Density of Classical Points in Eigenvarieties

In this short note, we study the geometry of the eigenvariety parametrizing $p$-adic automorphic forms for $\GL_1$ over a number field, 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 Grössencharacters 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 \emph{is} Zariski-dense in the formal scheme.

We also sketch the theory for $\GL_2$ over an imaginary quadratic field, following Calegari and Mazur, emphasizing the strong formal similarity with the case of $\GL_1$ over a general number field.

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Published 28 October 2011