PreprintCC BY 4.0arXiv
On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture
Abstract
We investigate how Heegner points detect the finiteness of the p-primary part of Tate-Shafarevich groups of semi-stable elliptic curves over the rationals of \emph{arbitrary} rank with any good reduction prime p ≥ 5 with irreducible mod p representation and partially recover Çiperiani--Wiles' theorem on solvable points on genus one curves. As a key ingredient of the proof of these results, we also give a concise proof of Kolyvagin's conjecture for semi-stable elliptic curves with supersingular reduction with a relevant choice of an imaginary quadratic field. Our approach is independent of the cyclotomic Iwasawa main conjecture for elliptic curves with supersingular reduction.
Related
- Variation of Shafarevich-Tate groups in Hida families
- Relative Tate-Shafarevich kernel of a reductive group over a global field
- Second derivatives of p-adic L-functions and the Shafarevich--Tate group of rank-two CM elliptic curves
- Golod--Shafarevich for arithmetic surfaces
- Kolyvagin's conjecture at non-ordinary primes
BibTeX
@misc{kim2026arithmetic,
title = {{On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture}},
author = {Chan-Ho Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15328},
eprint = {2609.15328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15328},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture
AU - Chan-Ho Kim
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2609.15328
UR - https://arxiv.org/abs/2609.15328
AB - We investigate how Heegner points detect the finiteness of the p-primary part of Tate-Shafarevich groups of semi-stable elliptic curves over the rationals of \emph{arbitrary} rank with any good reduction prime p ≥ 5 with irreducible mod p representation and partially recover Çiperiani--Wiles' theorem on solvable points on genus one curves. As a key ingredient of the proof of these results, we also give a concise proof of Kolyvagin's conjecture for semi-stable elliptic curves with supersingular reduction with a relevant choice of an imaginary quadratic field. Our approach is independent of the cyclotomic Iwasawa main conjecture for elliptic curves with supersingular reduction.
ER -