Non smooth classical point on eigenvarieties.
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.