On refined nonvanishing conjectures by Kurihara and Kolyvagin
Francesc Castella, Takamichi Sano
TL;DR
This paper extends refined nonvanishing conjectures of Kurihara and Kolyvagin by developing a determinant-based approach to Iwasawa main conjectures via Selmer complexes, enabling primes of any reduction type and inert primes in the imaginary quadratic field. It builds a Λ-adic Kato Kolyvagin system and relates its p-adic divisibility to Kurihara’s analytic quantities δ_n, establishing equivalences with the det-formulated IMC for the cyclotomic extension; it also develops a parallel, determinant-driven framework for the anticyclotomic setting using Heegner-point Kolyvagin systems and Perrin-Riou theory. The key contributions include descent computations that replace p-adic L-function arguments, a rigidity result for the divisibility index, and explicit formulas linking Selmer groups, Tamagawa factors, and L-values under refined conjectures. The work broadens the validity of refined nonvanishing statements to broader reduction types and inert primes, enhancing the conceptual and technical toolkit for understanding the arithmetic of elliptic curves via Iwasawa theory and Euler systems.
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 $Λ$-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.