Modular Forms and Related Topics

James一直在从理论和计算的角度研究整半整数权的模形式的傅里叶系数的算术性质。他的研究特别集中在与Birch和Swinnerton-Dyer猜想、Goldfeld猜想和朗特罗特猜想有关的问题上。他已经成功地证明了戈德菲尔德猜想和Birch和Swinnerton-Dyer猜想的弱形式的结果。此外，与K.Ono一起，他成功地证明了一个关于椭圆曲线的二次扭曲的Selmer群的行为的结果。最近，K.Murty和James从数值和统计的角度对Lang和Trotter猜想的某些推广进行了广泛的研究，James计划继续探索模形式和算术几何之间的联系。特别是，他希望更好地理解椭圆曲线的Tate-Shafarevich群的结构，以及整权和半整权的模形式的系数的分布。詹姆斯计划继续对这些问题进行数字和理论调查。

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