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Recipes to Fermat-type equations of the form \(x^r + y^r =Cz^p\)

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Abstract

We describe a strategy to attack infinitely many Fermat-type equations of signature \((r,r,p)\), where \(r \ge 7\) is a fixed prime and \(p\) is a prime allowed to vary. Indeed, to a solution \((a,b,c)\) of \(x^r + y^r =Cz^p\) we will attach several Frey curves \(E=E_{(a,b)}\) defined over totally real subfields of \(\mathbb {Q}(\zeta _r)\). We prove modularity of all the Frey curves and the exsitence of a constant constant \(M_r\), depending only on \(r\), such that for all \(p>M_r\) the representations \(\bar{\rho }_{E,p}\) are absolutely irreducible. Along the way, we also prove modularity of certain elliptic curves that are semistable at all \(v \mid 3\). Finally, we illustrate our methods by proving arithmetic statements about equations of signature \((7,7,p)\). Among which we emphasize that, using a multi-Frey technique, we show there is some constant \(M\) such that if \(p > M\) then the equation \(x^7 + y^7 = 3z^p\) has no non-trivial primitive solutions.

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Acknowledgments

My greatest thanks go to Luis Dieulefait for our numerous discussions. I also thank Samir Siksek and Panagiotis Tsaknias for their valuable suggestions. I am grateful to John Voight for his computions of Hilbert modular forms that were crucial to this work. I am indebted to Nicolas Billerey, Fred Diamond, Gabor Wiese, Sara Arias-de-Reyna and Xavier Guitart for helpful suggestions and comments. I also thank Gabor Wiese and Fred Diamond for having me as visitor at University of Luxembourg and King’s College London, respectively. Great progress was made on this work during these visits.

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  1. Mathematisches Institut, Universitat Bayreuth, 95440, Bayreuth, Germany

    Nuno Freitas

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  1. Nuno Freitas
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Correspondence to Nuno Freitas.

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This work was supported by the scholarship with reference \(SFRH/BD/44283/2008\) from Fundaçao para a Ciência e a Tecnologia, Portugal and also by a grant from Fundació Ferran Sunyer i Balaguer.

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Freitas, N. Recipes to Fermat-type equations of the form \(x^r + y^r =Cz^p\) . Math. Z. 279, 605–639 (2015). https://doi.org/10.1007/s00209-014-1384-5

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