Elliptic Curves and Lower Bounds for Class Numbers of Global Fields

Abstract

This thesis is concerned with effective lower bounds for class numbers of imaginary quadratic global fields. Finding such bounds is a classical problem in number theory with applications in e.g. cryptography and the theory of error correcting codes. Griffin and Ono (2020) have derived such bounds for number fields using elliptic curve ideal class pairings which improved on existing effective bounds for many quadratic fields. Here we generalise their construction to global function fields, yielding similar bounds for the class number of imaginary quadratic function fields. Over function fields, multiple constructions of elliptic curves with arbitrarily large rank are known. We examine in particular the class number bounds we obtain using such a family of elliptic curves constructed by Ulmer. Our results do not improve on those in the literature. Finally, as an auxiliary result, we bound the difference between the canonical height and the Weil height on an elliptic curve over a global field. All our results are for characteristic unequal to 2 or 3. We start in the first two chapters with an introduction to global function fields and height functions on elliptic curves over global function fields.