Classical weight one forms in Hida families: Hilbert modular case

TL;DR

This paper studies the conditions under which non-CM ordinary Hida families of Hilbert modular forms have classical weight one specializations, providing explicit bounds and clarifying the difference from CM cases.

Contribution

It offers an explicit upper bound on classical weight one specializations for non-CM Hida families, extending understanding of their classical points.

Findings

Non-CM Hida families have finitely many classical weight one specializations.

CM Hida families can have infinitely many classical weight one specializations.

Explicit bounds depend on the family's properties.

Abstract

The purpose of this paper is to investigate the number of classical weight one specializations of a non-CM ordinary Hida family of parallel weight Hilbert cusp forms. It is known that a specialization of a primitive ordinary Hida family at any arithmetic points of weight at least two is a classical holomorphic Hilbert cusp form. However, this is not always the case for weight one specializations. Balasubramanyam, Ghate and Vatsal proved that such a Hida family admits infinitely many classical weight one specializations if and only if it is of CM type. We give an explicit upper bound on the number of classical weight one specializations of a non-CM primitive ordinary Hida family.