Let $\mathcal{S}=S_2(\Gamma_0(N) \cap \mathbf{Z} [[ q ]]$ be the set of cusp forms of weight $2$ on $\Gamma_0(N)$ with integral coefficients.

Let $f \in \mathcal{S}$ be a normalized newform, so it has an associated elliptic curve $E_f$ by Eichler-Shimura, and denote by $L$ the orthogonal complement of $f$ in $\mathcal{S}$.

Define the congruence number $D$ of $f$ to be the exponent of the group $\mathcal{S} / (\mathbf{Z}f + L)$. This number is divisible by primes $p$ for which there exists another cusp form $g \in \mathcal{S}$ such that $f \equiv g \mod p$ in an appropriate sense.

There is an elliptic curve $E$ which is isogenous to $E_f$ such that the modular parameterization $\phi_E : X_0(N) \to E$ is of minimal degree. Let $M$ denote this minimal degree.

The following theorem is cited in many papers (e.g. "Bounds for Congruence Primes" by M.R. Marty, but I can give other examples):

Theorem. $M | D$, and the only primes dividing $D/M$ are primes dividing the conductor of $E$.

This theorem is attributed to Mazur and Ribet, but the proof has not been published.

There are many articles which prove and give refinements on the first part of the theorem (that $M | D$). I have been unable to find a proof of the second part of the theorem, that only primes dividing the conductor of $E$ may divide $D/M$.

Does anyone know a proof of the second part of this theorem?


This, and more, is all in Theorem 2.1 of Agashe-Ribet-Stein:



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