Modular Forms and Related Topics

Clemson University, Clemson SC

Investigators

Abstract

James has been exploring the arithmetic properties of the Fourier coefficients of modular forms of integral and half-integral weight from a theoretical and a computational point of view. His research has been especially focused on problems related to the Birch and Swinnerton-Dyer conjecture, Goldfeld's conjecture and to the Lang Trotter conjecture. He has been successful in proving results toward Goldfeld's conjecture and toward a weak form of the Birch and Swinnerton-Dyer conjecture. In addition, along with K. Ono he has been successful in proving a result concerning the behavior of the Selmer groups of quadratic twists of an elliptic curve. Recently, K. Murty and James have been conducting extensive investigations into certain generalizations of the Lang and Trotter conjecture from both a numerical and statistical point of view. James is planning to continue to explore the connections between modular forms and arithmetic geometry. In particular, he hopes to better understand the structure of Tate-Shafarevich groups of elliptic curves and the distribution of the coefficients of modular forms of integral and half-integral weight. James plans to proceed with both numerical and theoretical investigations into these matters