Kolyvagin Systems by Barry Mazur PDF

Algebraic Geometry

By Barry Mazur

On the grounds that their creation via Kolyvagin, Euler structures were utilized in numerous vital purposes in mathematics algebraic geometry. For a $p$-adic Galois module $T$, Kolyvagin's equipment is designed to supply an higher certain for the dimensions of the Selmer crew linked to the Cartier twin $T^*$. Given an Euler method, Kolyvagin produces a set of cohomology sessions which he calls 'derivative' periods. it's those by-product sessions that are used to certain the twin Selmer team. the start line of the current memoir is the commentary that Kolyvagin's structures of by-product periods fulfill greater interrelations than have formerly been famous. We name a process of cohomology sessions enjoyable those more suitable interrelations a Kolyvagin system.We exhibit that the additional interrelations supply Kolyvagin structures an enticing inflexible constitution which in lots of methods resembles (an enriched model of) the 'leading time period' of an $L$-function. via employing the additional pressure we additionally end up that Kolyvagin structures exist for lots of fascinating representations for which no Euler process is understood, and extra that there are Kolyvagin structures for those representations which offer upward push to distinct formulation for the dimensions of the twin Selmer workforce, instead of simply higher bounds.

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This argument proves (iii), and when m = 1 it proves (ii). 10. Suppose that n ∈ N is a core vertex. Then HF1 (n) (Q, T ) and HF1 (n)∗ (Q, T ∗ ) are free R-modules. The ranks of these modules are independent of the choice of core vertex n, and one of them is zero. Proof. 5. 11. The core Selmer rank of T is the rank of the free R-module HF1 (n) (Q, T ) for any core vertex n. We will denote the core rank of T by χ(T ). Similarly we define χ(T ∗ ) = rankR (HF1 (n)∗ (Q, T ∗ )) for any core vertex n.

We give Gal(FC /F ) the structure of an R-module by identifying it with its image in HomR (C, T ). 1)) is equal to T /mT . With this definition, the composition (8) factors through an injection of Rmodules C → HomR[[GQ ]] (Gal(FC /F ), T ). Now using the fact that R is artinian and principal we have HomR[[G ]] (T /mT, T ) = HomR[[G ]] (T /mT, T [m]) ∼ = HomR[[G Q Q Q ]] (T /mT, T /mT ) which is free of rank one over k since T /mT is absolutely irreducible. Hence we see by induction on the length of Gal(FC /F ) that length(C) ≤ length(HomR[[GQ ]] (Gal(FC /F ), T )) ≤ length(Gal(FC /F )) .

Iii) If κ ∈ Γ(S), and if u is a vertex such that κu = 0 and κu generates mi S(u) for some i ∈ Z+ , then κw generates mi S(w) for every vertex w. Proof. For every vertex w fix a surjective path Pw from v to w. If κ ∈ Γ(S) then κw = ψPw (κv ) for every w, so the map fv of (i) is injective. Now fix c ∈ S(v) and for every vertex w define κw = ψPw (c). If S has trivial monodromy then this is independent of the choice of Pw , and defines a global section κ. Thus c is in the image of fv and hence fv is surjective as well.

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