MATH 488/489: BD Kim's Project

Ranks of Elliptic Curves

In light of the Birch and Swinnerton-Dyer Conjecture, the rank of an elliptic curve is equal to the order of its Hasse-Weil L-function, and thus finding the ranks of elliptic curves is an interesting problem. You can go in many different directions: You may try to find elliptic curves with large ranks numerically (Elkies, Mestre, et. al.), find the ranks of a fixed elliptic curve as you vary the base fields (Mazur-Rubin, et. al.), and find the average size of the rank with a possible contribution from the Shafarevich-Tate group (Bhargava-Shankar, et. al.) Some of these questions do not require a large degree of advanced knowledge.

References:

M. Bhargava; A. Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, to appear in Annals of Mathematics.
N. D. Elkies, Three lectures on elliptic surfaces and curves of high rank, Lecture notes, Oberwolfach, 2007, arXiv:0709.2908.
B. Mazur; K. Rubin, Ranks of twists of elliptic curves and Hilbert's Tenth Problem. Inventiones Mathematicae 181 (2010) 541-575.
J.-F. Mestre, Construction d'une courbe elliptique de rang ≥ 12, C. R. Acad. Sci. Paris Ser I Math. 295 (1982), 643-644.