全实域的全虚二次扩域上的$p$部分Birch和Swinnerton-Dyer公式
A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields
浏览论文内容
中文总结 AI 辅助
本文在Iwasawa主猜想及其他假设下，证明了全实域的全虚二次扩域上，模半稳定椭圆曲线$E$的Birch和Swinnerton-Dyer公式的$p$部分变体，给出了相关$p$进单位范围内的等式关系。
中文摘要 AI 辅助
本文研究全实域$F$上的模半稳定椭圆曲线$E$，其在基变换至全虚二次扩域$K$后解析秩为1。在假设Iwasawa主猜想及大量其他条件下，我们证明了$K$上Birch和Swinnerton-Dyer公式的$p$部分的一个变体，其中$p$为奇素数。更精确地，在差一个$p$进单位的范围内，有$$ \frac{L'(E/K,1)}{\Omega^{\mathrm{cong}}_{\mathbf{f}} \operatorname{Reg}(E/K)} = \\# Sha(E/K)[p^\infty]\prod_{u} c_u(E/K), $$其中$\Omega^{\mathrm{cong}}_{\mathbf{f}}$是通过模性猜想与$E$关联的Hilbert模形式$\mathbf{f}$的同余周期。
英文摘要
This article studies a modular semistable elliptic curve $E$ over a totally real number field $F$ such that, upon base change to a totally imaginary quadratic extension $K$, it has analytic rank one. Assuming the Iwasawa main conjecture, along with a substantial number of assumptions, we prove a variant of the $p$-part of the Birch and Swinnerton-Dyer formula over $K$, where $p$ is an odd prime. More precisely, up to a $p$-adic unit, we have $$ \frac{L'(E/K,1)}{Ω^{\mathrm{cong}}_{\mathbf{f}} \operatorname{Reg}(E/K)} = \# Sha(E/K)[p^\infty]\prod_{u} c_u(E/K), $$ where $Ω^{\mathrm{cong}}_{\mathbf{f}}$ is the congruence period of the Hilbert modular form $\mathbf{f}$ associated to $E$ via the modularity conjecture.
