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

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.

Theorems & Definitions (17)

• Conjecture 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• proof
• Corollary 2.3