Research Fellows Directory
Dr Tim Dokchitser
University of Bristol
One of the most fascinating subjects in mathematics is number theory. It has a long and rich history, from the first attempts by the ancient Greeks to understand what numbers are to the modern methods like transcendental number theory and arithmetic geometry. Especially in recent years, the progress has been enormous, and some of the results in number theory found applications not only in other branches of mathematics, but also in cryptography, computer science and physics.
Some of the most important conjectures in number theory concern the so-called Lfunctions. An L-function is associated to a number-theoretic equation or a system of equations and encodes the most essential information for these equations. Consequently, many problems in number theory and geometry, old and new, admit a natural formulation in terms of these L-functions. It suffices to mention that out of seven Clay Institute Millenium Problems which represent milestones in all of mathematics, two directly concern L-functions (the Birch-Swinnerton-Dyer conjecture and the Riemann hypothesis) and one is strongly related to them (the Hodge conjecture).
My research concentrates on the arithmetic invarints associated to elliptic curves and their L-functions. The work I do relates to the famous conjecture of Birch and Swinnerton-Dyer, which is perhaps the most important unsolved problem remaining in number theory. Two of its consequences, the so-called Selmer parity for elliptic curves over the rationals, and the `weak parity conjecture' over all number fields remained a conjecture for over 40 years. In the last few years Vladimir Dokchitser and myself have finally completed their proof. We hope that the method that we invented (`regulator constants') is something that makes us understand elliptic curves better, and might bring us closer to settling the whole Birch-Swinnerton-Dyer conjecture.