Twists of X(7) and primitive solutions to x^2+y^3=z^7

By Bjorn Poonen, Edward F. Schaefer, Michael Stoll. 2005

107 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/cae78bc53a2b52aadd98d468ba10f52651b45312

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Citations in the corpus

Generalized Fermat equations: A miscellany (2015, background): “Since all such forms are one dimensional and have a3(f) ∈ {0,±2,±3}, it follows that n = 7, contradicting the main result of [64].”
Quadratic Chabauty and rational points, I: p-adic heights (2016, background): “It was introduced by Scharaschkin [40]; further information on the case K = Q can be found, for instance, in [14] and [38].”