Solving Fermat-type equations $x^4 + d y^2 = z^p$ via modular Q-curves over polyquadratic fields

Details Solving Fermat-type equations $x^4 + d y^2 = z^p$ via modular Q-curves over polyquadratic fields Solving Fermat-type equations $x^4 + d y^2 = z^p$ via modular Q-curves over polyquadratic fields

2006-11-21

arxiv.org/abs/math/0611663v2

Mathematics

Number Theory

better lower bounds and other minor changes

Scientific paper

We solve the diophantine equations x^4 + d y^2 = 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 x^4 + y^2 = 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.

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