Hida once described his theory of families of ordinary p-adic
modular eigenforms as obtained from cutting “the clear surface
out of the pitch-dark well too deep to see through” of the space
of all elliptic modular forms. In this colloquium-style talk, we
shall peer into the well of Drinfeld modular forms instead of
classical modular forms. More precisely, we shall explain how to
construct families of finite slope Drinfeld modular forms over
Drinfeld modular varieties of any dimension. In the ordinary case
(the “clear surface”), we show that the weight may vary p-adically
in families of Drinfeld modular forms (a direct analogue of Hida’s
Vertical Control Theorem). In the deeper & murkier waters of positive
slope, the situation is more subtle: the weight may indeed vary
continuously, but not analytically, thereby contrasting markedly
with Coleman’s well-known p-adic theory.
Joint work with G. Rosso (Cambridge University / Concordia University (Montréal)).