A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields

Abstract: 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 $$ \frac{L&#39;(E/K,1)}{Ω^{{\mathrm{cong}}_{\mathbf{f}}} \operatorname{Reg}(E/K)} = # Sha(E/K)[p^{\infty]\prod_{u}} c_u(E/K), $$ where $Ω<sup>{\mathrm{cong}}_{\mathbf{f}}$ is the congruence period of the Hilbert modular form $\mathbf{f}$ associated to $E$ via the modularity conjecture.