On the $p$-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{-3})$

Abstract

We study infinite families of quadratic and cubic twists of the elliptic curve $E = X_0(27)$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the value of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.