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#### Tamagawa number divisibility of central L-values of twists of the Fermat elliptic curve

##### External Resource

https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1183/

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Kezuka_Tamagawa number divisibility of central L-values of twists of the Fermat elliptic curve_2021.pdf

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##### Citation

Kezuka, Y. (2021). Tamagawa number divisibility of central L-values of twists of the
Fermat elliptic curve.* Journal de Théorie des Nombres de Bordeaux,* *33*(3.2),
945-970. doi:10.5802/jtnb.1183.

Cite as: https://hdl.handle.net/21.11116/0000-0009-EEBC-8

##### 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.

$x^3+y^3=N$. 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.