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

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Kezuka,  Yukako
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Kezuka, Y. (2023). On central L-values and the growth of the 3-part of the Tate-Shafarevich group. International Journal of Number Theory, 19(4), 785-802. doi:10.1142/S1793042123500392.


Cite as: https://hdl.handle.net/21.11116/0000-000D-3248-A
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.