A characterization of ordinary modular eigenforms with CM

Publication date: 28 February 2017

Abstract: For a rational prime

p g e q 3

we show that a

p

-ordinary modular eigenform

f

of weight

k g e q 2

, with

p

-adic Galois representation

h o_{f}

, mod

p^{m}

reductions

h o_{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 g e q 1

in the sense that

h o_{f, m} s i m h o_{h_{m}, m} o t i m e s c h i^{k - 1}

, where

c h i

is the

p

-adic cyclotomic character.