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15 March 2007 Twists of X(7) and primitive solutions to x2+y3=z7
Bjorn Poonen, Edward F. Schaefer, Michael Stoll
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Duke Math. J. 137(1): 103-158 (15 March 2007). DOI: 10.1215/S0012-7094-07-13714-1

Abstract

We find the primitive integer solutions to x2+y3=z7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7) and use modularity of elliptic curves and level lowering. This leaves 10 genus 3 curves, whose rational points are found by a combination of methods

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Bjorn Poonen. Edward F. Schaefer. Michael Stoll. "Twists of X(7) and primitive solutions to x2+y3=z7." Duke Math. J. 137 (1) 103 - 158, 15 March 2007. https://doi.org/10.1215/S0012-7094-07-13714-1

Information

Published: 15 March 2007
First available in Project Euclid: 8 March 2007

zbMATH: 1124.11019
MathSciNet: MR2309145
Digital Object Identifier: 10.1215/S0012-7094-07-13714-1

Subjects:
Primary: 11D41
Secondary: 11G10 , 11G18 , 11G30 , 14G05

Rights: Copyright © 2007 Duke University Press

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Vol.137 • No. 1 • 15 March 2007
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