# My research

### 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 softwares (such as NumPy, SciPi, matplotlib, etc).

My most important contributions includes:

- Implemention of quasimodular forms. See my blog;
- Contributions to the Drinfeld modules implementation. See PR #36325, #35991, #35260, and #35057. See also this paper;
- Currently implementing Drinfeld modular forms. See PR #36538.

For a more detailed list of my contributions:

- Github Issues;
- Open and closed Pull Requests;
- List of merged commits.

### Mathematics

My PhD research was about 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, in my thesis, I proved a function field analogue of a theorem by Shimura about the special values of Drinfeld modular forms at CM points. I also developed a pip-installable SageMath package named `drinfeld-modular-forms`

, see the documentation. As mentionned above, I am currently integrating this package directly in SageMath.

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 undergraduate 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.

## Publications

- Ayotte, D (2023), Arithmetic and computational aspects of modular forms over global fields, PhD thesis
- Ayotte, D., Caruso, X., Leudière, A., & Musleh, J. (2023). Drinfeld modules in SageMath. ACM Communications in Computer Algebra, Vol. 57, No. 2 (ISSAC’23 Software presentation). April 2023. ACM arXiv:2305.00422.
- Ayotte, D. (2019), Relations entre le nombre de classes et les formes modulaires, MSc thesis, 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