Θ(τ, z) and Transcendence

Abstract

At inter affectus aequationum modularium id maxime memorabile ac singulare videor animadvertere, quod eidem omnes aequationi differentiali tertii ordinis satisfaciunt. C. Jacobi, Fundamenta nova, §32 (1829)

The first two sections of this Chapter are devoted to the differential properties of modular forms on which Nesterenko’s theorem on the values of Eisenstein series (see Chapter 3, Theorem 1.1 and [Nes9] is based. The emphasis is on purely modular arguments, but we also recall how to establish them via elliptic functions. Similarly, Section 3 describes modular and elliptic proofs of the algebraic relations which connect their singular values (i.e. values at CM points), and thanks to which [Nes9] becomes a statement on the exponential and the gamma functions.