Density of Classical Points in Eigenvarieties
نویسندگان
چکیده
In this short note, we study the geometry of the eigenvariety parametrising p-adic automorphic forms for GL1 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 is Zariski-dense in the formal scheme. We also sketch the theory for GL2 over an imaginary quadratic field, following Calegari and Mazur, emphasising the strong formal similarity with the case of GL1 over a general number field.
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