On the Modular Function and Its Importance for Arithmetic

Abstract

The modular function

$$ j(\tau ) = \exp ( - 2i\pi \tau ) + 744 + \sum\limits_{n = 1}^\infty {a_n \exp (2i\pi n\tau )} , a_n \in Z, $$

automorphic with respect to the action of SL(2,Z) on the Poincaré upper half plane of those τ ∈ C with positive imaginary part, is very important for the theory of elliptic curves and of modular forms. Indeed, the values of j parametrise the isomorphism classes over C of elliptic curves. In this lecture, we give an introduction to the modular function, and explain in particular a celebrated result of Th. Schneider (1937) which says that the j function takes an algebraic value at an algebraic point τ if and only if τ is imaginary quadratic, that is the associated class of elliptic curves has complex multiplication. We also discuss some more recent results.