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

TL;DR

This paper investigates the $p$-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by $Q( oot 3 hickspace ext{of} hickspace -3)$, establishing bounds on valuations of special $L$-values.

Contribution

It provides explicit lower bounds for the $p$-adic valuations of algebraic parts of $L$-values for specific twists of a CM elliptic curve, matching BSD conjecture predictions.

Findings

Lower bounds for 2-adic valuations in quadratic twists

Lower bounds for 3-adic valuations in cubic twists

Results align with BSD conjecture predictions

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.