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Diagonal genus 5 curves, elliptic curves over ℚ(t), and rational diophantine quintuples

Title data

Stoll, Michael:
Diagonal genus 5 curves, elliptic curves over ℚ(t), and rational diophantine quintuples.
In: Acta Arithmetica. Vol. 190 (2019) Issue 3 . - pp. 239-261.
ISSN 0065-1036
DOI: https://doi.org/10.4064/aa180416-4-10

Abstract in another language

The problem of finding all possible extensions of a given rational diophantine quadruple to a rational diophantine quintuple is equivalent to the determination of the set of rational points on a certain curve of genus 5 that can be written as an intersection of three diagonal quadrics in P4. We discuss how one can (try to) determine the set of rational points on such a curve. We apply our approach to the original question in several cases. In particular, we show that Fermat’s diophantine quadruple (1,3,8,120) can be extended to a rational diophantine quintuple in only one way, namely by 777480/8288641.

We then discuss a method that allows us to find the Mordell–Weil group of an elliptic curve E defined over the rational function field Q(t) when E has full Q(t)-rational 2-torsion. This builds on recent results of Dujella, Gusić and Tadić. We give several concrete examples to which this method can be applied. One of these results implies that there is only one extension of the diophantine quadruple (t−1,t+1,4t,4t(4t2−1)) over Q(t).

Further data

Item Type: Article in a journal
Refereed: Yes
Keywords: diophantine quintuples; rational points; elliptic curves
Subject classification: 2010 Mathematics Subject Classification: Primary 11D09, 11G05, 11G30, 14G05, 14H40; Secondary 11Y50, 14G25, 14G27, 14H25, 14H52
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
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 19 Aug 2019 08:09
Last Modified: 19 Aug 2019 08:09
URI: https://eref.uni-bayreuth.de/id/eprint/51926