Newforms and Optimal Quotients

The Hecke algebra

$$\mathbf{T}=\mathbf{Z}[\ldots T_n\ldots]\subset\End(J_0(N))$$

is a commutative ring of endomorphisms of $J_0(N)$ of $\mathbf{Z}$-rank equal to the dimension of $J_0(N)$. The character group $X_{J,p}$ of $J_0(N)$ at $p$ is equipped with a functorial action of  $\mathbf{T}$. The Hecke algebra  $\mathbf{T}$ also acts on the complex vector space $S = S_2(\Gamma_0(N),\mathbf{C})$ of cusp forms.

Let $f$ be a newform, and associate to $f$ the ideal $I_f$ of the Hecke algebra  $\mathbf{T}$ of elements which annihilate $f$. Then $\mathcal{O}_f=\mathbf{T}/I_f$ is an order in the ring of integers of the totally real number field $K_f$ obtained by adjoining the Fourier coefficients of $f$ to  $\mathbf{Q}$. The quotient

$$A_f = J_0(N)/ I_f J_0(N)$$

is an optimal quotient of $J_0(N)$ of dimension equal to $[K_f:\mathbf{Q}]$. As discussed in the previous section, $A_f$ is purely toric at $p$.