Supervisor: Professor Shaun Stevens
My research is in the area of modular forms and Diophantine equations.
For P a non-torsion point on an elliptic curve E defined over a field K, we look at the denominators, B_n, of the coordinates of P by the multiplacion by n map. These form an elliptic divisibility sequence.
Faltings' theorem implies then that there are finitely many points P in E(K) with B_n an S-integer raised to some power f>1. The ring of S-integers arise from some finite subset of archimedean valuations on K.
Faltings' proof is ineffective in the sense that it does not allow us to construct (infinite) families of examples. As such, presently there is no elliptic curve of positive rank for which all such points are known. By using the modular method we hope to make some progress in this area, and thereby allow us to examine the occurrence of primes in the elliptic divisibility sequence B_n.
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