Higher Eisenstein elements in weight 2 and prime level

Abstract: In his classical work, Mazur considers the Eisenstein ideal $I$ of the Hecke algebra $\mathbb{T}$ acting on cusp forms of weight $2$ and level $\Gamma_0(N)$ where $N$ is prime. When $p$ is an Eisenstein prime, i.e. $p$ divides the numerator of $\frac{N-1}{12}$, denote by $\mathbf{T}$ the completion of $\mathbb{T}$ at the maximal ideal generated by $I$ and $p$. This is a $\mathbf{Z}_p$-algebra of finite rank $g_p \geq 1$ as a $\mathbf{Z}_p$-module. Mazur asked what can be said about $g_p$. Merel was the first to study $g_p$. Assume for simplicity that $p \geq 5$. Let $\log : (\mathbf{Z}/N\mathbf{Z})^{\times} \rightarrow \mathbf{F}_p$ be a surjective morphism. Then Merel proved that $$g_p \geq 2$$ if and only if $$ \sum_{k=1}^{\frac{N-1}{2}} k \cdot \log(k) \equiv 0 \text{ (modulo } p\text{).}$$ We prove that we have $g_p \geq 3$ if and only if $$ \sum_{k=1}^{\frac{N-1}{2}} k \cdot \log(k) \equiv \sum_{k=1}^{\frac{N-1}{2}} k \cdot \log(k)^2 \equiv 0 \text{ (modulo } p\text{).}$$ We also give a more complicated criterion to know when $g_p \geq 4$. Moreover, we prove higher Eichler formulas. More precisely, let $$H(X) = \sum_{k=0}^{\frac{N-1}{2}} {\frac{N-1}{2} \choose k}^2 \cdot X^k \in \mathbf{F}_N[X]$$ be the classical Hasse polynomial. It is well-known that the roots of $H$ are simple and in $\mathbf{F}_{N^2}^{\times}$. Let $L$ be this set of roots. We prove that $$\sum_{\lambda \in L} \log(H'(\lambda)) \equiv 4\cdot \sum_{k=1}^{\frac{N-1}{2}} k\cdot \log(k) \text{ (modulo }p \text{)}$$ and, if $g_p \geq 2$, $$\sum_{\lambda \in L} \log(H'(\lambda))^2 \equiv 4\cdot \sum_{k=1}^{\frac{N-1}{2}} k\cdot \log(k)^2\text{ (modulo }p \text{)} \text{ .}$$