Heegner points: the beginnings

Dick Gross and I were invited to talk about Heegner points from a historical point of view, and we agreed that I should talk first, dealing with the period before they became well known. I felt encouraged to indulge in some personal reminiscence of that period, particularly where I can support it by documentary evidence. I was fortunate enough to be working on the arithmetic of elliptic curves when comparatively little was known, but when new tools were just be-coming available, and when forgotten theories such as the theory of automorphic function were being rediscovered. At that time, one could still obtain exciting new results without too much sophisticated apparatus: one was learning exciting new mathematics all the time, but it seemed to be less difficult! To set the stage for Heegner points, one may compare the state of the theory of elliptic curves over the rationals, E/Q for short, in the 1960’s and in the 1970’s; Serre [15] has already done this, but never mind! Lest I forget, I should stress that when I say “elliptic curve ” I will always mean “elliptic curve defined over the rationals”. In the 1960’s, we were primarily interested in the problem of determining the Mordell-Weil group E(Q), though there was much other interesting apparatus waiting to be investigated (cf Cassels ’ report [7]). There was a good theory of descent, Selmer and Tate-Shafarevich groups, and so forth: plenty of algebra. But there was hardly any useful analytic theory, unless the elliptic curve had complex multiplication; E(C) was a complex torus, beautiful maybe, but smooth and featureless, with nothing to get hold of. One could define the L-function LS(E, s) ∼ p/∈S (1 − app−s + p1−2s)−1 (where S is a set of “bad ” primes), and Hasse had conjectured that this is an analytic function with a good functional equation; but most of us could only prove this when the curve had complex multiplication. 1 www.cambridge.org © Cambridge University Pres