Elliptic curves with complex multiplication: history and perspectives
The theory of complex multiplication on curves is very old and rich, going back at least to Klein. Since then, many authors have been developing the theory, in parallel with quite a heavy load of computations and formulas (by hand!). Soon after Schoof’s 1985 major article, reduction of curves with complex multiplication over finite fields were used to prove the primality of special or general numbers, and the corresponding algorithms are still in use today. As a result, this led to the emergence of the so-called CM-method to build curves with prescribed properties. The talk will present some parts of this history, concentrating on explicit computations and applications of the CM theory to some old and new problems.