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Rational points on Erdős–Selfridge superelliptic curves

Part of: Diophantine equations Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  14 July 2016

Michael A. Bennett  and
Samir Siksek
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, CanadaV6T 1Z2 email bennett@math.ubc.ca
Samir Siksek
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email S.Siksek@warwick.ac.uk
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Abstract

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Given $k\geqslant 2$, we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$.

Keywords

superelliptic curvesGalois representationsFrey curvemodularitylevel lowering

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11D41: Higher degree equations; Fermat's equation 11F80: Galois representations 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 11 , November 2016 , pp. 2249 - 2254
Copyright
© The Authors 2016 

References

Bennett, M. A., Bruin, N. B., Győry, K. and Hajdu, L., Powers from products of consecutive terms in arithmetic progression , Proc. Lond. Math. Soc. (3) 92 (2006), 273–306.Google Scholar
Bennett, M. A. and Dahmen, S., Klein forms and the generalized superelliptic equation , Ann. of Math. (2) 177 (2013), 171–239.Google Scholar
Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over ℚ: wild 3-adic exercises , J. Amer. Math. Soc. 14 (2001), 843–939.Google Scholar
Darmon, H. and Merel, L., Winding quotients and some variants of Fermat’s last theorem , J. Reine Angew. Math. 490 (1997), 81–100.Google Scholar
Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power , Illinois J. Math. 19 (1975), 292–301.Google Scholar
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpen , Invent. Math. 73 (1983), 349–366.Google Scholar
Győry, K., Hajdu, L. and Pintér, Á., Perfect powers from products of consecutive terms in arithmetic progression , Compositio Math. 145 (2009), 845–864.Google Scholar
Győry, K., Hajdu, L. and Saradha, N., On the diophantine equation n (n + d)⋯n + (k - 1d) = by l , Canad. Math. Bull. 47 (2004), 373–388.CrossRefGoogle Scholar
Kraus, A., Majorations effectives pour l’équation de Fermat généralisée , Canad. J. Math. 49 (1997), 1139–1161.Google Scholar
Lakhal, M. and Sander, J. W., Rational points on the superelliptic Erdős–Selfridge curve of fifth degree , Mathematika 50 (2003), 113–124.Google Scholar
Martin, G., Dimensions of the spaces of cuspforms and newforms on 𝛤0(N) and 𝛤1(N) , J. Number Theory 112 (2005), 298–331.Google Scholar
Mazur, B., Rational isogenies of prime degree , Invent. Math. 44 (1978), 129–162.Google Scholar
Ribet, K., On modular representations of Gal(ℚ/ℚ) arising from modular forms , Invent. Math. 100 (1990), 431–476.Google Scholar
Sander, J. W., Rational points on a class of superelliptic curves , J. Lond. Math. Soc. (2) 59 (1999), 422–434.Google Scholar
Schinzel, A. and Tijdeman, R., On the equation y m = P (x) , Acta Arith. XXXI (1976), 199–204.Google Scholar
Schoenfeld, L., Sharper bounds for the Chebyshev functions 𝜃(x) and 𝜓(x) II , Math. Comp. 30 (1976), 337–360.Google Scholar
Siksek, S., The modular approach to Diophantine equations , in Explicit Methods in Number Theory: Rational Points and Diophantine Equations, Panoramas et Syntheses, vol. 36, eds Belabas, K., Beukers, F., Gaudry, P., McCallum, W., Poonen, B., Siksek, S., Stoll, M. and Watkins, M. (Société Mathématique de France, Paris, 2012), 151–179.Google Scholar