Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
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.
Motivation.
 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,
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.
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 formula_1 generated by "h""p"(ω"i","T") and "G""p"(ω1–"i","T") are equal.
Notes.
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Sources.
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