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**(&mu_{pn})
generated by *p*^{n}-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 *L*_{p,an}(*s*).
He also defined an algebraic *p*-adic *L*-function
*L*_{p,alg}(*s*) in terms of the
limit of the
ideal class groups of
**Q**(&mu_{pn}) 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**(&mu_{pn}),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

then for any primey^{2}+y=x^{3}-x^{2}- 10x- 20

with &tau(a(_{p}E) ≡ &tau(p) (mod 11)