The Fermat-type equations \(x^5+y^5=2z^p\) or \(3z^p\) solved through \(\mathbb Q\)-curves (Q2871191)

The paper under review tackles the issue of studying the Fermat-type Diophantine equation \(x^5+y^5=dz^p \) for a positive integer \(d\) and a prime number \(p\).NEWLINENEWLINENEWLINEThis type of equation has previously been considered by \textit{N. Billerey} [Bull. Aust. Math. Soc. 76, No. 2, 161--194 (2007; Zbl 1148.11016)] and later by \textit{N. Billerey} and \textit{L. V. Dieulefait} [Math. Comput. 79, No. 269, 535--544 (2010; Zbl 1227.11053)] where the authors show that this equation has no non-trivial primitive solutions for infinitely many values of \(d\). More precisely, relying on a classic modular approach using elliptic curves over \(\mathbb{Q}\), their method is able to deal with \(d=2^{\alpha}3^{\beta}5^{\gamma}\) where \(\alpha\geq 2\) or \(d=7\) and \(p\geq 13\) and with \(d=13\) and \(p\geq 19\). But, as explained in [Zbl 1227.11053], the method fails to provide any result for \(d=2\) or \(d=3\). The paper under review then deals with with a new infinite set of values for \(d\), including \(2\) and \(3\), for a set of primes of density \(3/4\).NEWLINENEWLINEWe will say that a triple \((a,b,c) \in \mathbb{Z}^3\) satisfying \(a^5+b^5=dc^p\) is primitive if \(\gcd(a,b)=1\) and that it is trivial if \(|abc|\leq 1\). The two principal results of the paper are the following. Let \(\beta\) be an integer divisible only by primes \(\ell\not \equiv 1\pmod 5\). Then, if \(p>13\) such that \(p \equiv 1\pmod 4\) or \(p\equiv \pm 1\pmod 5\), the equation \(x^5+y^5=2\beta z^p\) has no non-trivial primitive solutions and if \(p>73\) such that \(p \equiv 1\pmod 4\) or \(p \equiv \pm 1\pmod 5\), the equation \(x^5+y^5=3\beta z^p\) has no non-trivial primitive solutions.NEWLINENEWLINENEWLINEThe method relies on generalized modular techniques and more particularly on the multi-Frey technique developed by Siksek. In the paper, the authors are using two elliptic curves attached to a non-trivial primitive solutions, that are actually \(\mathbb{Q}\)-curves, and a new method to eliminate newforms. More precisely, they find twists of these two elliptic curves such that the Weil restrictions decompose as a product of two abelian surfaces of type \(\mathrm{GL}_2\) with endomorphism algebra \(\mathbb{Q}(i)\). Then, they show that a residual representations \(\overline{\rho}\) of one of these abelian surfaces with respect to a prime \(\lambda\) above \(p\) in \(\mathbb{Q}(i)\) is isomorphic, for each of the two elliptic curves, to the residual representation of some newform of level among finitely many explicit possibilities. Finally, using a software, they compute every such possible newforms and reach a contradiction in each case.