Hilbert Modular Forms

Corresponds to §1.0, §1.1, §1.2 and §1.3. The goal is to introduce the theory of Hilbert modular forms. An interesting result proved by Ribet and used in §1.2.14 to provide the q-expansion principle, is the fact that the geometric fibres of M(c, Γ00(N)) over Spec(Z) are geometrically irreducible. This result was proved by Ribet and has an interesting historical importance: In december1973, Deligne explained to Serre a program to construct abelian p-adic L-functions for totally real number fields using a p-adic theory of Hilbert modular forms. Moreover, Deligne pointed out that this p-adic theory would be possible if enough was known about the moduli space of HBAV (i.e. M(c, Γ00(N)) which was introduced by Rappoport). Finally, Ribet’s result and Rapoport’s thesis
removed the obstacles to Deligne’s program and then the paper [DR80] was written.