Since their introduction by Kolyvagin, 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 the Selmer group associated
to the Cartier dual $T^*$.

Given an Euler system, Kolyvagin produces a collection of
cohomology classes which he calls “derivative” classes. It is
these derivative classes which are used to bound the dual Selmer group.

The starting point of the present memoir is the observation
that Kolyvagin's systems of derivative classes satisfy stronger
interrelations than have previously been recognized. We call a system of
cohomology classes satisfying these stronger interrelations a *Kolyvagin
system*. We show that the extra interrelations give Kolyvagin systems an
interesting rigid structure which in many ways resembles (an enriched version
of) the “leading term” of an $L$-function.

By making use of the extra rigidity we also prove 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.

Readership

Graduate students and research mathematicians interested in
number theory.