Duke Mathematical Journal
- Duke Math. J.
- Volume 137, Number 1 (2007), 103-158.
Twists of and primitive solutions to
We find the primitive integer solutions to . A nonabelian descent argument involving the simple group of order reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve . To restrict the set of relevant twists, we exploit the isomorphism between and the modular curve and use modularity of elliptic curves and level lowering. This leaves genus curves, whose rational points are found by a combination of methods
Duke Math. J. Volume 137, Number 1 (2007), 103-158.
First available in Project Euclid: 8 March 2007
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Digital Object Identifier
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