Innokentij Zotov : The relationship between modular forms modulo p and supersingular modular forms

Abstract: Serre showed in his 1987 letter to Tate that the systems of Hecke eigenvalues arising from modular forms modulo p coincide with those arising from supersingular modular forms. Starting from the classical theory of modular forms, we motivate the geometric interpretation of modular forms as sections of an invertible sheaf on the modular curve. We then describe the theory of modular forms modulo p and treat in detail the proof of Serre's result.