Grant: Discovery Grants Program - Individual

My research program is in the area of number theory. This theory is linked to all mathematics, from the purest abstraction to fundamental applications in codes and cryptography. From the theoretical point of view, the last decades have seen spectacular advances around the work of Wiles which led to the solution of Fermat's theorem, as well as advances in analytic number theory. There are also applications in theoretical physics and in cryptographic schemes against quantum computers. Number theory is also a powerful tool in developing computing methods. In this proposal I study a kind of differential equations that are closely related to automorphic forms and are called automorphic differential equations. In nature, they are types of Schrodinger equations with an automorphic potential in the same way that the Lamé equations are Schrodinger equations with an elliptic potential. The automorphic potential is in fact an automorphic form of weight four. Since the space of these forms for the modular group is one-dimensional, our equations depend on a complex parameter. For each class of these parameters, the solutions are completely different in the sense that they possess different kinds of symmetry. However, they all share one common property: They are all equivariant with respect to a representation of the modular group. Thus the classification of these representations yields various types of solutions to the automorphic differential equations. Various tools from complex analysis, algebraic geometry and representation theory are used. This has many applications, the most relevant of which to this proposal is about making progress toward solving a conjecture that states that the zeros of the weight two Eisenstein series are transcendental numbers. This is an important conjecture which turns out to be a particular case of two famous conjectures by Grothendieck/André and by Nesterenko/Bertrand. This research program will also provide high-quality training to HQP allowing them to develop the skills that will make them competitive candidates for academic and non-academic jobs.