Structure of Hecke Algebras and Applications

The compact algebra generated by the action of Hecke operators on the space of modular forms modulo a prime p, of level 1 but arbitrary weight, was introduced by Mazur, Tate and Serre in the 70’s in relation to the theory of congruences between modular forms. Until a very recent work of Nicolas and Serre concerning the case p=2, the Krull dimension, let alone the precise structure of this Hecke algebra, was not known in any case. I shall explain a joint work with Chandrashekhar Khare which determines the structure of this Hecke algebra in almost all cases, and give some applications to the asymptotics of the coefficients of modular forms modulo p.