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

Yukako Kezuka

doi:10.5802/jtnb.1183

Tamagawa number divisibility of central

L

-values of twists of the Fermat elliptic curve

Yukako Kezuka ¹

¹ Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn, Germany

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 945-970.

Abstract
Résumé

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 $ℚ$ 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 $K = ℚ (\sqrt{- 3})$ . 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.

Étant donné un entier $N > 1$ premier à $3$ , on désigne par $C_{N}$ la courbe elliptique $x^{3} + y^{3} = N$ . On étudie d’abord la valuation $3$ -adique de la partie algébrique de la valeur en $s = 1$ de la fonction $L$ de Hasse–Weil $L (C_{N}, s)$ de $C_{N}$ sur $ℚ$ et on établit une relation entre la partie de $3$ -torsion de son groupe de Tate–Shafarevich et le nombre de diviseurs premiers distincts de $N$ qui sont inertes dans le corps quadratique imaginaire $K = ℚ (\sqrt{- 3})$ . Dans la cas où $L (C_{N}, 1) \neq 0$ et $N$ est un produit de nombres premiers décomposés dans $K$ , on montre que l’ordre du groupe de Tate–Shafarevich, comme prédit par la conjecture de Birch et Swinnerton-Dyer, est un carré parfait.

Received: 2020-03-09
Revised: 2020-08-12
Accepted: 2020-09-18
Published online: 2022-01-26

DOI: 10.5802/jtnb.1183

Classification: 14H52, 11R23
Keywords: Elliptic curves, Complex multiplication, Tate–Shafarevich group, $L$-functions

Author's affiliations:

Yukako Kezuka ¹

¹ Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn, Germany

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Yukako Kezuka},
     title = {Tamagawa number divisibility of central $L$-values of twists of the {Fermat} elliptic curve},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {945--970},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {3.2},
     year = {2021},
     doi = {10.5802/jtnb.1183},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1183/}
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Yukako Kezuka. Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 945-970. doi : 10.5802/jtnb.1183. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1183/

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