On ordinary forms and ordinary Galois representations (Q2765410)

The following is one result of this paper. Let \(\rho: \text{Gal}(\overline{\mathbb{Q}}/{\mathbb{Q}}) \rightarrow \text{GL}_2(\overline{\mathbb{F}}_p)\) be an odd representation which is ordinary, i.e.\ its restriction to some decomposition group at \(p\) is reducible and has an unramified one-dimensional quotient. Assume that \(\rho\) is modular, belonging to some modular form \(f\). Then there is a modular form \(g\) congruent to \(f\) modulo some place \(\mathcal P\) above \(p\) such that \(g\) is ordinary at \(\mathcal P\), i.e.\ the eigenvalue of \(g\) under the Hecke operator \(T_p\) (resp.\ \(U_p\) if \(p\) divides the level of \(g\)) is a \({\mathcal P}\)-adic unit.