Elliptic Curves — The Crossroads of Theory and Computation

Abstract

The interplay between theory and computation has been a vital force for progress throughout the long history of the arithmetic of elliptic curves. I have been fortunate to see at fairly close hand two marvellous examples of this interplay. Firstly, I remember my amazement as a student in Canberra and Paris in the mid 1960’s to see the conjecture of Birch and Swinnerton-Dyer evolve from a series of brilliant numerical experiments, which revolutionized arithmetical algebraic geometry. Secondly, I remember my fascination as a young post-doc at Harvard in the beginning of the 1970’s to see John Tate work on a daily basis by always mixing sophisticated theory with hand calculations of numerical examples. Of course, since this time, numerical computations have been greatly changed by the advent of ever faster computers, and the discovery of important practical applications via cryptography. Computational mathematics has rightly become a branch of mathematics in its own right. Nevertheless, the theme I want to stress in my lecture is that the ancient union between theory and computation is as potent a force as ever today. It is my strong personal view that the best computations on elliptic curves are those that lead to new insights for attacking the unsolved theoretical problems. Equally, I firmly believe that no abstract theorem about the arithmetic of elliptic curves is worth its salt unless illuminating numerical examples of it can be given.