Elliptic Curves Over Function Fields: A Numerical Investigation of Lower Bounds for Ulmer Curves

Abstract

This thesis investigates the ranks of Ulmer curves over the function fields F_p(t), p a prime, with a focus on computational techniques to estimate their group structure. Using SageMath, we implement point-generation algorithms, discriminant checks, and height-pairing computations to produce numerical evidence supporting predicted ranks. We combine brute-force and probabilistic sampling methods, enabling point generation and verification across a range of parameters. These results illustrate the computational challenges in large rank detection, suggest refinements, and contribute to the broader study of function fields.