[proofplan] We invoke Deligne's construction theorem for normalized cuspidal Hecke eigenforms of weight at least $2$, in the arithmetic Frobenius convention used in the statement. The theorem constructs a two-dimensional $\lambda$-adic representation from the $f$-isotypic part of étale cohomology of modular curves with the algebraic local system of weight $k-2$, and the Eichler-Shimura congruence identifies arithmetic Frobenius characteristic polynomials away from $N\ell$ with Hecke polynomials. Passing to the semisimplification gives the representation required in the statement while preserving the trace and determinant at unramified arithmetic Frobenius elements. [/proofplan]

[step:Apply Deligne's construction theorem to the given eigenform]Let $K_f$ denote the coefficient field generated over $\mathbb{Q}$ by the Hecke eigenvalues of $f$ and the values of the nebentypus character $\varepsilon$, and let $K_{f,\lambda}$ be the completion of $K_f$ at the finite place $\lambda$. Let $G_{\mathbb{Q}}:=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute [Galois group](/page/Galois%20Group) of $\mathbb{Q}$, equipped with its profinite topology. Deligne's $\ell$-adic construction theorem for modular eigenforms, together with the Eichler-Shimura relation in the arithmetic Frobenius convention, states that if $f \in S_k(\Gamma_1(N),\varepsilon)$ is a normalized cuspidal Hecke eigenform with $k\geq 2$ and $\lambda\mid \ell$ is a finite place of $K_f$, then there exists a continuous representation \begin{align*} r_{f,\lambda}:G_{\mathbb Q}&\to GL_2(K_{f,\lambda}) \end{align*} obtained from the $f$-isotypic quotient of étale cohomology of modular curves with the algebraic local system of weight $k-2$. Here the algebraic local system of weight $k-2$ is the local system associated to the $(k-2)$-nd symmetric power representation \begin{align*} \operatorname{Sym}^{k-2}:GL_2&\to GL(\operatorname{Sym}^{k-2} K_{f,\lambda}^2) \end{align*} on homogeneous polynomials of degree $k-2$ in two variables. The hypotheses of Deligne's theorem are exactly the hypotheses in the statement: $f$ is normalized, cuspidal, a Hecke eigenform, has weight $k\geq 2$, has level $\Gamma_1(N)$ and nebentypus $\varepsilon$, and $\lambda$ lies over the rational prime $\ell$.[/step]

[guided]The first point is to identify precisely which deep input is being used and which Frobenius convention it uses. Deligne's theorem, combined with the Eichler-Shimura relation in the arithmetic Frobenius convention, applies to a normalized cuspidal Hecke eigenform \begin{align*} f&\in S_k(\Gamma_1(N),\varepsilon) \end{align*} with $k\geq 2$ and to a finite place $\lambda$ of its coefficient field $K_f$ lying above a rational prime $\ell$. Here $K_f$ is the field generated over $\mathbb{Q}$ by the Hecke eigenvalues of $f$ together with the values of $\varepsilon$, so the determinant values $\varepsilon(p)p^{k-1}$ lie in $K_f$. These are exactly the assumptions of the theorem statement, so no additional hypothesis is being introduced. The output of Deligne's theorem is a continuous two-dimensional $\lambda$-adic representation \begin{align*} r_{f,\lambda}:G_{\mathbb Q}&\to GL_2(K_{f,\lambda}), \end{align*} where $G_{\mathbb Q}:=\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$ is the absolute Galois group with its profinite topology and $K_{f,\lambda}$ is the completion of $K_f$ at $\lambda$. The cohomological construction uses the algebraic local system of weight $k-2$. Concretely, this is the local system attached to the representation \begin{align*} \operatorname{Sym}^{k-2}:GL_2&\to GL(\operatorname{Sym}^{k-2} K_{f,\lambda}^2), \end{align*} where $\operatorname{Sym}^{k-2} K_{f,\lambda}^2$ is the [vector space](/page/Vector%20Space) of homogeneous polynomials of degree $k-2$ in two variables. This defines the notation $\operatorname{Sym}^{k-2}$ before it is used in the construction.[/guided]

[step:Read the Frobenius characteristic polynomial from the Eichler-Shimura relation]For a prime $p\nmid N\ell$, Deligne's theorem gives that $r_{f,\lambda}$ is unramified at $p$. Let $\operatorname{Frob}_p\in G_{\mathbb Q}$ denote an arithmetic Frobenius element at $p$, well-defined up to conjugacy in the quotient by inertia. Since $r_{f,\lambda}$ is unramified at $p$, the conjugacy class of $r_{f,\lambda}(\operatorname{Frob}_p)$ is well-defined in $GL_2(K_{f,\lambda})$. The Eichler-Shimura relation in Deligne's construction, with this arithmetic Frobenius convention, identifies the characteristic polynomial of $r_{f,\lambda}(\operatorname{Frob}_p)$ with the Hecke polynomial of $f$ at $p$: \begin{align*} \det\bigl(1-X\,r_{f,\lambda}(\operatorname{Frob}_p)\bigr) &=1-a_p(f)X+\varepsilon(p)p^{k-1}X^2. \end{align*} For a two-dimensional linear operator $A$ over $K_{f,\lambda}$, the identity \begin{align*} \det(1-XA)&=1-\operatorname{tr}(A)X+\det(A)X^2 \end{align*} shows, after comparing coefficients, that \begin{align*} \operatorname{tr}(r_{f,\lambda}(\operatorname{Frob}_p))&=a_p(f),\\ \det(r_{f,\lambda}(\operatorname{Frob}_p))&=\varepsilon(p)p^{k-1}. \end{align*}[/step]

[guided]Now we use the part of Deligne's theorem that compares Galois action with Hecke action. Fix a prime $p\nmid N\ell$. The condition $p\nmid N\ell$ is the good-prime condition: $p\nmid N$ means the modular curve and the Hecke correspondence have good reduction at $p$, and $p\ne \ell$ is required for the $\lambda$-adic étale cohomology representation to have a well-defined unramified Frobenius action at $p$. Deligne's theorem says that $r_{f,\lambda}$ is unramified at such a prime $p$. Therefore an inertia subgroup at $p$ acts as the identity, and the image of an arithmetic Frobenius element $\operatorname{Frob}_p$ is well-defined up to conjugacy. Trace and determinant are conjugacy-invariant functions, so their values at $r_{f,\lambda}(\operatorname{Frob}_p)$ are well-defined. The Eichler-Shimura relation identifies the arithmetic Frobenius characteristic polynomial with the Hecke polynomial attached to $f$: \begin{align*} \det\bigl(1-X\,r_{f,\lambda}(\operatorname{Frob}_p)\bigr) &=1-a_p(f)X+\varepsilon(p)p^{k-1}X^2. \end{align*} For any two-dimensional $K_{f,\lambda}$-linear operator $A$, its characteristic identity in the variable $X$ is \begin{align*} \det(1-XA)&=1-\operatorname{tr}(A)X+\det(A)X^2. \end{align*} Applying this with $A=r_{f,\lambda}(\operatorname{Frob}_p)$ and comparing the coefficients of $X$ and $X^2$ gives \begin{align*} \operatorname{tr}(r_{f,\lambda}(\operatorname{Frob}_p))&=a_p(f),\\ \det(r_{f,\lambda}(\operatorname{Frob}_p))&=\varepsilon(p)p^{k-1}. \end{align*}[/guided]

[step:Pass to the semisimplification without changing Frobenius traces or determinants]Define \begin{align*} \rho_{f,\lambda}:G_{\mathbb Q}&\to GL_2(K_{f,\lambda}) \end{align*} to be the semisimplification of $r_{f,\lambda}$. Semisimplification of a finite-dimensional continuous representation over $K_{f,\lambda}$ is semisimple by definition. Since trace and determinant depend only on the Jordan-Hölder constituents, $\rho_{f,\lambda}$ and $r_{f,\lambda}$ have the same trace and determinant at every element of $G_{\mathbb Q}$. Continuity is preserved for this semisimplification in the usual $\lambda$-adic representation category, and unramifiedness at $p\nmid N\ell$ is also preserved because inertia acts as the identity on $r_{f,\lambda}$ and hence on its subquotients. Therefore, for every prime $p\nmid N\ell$, \begin{align*} \operatorname{tr}(\rho_{f,\lambda}(\operatorname{Frob}_p))&=a_p(f),\\ \det(\rho_{f,\lambda}(\operatorname{Frob}_p))&=\varepsilon(p)p^{k-1}. \end{align*} Thus $\rho_{f,\lambda}$ is a continuous semisimple representation, unramified at every prime $p\nmid N\ell$, with the required trace and determinant formulas.[/step]

[guided]The representation produced directly by cohomology is usually denoted here by $r_{f,\lambda}$. The theorem statement asks for a semisimple representation, so we define \begin{align*} \rho_{f,\lambda}:G_{\mathbb Q}&\to GL_2(K_{f,\lambda}) \end{align*} to be the semisimplification of $r_{f,\lambda}$. This operation has three properties needed here. First, $\rho_{f,\lambda}$ is semisimple by construction. Second, trace and determinant are unchanged by semisimplification: for each $g\in G_{\mathbb Q}$, the characteristic polynomial of the action on a finite-dimensional representation equals the product of the characteristic polynomials on its Jordan-Hölder constituents, and the semisimplification has exactly the same constituents. Hence \begin{align*} \operatorname{tr}(\rho_{f,\lambda}(g))&=\operatorname{tr}(r_{f,\lambda}(g)),\\ \det(\rho_{f,\lambda}(g))&=\det(r_{f,\lambda}(g)). \end{align*} Third, unramifiedness is preserved: if $p\nmid N\ell$, inertia at $p$ acts as the identity on $r_{f,\lambda}$, so it acts as the identity on every subquotient and therefore on the semisimplification. Applying these facts to $g=\operatorname{Frob}_p$ and using the formulas already obtained for $r_{f,\lambda}$ gives \begin{align*} \operatorname{tr}(\rho_{f,\lambda}(\operatorname{Frob}_p))&=a_p(f),\\ \det(\rho_{f,\lambda}(\operatorname{Frob}_p))&=\varepsilon(p)p^{k-1}. \end{align*} This is exactly the existence, continuity, semisimplicity, unramifiedness, trace formula, and determinant formula required in the statement.[/guided]