Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics

In this thesis, we investigate a certain kind of progressions on elliptic curves, namely, sequences of consecutive squares. Let C be an elliptic curve de{uFB01}ned over Q by the equation y² = f(x), where f(x) {u2208} Q[x] is a polynomial of degree 3 or 4. A sequence of rational points (x₁, y₂) {u2208} C(Q), i =1, 2, . . . , is said to form a sequence of consecutive squares on C if the sequence of x-coordinates, xi, i =1, 2, . . ., consists of consecutive rational squares. We produce an in{uFB01}nite family of elliptic curves de{uFB01}ned by the equation C : y² =x³+A₂+B with a 5-term sequence of consecutive squares and we prove that the rational points in this sequence are independent. Moreover, we display an in{uFB01}nite family of elliptic curves de{uFB01}ned by the equation E : y² =Ax⁴+Bx²+C possessing a 6-term sequence of consecutive squares and prove that these six terms are independent in E(Q). On the other hand, we investigate the rank of elliptic curves with torsion group isomorphic to one of the groups Z₇, Z₉, Z₁₀, Z₁₂ or Z₂{u₀₀D₇}Z₈. For elliptic curves with torsion group isomorphic to Z₇, we produce an in{uFB01}nite family of such elliptic curves with rank 2. Furthermore, for each group of Z₉, Z₁₀, Z₁₂ and Z₂{u₀₀D₇}Z₈, we construct an in{uFB01}nite family of elliptic curves of rank {u2265}1 and torsion group isomorphic to one of the aforementioned groups. All the constructed elliptic curves are parameterized by rational points of an elliptic curve with positive rank

Issued also as CD