Elliptic Functions and Transcendence

Abstract

Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like $\pi$ and $e^\pi$ which are related with the classical exponential function, it turns out that elliptic functions are required (so far -- this should not last forever!) to prove transcendence results and get a better understanding of the situation. First we briefly recall some of the basic transcendence results related with the exponential function. Next, we survey the main properties of elliptic functions that are involved in transcendence theory. We survey transcendence theory of values of elliptic functions, linear independence and algebraic independence. This splitting is somewhat artificial but convenient. Moreover, we restrict ourselves to elliptic functions, even when many results are only special cases of statements valid for abelian functions. A number of related topics are not considered here (e.g. heights, $p$-adic theory, theta functions, diophantine geometry on elliptic curves...).