Classical forms of weight one in ordinary families

TL;DR

This paper introduces a new approach to studying low weight specializations in p-adic families of ordinary modular forms, providing a novel proof of a classical result relating weight one forms and complex multiplication.

Contribution

It presents a new proof technique that avoids relying on known properties of Galois representations and Ramanujan conjecture for weight one forms, with potential extensions to Hilbert modular forms.

Findings

Hida family contains infinitely many classical eigenforms of weight one iff it has complex multiplication

New proof avoids using Galois representation finiteness and Ramanujan conjecture for weight one forms

Strategy may extend to partial weight one Hilbert modular forms

Abstract

We develop a new strategy for studying low weight specializations of $p$-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate--Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be used to prove similar statement in the case of partial weight one Hilbert modular forms, given a suitable development of Hida theory in that setting.