David Ayotte

David Ayotte

PhD candidate in Mathematics

My research

I love mathematics with a hint of computer experiments. In this page, I explain briefly some of my ongoing/past projects.

Mathematics

Currently, my Ph. D. research is on the arithmetic aspects of Drinfeld modules and Drinfeld modular forms. These objects are analogues of elliptic curves and classical modular forms respectively but for global field of finite characteristic. More precisely, I’m interested in the special values of Drinfeld modular forms of arbitrary rank at CM points. My advisor is Giovanni Rosso.

In the past, for my master thesis, I studied modular forms and their links with mysterious algebraic objects, such as the class number of a number field. In particular, I generalized a result about dihedral congruences for the coefficients of modular forms and did some computations with PARI/gp in order to formulate a conjecture (see the scripts).

During my bachelor degree, I did multiple summer research projects supervised by Antonio Lei. One of them led to the publication of a paper. The goal of this research was to understand some special properties of a specific class of polynomials.

Programming

In my free time, I contribute to the open-source mathematical software SageMath. This software is built on top of multiple already existing open-source computing software (such as NumPy, SciPi, matplotlib, etc).

In relation with my PhD, I’m currently developing a SageMath package named drinfeld_modular_forms. This package implements rings of Drinfeld modular forms and allows to manipulate modular forms as polynomials in the generators of the ring. In the rank two case, I use the theory of non-standard A-expansions of Drinfeld modular forms developed by Lopez and Petrov in order to compute their t-expansions. The documentation for this project is hosted here and the source code is located at this github repo.

Over the course of summer 2021, I participated in the program Google Summer of Code. The goal of my project was to implement the graded ring of quasimodular forms into SageMath. I blogged about my experience here on my website (see the section GSoC). I was mentored by Vincent Delecroix.

Since then, I continue to contribute to SageMath on a voluntary basis. My contributions includes new enhancements, bug fixes and code reviews. Here’s a list of my contributions: issues, PR and commits.

Publications

  • Ayotte, D., Caruso, X., Leudière, A., & Musleh, J. (2023). Drinfeld modules in SageMath. arXiv preprint arXiv:2305.00422.
  • Ayotte, D. (2019), Relations entre le nombre de classes et les formes modulaires, Mémoire de maitrise, hdl.handle.net/20.500.11794/37368
  • Ayotte, D., Lei, A. & Rondy-Turcotte, JC. (2016). On the parity of supersingular Weil polynomials. Arch. Math. 106, 345–353. doi:10.1007/s00013-016-0888-0