In this proposal, the principal investigator (PI) plans to study the second part of the Birch and Swinnerton-Dyer conjecture for modular abelian varieties (including elliptic curves). This conjecture gives a formula that relates certain invariants attached to the abelian variety, including the orders of the Shafarevich-Tate group, the torsion subgroup, and the component groups. To start with, the PI will study the possible cancellations between the orders of the torsion and component groups in the formula, by extending work of Mazur and Emerton. The PI also plans to use the notion of visibility to show that the conjectural order of the Shafarevich-Tate group divides the actual order (in certain cases), assuming the first part of the Birch and Swinnerton-Dyer conjecture. In the course of the work, the PI hopes to prove results that should be of interest independent of the Birch and Swinnerton-Dyer conjecture, including generalizations of several parts of Mazur's seminal paper "Modular curves and the Eisenstein ideal" to non-prime level.
This proposal falls in the area of arithmetic geometry, which concerns the application of algebraic geometry to number theory. Number theory is one of the most ancient branches of mathematics, and deals with questions regarding integers or rational numbers and their properties. Algebraic geometry is the study of solutions of polynomial equations using geometric techniques. Thus arithmetic geometry can be thought of as studying integer or rational number solutions to systems of polynomials. Elliptic curves are certain polynomial equations whose solutions can be added in a certain way, similar to usual addition. The work of the PI will enrich our understanding of elliptic curves; for example, in finding the number of solutions to the equation defining an elliptic curve, provided this number is finite. Elliptic curves are important objects of study in number theory, and have applications both within and outside mathematics. For example, they played an important role in Wiles's proof of Fermat's Last Theoerem, and are being applied to cryptography. In particular, the work of the PI has potential long-term applications to real-life problems.
