PhD Oral Exam - David Ayotte, Mathematics

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

This thesis consists of two parts. In the first part, we present a positive characteristic analogue of Shimura's theorem on the special values of modular forms at CM points. More precisely, we show using Hayes' theory of Drinfeld modules that the special value at a CM point of an arithmetic Drinfeld modular form of arbitrary rank lies in the Hilbert class field of the CM field up to a period, independent of the chosen modular form. This is achieved via Pink's realisation of Drinfeld modular forms as sections of a sheaf over the compactified Drinfeld modular curve.

In the second part of the thesis, we present various computational and algorithmic aspects both for the classical theory (over C) and function field theory. First, we implement the rings of quasimodular forms in SageMath and give some applications such as the symbolic calculation of the derivative of a classical modular form. Second, we explain how to compute objects associated with a Drinfeld modules such as the exponential, the logarithm, and Potemine's set of basic J-invariants. Lastly, we present a SageMath package for computing with Drinfeld modular forms and their expansion at infinity using the nonstandard A-expansion theory of López and Petrov.