### A note on cyclotomic Euler systems and the double complex method, by Greg W. Anderson and Yi Ouyang

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
$\kappa_r\in{\mathbb F}^{\times}/{\mathbb F}^{\times\, M}$ satisfies a
remarkable and crucial recursion which for each prime number $\ell$
dividing $r$ determines the order of vanishing of $\kappa_r$ at each place
of $\mathbb F$ above $\ell$ in terms of $\kappa_{r/\ell}$. 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 {\em \`{a} 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.

Greg W. Anderson and Yi Ouyang <gwanders@math.umn.edu, youyang@math.toronto.edu>