Ranks of Elliptic Curves in Families of Quadratic Twists

In this paper we reformulate the question of whether the ranks of the quadratic twists of an elliptic curve over Q are bounded, into the question of the whether certain infinite series converge. Our results were inspired by ideas in a paper of Gouvêa and Mazur [2]. Fix a, b, c ∈ Z such that f(x) = x + ax + bx+ c has 3 distinct complex roots, and let E be the elliptic curve y = f(x). For D ∈ Z−{0}, let E be the elliptic curve Dy = f(x). For every rational number x which is not a root of f(x), there are a unique squarefree integer D and rational number y such that (x, y) ∈ E(Q). For all but finitely many x, the point (x, y) has infinite order on the elliptic curve E. In [2], Gouvêa and Mazur count the number of D that occur this way as x varies, and thereby obtain lower bounds for the number of D in a given range for which E(Q) has positive rank. Building on their idea, in this paper we keep track not only of the number of D which occur, but also how often each D occurs. The philosophy is that the greater the rank of E, the more often D should occur, i.e., curves of high rank should “rise to the top”. By implementing our approach, Rogers [10] found a curve of rank 6 in the family Dy = x − x. Let