×

Ternary Diophantine equations via Galois representations and modular forms. (English) Zbl 1053.11025

Let \(n\) be an integer \(\geq 7\). The authors study ternary Diophantine equations of the form \[ Ax^n+ By^n= Cz^2,\tag{1} \] with \(A\), \(B\) and \(C\) fixed nonzero pairwise coprime integers. They define a primitive solution to the equation (1) as being an element \((x,y,z)\in\mathbb{Z}^3\) satisfying (1), with \(xA\), \(yB\) and \(zC\) nonzero and pairwise coprime. They are interested in the description of the primitive solutions to (1). The case \(A= B= C= 1\) has been treated by H. Darmon and L. Merel in 1997. For various choices of the parameters \(A\), \(B\) and \(C\), the authors describe the set of the primitive solutions to (1). They use arguments of modular nature which are consequences of the works of A. Wiles, K. Ribet and G. Frey on Galois modular representations. As an application of their results, they solve equations of Ramanujan-Nagell type.
They get results in case \(ABC\) is of the shape \(2^\alpha\ell^\beta_1 \ell^\gamma_2\) for prime numbers \(\ell_1\) and \(\ell_2\) less than 80. For instance, they prove that the primitive solutions to the equation \(x^n+ y^n= 2z^2\) correspond to the equality \(xyz= \pm 1\) (this statement has been proved independently by W. Ivorra in [Acta Arith. 108, No. 4, 327–338 (2003; Zbl 1026.11035)]. As a consequence of this result and a research of the integer points on an elliptic curve, the authors deduce the following statement:
let \(x\), \(y\) and \(n\) be natural integers such that \(2x^2- 1= y^n\), \(x\geq 1\), \(y\geq 2\) and \(n\geq 3\). Then, we have \((x,y,n)= (78, 23, 3)\).
They also make remarks about equations of the form \[ x^2+ D= y^n,\tag{2} \] for a fixed natural integer \(D\equiv 7\text{\,mod\,}8\). Combining modular arguments and old results already proved on that equation, they obtain that if \(n\geq 3\) and \(D= 55,95\), the only positive integral solutions \(x\), \(y\) to (2) are given by \((x,y,n,D)= (3,2,6,55)\), \((3, 4, 3, 55)\), \((419, 56, 3, 55)\), \((11, 6, 3, 95)\) and \((529,6, 7, 95)\).

MSC:

11D41 Higher degree equations; Fermat’s equation
11F11 Holomorphic modular forms of integral weight
11G05 Elliptic curves over global fields

Citations:

Zbl 1026.11035
PDFBibTeX XMLCite
Full Text: DOI