A Note on Cyclotomic Euler Systems and the Double Complex Method

Abstract

AbstractLetbe a finite real abelian extension of ℚ. LetMbe an odd positive integer. For every squarefree positive integerrthe prime factors of which are congruent to 1 moduloMand split completely in, the corresponding Kolyvagin classsatisfies a remarkable and crucial recursion which for each prime number ℓ dividingrdetermines the order of vanishing of κrat each place ofabove ℓ in terms of κr/ℓ. 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 satis_ed by Kolyvagin classes is the specialization of a universal recursion independent ofsatisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distributionà laKubert tensored with ℤ=Mℤ. 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.