Duke Mathematical Journal

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

Bjorn Poonen, Edward F. Schaefer, and Michael Stoll

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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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Duke Math. J. Volume 137, Number 1 (2007), 103-158.

First available in Project Euclid: 8 March 2007

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Mathematical Reviews number (MathSciNet)

Primary: 11D41: Higher degree equations; Fermat's equation
Secondary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx] 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25] 14G05: Rational points


Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael. Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$. Duke Math. J. 137 (2007), no. 1, 103--158. doi:10.1215/S0012-7094-07-13714-1. https://projecteuclid.org/euclid.dmj/1173373452

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