Divisibility Of Rational Points On Elliptic Curves And Arithmetic Progressions In Polynomial Dynamical Systems

Yesin Elsheikh, Emine Tuğba (2023) Divisibility Of Rational Points On Elliptic Curves And Arithmetic Progressions In Polynomial Dynamical Systems. [Thesis]

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Let K be a number field and E be an elliptic curve described by the Weierstrass equation over K. As a result of 2-descent Theorem on elliptic curves, a criterion for the divisibility-by-2 of a rational point on E is obtained previously. This divisibility criterion has been used to study rational D(q)-m-tuples. In this thesis, we investigate smooth genus one curves C described by a quartic polynomial equation over the rational field Q together with P ∈ C(Q). We give an analogous divisibility-by-2 criterion for rational points in C(Q). We also show how this criterion might be used to study extensions of rational D(q)-quadruples to quintuples. The existence of consecutive squares in arithmetic progression is a classical problem. Fermat claimed that there does not exist an arithmetic progression of four rational squares; and Euler proved this claim. In this thesis, we give a dynamical analogue of Fermat’s Squares Theorem. More precisely, given a polynomial f(x) and a rational point a, we ask how many consecutive squares can be there in the orbit {a,f(a),f 2 (a),...,fn (a),...}? In fact, we give explicit constructions of quadratic polynomials with orbits containing three consecutive squares. Finally, we investigate the question of covering the latter orbit using finitely many arithmetic progressions. We establish a connection between the answer to the latter question and the existence of primitive divisors in the orbit.
Item Type: Thesis
Uncontrolled Keywords: elliptic curves, quartic models, divisibility-by-2, Diophantine quintuples, dynamical systems, polynomial orbits, arithmetic progressions. -- eliptik eğriler, dördüncü dereceden modeller, 2 ile bölünebilme, Diophantine beşlileri, dinamik sistemler, polinom yörüngeleri, aritmetik diziler.
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics
Faculty of Engineering and Natural Sciences
Depositing User: Dila Günay
Date Deposited: 21 Dec 2023 14:20
Last Modified: 21 Dec 2023 14:20
URI: https://research.sabanciuniv.edu/id/eprint/48859

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