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On the number of rational iterated preimages of the origin under quadratic dynamical systems

Title data

Faber, Xander ; Hutz, Benjamin ; Stoll, Michael:
On the number of rational iterated preimages of the origin under quadratic dynamical systems.
In: International Journal of Number Theory. Vol. 7 (2011) Issue 7 . - pp. 1781-1806.
ISSN 1793-0421
DOI: https://doi.org/10.1142/S1793042111004162

Official URL: Volltext

Project information

Project financing: The first author was supported by a National Science Foundation Postdoctoral Research Fellowship during part of this work.

Abstract in another language

For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article "Uniform bounds on pre-images under quadratic dynamical systems," by two of the present authors and five others, it was shown that the number of rational iterated preimages of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the L-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated preimages is six. We also provide further insight into the geometry of the "preimage curves."

Further data

Item Type: Article in a journal
Refereed: Yes
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - Univ.-Prof. Dr. Michael Stoll
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 28 Nov 2014 11:11
Last Modified: 10 Dec 2014 12:27
URI: https://eref.uni-bayreuth.de/id/eprint/4262