= Kolyvagin derivative operator for a cyclic group
{c}
{title2=$D=\sum_{i=1}^{m-1}i\sigma^i,\quad(\sigma-1)D=m-N$}
For a cyclic <group> of order $m$ with generator $\sigma$, let $N=\sum_{i=0}^{m-1}\sigma^i$. Subtracting adjacent coefficients and using $\sigma^m=1$ proves the displayed group-ring identity. If $p^r\mid m$, then $(\sigma-1)D\equiv-N\pmod{p^r}$. Together with the norm relations of a <root-of-unity Euler system>, this makes derivative classes invariant modulo p-power powers and permits Kummer descent, with roots-of-unity corrections where necessary. The congruence axiom controls the associated residue and valuation maps.
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