Iftikhar A. Burhanuddin Research Statement

Astronomers and biologists have had telescopes and microscopes respectively to aid in their research. With the advent of the computer, mathematicians acquired a powerful tool, using which they could generate data, make conjectures and try turning them into theorems — this was the dawn of the golden age of experimental mathematics. My research sits at the crossroads of number theory, algorithms and computation. The problems and conjectures which pique my interest are the ones which have the potential of giving us an insight into the computational complexity of problems and the underlying mathematics. A fondness for the building blocks of modern numbertheoretic cryptography — integer factoring and computing discrete logarithms — also influence the problems I choose to investigate. For instance, my excursion into number theory had a cryptographic motivation, namely trying to understand the Semaev-Smart-Satoh-Araki attack on the elliptic curve discrete logarithm problem [22]. This naturally lead to the question of deciding whether the p-part of a certain group is nontrivial — see §3.2. Proceeding from the above to computing elliptic curve rational torsion and in turn to the BSD conjecture has been a wonderful introduction to a world where conjectures abound and computations are indispensable. This document captures past and current work and outlines my plans for future research. My thesis revolves around the BSD (Birch and Swinnerton-Dyer) conjecture for elliptic curves defined over the rational numbers, a famous problem that has been open for over forty years and one of the seven Millennium Prize problems [20]. This conjecture is considered to be the first nontrivial number theoretic problem put forth as a result of explicit machine computation — in the late ’50s at Cambridge University. The BSD conjecture relates the rank of the Mordell-Weil group, the group of rational points of an elliptic curve, a quantity which seems to be difficult to pin down, to the order of vanishing of the L-series of the elliptic curve at its central point. The problems I investigate in my thesis are motivated by viewing this conjecture and its formula — which is a bridge which connects algebraic objects to complex analytic ones — from a computational perspective. In what follows, I will share with the reader a glimpse of the contents of my thesis on a chapter-by-chapter basis — §2–§5, §8 — and proceed to enumerate questions which arise and problems I plan to work on including ones not related to my thesis. Section 2 states the conjecture of Birch and Swinnerton-Dyer and sets up notation and definitions for the rest of this document. Section 3 presents an efficient randomized algorithm for elliptic curve rational torsion computation. Section 4 concerns a family of certain quartic twists of the elliptic curve y2 = x3−x, which raises interesting questions about integer factoring and heights of rational points. The latter we address in §5 by comparing the situation to the multiplicative group scenario and the Brauer-Siegel theorem. Next, in §6, projects involving Heegner point machinery are discussed. Section 7 states initial work involving modular Galois representations and the method of graphs algorithm. My involvement with the SAGE project is sketched in §8. Finally, §9 elaborates on my career plans. Note. Elliptic curves will be defined over Q; p, q and l will denote primes numbers, unless stated otherwise. In time complexity analysis we present, a time step is a bit operation. Õ notation is defined to be the terms appearing in the Big-Oh notation modulo factors sublinear in the length of input. Everything is joint work with Ming-Deh Huang, unless stated otherwise.