The PI proposes to research algorithms for cohomological automorphic forms and the arithmetic of curves parametrized by congruence semi-arithmetic groups. This proposal interrelates abstract theory and practical computation, linking research with the development of tools and concrete projects which will be integrated into undergraduate and graduate education. In the first proposed activity, the PI will prove that there exists an algorithm to compute the cohomology of an arithmetic group as a Hecke module. The PI will implement the proposed algorithm robustly in a computer algebra system, collect and analyze data, and conjecture and prove results based on the discovered phenomena. In the second activity, the PI will study arithmetic applications of Galois Belyi covers that arise from a modular embedding of a triangle group into a quaternionic unit group. The PI will investigate questions of modularity, parametrization of elliptic curves, and special points arising from this construction. The proposed research will allow explicit investigation of the Langlands correspondence--the deep connection between automorphic forms, algebraic groups, and Galois representations--in both classical and novel settings, exploring exciting new ground.
Classical unsolved problems often serve as the genesis for the formulation of a rich and unified mathematical fabric. Diophantus of Alexandria first sought solutions to algebraic equations in integers almost two thousand years ago. Today, mathematicians recognize that geometric properties often govern the behavior of arithmetic objects. Furthermore, computational tools provide a means to test conjectures and can sometimes furnish partial solutions; at the same time, theoretical advances fuel dramatic improvements in computation. The theory, design, and implementation of algorithms in arithmetic geometry is a burgeoning area, and there are many exciting applications of these methods to diverse fields. The PI further proposes many specific projects at the undergraduate and graduate level, and the proposed algorithmic tools will be integrated into teaching, training, and learning. The PI will continue undergraduate outreach, mentoring of undergraduate and graduate students, and expository writing and extensive lectures aimed at communicating high level mathematics to a student audience. Finally, the PI will design a course on the mathematics of cryptography for teacher leaders who have graduated from the Vermont Mathematics Initiative (VMI)
