Modular Forms for SL2(ℤ)

Abstract

The notion of a modular form is introduced. The dimension of the space of modular forms is determined and Fourier-expansions of Eisenstein series are determined leading to the first number theoretical applications. The Fourier-coefficients of a modular form are fed into a Dirichlet series thus forming the associated L-function, which is shown to extend to an entire function. Hecke operators acting on modular forms are introduced and it is shown that L-functions of Hecke eigenforms admit Euler products. Finally, non-holomorphic Eisenstein series and Maaß-wave forms, the real-analytic counterparts of modular forms, and there L-functions, are introduced.