Geodesic
A Note on Cyclotomic Euler Systems and the Double Complex Method
Canadian journal of mathematics, Tome 55 (2003) no. 4, pp. 673-692

Voir la notice de l'article provenant de la source Cambridge University Press

Let $\mathbb{F}$ be a finite real abelian extension of $\mathbb{Q}$ . Let $M$ be an odd positive integer. For every squarefree positive integer $r$ the prime factors of which are congruent to 1 modulo $M$ and split completely in $\mathbb{F}$ , the corresponding Kolyvagin class ${{k}_{r}}\in {{\mathbb{F}}^{\times }}/{{\mathbb{F}}^{\times M}}$ satisfies a remarkable and crucial recursion which for each prime number $\ell $ dividing $r$ determines the order of vanishing of ${{k}_{r}}$ at each place of $\mathbb{F}$ above $\ell $ in terms of ${{k}_{r/\ell }}$ . In this note we give the recursion a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of $\mathbb{F}$ satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution à la Kubert tensored with $\mathbb{Z}/M\mathbb{Z}$ . Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.
DOI : 10.4153/CJM-2003-028-5
Mots-clés : Primary:, 11R18, 11R23, secondary:, 11R34 
Anderson, Greg W.; Ouyang, Yi. A Note on Cyclotomic Euler Systems and the Double Complex Method. Canadian journal of mathematics, Tome 55 (2003) no. 4, pp. 673-692. doi: 10.4153/CJM-2003-028-5
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[1] [1] Anderson, G. W., A double complex for computing the sign-cohomology of the universal ordinary distribution. Contemp. Math. 224(1999), 1–27. Google Scholar

[2] [2] Das, P., Algebraic Gamma monomials and double coverings of cyclotomic fields. Trans. Amer.Math. Soc. 352(2000), 3557–3594. Google Scholar

[3] [3] Kubert, D. S., The universal ordinary distribution. Bull. Soc. Math. France 107(1979), 179–202. Google Scholar

[4] [4] Lang, S., Cyclotomic Fields I and II, combined 2nd edition. Grad. Texts in Math. 121, Springer Verlag, 1990. Google Scholar

[5] [5] Ouyang, Y., Group cohomology of the universal ordinary distribution. J. Reine Angew. Math. 537(2001), 1–32. Google Scholar

[6] [6] Rubin, K., Euler systems. Ann. of Math. Stud. 147, Princeton University Press, 2000. Google Scholar

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