Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1065.11014
Herrmann, Emanuel; Pethö, Attila
$S$-integral points on elliptic curves -- notes on a paper of B. M. M. de Weger. (English)
[J] J. Théor. Nombres Bordx. 13, No. 2, 443-451 (2001). ISSN 1246-7405

{\it B. M. M. de Weger} solved the Diophantine equation $y^2=x^3-228x+848$ (*) [J. Théor. Nombres Bordeaux 9, 281--301 (1997; Zbl 0898.11008)] in $S$-integers with $S=\{2,\infty\}$, by using tools from Algebraic Number Theory and lower estimates for linear forms in complex and $q$-adic logarithms of algebraic numbers. In the present paper a shorter solution for the $S$-solutions to (*) is given, for a larger set $S$, namely $S=\{2,3,5,7,\infty\}$. Now, linear forms in elliptic logarithms are used and the basic tool for computing lower bounds for such linear forms is a theorem of {\it G. Rémond} and {\it F. Urfels} [J. Number Theory 57, No. 1, 133--160 (1996; Zbl 0853.11055)], which applies to elliptic curves of rank at most 2, hence to the elliptic curve defined by (*). It should be noted that the same authors in cooperation with {\it J. Gebel} and {\it H. G. Zimmer} have given an alternative approach to (*), which avoids linear forms in $q$-adic elliptic logarithms [Math. Proc. Camb. Philos. Soc. 127, No. 3, 383--402 (1999; Zbl 0949.11033)], but the bounds resulting there are much larger than those of the present paper which result from the theorem of Rémond and Urfels.
[Nikos Tzanakis (Iraklion)]

MSC 2000:: *11D25 Cubic and quartic diophantine equations
11J86 Linear forms in logarithms; Baker's method
11G05 Elliptic curves over global fields

Keywords: $S$-integral point; elliptic curve linear form in elliptic logarithms; $q$-adic elliptic logarithm

Citations: Zbl 0898.11009; Zbl 0949.11033; Zbl 0853.11055; Zbl 0898.11008

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]