By Barry Mazur

Seeing that their creation by way of Kolyvagin, Euler structures were utilized in a number of very important functions in mathematics algebraic geometry. For a $p$-adic Galois module $T$, Kolyvagin's equipment is designed to supply an higher sure for the scale of the Selmer team linked to the Cartier twin $T^*$. Given an Euler approach, Kolyvagin produces a set of cohomology periods which he calls 'derivative' sessions. it's those by-product periods that are used to certain the twin Selmer team. the place to begin of the current memoir is the remark that Kolyvagin's structures of spinoff sessions fulfill superior interrelations than have formerly been famous. We name a approach of cohomology sessions enjoyable those improved interrelations a Kolyvagin system.We express that the additional interrelations supply Kolyvagin structures an attractive inflexible constitution which in lots of methods resembles (an enriched model of) the 'leading time period' of an $L$-function. by way of using the additional tension we additionally end up that Kolyvagin platforms exist for lots of attention-grabbing representations for which no Euler approach is understood, and extra that there are Kolyvagin structures for those representations which offer upward thrust to distinctive formulation for the scale of the twin Selmer crew, instead of simply higher bounds.