Fix a number field and p a prime. Define where are the -th roots of unity and let . Let be the class group of and let be the p-part of . Finally, let be the Galois group of .
Iwasawa theory broadly describes the structure of and the action of on it. For instance, Iwasawa proved in 1959 that the size of is equal to for large .
There are also various results relating to the p-adic properties of special values of the zeta function of . For instance, taking , let be the usual zeta function. For negative odd integers n, is always a rational number and writing these in reduced form, Kummer proved that divides one of the numerators of if and only if is non trivial. The Herbrand-Ribet theorem provides a generalization of this statement and also describes the action of the Galois group on .
While very beautiful and striking, these results might seem to come out of nowhere. I would like to describe Iwasawa’s original motivation in the reminder of this post.
The analogy with Function Fields:
Consider a function field over a finite field or equivalently, a projective curve over a finite . There are strong similarities between these and number fields and the correct analog of the Ideal Class group is the group of divisor classes of degree 0. This group can be given the structure of an Abelian variety and is called the Jacobian of and is denoted . It has dimension equal to the genus of the curve .
We understand this group well. In particular, if we take p to be a prime different from the characteristic of the base field, then the p-torsion of J(C) (over an algebraic closure ) is isomorphic to .
However, we are interested in the points of . Luckily, the image of the natural embedding is contained in the torsion subgroup which is well understood.
Since is obtained from by adjoining all the roots of unity, one might hope that a similar procedure might help us on the number theoretic side. Unfortunately, such a field is too large and loses too much arithmetic information.
However, since the ideal class group is a finite abelian group for a number field, it is determined by it’s p-Sylow subgroups. Fixing a prime p as before, let denote the p-part of . It turns out that we can study by adjoining all the p-th roots of unity.
That is, in the notation from the beginning of the post, if we define as a limit of directed sets, then it can be shown that
Here, is a non-negative integer, is a finite abelian group of order some power of p, the direct sum is of an infinite copies of and is a finite subgroup of .
It is conjectured that in general, does not appear and we can prove this in the case that is an abelian extension.
The Frobenius and the action of the Galois Group:
In the case of function fields, we have a Frobenius action on and on whose fixed points are precisely the points. The action of the Frobenius on induces an endomorphism on it which may be represented by a matrix . Importantly, the zeta function of over can be expressed in terms of the characteristic polynomial of . This is the content of the Weil Conjectures.
Let us come back to number fields and for simplicity. As mentioned in the last section, . The Galois group acts on $A_\infty$ naturally but unlike the function field case, the Galois group does not have a generator. Since , we have .
We have . The second factor is isomorphic to and we take a topological generator of it.
If we believe the analogy with function fields, then we would expect a relation between the characteristic polynomial of and some zeta function. The answer is, amazingly enough, yes! This zeta function is called the Kubota-Leopolft p-adic L function derived from the special values of the ordinary L-functions at integers.
The Iwasawa Main Conjecture:
To make everything above more precise, define the Teichmuller character:
by the canonical splitting . Extend to be a map on through the quotient . For an integer in the range , define:
By general theory, we have:
Write . Our topological generator acts on and the action of can be represented by a matrix of degree . Let be the characteristic polynomial of .
The Iwasawa main conjecture states the relation between the polynomial defined as above and the p-adic Riemann function:
Theorem: Let be an odd integer in the range .
- For any negative integer satisfying , there exists a unique formal power series such that where is the Riemann function and $latex $ u is our topological generator as before. The function is a p-adic continuous function from to and is the p-adic L-function we alluded to before.
- (Iwasawa Main Conjecture, Theorem of Mazur-Wiles) The characteristic polynomial defined as above and coincides up to unit in .
This was proven in 1984 by Mazur and Wiles for and for totally real number fields in 1990 by Wiles.