On the modularity of certain \(\text{GL}_2(\mathbb{F}_7)\) Galois representations (Q1598404)

Let \(\overline\rho\) be an absolutely irreducible continuous representation of the absolute Galois group of the rationals to \(\text{GL}_2(\mathbb{F}_7)\). Assume that any complex conjugation has image under \(\rho\) with eigenvalues 1 and \(-1\). Then, according to a conjecture of Serre, \(\overline\rho\) should be modular. Here, the author imposes further conditions on \(\overline\rho\) concerning the decomposition groups at 3 and 7. Then he can show modularity of \(\overline\rho\). The proof goes along the lines of \textit{R. Taylor}'s paper [On icosahedral Artin representations. II, Am. J. Math. 125, No. 3, 549--566 (2003; Zbl 1031.11031)]. It uses a concrete description of the moduli space \(X(7)\) of elliptic curves with prescribed level 7 structure in order to find an elliptic curve (defined over some totally real field \(F\)) whose \(G_F\)-representation on \(E[7]\) is equivalent to \(\overline\rho|_{G_F}\) and which thereby assists in finishing the proof.