On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group

Abstract: Given any cube-free integer $\lambda>0$, we study the $3$-adic valuation of the algebraic part of the central $L$-value of the elliptic curve $$X^3+Y^3=\lambda Z^3.$$ We give a lower bound in terms of the number of distinct prime factors of $\lambda$, which, in the case $3$ divides $\lambda$, also depends on the power of $3$ in $\lambda$. This extends an earlier result of the author in which it was assumed that $3$ is coprime to $\lambda$. We also study the $3$-part of the Tate-Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate-Shafarevich group.