发表机构
Korea Institute for Advanced Study; Jeonbuk National University（韩国高等研究院; 全北国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Heegner点研究半稳定椭圆曲线Tate-Shafarevich群p-primary部分的有限性，给出Kolyvagin猜想的简洁证明，并部分恢复Çiperiani--Wiles定理，方法独立于分圆Iwasawa主猜想。
AI 中文摘要
我们研究了Heegner点如何检测半稳定椭圆曲线（定义在有理数上、具有任意秩且存在不可约模p表示的好约化素数p≥5）的Tate-Shafarevich群的p-primary部分的有限性，并部分恢复了Çiperiani--Wiles关于亏格一曲线上的可解点的定理。作为这些结果证明的关键要素，我们还给出了Kolyvagin猜想对于具有超奇异约化的半稳定椭圆曲线（并适当选择虚二次域）的一个简洁证明。我们的方法独立于椭圆曲线超奇异约化的分圆Iwasawa主猜想。
英文摘要
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 $p \geq 5$ 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.
CommentsVersion of January 2026; comments welcome
