Power Integral Points on Elliptic Curves

Buck, Daniel (2014) Power Integral Points on Elliptic Curves. Doctoral thesis, University of East Anglia.

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Abstract

This thesis looks at some of the modern approaches towards the solution of Diophantine equations, and utilizes them to display the nonexistence of perfect powers occurring in certain types of sequences. In particular we look at the denominator divisibility sequences (Bn) formed by Mordell elliptic curves ED : y2 = x3+D. For the curve-point pair (E−2, P), where E−2 : y2 = x3 −2, and P = (3, 5) is a nontorsion
point, we prove that no term Bn is a perfect 5th power, and we give the explicit bound p � 137 for any term in the associated elliptic denominator sequence to be a perfect power Bn = Zpn for 1 < n < 113762879. We then look at obtaining upper bounds on p for the seventy-two rank 1 Mordell curves in the range |D| < 200 to possess a pth perfect power. This is done by consideration of the finite number of
rational and irrational newforms corresponding to an also finite number of levels of these newforms: in thirty cases we give a bound via examination of both the rational and irrational cases, and for the remaining forty-two cases our bound is merely for the rational case due to computational limitations.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Mia Reeves
Date Deposited: 19 Jun 2015 10:53
Last Modified: 06 Oct 2015 09:48
URI: https://ueaeprints.uea.ac.uk/id/eprint/53372
DOI:

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