Un analogue du calcul de Heegner. (An analogue of the Heegner calculus)
type: Item
instance of: scholarly article
title: Un analogue du calcul de Heegner. (An analogue of the Heegner calculus)
zbMATH Open document ID: 0616.14023
DOI: 10.1007/BF01389425
author: Philippe Satgé
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
review text: In this interesting paper, the author extends to cubic twists of an elliptic curve Heegner's modular construction of a point of infinite order on quadratic twists. He shows in particular that if p is an odd prime congruent to 2 (resp. 5) mod 9, then, the elliptic curve \(X^ 3+Y^ 3=2pZ^ 3\) (resp. 2p\({}^ 2Z^ 3)\) has infinitely many rational points. The main point in the proof is the construction of a point on an associated curve of genus \(1,\) whose field of definition has degree prime to 3, and which is obtained by evaluating certain modular forms at quadratic integers with large conductors. The end of the proof follows Selmer's descent argument [see \textit{J. W. S. Cassels}, J. Lond. Math. Soc. 41, 193-291 (1966; Zbl 0138.270)].
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 3997978
zbMATH Keywords: modular forms
zbMATH Keywords: Heegner points
zbMATH Keywords: cubic twists of an elliptic curve
zbMATH Keywords: infinitely many rational points
reviewed by: Daniel Bertrand
MaRDI profile type: Publication
cites work: Q5631241
cites work: Q4078161
cites work: Diophantine Equations with Special Reference To Elliptic Curves
cites work: Heegner points and derivatives of \(L\)-series
cites work: Diophantine analysis and modular forms
cites work: Modular curves and the Eisenstein ideal
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cites work: Groupes de Selmer et corps cubiques. (Selmer group and cubic fields)
cites work: The Diophantine equation \(ax^3+by^3+cz^3=0\)
cites work: Q5625265
description: scientific article; zbMATH DE number 3997978