Rational Points On, and the Arithmetic Of, Elliptic Curves: a Tale of Two Books (and an Article)

Our tale begins in 1961, when Professor John Tate was invited by John Solomon to deliver a series of lectures at Haverford College on the subject of “Rational Points on Cubic Curves” [8]. Quoting from the preface to [6], “these lectures, intended for junior and senior mathematics majors, were recorded, transcribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and portions have appeared in various textbooks . . . In view of the recent interest in the theory of elliptic curves . . . it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience.” Several generations of students, myself included, received their first introduction to the arithmetic of elliptic curves from Tate’s Haverford lecture notes, supplemented by his later advanced survey article [9]. After receiving my PhD in 1982 under Tate’s supervision, one of my first teaching assignments was an undergraduate course in abstract algebra at Brown University during the 1989–1990 academic year. The first semester always covered groups, rings, and the start of field theory, while the second semester generally included Galois theory and one or more additional topics. I decided that my added topic would be elliptic curves taught from the Haverford notes, and rather than simply photocopying my own barely legible photocopy, I decided to retype the lectures using (plain) TEX, with some minor editing and the addition of exercises. After teaching the course, it seemed worthwhile to expand the notes and publish them in a more permanent format. So I proposed to Tate that I add material on Lenstra’s elliptic curve factorization