The Birch and Swinnerton-Dyer conjectural formula Project description

This proposal falls broadly in the area of number theory and more specifically in arithmetic geometry. It is concerned with a part of the Birch and Swinnerton-Dyer (BSD) conjecture on elliptic curves and abelian varieties. A fundamental problem of number theory is: given a set of polynomial equations with rational coefficients, find all of its rational solutions and investigate their structure. In many cases, the BSD conjecture predicts the existence of such solutions and describes some of their structure without actually finding the solutions. The importance and centrality of this conjecture in mathematics is underscored by the fact that a part of the conjecture was selected as one of the seven millennium prize problems by the Clay Mathematical Institute. We study the second part of the BSD conjecture, which is a formula that relates several fundamental invariants of the elliptic curve or abelian variety. In particular, the conjecture gives a computable formula for the order of the Shafarevich-Tate group of the abelian variety, a mysterious invariant that arises in the calculation of the rational points on the abelian variety, and is an analog of the ideal class group. The theory of visibility has recently been used to give new evidence for this conjectural formula, mainly in specific examples. The PI proposes to use the theory of visibility to show theoretically that the order of the Shafarevich-Tate group predicted by the BSD conjecture divides the actual order, assuming the first part of the BSD conjecture on rank. The PI will also investigate certain other arithmetic invariants appearing in the BSD formula, viz., the orders of the torsion and component groups of an abelian variety. These groups are of independent interest – the torsion group addresses part of the problem of finding rational solutions to equations, and component groups play an important role in the study of abelian varieties (e.g., in Ribet’s proof that the Shimura-Taniyama-Weil conjecture implies Fermat’s last theorem). The PI proposes to extend techniques of Mazur and Emerton in order to characterize the primes that can divide the orders of the torsion and component groups. In the next section (Section 2), we give the precise definitions of the objects we are interested in and give a more technical overview of the proposal. The research part of the proposal consists of two parts: Section 3 concerns the orders of the torsion and component groups and Sections 4–7 are devoted to the application of the theory of visibility to study the Shafarevich-Tate group. The two parts can be read more or less independently of each other (after reading Section 2), although there is some cross-referencing. In any case, the two parts fit together nicely to provide a bigger picture for the BSD formula. While working in arithmetic geometry, the PI is also involved in applications of elliptic curves to cryptography [ALV04], which has broader applications to society. He has taught graduate courses on the topic, and served as an advisor for an undergraduate reading course as well as a Master’s project in the the applications of number theory to cryptography. The PI is currently advising one graduate student in cryptography, and some of the funding will be used to support his research and provide travel money for students to attend conferences. We also plan to use the funds to invite outside speakers to the weekly Algebra seminar at Florida State University.