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

Title data

Poonen, Bjorn ; Schaefer, Edward F. ; Stoll, Michael:
Twists of X(7) and primitive solutions to x²+y³=z⁷.
In: Duke Mathematical Journal. Vol. 137 (2007) Issue 1 . - pp. 103-158.
ISSN 0012-7094
DOI: https://doi.org/10.1215/S0012-7094-07-13714-1

Official URL: Volltext

Abstract in another language

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

Further data

Item Type: Article in a journal
Refereed: Yes
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - Univ.-Prof. Dr. Michael Stoll
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: No
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 30 Jan 2015 07:53
Last Modified: 12 Feb 2015 11:59
URI: https://eref.uni-bayreuth.de/id/eprint/6128