Elliptic modularity for octahedral Galois representations (Q1815212)

Let \(\rho: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\overline{\mathbb{F}}_p)\) be an odd irreducible Galois representation. According to a conjecture of Serre, \(\rho\) should be modular. This is known if \(p=2\) and \(p=3\), due to Hecke and a theorem of \textit{R. P. Langlands} [Base change for \(GL_2\), Ann. Math. Stud. 96, Princeton Univ. Press (1980; Zbl 0444.22007)] and \textit{J. Tunnell} [Bull. Am. Math. Soc., New Ser. 5, 173-175 (1981; Zbl 0475.12016)]. According to the authors, Shepherd-Barron and Taylor recently have also proved Serre's conjecture for icosahedral representations of \(GL_2(\overline{\mathbb{F}}_4)\) and \(GL_2(\overline{\mathbb{F}}_5)\) with some conditions on ramification. The main result of the present paper states: Let \(\rho: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\overline{\mathbb{F}}_3)\) be a Galois representation with cyclotomic determinant and splitting field of degree at least 16. Then there is a modular elliptic curve \(E/\mathbb{Q}\) such that \(\rho\) is isomorphic to the representation \(\rho_{E,3}\) arising from the action of the Galois group on the 3-division points of \(E\). The proof studies in detail the moduli space \({\mathcal {PX}}(3,\overline{\rho})\), classifying isomorphism classes of elliptic curves whose projective Galois representation attached to the 3-torsion module is isomorphic to \(\overline{\rho}\), the projective representation attached to \(\rho\). The modular elliptic curve \(E\) then is produced by proving the existence of a modular point in \({\mathcal {PX}}(3,\overline{\rho})\).