Joel Bellaiche

Non smooth classical point on eigenvarieties.

Abstract: Eigenvarieties are "moduli spaces" of (overconvergent, finite slope) p-adic automorphic forms for a given reductive groups G. The points on eigenvarieties that corresponds to true, algebraic (as opposed to p-adic) automorphic forms are called classical points, and are Zariski dense when G has no discrete series. In favorable cases, we can interpolate the Galois representations attached to those classical forms in order to construct a canonical family of Galois representation on an eigenvariety, which is the main feature that make those varieties very useful in arithmetic.
Irreducible components of the Eigencurve (that is the eigenvariety of dimension 1, attached to Gl₂, constructed By Coleman and Mazur) are known to be smooth (at least at classical points) and to cross only at non-classical points. In contrast with this, we will show that irreducible components of higher dimensional eigenvarieties attached to unitary groups are non smooth at some well-chosen classical points. Our proof will use, and illustrate, some important properties of the families of Galois representations carried by those eigenvarieties.