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arXiv · 2110.05521

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

Yukako Kezuka
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Abstract

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.

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Yukako Kezuka. 2024-06-06. On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group. https://doi.org/10.1142/s1793042123500392

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