Skinner and Eric Urban proved several cases of the main conjectures for a large class of modular forms.Template:Sfnp 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.Template:Sfnp Template:Sfnp

Statement

p is a prime number.
F_n 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.
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_∞⊗_{Z_p}Q_p.
ω is the Teichmüller character.
Vⁱ is the ωⁱ eigenspace of V.
h_p(ωⁱ,T) is the characteristic polynomial of γ acting on the vector space Vⁱ.
L_p is the p-adic L function with L_p(ωⁱ,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(ωⁱ,u^s–1) = L_p(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 $𝐙_{p} [[T]]$ generated by h_p(ωⁱ,T) and G_p(ω^1–i,T) are equal.