By Barry Mazur
Because their creation through Kolyvagin, Euler structures were utilized in a number of vital 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 staff linked to the Cartier twin $T^*$. Given an Euler method, Kolyvagin produces a set of cohomology sessions which he calls 'derivative' sessions. it truly is those spinoff periods that are used to certain the twin Selmer crew. the place to begin of the current memoir is the statement that Kolyvagin's platforms of spinoff sessions fulfill better interrelations than have formerly been well-known. We name a process of cohomology sessions fulfilling those greater interrelations a Kolyvagin system.We exhibit that the additional interrelations supply Kolyvagin structures an engaging inflexible constitution which in lots of methods resembles (an enriched model of) the 'leading time period' of an $L$-function. by way of utilising the additional stress we additionally end up that Kolyvagin platforms exist for lots of attention-grabbing representations for which no Euler approach is understood, and additional that there are Kolyvagin platforms for those representations which provide upward thrust to specific formulation for the dimensions of the twin Selmer workforce, instead of simply higher bounds.
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