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

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.

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Item Permalink: https://hdl.handle.net/21.11116/0000-0009-EEBC-8 Version Permalink: https://hdl.handle.net/21.11116/0000-000E-4A9A-2
Genre: Journal Article
Latex : Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve

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© Société Arithmétique de Bordeaux, 2021, tous droits réservés. L’accès aux articles de la revue « Journal de Théorie des Nom- bres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
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https://doi.org/10.48550/arXiv.2003.02772 (Preprint)
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 Creators:
Kezuka, Yukako1, Author           
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Number Theory
 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.

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Language(s): eng - English
 Dates: 2021
 Publication Status: Issued
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 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 2003.02772
DOI: 10.5802/jtnb.1183
 Degree: -

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Title: Journal de Théorie des Nombres de Bordeaux
Source Genre: Journal
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Publ. Info: Université Bordeaux 1
Pages: - Volume / Issue: 33 (3.2) Sequence Number: - Start / End Page: 945 - 970 Identifier: -