Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

Abstract

Let N and p be primes such that p divides the numerator of $$\frac{N-1}{12}$$ . 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 $$\Gamma _0(N)$$ at the p-maximal Eisenstein ideal. We give in particular an explicit criterion to know if $$g_p \ge 3$$ , 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.