[proofplan] We prove the statement by invoking the Khare-Wintenberger modularity theorem in its standard formulation: every continuous, absolutely irreducible, odd two-dimensional residual representation of $G_{\mathbb Q}$ over $\overline{\mathbb F}_p$ is modular. The proof checks that the representation in the statement satisfies exactly those hypotheses, including the determinant condition at complex conjugation when $p$ is odd and the standard convention when $p=2$. The cited theorem then supplies a normalized cuspidal Hecke eigenform and a place above $p$, and its conclusion is precisely the asserted semisimple isomorphism. [/proofplan] [step:State the Khare-Wintenberger modularity theorem in the needed form] We use the Khare-Wintenberger modularity theorem in the following form. Let $q$ be a prime number, let $\overline{\mathbb F}_q$ be an [algebraic closure](/page/Algebraic%20Closure) of the finite field $\mathbb F_q$, and let \begin{align*} \sigma: G_{\mathbb Q} &\to GL_2(\overline{\mathbb F}_q) \end{align*} be a continuous, absolutely irreducible representation. If $q$ is odd, assume that $\det(\sigma(c))=-1$ for a complex conjugation element $c \in G_{\mathbb Q}$; if $q=2$, use the standard mod $2$ convention under which the oddness condition is automatic. Then there exist a normalized cuspidal Hecke eigenform $g$, its coefficient field $K_g$, and a finite place $\mu$ of $K_g$ with residue characteristic $q$ such that \begin{align*} \sigma^{\mathrm{ss}} \cong \bar\rho_{g,\mu}^{\mathrm{ss}}. \end{align*} Here $\sigma^{\mathrm{ss}}$ denotes the semisimplification of $\sigma$, and $\bar\rho_{g,\mu}$ denotes the residual Galois representation attached to $g$ at $\mu$. [guided] The external input is a precise theorem, not merely the name of a program. The Khare-Wintenberger modularity theorem says that a two-dimensional residual Galois representation of the absolute [Galois group](/page/Galois%20Group) of $\mathbb Q$ is modular once it satisfies the three structural hypotheses appearing in Serre's conjecture: continuity, absolute irreducibility, and oddness. In the notation needed here, the theorem applies to a map \begin{align*} \sigma: G_{\mathbb Q} &\to GL_2(\overline{\mathbb F}_q), \end{align*} where $q$ is a prime number. For odd $q$, oddness means that a complex conjugation element $c \in G_{\mathbb Q}$ has determinant $-1$ under $\sigma$: \begin{align*} \det(\sigma(c))=-1. \end{align*} For $q=2$, this determinant test cannot distinguish $1$ from $-1$ in $\overline{\mathbb F}_2$, so the theorem is stated with the standard mod $2$ convention. The conclusion of the Khare-Wintenberger theorem is exactly a modularity conclusion. It gives a normalized cuspidal Hecke eigenform $g$, its coefficient field $K_g$, and a finite place $\mu$ of $K_g$ whose residue characteristic is $q$, such that the semisimplified residual representation attached to $g$ agrees with the given representation: \begin{align*} \sigma^{\mathrm{ss}} \cong \bar\rho_{g,\mu}^{\mathrm{ss}}. \end{align*} The semisimplification is included because residual representations attached to eigenforms and arbitrary residual representations are compared up to semisimplification in this formulation. [/guided] [/step] [step:Verify that the given representation satisfies the hypotheses of the theorem] Apply the preceding theorem with $q=p$ and $\sigma=\bar\rho$. The theorem statement gives a map \begin{align*} \bar\rho: G_{\mathbb Q} &\to GL_2(\overline{\mathbb F}_p), \end{align*} and assumes that it is continuous and absolutely irreducible. If $p$ is odd, the statement assumes that $\det(\bar\rho(c))=-1$ for complex conjugation $c$, which is precisely the oddness hypothesis. If $p=2$, the statement explicitly adopts the standard mod $2$ convention, which is precisely the convention used in the cited theorem. Therefore every hypothesis of the Khare-Wintenberger modularity theorem is satisfied by $\bar\rho$. [guided] We now match the objects in the theorem we want to prove with the objects in the Khare-Wintenberger theorem. The cited theorem uses a prime $q$ and a representation $\sigma$. We set \begin{align*} q &= p, & \sigma &= \bar\rho. \end{align*} This is legitimate because the statement begins with a representation \begin{align*} \bar\rho: G_{\mathbb Q} &\to GL_2(\overline{\mathbb F}_p), \end{align*} which has exactly the domain and codomain required by the cited theorem. Next we check the hypotheses one by one. The cited theorem requires continuity; this is one of the stated hypotheses on $\bar\rho$. It requires absolute irreducibility; this is also one of the stated hypotheses on $\bar\rho$. It requires oddness. When $p$ is odd, the statement gives a complex conjugation element $c \in G_{\mathbb Q}$ and assumes \begin{align*} \det(\bar\rho(c))=-1, \end{align*} which is exactly the oddness condition in the cited theorem. When $p=2$, the equality $-1=1$ in $\overline{\mathbb F}_2$ makes that determinant condition vacuous, and the statement explicitly says that the theorem is read with the standard mod $2$ convention. This is the same convention in the cited theorem. Hence no hypothesis is missing. [/guided] [/step] [step:Extract the modular form and identify the residual representation] By the Khare-Wintenberger modularity theorem applied to $\bar\rho$, there exist a normalized cuspidal Hecke eigenform $f$, its coefficient field $K_f$, and a finite place $\lambda$ of $K_f$ with $\lambda \mid p$ such that \begin{align*} \bar\rho^{\mathrm{ss}} \cong \bar\rho_{f,\lambda}^{\mathrm{ss}}. \end{align*} This is exactly the assertion that $\bar\rho$ is modular after semisimplification. Thus the theorem follows. [/step]