A characterization of ordinary modular eigenforms with CM

Abstract

For a rational prime $p\ge 3$ we show that a $p$-ordinary modular eigenform $f$ of weight $k\ge 2$, with $p$-adic Galois representation ${\rho }_f$, mod $p^m$ reductions ${\rho }_{f,m}$, and with complex multiplication (CM), is characterized by the existence of $p$-ordinary CM companion forms $h_m$ modulo $p^m$ for all integers $m\ge 1$ in the sense that ${\rho }_{f,m}\sim {\rho }_{h_m,m}\otimes {\chi }^{k-1}$, where $\chi$ is the $p$-adic cyclotomic character.

9 pages. Elementary proof of the main theorem added in Section 5