[proofplan] We reduce the statement to Mazur's prime-level Eisenstein ideal theorem by matching the Hecke algebra and Eisenstein ideal in the theorem statement with Mazur's notation, keeping the proofplan tags on their own lines. [Mazur's theorem](/theorems/985) gives that the Eisenstein quotient of the Hecke algebra is finite cyclic and computes its order as the numerator of $(p-1)/12$. The final step translates divisibility of this index by a prime $\ell$ into the existence of a mod-$\ell$ cuspidal Hecke eigensystem congruent to the Eisenstein eigensystem. [/proofplan] [step:Identify the Eisenstein ideal with Mazur's prime-level Eisenstein ideal] Let $S_2(\Gamma_0(p))$ denote the complex [vector space](/page/Vector%20Space) of weight $2$ cusp forms of level $\Gamma_0(p)$. The phrase "the Hecke algebra acting on the weight $2$ cuspidal quotient of level $p$" means the integral Hecke algebra acting faithfully on the cuspidal part, equivalently the $\mathbb{Z}$-subalgebra \begin{align*} \mathbb{T} \subseteq \operatorname{End}_{\mathbb{C}}(S_2(\Gamma_0(p))) \end{align*} generated by the Hecke operators $T_q$ for primes $q \ne p$ together with the level-$p$ operator, denoted here by $U_p$. This is the same prime-level weight $2$ Hecke algebra used in Mazur's theorem. Define the Eisenstein ideal $I_E \trianglelefteq \mathbb{T}$ by \begin{align*} I_E := \left(\, T_q - (1+q) \text{ for primes } q \ne p,\; U_p - 1 \,\right). \end{align*} This is the standard prime-level Eisenstein ideal in Mazur's notation for level $p$. [guided] The first task is only a comparison of notation. The theorem statement describes the Hecke algebra acting on the weight $2$ cuspidal quotient of level $p$ and says that the Eisenstein ideal is generated by $T_q-(1+q)$ for $q \ne p$ together with the appropriate level-$p$ operator. We make that explicit as follows. Let $S_2(\Gamma_0(p))$ be the complex vector space of weight $2$ cusp forms of level $\Gamma_0(p)$. The cuspidal quotient formulation and the cusp-form formulation give the same integral Hecke algebra in this setting: namely the faithful image of the abstract Hecke algebra on the weight $2$ cuspidal part. We write this algebra as the $\mathbb{Z}$-subalgebra \begin{align*} \mathbb{T} \subseteq \operatorname{End}_{\mathbb{C}}(S_2(\Gamma_0(p))) \end{align*} generated by the Hecke operators $T_q$ for primes $q \ne p$ and by the level-$p$ operator $U_p$. In the prime-level Eisenstein setting, the Eisenstein eigensystem has eigenvalue $1+q$ for $T_q$ when $q \ne p$ and eigenvalue $1$ for $U_p$. Therefore the ideal cutting out this eigensystem is \begin{align*} I_E := \left(\, T_q - (1+q) \text{ for primes } q \ne p,\; U_p - 1 \,\right) \trianglelefteq \mathbb{T}. \end{align*} This is precisely the ideal convention used in Mazur's prime-level Eisenstein ideal theorem. [/guided] [/step] [step:Apply Mazur's computation of the Eisenstein quotient] We use Mazur's prime-level Eisenstein ideal theorem in the following precise form: for the integral Hecke algebra acting on the weight $2$ cuspidal quotient of level $p$, with Eisenstein ideal generated by $T_q-(1+q)$ for primes $q \ne p$ and by $U_p-1$, if $p \ge 5$ is prime, then the quotient ring $\mathbb{T}/I_E$ is finite cyclic as an abelian group and \begin{align*} \#(\mathbb{T}/I_E) = \operatorname{num}\left(\frac{p-1}{12}\right), \end{align*} where $\operatorname{num}(a/b)$ denotes the positive numerator of the rational number $a/b$ written in lowest terms. The hypotheses of Mazur's theorem are exactly satisfied: the integer $p$ is prime with $p \ge 5$, $\mathbb{T}$ is the prime-level weight $2$ Hecke algebra, and $I_E$ is the Eisenstein ideal identified above. Hence $I_E$ has finite index in $\mathbb{T}$, and that index is governed by the numerator of $(p-1)/12$. [guided] We now use the substantive arithmetic input. Mazur's prime-level Eisenstein ideal theorem says: for a prime $p \ge 5$, the quotient of the integral Hecke algebra acting on the weight $2$ cuspidal quotient of level $p$ by the Eisenstein ideal generated by $T_q-(1+q)$ for primes $q \ne p$ and by $U_p-1$ is a finite cyclic abelian group, and its order is \begin{align*} \operatorname{num}\left(\frac{p-1}{12}\right), \end{align*} where $\operatorname{num}(a/b)$ means the numerator after reducing the rational number $a/b$ to lowest terms with positive denominator. We verify the hypotheses before applying it. The theorem statement assumes $p \ge 5$ is prime. The algebra $\mathbb{T}$ in the statement is the Hecke algebra acting on the weight $2$ cuspidal quotient of level $p$, which is the prime-level weight $2$ Hecke algebra in Mazur's theorem. The preceding step identified $I_E$ with the standard Eisenstein ideal generated by $T_q-(1+q)$ for primes $q \ne p$ and $U_p-1$. Therefore Mazur's theorem applies and gives \begin{align*} \#(\mathbb{T}/I_E) = \operatorname{num}\left(\frac{p-1}{12}\right) < \infty. \end{align*} Thus $I_E$ has finite index in $\mathbb{T}$, and the numerical value of that index is controlled exactly by the numerator appearing in the theorem statement. [/guided] [/step] [step:Translate prime divisibility of the index into Eisenstein congruences] Let $N := \operatorname{num}((p-1)/12)$. Let $\ell$ be a prime, let $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements, and let $\overline{\mathbb{F}}_\ell$ denote an [algebraic closure](/page/Algebraic%20Closure) of $\mathbb{F}_\ell$. The condition $\ell \mid N$ is equivalent to \begin{align*} (\mathbb{T}/I_E) \otimes_{\mathbb{Z}} \mathbb{F}_\ell \ne 0, \end{align*} because $\mathbb{T}/I_E$ is finite cyclic of order $N$. This nonzero [tensor product](/page/Tensor%20Product) is equivalent to the existence of a ring homomorphism \begin{align*} \lambda: \mathbb{T} &\to \overline{\mathbb{F}}_\ell \end{align*} whose kernel contains $I_E$. The containment $I_E \subset \ker(\lambda)$ means precisely that \begin{align*} \lambda(T_q) &= 1+q \pmod{\ell} \quad \text{for every prime } q \ne p, \\ \lambda(U_p) &= 1 \pmod{\ell}. \end{align*} Thus the mod-$\ell$ Hecke eigensystem defined by $\lambda$ is congruent to the Eisenstein eigensystem. Because $\mathbb{T}$ is the faithful Hecke algebra acting on the cuspidal quotient, a character $\lambda: \mathbb{T} \to \overline{\mathbb{F}}_\ell$ is realized on a nonzero simultaneous generalized eigenspace after extending the cuspidal quotient modulo $\ell$ to $\overline{\mathbb{F}}_\ell$; choosing an eigenvector in the corresponding simple quotient gives a nonzero mod-$\ell$ cuspidal Hecke eigenclass, equivalently the representative mod-$\ell$ cuspidal eigensystem. Conversely any such congruence gives a nonzero quotient of $(\mathbb{T}/I_E) \otimes_{\mathbb{Z}} \mathbb{F}_\ell$. Therefore the Eisenstein congruence primes in this representative setting are exactly the primes dividing $N$, as claimed. [guided] It remains to explain why the index computation is the same thing as the congruence-prime statement. Define \begin{align*} N := \operatorname{num}\left(\frac{p-1}{12}\right). \end{align*} Let $\ell$ be a prime, let $\mathbb{F}_\ell$ denote the finite field with $\ell$ elements, and let $\overline{\mathbb{F}}_\ell$ denote an algebraic closure of $\mathbb{F}_\ell$. Mazur's theorem gives that $\mathbb{T}/I_E$ is a finite cyclic abelian group of order $N$. For a prime $\ell$, a finite cyclic group of order $N$ has nonzero reduction modulo $\ell$ exactly when $\ell$ divides $N$; equivalently, \begin{align*} \ell \mid N \quad \Longleftrightarrow \quad (\mathbb{T}/I_E) \otimes_{\mathbb{Z}} \mathbb{F}_\ell \ne 0. \end{align*} A nonzero finite-dimensional algebra over $\mathbb{F}_\ell$ admits a homomorphism to an algebraic closure after quotienting by a maximal ideal. Hence the right-hand condition is equivalent to the existence of a ring homomorphism \begin{align*} \overline{\lambda}: (\mathbb{T}/I_E) \otimes_{\mathbb{Z}} \mathbb{F}_\ell &\to \overline{\mathbb{F}}_\ell. \end{align*} Composing $\overline{\lambda}$ with the quotient map from $\mathbb{T}$ gives a Hecke eigensystem \begin{align*} \lambda: \mathbb{T} &\to \overline{\mathbb{F}}_\ell \end{align*} whose kernel contains $I_E$. Since $I_E$ is generated by $T_q-(1+q)$ for $q \ne p$ and by $U_p-1$, this containment is exactly the pair of congruences \begin{align*} \lambda(T_q) &= 1+q \pmod{\ell} \quad \text{for every prime } q \ne p, \\ \lambda(U_p) &= 1 \pmod{\ell}. \end{align*} These are precisely the Eisenstein congruence conditions. We also need to connect the algebra homomorphism to an actual cuspidal object. Since $\mathbb{T}$ acts faithfully on the cuspidal quotient, after tensoring that cuspidal quotient with $\overline{\mathbb{F}}_\ell$, the character $\lambda$ cuts out a nonzero simultaneous generalized eigenspace. Passing to the corresponding simple quotient gives a nonzero mod-$\ell$ cuspidal Hecke eigenclass with eigensystem $\lambda$, so the congruence is realized cuspidally. Conversely, any mod-$\ell$ cuspidal eigensystem satisfying these congruences kills every generator of $I_E$, so it factors through $(\mathbb{T}/I_E) \otimes_{\mathbb{Z}} \mathbb{F}_\ell$, which must then be nonzero. Therefore a prime $\ell$ is an Eisenstein congruence prime exactly when $\ell$ divides $N$. [/guided] [/step]