📝 SummarXiv

Abstract

We develop a theory of Euler and Kolyvagin systems relative to the Nekov\'{a}\v{r}-Selmer complexes of $p$-adic representations over local complete Gorenstein rings. This is simultaneously finer, and requires weaker hypotheses, than the theory of Kolyvagin systems developed by Mazur and Rubin over discrete valuation rings and then by Sakamoto, Sano and the second author over Gorenstein rings. To illustrate its advantages, we prove new cases of Kato's generalised Iwasawa main conjecture for $\mathbb{Z}_p(1)$ and the $p$-adic Tate modules of rational elliptic curves, as well as of the Quillen-Lichtenbaum conjecture, and we also strengthen existing results on the Birch-Swinnerton-Dyer conjecture for CM elliptic curves.