Another profitable way to study class number formulae is via Iwasawa theory. Fix a prime number p. Kenkichi Iwasawa considered the tower of cyclotomic fields Q(&mupn) generated by pn-th roots of unity for all n ≥ 1. His main conjecture states that the Dirichlet class number formulae for these fields should interpolate to a p-adic class number formula. More precisely, he showed that the special values of the relevant L-functions interpolate to define a p-adic analytic function Lp,an(s). He also defined an algebraic p-adic L-function Lp,alg(s) in terms of the limit of the ideal class groups of Q(&mupn) as n goes to infinity. The main conjecture states that these two functions are equal (up to a unit function). It was proven by Barry Mazur and Andrew Wiles via the arithmetic of modular curves.

It was observed by Mazur that Iwasawa's ideas can be extended to elliptic curves, and by Ralph Greenberg that they can be further extended to modular forms and Galois representations. For elliptic curves, the resulting main conjecture predicts that the knowledge of a certain infinite collection of special values of the L-function of the elliptic curve determines the Selmer groups Sel(Q(&mupn),E[p&infin]) for all n ≥ 1; the statement for modular forms is analogous. Recent spectacular work of Kazuya Kato has proven this main conjecture for many, but not all, elliptic curves and modular forms.

A striking aspect of the theory of modular forms is the existence of congruences between different modular forms. For example, if E is the elliptic curve

y2 + y = x3 - x2 - 10x - 20
then for any prime p different from 11 we have
ap(E) ≡ &tau(p) (mod 11)
with &tau(p) the Ramanujan τ-function. The 11-adic main conjecture for E is relatively easy to establish with Kato's results. A recent result of Matthew Emerton, Robert Pollack, and myself (Variation of Iwasawa invariants in Hida families) then allows one to deduce the 11-adic main conjecture for Δ as well. More recently I have shown in Iwasawa invariants of Galois deformations that similar techniques apply for many Galois representations as well.

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