Solving Fermat-type equations x4 + d y2 = zp via modular Q-curves over polyquadratic fields
Abstract
We solve the diophantine equations x4 + d y2 = 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.
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