| dc.description.abstract | Our results are divided into two main parts, both related to a conjecture by Watkins. In 2002, Watkins conjectured that the rank of an elliptic curve defined over Q is at most the 2-adic valuation of its modular degree. The first part is related to presenting some approaches to Watkins’s conjecture in its original version. We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2, provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction. In addition, we show that this conjecture is satisfied when E is any quadratic twist of an elliptic curve with non-trivial rational 2-torsion and prime power conductor, in particular, for the congruent number elliptic curves. In the second part, we consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semistable elliptic curve over Fq(T) after extending constant scalars and every quadratic twist of a modular elliptic curve over Fq(T) by a polynomial with sufficiently many prime factors satisfy this version of Watkins’s conjecture. Additionally, we prove the analogue of Watkins’s conjecture for a well-known family of elliptic curves with unbounded rank due to Ulmer. In addition, we include a final appendix describing joint work with Hector Pasten [16] on a generalization of the Chabauty-Coleman bound for surfaces. While this is not directly related to the core of the thesis, it is a report on work that was performed during my time as a Ph.D. student. | |
