A characterization of ordinary modular eigenforms with CM

TL;DR

The paper addresses the problem of characterizing $p$-ordinary CM modular eigenforms through higher congruence companions modulo powers of $p$. It develops two complementary approaches: (i) a main theorem proving the existence, for every $m\ge1$, of $p$-ordinary CM companion forms $h_m$ to a given $p$-ordinary CM form $f$ modulo $p^m$, obtained by embedding CM forms into a $p$-adic CM family and using a weight-shift relation $a_n(f)=n^{k-1}a_n(h_{2-k})$; and (ii) an elementary Hecke-character-based method that yields companions modulo odd moduli $M$ under a class-number coprimality condition. The results show that a $p$-ordinary CM form has CM if and only if it possesses CM companions for all $m$, providing a complete arithmetic characterization and linking CM forms to both Hida-family and Hecke-character constructions through $p$-adic interpolation and infinity-type adjustments.

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 $ρ_f$, mod ${p^m}$ reductions $ρ_{f,m}$, 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 $ρ_{f,m}\sim ρ_{h_m,m}\otimesχ^{k-1}$, where $χ$ is the $p$-adic cyclotomic character.

Theorems & Definitions (11)

• Definition 2.1
• Remark 2.2
• Definition 3.1
• Theorem 3.2
• Definition 3.3
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 5.1