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Licensed Unlicensed Requires Authentication Published by De Gruyter June 16, 2009

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

  • Luis Dieulefait and Jorge Jiménez Urroz


We solve the diophantine equations x4 + dy2 = zp 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 = zp, 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

© Walter de Gruyter Berlin · New York 2009

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