 
Summary: A NOTE ON CYCLOTOMIC EULER SYSTEMS AND THE DOUBLE
COMPLEX METHOD
GREG W. ANDERSON AND YI OUYANG
ABSTRACT. Let F be a finite real abelian extension of 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 F, the corresponding Kolyvagin class r F×/F×M satisfies a
remarkable and crucial recursion which for each prime number dividing r determines the
order of vanishing of r at each place of F above in terms of r/ . 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 F satisfied by universal Kolyvagin classes in the group cohomology of the
universal ordinary distribution `a la Kubert 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.
1. INTRODUCTION
Let F be a finite real abelian extension of 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 F, the corresponding Kolyvagin class
