AI 中文总结
本文将Burungale-Castella-Skinner两变量主猜想特例化到$\boldsymbol{\rm Z}_p$-线，证明例外线仅有限条，反分圆线为唯一例外线当$E(K)$秩为1时。
AI 中文摘要
设$p>3$为素数，$E/\boldsymbol{\rm Q}$为椭圆曲线，$K$为满足Burungale-Castella-Skinner假设的虚二次域，记$L/K$为唯一的$\boldsymbol{\rm Z}_p^2$-扩张。本文中，我们结合该团队的整系数两变量主猜想与第一作者的特例化公式，得到$L/K$中每条$\boldsymbol{\rm Z}_p$-线上含显式局部因子的特征理想等式。将非挠模的特征理想定义为零，该等式也适用于两变量Perrin-Riou元特例化为零的情形，我们称这类线为例外线并证明其仅有限多条。此外，我们证明分圆线为非例外线且局部因子平凡，反分圆线为例外线；最后将例外线数目以分圆增广阶数为界，且若$K$上的$p$-主Tate-Shafarevich群有限、分圆$p$-进高度配对非退化，则该数目至多为${\rm rank}(E(K))$。特别地，当$E(K)$秩为1时，反分圆线是唯一的例外线。
英文摘要
Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension. In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor. With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero. We call such lines exceptional and prove that only finitely many occur. We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional. Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$. In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.
Comments14 pages
