A unified scheme of approach to the Ramanujan conjecture

Abstract

The Ramanujan conjecture for modular forms of holomorphic type was proved by Deligne almost half a century ago: the proof, based on his earlier proof of Weil's conjectures, was an achievement of algebraic geometry. We give here a short analytic proof of the Ramanujan-Deligne theorem, and we shall indicate at the end the close analogy of the proof with that of the Ramanujan-Petersson conjecture for Maass forms [8].

This is a complete revision of the former (incorrect at some point) version. The present version follows closely the proof of the Ramanujan-Petersson conjecture for Maass forms arXiv:2001.10956. There are however interesting differences