发表机构
University of Ottawa（渥太华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为潜在超奇异素数处的CM椭圆曲线建立正负Iwasawa理论，构造局部点以研究Selmer群，构造p-adic L-函数，证明主猜想，并推导Tate-Shafarevich群的渐近公式。
AI 中文摘要
我们为在素数$p\ge5$处具有潜在超奇异约化的CM椭圆曲线发展了正负Iwasawa理论。我们利用Honda--Demchenko理论，应用于局部域上小分歧度的高度二形式群，构造了满足合适“跳跃迹”关系的局部点。这些局部点使我们能够研究CM椭圆曲线在分圆$\mathbb{Z}_p$-扩张上的正负Selmer群，构造相应的正负$p$-adic $L$-函数，并证明联系这些对象的Iwasawa主猜想。作为应用，我们获得了Tate--Shafarevich群的$p$-primary部分的新近增长公式。
英文摘要
We develop a plus and minus Iwasawa theory for CM elliptic curves with potentially supersingular reduction at a prime $p\ge5$. We construct local points that satisfy suitable "jumping trace" relations using the Honda--Demchenko theory applied to height-two formal groups over local fields with small ramification degree. These local points allow us to study plus and minus Selmer groups of CM elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension, to construct the corresponding plus and minus $p$-adic $L$-functions, and to prove an Iwasawa main conjecture relating these objects. As an application, we obtain asymptotic growth formulae for the $p$-primary part of the Tate--Shafarevich groups.
