Louis Lecru: Modular forms and L function.

Abstract : The goal of this project is to prove the fact that the L function associated to a cusp form f satisfies a functional equation, and has an analytic continuation. We will first recall some basic results of analysis to justify clearly the convergence of the object we are going to play with. Then we will prove the analytic continuation of the Riemann ζ function, a theorem which necessites the Poisson summation formula. This will give a first example of a particular L function satisfying a functional equation. Then we will present modular forms and cusp forms, and discuss some examples, namely Eisenstein series and the discriminant function. Then we will discuss Dirichlet’s characters which will allow a decomposition of the space of modular form in a direct sum. This philosophy often occurs in mathematics, split a complex object into several smaller ones which are easier to study, and then state more general results about the bigger one. We will also introduce the Hecke operators, which are linear operators on the modular form space, and we will see that they all commute, and so can give a spectral and orthogonal decomposition of the modular space. Then we will discuss Dirichlet’s series, and see the link between modular forms, Dirichlet’s characters and Euler product expansion. And finally go in the direction of the main theorem.