[proofplan] The general Langlands reciprocity principle is not proved here as a theorem, because in this breadth it is conjectural. The mathematical assertion proved from the course construction is the holomorphic normalized cuspidal Hecke eigenform case for $GL_2/\mathbb Q$: such an eigenform gives a weakly compatible system of $\lambda$-adic Galois representations. We isolate Deligne's precise input, including the arithmetic Frobenius convention, verify that its hypotheses match the sharpened statement, and then read off the equality of unramified local Euler factors with the Hecke polynomials at all primes outside the stated exceptional set. [/proofplan] [step:Separate the conjectural principle from the classical theorem used here] The first sentence of the statement is a formulation of the Langlands reciprocity principle: it predicts a correspondence between algebraic automorphic representations and compatible systems of Galois representations. This part is not a proven theorem in the stated generality, so it is treated as a principle rather than as a conclusion established in this proof. The theorem-level claim available in the present setting is the holomorphic normalized cuspidal Hecke eigenform case for $GL_2/\mathbb Q$. Let $f \in S_k(\Gamma_0(N), \varepsilon)$ denote a holomorphic normalized cuspidal Hecke eigenform of weight $k \ge 2$, level $N \in \mathbb N$, and nebentypus Dirichlet character $\varepsilon: (\mathbb Z/N\mathbb Z)^\times \to \mathbb C^\times$. For each rational prime $p \nmid N$, let $a_p(f) \in \mathbb C$ denote the eigenvalue of the Hecke operator $T_p$ acting on $f$, and let \begin{align*} E_f := \mathbb Q(a_p(f) : p \nmid N) \subset \mathbb C \end{align*} denote the Hecke field of $f$. For each finite place $\lambda$ of $E_f$, let $E_{f,\lambda}$ denote the completion of $E_f$ at $\lambda$, and let $\ell$ denote the rational prime below $\lambda$. A weakly compatible system here means that, for each rational prime $p \nmid N$, the Frobenius characteristic polynomial at $p$ has coefficients in the common field $E_f$ and is independent of $\lambda$ for all finite places $\lambda$ with $\lambda \nmid p$. [/step] [step:Invoke Deligne's construction for the holomorphic eigenform] By Deligne's construction theorem for holomorphic cuspidal eigenforms, in the form recorded in [Frobenius Polynomial And The Good Euler Factor](/theorems/41), applied to the normalized cuspidal Hecke eigenform $f \in S_k(\Gamma_0(N), \varepsilon)$ with $k \ge 2$, there exists a continuous semisimple representation \begin{align*} \rho_{f,\lambda}: \operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q) &\to GL_2(E_{f,\lambda}) \end{align*} for every finite place $\lambda$ of $E_f$. The theorem is used with the arithmetic Frobenius convention: for each rational prime $p \nmid N\ell$, $\rho_{f,\lambda}$ is unramified at $p$ and the polynomial $\det(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p))$ is computed using arithmetic Frobenius. Its hypotheses are satisfied because $f$ is holomorphic, cuspidal, normalized, a simultaneous eigenvector for the Hecke operators away from $N$, has nebentypus $\varepsilon$, has algebraic Hecke eigenvalues contained in $E_f$, and has weight $k \ge 2$. [guided] The general reciprocity principle becomes a theorem in this proof only after we specialize to the holomorphic modular-form setting. We therefore use Deligne's construction theorem for a normalized cuspidal Hecke eigenform $f \in S_k(\Gamma_0(N), \varepsilon)$ of weight $k \ge 2$, level $N$, and nebentypus Dirichlet character $\varepsilon$. The coefficient field is the number field $E_f \subset \mathbb C$ generated by the Hecke eigenvalues $a_p(f)$ for primes $p \nmid N$. For each finite place $\lambda$ of $E_f$, the completion $E_{f,\lambda}$ is a finite extension of $\mathbb Q_\ell$, where $\ell$ is the rational prime below $\lambda$. Deligne's theorem, in the form recorded in [Frobenius Polynomial And The Good Euler Factor](/theorems/41), applies because $f$ is holomorphic, cuspidal, normalized, a simultaneous Hecke eigenform, has nebentypus $\varepsilon$, has algebraic Hecke eigenvalues in $E_f$, and has weight $k \ge 2$. Its conclusion is the existence of a continuous semisimple representation \begin{align*} \rho_{f,\lambda}: \operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q) &\to GL_2(E_{f,\lambda}). \end{align*} The same theorem fixes the local convention used below: $\operatorname{Frob}_p$ denotes arithmetic Frobenius, and for primes $p \nmid N\ell$ the displayed characteristic polynomial is $\det(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p))$. This is the representation denoted $\rho_{f,\lambda}$ in the statement. [/guided] [/step] [step:Compare Frobenius characteristic polynomials with Hecke polynomials] Let $p$ be a rational prime such that $p \nmid N\ell$, where $\ell$ is the rational prime below $\lambda$. Let $\operatorname{Frob}_p \in \operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$ denote an arithmetic Frobenius element at $p$. The same Deligne construction gives that $\rho_{f,\lambda}$ is unramified at $p$ and satisfies \begin{align*} \det\left(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= 1 - a_p(f)T + \varepsilon(p)p^{k-1}T^2. \end{align*} Define the Galois local Euler factor at $p$ by \begin{align*} L_p(\rho_{f,\lambda}, T) := \det\left(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p)\right)^{-1}. \end{align*} The denominator on the right-hand side is exactly the unramified Hecke polynomial of $f$ at $p$. Hence the Galois local Euler factor of $\rho_{f,\lambda}$ at $p$ matches the automorphic local $L$-factor of $f$ at $p$. [guided] We now verify the promised matching of local factors. Let $p$ be a rational prime with $p \nmid N\ell$, where $\ell$ is the rational prime lying below the finite place $\lambda$ of $E_f$. This exclusion is exactly the usual finite exceptional set: primes dividing the level may be ramified on the automorphic side, and the prime $\ell$ is excluded because the representation is $\ell$-adic. Let $a_p(f) \in E_f$ be the eigenvalue of the Hecke operator $T_p$ acting on $f$. Let $\operatorname{Frob}_p$ be an arithmetic Frobenius element at $p$. Deligne's theorem states not only that $\rho_{f,\lambda}$ exists, but also that it is unramified at every such $p$ and that the characteristic polynomial in the arithmetic Frobenius convention is \begin{align*} \det\left(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= 1 - a_p(f)T + \varepsilon(p)p^{k-1}T^2. \end{align*} We define the Galois local Euler factor by \begin{align*} L_p(\rho_{f,\lambda}, T) := \det\left(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p)\right)^{-1}. \end{align*} This denominator is the Hecke polynomial defining the unramified local factor of the automorphic representation attached to $f$. Therefore the Galois local Euler factor and the automorphic local $L$-factor agree at every prime $p \nmid N\ell$. [/guided] [/step] [step:Conclude weak compatibility and the precise reciprocity realization] Fix a rational prime $p \nmid N$. For every finite place $\lambda$ of $E_f$ with $\lambda \nmid p$, the rational prime below $\lambda$ is different from $p$, so Deligne's comparison applies at $p$ and gives \begin{align*} \det\left(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= 1 - a_p(f)T + \varepsilon(p)p^{k-1}T^2. \end{align*} The polynomial on the right belongs to $E_f[T]$ and does not depend on $\lambda$. Thus the family $(\rho_{f,\lambda})_\lambda$ is weakly compatible: for each prime $p \nmid N$, all finite places $\lambda \nmid p$ give the same Frobenius characteristic polynomial in the common coefficient field $E_f$. Therefore the representations $\rho_{f,\lambda}$ realize the Langlands reciprocity prediction in the precise holomorphic normalized cuspidal Hecke eigenform case of weight $k \ge 2$ on $GL_2/\mathbb Q$. This proves the formal statement while preserving the distinction that the fully general reciprocity principle remains conjectural. [guided] We finish by checking compatibility with the correct order of quantifiers. Fix a rational prime $p$ such that $p \nmid N$. Now let $\lambda$ be any finite place of $E_f$ with $\lambda \nmid p$. If $\ell$ is the rational prime below $\lambda$, then $p \ne \ell$, so $p \nmid N\ell$. Hence Deligne's theorem applies to the pair $(p, \lambda)$ and gives \begin{align*} \det\left(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= 1 - a_p(f)T + \varepsilon(p)p^{k-1}T^2. \end{align*} The right-hand side is built only from the Hecke eigenvalue $a_p(f)$, the nebentypus value $\varepsilon(p)$, the prime $p$, and the weight $k$. All of these are independent of $\lambda$, and the coefficients lie in $E_f$ by the definition of the Hecke field and because $\varepsilon(p)$ is among the algebraic coefficient data of the eigenform. Therefore, for this fixed $p \nmid N$, every $\lambda \nmid p$ gives the same Frobenius characteristic polynomial in $E_f[T]$. This is the weak compatibility condition for the system $(\rho_{f,\lambda})_\lambda$. Combining this compatibility with the local Euler-factor equality proved above shows exactly the classical reciprocity realization asserted in the sharpened theorem statement: holomorphic normalized cuspidal Hecke eigenforms of weight $k \ge 2$ on $GL_2/\mathbb Q$ produce compatible Galois representations with matching unramified local factors. The proof does not claim the still-conjectural general Langlands reciprocity principle. [/guided] [/step]