AI 中文总结
该研究针对CM域中分歧素数处的CM椭圆曲线，构建反分圆岩泽理论的整框架，定义带符号塞尔默群，证明相关岩泽主猜想，填补了无非三角几何形变的对应理论空白。
AI 中文摘要
我们针对在CM域中分歧的素数$p$处的CM椭圆曲线$E$，构建了反分圆岩泽理论的整框架。相关$p$进共轭辛自对偶形变的几何特殊化的$\varepsilon$常数，在反分圆塔的每一层内于$\pm 1$之间等分布，且所有几何特殊化在$p$处均非三角的。我们借助前期工作\textit{\cite{BKNO}}中建立的局部符号分解所导出的拉格朗日局部条件，定义了带符号塞尔默群；核心成果是表述并证明了整岩泽主猜想，该猜想将其中一个带符号塞尔默群与该文中构造的$p$进$L$函数$\mathscr{L}_p(E)$关联起来。我们进一步证明，$\mathscr{L}_p(E)$插值了具有$\varepsilon$常数$+1$的扭的中心赫克$L$值，包括任意无穷型的扭，并将其在具有$\varepsilon$常数$-1$的扭处的值与某些塞尔默元素的$p$进对数相关联。这是首个针对无非三角几何特殊化的$p$进形变，以$p$进$L$函数与塞尔默群表述的岩泽主猜想。证明依赖于我们对底层局部形变的鲁宾型猜想的解决，以及基于高斯《算术研究》中的高斯正负分圆多项式建立的、沿反分圆塔的正负局部点理论。
英文摘要
We propose an integral framework for the anticyclotomic Iwasawa theory of CM elliptic curves $E$ at primes $p$ ramified in the CM field. The $\varepsilon$-constants of the geometric specialisations of the associated $p$-adic conjugate symplectic self-dual deformation equidistribute between $\pm 1$ within every layer of the anticyclotomic tower, and none of the geometric specialisations are trianguline at $p$. We define signed Selmer groups via the Lagrangian local conditions arising from the local sign decomposition established in the prequel \cite{BKNO}, and our central result is the formulation and proof of an integral Iwasawa main conjecture relating one of them to the $p$-adic $L$-function $\mathscr{L}_p(E)$ constructed there. We further show that $\mathscr{L}_p(E)$ interpolates the central Hecke $L$-values of the twists with $\varepsilon$-constant $+1$, including twists of arbitrary infinity type, and relate its values at twists with $\varepsilon$-constant $-1$ to the $p$-adic logarithm of certain Selmer elements. This provides the first Iwasawa main conjecture in terms of a $p$-adic $L$-function and Selmer groups for a $p$-adic deformation admitting no trianguline geometric specialisation. The proofs rest on our resolution of a Rubin-type conjecture for the underlying local deformation, together with a theory of plus/minus local points along the anticyclotomic tower, based on the Gaussian plus/minus cyclotomic polynomials rooted in Gauss' Disquisitiones Arithmeticae.
