Dear MO Community,

Let $X = X_0(N)_{/\mathbb{Q}}$, and $J$ its jacobian. Mazur defines the Eisenstein quotient of $J$, denoted $\widetilde{J}$, as

\[ 0 \rightarrow \gamma_IJ \rightarrow J \rightarrow \widetilde{J} \rightarrow 0. \]

Here, $I$ is the Eisenstein ideal of the Hecke algebra $\mathbb{T}$, $\gamma_I$ is the kernel of the map $\mathbb{T} \rightarrow \lim_{\leftarrow_m} \mathbb{T}/I^m$, and $\gamma_IJ$ is the sub-abelian variety generated by the images $\alpha J$ for $\alpha \in \gamma_I$.

My question is: why is it actually a quotient of $J^{new}_{/\mathbb{Q}}$, the new part of the Jacobian?

Mazur does prove that $\widetilde{J}$ is actually a quotient of $J^- = J/(1 + w)J$; is this the key?

References:

Mazur, B. "Modular curves and the Eisenstein Ideal", Publications MathÃ©matiques de l'IHES, 1977

Mazur, B. "Rational isogenies of prime degree", Inventiones mathematicae, 1978