One aim of this course.

We want to start at a basic level, and develop a few of the powerful techniques that are currently being used in the arithmetic of elliptic curves and abelian varieties.

One eventual focus for us, will be the method of Kolyvagin, which has so spectacularly been used to establish deep results. But we will get to this only after we have adequately covered a certain amount of preliminary material. Kolyvagin’s method requires familiarity with Galois cohomology, and various local and global duality theorems. Nevertheless, as a preview of coming attractions, it may be fun to give a taste of how Kolyvagin’s approach treats a classical object in number theory: the ideal class group of a real quadratic field. I think that the “ flavor” of Kolyvagin’s theory can already be appreciated in this case without too much technical preliminaries. In any event, we will try to cover this case in fair detail, later on in the course. Here is a first introduction to it.

The ideal class group.

Let K be a number field, which for no important reason except to cut a few words out of this discussion I’ll assume to be totally complex. Let Cl(K) denote the ideal class group of K. The ideal class group can be thought of as

\frac{⨁_{v} Z_{\{v\}}}{[(f) for f ∈ K^{∗}]},

(where the summation in the numerator is of copies of Z for each of nonarchimedean primes of K and is canonically isomorphic to the group of all fractional ideals in K, and the denominator describes the subgroup of all nonzero principal ideals. I’ve written it this way so that the ideal class group is being given by a concoction of local objects (the numerator) and then subjecting this concoction to global constraints (i.e., dividing by the denominator, which is given by global functions).

One of the most basic consequences of Class Field Theory is that we have a canonical identification of Cl(K) with

H = Gal(K^{unram,ab}/K)

(the Galois group of the maximal everywhere unramified abelian extension of K. So H is the Galois group of the maximal abelian extension K^{ab} over K (which I think of as a “global object”) but then subjected to local constraints (i.e., unramified at all primes, which means that we have divided by the image of all local inertia groups).

We therefore can think of the above as two ways of studying “the same group”. You can study ideals and their equivalence classes, or you can study abelian Galois extensions.

Kolyvagin’s approach to the study of the ideal class groups of real quadratic fields.

In the case of ideal class groups of real quadratic fields, Kolyvagin gives yet a third way, or at least a serious variant of the either of the above ways, which sometimes provides information that neither of the two ways directly provides. Let F = Q(\sqrt{d}) be the real quadratic field. Fix a power of a prime number p^{N} and, following Kolyvagin (who in the context of ideal class groups was partially anticipated by Kummer, Stickelberger, Thaine, Rubin) we will aim to “compute”

CL(F)/p^{N}CL(F).

The Kolyvagin elements. First discussion which is slightly inaccurate! Let L denote the set of odd prime numbers ℓ ≡ 1 mod p^{N} and which split in F. Let R denote the set of squarefree positive integer r whose prime divisors are in L. For each such r Kolyvagin will essentially give you an element (let us call it κ_{r}) in the multiplicative group F^{∗} of the field F, modulo p^{N}-th powers. I say essentially above because to speak accurately you have to keep account of certain twists. (You can get a hint of them by turning to the tables below.) In the next instalment of these notes I will discuss these twists. We will be reviewing his construction (which depends upon the theory of cyclotomic units) in the course of the seminar. But for today let us accept the fact that we get such a collection of elements. And let me signal two facts about these elements:

1 If μ is a prime of F which does not divide r, then ord_{μ}(κ_{r}) (which is well-defined in Z/p^{N}Z) is zero. I.e., κ_{r} “involves only primes of F dividing r.

2 κ_{r} ∈ F^{∗−}, the minus eigenspace for the conjugation involution of F over Q.

Now I want to define a group (which let us provisionally call the “Kolyvagin-ideal class group mod p^{N} of F ”).

First form a group which reminiscent of the “numerator” in our discussion of the ideal class group. Consider:

D_{L} := (⨁_{λ∈L} Z/p^{N}Z_{{λ}})^{−},

where the summation is of copies of Z/p^{N}Z for each λ ∈ L and the minus means we are taking the minus eigenspace for the (evident) conjugation involution (which switches λ and λ̄). We can think of this group D_{L} as the group of Z/p^{N}Z-valued divisors concentrated in L and in the minus eigenspace of the evident conjugation involution. For each r ∈ calR, define D_{r} ⊂ D_{L} to be the subgroup consisting of divisors with support only on primes of F dividing r. For each κ_{r} for r ∈ R, we can form the “divisor” (which coefficients modulo p^{N}) associated to κ_{r}. The coefficient of λ in this divisor being simply ord_{μ}(κ_{r}). We then can form the subgroup of D_{L} generated by the divisors of all Kolyvagin elements (call them “Kolyvagin divisors”) .

Define KCL(F; Z/p^{N}Z) to be the quotient of D_{L} by the subgroup of Kolyvagin divisors. We have a natural mapping

KCL(F; Z/p^{N}Z) → CL(F)/p^{N}CL(F).

In an attempt to give a streamlined version, and exposition, of Kolyvagin’s theory, Karl Rubin and I set ourselves the task of

a. proving that the above mapping is an isomorphism. b. giving clean criteria satisfied by the κ_{r}’s which guarantee that the subgroup D_{r}

projects surjectively to CL(F)/p^{N}CL(F) (i.e., for starters it is good to have κ_{r} a unit mod p^{N}-th powers).

c. computing the κ_{r}’s efficiently, and studying their airthmetic properties!

We haven’t completely done any of the three projects listed above, except in very special cases, but these cases are quite illuminating (to me) and I will try to explain some of it, in the course of the seminar.