The Birch and Swinnerton-Dyer conjectural formula and visibility Project description

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 Birch and Swinnerton-Dyer conjecture (henceforth abbreviated 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. While much work has been done on the BSD conjecture for elliptic curves (for a summary, see [Sil92, §16] and [IR90, §20.5]), the principal investigator (PI) of this project was one of the first [Aga00] to do calculations regarding this conjecture for abelian varieties of arbitrary dimension. Working with L. Merel and W. Stein, the PI obtained partial results and computational evidence towards this conjecture for certain quotients of Jacobians of modular curves. These quotients form a large class of abelian varieties (which includes elliptic curves) that are of great importance in arithmetic geometry. For example, the proof of the celebrated Fermat’s Last Theorem relies on properties of such quotients. For these abelian varieties, we study the second part of the BSD conjecture, which is a conjectural formula that relates several fundamental invariants of the 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 (e.g., see [Sil92, Chapter X]), and elsewhere. The theory of visibility has recently been used to give new evidence for this conjectural formula. As a concrete example, consider the elliptic curve y2+xy = x3+x2−1154x−15345 (denoted 681B1 in [Cre97]). The BSD conjectural formula predicts that the order of the Shafarevich-Tate group of this elliptic curve is 9. Using the idea of visibility, Cremona and Mazur gave an explicit construction of a subgroup of the Shafarevich-Tate group of order 9 (see [CM00, p. 22], and [AS05, Appendix]). The PI proposes to generalize the techniques used in this example (and several others) with the goal of showing that for certain quotients of Jacobians of modular curves, 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. In fact our methods do more: they give explicit constructions of the conjectured elements of the Shafarevich-Tate group. The PI will also investigate certain other arithmetic invariants appearing in the BSD formula, viz., the orders of the torsion group and the component groups of an abelian variety. These groups are of interest independent of the BSD formula – 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). Thanks to work of Mazur and Emerton, when the level of the modular curve is prime, the torsion and component groups are well understood. The PI proposes