## A characterization of ordinary modular eigenforms with CM

### Abstract

For a rational prime $p \geq 3$ we show that a $p$-ordinary modular eigenform $f$ of weight $k\geq 2$, 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 $m \geq 1$ in the sense that $\rho_{f,m}\sim \rho_{h_m,m}\otimes\chi^{k-1}$, where $\chi$ is the $p$-adic cyclotomic character.Comment: 9 pages. Elementary proof of the main theorem added in Section

Topics: Mathematics - Number Theory, 11F33, 11F80
Year: 2013
OAI identifier: oai:arXiv.org:1206.0177

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