Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields

Luis Dieulefait; Jorge Jiménez Urroz

doi:10.1515/CRELLE.2009.064

Published by De Gruyter June 16, 2009

Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields

Luis Dieulefait and Jorge Jiménez Urroz

From the journal Journal für die reine und angewandte Mathematik

https://doi.org/10.1515/CRELLE.2009.064

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Abstract

We solve the diophantine equations x⁴ + dy² = z^p for d = 2 and d = 3 and any prime p > 349 and p > 131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermat's Last Theorem, and by Ellenberg in the solution of the equation x⁴ + y² = z^p, and we use ℚ-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations for other values of d.

Received: 2007-03-12

Revised: 2008-04-15

Published Online: 2009-06-16

Published in Print: 2009-August

Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields

Abstract

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