New examples of $p$-adically rigid automorphic forms

Joël Bellaïche
2010 Mathematical Research Letters  
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
more » ... 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 [E] (or [BC2]), is not equi-dimensional. Hence the situation for the quasisplit 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 in any degree.
doi:10.4310/mrl.2010.v17.n4.a15 fatcat:4gmelvjfnrahhi5gxtrhwmqtk4