On refined nonvanishing conjectures by Kurihara and Kolyvagin

In this paper, we extend the results of \cite{BCGS} on refined conjectures by Kurihara and Kolyvagin, allowing primes of any reduction type in the case of Kurihara’s conjectures, and inert primes in the underlying imaginary quadratic field in the case of Kolyvagin’s. The key innovation is a new approach to the computation of the $p$-divisibility index of certain special elements in Galois cohomology (the bottom class of a $\mathrm{\Lambda }$-adic Euler system twisted by a character sufficiently close to the trivial character) based on a reformulation of the Iwasawa Main Conjectures in terms of determinants of Selmer complexes.