Density of classical points in eigenvarieties

TL;DR

This paper investigates the geometric distribution of classical automorphic points within eigenvarieties for GL(1) over certain number fields, revealing non-density in the rigid space but density in the formal scheme model.

Contribution

It demonstrates the non-Zariski-density of classical points in the eigenvariety for specific number fields and establishes their density in the formal scheme, extending the understanding to GL(2) over imaginary quadratic fields.

Findings

Classical points are not Zariski-dense in the eigenvariety for non-totally real fields.

Classical points are Zariski-dense in the formal scheme model.

The theory for GL(2) over imaginary quadratic fields parallels the GL(1) case.

Abstract

In this short note, we study the geometry of the eigenvariety parametrising p-adic automorphic forms for GL(1) over a number field K, as constructed by Buzzard. We show that if K is not totally real and contains no CM subfield, points in this space arising from classical automorphic forms (i.e. algebraic Grossencharacters of K) are not Zariski-dense in the eigenvariety (as a rigid space); but the eigenvariety posesses a natural formal scheme model, and the set of classical points is Zariski-dense in the formal scheme. We also sketch the theory for GL(2) over an imaginary quadratic field, following Calegari and Mazur, emphasising the strong formal similarity with the case of GL(1) over a general number field.