Main conjecture of Iwasawa theory
Template:Short description Template:Infobox mathematical statement
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 Template:Harvs. 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
Template:Harvtxt 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 Template:Harvtxt for Q, and for all totally real number fields by Template:Harvtxt. 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 Template:Harvtxt and Template:Harvtxt, and later proved other generalizations of the main conjecture for imaginary quadratic fields.Template:Sfn
In 2014, Christopher Skinner and Eric Urban proved several cases of the main conjectures for a large class of modular forms.Template:Sfn 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:SfnTemplate:Sfn
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 generated by hp(ωi,T) and Gp(ω1–i,T) are equal.