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  Central values of L-functions of cubic twists

Roşu, E. (2020). Central values of L-functions of cubic twists. Mathematische Annalen, 378(3-4), 1327-1370. doi:10.1007/s00208-020-02018-0.

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Genre: Journal Article
Abbreviation : Central values of $L$-functions of cubic twists

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This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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 Creators:
Roşu, Eugenia1, 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: We are interested in finding for which positive integers $D$ we have rational solutions for the equation $x^3+y^3=D.$ The aim of this paper is to compute the value of the $L$-function $L(E_D, 1)$ for the elliptic curves $E_D: x^3+y^3=D$. For the case of $p$ prime $p\equiv 1\mod 9$, two formulas have been computed by Rodriguez-Villegas and Zagier. We have computed formulas that relate $L(E_D,
1)$ to the square of a trace of a modular function at a CM point. This offers a criterion for when the integer $D$ is the sum of two rational cubes. Furthermore, when $L(E_D, 1)$ is nonzero we get a formula for the number of
elements in the Tate-Shafarevich group and we show that this number is a square when $D$ is a norm in $\mathbb{Q}[\sqrt{-3}]$.

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Language(s): eng - English
 Dates: 2020
 Publication Status: Issued
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 Rev. Type: Peer
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Title: Mathematische Annalen
  Abbreviation : Math. Ann.
Source Genre: Journal
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Publ. Info: Springer
Pages: - Volume / Issue: 378 (3-4) Sequence Number: - Start / End Page: 1327 - 1370 Identifier: -