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By Barry Mazur

Because their advent via Kolyvagin, Euler platforms were utilized in a number of vital 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 staff linked to the Cartier twin $T^*$. Given an Euler process, Kolyvagin produces a suite of cohomology sessions which he calls 'derivative' sessions. it's those spinoff periods that are used to certain the twin Selmer workforce. the start line of the current memoir is the remark that Kolyvagin's platforms of by-product sessions fulfill more desirable interrelations than have formerly been well-known. We name a method of cohomology sessions pleasing those more advantageous interrelations a Kolyvagin system.We convey that the additional interrelations supply Kolyvagin structures a fascinating inflexible constitution which in lots of methods resembles (an enriched model of) the 'leading time period' of an $L$-function. through utilising the additional pressure we additionally end up that Kolyvagin platforms exist for plenty of fascinating representations for which no Euler method is understood, and additional that there are Kolyvagin platforms for those representations which offer upward push to distinctive formulation for the dimensions of the twin Selmer crew, instead of simply higher bounds.

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4 (and increasing the ring R as necessary, to include the values of ρ), an Euler system for T gives rise to a Kolyvagin system for T ⊗ ρ for every character ρ of finite order of Gal(K/Q). It is not difficult to show that the Kolyvagin systems obtained in this way “interpolate”, in the sense that if ρ ≡ ρ (mod mk ), then the induced Kolyvagin systems coincide in KS((T /mk T ) ⊗ ρ) = KS((T /mk T ) ⊗ ρ ). 6. 4 above, we require that K contains the cyclotomic Zp -extension Q∞ of Q. In other words, the Euler system “extends in the p-direction”, and in particular each class cF is a universal norm from F Q∞ .

Then there is a path in X 0 from n to n . Proof. If n and n are connected by an edge there is nothing to prove. , the localization map HF1 (n) (Q, T¯) → Hf1 (Q , T¯) is zero. 8 we have HF1 (n) (Q, T¯) = HF1 (n ) (Q, T¯). 6 we have c = 0, so c∗ = 1. By definition of X 0 we have HF1 (n)∗ (Q, T¯∗ ) = 0, so we deduce that dimk HF1 (n)∗ (Q, T¯∗ ) = 1. 3. THE SHEAF OF STUB SELMER MODULES 43 ∼ (a) HF1 (n) (Q, T¯) = HF1 (n ) (Q, T¯) − → Hf1 (Qq , T¯), ∼ ¯∗ → H 1 (Qq , T¯∗ ) (b) H 1 ∗ (Q, T ) − F (n) f are both isomorphisms.

3. 3. 2 as follows. 1. The sheaf of stub Selmer modules H = H(T,F ,P) ⊂ HT is the subsheaf of H defined by ∗ ∗ • H (n) = mλ(n,T ) H(n) = mλ(n,T ) HF1 (n) (Q, T ) ⊗ Gn ⊂ H(n) if n ∈ N , • H (e) is the image of H (n) in H(e) = Hs1 (Q , T ) ⊗ Gn under the vertexto-edge map of H, if e is an edge joining n and n , and the vertex-to-edge maps are the restrictions of those of the sheaf H: • H (n) → H (e) is localization at followed by φfs , • H (n ) → H (e) is localization at . 7(iii). Clearly we have Γ(H ) ⊂ Γ(H).

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