Higher Eisenstein Elements, Higher Eichler Formulas and Rank of Hecke Algebras

Let N and p be primes such that p divides the numerator of N-1/12 . In this paper, we study the rank g_p of the completion of the Hecke algebra acting on cuspidal modular forms of weight 2 and level Γ _0(N) at the p -maximal Eisenstein ideal. We give in particular an explicit criterion to know if g_p ≥ 3 , thus answering partially a question of Mazur. In order to study g_p , we develop the theory of higher Eisenstein elements , and compute the first few such elements in four different Hecke modules. This has applications such as generalizations of the Eichler mass formula in characteristic p .