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Abstract

Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve
$x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the
value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $\mathbb{Q}$ at
$s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich
group and the number of distinct prime divisors of $N$ which are inert in the
imaginary quadratic field $K=\mathbb{Q}(\sqrt{-3})$. In the case where
$L(C_N,1)\neq 0$ and $N$ is a product of split primes in $K$, we show that the
order of the Tate-Shafarevich group as predicted by the conjecture of Birch and
Swinnerton-Dyer is a perfect square.