New examples of $p$-adically rigid automorphic forms

In this paper, we prove two $p$-adic rigidity results for automorphic forms for the quasi-split unitary group in three variables $\U(2,1)$ attached to a quadratic imaginary field. We show first that the discrete automorphic forms for this group that are cohomological in degree $1$ (and refined, with a non semi-ordinary refinement) are rigid, in the sense that they can not be interpolated in a positive dimensional $p$-adic family, even though the set of Hodge-Tate weights of all such forms is not $p$-adically discrete. This results implies that the eigenvariety of $\U(2,1)$ in cohomological degree $1$, if it exists in the sense of \cite{E} (or \cite{BCbook}), is not equi-dimensional. Hence the situation for the quasi-split unitary group is in striking contrast with the one for its definite inner form $\U(3)$ and more generally any definite reductive group. We then show that some of the automorphic forms considered above, namely the ones that are minimally ramified in their $A$-packet and attached to an Hecke character whose $L$-function does not vanish at the center of its functional equation, are even rigid in the stronger sense that they can not be put in a non trivial family interpolating cohomological automorphic forms {\it in any degree}.

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Published 1 January 2010