Main conjecture of Iwasawa theory

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 for Q, and for all totally real number fields by . 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 and , and later proved other generalizations of the main conjecture for imaginary quadratic fields. In 2014, Christopher Skinner and Eric Urban proved several cases of the main conjectures for a large class of modular forms. 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. ==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 p-adic L function with Lp(ωi,1–k) = –Bk(ωi–k)/k, where B is a 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 \mathbf Z_pT generated by hp(ωi,T) and Gp(ω1–i,T) are equal. ==Notes==