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.

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 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=(-3). In the case where L(C N ,1)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 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=(-3). Dans la cas où L(C N ,1)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.

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Accepted:
Published online:
DOI: 10.5802/jtnb.1183
Classification: 14H52,  11R23
Keywords: Elliptic curves, Complex multiplication, Tate–Shafarevich group, L-functions
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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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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