Introduction to Kolyvagin systems

Since their introduction by Kolyvagin in [Ko], Euler systems have been used in several important applications in arithmetic algebraic geometry. For a p-adic Galois module T , Kolyvagin’s machinery is designed to provide an upper bound for the size of a Selmer group associated to the Cartier dual of T . Kolyvagin’s method proceeds in three steps. The first step is to establish an Euler system as input to the machine. The second step gives as intermediate output a new collection of cohomology classes, which Kolyvagin calls “derivative” classes, with coefficients in certain quotient Galois modules. The third step uses this system of derivative classes to obtain an upper bound on the size of the dual Selmer group. In [MR] we showed that Kolyvagin’s systems of derivative classes satisfy even stronger interrelations than had previously been recognized. A system of cohomology classes satisfying these stronger interrelations, which we call a Kolyvagin system, has an interesting rigid structure which in many ways resembles (an enriched version of) the “leading term” of an L-function. See [MR], especially the introduction, for an explanation of what we mean by this. By making use of the extra rigidity, we prove in [MR] that Kolyvagin systems exist for many interesting representations for which no Euler system is known, and further that there are Kolyvagin systems for these representations which give rise to exact formulas for the size of the dual Selmer group, rather than just upper bounds. The purpose of this paper is to present an introduction to the theory of Kolyvagin systems by describing in detail one of its simplest and most concrete settings. Namely, we take the Galois module T to be a twist of the group μpk of p-th roots of unity by a Dirichlet character of conductor p, and then the dual Selmer group is a Galois-eigenspace in the ideal class group of the cyclotomic field Q(μp). For this T there is an Euler system made from cyclotomic units, and we will see that every Kolyvagin system is a multiple of the one produced by Kolyvagin’s machinery, a fact essentially equivalent to Iwasawa’s main conjecture. We hope that removing the extra layers of notation and hypotheses that occur in the general case will make