Twists of X(7) and primitive solutions to x²+y³=z⁷

Titelangaben

Poonen, Bjorn ; Schaefer, Edward F. ; Stoll, Michael:
Twists of X(7) and primitive solutions to x²+y³=z⁷.
In: Duke Mathematical Journal. Bd. 137 (2007) Heft 1 . - S. 103-158.
ISSN 1547-7398
DOI: https://doi.org/10.1215/S0012-7094-07-13714-1

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Abstract

We find the primitive integer solutions to x^2+y^3=z^7. 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