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By Barry Mazur

Given that their creation via Kolyvagin, Euler structures were utilized in numerous very important purposes in mathematics algebraic geometry. For a $p$-adic Galois module $T$, Kolyvagin's equipment is designed to supply an top certain for the scale of the Selmer workforce linked to the Cartier twin $T^*$. Given an Euler method, Kolyvagin produces a suite of cohomology sessions which he calls 'derivative' periods. it truly is those spinoff sessions that are used to sure the twin Selmer team. the start line of the current memoir is the commentary that Kolyvagin's platforms of by-product sessions fulfill more suitable interrelations than have formerly been famous. We name a procedure of cohomology sessions pleasant those enhanced interrelations a Kolyvagin system.We exhibit that the additional interrelations provide Kolyvagin structures an engaging inflexible constitution which in lots of methods resembles (an enriched model of) the 'leading time period' of an $L$-function. via utilizing the additional pressure we additionally turn out that Kolyvagin platforms exist for plenty of fascinating representations for which no Euler process is understood, and additional that there are Kolyvagin structures for those representations which provide upward push to distinctive formulation for the dimensions of the twin Selmer staff, instead of simply top bounds.

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