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

Official URL: https://risc01.sabanciuniv.edu/record=b3205724

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.