2 - Elliptic functions

Abstract

This chapter provides an idea of the circumstances of the birth and rebirth of elliptic functions. The chapter will use the theory of functions of one complex variable—a theory developed around the time of the second birth of elliptic functions—to develop the standard construction due to Weierstrass, which is very fashionable today. The chapter introduces some theorems such as Liouville theorems and Abel's theorem. The chapter also backtracks and gives a second construction, which becomes equivalent to the standard construction over the field of complex numbers. This construction was developed in particular by Rausenberger, and has the advantage of passing meaningfully to the ultrametric framework. It leads to the construction of a “universal elliptic curve” known as the Tate curve Eq.