On refined nonvanishing conjectures by Kurihara and Kolyvagin

TL;DR

This paper advances the understanding of refined nonvanishing conjectures by extending previous results to include primes of any reduction type and inert primes in imaginary quadratic fields, using a novel cohomological approach.

Contribution

It introduces a new method to compute the $p$-divisibility index of special Galois cohomology elements, reformulating Iwasawa Main Conjectures via determinants of Selmer complexes.

Findings

Extended results to primes of any reduction type for Kurihara's conjectures.

Included inert primes in the case of Kolyvagin's conjectures.

Developed a new approach based on determinants of Selmer complexes.

Abstract

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 $\Lambda$-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.