English

On refined nonvanishing conjectures by Kurihara and Kolyvagin

Number Theory 2026-01-22 v1

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 pp-divisibility index of certain special elements in Galois cohomology (the bottom class of a Λ\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.

Keywords

Cite

@article{arxiv.2601.14504,
  title  = {On refined nonvanishing conjectures by Kurihara and Kolyvagin},
  author = {Francesc Castella and Takamichi Sano},
  journal= {arXiv preprint arXiv:2601.14504},
  year   = {2026}
}

Comments

18 pages, comments very welcome!