A characterization of ordinary modular eigenforms with CM

TL;DR

This paper characterizes ordinary modular eigenforms with complex multiplication by their associated $p$-ordinary CM companion forms modulo powers of $p$, linking Galois representations and CM properties.

Contribution

It provides a new characterization of CM modular eigenforms through the existence of compatible CM companion forms modulo all powers of $p$.

Findings

Characterization of CM eigenforms via CM companion forms.

Equivalence of Galois representations up to twist with CM forms.

Applicable for all integers m ≥ 1.

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.