Research plan

In this document, I briefly describe the projects that I am currently working on. More technical details may be found in my grant proposal to the National Science Foundation, which was approved, and may be found at: http://www.math.fsu.edu/~agashe/info.html. My past research is summarized in the document “Past research”, which may be found at the same website. My current research is in Arithmetic geometry, which falls under the broad area of Algebra and overlaps most with Number theory and Algebraic geometry. Arithmetic geometry originates in solving certain number theoretic problems, which are usually diophantine in nature, i.e., they concern questions about integer or rational solutions to sets of polynomials with integer or rational coefficients. Perhaps the most famous example of such a problem is Fermat’s last theorem (proved recently by Wiles), which says that if n is an integer greater than two, then there is no solution to xn + yn = zn in non-zero integers. Arithmetic geometry is concerned with building and using tools primarily from algebraic geometry to attack such problems. One also uses techniques and ideas from several areas in algebra, as well as from analysis and topology. Thus in some sense, arithmetic geometry is an interdisciplinary area within pure mathematics. Most of my ongoing research revolves around the Birch and Swinnerton-Dyer (BSD) conjecture, which is one of the main outstanding problems in arithmetic geometry, and part of which is one of the seven Clay millennium prize problems. My research is currently supported by NSF grant no. 0603668. While I describe below the projects that I am currently working on, I would like to emphasize that my research interests are not limited to the projects mentioned below. I am very much interested in other topics in the broader areas of number theory (e.g., work related to the Riemann hypothesis) and algebraic geometry (e.g., bad reduction of modular curves), as well as adjoining areas such as representation theory (e.g., in the context of the Langlands program) and applications (e.g., cryptography, in which I have a publication [ALV04]). The projects mentioned below are the ones where I have some strong leads as of now, and where I expect to prove results in the immediate future. Two of these projects are outside the area of arithmetic geometry, and are mentioned at the very end of this article.