From the classical to the noncommutative Iwasawa theory (for totally real number fields)

Summary

Introduction

The conjectures of Deligne [17], Beilinson [3] and Bloch–Kato [4] are a vast generalisation of the Dirichlet–Dedekind class number formula and Birch–Swinnerton-Dyer conjecture. They predict the order of arithmetic objects (such as class groups, Tate–Shafarevich groups, etc.) in terms of special values of L-functions. On the other hand, the aim of Iwasawa theory is to understand the Galois module structure of these arithmetic objects in terms of L-values. We roughly explain what may now be called classical Iwasawa theory. Let p be a prime. Let ℚ^cyc be the cyclotomic ℤ_p-extension of ℚ (see section 2). Let M be a motive over ℚ. We assume that M is critical in the sense of Deligne (this means that the Euler factor at infinity L_∞(M, s) and L_∞(M^*(1), -s) are both holomorphic at s = 0, where M^* is the dual motive. For details see [14]). Assume that p is a good ordinary prime for M (in the sense of Greenberg [25]. This just means that the p-adic realisation of M has a finite decreasing filtration such that the action of inertia on the ith graded piece is via the ith power of the p-adic cyclotomic character). Let Γ = Gal(ℚ^cyc/ℚ) ≅ ℤ_p and let ∧(Γ) be the Iwasawa algebra ℤ_p[[Γ]] (see end of Section 2). Fix a topological generator γ of Γ. Then the Iwasawa algebra and;(Γ) is isomorphic to the power series ring ℤ_p[[T]].