The PI proposes to investigate three problems in number theory. In the first project the PI proposes to use the arithmetic of elliptic curves and the function fields analog of the classical class number relations to study divisibility and indivisibility of class number of quadratic function fields. The second project is about three-dimensional Artin representations, specificially their group-theoretical classifications, their Artin conductors, their infinity types, and analytic applications. The third problem is about a question of Henri Cohen concerning diagonal Fermat equations over finite fields with no points.
This proposal is in the area of mathematics called Number Theory, which includes the study of integer and rational solutions of diophantine equations. In recent years researchers have discovered many important applications of Number Theory to other areas of science and engineering, most notably in cryptography and telecommunications. For example, Mordell-Weil lattices of elliptic curves have been used to construct error-correcting codes with good properties, and the PI's projects on twisted $L$-functions could have implications in this direction. This interplay between applications and intellectual beauty is very effective in attracting many students to the subject. For example, the PI has recently finished directing the Honors Thesis on coding theory of an undergraduate who was attracted to the subject after he had taken a class from the PI. The PI will continue to foster an atmosphere in his classes that encourages women and minorities to pursue further study in Mathematics.
