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| Main Author: | |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2110.05521 |
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Table of Contents:
- Given any cube-free integer $λ>0$, we study the $3$-adic valuation of the algebraic part of the central $L$-value of the elliptic curve $$X^3+Y^3=λZ^3.$$ We give a lower bound in terms of the number of distinct prime factors of $λ$, which, in the case $3$ divides $λ$, also depends on the power of $3$ in $λ$. This extends an earlier result of the author in which it was assumed that $3$ is coprime to $λ$. 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.