The Zeros of Elliptic Curve L-functions: Analytic Algorithms with Explicit Time Complexity

Abstract

Elliptic curves are central objects of study in modern-day algebraic number theory. The problem of how to determine the rank of a rational elliptic curve is a difficult one, and at the time of the writing of this thesis an unconditional general method for doing so is not known. It has been known for decades that contingent on the Birch and Swinnerton-Dyer Conjecture, an algorithm to compute rank exists, but this algorithm has unknown time complexity. In the first part of this thesis we prove that, assuming standard conjectures, an effective algorithm exists to compute rank with time complexity that is polynomial in the curve's conductor. This method involves evaluating the L-function of the curve in question, and as such is practical for curves with conductors up to about 10^16 on current computer architecture. The second part of this work addresses the question of what can be done when the conductor is too large for the above method to be practical. To this end we exhibit an analytic method to bound rank from above that doesn't rely on directly evaluating an elliptic curve's L-function, and as such can be used on curves with very large conductors. Because this method involves sums over the imaginary parts of the zeros of an elliptic curve L-function, we also include results concerning the locations thereof, and an exposition of related quantities.