Pairs of mod 3 and mod 5 representations arising from elliptic curves
Fix an elliptic curve \(E\) over \(\mathbb{Q}\) and consider the corresponding mod-5-representation \(\rho_5\) of the absolute Galois group \(G\) of \(\mathbb{Q}\). Let \(D\) be a decomposition group for the prime 3. As \(E\) varies (with fixed \(\rho_5\)) we get a series of mod-3-representations of \(D\). The author describes this series rather explicitely, in particular he gives all representations occuring up to twist by the unramified quadratic character of \(D\). The main step of the proof is to reduce to the local situation: if \(E_1\) is an elliptic curve over \(\mathbb{Q}_3\) with mod-5-representation equivalent to \(\rho_5|_D\), then there is an elliptic curve over \(\mathbb{Q}\) whose mod-3-representation (restricted to \(D\)) is isomorphic to that of \(E_1\). The author's result is of interest as \textit{C. Breuil, B. Conrad, F. Diamond} and \textit{R. Taylor} use it in their proof that every elliptic curve over \(\mathbb{Q}\) is modular [J. Am. Math. Soc. 14, 843-939 (2001; Zbl 0982.11033)] by finding (for given \(E\)) an elliptic curve which is known to be modular, has the same mod-5-representation, and a nicer mod-3-representation.
scientific article; zbMATH DE number 1489518