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Fermat-type equations of signature \((13,13,p)\) via Hilbert cuspforms

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Abstract

In this paper we prove that for \(p > 13649\) equations of the form \(x^{13} + y^{13} = Cz^{p}\) have no non-trivial primitive solutions \((a,b,c)\) such that \(13 \not \mid c\) for an infinite family of values for \(C\). Our method consists on relating a solution \((a,b,c)\) to the previous equation to a solution \((a,b,c_1)\) of another Diophantine equation with coefficients in \(\mathbb Q (\sqrt{13})\). Then we attach to \((a,b,c_1)\) a Frey curve \(E_{(a,b)}\) defined over \(\mathbb Q (\sqrt{13})\) that is not a \(\mathbb Q \)-curve. We prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at \(3\). This theorem, in particular, implies modularity of \(E_{(a,b)}\). This enables us to use level lowering results and apply the modular approach via Hilbert cuspforms over \(\mathbb Q (\sqrt{13})\) to prove the non-existence of \((a,b,c_1)\) and, consequently, of \((a,b,c)\).

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Acknowledgments

The authors want to thank John Voight for performing multiple computations that were fundamental to finish this work and John Cremona for providing a list of elliptic curves that was useful to test our strategy. We also want to thank the anonymous referee for many comments and suggestions. The second author is also indebted to Ariel Pacetti for helpful conversations.

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Authors and Affiliations

  1. Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 , Barcelona, Spain

    Luis Dieulefait & Nuno Freitas

Authors
  1. Luis Dieulefait
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  2. Nuno Freitas
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Correspondence to Nuno Freitas.

Additional information

L. Dieulefait supported by project MICINN MTM2009-07024 from MECD, Spain, and ICREA Academia Research Prize.

N. Freitas supported by the scholarship with reference \(SFRH/BD/44283/2008\) from Fundaçao para a Ciência e a Tecnologia, Portugal.

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Dieulefait, L., Freitas, N. Fermat-type equations of signature \((13,13,p)\) via Hilbert cuspforms. Math. Ann. 357, 987–1004 (2013). https://doi.org/10.1007/s00208-013-0920-7

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